https://learning.subwiki.org/w/api.php?action=feedcontributions&user=Issa+Rice&feedformat=atomLearning - User contributions [en]2019-04-22T10:15:34ZUser contributionsMediaWiki 1.29.2https://learning.subwiki.org/w/index.php?title=Self-explanation&diff=769Self-explanation2019-04-14T09:35:35Z<p>Issa Rice: </p>
<hr />
<div>'''Self-explanation''' is a [[learning technique]] where the [[learner]] explains the steps they take in solving a problem or their processing of new information to themselves.<br />
<br />
==History==<br />
<br />
Dunlosky et al. (2013) [http://www.indiana.edu/~pcl/rgoldsto/courses/dunloskyimprovinglearning.pdf] calls a 1983 study by Berry "the seminal study on self-explanation".<br />
<br />
==Software engineering==<br />
<br />
Closely related to self-explanation is a technique called ''rubber duck debugging'' (or ''rubber ducking''), where a programmer explains a software problem to themselves (or someone who knows nothing about programming) to help them debug code.<br />
<br />
==External links==<br />
<br />
* https://siderea.dreamwidth.org/1368412.html</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Learning_through_osmosis&diff=768Learning through osmosis2019-04-03T04:17:37Z<p>Issa Rice: /* Notes */</p>
<hr />
<div>'''Learning through osmosis''' (also called '''learning by osmosis''', '''learning via osmosis''', and '''osmotic learning''') is the idea that one can learn things through a mysterious/not-well-understood method where one immerses oneself in some environment.<br />
<br />
==Notes==<br />
<br />
Is this how people learn their native language?<br />
<br />
In math:<br />
<br />
<blockquote>Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier.<ref>Ravi Vakil. [http://math.stanford.edu/~vakil/potentialstudents.html "For potential Ph.D. students"].</ref></blockquote><br />
<br />
https://www.greaterwrong.com/search?q=osmosis<br />
<br />
https://www.greaterwrong.com/posts/zLZDxXbcXP3hdM3sh/osmosis-learning-a-crucial-consideration-for-the-craft<br />
<br />
https://www.greaterwrong.com/posts/9SaAyq7F7MAuzAWNN/teaching-the-unteachable<br />
<br />
https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/#comment-1782<br />
<br />
==See also==<br />
<br />
* [[Importance of struggling in learning]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=759Teaching for understanding versus teaching for creation2019-02-20T23:09:40Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) (note: this does ''not'' mean that the ''act of teaching itself'' has been systematized; linear algebra is systematized even if people have not figured out how to teach it) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative example]]s are available || Positive examples are hard to convey, while negative examples are available<br />
|}<br />
<br />
The meta levels are somewhat confusing, so let me try listing them:<br />
<br />
# object level (linear algebra): this is what a typical student taking a linear algebra course does<br />
# (how to invent linear algebra): this is what the people who invented linear algebra did, or what a highly-above-average student taking a linear algebra course might do, if they were trying to really understand the subject<br />
# (how to teach linear algebra): this is what a graduate student figuring out how to teach a linear algebra course does<br />
# (how to teach how to invent linear algebra): this is what Jeffreyssai (i.e. someone who wants to teach his students how to invent) must figure out<ref>https://wiki.lesswrong.com/wiki/Beisutsukai</ref><br />
# (how to invent how to teach linear algebra): what an unusual instructor of linear algebra does, if they want to figure out how to best teach linear algebra<br />
<br />
==Differential teaching strategies==<br />
<br />
Why does the "teaching for understanding" vs "teaching for creation" distinction matter? One reason is that depending on the audience/goal, it makes sense to alter the teaching strategy.<br />
<br />
For example, if the goal is to create, it makes sense to prove as many theorems as possible without looking at the proofs in the book. It might make sense (after empirical investigation) to also do this even if the goal is just to understand the material (see [[pre-testing effect]]).<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=758Teaching for understanding versus teaching for creation2019-02-20T23:05:49Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) (note: this does ''not'' mean that the ''act of teaching itself'' has been systematized; linear algebra is systematized even if people have not figured out how to teach it) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative example]]s are available || Positive examples are hard to convey, while negative examples are available<br />
|}<br />
<br />
The meta levels are somewhat confusing, so let me try listing them:<br />
<br />
# object level (linear algebra): this is what a typical student taking a linear algebra course does<br />
# (how to invent linear algebra): this is what the people who invented linear algebra did, or what a highly-above-average student taking a linear algebra course might do, if they were trying to really understand the subject<br />
# (how to teach linear algebra): this is what a graduate student figuring out how to teach a linear algebra course does<br />
# (how to teach how to invent linear algebra): this is what Jeffreyssai (i.e. someone who wants to teach his students how to invent) must figure out<ref>https://wiki.lesswrong.com/wiki/Beisutsukai</ref><br />
# (how to invent how to teach linear algebra): what an unusual instructor of linear algebra does, if they want to figure out how to best teach linear algebra<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=757Teaching for understanding versus teaching for creation2019-02-20T23:05:31Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) (note: this does ''not'' mean that the ''act of teaching itself'' has been systematized; linear algebra is systematized even if people have not figured out how to teach it) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative example]]s are available || Positive examples are hard to convey, while negative examples are available<br />
|}<br />
<br />
The meta levels are somewhat confusing, so let me try listing them:<br />
<br />
# object level (linear algebra): this is what a typical student taking a linear algebra course does<br />
# (how to invent linear algebra): this is what the people who invented linear algebra did, or what a highly-above-average student taking a linear algebra course might do, if they were trying to really understand the subject<br />
# (how to teach linear algebra): this is what a graduate student figuring out how to teach a linear algebra course does<br />
# (how to teach how to invent linear algebra): this is what Jeffreyssai (i.e. someone who wants to teach his students how to invent) must figure out<ref>https://wiki.lesswrong.com/wiki/Beisutsukai</ref><br />
# (how to invent how to teach linear algebra): what an unusual instructor of linear algebra does, if they want to figure out how to best teach linear algebra</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=756Teaching for understanding versus teaching for creation2019-02-20T22:59:21Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) (note: this does ''not'' mean that the ''act of teaching itself'' has been systematized; linear algebra is systematized even if people have not figured out how to teach it) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative example]]s are available || Positive examples are hard to convey, while negative examples are available<br />
|}<br />
<br />
The meta levels are somewhat confusing, so let me try listing them:<br />
<br />
# object level (linear algebra): this is what a typical student taking a linear algebra course does<br />
# (how to invent linear algebra): this is what the people who invented linear algebra did, or what a highly-above-average student taking a linear algebra course might do, if they were trying to really understand the subject<br />
# (how to teach linear algebra): this is what a graduate student figuring out how to teach a linear algebra course does<br />
# (how to teach how to invent linear algebra)<br />
# (how to invent how to teach linear algebra)</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=755Teaching for understanding versus teaching for creation2019-02-20T22:52:46Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) (note: this does ''not'' mean that the ''act of teaching itself'' has been systematized; linear algebra is systematized even if people have not figured out how to teach it) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative example]]s are available || Positive examples are hard to convey, while negative examples are available<br />
|}</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=754Teaching for understanding versus teaching for creation2019-02-20T22:51:02Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative example]]s are available || Positive examples are hard to convey, while negative examples are available<br />
|}</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=753Teaching for understanding versus teaching for creation2019-02-20T22:50:40Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|-<br />
| Both [[positive and negative examples]] are available || Positive examples are hard to convey, while negative examples are available<br />
|}</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Teaching_for_understanding_versus_teaching_for_creation&diff=752Teaching for understanding versus teaching for creation2019-02-20T22:48:38Z<p>Issa Rice: Created page with "(there might be a more standard term for this distinction) '''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a learne..."</p>
<hr />
<div>(there might be a more standard term for this distinction)<br />
<br />
'''Teaching for understanding versus teaching for creation''' refers to the distinction between teaching a [[learner]] to simply understand the material (which allows them to use the material in simple applications) versus teaching the learner to create new ideas in the subject.<br />
<br />
Here is a rough categorization (not necessarily very accurate):<br />
<br />
{| class="wikitable"<br />
|-<br />
! Teaching for understanding !! Teaching for creation<br />
|-<br />
| Undergraduate curriculum (teaches standard topics in a field) || Graduate school (is supposed to teach students to advance the field)<br />
|-<br />
| Teaching the object-level skill/material || Teaching a meta-level skill (note: there is more than one way to "go meta" from the object level, e.g. one could also "go meta" by learning about how to learn, rather than learning how to create)<br />
|-<br />
| Teaching of material that has been systematized (e.g. linear algebra has been systematized and is well-understood) || Teaching of material/skills that have not been systematized (e.g. the act of inventing linear algebra from scratch has ''not'' been systematized, and is not well-understood)<br />
|}</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=751List of problems in mathematical notation2019-02-19T04:14:30Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability. 0 in linear algebra to mean the scalar 0 as well as the 0 vector of any dimension.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law, "same object denotes different things" || in probability, where <math>\Pr(X=x)</math> is abbreviated <math>\Pr(x)</math>. Then if we take something like <math>x:=3</math>, we would have <math>\Pr(3)</math>, but now we have lost the information about what random variable we are talking about.<br />
|-<br />
| "different objects denoted by the same thing" || see e.g. [https://machinelearning.subwiki.org/wiki/User:IssaRice/Type_checking_Pearl's_belief_propagation_notation here], where <math>M_{y\,|\,x}</math> can mean two different things on each side of an equation.<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added, which can sometimes [[wikipedia:Riemann series theorem|matter]]) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|-<br />
| Ambiguous order of operations || e.g. <math>a/bc</math><br />
|-<br />
| Undefined operations || e.g. I often see in mathematical logic things like <math>\Gamma \cup \phi</math> to mean <math>\Gamma \cup \{\phi\}</math>, where <math>\Gamma</math> is a set of formulas and <math>\phi</math> is a formula.<br />
|-<br />
| strange reorderings based on context || e.g. https://machinelearning.subwiki.org/wiki/User:IssaRice/Minus_notation_in_game_theory<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=750List of problems in mathematical notation2019-02-19T04:11:07Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability. 0 in linear algebra to mean the scalar 0 as well as the 0 vector of any dimension.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law, "same object denotes different things" || in probability, where <math>\Pr(X=x)</math> is abbreviated <math>\Pr(x)</math>. Then if we take something like <math>x:=3</math>, we would have <math>\Pr(3)</math>, but now we have lost the information about what random variable we are talking about.<br />
|-<br />
| "different objects denoted by the same thing" || see e.g. [https://machinelearning.subwiki.org/wiki/User:IssaRice/Type_checking_Pearl's_belief_propagation_notation here], where <math>M_{y\,|\,x}</math> can mean two different things on each side of an equation.<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added, which can sometimes [[wikipedia:Riemann series theorem|matter]]) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|-<br />
| Ambiguous order of operations || e.g. <math>a/bc</math><br />
|-<br />
| Undefined operations || e.g. I often see in mathematical logic things like <math>\Gamma \cup \phi</math> to mean <math>\Gamma \cup \{\phi\}</math>, where <math>\Gamma</math> is a set of formulas and <math>\phi</math> is a formula.<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=749List of problems in mathematical notation2019-02-19T04:08:00Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability. 0 in linear algebra to mean the scalar 0 as well as the 0 vector of any dimension.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law || in probability, where <math>\Pr(X=x)</math> is abbreviated <math>\Pr(x)</math>. Then if we take something like <math>x:=3</math>, we would have <math>\Pr(3)</math>, but now we have lost the information about what random variable we are talking about.<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added, which can sometimes [[wikipedia:Riemann series theorem|matter]]) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|-<br />
| Ambiguous order of operations || e.g. <math>a/bc</math><br />
|-<br />
| Undefined operations || e.g. I often see in mathematical logic things like <math>\Gamma \cup \phi</math> to mean <math>\Gamma \cup \{\phi\}</math>, where <math>\Gamma</math> is a set of formulas and <math>\phi</math> is a formula.<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=748List of problems in mathematical notation2019-02-19T04:04:57Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability. 0 in linear algebra to mean the scalar 0 as well as the 0 vector of any dimension.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added, which can sometimes [[wikipedia:Riemann series theorem|matter]]) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|-<br />
| Ambiguous order of operations || e.g. <math>a/bc</math><br />
|-<br />
| Undefined operations || e.g. I often see in mathematical logic things like <math>\Gamma \cup \phi</math> to mean <math>\Gamma \cup \{\phi\}</math>, where <math>\Gamma</math> is a set of formulas and <math>\phi</math> is a formula.<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=747List of problems in mathematical notation2019-02-19T04:01:32Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added, which can sometimes [[wikipedia:Riemann series theorem|matter]]) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|-<br />
| Ambiguous order of operations || e.g. <math>a/bc</math><br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=746List of problems in mathematical notation2019-02-19T03:52:59Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added, which can sometimes [[wikipedia:Riemann series theorem|matter]]) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=745List of problems in mathematical notation2019-02-19T03:51:51Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index || Writing things like <math>\sum_i f(x_i)</math> (unclear what set the index ranges over, or the order in which the terms are added) or even just <math>\sum f(x_i)</math> (unclear which variable is the index).<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=744List of problems in mathematical notation2019-02-19T03:44:35Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation || This happens when the same symbol is used for different purposes. For instance, parentheses are used to group expressions, for writing tuples, for function calls, in superscripts or subscripts with various meanings (<math>f^{(i)}</math> for the <math>i</math>th derivative or the <math>i</math>th function in some sequence), and so forth. The equals sign is often used for equality both in the object language and the metalanguage in mathematical logic, and used as part of the expression <math>X=x</math> to define an event in probability.<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=743List of problems in mathematical notation2019-02-19T03:40:13Z<p>Issa Rice: </p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index<br />
|}<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=List_of_problems_in_mathematical_notation&diff=742List of problems in mathematical notation2019-02-19T03:39:55Z<p>Issa Rice: Created page with "This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conv..."</p>
<hr />
<div>This page gives a '''list of problems in mathematical notation'''. The list focuses on problems with the symbolism of mathematics rather than other communication problems/conventions (such as using "if" to mean "iff" in definitions, not introducing variables, etc.).<br />
<br />
{| class="sortable wikitable"<br />
|-<br />
! Name !! Description !! How to avoid<br />
|-<br />
| Overloading/abuse of notation<br />
|-<br />
| Type error<br />
|-<br />
| Failure of Leibniz law<br />
|-<br />
| Omission of index<br />
|}</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=741Examples in mathematics2019-02-19T03:23:11Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have a different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
There is a tendency in human thinking to leave ideas merely at the verbal level, i.e. at a level where the ideas don't constrain anticipation.<ref>https://www.readthesequences.com/A-Technical-Explanation-Of-Technical-Explanation</ref> Giving surprising examples and non-examples is one way to catch people's fuzzy thinking and to correct them.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example.<br />
| A surprising non-example. False positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. False negatives, also known as type II errors.<br />
| An obvious non-example.<br />
|}<br />
<br />
===Obvious examples===<br />
<br />
An "obvious" example, or central example.<br />
<br />
Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
<br />
===Surprising non-examples===<br />
<br />
A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br />
<br />
As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br />
<br />
The examples in this cell are false positives, also known as type I errors.<br />
<br />
===Surprising examples===<br />
<br />
A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br />
<br />
Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br />
<br />
As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
<br />
The examples in this cell are false negatives, also known as type II errors.<br />
<br />
===Obvious non-examples===<br />
<br />
An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=740Examples in mathematics2019-02-19T03:21:51Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have a different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
There is a tendency in human thinking to leave ideas merely at the verbal level, i.e. at a level where the ideas don't constrain anticipation.<ref>https://www.readthesequences.com/A-Technical-Explanation-Of-Technical-Explanation</ref> Giving surprising examples and non-examples is one way to catch people's fuzzy thinking and to correct them.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example.<br />
| A surprising non-example.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example.<br />
| An obvious non-example.<br />
|}<br />
<br />
===Obvious examples===<br />
<br />
An "obvious" example, or central example.<br />
<br />
Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
<br />
===Surprising non-examples===<br />
<br />
A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br />
<br />
As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br />
<br />
The examples in this cell are false positives, also known as type I errors.<br />
<br />
===Surprising examples===<br />
<br />
A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br />
<br />
Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br />
<br />
As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
<br />
The examples in this cell are false negatives, also known as type II errors.<br />
<br />
===Obvious non-examples===<br />
<br />
An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=739Examples in mathematics2019-02-19T03:21:40Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have a different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
There is a tendency in human thinking to leave ideas merely at the verbal level, i.e. at a level where the ideas don't constrain anticipation.<ref>https://www.readthesequences.com/A-Technical-Explanation-Of-Technical-Explanation</ref> Giving surprising examples and non-examples is one way to catch people's fuzzy thinking and to correct them.<br />
<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example.<br />
| A surprising non-example.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example.<br />
| An obvious non-example.<br />
|}<br />
<br />
===Obvious examples===<br />
<br />
An "obvious" example, or central example.<br />
<br />
Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
<br />
===Surprising non-examples===<br />
<br />
A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br />
<br />
As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br />
<br />
The examples in this cell are false positives, also known as type I errors.<br />
<br />
===Surprising examples===<br />
<br />
A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br />
<br />
Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br />
<br />
As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
<br />
The examples in this cell are false negatives, also known as type II errors.<br />
<br />
===Obvious non-examples===<br />
<br />
An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=738Examples in mathematics2019-02-19T03:13:19Z<p>Issa Rice: </p>
<hr />
<div>'''Examples in mathematics''' have a different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
There is a tendency in human thinking to leave ideas merely at the verbal level, i.e. at a level where the ideas don't constrain anticipation.<ref>https://www.readthesequences.com/A-Technical-Explanation-Of-Technical-Explanation</ref> Giving surprising examples and non-examples is one way to catch people's fuzzy thinking and to correct them.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=737Examples in mathematics2019-02-19T03:12:40Z<p>Issa Rice: </p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
There is a tendency in human thinking to leave ideas merely at the verbal level, i.e. at a level where the ideas don't constrain anticipation.<ref>https://www.readthesequences.com/A-Technical-Explanation-Of-Technical-Explanation</ref> Giving surprising examples and non-examples is one way to catch people's fuzzy thinking and to correct them.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
==References==<br />
<br />
<references/><br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=736Examples in mathematics2019-02-19T03:12:22Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
There is a tendency in human thinking to leave ideas merely at the verbal level, i.e. at a level where the ideas don't constrain anticipation.<ref>https://www.readthesequences.com/A-Technical-Explanation-Of-Technical-Explanation</ref> Giving surprising examples and non-examples is one way to catch people's fuzzy thinking and to correct them.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=735Examples in mathematics2019-02-19T03:08:16Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing [[wikipedia:Unit testing|unit tests]] in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=734Examples in mathematics2019-02-19T03:07:55Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
One can view the giving of examples as analogous to writing unit tests in programming. It is good to have some obvious examples, but one also wants to test the software on surprising cases (called "edge cases") to make sure the software really works.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=733Examples in mathematics2019-02-19T03:04:52Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
In giving examples, it is particularly important to give examples in the places where intuition and the formal definition disagree. By default, the [[learner]] may have a tendency to [[wikipedia:Peter Cathcart Wason#Wason and the 2-4-6 Task|search only for positive examples]].<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=732Examples in mathematics2019-02-19T02:58:21Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br>The examples in this cell are false positives, also known as type I errors.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br>The examples in this cell are false negatives, also known as type II errors.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=731Examples in mathematics2019-02-19T02:54:49Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
| An obvious non-example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = x/0</math>. This does ''not'' define a function because division by zero is undefined. Someone familiar with division by zero would recognize this, and correctly reject this example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=730Examples in mathematics2019-02-19T02:50:56Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see the next cell in this table) might conflate this example with it, and think that this is a function.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=729Examples in mathematics2019-02-19T02:49:53Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator). Someone who thought that a function is "something that is defined by a formula" might mistakenly think "this thing is defined by a formula, so must be a function".<br>As another example, let <math>f : A \to \emptyset</math> be a function where <math>A \ne \emptyset</math>. This does ''not'' define a function. To see this, note that since <math>A\ne \emptyset</math>, we must have some <math>a \in A</math>. By the definition of function, we would have <math>f(a) \in \emptyset</math>, which is a contradiction since <math>\emptyset</math> is empty. Someone who was familiar with the empty function (see "Is not an example according to intuition" cell in this table) might conflate this example with it, and think that this is a function.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=728Examples in mathematics2019-02-19T02:44:53Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. Let <math>f : \mathbf Q \to \mathbf Z</math> be defined by <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction). This does ''not'' define a function. To see this, note that <math>f(1/2) = 1</math> and <math>f(3/6) = 3</math>. But <math>1/2=3/6</math> so we must have <math>f(1/2)=f(3/6)</math> (a function must output a unique object for any given object), but <math>1\ne3</math>, so something has gone wrong. It turns out that each fraction has many different representations, and the idea of taking "the" numerator does not make sense, unless we constrain the representation somehow (e.g. by reducing the fraction and always putting any minus sign in the numerator).<br> a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=727Examples in mathematics2019-02-19T02:40:32Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function. In this example, it is not the intuitive notion of "function" that is getting in the way, but rather, a different technical concept (i.e., that of a computable function) that is getting in the way.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=726Examples in mathematics2019-02-19T02:39:07Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>. This does define a function, although the function is not ''computable''. Someone familiar with the halting problem might substitute "is a well-defined function" with "is a computable function" and say that this is not a function.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=725Examples in mathematics2019-02-19T02:36:51Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N \to \{\text{true}, \text{false}\}</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=724Examples in mathematics2019-02-19T02:36:24Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br>As a third example, let <math>\mathcal M</math> be the set of all Turing machines, and let <math>f : \mathcal M \times \mathbf N</math> be defined by <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math>.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=723Examples in mathematics2019-02-19T02:35:19Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions (or someone who isn't used to working with the empty set or vacuous conditions) might dismiss this example.<br><math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math><br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=722Examples in mathematics2019-02-19T02:33:37Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. Let <math>f : \mathbf N \to \mathbf N</math> be defined by <math>f(n) = n\text{th digit of }\pi</math>. This does define a function, but someone who thought that a function is "something that is defined by a formula" wouldn't think it is a function.<br>Another example is the empty function <math>f : \emptyset \to A</math> for any set <math>A</math>. This does define a function, but the function doesn't "do" anything. Since it is an "extreme" example of a function, someone who was only used to dealing with "normal-looking" functions might dismiss this example<br><math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math><br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=721Examples in mathematics2019-02-19T02:30:18Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. Let <math>f : \mathbf R \to \mathbf R</math> be defined by <math>f(x) = 2x^2 - 3x + 5</math>. This does define a function, and someone who thought that a function is "something that is defined by a formula" would think that this is a function.<br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. <math>f(n) = n\text{th digit of }\pi</math>; the empty function; <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math><br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=720Examples in mathematics2019-02-19T02:29:13Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. <math>f(x) = 2x^2 - 3x + 5</math><br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. <math>f(n) = n\text{th digit of }\pi</math>; the empty function; <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math><br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=719Examples in mathematics2019-02-19T02:22:30Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. <math>f(x) = 2x^2 - 3x + 5</math><br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \emptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. <math>f(n) = n\text{th digit of}\pi</math>; the empty function; <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math><br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=718Examples in mathematics2019-02-19T02:22:12Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
Take the example of a definition of function. A function is some object that takes each object in some set to a unique object in another set. Someone who was not familiar with the formal definition might mistakenly think of a function as "something that is defined by a formula".<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example. <math>f(x) = 2x^2 - 3x + 5</math><br />
| A surprising non-example. <math>f(a/b) = a</math> (i.e. a function that outputs the numerator of a fraction); a function <math>f : A \to \meptyset</math> where <math>A \ne \emptyset</math>.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example. <math>f(n) = n\text{th digit of}\pi</math>; the empty function; <math>f(M,n) = \text{Turing machine }M\text{ halts on input }n</math><br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=717Examples in mathematics2019-02-19T02:08:32Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
A common problem in math is that one comes in with some preconceived idea of what an object should "look like" which is different from what the definition says. In other words, there is a mismatch between one's intuitive notion and the definition.<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example.<br />
| A surprising non-example.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=716Examples in mathematics2019-02-19T02:06:58Z<p>Issa Rice: /* Unit testing and examples */</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
{| class="wikitable"<br />
|-<br />
!<br />
! Is an example according to definition<br />
! Is not an example according to definition<br />
|-<br />
! Is an example according to intuition<br />
| An "obvious" example, or central example.<br />
| A surprising non-example.<br />
|-<br />
! Is not an example according to intuition<br />
| A surprising example.<br />
| An obvious non-example.<br />
|}<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=715Examples in mathematics2019-02-19T02:01:23Z<p>Issa Rice: </p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).<br />
<br />
[[Category:Mathematics]]</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Examples_in_mathematics&diff=714Examples in mathematics2019-02-19T02:01:09Z<p>Issa Rice: Created page with "'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because definitions in mathematics are different from definitions..."</p>
<hr />
<div>'''Examples in mathematics''' have different flavor than examples in other disciplines. This is probably because [[definitions in mathematics]] are different from definitions in other disciplines (mathematical definitions are exact). Some [https://www.readthesequences.com/The-Cluster-Structure-Of-Thingspace common] [https://wiki.lesswrong.com/wiki/How_an_algorithm_feels problems] of deciding whether something is or is not an example do not appear in mathematics. Instead, there are other problems.<br />
<br />
==Unit testing and examples==<br />
<br />
==Hierarchical nature of examples==<br />
<br />
Something can be considered "concrete" or "abstract" depending on the context. Consider a term like "metric space". One can give examples of metric spaces. On the other hand, a metric space is itself an example (of a structured space, of a topological space).</div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=713Combinatorial explosion of questions and errors2019-02-16T00:02:55Z<p>Issa Rice: /* Ways to mitigate */</p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==Ways to mitigate==<br />
<br />
For the [[explainer]]:<br />
<br />
* test the explanation on many readers to catch potential errors/questions, so that these can be cached<br />
* leave the explanation for a while to make it fresher in your mind<br />
* move to a more interactive format<br />
* allow comments (if it's a blog post)<br />
* explain more of the background material to "uniformize" the audience, to make the audience more predictable<br />
* stipulate conditions on the audience (e.g. say that X and Y are required background reading, or that this is intended for intermediate students)<br />
<br />
For the [[learner]]:<br />
<br />
* make use of peers/tutors/TAs/teachers/question-and-answer-sites (e.g. if you get stuck reading a textbook, ask a peer)<br />
* [[Learning from multiple sources|make use of multiple explanations]]<br />
<br />
==Related phenomena==<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
* in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of ''potential'' arguments each side could make, even if the ''actual'' path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Ricehttps://learning.subwiki.org/w/index.php?title=Combinatorial_explosion_of_questions_and_errors&diff=712Combinatorial explosion of questions and errors2019-02-16T00:02:11Z<p>Issa Rice: </p>
<hr />
<div>(there might be a more standard term for this)<br />
<br />
Non-interactive explanations (e.g. textbooks, blog posts, and YouTube videos, in contrast to interactive explanations like classrooms and tutoring) face the problem of '''combinatorial explosion of questions and errors'''. As the length of the explanation increases, there will be more and more potential questions a [[learner]] could ask, as well as more and more potential errors in reasoning the learner could make (or misconceptions they could have). As the explanation is non-interactive, the [[explainer]] must anticipate in advance which questions and errors are most likely for the intended audience, and must decide how extensive the explanation will be.<br />
<br />
Consider the experience of reading a mathematical proof. Each step in the proof is an opportunity for the reader to become confused, as they might not understand a calculation or reasoning that is being done.<br />
<br />
Here is an example of this sort of thing, as related by Nate Soares:<ref>https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things</ref><br />
<br />
<blockquote>The problem is, most of the time that I get stuck, I get stuck on something incredibly stupid. I’ve either misread something somewhere or misremembered a concept from earlier in the book. Usually, someone looking over my shoulder could correct me in ten seconds with three words.<br><br>“Dude. Disjunction. ''Dis''junction.”<br><br>These are the things that eat my days.</blockquote><br />
<br />
==Ways to mitigate==<br />
<br />
For the [[explainer]]:<br />
<br />
* test the explanation on many readers to catch potential errors/questions, so that these can be cached<br />
* leave the explanation for a while to make it fresher in your mind<br />
* move to a more interactive format<br />
* allow comments (if it's a blog post)<br />
* explain more of the background material to "uniformize" the audience, to make the audience more predictable<br />
* stipulate conditions on the audience (e.g. say that X and Y are required background reading, or that this is intended for intermediate students)<br />
<br />
For the [[learner]]:<br />
<br />
* make use of peers/tutors/TAs/teachers/question-and-answer-sites (e.g. if you get stuck reading a textbook, ask a peer)<br />
* make use of multiple explanations<br />
<br />
==Related phenomena==<br />
<br />
* the thing where course instructors reuse old exam/homework problems it's hard to come up with new problems (this problem also arises from the static/non-interactive nature of learning material)<br />
* the problem with storing static example problems in [[spaced repetition software]], because the user will just memorize the answer instead of treating it like a new problem (this problem also arises from the static/non-interactive nature of learning material)<br />
* inferential distance/lack of uniform background of learners (this problem also arises from the difficulty of anticipating the identity of the learner or how they will react); see e.g. "“There are 10 pre-requisites for understanding concept X. Most people have 6 or seven, and then I write a blog post for each of the 10. Most people, most of the time, feel like they’re reading a thing they already know, yet I did have to write all 10 to be able to get everyone to take the step forward together.”"<ref>https://www.greaterwrong.com/posts/Q924oPJzK92FifuFg/write-a-thousand-roads-to-rome/comment/zCSb8aGbfuLeKKwBP</ref><br />
* there is a kind of "reverse" or "ironic" problem that happens where the explainer ''did'' correctly anticipate a question, but the reader fails to anticipate that the explainer anticipates this, so the reader stops reading as soon as they become confused, when in fact in the next paragraph (or next section or whatever) the question is answered. Some authors try to prevent this by saying things like "more on this soon" or "see section 4.5.3 for details".<br />
* in arguments/debates (which are a special kind of explanation), there is a combinatorial explosion of ''potential'' arguments each side could make, even if the ''actual'' path of arguments is just a single path through this tree. See https://arxiv.org/abs/1805.00899 for more on this.<br />
<br />
==See also==<br />
<br />
* [[wikipedia:Combinatorial explosion]]<br />
<br />
==References==<br />
<br />
<references/></div>Issa Rice