Examples in mathematics - Revision history

Issa Rice: /* Unit testing and examples */

2021-10-28T20:13:50Z

Unit testing and examples

← Older revision		Revision as of 20:13, 28 October 2021
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	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.		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.

	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~~.		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 it.

	(I think there's an analogy to overfitting that would be good to explain.)		(I think there's an analogy to overfitting that would be good to explain.)

Issa Rice: /* Unit testing and examples */

2019-09-28T08:13:04Z

Unit testing and examples

← Older revision		Revision as of 08:13, 28 September 2019
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	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.		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.

			(I think there's an analogy to overfitting that would be good to explain.)

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Issa Rice: /* Unit testing and examples */

2019-02-19T03:23:11Z

Unit testing and examples

← Older revision		Revision as of 03:23, 19 February 2019
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	! Is an example according to intuition		! Is an example according to intuition
	\| An "obvious" example, or central example.		\| An "obvious" example, or central example.
	\| A surprising non-example.		\| A surprising non-example. False positives, also known as type I errors.
	\|-		\|-
	! Is not an example according to intuition		! Is not an example according to intuition
	\| A surprising example.		\| A surprising example. False negatives, also known as type II errors.
	\| An obvious non-example.		\| An obvious non-example.
	\|}		\|}

Issa Rice: /* Unit testing and examples */

2019-02-19T03:21:51Z

Unit testing and examples

← Older revision		Revision as of 03:21, 19 February 2019
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	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.		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.


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Issa Rice: /* Unit testing and examples */

2019-02-19T03:21:40Z

Unit testing and examples

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 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.
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 |-
 ! Is an example according to intuition
-| 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.
+| An "obvious" example, or central example.
-| 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.
+| A surprising non-example.
 |-
 ! Is not an example according to intuition
-| 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.
+| A surprising example.
-| 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.
+| An obvious non-example.
 |}
 ==Hierarchical nature of examples==

Issa Rice at 03:13, 19 February 2019

2019-02-19T03:13:19Z

← Older revision		Revision as of 03:13, 19 February 2019
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	'''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.		'''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.

	==Unit testing and examples==		==Unit testing and examples==

Issa Rice at 03:12, 19 February 2019

2019-02-19T03:12:40Z

← Older revision		Revision as of 03:12, 19 February 2019
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	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).		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).

			==References==

			<references/>

	[[Category:Mathematics]]		[[Category:Mathematics]]

Issa Rice: /* Unit testing and examples */

2019-02-19T03:12:22Z

Unit testing and examples

← Older revision		Revision as of 03:12, 19 February 2019
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	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.		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.

			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.

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Issa Rice: /* Unit testing and examples */

2019-02-19T03:08:16Z

Unit testing and examples

← Older revision		Revision as of 03:08, 19 February 2019
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	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]].		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]].

	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.		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.

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Issa Rice: /* Unit testing and examples */

2019-02-19T03:07:55Z

Unit testing and examples

← Older revision		Revision as of 03:07, 19 February 2019
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	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]].		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]].

			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.

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