Understanding mathematical definitions - Revision history

Issa Rice: /* List of steps */

2024-07-23T05:37:21Z

List of steps

← Older revision		Revision as of 05:37, 23 July 2024
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	\| Understanding the kind of definition \|\| Do this for every definition. \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| This is good to keep in mind so that you don't get confused about what the definition is trying to do. \|\|		\| Understanding the kind of definition \|\| Do this for every definition. \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| This is good to keep in mind so that you don't get confused about what the definition is trying to do. \|\|
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	\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.		\| Checking well-definedness \|\| If the definition defines an operation \|\| Carry out the standard procedure for checking well-definedness of operations. For a binary operation <math></math>, you must check that if <math>x,x'</math> are equivalent objects and <math>y,y'</math> are equivalent objects, then <math>xy</math> and <math>x'*y'</math> are equivalent objects (there are slightly different ways to verify the same thing, and you can generalize to non-binary operations). \|\| If an operation is not well-defined, then there is no point in talking about it. \|\| Checking that addition on the integers is well-defined.
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	\| Checking consistency with existing definition \|\| If the definition supersedes an older definition or it clobbers up a previously defined notation \|\| \|\| \|\| Addition on the reals after addition on the rationals has been defined.<br/><br/>For any function <math>f:X\to Y</math> and <math>U\subset Y</math>, the inverse image <math>f^{-1}(U)</math> is defined. On the other hand, if a function <math>f : X\to Y</math> is a bijection, then <math>f^{-1} : Y \to X</math> is a function, so its forward image <math>f^{-1}(U)</math> is defined given any <math>U\subset Y</math>. We must check that these two are the same set (or else have some way to disambiguate which one we mean). (This example is mentioned in both Tao's ''Analysis I'' and in Munkres's ''Topology''.)		\| Checking consistency with existing definition \|\| If the definition supersedes an older definition or it clobbers up a previously defined notation \|\| \|\| \|\| Addition on the reals after addition on the rationals has been defined.<br/><br/>For any function <math>f:X\to Y</math> and <math>U\subset Y</math>, the inverse image <math>f^{-1}(U)</math> is defined. On the other hand, if a function <math>f : X\to Y</math> is a bijection, then <math>f^{-1} : Y \to X</math> is a function, so its forward image <math>f^{-1}(U)</math> is defined given any <math>U\subset Y</math>. We must check that these two are the same set (or else have some way to disambiguate which one we mean). (This example is mentioned in both Tao's ''Analysis I'' and in Munkres's ''Topology''.)

Issa Rice: /* List of steps */

2024-07-23T05:31:41Z

List of steps

← Older revision		Revision as of 05:31, 23 July 2024
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	\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).		\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).
	\|-		\|-
	\| Understanding the kind of definition \|\| Do this for every definition. \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| \|\|		\| Understanding the kind of definition \|\| Do this for every definition. \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| This is good to keep in mind so that you don't get confused about what the definition is trying to do. \|\|
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	\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.		\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.

Issa Rice: /* List of steps */

2024-07-23T05:29:52Z

List of steps

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	\| Coming up with examples \|\| This should be done for definitions that define a class of objects or a new kind of object. \|\| Come up with some examples of objects that fit the definition. Emphasize edge cases. \|\| Examples help to train your intuition of what the object "looks like". \|\| For monotone increasing functions, an edge case would be the constant function. Such an edge case is useful because it doesn't look like a "typical" object of its class, and so one might naively not think it's an example; explicitly noting the edge case helps build intuition that it's not just strictly increasing functions or functions that increase at all are covered by the definition.		\| Coming up with examples \|\| This should be done for definitions that define a class of objects or a new kind of object. \|\| Come up with some examples of objects that fit the definition. Emphasize edge cases. \|\| Examples help to train your intuition of what the object "looks like". \|\| For monotone increasing functions, an edge case would be the constant function. Such an edge case is useful because it doesn't look like a "typical" object of its class, and so one might naively not think it's an example; explicitly noting the edge case helps build intuition that it's not just strictly increasing functions or functions that increase at all are covered by the definition.
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	\| Coming up with counterexamples \|\| This should be done for definitions that define a class of objects or a new kind of object. \|\| Come up with examples of objects that ''don't'' satisfy the definition. Emphasize objects that one might intuitively think would fit the definition but in reality don't. \|\| As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. [https://en.wikipedia.org/wiki/Confirmation_bias#Wason's_research_on_hypothesis-testing] \|\| For monotone increasing functions, a counterexample is the function <math>f : \mathbf R \to \mathbf R</math> defined by <math>f(x) = x^2</math>. Even though this function ''is'' monotone on the non-negative reals, if we consider this function on all of <math>\mathbf R</math> then it stops being monotone increasing.		\| Coming up with counterexamples \|\| This should be done for definitions that define a class of objects or a new kind of object. \|\| Come up with examples of objects that ''don't'' satisfy the definition. Emphasize objects that one might intuitively think would fit the definition but in reality don't. \|\| As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. [https://en.wikipedia.org/wiki/Confirmation_bias#Wason's_research_on_hypothesis-testing] \|\| For monotone increasing functions, a counterexample is the function <math>f : \mathbf R \to \mathbf R</math> defined by <math>f(x) = x^2</math>. Even though this function ''is'' monotone on the non-negative reals, if we consider this function on all of <math>\mathbf R</math> then it stops being monotone increasing. This counterexample also emphasizes the idea that the property of being "monotone increasing" doesn't just depend on the symbolic definition/behavior of the function, but also on its domain.
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	\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).		\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).

Issa Rice: /* List of steps */

2024-07-23T05:28:27Z

List of steps

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	\| Checking assumptions of objects introduced \|\| \|\| Remove or alter each assumption of the objects that have been introduced in the definition to see why they are necessary. \|\| Generally you want definitions to be "expansive" in the sense of applying to many different objects. But each assumption you introduce whittles down the number of objects the definition applies to. In other words, there is tension between (1) trying to have expansive definitions, and (2) adding in assumptions/restrictions in a definition. So you want to make sure each assumption [https://wiki.lesswrong.com/wiki/Making_beliefs_pay_rent pays its rent] so that you don't make a definition narrower than it needs to be. \|\| In the definition of convergence of a function at a point, Tao requires that <math>x_0</math> must be adherent to <math>E</math>. He then says that it is not worthwhile to define convergence when <math>x_0</math> is not adherent to <math>E</math>. (The idea is for the reader to make sure they understand why this assumption is good to have.)		\| Checking assumptions of objects introduced \|\| \|\| Remove or alter each assumption of the objects that have been introduced in the definition to see why they are necessary. \|\| Generally you want definitions to be "expansive" in the sense of applying to many different objects. But each assumption you introduce whittles down the number of objects the definition applies to. In other words, there is tension between (1) trying to have expansive definitions, and (2) adding in assumptions/restrictions in a definition. So you want to make sure each assumption [https://wiki.lesswrong.com/wiki/Making_beliefs_pay_rent pays its rent] so that you don't make a definition narrower than it needs to be. \|\| In the definition of convergence of a function at a point, Tao requires that <math>x_0</math> must be adherent to <math>E</math>. He then says that it is not worthwhile to define convergence when <math>x_0</math> is not adherent to <math>E</math>. (The idea is for the reader to make sure they understand why this assumption is good to have.)
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	\| Coming up with examples \|\| This should be done for ~~basically every definition~~. \|\| Come up with some examples of objects that fit the definition. Emphasize edge cases. \|\| Examples help to train your intuition of what the object "looks like". \|\| For monotone increasing functions, an edge case would be the constant function. Such an edge case is useful because it doesn't look like a "typical" object of its class, and so one might naively not think it's an example; explicitly noting the edge case helps build intuition that it's not just strictly increasing functions or functions that increase at all are covered by the definition.		\| Coming up with examples \|\| This should be done for definitions that define a class of objects or a new kind of object. \|\| Come up with some examples of objects that fit the definition. Emphasize edge cases. \|\| Examples help to train your intuition of what the object "looks like". \|\| For monotone increasing functions, an edge case would be the constant function. Such an edge case is useful because it doesn't look like a "typical" object of its class, and so one might naively not think it's an example; explicitly noting the edge case helps build intuition that it's not just strictly increasing functions or functions that increase at all are covered by the definition.
	\|-		\|-
	\| Coming up with counterexamples \|\| \|\| \|\| As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. [https://en.wikipedia.org/wiki/Confirmation_bias#Wason's_research_on_hypothesis-testing] \|\|		\| Coming up with counterexamples \|\| This should be done for definitions that define a class of objects or a new kind of object. \|\| Come up with examples of objects that ''don't'' satisfy the definition. Emphasize objects that one might intuitively think would fit the definition but in reality don't. \|\| As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. [https://en.wikipedia.org/wiki/Confirmation_bias#Wason's_research_on_hypothesis-testing] \|\| For monotone increasing functions, a counterexample is the function <math>f : \mathbf R \to \mathbf R</math> defined by <math>f(x) = x^2</math>. Even though this function ''is'' monotone on the non-negative reals, if we consider this function on all of <math>\mathbf R</math> then it stops being monotone increasing.
	\|-		\|-
	\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).		\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).

Issa Rice: /* List of steps */

2024-07-23T05:22:08Z

List of steps

← Older revision		Revision as of 05:22, 23 July 2024
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	\| Checking assumptions of objects introduced \|\| \|\| Remove or alter each assumption of the objects that have been introduced in the definition to see why they are necessary. \|\| Generally you want definitions to be "expansive" in the sense of applying to many different objects. But each assumption you introduce whittles down the number of objects the definition applies to. In other words, there is tension between (1) trying to have expansive definitions, and (2) adding in assumptions/restrictions in a definition. So you want to make sure each assumption [https://wiki.lesswrong.com/wiki/Making_beliefs_pay_rent pays its rent] so that you don't make a definition narrower than it needs to be. \|\| In the definition of convergence of a function at a point, Tao requires that <math>x_0</math> must be adherent to <math>E</math>. He then says that it is not worthwhile to define convergence when <math>x_0</math> is not adherent to <math>E</math>. (The idea is for the reader to make sure they understand why this assumption is good to have.)		\| Checking assumptions of objects introduced \|\| \|\| Remove or alter each assumption of the objects that have been introduced in the definition to see why they are necessary. \|\| Generally you want definitions to be "expansive" in the sense of applying to many different objects. But each assumption you introduce whittles down the number of objects the definition applies to. In other words, there is tension between (1) trying to have expansive definitions, and (2) adding in assumptions/restrictions in a definition. So you want to make sure each assumption [https://wiki.lesswrong.com/wiki/Making_beliefs_pay_rent pays its rent] so that you don't make a definition narrower than it needs to be. \|\| In the definition of convergence of a function at a point, Tao requires that <math>x_0</math> must be adherent to <math>E</math>. He then says that it is not worthwhile to define convergence when <math>x_0</math> is not adherent to <math>E</math>. (The idea is for the reader to make sure they understand why this assumption is good to have.)
	\|-		\|-
	\| Coming up with examples \|\| \|\| Come up with some examples of objects that fit the definition. Emphasize edge cases. \|\| Examples help to train your intuition of what the object "looks like". \|\| For monotone increasing functions, an edge case would be the constant function.		\| Coming up with examples \|\| This should be done for basically every definition. \|\| Come up with some examples of objects that fit the definition. Emphasize edge cases. \|\| Examples help to train your intuition of what the object "looks like". \|\| For monotone increasing functions, an edge case would be the constant function. Such an edge case is useful because it doesn't look like a "typical" object of its class, and so one might naively not think it's an example; explicitly noting the edge case helps build intuition that it's not just strictly increasing functions or functions that increase at all are covered by the definition.
	\|-		\|-
	\| Coming up with counterexamples \|\| \|\| \|\| As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. [https://en.wikipedia.org/wiki/Confirmation_bias#Wason's_research_on_hypothesis-testing] \|\|		\| Coming up with counterexamples \|\| \|\| \|\| As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. [https://en.wikipedia.org/wiki/Confirmation_bias#Wason's_research_on_hypothesis-testing] \|\|

Issa Rice: /* List of steps */

2024-07-22T21:23:12Z

List of steps

← Older revision		Revision as of 21:23, 22 July 2024
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	\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).		\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).
	\|-		\|-
	\| Understanding the kind of definition \|\| \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| \|\|		\| Understanding the kind of definition \|\| Do this for every definition. \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| \|\|
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	\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.		\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.

Issa Rice at 21:21, 22 July 2024

2024-07-22T21:21:39Z

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	\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).		\| Writing out a wrong version of the definition \|\| \|\| \|\| \|\| See [https://gowers.wordpress.com/2011/09/30/basic-logic-quantifiers/ this post] by Tim Gowers (search "wrong versions" on the page).
	\|-		\|-
	\| Understanding the kind of definition \|\| \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition'').		\| Understanding the kind of definition \|\| \|\| Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a ''function''); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate ''even''); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of ''addition''). See also [https://qchu.wordpress.com/2013/05/26/constructions-vs-specifications/ this post]. \|\| \|\|
	\|-		\|-
	\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.		\| Checking well-definedness \|\| If the definition defines an operation \|\| \|\| \|\| Checking that addition on the integers is well-defined.

Issa Rice: /* External links */

2020-04-26T21:39:10Z

External links

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	* http://www.abstractmath.org/MM/MMDefs.htm		* http://www.abstractmath.org/MM/MMDefs.htm
	* https://www.maa.org/node/121566 lists some other steps for both theorems and definitions		* https://www.maa.org/node/121566 lists some other steps for both theorems and definitions
	* https://en.wikipedia.org/wiki/Reverse_mathematics -- this one is more important for [[understanding theorems]]. But the idea is to think, for each theorem, its place in the structure of the theory/relationship to other theorems. see for example https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Forms_of_completeness and https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents and https://en.wikipedia.org/wiki/Mathematical_induction#Equivalence_with_the_well-ordering_principle John Stillwell (who also wrote ''Mathematics and Its History'') has a book called ''Reverse Mathematics'' that might explain this at an accessible level.		* https://en.wikipedia.org/wiki/Reverse_mathematics -- this one is more important for [[understanding theorems]]. But the idea is to think, for each theorem, its place in the structure of the theory/relationship to other theorems. see for example https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Forms_of_completeness and https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents and https://en.wikipedia.org/wiki/Mathematical_induction#Equivalence_with_the_well-ordering_principle John Stillwell (who also wrote ''Mathematics and Its History'') has a book called ''Reverse Mathematics'' that might explain this at an accessible level. See also Michael J. Schramm's ''Introduction to Real Analysis'', which has the most complete "implication structure" between all the properties of the real line I've seen in a textbook. See also James Propp's paper [https://arxiv.org/abs/1204.4483 "Real Analysis in Reverse"].
	* https://gowers.wordpress.com/2011/10/23/definitions/		* https://gowers.wordpress.com/2011/10/23/definitions/
	* https://gowers.wordpress.com/2011/10/25/alternative-definitions/		* https://gowers.wordpress.com/2011/10/25/alternative-definitions/

Issa Rice: /* List of steps */

2019-07-22T00:22:09Z

List of steps

← Older revision		Revision as of 00:22, 22 July 2019
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	\| Googling around/reading alternative texts \|\| \|\| Sometimes a definition is confusingly written (in one textbook) or the concept itself is confusing (e.g. because it is too abstract). It can help to look around for alternative expositions, especially ones that try to explain the intuitions/historical motivations of the definition. See also [[learning from multiple sources]]. \|\| \|\| In mathematical logic, the terminology for formal languages is a mess: some books define a structure as having a domain and an interpretation (so structure = (domain, interpretation)), while others define the same thing as interpretation = (domain, denotations), while still others define it as structure = (domain, signature, interpretation). The result is that in order to not be confused when e.g. reading an article online, one must become familiar with a range of definitions/terminology for the same concepts and be able to quickly adjust to the intended one in a given context.<br><br>To give another example from mathematical logic, there is the expresses vs captures distinction. But different books use terminology like arithmetically defines vs defines, represents vs expresses, etc. So again things are a mess.		\| Googling around/reading alternative texts \|\| \|\| Sometimes a definition is confusingly written (in one textbook) or the concept itself is confusing (e.g. because it is too abstract). It can help to look around for alternative expositions, especially ones that try to explain the intuitions/historical motivations of the definition. See also [[learning from multiple sources]]. \|\| \|\| In mathematical logic, the terminology for formal languages is a mess: some books define a structure as having a domain and an interpretation (so structure = (domain, interpretation)), while others define the same thing as interpretation = (domain, denotations), while still others define it as structure = (domain, signature, interpretation). The result is that in order to not be confused when e.g. reading an article online, one must become familiar with a range of definitions/terminology for the same concepts and be able to quickly adjust to the intended one in a given context.<br><br>To give another example from mathematical logic, there is the expresses vs captures distinction. But different books use terminology like arithmetically defines vs defines, represents vs expresses, etc. So again things are a mess.
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	\| Drawing a picture \|\| \|\| \|\| \|\| Pugh's ''Real Mathematical Analysis'', Needham's ''Visual Complex Analysis''.		\| Drawing a picture \|\| Ideally ask this about every definition. But some subfields of math (e.g. analysis) are a lot more visual than others (e.g. mathematical logic). \|\| \|\| \|\| Pugh's ''Real Mathematical Analysis'', Needham's ''Visual Complex Analysis''.
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	\| Chunking/processing level by level \|\| If a definition involves multiple layers of quantifiers. \|\| \|\| \|\| See Tao's definitions for <math>\varepsilon</math>-close, eventually <math>\varepsilon</math>-close, <math>\varepsilon</math>-adherent, etc.		\| Chunking/processing level by level \|\| If a definition involves multiple layers of quantifiers. \|\| \|\| \|\| See Tao's definitions for <math>\varepsilon</math>-close, eventually <math>\varepsilon</math>-close, <math>\varepsilon</math>-adherent, etc.

Issa Rice: /* Relying on experience and intuition */

2019-07-22T00:18:17Z

Relying on experience and intuition

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	===Relying on experience and intuition===		===Relying on experience and intuition===

	Eventually, after one has studied a lot of mathematics, understanding definitions becomes more automatic. One can gain an intuition of which steps are important for a particular definition, or when to spend some time and when to move quickly.		Eventually, after one has studied a lot of mathematics, understanding definitions becomes more automatic. One can gain an intuition of which steps are important for a particular definition, or when to spend some time and when to move quickly. One naturally asks the important questions, and can let curiosity guide one's exploration.

	==When reading textbooks==		==When reading textbooks==