Platonism - Revision history

Lfox: /* Aristotle's refutation */

2024-04-30T23:02:46Z

Aristotle's refutation

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	=== Aristotle's refutation ===		=== Aristotle's refutation ===
	[TODO I heard this argument once and it was great, but I forget it. Something about how the idea of objects "participating in" a form leads to an infinite regress.		[TODO I heard this argument once and it was great, but I forget it. Something about how the idea of objects "participating in" a form leads to an infinite regress. Something like: if an object participates in its form, what sort of thing is the participation? There needs to be another form of that type of object participating in that type of form. And so on.

	== References ==		== References ==

Lfox: /* Examples */

2024-02-11T04:15:15Z

Examples

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	From a book by Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:		From a book by Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:

	<blockquote>For larger values of <math>n</math>, even the language of stacks is '''not sufficient to describe the nature of''' the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From an interview with Peter Scholze<ref>''Interview with Peter Scholze''. https://www.youtube.com/watch?v=HYZ3reRcVi8 beginning at 7:46.</ref>:<blockquote>So actually, I'm not a creative person at all. I'm a hardcore mathematical Platonist, so I believe that there is a mathematical world for us to discover. And in this way, I'm not creative. It's like, when I am trying to learn math, I am trying to learn what's there. It's that I'm trying to give it names, but that's all.</blockquote>		<blockquote>For larger values of <math>n</math>, even the language of stacks is '''not sufficient to describe the nature of''' the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From a memoir by Alexandre Grothendieck, ''Récoltes et Semailles''<ref>Grothendieck, Alexandre. ''Récoltes et Semailles'', 1983-1986. https://agrothendieck.github.io/divers/ReS.pdf pp 346.</ref>:<blockquote>À vrai dire, c’est la “tour de Teichmüller” dans laquelle la famille de toutes ces multiplicités s’insère, et le paradigme discret ou profini de cette tour en termes de groupoïdes fondamentaux, qui constitue '''l’objet unique le plus riche, le plus fascinant que j’aie rencontré en mathématique'''. Le groupe Sl(2), avec la structure “arithmétique” du compactifié profini de Sl(2,Z), (consistant en l’opération du groupe de Galois <math display="inline">\text{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})</math> sur celui-ci), peut être considéré comme '''la principale pierre de construction''' pour la “version profinie” de cette tour.</blockquote>From an interview with Peter Scholze<ref>''Interview with Peter Scholze''. https://www.youtube.com/watch?v=HYZ3reRcVi8 beginning at 7:46.</ref>:<blockquote>So actually, I'm not a creative person at all. I'm a hardcore mathematical Platonist, so I believe that there is a mathematical world for us to discover. And in this way, I'm not creative. It's like, when I am trying to learn math, I am trying to learn what's there. It's that I'm trying to give it names, but that's all.</blockquote>

	== Refutations ==		== Refutations ==

Lfox: /* Examples */

2024-02-04T06:57:23Z

Examples

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	== Examples ==		== Examples ==
	In this section, I will show some examples of ~~implicitly~~ Platonic things written by ~~working~~ mathematicians.		In this section, I will show some examples of Platonic things written by leading 20th or 21st-century mathematicians.

	The purpose of this section is to justify my claim that the premises of Platonism are widespread.		The purpose of this section is to justify my claim that the premises of Platonism are widespread.

	It is easy to find examples, because almost every single sentence of modern math is ~~Platonic~~.		From one perspective, it is easy to find examples, because almost every single sentence of a modern math paper sounds Platonic. From another perspective, it is difficult to find examples, because it's hard to know if any given mathematician literally believes what he's saying, or if he just sees it as more of a convention to which he is adhering.

	On the ~~[[Intuitionism]] and [[Formalism/Logicism]] pages~~, I ~~will quote from ''the same'' papers written by ''the same'' mathematicians~~.		In some cases, for the benefit of the reader, I have bolded words which I think reflect Platonism.

	~~At first glance~~, ~~these quotes will appear to be innocuous~~. ~~They have the form of statements about real things~~. ~~They are Platonic because they are not about real things~~.		From a book by Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:

	~~In the quotes, I have bolded some of the words which I think reflect Platonism~~		<blockquote>For larger values of <math>n</math>, even the language of stacks is '''not sufficient to describe the nature of''' the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From an interview with Peter Scholze<ref>''Interview with Peter Scholze''. https://www.youtube.com/watch?v=HYZ3reRcVi8 beginning at 7:46.</ref>:<blockquote>So actually, I'm not a creative person at all. I'm a hardcore mathematical Platonist, so I believe that there is a mathematical world for us to discover. And in this way, I'm not creative. It's like, when I am trying to learn math, I am trying to learn what's there. It's that I'm trying to give it names, but that's all.</blockquote>

	~~From Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:~~

	<blockquote>For larger values of <math>n</math>, even the language of stacks is '''not sufficient to describe the nature of''' the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From ~~[TODO]~~

	== Refutations ==		== Refutations ==

Lfox at 23:04, 30 January 2024

2024-01-30T23:04:06Z

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	'''Platonism''' is a philosophy of mathematics, which holds that mathematical concepts are objects ("''forms"''), existing independently of man's mind, that cannot be perceived through the senses.		'''Platonism''' is a philosophy of mathematics, which holds that mathematical concepts are objects ("''forms"''), existing independently of man's mind, that cannot be perceived through the senses.

	Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally.		Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally, which applies to much more than just mathematics.

	Platonism is one of the ~three philosophies which dominate modern mathematics, the others being [[Intuitionism]] and [[Formalism\|Formalism/Logicism]]. ~~Few if any~~ mathematicians adhere consistently to any of those three philosophies.		Platonism is one of the three major philosophies which dominate the minds of modern mathematics, the others being [[Intuitionism]] and [[Formalism\|Formalism/Logicism]]. Very few working mathematicians adhere consistently to any of those three philosophies. Almost all of them tacitly accept the (mutually inconsistent) premises of all three.

	== Examples ==		== Examples ==

Lfox at 19:45, 24 January 2024

2024-01-24T19:45:49Z

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	Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally.		Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally.

	Platonism is one of the ~three philosophies which dominate modern mathematics, the others being [[Intuitionism]] and [[Formalism\|Formalism/Logicism]]. ~~However, few~~ if any mathematicians adhere consistently to any of those three philosophies.		Platonism is one of the ~three philosophies which dominate modern mathematics, the others being [[Intuitionism]] and [[Formalism\|Formalism/Logicism]]. Few if any mathematicians adhere consistently to any of those three philosophies.

	== Examples ==		== Examples ==
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	From Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:		From Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:

	<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From [TODO]		<blockquote>For larger values of <math>n</math>, even the language of stacks is '''not sufficient to describe the nature of''' the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From [TODO]

	== Refutations ==		== Refutations ==

Lfox at 22:50, 23 January 2024

2024-01-23T22:50:12Z

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	<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From [TODO]		<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From [TODO]

			== Refutations ==

			=== Aristotle's refutation ===
			[TODO I heard this argument once and it was great, but I forget it. Something about how the idea of objects "participating in" a form leads to an infinite regress.

	== References ==		== References ==

Lfox: /* Examples */

2024-01-23T21:22:10Z

Examples

← Older revision		Revision as of 21:22, 23 January 2024
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	At first glance, these quotes will appear to be innocuous. They have the form of statements about real things. They are Platonic because they are not about real things.		At first glance, these quotes will appear to be innocuous. They have the form of statements about real things. They are Platonic because they are not about real things.

			In the quotes, I have bolded some of the words which I think reflect Platonism

	From Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:		From Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:

Lfox at 21:18, 23 January 2024

2024-01-23T21:18:34Z

← Older revision		Revision as of 21:18, 23 January 2024
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	At first glance, these quotes will appear to be innocuous. They have the form of statements about real things. They are Platonic because they are not about real things.		At first glance, these quotes will appear to be innocuous. They have the form of statements about real things. They are Platonic because they are not about real things.

	<~~blockquote>For larger values of <math>n</math~~>, ~~even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>~~. ~~To address the situation~~, ~~Grothendieck proposed [~~...~~] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0~~</~~math~~>.		From Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref>:

			<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.</blockquote>From [TODO]
	~~-- Jacob Lurie~~, ~~''Higher Topos Theory''~~<~~ref~~>~~Lurie, Jacob~~. ~~''Higher Topos Theory''. Princeton~~, N.J.~~, Princeton University Press, 2009~~.</~~ref~~></blockquote>

	== References ==		== References ==

Lfox: /* Examples */

2024-01-23T21:17:38Z

Examples

← Older revision		Revision as of 21:17, 23 January 2024
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	<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.		<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.


	-- Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref></blockquote>		-- Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref></blockquote>

	== References ==		== References ==

Lfox: /* Examples */

2024-01-23T21:17:16Z

Examples

← Older revision		Revision as of 21:17, 23 January 2024
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	'''Platonism''' is a philosophy of mathematics, which holds that mathematical concepts are objects ("''forms"''), existing independently of man's mind, that cannot be perceived through the senses.		'''Platonism''' is a philosophy of mathematics, which holds that mathematical concepts are objects ("''forms"''), existing independently of man's mind, that cannot be perceived through the senses.

			Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally.

	Platonism is one of the ~three philosophies which dominate modern mathematics, the others being [[Intuitionism]] and [[Formalism\|Formalism/Logicism]]. However, few if any mathematicians adhere consistently to any of those three philosophies.		Platonism is one of the ~three philosophies which dominate modern mathematics, the others being [[Intuitionism]] and [[Formalism\|Formalism/Logicism]]. However, few if any mathematicians adhere consistently to any of those three philosophies.

	~~Platonism dates back to Plato and Socrates. The term "Platonism" is also sometimes used to refer to the philosophy of Plato generally.~~

	== Examples ==		== Examples ==
	In this section, I will show some examples of implicitly Platonic things written by working mathematicians.		In this section, I will show some examples of implicitly Platonic things written by working mathematicians.

	The purpose of this section is to justify my claim that the premises of Platonism are widespread		The purpose of this section is to justify my claim that the premises of Platonism are widespread.

			It is easy to find examples, because almost every single sentence of modern math is Platonic.

			On the [[Intuitionism]] and [[Formalism/Logicism]] pages, I will quote from ''the same'' papers written by ''the same'' mathematicians.

			At first glance, these quotes will appear to be innocuous. They have the form of statements about real things. They are Platonic because they are not about real things.

			<blockquote>For larger values of <math>n</math>, even the language of stacks is not sufficient to describe the nature of the sheaf <math>\mathcal{F}</math> associated to the fibration <math>\tilde{X} \rightarrow X</math>. To address the situation, Grothendieck proposed [...] that there should be a theory of <math>n</math>-stacks on <math>X</math> for every integer <math>n \geq 0</math>.

	~~modern mathematicians~~		-- Jacob Lurie, ''Higher Topos Theory''<ref>Lurie, Jacob. ''Higher Topos Theory''. Princeton, N.J., Princeton University Press, 2009.</ref></blockquote>

	~~saying things that implicit~~		== References ==