更改

The fact that renowned theoretical physicists like Poincaré, Boltzmann or Gibbs were not awarded the Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory.<ref>{{cite journal|last1=Crawford |first1= Elizabeth |title=Nobel: Always the Winners, Never the Losers|journal=[[Science (journal)|Science]]|date=13 November 1998|volume=282|issue=5392|pages=1256–1257|doi=10.1126/science.282.5392.1256|bibcode = 1998Sci...282.1256C |s2cid= 153619456 }}{{dead link|date=July 2016}}</ref><ref>{{cite journal|last1=Nastasi|first1=Pietro|title=A Nobel Prize for Poincaré? |journal=Lettera Matematica|date=16 May 2013|volume=1|issue=1–2|pages=79–82|doi=10.1007/s40329-013-0005-1 |url= |accessdate=|doi-access=free}}</ref> In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique.<ref name="gray-biography"/>

The fact that renowned theoretical physicists like Poincaré, Boltzmann or Gibbs were not awarded the Nobel Prize is seen as evidence that the Nobel committee had more regard for experimentation than theory.<ref>{{cite journal|last1=Crawford |first1= Elizabeth |title=Nobel: Always the Winners, Never the Losers|journal=[[Science (journal)|Science]]|date=13 November 1998|volume=282|issue=5392|pages=1256–1257|doi=10.1126/science.282.5392.1256|bibcode = 1998Sci...282.1256C |s2cid= 153619456 }}{{dead link|date=July 2016}}</ref><ref>{{cite journal|last1=Nastasi|first1=Pietro|title=A Nobel Prize for Poincaré? |journal=Lettera Matematica|date=16 May 2013|volume=1|issue=1–2|pages=79–82|doi=10.1007/s40329-013-0005-1 |url= |accessdate=|doi-access=free}}</ref> In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique.<ref name="gray-biography"/>

==Philosophy==

==Philosophy哲学==

On algebraic topology:

On algebraic topology:

Poincaré had philosophical views opposite to those of [[Bertrand Russell]] and [[Gottlob Frege]], who believed that mathematics was a branch of [[logic]]. Poincaré strongly disagreed, claiming that [[intuition (knowledge)|intuition]] was the life of mathematics. Poincaré gives an interesting point of view in his book ''[[Science and Hypothesis]]'':

Poincaré had philosophical views opposite to those of [[Bertrand Russell]] and [[Gottlob Frege]], who believed that mathematics was a branch of [[logic]]. Poincaré strongly disagreed, claiming that [[intuition (knowledge)|intuition]] was the life of mathematics. Poincaré gives an interesting point of view in his book ''[[Science and Hypothesis]]'':

 

| url=http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf}}. The first systematic study of topology.

| url=http://www.maths.ed.ac.uk/~aar/papers/poincare2009.pdf}}. The first systematic study of topology.

{{quote|text=For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule.}}

{{quote|text=For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule.}}

 

On celestial mechanics:

On celestial mechanics:

Poincaré believed that [[arithmetic]] is [[Analytic/synthetic distinction|synthetic]]. He argued that [[Peano's axioms]] cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is ''[[A priori and a posteriori|a priori]]'' synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of [[Immanuel Kant]] (Kolak, 2001, Folina 1992). He strongly opposed Cantorian [[set theory]], objecting to its use of [[Impredicativity|impredicative]] definitions{{Citation needed|date=March 2018}}.

Poincaré believed that [[arithmetic]] is [[Analytic/synthetic distinction|synthetic]]. He argued that [[Peano's axioms]] cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is ''[[A priori and a posteriori|a priori]]'' synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of [[Immanuel Kant]] (Kolak, 2001, Folina 1992). He strongly opposed Cantorian [[set theory]], objecting to its use of [[Impredicativity|impredicative]] definitions{{Citation needed|date=March 2018}}.

 

However, Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure of [[Non-Euclidean geometry|non-Euclidean space]] can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "[[conventionalism]]".<ref>Yemima Ben-Menahem, ''Conventionalism: From Poincare to Quine'', Cambridge University Press, 2006, p. 39.</ref> Poincaré believed that [[Newton's first law]] was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012).<ref>{{Citation|author=Gargani Julien|title=Poincaré, le hasard et l'étude des systèmes complexes|publisher=L'Harmattan|year=2012|page=124|url=http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754|access-date=5 June 2015|archive-url=https://web.archive.org/web/20160304140554/http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754|archive-date=4 March 2016|url-status=dead}}</ref> He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to [[Euclidean geometry]] that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.<ref>{{Citation|title=Science and Hypothesis|first1=Henri |last1=Poincaré |publisher=Cosimo, Inc. Press|year=2007|isbn=978-1-60206-505-5 |page=50

However, Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure of [[Non-Euclidean geometry|non-Euclidean space]] can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "[[conventionalism]]".<ref>Yemima Ben-Menahem, ''Conventionalism: From Poincare to Quine'', Cambridge University Press, 2006, p. 39.</ref> Poincaré believed that [[Newton's first law]] was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012).<ref>{{Citation|author=Gargani Julien|title=Poincaré, le hasard et l'étude des systèmes complexes|publisher=L'Harmattan|year=2012|page=124|url=http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754|access-date=5 June 2015|archive-url=https://web.archive.org/web/20160304140554/http://www.editions-harmattan.fr/index.asp?navig=catalogue&obj=livre&no=38754|archive-date=4 March 2016|url-status=dead}}</ref> He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to [[Euclidean geometry]] that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.<ref>{{Citation|title=Science and Hypothesis|first1=Henri |last1=Poincaré |publisher=Cosimo, Inc. Press|year=2007|isbn=978-1-60206-505-5 |page=50

|url=https://books.google.com/books?id=2QXqHaVbkgoC&pg=PA50}}</ref>

|url=https://books.google.com/books?id=2QXqHaVbkgoC&pg=PA50}}</ref>

 

On the philosophy of mathematics:

On the philosophy of mathematics:

关于数学哲学:

关于数学哲学:

===Free will===

===Free will自由意志===

Poincaré's famous lectures before the Société de Psychologie in Paris (published as ''[[Science and Hypothesis]]'', ''[[The Value of Science]]'', and ''Science and Method'') were cited by [[Jacques Hadamard]] as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.<ref>Hadamard, Jacques. ''An Essay on the Psychology of Invention in the Mathematical Field''. Princeton Univ Press (1945)</ref>

Poincaré's famous lectures before the Société de Psychologie in Paris (published as ''[[Science and Hypothesis]]'', ''[[The Value of Science]]'', and ''Science and Method'') were cited by [[Jacques Hadamard]] as the source for the idea that creativity and invention consist of two mental stages, first random combinations of possible solutions to a problem, followed by a critical evaluation.<ref>Hadamard, Jacques. ''An Essay on the Psychology of Invention in the Mathematical Field''. Princeton Univ Press (1945)</ref>

 

Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves [[Randomness|chance]].

Although he most often spoke of a deterministic universe, Poincaré said that the subconscious generation of new possibilities involves [[Randomness|chance]].

<blockquote>

<blockquote>

It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.<ref>{{cite book|title =Science and Method|chapter= 3: Mathematical Creation|date= 1914|chapter-url = https://ebooks.adelaide.edu.au/p/poincare/henri/science-and-method/book1.3.html|first = Henri|last =Poincaré }}</ref>

It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.<ref>{{cite book|title =Science and Method|chapter= 3: Mathematical Creation|date= 1914|chapter-url = https://ebooks.adelaide.edu.au/p/poincare/henri/science-and-method/book1.3.html|first = Henri|last =Poincaré }}</ref>

</blockquote>

</blockquote>

Poincaré's two stages—random combinations followed by selection—became the basis for [[Daniel Dennett]]'s two-stage model of free will.<ref>Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293</ref>

Poincaré's two stages—random combinations followed by selection—became the basis for [[Daniel Dennett]]'s two-stage model of free will.<ref>Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on Mind and Psychology. The MIT Press, p.293</ref>

 

==Bibliography==

==Bibliography==