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Archive 1 |
The joy of wikipedia is the teaching of feverent independent thinkers to honor disagreements also of a deep kind.
...then why is it still a mere shadow of philosophy of mathematics? This deserves to be remedied, especially as this is not tucked away in an obscure corner but can be linked to directly from the mathematics portal (care of the rather nifty Topics in Mathematics table).
My opinion is that attention should be focused on those programs various mathematicians and logicians have advanced to supply a formal foundation (or more generally something along the way towards one). The main lines have been set theory (Cantor -> Zermelo -> ZFS and others) and type theory (Frege -> Russell -> Ramsey and Church -> ... -> Martin-Löf) with assorted logical calculi (often in league with type theory) and more recently category and topos theory too. This work has begun to bear fruit in the form of programs for computer-assisted proof and automated theorem proving, which I think are areas that will grow in importance as the "heroic" proofs get longer and longer and more difficult to check for errors.
Abt 12 04:57, 24 March 2006 (UTC)
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 04:00, 10 November 2007 (UTC)
The statement that:
"In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided."
leads one to believe that the formal foundations of modern mathematics contains logical paradoxes. This is not generally beleoved to be the case. Unless the author is able to point to logical paradoxes present in what is considered to be the formal foundations of mathematics, I suggest that this line be changed to:
"In most of mathematics as it is practiced, the incompleteness of the underlying formal theories never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be treated carefully."
71.148.59.179 ( talk) 04:07, 3 August 2008 (UTC)
Clarifying, about least upper bounds: of course the definition "the upper bound that is less than or equal to any other upper bound" is impredicative. But x is the l.u.b. of a set A if and only if every element of A is less than or equal to x, and for every natural number n there is an element of A greater than x - 1/n. That definition seems perfectly predicative to me, athough Weyl would not accept it as such. Predicative subsystems of second-order arithmetic, such as ACA0, can prove things like "every bounded set of rational numbers has a least upper bound" and "every bounded open set of real numbers has a least upper bound". — Carl ( CBM · talk) 01:43, 24 September 2008 (UTC)
I was once told that mathematics had several roots, a set of assumptions, such as set theory, peano arithmetic or lambda calculus. From any of these roots it was possible to derive the same theorems. Is this a widely held view? If so, perhaps it could be made clearer in the article. pgr94 ( talk) 18:00, 17 February 2010 (UTC)
The section Platonism ends with a (cited) question out of the blue:
It would be profitable if this question be put in a context, such as reference [3] being a critic of the platonic view, otherwise I cannot perceive the question as especially "obvious" since it is not too far fetched to perceive our brains as an "eye" that can be taught to look straight into that real world of mathematical objects, and also into other worlds. Another more platonic answer would of course be, that since we are shadowy copies of that mathematical plane, some of us would retain a semblance of the mathematical plane. Rursus dixit. ( mbork3!) 20:18, 29 June 2010 (UTC)
First of all, it’s nothing but a blurb of personal opinion in comment form, and should go on the discussion page. Not the actual article. Second of all, it contains a blatant error in logic, by stating “But the working number theorist is concerned with proving theorems from initial assumptions which are obviously true using proof methods which are obviously correct, not with any particular logical system.” [emphasis mine]. “Obviously” is a weasel word, stating nothing about actual truth, and showing that the author has not understood, that this exact thing (“what do you mean with ‘obvious’? what is actually true?”) is the exact problem that the foundational schools are trying to solve. For example: Someone might state, that it is “obvious”, that the sun is yellow. Until he is told to look at it at dusk. Meaning that humans can not physically state objective things anyway, since all input their brain receives, is already subjective (processed by senses and other sources), and the brain additionally makes it even more subjective, since processing only happens relative to previous experiences. (You can look up how neural networks work.)
Since I find deletion of the opinions of others in a public spaces (which in my eyes includes Wikipedia) to be morally wrong and a violation of free speech, I have not deleted but moved the section right below this one. —Preceding unsigned comment added by 188.100.192.146 ( talk) 09:17, 19 August 2010 (UTC)
The philosophy of math article is getting bigger, the stuff about all this "problems with the foundations of maths" and "Godel proves that all truth is relative" crap is getting a little tighter.
FPOM and FOM should probably be merged to FPOM, not the other way around though. ZF(C) is A foundation of math, but the foundations PROBLEM raised by Hilbert, and the Godel and co. are the meat.
Thoughts? -- M a s 18:38, 22 May 2006 (UTC)
JA: The main thing is that the current split is a WikiPeculiar Neologism that does not really exist in the Literatures, as all of these questions have always been discussed under ∃! umbrella, to wit, FOM. Jon Awbrey 18:48, 22 May 2006 (UTC)
JA: The term "foundations of math", as with the longstanding FOM discussion list, is generally taken to include the consideration of problems associated with foundational questions. If I don't hear any strong objections in the next day or so, I'll go ahead and do the merge. Jon Awbrey 21:46, 29 May 2006 (UTC)
JA: I don't see much prospect of folding this bit of Foma back into the Philosophy of mathematics article, as things are more likely to spin the other way sooner or later. In math as in architecture, as any home-owner knows, problems with foundations just go with the territory, so I think it's still the best course to combine those two. Jon Awbrey 13:40, 31 May 2006 (UTC)
JA: I begin to think that it should be called the "Identity crisis in some people's mathematics", as there seems to be a need to change the name of the article on a recursèd basis. Well, enough of that not-so-abstract non-sense. Jon Awbrey 15:54, 31 May 2006 (UTC)
In his Chapter 39. Foundations, Anglin has three sections titled “Platonism”, “Formalism” and “Intuitionism”. I will quote most of “Intuitionism”. Anglin begins each as follows:
wvbailey Wvbailey 16:01, 27 June 2006 (UTC)
I am just as confused as everyone else above as to exactly what is meant by "foundations" -- is it "philosphy of approach" re the existence of mathematical "objects", or what? (I usually think of it as the core logic and axioms especially with regard to arithmetic (numbers) -- but clearly a specific choice of axioms/methods is made within a philosophic framework). So I quote Anglin above -- he wasn't confused cf his chapter 39. At least Anglin produces some "quotable objects" in the spirit of inline citation. I will work on this more when I get back to my books. wvbailey Wvbailey 18:01, 6 October 2006 (UTC)
I cc'd this over from the Intuitionism talk page. The question I have after reading all of the above: what is the "modern" take on "foundations"? We still have the Platonists (Roger Penrose for example; his Emperor's New Mind really surprised me) and the Logicists (almost everyone else who isn't a computer scientist or a Platonist [?]) and the Intuitionists (aka computer scientists, perhaps with their "IF then ELSE" constructions)... but what is going on in "modern" thought?. The following I thought was interesting: (wvbailey Wvbailey 18:14, 8 October 2006 (UTC))
Stewart's reference in E.B.:
wvbailey Wvbailey 13:10, 15 June 2006 (UTC)
OK, there's some misinformation in the remarks by wvbailey, which I'd like to address. First off, hardly anyone claims to be a "logicist" these days. There are a few serious thinkers who will call themselves "neo-logicist" or some such. But logicism in its original Frege–Russell form (mathematical truth can be derived from logic alone, without any assumptions not justifiable by logic) is pretty hard to sustain in the wake of the Russell paradox (which refuted the logicist version of set theory) and the Gödel theorems (which made a logicist take even on arithmetic seem very problematic, though they may not have refuted such an account outright).
What I suspect wvbailey means by "logicist" is really more like "formalist" (the content of mathematics is what we can formally prove from well-specified assumptions).
Formalism is probably the dominant view (or at least publicly professed view) among, to put it bluntly, those mathematicians who haven't really thought about it very much and don't really want to. Since that's most of them, it comes out dominant overall. That's not to say there aren't serious thinkers who are formalist. But it is to say that it's a very appealing position to those who don't want to spend much time on it. They dismiss the question and move on, before the harder questions show up, like: "If mathematics is about formal theorems, then why have you never proved a nontrivial theorem formally?" and "If you really think that's what math is about, then why on Earth do you care about math?".
As for intuitionism, the above discussion of motivations does not apply only to them, nor to all of them. The defining feature of an intuitionist is that he doesn't accept excluded middle. Lots of formalists and fictionalists and instrumentalists (who might recognize themselves in the above discussion) accept excluded middle (and thus are not intuitionists). And plenty of intuitionists are very realist (that is, Platonist) about the natural numbers individually, even if not about the totality of natural numbers as a completed whole. --
Trovatore 15:57, 10 October 2006 (UTC)
Re Constructivism as a foundational "movement": "computer science" is necessarily constructive (hmm... I wonder why... pure syntax?). Has there been any "foundations" discussions/papers re the effects of "computer science" on "foundations?" Wasn't the four-color theorem finally proved by exhaustive search? This must have "rippled thru" the mathematics community. In a similar vein, the Turing test debates of the past 60 years must have had some effect on "foundations", it certainly roiled the philosphers-of-mind:
Re philosophy of mind playing a role in foundations: I am currently reading parts of a book by the philosopher-of-mind John R. Searle (2002), Consciousness and Language, Cambridge University Press, Cambridge England. He comes back to, again and again, the notion that a mind adding 2+2 as different from a computer adding 2+2 because the latter (computer, calculator) case is one of syntax only (symbols in relation to symbols) but the former "has semantic content" ( i.e. "meaning"-- whatever that means...) as well (and he keeps bringing up intentionality... but I'm not convinced ...). In the machine's case "the information" is defined exterior to the machine, and becomes "information" only in the eye of the machine's beholder (e.g. the computer programmer -- who serves as the homunculus for the machine (cf he states this explicitly p. 122)):
The last paragraph of the "foundations" article touches on this notion of "mind of the mathematician" (without references). This Searle philosophy seems germane to the article but how hasn't crystallized yet. Maybe this will trigger some other folks' thinking. wvbailey Wvbailey 18:55, 12 October 2006 (UTC)
Searle doesn't have too much good to say about "cognitive science" (of any sort, mathematical or otherwise) and its attempts to devolve everything into "algorithms" (cf pp. 108 ff) -- in his view "algorithms" are just "syntax in motion" -- sound and fury implying nothing. wvbailey Wvbailey 19:02, 12 October 2006 (UTC)
The section on formalism is currently dominated by quotations from Weyl. This may be excessive not only because Weyl is highly critical of formalism here, but also because Weyl himself rejected intuitionism and retracted most of these views, merely a few years later. Perhaps some quotes from Bernays would be more appropriate. The way it stands now, we learn more about Weyl's somewhat vague philosophical speculations about formalism than about formalism itself. The main point that needs to be emphasized is that Hilbert saw formalism as a way of answering paradoxes of set theory. It's not that mathematics is a game, but merely a way of grounding mathematics in finitistic procedures so as to minimize risk of paradox. Formalism is not a denial that mathematics is meaningful, but rather an attempt to steer clear of foundational paradoxes. Tkuvho ( talk) 17:33, 8 December 2011 (UTC)
--
Actually I'd like to see more of the nasty dialog in this article (ie. the "crisis" part of it) -- the criticisms and counterclaims/rebuttals help us understand the issues. More on this to follow in another section.
I'm trying to broaden us out, pulling the lens back so we get a wider-angle view of "foundations of mathematics" and what "foundations" is all about/what it means. I'm using as references the following books, the 4th being dubious (but it has Dedekind's and Cantor's works). Notice that the years stop at about 1936-7 with Turing. So there's lots of stuff missing:
I. Grattan-Guinness 2000 The Search for Mathematical Roots 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Canotr Through Russell to Goedel, Princeton University Press, Princeton NJ, ISBN 0-691-05858-X (pbk.: alk. paper)
Paolo Mancosu 1998 From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, NY, ISBN 0-19-509632-0 pb.
Jean van Heijenoort, 1967, From Frege to Goedel: A Source Book in Mathematical Logic 1879-1931, 3rd Printing 1976, Harvard University Press, cambridge, MA, ISBN 0-674-32449-8 (pbk).
Stephen Hawking ed., 2005, And God Created the Integers: The Mathematical breakthrougs that Changed History, Running Press, Philadelphia PA, ISBN-13: 978-0-7624-1922-7.
Howard Eves 1990 Foundations and Fundamental Concepts of Mathematics: Third Edition, Dover Publications, Inc, Mineola, NY, ISBN: 0-486-69609-X (pbk.)
Charles Seife 2000 Zero: the Biography of a Dangerous Idea, Penguin Books, NY, NY, ISBN 0 14 02.9647 6 (pbk).
W. S. Anglin 1994 Mathematics: A Concise History and Philosophy, Springer-Verlag, New York, ISBN: 0-387-94280-7.
Ian Stewart 2007 Taming the Infinite: The story of mathematics from the first numbers to chaos theory, Quercus Publishing Plc, London UK, ISBN: 978 1 84724 7 68 1.
The book by Eves, above, offers a list of three "foundational crises". The advantage of using this is that it is clearly not OR. The disadvantage of sticking to it is Eves has missed some "difficulties" in ancient and medieval foundations that spurred on the development of modern mathematics. Some of these can be inferred from the topics in the other books above (Eves has some but not all of these): symbolization, 0, the infinite, negative numbers; roots of equations and the square root of a negative number; convergence, limits, continuity; inadequacy of Aristotelian logic and the school-men forms (e.g. failure of Euler diagrams), etc. This is not a complete list, just an example.
Eves identifies 3 major “profoundly disturbing” crises of foundations (pages 262-263):
My intention is to work up a list of "difficulties". Then move to a list of "foundations" (ancient, medieval, early modern, modern) e.g. topics like axiomatization, proof; algebraic generalization and symbol-manipulation; calculus (analysis), etc. Any thoughts? Bill Wvbailey ( talk) 22:34, 12 December 2011 (UTC)
An editor recently commented that "Actually I'd like to see more of the nasty dialog in this article (ie. the "crisis" part of it) -- the criticisms and counterclaims/rebuttals help us understand the issues". My personal preference would be to have succinct sympathetic statements of the positions of the various schools. Other editors are welcome to chip in. Tkuvho ( talk) 12:47, 14 December 2011 (UTC)
---
Here are some examples of "the nasty dialog" so editors know what this is about. Even this small selection either directly covers or hints at a wide range of foundational issues -- the nature of mathematical thought, the completed infinite, the continuum, impredicativity, tacit contentual assumptions vs formalism, the nature of mathematical induction, the various philosophic "-isms" (Finitism/constructivism/intuitionism, Logicism, Formalism):
Strict finitism vs set theorists: Those who were mishandled by Kronecker, in particular: Dedekind and Cantor: Here's Dedekind 1887:
The infinite taken as a completed whole -- Cantor: There's a whole chapter to be written here about how badly treated he was (or felt he was) by Kronecker and Kronecker's Crelle's Journal.
The continuum: Analysis and Dedekind cuts, Zermelo's set theory: Apparently any "contentual assumptions" in a theory were an anathema to Hilbert. Here's Bernays 1922: "In Dedekind's grounding of analysis what is taken as a basis is the system of the elements of the continuum, and Zermelo's construction of set theory it is the domain of operations B" (p. 215 in Mancosu 1998). After a glowing tribute to the ideas of Zermelo, Hilbert adds this zinger:
Impredicative definitions and Zermolo's set theory: In a section b. of his 1908 The Possibility of a Well-Ordering, Zermolo states:
The following is a very important quote, e.g. taken up in detail by Goedel 1944. This is Zermelo
Hilbert criticises everyone:
Re the completed infinite, the LoEM -- Hilbert's famous boxer quote:
Bill Wvbailey ( talk) 16:23, 14 December 2011 (UTC)
My essay about problems related to this topic in Wikipedia. A help to improve the quality of articles on mathematical logic, currently being unsatisfactory, is appreciated. Incnis Mrsi ( talk) 09:10, 5 March 2012 (UTC)
I removed the reference of the book by W.S. Anglin as I read this critique. I did not see the book however. Can someone who saw the book confirm or infirm the validity of this critique ? If it is valid, can someone justify to include the reference of a book like this in this article ? I mean, it did not seem to me that the quotations from this book had any specially unique value here. I guess that if any reference is needed, other sources can be found as well in replacement. Sorry I did not make a search on the issue. As for Gödel, I mention him for his set-theoretical platonism later in the article where modern mathematics are concerned. Quoting him in a section about the Ancient Greek philosophy would be anachronistic, and there is no need to quote Anglin about Gödel for giving a definition of Platonism. Spoirier ( talk) 22:00, 16 September 2012 (UTC)
The following entry is in the more Paradoxes section of the article, but doesn't seem to be part of mathematics and further it isn't a paradox. Any reason not to remove it? RJFJR ( talk) 19:02, 16 January 2014 (UTC)
The comment(s) below were originally left at Talk:Foundations of mathematics/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
This article is just an outline, but it could be expanded to a more thorough description. That is, there are a lot of references but the content is shallow. CMummert - 5 Oct 2006 |
Last edited at 20:47, 16 April 2007 (UTC). Substituted at 14:48, 1 May 2016 (UTC)
Under "Group theory", Wantzel gets the credit for the work of Hermite and Lindemann. — Preceding unsigned comment added by 81.171.52.6 ( talk) 11:49, 2 September 2014 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
The joy of wikipedia is the teaching of feverent independent thinkers to honor disagreements also of a deep kind.
...then why is it still a mere shadow of philosophy of mathematics? This deserves to be remedied, especially as this is not tucked away in an obscure corner but can be linked to directly from the mathematics portal (care of the rather nifty Topics in Mathematics table).
My opinion is that attention should be focused on those programs various mathematicians and logicians have advanced to supply a formal foundation (or more generally something along the way towards one). The main lines have been set theory (Cantor -> Zermelo -> ZFS and others) and type theory (Frege -> Russell -> Ramsey and Church -> ... -> Martin-Löf) with assorted logical calculi (often in league with type theory) and more recently category and topos theory too. This work has begun to bear fruit in the form of programs for computer-assisted proof and automated theorem proving, which I think are areas that will grow in importance as the "heroic" proofs get longer and longer and more difficult to check for errors.
Abt 12 04:57, 24 March 2006 (UTC)
This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 04:00, 10 November 2007 (UTC)
The statement that:
"In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided."
leads one to believe that the formal foundations of modern mathematics contains logical paradoxes. This is not generally beleoved to be the case. Unless the author is able to point to logical paradoxes present in what is considered to be the formal foundations of mathematics, I suggest that this line be changed to:
"In most of mathematics as it is practiced, the incompleteness of the underlying formal theories never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be treated carefully."
71.148.59.179 ( talk) 04:07, 3 August 2008 (UTC)
Clarifying, about least upper bounds: of course the definition "the upper bound that is less than or equal to any other upper bound" is impredicative. But x is the l.u.b. of a set A if and only if every element of A is less than or equal to x, and for every natural number n there is an element of A greater than x - 1/n. That definition seems perfectly predicative to me, athough Weyl would not accept it as such. Predicative subsystems of second-order arithmetic, such as ACA0, can prove things like "every bounded set of rational numbers has a least upper bound" and "every bounded open set of real numbers has a least upper bound". — Carl ( CBM · talk) 01:43, 24 September 2008 (UTC)
I was once told that mathematics had several roots, a set of assumptions, such as set theory, peano arithmetic or lambda calculus. From any of these roots it was possible to derive the same theorems. Is this a widely held view? If so, perhaps it could be made clearer in the article. pgr94 ( talk) 18:00, 17 February 2010 (UTC)
The section Platonism ends with a (cited) question out of the blue:
It would be profitable if this question be put in a context, such as reference [3] being a critic of the platonic view, otherwise I cannot perceive the question as especially "obvious" since it is not too far fetched to perceive our brains as an "eye" that can be taught to look straight into that real world of mathematical objects, and also into other worlds. Another more platonic answer would of course be, that since we are shadowy copies of that mathematical plane, some of us would retain a semblance of the mathematical plane. Rursus dixit. ( mbork3!) 20:18, 29 June 2010 (UTC)
First of all, it’s nothing but a blurb of personal opinion in comment form, and should go on the discussion page. Not the actual article. Second of all, it contains a blatant error in logic, by stating “But the working number theorist is concerned with proving theorems from initial assumptions which are obviously true using proof methods which are obviously correct, not with any particular logical system.” [emphasis mine]. “Obviously” is a weasel word, stating nothing about actual truth, and showing that the author has not understood, that this exact thing (“what do you mean with ‘obvious’? what is actually true?”) is the exact problem that the foundational schools are trying to solve. For example: Someone might state, that it is “obvious”, that the sun is yellow. Until he is told to look at it at dusk. Meaning that humans can not physically state objective things anyway, since all input their brain receives, is already subjective (processed by senses and other sources), and the brain additionally makes it even more subjective, since processing only happens relative to previous experiences. (You can look up how neural networks work.)
Since I find deletion of the opinions of others in a public spaces (which in my eyes includes Wikipedia) to be morally wrong and a violation of free speech, I have not deleted but moved the section right below this one. —Preceding unsigned comment added by 188.100.192.146 ( talk) 09:17, 19 August 2010 (UTC)
The philosophy of math article is getting bigger, the stuff about all this "problems with the foundations of maths" and "Godel proves that all truth is relative" crap is getting a little tighter.
FPOM and FOM should probably be merged to FPOM, not the other way around though. ZF(C) is A foundation of math, but the foundations PROBLEM raised by Hilbert, and the Godel and co. are the meat.
Thoughts? -- M a s 18:38, 22 May 2006 (UTC)
JA: The main thing is that the current split is a WikiPeculiar Neologism that does not really exist in the Literatures, as all of these questions have always been discussed under ∃! umbrella, to wit, FOM. Jon Awbrey 18:48, 22 May 2006 (UTC)
JA: The term "foundations of math", as with the longstanding FOM discussion list, is generally taken to include the consideration of problems associated with foundational questions. If I don't hear any strong objections in the next day or so, I'll go ahead and do the merge. Jon Awbrey 21:46, 29 May 2006 (UTC)
JA: I don't see much prospect of folding this bit of Foma back into the Philosophy of mathematics article, as things are more likely to spin the other way sooner or later. In math as in architecture, as any home-owner knows, problems with foundations just go with the territory, so I think it's still the best course to combine those two. Jon Awbrey 13:40, 31 May 2006 (UTC)
JA: I begin to think that it should be called the "Identity crisis in some people's mathematics", as there seems to be a need to change the name of the article on a recursèd basis. Well, enough of that not-so-abstract non-sense. Jon Awbrey 15:54, 31 May 2006 (UTC)
In his Chapter 39. Foundations, Anglin has three sections titled “Platonism”, “Formalism” and “Intuitionism”. I will quote most of “Intuitionism”. Anglin begins each as follows:
wvbailey Wvbailey 16:01, 27 June 2006 (UTC)
I am just as confused as everyone else above as to exactly what is meant by "foundations" -- is it "philosphy of approach" re the existence of mathematical "objects", or what? (I usually think of it as the core logic and axioms especially with regard to arithmetic (numbers) -- but clearly a specific choice of axioms/methods is made within a philosophic framework). So I quote Anglin above -- he wasn't confused cf his chapter 39. At least Anglin produces some "quotable objects" in the spirit of inline citation. I will work on this more when I get back to my books. wvbailey Wvbailey 18:01, 6 October 2006 (UTC)
I cc'd this over from the Intuitionism talk page. The question I have after reading all of the above: what is the "modern" take on "foundations"? We still have the Platonists (Roger Penrose for example; his Emperor's New Mind really surprised me) and the Logicists (almost everyone else who isn't a computer scientist or a Platonist [?]) and the Intuitionists (aka computer scientists, perhaps with their "IF then ELSE" constructions)... but what is going on in "modern" thought?. The following I thought was interesting: (wvbailey Wvbailey 18:14, 8 October 2006 (UTC))
Stewart's reference in E.B.:
wvbailey Wvbailey 13:10, 15 June 2006 (UTC)
OK, there's some misinformation in the remarks by wvbailey, which I'd like to address. First off, hardly anyone claims to be a "logicist" these days. There are a few serious thinkers who will call themselves "neo-logicist" or some such. But logicism in its original Frege–Russell form (mathematical truth can be derived from logic alone, without any assumptions not justifiable by logic) is pretty hard to sustain in the wake of the Russell paradox (which refuted the logicist version of set theory) and the Gödel theorems (which made a logicist take even on arithmetic seem very problematic, though they may not have refuted such an account outright).
What I suspect wvbailey means by "logicist" is really more like "formalist" (the content of mathematics is what we can formally prove from well-specified assumptions).
Formalism is probably the dominant view (or at least publicly professed view) among, to put it bluntly, those mathematicians who haven't really thought about it very much and don't really want to. Since that's most of them, it comes out dominant overall. That's not to say there aren't serious thinkers who are formalist. But it is to say that it's a very appealing position to those who don't want to spend much time on it. They dismiss the question and move on, before the harder questions show up, like: "If mathematics is about formal theorems, then why have you never proved a nontrivial theorem formally?" and "If you really think that's what math is about, then why on Earth do you care about math?".
As for intuitionism, the above discussion of motivations does not apply only to them, nor to all of them. The defining feature of an intuitionist is that he doesn't accept excluded middle. Lots of formalists and fictionalists and instrumentalists (who might recognize themselves in the above discussion) accept excluded middle (and thus are not intuitionists). And plenty of intuitionists are very realist (that is, Platonist) about the natural numbers individually, even if not about the totality of natural numbers as a completed whole. --
Trovatore 15:57, 10 October 2006 (UTC)
Re Constructivism as a foundational "movement": "computer science" is necessarily constructive (hmm... I wonder why... pure syntax?). Has there been any "foundations" discussions/papers re the effects of "computer science" on "foundations?" Wasn't the four-color theorem finally proved by exhaustive search? This must have "rippled thru" the mathematics community. In a similar vein, the Turing test debates of the past 60 years must have had some effect on "foundations", it certainly roiled the philosphers-of-mind:
Re philosophy of mind playing a role in foundations: I am currently reading parts of a book by the philosopher-of-mind John R. Searle (2002), Consciousness and Language, Cambridge University Press, Cambridge England. He comes back to, again and again, the notion that a mind adding 2+2 as different from a computer adding 2+2 because the latter (computer, calculator) case is one of syntax only (symbols in relation to symbols) but the former "has semantic content" ( i.e. "meaning"-- whatever that means...) as well (and he keeps bringing up intentionality... but I'm not convinced ...). In the machine's case "the information" is defined exterior to the machine, and becomes "information" only in the eye of the machine's beholder (e.g. the computer programmer -- who serves as the homunculus for the machine (cf he states this explicitly p. 122)):
The last paragraph of the "foundations" article touches on this notion of "mind of the mathematician" (without references). This Searle philosophy seems germane to the article but how hasn't crystallized yet. Maybe this will trigger some other folks' thinking. wvbailey Wvbailey 18:55, 12 October 2006 (UTC)
Searle doesn't have too much good to say about "cognitive science" (of any sort, mathematical or otherwise) and its attempts to devolve everything into "algorithms" (cf pp. 108 ff) -- in his view "algorithms" are just "syntax in motion" -- sound and fury implying nothing. wvbailey Wvbailey 19:02, 12 October 2006 (UTC)
The section on formalism is currently dominated by quotations from Weyl. This may be excessive not only because Weyl is highly critical of formalism here, but also because Weyl himself rejected intuitionism and retracted most of these views, merely a few years later. Perhaps some quotes from Bernays would be more appropriate. The way it stands now, we learn more about Weyl's somewhat vague philosophical speculations about formalism than about formalism itself. The main point that needs to be emphasized is that Hilbert saw formalism as a way of answering paradoxes of set theory. It's not that mathematics is a game, but merely a way of grounding mathematics in finitistic procedures so as to minimize risk of paradox. Formalism is not a denial that mathematics is meaningful, but rather an attempt to steer clear of foundational paradoxes. Tkuvho ( talk) 17:33, 8 December 2011 (UTC)
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Actually I'd like to see more of the nasty dialog in this article (ie. the "crisis" part of it) -- the criticisms and counterclaims/rebuttals help us understand the issues. More on this to follow in another section.
I'm trying to broaden us out, pulling the lens back so we get a wider-angle view of "foundations of mathematics" and what "foundations" is all about/what it means. I'm using as references the following books, the 4th being dubious (but it has Dedekind's and Cantor's works). Notice that the years stop at about 1936-7 with Turing. So there's lots of stuff missing:
I. Grattan-Guinness 2000 The Search for Mathematical Roots 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Canotr Through Russell to Goedel, Princeton University Press, Princeton NJ, ISBN 0-691-05858-X (pbk.: alk. paper)
Paolo Mancosu 1998 From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, NY, ISBN 0-19-509632-0 pb.
Jean van Heijenoort, 1967, From Frege to Goedel: A Source Book in Mathematical Logic 1879-1931, 3rd Printing 1976, Harvard University Press, cambridge, MA, ISBN 0-674-32449-8 (pbk).
Stephen Hawking ed., 2005, And God Created the Integers: The Mathematical breakthrougs that Changed History, Running Press, Philadelphia PA, ISBN-13: 978-0-7624-1922-7.
Howard Eves 1990 Foundations and Fundamental Concepts of Mathematics: Third Edition, Dover Publications, Inc, Mineola, NY, ISBN: 0-486-69609-X (pbk.)
Charles Seife 2000 Zero: the Biography of a Dangerous Idea, Penguin Books, NY, NY, ISBN 0 14 02.9647 6 (pbk).
W. S. Anglin 1994 Mathematics: A Concise History and Philosophy, Springer-Verlag, New York, ISBN: 0-387-94280-7.
Ian Stewart 2007 Taming the Infinite: The story of mathematics from the first numbers to chaos theory, Quercus Publishing Plc, London UK, ISBN: 978 1 84724 7 68 1.
The book by Eves, above, offers a list of three "foundational crises". The advantage of using this is that it is clearly not OR. The disadvantage of sticking to it is Eves has missed some "difficulties" in ancient and medieval foundations that spurred on the development of modern mathematics. Some of these can be inferred from the topics in the other books above (Eves has some but not all of these): symbolization, 0, the infinite, negative numbers; roots of equations and the square root of a negative number; convergence, limits, continuity; inadequacy of Aristotelian logic and the school-men forms (e.g. failure of Euler diagrams), etc. This is not a complete list, just an example.
Eves identifies 3 major “profoundly disturbing” crises of foundations (pages 262-263):
My intention is to work up a list of "difficulties". Then move to a list of "foundations" (ancient, medieval, early modern, modern) e.g. topics like axiomatization, proof; algebraic generalization and symbol-manipulation; calculus (analysis), etc. Any thoughts? Bill Wvbailey ( talk) 22:34, 12 December 2011 (UTC)
An editor recently commented that "Actually I'd like to see more of the nasty dialog in this article (ie. the "crisis" part of it) -- the criticisms and counterclaims/rebuttals help us understand the issues". My personal preference would be to have succinct sympathetic statements of the positions of the various schools. Other editors are welcome to chip in. Tkuvho ( talk) 12:47, 14 December 2011 (UTC)
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Here are some examples of "the nasty dialog" so editors know what this is about. Even this small selection either directly covers or hints at a wide range of foundational issues -- the nature of mathematical thought, the completed infinite, the continuum, impredicativity, tacit contentual assumptions vs formalism, the nature of mathematical induction, the various philosophic "-isms" (Finitism/constructivism/intuitionism, Logicism, Formalism):
Strict finitism vs set theorists: Those who were mishandled by Kronecker, in particular: Dedekind and Cantor: Here's Dedekind 1887:
The infinite taken as a completed whole -- Cantor: There's a whole chapter to be written here about how badly treated he was (or felt he was) by Kronecker and Kronecker's Crelle's Journal.
The continuum: Analysis and Dedekind cuts, Zermelo's set theory: Apparently any "contentual assumptions" in a theory were an anathema to Hilbert. Here's Bernays 1922: "In Dedekind's grounding of analysis what is taken as a basis is the system of the elements of the continuum, and Zermelo's construction of set theory it is the domain of operations B" (p. 215 in Mancosu 1998). After a glowing tribute to the ideas of Zermelo, Hilbert adds this zinger:
Impredicative definitions and Zermolo's set theory: In a section b. of his 1908 The Possibility of a Well-Ordering, Zermolo states:
The following is a very important quote, e.g. taken up in detail by Goedel 1944. This is Zermelo
Hilbert criticises everyone:
Re the completed infinite, the LoEM -- Hilbert's famous boxer quote:
Bill Wvbailey ( talk) 16:23, 14 December 2011 (UTC)
My essay about problems related to this topic in Wikipedia. A help to improve the quality of articles on mathematical logic, currently being unsatisfactory, is appreciated. Incnis Mrsi ( talk) 09:10, 5 March 2012 (UTC)
I removed the reference of the book by W.S. Anglin as I read this critique. I did not see the book however. Can someone who saw the book confirm or infirm the validity of this critique ? If it is valid, can someone justify to include the reference of a book like this in this article ? I mean, it did not seem to me that the quotations from this book had any specially unique value here. I guess that if any reference is needed, other sources can be found as well in replacement. Sorry I did not make a search on the issue. As for Gödel, I mention him for his set-theoretical platonism later in the article where modern mathematics are concerned. Quoting him in a section about the Ancient Greek philosophy would be anachronistic, and there is no need to quote Anglin about Gödel for giving a definition of Platonism. Spoirier ( talk) 22:00, 16 September 2012 (UTC)
The following entry is in the more Paradoxes section of the article, but doesn't seem to be part of mathematics and further it isn't a paradox. Any reason not to remove it? RJFJR ( talk) 19:02, 16 January 2014 (UTC)
The comment(s) below were originally left at Talk:Foundations of mathematics/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
This article is just an outline, but it could be expanded to a more thorough description. That is, there are a lot of references but the content is shallow. CMummert - 5 Oct 2006 |
Last edited at 20:47, 16 April 2007 (UTC). Substituted at 14:48, 1 May 2016 (UTC)
Under "Group theory", Wantzel gets the credit for the work of Hermite and Lindemann. — Preceding unsigned comment added by 81.171.52.6 ( talk) 11:49, 2 September 2014 (UTC)