Talk:Foundations of mathematics

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The recently-added text has the bias that a foundation of mathematics is actually possible. Before that the text was balanced and ended with 'the matter remains controversial' without advocating any particular approach. The subject matter after that line belongs in philosophy of mathematics under whatever old school (or new school) believes in it.

The majority of modern mathematicians seem to believe that foundations are either non-existent or unnecessary. They accept neither the positions of the older schools, nor do they accept the cognitive or social foundations theories. Hilary Putnam pointed out that one can accept a sort of weak realism without accepting any form of Platonism tied to Plato's ontology <-- apologies for the lousy condition of that article, it was written by a local sacred idiot.

Article now more complete. Papers at end are critically important to this field and deserve summary treatment as articles, each of them. Particularly as there are fools out there who think realist=Platonist which is just wrong. Calling Putnam in particular a Platonist would piss him off to no end, dude.

Foundations search itself may be scientism. Is that view too contoversial?

Scientism is defined on that page as:
Scientism is the acceptance of scientific theory and scientific methods as applicable in all fields of inquiry, including morality.
Personally I don't see the connection between a search for a Foundation for Mathematics, and say, morality; so I would call it controversial. Chas zzz brown 21:16 Jan 20, 2003 (UTC)
Fine, we'll leave it out. Various arguments about this connection are in Talk:Philosophy_of_mathematic, which I notice you've found, so discuss it there.

Hilary Putnam raised an extremely important point in What Is Mathematical Truth? - which is, that you must consider foundations of mathematics as a quite separate issue from an ontology, which most philosophy of mathematics theories fail to do, as they confuse the two quite hopelessly. Specifically, Putnam said he was a realist but not a Platonist, and was clear that one could believe that mathematical ideas were 'real' without believing in any aspect of Plato's ontology. What he meant was foundation ontology or cosmology it seems, but that aspect or usage of an ontology is not in the ontology article due to it being authored by, as the above claims, a local sacred idiot. According to the existing article there, a foundation ontology is just some kind of comp sci thing, and cosmology is only about physics and religion.

At some point this sacred idiocy must be challenged and Putnam's disctinction accepted and noted in either or both of the ontology and philosophy of mathematics articles, plus here in foundations of mathematics. As it stands, this article is trapped in about 1980 and those are trapped in about 1959, without even acknowledging Wigner's points. This is a bit confusing for anyone who actually knows the topic, and may discourage intelligent contributors, leaving the articles sadly in the hands of local sacred idiots.

Thank you Axel. Chas zzz brown 18:00 Feb 10, 2003 (UTC)

Anyone, a mathematician especially, who appreciates the “unreasonable effectiveness of mathematics” and the “unreasonable ineffectiveness of philosophy" to scientific endeavors must recognize the dangers of letting "philosophy of math" ride roughshod over "foundations of math" and as a last line of defense, of letting "philosophy and foundations of math" ride roughshod over proper pure and applied maths.

Just look at the talk page for "philosophy of math"! What a mess. Note that some of these people actually believe the destiny of science can be mastered thru verbose semantics, concepts, schema, arguments, etc. The last time I looked, the language of science was still written in mathematics. Fortunately, bullshit had not yet taken over in the math journals.

Specialists in foundations and/or philosophy of math often over-estimate the importance of their work to those in other specialties. In fact, few mathematicians are typically concerned on a daily, working basis over logicism, formalism or any other philosophical position. Instead, their primary concern is that the mathematical enterprise as a whole always remains productive ... as evident by the work they are doing at the moment.

Typically, they see this as insured by remaining open-minded, practical and busy; as potentially threatened by becoming overly-ideological, fanatically reductionistic or lazy. I do not know of a name for this “philosophical position” but it may be the one most mathematicians adhere to more strongly than any traditional, philosophical position they also favor or agree with (if any).

I think it would be choice to boldly, truthfully publish that the nihilistic and productive "philosophical positions" are by far the most populus amongst modern mathematicians on the "philosophy of math" and "foundations of math" pages. Let the silent majority be heard at last. Let the fanatical minority, steeped with all of their dreaded, formidable philosophical arguments, just go nuts (as usual). We have nothing to fear from these people. Some of them are just "naked emperors". Some of them are qualified mathematicians but with a strange psychological affliction involving philosophy and/or religion.

OmegaMan

Contents

[edit] Sustain problems of foundations as separate article

The joy of wikipedia is the teaching of feverent independent thinkers to honor disagreements also of a deep kind.

  • Removing merge tag. Kerowyn 04:35, 14 January 2006 (UTC)

[edit] So if this remains as a separate article...

...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)

[edit] Proposed Merge : FPIM => FOM

[edit] I concur with a merge

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)

The present article Foundational crisis of mathematics is rather specifically about the historical quest and fight that took place about 85 years ago. In the Foundations of mathematics article I see mainly philosophical issues that are treated in more depth at Philosophy of mathematics. So I'd propose to redirect this article to there, instead of merging the Foundational crisis to here, and expend further effort in improving the Philosophy article (by which I do not mean making it fatter than it already is, although there could be a spinout covering the material of Hilbert's program#Hilbert's program after Gödel as far as it reflects current common thinking in more depth). --LambiamTalk 02:27, 31 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)

[edit] W.S. Anglin (1994) Mathematics: A Concise History and Philosophy, Chapter 39 Foundations

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:

”Platonism
”Platonists, such as Kurt Gödel, hold that numbers are abstract, necessarily existing objects, independent of the human mind” (p. 218)
”Formaism
“Formalists, such as David Hilbert (1862-1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...” (p. 218)
”Intuitionism
”Intuitionists, such as L. E. J. Brouwer (1882-1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them.
”Intuitionism is a philosophy in the tradition of Kantian subjectivism where, at least for all practical purposes, there are not externally existing objects at all: everything, including mathematics, is just in our minds. Since, in this tradition, a statement p does not acquire its truth or falsity from a correspondence or noncorrespondence with an objective reality, it may fail to be ‘true or false’. Thus intuitionists can, and do, deny that, for any mathematical statement p, it is a logical truth that ‘either p or not p’.
”Since intuitionists reject objective existence in mathematics, they are not necessarily convinced by the reasoning of the form:
”If there is not mathematical object A, then there is a contradiction; hence there is an A.
[I found this confusing. Does he mean: There exists B: (~A -> ~B) & B -> A [?]]
"If the details of the reasoning provide a way of imagining or conceiving an A, in a way open to ordinary human beings by, say, calculating it in a finite number of steps), then the intuitionist will agree that there is, indeed, an A. However, if the details of the reasoning do not provide this, the intuitionist will remain skeptical. For a Platonist, it is of interest that there is an A even if it is only God who can conceive it. For an intuitionist, however, a mathematical object is meaningless unless it can be somehow ‘constructed’ and ‘intuited’ by a human being.
"For the intuitionist, the human mind is basically finite, and Cantor’s hierarchy of infinites is just so much fantasy. Intuitionists thus reject any mathematics which is based on it, including most of calculus and most of topology.
"... To ... objections, the intuitionist can replay: (1) it does not make sense for human minds to try to conceive a world without human minds, and (2) it is better to have a small amount of mathematics all of which is solid and reliable than to have a large amount of mathematics, most of which is nonsense”(p. 219)

wvbaileyWvbailey 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. wvbaileyWvbailey 18:01, 6 October 2006 (UTC)

It's one of those words with lots of different meanings. One of them is just "mathematical logic plus set theory" (or "mathematical logic including set theory" if you think of set theory as part of logic; in that case it's just a synonym of "mathematical logic"). Another is the more philosophical meaning -- the question of what, if anything, mathematics is based upon. There's a book I want to get around to reading called Foundations without Foundationalism, which presumably advocates studying mathematical logic and set theory without assuming that mathematics is "founded" on them in some Euclidean sense. --Trovatore 18:43, 6 October 2006 (UTC)
Foundations without foundationalism (by Shapiro) focuses on second order logic and second order semantics as a non-"foundational" way to study math. The mathscinet review says that it discusses second order set theory, but I don't remember the book spending a long time on that. The "without foundationalism" might refer to the problem that second order semantics are often criticized by formalists as being really set theory, whearas they are defended by others (such as Shapiro, I believe, and by several category theorists) as capturing intuitive notions better than first order logic. An interesting thread on FOM entitled "Second order logic is a myth" had interesting opinions (for example Simpson [1] and Black [2]). CMummert 18:58, 6 October 2006 (UTC)
I'm pretty sure the original meaning was: the base on which the mathematical edifice rests. If the foundations are not sound, then neither is the superstructure. The issue was: Is there an incontrovertible justification for the leap from "P has been proved (by a mathematical proof)" to "P is true"? Obviously, you can't consider this without going into both the nature of mathematical proof and the nature of mathematical truth. Since mathematicians have learned how to skirt paradoxes and antinomies, this is generally no longer considered a burning question.  --LambiamTalk 20:51, 6 October 2006 (UTC)
I think your phrasing "if the foundations are not sound, then neither is the superstructure" captures excellently what I would call the foundationalst error. The fact of the matter is that the so-called foundations are generally developed and understood after the things that supposedly "rest" on them, and this does not invalidate what came before. To take a particular case, the calculus of Newton and Leibniz was, in my view, just fine, even before the work of Dedekind and Cauchy and so on. The latter work was extremely valuable, but not for the reasons often thought; it was valuable in that it provided ways to analyze corner cases where it's not clear what the Newtonian/Leibnizian intuitions even said, and in that it provided a way of connecting it better to a larger mathematical framework.
The Euclidean foundationalist supposes that there is some way of providing an unshakeable base, once and for all, that will provide apodeictic certainty to that which rests upon it. He's just wrong. No such certainty is available to us, even in mathematics. Luckily, we don't really need it. --Trovatore 21:02, 6 October 2006 (UTC)
Whether we now believe we need it or not, the point is that mathematicians in the first half of the 20th century, or at least those that cared, sure thought they did. If ZFC was strongly suspected to be inconsistent, perhaps present-day mathematicians would also feel a bit queasy.  --LambiamTalk 00:34, 7 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: (wvbaileyWvbailey 18:14, 8 October 2006 (UTC))

[edit] Ian Stewart for Encyclopedia Britannica 2006

Stewart's reference in E.B.:

"Errett Bishop and Douglas Bridges, Constructive Analysis (1985), offers a fairly accessible introduction to the ideas and methods of constructive analysis."
"Constructive analysis
"One philosophical feature of traditional analysis, which worries mathematicians whose outlook is especially concrete, is that many basic theorems assert the existence of various numbers or functions but do not specify what those numbers or functions are. For instance, the completeness property of the real numbers tells us that every Cauchy sequence converges, but not what it converges to. A school of analysis initiated by the American mathematician Errett Bishop .... This philosophy has its origins in the earlier work of the Dutch mathematician-logician L.E.J. Brouwer, who criticized “mainstream” mathematical logicians for accepting proofs that mathematical objects exist without there being any specific construction of them (for example, a proof that some series converges without any specification of the limit which it converges to). Brouwer founded an entire school of mathematical logic, known as intuitionism, to advance his views [italics and boldface added]
"However, constructive analysis remains on the fringes of the mathematical mainstream... Nevertheless, constructive analysis is very much in the same algorithmic spirit as computer science, and in the future there may be some fruitful interaction with this area" (Britannica Analysis/Constructive Analysis)

wvbaileyWvbailey 13:10, 15 June 2006 (UTC)

What the "modern take" is will depend on who you ask. I had a look at how other-language Wikipedias deal with the subject.
  • Class A: It is about the question of providing semantics and a criterion for the truth of mathematical statements: French, German;
  • Class B: It is about mathematical logic, axiomatization, model theory, category theory, ...: Slovenian, Spanish, Turkish;
  • Class C: <? could not decipher>: Bulgarian (which has, in any case, a historic account from Aristotle to Church), Chinese (has the same external links as our article), Japanese (but I can see they go from Frege to von Neumann and Turing).
 --LambiamTalk 21:44, 8 October 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).

Correct. I used the word synonymously because for me it is easier to remember and more expressive -- all good math is "formal" -- Platonism, Formalism, Intuitionism. But I agree with your distinction. wvbaileyWvbailey 18:23, 10 October 2006 (UTC)

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)

Sorry, the last paragraph doesn't correspond very well to what's described above for intuitionism, now that I reread it. I'm not sure where the text is that I thought I was responding to. The other two paragraphs stand. --Trovatore 16:09, 10 October 2006 (UTC)
What the heck are "fictionalists" and "instrumentalists"? Re proving theorems, there is this mysterious component of "randomness" (I've never proved a theorem but I've done lots of other creative things)... you suppress your rational (logicist, formalist) self and let "the bubbles" percolate to the surface -- dreams, images, doodles, impulses to follow leads even if they seem misdirected -- but all this will be useful only after much training so that the (logicist, formalist self) can recognize any value to be found in "the bubbles". (Hence to "sleep on it" can be a creative act. I used to get ideas while jogging, when my mind was totally wiped and "off-task"). In a lecture I attended some years ago the philosopher Daniel Dennett proposed the importance of "randomness" in creative thought. The idea is not too novel, really. But if I read between the lines above Travatore seems to be daring the mathematicians to admit they are really closet [what? Platonists? "creators"?] -- while they may be using the formal tools and techniques of Formalism in their work, secretly they are relying on their clever creative natures that cannot be described in Formalism. Because Formalism does not admit cleverness (randomness + "value-recognition"), Ergo: Formalism is incomplete, insufficient. wvbaileyWvbailey 18:23, 10 October 2006 (UTC)
If I get started here I might never stop, and this sort of discussion, fun as it is, is not what talk pages are for. So briefly: fictionalists/instrumentalists hold that mathematical objects, particularly completed infinities, are "useful fictions" that help us to make correct predictions, and as long as the predictions are correct, we needn't worry about whether there's any underlying reality to them. They differ from formalists in that they put the emphasis on reasoning with the fictional objects, rather than uninterpreted strings in a formal language. Your "reading between the lines" is mostly correct, except that I don't think real-life mathematicians even use the tools that formalism theoretically exalts. Formal proofs are almost never seen except for the most trivial theorems. Instead people write prose proofs that, allegedly, can "in principle" be reduced to formal proofs, but this reduction is almost never carried out in practice. --Trovatore 15:58, 11 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)):

"...information processing and symbol manipulation are observer-relative ... the addition in the calculator is not intrinsic to the circuit, the addition in me is intrinsic to my mental life..." (p. 34-35)
"For the commercial computer we are the homunculi who make sense of the whole operation" (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. wvbaileyWvbailey 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. wvbaileyWvbailey 19:02, 12 October 2006 (UTC)

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