Science Fair Project Encyclopedia
Talk:Philosophy of mathematics
Older discussion: see /Archive 1.
More peaceful discussion
- ...mathematics is studied in a markedly different way than other languages. The capacity to acquire mathematics, and competence in it, called numeracy, is seen as separate from literacy and the acquisition of language. Some argue that this is due to failures not of the philosophy of mathematics, but of linguistics and the study of natural grammar.
What thinkers are making these "linguistic" objections? Also, the half-implicit idea that human language and math are fundamentally the same and should be studied in the same way and by the same specialists is not NPOV and perhaps absurd. While there are undeniably commonalities between natural language and math, there are many differences; perhaps the most fundamental is that people automatically learn to be extremely competent speakers of their native languages without any formal instruction, while even the brightest mind will not get very far at all if it has to start without any math education. --Ryguasu 07:28 Feb 21, 2003 (UTC)
I've just made a few additions to the page (and one embarrassing error, quickly corrected - I'm sure someone else can find any others). There is now a bit more on logicism, realism and formalism. I think logicism could do with alot more, I only put in the bare minimum.
Also, a little history at the start might be a good idea. Given kant's huge influence on the subject someone should really put something in about him. I put a brief bit in alluding to the paradoxes at the start of the 20th century, and problems in founding analysis since it was so important, and isn't really covered under any of the subsections.
[sorry for my bad English!] Quoting form the page:
- The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez. (Since this book was first published in the year 2000, it may still be one of the only treatments of this perspective.) For more on the science that inspired this perspective, see cognitive science of mathematics.
If I understand well, this may well be what the French biologist Jean-Pierre Changeux says in his book "Matière à pensée" -- which is a dialog with Alain Connes. I don't think it has been translated into English.
I often wonder why there is such a debate over whether or not mathematics originated as a tool of mankind or if it existed before humans. Clearly the answer is--both. Mathematics as language is Humankind's neuro-linguistic interpretation of geometric phenomena that existed in such forms as crystals before humans came into being. A human tool that arises from the hominid-style perception of common phenomenal activity in 1st to 4th dimensional space-time that occurs in apparently logical and interrelated patterns. Khranus
On the absurdity of modern philosophy of mathematics (a rant)
Modern debates over both the nature of language and mathematics are very dumb, and shows why people hate philosophy and think of it as pointless semantic argument. I'm a double major in philosophy and cognitive science at Berkeley, and have had a class in which Lakoff taught; and I find many logical errors in his books. At any rate, nobody who is a realist, is suggesting that there is a number "one" floating out in some ethereal realm. Hopefully, those who are not realists would say that mathematics does have something to do with reality. Just like there are pseudo-sciences, cognitive science acts like a pseudo-philosophy and is philosophically naive. It talks as if there is no permanent relation between the "embodied mind" and what we call objective reality, that the mind doesnt come from reality (ie it talks about 'cognitive structures' in a mysterious way, as if the cognitive structures arent shaped by objective forces); as if other species with different metaphors couldn't approach the world using our metaphors preserving the truth of them (ie it doesnt realize language is the context its used in, and so it is meaningful universally); as if there is an "objective reality" at all (many philosophers since David Hume have tried to show this is wrong, there is nothing outside of experience; Hegel tried to argue Kant's conception of unknowable things-in-themselves that lied out of reality was wrong and subjectivity is objectivity); as if its possible for a species to exist compatibly living with us at the same time with a cognitive structure incompatible with ours (cognitive science professor Sweetzer at Berkeley went into some extravagant example that an observer the size of an atom would experience the world differently--as if its possible for such a thing to exist--[note, I hate thought experments in philosophy that assume things like radically different minds or worlds can exist, they always turn out to have a fundamental flaw]), etc. Hopefully, someone in philosophy will prove once and for all that epistemology is grounded in an ontology, and the relation between language and the world. This may mean returning to metaphysics and idealism. Brianshapiro
Questions about formalism (and its problems)
Hey. This article is actually pretty good. I don't know anything about the field and I found it interesting, at least.
It seems to creep just a little away from NPOV against formalism. (Keep in mind I have no idea what I'm talking about...)
It says: "The main problem with formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the minute string manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in terms of these games, the rules are certainly not substantial to the initial creation of those proofs." That just sounds like carping. It's really an attack on reductionism in general. When I as a programmer use a double, I generally don't have CMOS logic in mind, much less floating-point arithmetic, but that's hardly a "problem" for my reductionist "philosophy of how computers work". Or is it?
The last sentence of that paragraph is: "Formalism is also silent to the question of which axiom systems ought to be studied." This is a separate objection. I don't understand it. To the outside world, most of mathematics looks like basically unmotivated poking around; like pure science, it might yield rich practical applications at any moment, but if it doesn't, it's still considered valuable in its own right. Are all mathematicians that pursue random axiomatic systems purely for the hell of it formalists? Is John Conway a formalist? (This is a serious question; I actually don't know.) Is this really considered a "problem" in philosophical circles?
I mean, the real problem with formalism (stop me if I'm missing the point) is that it completely ignores the relationship between mathematics and nature. Right?
In general, nice work. Jorend 21:48, 4 Apr 2004 (UTC)
Given that logicism has its own article (for good reason), the section on that topic in this article should be summed up in a sentence or two and the material in that section moved/referred to the article. B 00:56, Apr 26, 2004 (UTC)
Commentary on David Corfield's The Philosophy of Real Mathematics Page
The anonymous user who added the David Corfield link to the main page also added the following commentary and quotations :-
- This page talks about to an event for the philosophy.
- It is happening an event in mathematics, that is important for the philosophy.
- In Internet (Báez...) and for example in the book of Corfield you will be able to see some track.
- Talking about Higher-dimensional algebra: (:"Many important constructions may profitably be seen as the “categorification” of (...)" .
- "It provides a way of organising a considerable proportion of mathematics. (...)"
- " These constructions have applications in mathematics, computer science, and physics. (...)"
- "Higher-dimensional algebra blurs the distinction between topology and algebra. (...)"
Hmmm. This article starts with some highly dubious stuff, if you ask me (first section after intro). Also, high time to archive some older discussion.
Charles Matthews 09:36, 17 Jul 2004 (UTC)
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