One of my favorite discussions of idealism/constructivism in mathematics is by Brian Rotman, both in his general work on the semiotics of mathematics and some particular stuff on infinities. He is a mathematician-philosopher, and pretty good I think. Broadly, his position is that math has to be done by somebody, so it's always a finite, constructive, material, semiotic practice.
On the other hand I do know some very thoughtful people, with natural science backgrounds, who've come late in life to conclude that Forms are "real" in some non-material, but still relevant sense. It's not quite Platonism, but more a sort of notion of immanence in place of transcendance, and the core argument goes back to the Big Bang and the Inflation, and all has to do with how and what kinds of formal patterns did and did not appear in our universe as opposed to other possible universes. I don't happen to agree, but it is an interesting idea. The corollary for our present discussion would be that the mathematics that works in our universe is not the only possible mathematics, but it is somehow built into the structure of possibilities that makes our universe ours and not some other one.
I do think that's going a bit far. I might believe that our universe has some properties that make some kinds of patterns more likely than others, or maybe even inevitable, or at least make other patterns no- shows. But that's a long way from an argument that there can be only one workable mathematics (i.e. that all workable maths can be put into 1-1 correspondence with one another, somehow). It does seem that Goedel's arguments imply that if the 1-1 is based on what theorems can be proven true within some mathematics, then you don't necessarily get a 1-1 even within a single mathematics, much less from one to another.
So yes, any human cultural system of mathematics (or any alien system in our universe) is the product of a lot of human/alien choices, only SOME of which were strongly constrained by the workability of their material applications. Now, historically, it seems reasonable that a lot of today's mathematics, even across cultures, started out long ago being highly adapted to practical applications: accounting, surveying, calendaring, astrologizing. Hence a lot of similarities and a lot of social determinism. Also a lot of cultural determinism in the sense that religious and philosophical beliefs played a big role in what kinds of mathematical practices people found acceptable vs. taboo or ridiculous; from things that had to work out (say the complementarity of yin and yang in Chinese group theory, aka the I Ching) to things that couldn't be allowed (e.g. the rejection of irrational length ratios ala Pythagoras, of later of negative numbers, imaginaries, transfinites, etc.), with attitudes changing often over historical time. It's quite amazing how many formerly "impossible things" we can now think before breakfast. And I doubt that we have yet witnessed the end of history in this respect.
We exercise a freedom to accept or reject, to develop in this direction or that direction for new mathematics. In this sense the point we arrive at in any historical moment is "arbitrary" out of all the possible current states of mathematics we might have found ourselves in. Arbitrary does not mean that any possible mathematics will have workable practical applications in our universe. It just means that we are not on a linear journey from ignorance to total understanding of the one-and-only possible Mathematics. As likewise for Physics, or Politics.
One final point, just so the social constructivist realists don't get too complacent. I find Bruno Latour's arguments very sensible and persuasive when he makes the case that all reals are real each in their own way (i.e. by their own criteria of determination of their existence, or reality) and real in a different sense than every other real (or maybe class of reals is more realistic). In this ontology, all sorts of things can be real in their own way, including gods, imaginary numbers, ghosts, etc. Only so long as their reality can be defined by some criteria based on what we would call practices in a network of interdependencies, though his version does not necessarily require people and certainly does not privilege our role. In a somewhat simplified version, formal patterns and abstract or formal mathematical objects are real if the society of people-and-things behaves as if they are real. But remember that here "real" is not some one single universal predicate; it's a cover term for a very large number of different ways of being "real".
In such a network ontology, mathematical objects are real _because_ they are constructed.
JAY. Jay Lemke Professor Educational Studies University of Michigan Ann Arbor, MI 48109 www.umich.edu/~jaylemke On Apr 30, 2009, at 7:03 PM, Ng Foo Keong wrote:
the reason why I brought up this topic is that i have been brought up (indoctrinated?) with a Platonic view of mathematics, a view held by a majority of mathematicians. however, while studying learning-sciences and maths education, i encounter post-modernist views including social constructivism, which gave me the cognitive dissonance that i felt i need to get off my chest. so i analysed the issue into two parts:- (P) "if humans exist, then they create mathematics" (C) "if mathematics exists, then it must have been created by humans" I am definitely convinced about (P), but not so sure about (C), the converse of (P). reading literature trying to convince me of (C) by citing examples of (P) is not going to cut it. prior to coming to xmca i did not find many knowledgable people to discuss/debate this with, so I began debating with myself. Like Andy, I'm still very much tainted with the Platonic view, although as an educator I recognise the need for different learners to make personal sense of maths and share their understandings socially. i find the pointer to Derrida's phenomenological "différAnce" (why must he invent such a new bombastic word?) analysis and "deconstruction" interesting, although if i were Derrida, i would have invented the word "égalEte" (the other way round from Derrida, using an analysis of equitable equalities to study deferring differences, much like algebraic topologists using homotopy to distinguish a doughnut from a figure of '8'). the reason i brought up my three examples viz. (1) the non-Riemannian geometries (vs Riemannian Geometry) [i meant non-Euclidean, but it doesn't matter. Euclidean geometry was historically discovered/invented first. Riemannian is just one of the "alternatives". if Riemannian had come first, then Euclidean might have been named as one of the "alternatives". ] (2) "non-Standard" Analysis (vs Standard Analysis, the dorminant brand of higher calculus, thanks to Cauchy and other French mathematicians), (3) Henstock/Daniell integration (vs Lebesgue integration, the dominant brand of modern integration theory)is to consider the other side of the argument (i.e. against an objectivemathematics). Mathematics can be thought of as a game with certain rules (like Jay Lemke hinted). Different rules have different consequences. Just like American football, soccer ("football") and vs basketball. What is allowed in one ball game (e.g. using your feet to handle the ball) is not allowed in another. All socially constructed. People happen agree to play by those rules. [BTW, logicians/mathematicians working at the foundations of mathematics explore set theory with and without Axiom of Choice. others advocate using Category Theory as a foundation for all mathematics.] So, it seems there are no "right" or "wrong" rules. Different rules, different games. What is wrong (if any) with this counter-argument? F.K.p/s: I have transfered Andy's side note for discussion in the other thread.From: "Andy Blunden" <firstname.lastname@example.org>To: "eXtended Mind, Culture, Activity" <email@example.com> Sent: Wednesday, April 29, 2009 10:59 PMSubject: Re: [xmca] In what sense(s) is mathematics a social construction.?I am not familiar with all of these theories FK, but let's keep it in the "public domain": If someone had decided that a minus times a minus was a plus, then they could do that, but such an arithmetic would have had little practical use, and sooner or later, most likely sooner, someone would have discovered something (say "negus") which looked very much like a minus in every way except when negus is times by itself it gave a plus. And then everyone would have been learning about negus in school and Mike's granddaughter would be asking him why negus times negus = plus. Famously of course, Riemann discovered his mathematics before Einstein found a use for it, otherwise it may still be rotting in the back room of some library. Does someone (Jay?) know how Einstein found Riemann's paper? On a side note, a lot of people calling on various metaphors to justify -x-=+ have never addressed the question a kid might ask as to why the example given doesn't prove that a - when **added* to a - gives a +. I certainly had kids confront me with that one. It is very easy to skate over the hidden equation of multiplication with intersection and compounding and so on which to a lot of non-mathematicians looks much more like addition. The link between these operations is obviously NOT arbitrary, is it? But nor is it obvious, Andy_______________________________________________ xmca mailing list firstname.lastname@example.org http://dss.ucsd.edu/mailman/listinfo/xmca
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