Re: [xmca] In what sense(s) is mathematics a social construction.?

[Ng Foo Keong <lefouque@gmail.com>]

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 objective
mathematics).  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" <ablunden@mira.net>
> To: "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
> Sent: Wednesday, April 29, 2009 10:59 PM
> Subject: 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
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