[FOM] Advertising an approach to the philosophy of mathematics

Sun Sep 9 14:12:34 EDT 2007

thank you Feng for your stimulating questions.

my reaction is twofold:

1. on the finitistic approach to physics:

the monograph you reference starts out with:

`The world to which we apply mathematics is finite. Macroscopically, the 
universe is believed to be finite; microscopically, our physics theories 
describe only things above the Planck scale (about 10^(-35)m, 10^(-45)s 
etc.). Therefore, all applications of mathematics are essentially similar to 
the applications of mathematics in areas such economics, population study 
and so on, where the subject matter is obviously finite and discrete. 
Intuitively, infinity, continuity and so on are idealizations for glossing 
over microscopic details or for generalizing beyond an unknown finite limit, 
in order to get simplified mathematical models in simulating the finite and 
discrete phenomena.'

i do not know of any scientific proof that reality/ the universe/ the world 
is finite. nor of the opposite. the same holds mutatis mutandis for 
`discrete'. to me these questions seem rather important, since in my humble 
opinion any real answer that indicates that the universe is not continuously 
infinite, but for instance countably discrete, could impact heavily on the 
validation of mathematical models in physics.

however, i seldom seem to come across discussions where the usual classical 
mathematical models are questioned for their validity in a recursive, 
discrete or even finite setting.

to me it seems definitely ;) worthwile to study finitism / discretism / 
recursive mathematics as an alternative approach to physics. on the other 
hand i would hesitate to be so sure about the `real' mathematical nature of 
the universe.

for me, mathematics is one of the ways for us to model physical phenomena. 
the science therein has largely to do with the reproducibility of phenomenal 
causes-and-effects. this enables us to `predict' or `explain' whatever 
natural phenomena we encounter. for this, what usually happens is that some 
mathematical structure is seen to fit very well with experimental 
observations. mostly this structure is arrived at through a careful thinking 
process, also based on previous observations and experiments. in this way we 
arrive at explanatory theories.

it doesn't always seem to really matter whether we deeply understand WHY a 
certain mathematical structure fits so well with observations. this only 
becomes an issue once we try to link different theories on different scales, 
or when we encounter phenomena that do not match our `local' theory. 
fortunately physics is always in this situation...thus forcing us to observe 
better, to think more daringly etc.

in summary my reaction is that finitism, intuitionism, discretism, 
recursivism...to me all seem worthy alternative candidates for use in 
mathematical modeling of the physical world. perhaps one will find 
structures that match (some of) our observations better than the classical 
models in use today. but i would hesitate to claim any platonistic truth 
about the mathematical nature of the universe.

~~~

2. on your statement that it is or should be possible to give  a completely 
scientific description of human mathematical practices, including a 
scientific explanation of the
applicability of mathematics. :

i very much doubt this claim. there seems to be some circularity in it, 
which i formulate somewhat like this: mathematics is a part of science. can 
one give a completely scientific description of human scientific practices, 
including a scientific explanation of the applicability of science? to me it 
seems one should believe in the applicability of science in the first place, 
in order to even start such an undertaking.

but that might look to be only for science as a whole. however, i don't see 
how to justify that mathematics can be subjected to some sort of scientific 
validation process, but science as a whole remains exempt from such 
validation...

~~~

i hope to have contributed something,

greetings,

frank waaldijk
http://home.hetnet.nl/~sufra/mathematics.html

----- Original Message ----- 
From: "Feng Ye" <yefeng at phil.pku.edu.cn>
To: <fom at cs.nyu.edu>
Cc: <fengye_us2001 at yahoo.com>
Sent: 08 September, 2007 05:04
Subject: [FOM] Advertising an approach to the philosophy of mathematics(with 
some logical technical work)

> Dear FOMers,
>
> First, let me apologize for advertising personal researches here. I really
> like to see comments on this kind of approach to the philosophy of
> mathematics by logicians. It differs from those well-known philosophies of
> mathematics (from the old logicism, formalism, intuitionism Platonism and
> so, to the recent fictionalism, naturalism, neo-logicism etc.), and it
> contains some technical work as well, which might interest some logicians.
>
> The goal for the entire research is to explore a completely scientific
> account for human mathematical practices. The technical part of it is an
> attempt to offer a logical explanation of the applicability of mathematics
> to things in this strictly finite (!) universe (from the Planck scale up 
> to
> the cosmological scale). The idea is to show that all applications of
> classical mathematics in realistic sciences are in principle reducible to
> the applications of 'strict finitism', which is a fragment of the
> quantifier-free Primitive Recursive Arithmetic (PRA), with recognized
> functions restricted to elementary functions (the proper subclass of
> primitive recursive functions). Technical work on this topic mostly 
> consists
> of developing ordinary mathematics within strict finitism. So far (in the
> monograph cited below) it has contained some basics of calculus, metric
> spaces, complex analysis, Lebesgue integration and linear operators on
> Hilbert spaces. This technical work grew out of the early work in my
> dissertation. (F. Ye, 'Toward a constructive theory of unbounded linear
> operators on Hilbert spaces', Journal of Symbolic Logic, 65(2000), no. 1; 
> F.
> Ye, Strict Constructivism and the Philosophy of Mathematics, PhD
> dissertation, Princeton University, 2000.) Besides that fact that some
> impressive applied mathematics can be developed within strict finitism,
> there are other intuitive reasons supporting the conjecture (!) that all
> applications of classical mathematics in realistic sciences are in 
> principle
> reducible to the applications of strict finitism.
>
> The philosophical part of the research might also interest those who are 
> not
> professional philosophers, because its goal is to show that we can replace
> metaphysical speculations regarding mathematics (as in Platonism,
> intuitionism and so on) by a completely scientific description of human
> mathematical practices, including a scientific explanation of the
> applicability of mathematics. It will view human mathematical practices as
> human brains' cognitive activities and take a study of human mathematical
> practices as a continuation of cognitive sciences, for the specific 
> subject
> matter of human mathematical cognitive activities. That is, its goal is to
> reduce the traditional philosophical problems to ordinary scientific
> problems, whose answers will then be pursued by ordinary scientific 
> methods
> (e.g. by postulating human cognitive models) and evaluated by ordinary
> scientific standards. From that perspective, the research discusses
> philosophical issues regarding mathematics, such as the nature of human
> mathematical knowledge, objectivity in mathematics, apriority of logic and
> arithmetic and so on.
>
> A monograph 'Strict Finitism and the Logic of Mathematical Applications' 
> and
> several articles belonging to the research project are available online at
>
> http://www.phil.pku.edu.cn/cllc/people/fengye/index_en.html
>
> The monograph focuses on the logical explanation of applicability of
> mathematics. It should be accessible to logicians who are not familiar 
> with
> the contemporary debates in philosophy of mathematics among philosophers.
> Other articles are mostly on philosophical issues, including an 
> introduction
> to the research project, titled 'Introduction to a Naturalistic Philosophy
> of Mathematics'. These belong to the context of contemporary debates in
> philosophy of mathematics among philosophers in the analytic tradition.
>
> All comments are greatly appreciated, and I am very happy to answer any
> questions.
>
>
> Sincerely,
>
> Feng Ye (PhD, Princeton, 2000)
> Associate Professor
> Department of Philosophy
> Peking University, China
> yefeng at phil.pku.edu.cn
>
>
> 

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