Re: Real world and mathematics
- From: "porky_pig_jr@xxxxxxxxxxx" <porky_pig_jr@xxxxxxxxxxx>
- Date: Thu, 24 Jul 2008 18:20:31 -0700 (PDT)
On Jul 24, 4:48 pm, gandalf <gandalf1...@xxxxxxxxxxx> wrote:
John von Neumann:
"As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration."
V. I. Arnold also said pretty much the same things (http://pauli.uni-muenster.de/~munsteg/arnold.html).
Of course no professional mathematician took either of them seriously, even though both were/are highly accomplished mathematicians. How dare they break ranks?
With all due respect to Arnold, mathematics is *not* branch of physics
(*) . Furthermore, mathematics is not being concerned with
'resemblance to the real world'. What is the real world anyway? I
remember the post of one R. Finlayson complaining that "Cantor talks
about many infinities whereas the Universe has only one kind of
infinity". But what do we know about the kind (or kinds) of infinite a
'real universe' has? Nothing.
I do agree that certain branches of maths, like Analysis, take many
real world phenomenas as 'inspiration point, so to speak, but then
proceed with generalization and abstraction. Such as the 'impulse
function' which led to generalized functions. But there are branches
of mathematics like Algebra which mostly interact with other branches
(e.g., Algebraic Number Theory, Algebraic Topology, Algebraic
Geometry). That probably makes Algebra 'more abstract' in some
respect, than Analysis. Yet it will be silly to say that Analysis is
'better than' Algebra (or something like that).
(*) Well, Arnold is essentially a mathematical physicist, and his goal
is often to get some practical solution at the expense of rigor. See
his book "What is mathematics".
.
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