Re: the need for relevance
- From: Gonçalo Rodrigues <nospam@xxxxxxxxxxxx>
- Date: Mon, 14 Jan 2008 16:18:08 +0000
On Mon, 14 Jan 2008 12:50:58 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> fed this fish to the penguins:
[snip]
The mathematics community is very much aware of the need for
relevance.
But we stretch it ...
If theory A yields some insight into theory B, then at least in
principle, A applies to B. Of course, they might both apply to each
other to some degree, in which case we would have arrows from A to B
and also from B to A. If we have a path A to B to C, then indirectly,
A applies to C. Of course, this simple concept does not take into
account the nature and strength of the applicability.
The core theories relate to concrete things -- for example, numbers,
geometric objects. At the foundation, there is logic, which allows us
to define and analyze methods of deduction for all the other theories.
Mathematical theories are, by design, abstractions. Even the core
theories (numbers, geometry) are abstract, but less so. For example,
I'm sure Euclid was well aware that the concept of a straight line,
extending forever in both directions, is an abstraction, not
necessarily realizable in the physical universe. Even more so for the
concept of a point -- an "object" with a location but no size.
True. But these abstractions can be materialized into things in the real
world by ading a tiny bit of uncertainity to them. Once you add a little
bit of thickness to an abstract (read: ideal, hence invisible) straight
line, it readily becomes a line you can actually draw on say: a piece of
paper or a computer screen. This is an important feature to remember.
Uncertainity is the gateway from abstraction (idealization) to reality.
Huh? What does it mean to "add a little bit of thickness" or
"uncertainty" to say, the Hilbert spaces of quantum mechanics, the
SU(2) group for the spin of an electron, the curvature tensor for GR?
What do you mean by "materializing" into things in the real world?
This should be fun to read.
The point I'm trying to make is that Math does care about reality, but
readily allows further abstraction in order to get a handle on it.
Still, for any new theory, there is a sense of obligation to reveal,
however indirectly, something new about existing theories. In other
words, if some theory A sheds some new light on existing theories, and
if those existing theories lead, in some chain, down to the core
theories, then that provides some justification for A.
This has clearly happened to e.g. Set Theory. The _ZFC_ system contanins
many axioms which are nearly trivial (perhaps better: may be considered
instead as _theorems_) for _finite_ sets. What has happened next is that
infinite sets have be endowed with finite set like properties, in order
to make them look a great deal like the already well known finite sets.
So _theorems_ for finite sets have been adopted as _axioms_ for infinite
sets, for the simple reason that, otherwise, we would have no starting
point, at all, for the latter. But these choices seem rather arbitrary.
And consequently, there is more than _one_ set theory for infinite sets.
But the problem could have been solved otherwise, by simply not allowing
other than finite sets, together with a limit concept of some sort, for
the purpose of approaching infinite sets in "the calculus way".
I see that you acknowledge that the problem *was solved*. Could be
solved in any other way? And doesn't the very question tell you
anything? I mean, besides wishful thinking on your part, what sort of
evidence do you have that the problem *can* be solved by "simply not
allowing other than finite sets, together with a limit concept of some
sort, for the purpose of approaching infinite sets in "the calculus
way""?
Hint: half-ass half-baked set-theory like axiom systems with funny
names like "Implementable Set Theory" or something like it, do not
count as solutions.
And even if we assume you could do this, why would theoretical
physicists, that already have to spend a large amount of time
absorbing the current mathematical tools, would embark on *your* ship?
Just because Mr. de Bruijn is prejudiced against infinite sets?
And about real analysis, what exactly do you have in mind? Usual real
analysis uses mostly countable choice. As far as I know, non-standard
models use ultrafilters to build them (= full AC) and since they
contain the usual reals, they are as infinitary as them. You could
also go the axiomatic route, but somehow I do not think that that is
what you have in mind.
This should be fun to read.
However, based on past experience, the mathematics community is very
tolerant of, and in fact encourages, free exploration, with little or
no requirement to demonstrate relevance at the start, especially if
the structure of the new theory seems intuitively right. Play with it,
see where it leads. If it leads nowhere, it may die a natural death,
or if the structure still feels "just right", it may survive on its
own. That liberal "do whatever you want" credo is based on the
expectation that somewhere down the line (but not necessarily in the
current lifetime), there will be some tie back to existing theories.
But it's not a blank check.
What I'm trying to say is that mathematics combines art (math as a
beautiful form of reasoning) with science (attempts to model and
explain reality). Math is not oblivious to the need for relevance, but
allows a lot of freedom in that regard. Eventual, potential relevance
is usually sufficient.
I have a page which is called "Snippets of Pure Applicable Mathematics"
or alternatively "Purified Applied Mathematics". What you say is right:
_potential_ relevance is sufficient. I would like to add: is NECESSARY.
Mathematics must not be _applied_ per se, but it should be _applicable_,
i.e. it should NOT BE IMPOSSIBLE TO FIND AN APPLICATION for it, somehow.
And oh enlightened Mr. de Bruijn, why exactly is that a necessary
criterion for mathematics?
I've said this innumerous times, but I will say it again: Mathematics
is an *autonomous* discipline with a conceptual framework of its own.
It enters in relations with other disciplines (e.g. theoretical
physics) but is not bound by them -- that would be denying its
autonomy. It seems even absurd to say this, when mathematicians *do
work* on that assumption. So unless Mr. de Bruijn is ready to act as a
censor and order mathematicians to do otherwise, they will continue to
work autonomously.
Note: and please do not tell me that Mathematics was born from trying
to explain the "real world" (one of those expressions that without ""
is meaningless). Astronomy was born from astrology; you are not
suggesting that astronomers should be drawing horoscopes are you?
And assuming that that is a necessary criterion, how can you hope to
apply it? Given some mathematical theory how can you decide that it is
*impossible* to find an application for it?
This should be fun to read.
Mathematics must be pure. And if it is applied, it should be _purified_
first, in order to get rid of the non-mathematical details and terms.
Huh? What exactly are the "non-mathematical details and terms"?
It is strange practice that mathematics should be rigourous in its logic
but at the same time should enjoy complete and unrestricted freedom with
the choice of its axioms. This chain is as weak as its weakest link, and
quite vulnerable to the nonsensical: from first principles.
Unrestricted freedom in the choice of its axioms? Since the vast
majority of mathematicians in practice works within ZF(C), that hardly
counts as "unrestricted freedom". And, please, do not harp about the
nonsensical in mathematics, since you have been unable to show a
*single* example of contradiction.
Regards,
G. Rodrigues
.
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