Re: the need for relevance

quasi wrote:

On Wed, 09 Jan 2008 08:21:05 -0500, quasi <quasi@xxxxxxxx> wrote:

On Wed, 09 Jan 2008 13:48:38 +0100, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

Is mathematics an activity of economic value? Is it rewarded? If yes, then feel the responsability of producing something else than fantasy.

Hey, we're artists, not engineers.

Mathematics can be regarded as "the art of thinking abstractly".

Sure it's been applied, and sure it uses the "scientific method", but at heart, it's art, not science.

I'm not happy with my above response to Han de Bruijn.

There are elements of truth in what I said, but on the whole, it's not what's really going on.

I said "Hey, we're artists, not engineers", but I take that back.

We're more like architects.

In that sense we're artists _and_ engineers.

And as architects, we want the end product to be both structurally
sound and relevant, not just aesthetically satisfying.

Good! An I suppose you want to build your structure on solid ground in
the first place?

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.

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".

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.

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.

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.

Han de Bruijn

.

Relevant Pages

  • Re: the need for relevance
    ... Uncertainity is the gateway from abstraction to reality. ... there is more than _one_ set theory for infinite sets. ... no requirement to demonstrate relevance at the start, ... or alternatively "Purified Applied Mathematics". ...
    (sci.math)
  • Re: the need for relevance
    ... Uncertainity is the gateway from abstraction to reality. ... there is more than _one_ set theory for infinite sets. ... no requirement to demonstrate relevance at the start, ... I have a page which is called "Snippets of Pure Applicable Mathematics" ...
    (sci.math)
  • the need for relevance
    ... Mathematics can be regarded as "the art of thinking ... I said "Hey, we're artists, not engineers", but I take that back. ... readily allows further abstraction in order to get a handle on it. ... no requirement to demonstrate relevance at the start, ...
    (sci.math)
  • Re: the need for relevance
    ... mathematician is somehow obliged to justify the relevance of his work ... Mathematics stands somewhere in between the arts and the ... because art is not reducible to beauty and second, ... of beauty, even a profound sense of beauty, arises in virtually every ...
    (sci.math)
  • Re: Can time be separated from space?
    ... I do not recall calling s-t an abstraction, ... Math constructs are not abstractions in the sense that they are ... Mathematics is "the study of... ... Microsoft® Encarta® Reference Library 2005. ...
    (sci.physics)