the need for relevance

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.

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.

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.

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.

quasi
.

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)
  • Re: the need for relevance
    ... 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. ... Uncertainity is the gateway from abstraction to reality. ... no requirement to demonstrate relevance at the start, ...
    (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)
  • 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.relativity)