X-Message-Number: 10292
Subject: math sillyness
Date: Thu, 20 Aug 1998 13:49:23 -0400
From: "Perry E. Metzger" <>

> From: Thomas Donaldson <>
> To Perry Metzgar:
> Not all civilizations used formal systems as we do. Not only that, but
> constructivists in math simply do not accept the existence of entities
> which have no construction...

Who might these be?

I've never met such an individual.

> and it HAS been possible to create a mathematics on this basis.

Yeah, right. A math without number theory, calculus or most other
things, but I'm sure you could live with *something* perhaps.

> Yes, if we insist on constructivism a lot of mathematics with either
> disappear (or perhaps more interesting to the mathematician!) provide
> us with new and interesting problems: OK, so we must construct all 
> entities we use in our proofs. Just how far can we take this?

Not very far.

Many of the most fundamental results are inherently
non-constructable. Take, for instance, the prime number theorem. It is
inherently non-constructivist. Most of the proofs behind the calculus
are inherently of the same form. Limit theory would vanish in a puff.

Sure, if you are willing to give up most of the tools of math, you can 
live this way, but I don't think I'd be particularly interested.

> If you like formal systems which are NOT constructive, then no law of
> God or Man prevents you from thinking about them. And yes, a lot of 
> beautiful math can be developed with those assumptions. If you like it,
> then all a constructivist would ask is that you accept that it is an
> art form rather than science. 

"Constructivist" formal systems are equally subject to Godelization.

> As a mathematician by training, I've often asked myself what will happen
> to mathematics in the future. I do not see any special reason why
> formal systems will necessarily survive (other than possibly as a form
> of exposition for mathematics). Math was done before anyone tried to
> formalize it, even plane geometry, and it will continue to be done
> if formal systems are forgotten.

Whatever you say, Thomas.

If you get rid of formalism, you get rid of math in the end. The idea
of the formalism was inherent from Euclid's day and before -- they
just lacked the language to express it. Without formal reasoning, all
you have left is voodoo. There is nothing wrong with doing the work
without the formalism, but if you can't reduce the work to the
formalism you no longer have math.


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