[FOM] Resources on the empirical foundations of mathematics

[hendrik@topoi.pooq.com]

On Tue, Jan 10, 2006 at 03:02:12PM -0800, Richard Haney wrote:
> I am interested in studying the *empirical* foundations of mathematics.
>  In particular, I am interested in exploring the *empirical*
> foundations of logic, number systems, and geometry.  I am interested in
> exploring the possibility that conventional logic(s) may fail to yield
> empirically true conclusions when applied to *real-world* phenomena
> even though the premises may seem to be perfectly true.  I am also
> interested in exploring the possibility, for example, that the rules of
> arithmetic may regularly (or irregularly) fail for, say, sufficiently
> large numbers when applied to *real-world* phenomena.  I am interested,
> for example, in considering the possibility that assumptions of
> constructivists may be too strong.  And, for example, I am interested
> in exploring how much interesting, useful mathematics can be done by
> allowing oneself, when talking about natural numbers, to talk only
> about natural numbers less than or equal to some unspecified, rather
> large natural number.

Yessenin Volpin was working along these lines, but I never fully 
understood his methods.  I believe David Isles has been trying to 
explicate some of this material.

> I have also read that some theoretical physicists have theorized that
> space-time may be discrete (i.e., not continuous) at a sufficiently
> small, sub-atomic scale.

Based on numbers I found in a recent Scientific American article on loop 
quantum gravity, I estimated that the sugar I put in my tea (two 
teaspoons) contains about a googol such grains of space.  At last a 
practical use for that number name.

See the last few chapters of MotionMountain, the free on-line physics 
textbook (http://www.motionmountain.net/), where the author presents 
absolte limits on the large scale -- it turns out there are maxima on 
force, power, distance, time, and so forth.  This seems to suggest that 
there is a finite limit N on the number of distinguishable points of 
space in the entire universe; thus a limit on the number of countable, 
observable, real objects.  This need not be a bound on the size of 
numbers, of course.  If one were to define a number as a sequence of 
zeros and ones, one write numbers up to something like 2^N (probably a 
different but similarly large N, of course).  If one were to use Church 
numerals written as binary-encoded landa-expressions, one could get 
larger numbers still, but at the cost of huge gaps in the representable 
numbers.  Is this reminiscent of ordinal notations?  I suspect one of 
the reasons infinity is a useful thing to reason about is that is 
provides an analogue to the behaviour of the inconceivable large, and in 
the limit, we can round off a lot of tedious detail.


> This sort of thing, too, would be of interest
> as to the sort of mathematics this might give rise to, say, in place of
> the real numbers, especially considering that physical space on a
> straight line was the origin of the ancient Greek concept of numbers. 
> Insisting that every side of every idealized triangle had to have
> exactly one length (number) may have led the world down one historical
> path in mathematics, whereas alternatives might have been quite
> different.

The reasons we have considered these things so reasonable and obvious 
for so many millenia is, I believe, that they are approximately valid at 
the scales at which we have had to use our brains to interact with the 
environment -- we have simply evolved brains with all these assumptions 
built in.  We have been thinking for a long time, too, even longer, if 
one is to believe Pinker (How the Mind Works, the Language Instinct), 
than we have had spoken, learned languages.  So certain rules of logic 
are built in too.  As we applied them to things beyond the environment 
they evolved in, they have been found lacking.  Most notably, the 
evolution of logic in the last few thousand years have involved the 
application of logic to linguistically created entities. and the 
formulation and formalisation of logic has been linguistic (leading to 
the widely held belief that one cannot think without language).  This 
recent evolution has been cultural, not linguistic, and has involved 
recurring failures and recoveries.  The rules of logic have been 
debugged against paradoxes, such as the liar paradox, Zeno's paradoxes 
of motion, Newton's fascination with the ghosts of departed quantities 
(Berkeley's description of differentials, if I recall correctly), the 
paradoxes involved in infinite series approxomations, and so forth.

The interesting thing is that each generation of mathematicians appears 
to believe the mathematics they grew up on, and trust it as being 
*true*, without a real understanding of what that means, if anything.  
This makes it remarkably difficult to strike out in a different 
direction.  Physicicts have the advantage of experiments that falsify 
their theories.

> I don't pretend to be an expert on these things, but I
> would like to learn more about them.  As far as I can tell, these
> issues, with a focus on empirical relevance, seem to have gone
> unexplored at the level of serious research.
> 
> I imagine that there may be a great many other interesting, relevant
> issues and questions in the empirical foundations of mathematics but
> that I am simply unable to imagine or formulate them at the moment.

If. for example we actually had experience with actual infinities, we 
might find them behaving distressingly different from the way we now 
expect, based merly on extrapolation from the finite.
 
> So what I would like to do is to find some really good resources for
> research in this area.
> 
> Can anyone help me out with this?

Perhaps a good place to start would be to try to come up with 
more of the questions.

-- hendrik