Numbers and Nature

[Edward Grefenstette <pha04eg@sheffield.ac.uk>]

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Phew! I just had a grueling mathematics exam today (oh how I envy  
those of you who have their exams before the holidays), and normally  
the last thing I want to do is talk about maths. However while I was  
revising I was procrastinating by thinking around the topic of maths  
a bit, so here's my bit of catharsis...

Mathematicians are interesting people. You need to be a special type  
of crazy to be a mathematician. I mean, most jobs and academic  
domains are more or less empirical and down to earth. Philosophy,  
physics and the such can get pretty abstract, but most of the time  
you have at least some ground in 'reality', be it observing  
behaviour, or searching for practical applications of a theory. But  
theoretical mathematics doesn't ever seem to touch the ground.  
Generally speaking, it's about abstract theories which are applied to  
other abstract theories in order to find or prove some other abstract  
theory. To want to work on something so removed from reality, you  
need to love your numbers. And I love irking mathematicians about  
these same numbers.

You see, it has been my theory ever since the first time I argued  
about something with a maths teacher that it was all theoretical. I  
claim that numbers and mathematical theories exist only in the minds  
of those using them. I'd like to discuss my reasons why before  
telling you where I'm going with this.

First, let's talk about quantification. You have three apples on a  
table. How come there are three? Because I considered them as a  
group. Otherwise, for example if for some reason I couldn't see two  
of the apples on the table, I'd say that there was one apple. And if  
I could only see one at a time, I could say that there is one apple,  
and one apple, and one apple. Of course in that case, I could  
remember seeing two apples before the last one, and say that there  
are in total, three apples on the table. The important thing is that  
quantification of an group of objects requires knowledge of each  
element of the group (or does it? See much further below). Thus there  
is the need for an intentional relation between an observer (who, I  
argue, is necessary) and the objects quantified. What is there if  
there is no observer? Objects (or as Sartre would put it, just  
"Matter") with no discreet link between them, no "Universe" in which  
to be counted. Just things sitting in a spatial frame.

Now imagine I were to pick up an apple and cut it in three slices  
perfectly equal slices (let's assume this is possible). We therefore  
have 3 times 1/3 of an apple. Or three slices which are units of  
their own. If they were of unequal size, I'd still call them slices,  
so let's assume slices means "equal slices" (notice the fact that  
units are not discreetly defined, but indeed are relative qualities  
attributed by the observer). Now once again, in defining units and we  
have allowed for these slices to represent fractions of the apple.  
But does this concept of fraction exist in nature?
The objective existence of fractions is even harder to defend: they  
only exist when a unit and a type (or body) is defined, as opposed to  
just a type (or class), and both qualities are once again dependent  
on an arbitrary relationship defined by the observers. In short I  
believe that fractional quantification is justified by the state of  
division of an object which itself numerically justified... by  
fractional quantification. If these seems overly complicated to you,  
it's probably because I'm confusing myself and there actually isn't  
circularity. Note that whether or not the justification for fraction  
is a circular-justification fallacy or not, it's interesting to note  
that primitive cultures worked without it just fine, and managed to  
work mathematical operations (such as divisions) that require  
fractions in our modern system... without the use of fractions. One  
example is the glyph-counting system used in ancient Egypt (http:// 
www.math.buffalo.edu/mad/Ancient-Africa/ 
mad_ancient_egypt_arith.html). Any way, the problem of quantification  
and intentionality remains. That's essentially my problem with the  
whole objectivity of mathematics. Add the fact that all mathematics  
are based on axioms (then again, what isn't) and you've got yourself  
quite mission if you want to prove that mathematics exist in nature  
and outside of human (or animal?) experience.

Now while I just said is hotly debated by mathematicians, non  
mathematicians, the other part of my brain and hopefully will be by  
BUPS-Dis readers as well, it's interesting to consider it as an  
analogy for many other things we pragmatically view as objective  
truths for every day purposes: ethics and morality, freedom, rights,  
the idea that each body holds one person... etc. Essentially,  
problems of 'objective quantification' seem to support phenomenology  
quite well, in my opinion. But once again, the whole problem with  
numbers, and other things is difficult to outline and easily gets  
messy. So I turn to ye of clearer thought that I. Has anyone else  
given the issue any thought (or would like to right now)? Do you  
think there are discreet properties to things? That ethereal concepts  
float through the air (or are, like a certain account of souls,  
dimensionless entities) without the ontological necessity for an  
intentional observer? And finally, am I right in provoking my  
mathematically minded friends in such a way, or is the above just  
willful thinking on the part of a man who is comfortable with the  
idea that phenomenology explains a lot of things?

-- Edward.

PS: For your number related amusement. I ran into a Poem by Mike  
Keith parodying E. A. Poe's "Raven", which maps out 'pi' to the 740th  
decimal. An amazing piece of work: (http://users.aol.com/s6sj7gt/ 
mikerav.htm)
Perhaps numbers don't exist in nature. But when men make them exist,  
the sometimes do it quite well...


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