Re: Numbers and Nature

It seems to me that we can set up the '3 predicate' such that some things are three iff we can count them and we get to three, counting proceeding 'one,' 'two,' 'three,' and so on. (I don't know whether getting at numbers by counting is the best idea or not). You might say, "Hang on a minute. I agree that you can count three objects, but this requires that there be a you to count, and surely our concept of numerosity is such that there doesn't need to be a counter." This is true, yet for there to be three objects it is only required that someone could count "one, two, three," and not that someone does. Still though you might say, "OK, fair enough, but someone invented this predicate. It was set up by a human. If numerosity depends upon the predicate and the predicate depends upon a human, doesn't numerosity require some sort of agent to get going." I don't think so. Regardless of whether anyone ever uttered my '3 predicate' or not, and regardless even of whether anyone conceived of it, it is still true that things are three if you can count them thus: "one" "two" "three." This is true even if no one is there to count. A sycamore would still fulfil a tree predicate even if there were no agents to talk of them, or if there were agents who considered the world differently such that they had no analogous concepts to 'tree' or 'sycamore.'

On 16/01/06, Edward Grefenstette <pha04eg@sheffield.ac.uk> wrote:

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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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