An ancient

Greek word (απειρων), pronounced "ah - PAY - rohn," usually translated
'unlimited' or 'boundless', perhaps

*incalculable*. In the

Pythagorean "

table of the opposites"
it appears thus:

limited **apeiron**

one many

straight crooked

good bad

nodes freegel

(Unfortunately I don't have an authoritative source for that last pair.)
The Pythagoreans believed that, in the words of
Aristotle, "the whole heaven is a harmony and
a number" and this gives us a clue to what they
thought about *apeiron*. The ancient Greek concepts of number and harmony were very much bound up in their concept of **ratio**, in Greek:
*logos*. This has given us words to do with
systematic attempts to understand the world: logic,
rationality (the same metaphor filtered through
Latin.) *Apeiron*, therefore, can be understood
as standing in opposition to this Pythagorean good.

To see how the Pythagoreans might have understood the word, it helps to describe one of the Pythagorean struggles against *apeiron*. When Pythagoras invented his famous theorem about right-angled triangles, one of the immediate
consequences was a demonstration of the existence
of what we now call *irrational* numbers. The Greeks used the terms *alogos* (inexpressible) and *arratos* (without a ratio).

Remember, having no decimal point notation, they
must conceive fractional parts as ratios between
whole numbers. A simple proof shows that the
dividend in the ratio for sqrt(2) (the length
of the hypoteneuse for a right angled triangle
with other sides equal to one, by Pythagoras'
theorem) must be both odd and even. In other
words, sqrt(2) cannot be exactly represented
as a fraction. Even using the modern method, its
representation would require an infinite decimal
expansion: *apeiron*.

According to one story, the Pythagoreans took this so seriously that Hippasus, who wanted to go public with the information, was murdered at sea.

*Apeiron* can also be understood as chaotic, or
*infinitely complex* - the Pythagoreans thought
of a crumpled handkerchief as *apeiron* - and
indeed there is a close relationship
between the concepts of infinity and complexity.

One way of thinking of the task of defining larger
and larger numbers is as the task of
packing more and more
complexity into a finite
definition. But when it comes to the Absolute
Infinite, somewhere along the line we give up
the ghost.

To my mind, this Pythagorean denial of *apeiron*
is reminiscent of a modern trend in philosophy, which
I like to call physicalism. This holds that what
is real is what can be treated of by physics. As physics
is a formal discipline, this amounts to a denial of
*apeiron*, the undelimited, the informal. This is not to say that
physics is wrong, or even incomplete within its own
domain, just that the philosophical use of a formalism
such as physics as an ontological yardstick is undermined
by the implied rejection of the reality of *apeiron*.

It's quite easy to place the opposing concept of limit,
ratio, order, in our world: it's the regular way
that stuff behaves when it moves about: Information,
causality, natural law. *Apeiron* might seem
harder to spot. But it seems we're immersed in the informal;
it's just all the other stuff, the stuff that philosophers get blasted for ignoring! Emotion, values, consciousness, humour, the funny feelings you get when you're drifting off to sleep.. None of these are capable of a realistic treatment by latter-day Pythagoreans, because the anti-apeiron spectacles render them invisible. They must be reconstructed as regularities in behaviour, which, it seems, is precisely what they are *not*.

*See also:* Reflection principle.