Theories about Infinity

The science of nature is concerned with spatial magnitudes and motion and time.

Each of these at least is necessarily infinite or finite, even if some things dealt with by the science are not, e.g. a quality or a point-it is not necessary perhaps that such things should be put under either head.

Hence, the physicist should:

• discuss the infinite
• inquire whether there is such a thing or not, and, if there is, what it is.

The Pythagoreans Versus Plato

The Pythagoreans and Plato make the infinite a principle in the sense of a self-subsistent substance, and not as a mere attribute of some other thing.

Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these).

They assert that what is outside the heaven is infinite.

Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also.

The Pythagoreans identify the infinite with the even.

When it is cut off and shut in by the odd, it provides things with the element of infinity. This happens with numbers.

If the gnomons are placed around the one, and without the one, in the one construction the figure that results is always different, in the other it is always the same.

But Plato has 2 infinites:

• the Great
• the Small

The physicists, on the other hand, always regard the infinite as an attribute of a substance which is different from it and belongs to the class of the so-called elements-water or air or what is intermediate between them.

Those who make them limited in number never make them infinite in amount.

Anaxagoras Versus Democritus

Anaxagoras and Democritus make the elements infinite in number.

• They say that the infinite is continuous by contact.
• The infinite is compounded of the homogeneous parts according to the one, of the seed-mass of the atomic shapes according to the other.

Anaxagoras held that any part is a mixture in the same way as the All, on the ground of the observed fact that anything comes out of anything.

This is why he maintains that once upon a time all things were together.

There is a beginning of separation, not only for each thing, but for all.

Each thing that comes to be comes from a similar body, and there is a coming to be of all things, though not at the same time.

Hence there must also be an origin of coming to be.

One such source there is which he calls Mind.

• Mind begins its work of thinking from some starting-point.

So necessarily all things must have been together at a certain time, and then moved at another time.

Democritus asserts the contrary: no element arises from another element.

The common body is a source of all things, differing from part to part in size and in shape.

Aristotle’s Own Theory

All of these people make infinity a principle or source. We cannot say that the infinite has no effect, and the only effectiveness which we can ascribe to it is that of a principle.

Everything is either a source or derived from a source.

But there cannot be a source of the infinite or limitless, for that would be a limit of it.

• As it is a beginning, it is both uncreatable and indestructible.

For there must be a point at which what has come to be reaches completion, and also a termination of all passing away.

That is why there is no principle of this, but it is this which is held to be the principle of other things, and to encompass all and to steer all, as those assert who do not recognize, alongside the infinite, other causes, such as Mind or Friendship. Further they identify it with the Divine, for it is ‘deathless and imperishable’ as Anaximander says, with the majority of the physicists.

The belief in the existence of the infinite comes from 5 considerations:

1. From the nature of time-for it is infinite.
2. From the division of magnitudes-for the mathematicians also use the notion of the infinite.
3. If coming to be and passing away do not give out, it is only because that from which things come to be is infinite
4. Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself
5. Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody-not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.

The last fact (that what is outside is infinite) leads people to suppose that body also is infinite, and that there is an infinite number of worlds. Why should there be body in one part of the void rather than in another? Grant only that mass is anywhere and it follows that it must be everywhere.

Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be. But the problem of the infinite is difficult: many contradictions result whether we suppose it to exist or not to exist.

If it exists, how does it exist? As a substance or as the essential attribute of some entity?

If not, is there something which is infinite, or some things which are infinitely many?

The physicist’s main problem is to investigate whether there is a sensible magnitude which is infinite.

What does infinite mean?

1. What is incapable of being gone through, because it is not in its nature to be gone through (the sense in which the voice is ‘invisible’)
2. What admits of being gone through, the process however having no termination, or what scarcely admits of being gone through
3. What naturally admits of being gone through, but is not actually gone through or does not actually reach an end.

Everything that is infinite may be so in respect of addition or division or both.