To: Marcus S.
From: Geoffrey Klempner
Subject: Leucippus and Democritus on atoms and the void
Date: 30 August 2005 11:52

Dear Marcus,

Thank you for your email of 18 August, with your fourth essay for the Presocratics program in response to the question, "'All that exists is atoms and the void.' What is an atom? What is the void?'

[Correction: the question should have said, 'All that exists is atoms and the void.']

You have made a great go at the argument. Even though you say you found the essay 'very difficult' you must have enjoyed writing it. You have demonstrated your willingness to consider the question from first principles without any preconceptions - just as the Greek atomists did.

As you have interpreted the question, *if* all that exists is atoms and the void what *must* 'atoms' and 'the void' be? There are other ways of approaching the question: for example, by considering the historical context, viz. Parmenides views on Being and reactions to it. However, your approach is not only legitimate but also very fruitful.

You say that there must be at least two atoms. It is difficult to argue with that: 'atoms' means more than one atom.

Next, you consider three alleged alternatives: that atoms constitute less than half, half, or more than half of 'all that exists'. This seems to make logical sense. On the assumption that the void has no mass or weight (which, strictly speaking, if we were considering every aspect of the question deserves to be argued for and not just assumed), it would seem that the only measure is volume: i.e. atoms constitute less than half, half or more than half the total volume of 'all that exists'

However, if, as you go on to argue, the void is infinite (i.e. of infinite volume), then the only way that atoms could constitute half, or more than half, of all that exists is if either there are an infinite number of finite atoms, or there exists one or more infinite atoms.

- At this point we need to pause to consider the logic of infinity. The natural numbers 1,2,3,4,... constitute an infinite set. So do the even numbers, 2,4,6,8,... even though there are only 'half as many' even numbers. Mathematicians define an 'infinite set' as a set whose members can be put into a 1-1 correlation with a 'proper subset', i.e. some but not all of the members of that set. As an illustration of this, consider a galactic hotel with an infinite number of rooms. A party arrives consisting of an infinite number of new guests. That poses no problem. The occupant of room number 1 is asked to move to room number 2, the occupant of room number 2 is asked to move to room number 4 - and so on. Then the new party all take the odd numbered rooms.

See the Wikipedia article:

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Back to your essay.

You say, 'The expression 'all' then means the same thing as the 'void'. The 'void' is 'all.' Thus, the word 'all' is superfluous.' Suppose we are attempting an inventory of reality, which takes account of everything that exists. For example, the atomists' inventory would be very simple:

Atoms
Void

The problem with this is that in putting forward this list there is another statement, implied but not made: that this is indeed 'all'. As an illustration of this, suppose someone handed you a class list for your new course on Hamlet. The class list doesn't tell you everything you want to know: is this *all* the students who will be attending the class? in other words, is the list exhaustive, or is it possible that more students will be joining later?

So, in terms of the atomist theory:

There are atoms
There is one void
There is nothing else

In stating that there is nothing else, we are considering everything, both atoms and void. Any x that we come across must either be an atom or the void. The quantifier 'Any x' refers to an unspecified 'domain of quantification', rather than just to the void. (You can look up these terms in any modern introduction to logic.)

Moving on, in what sense can there be a 'clump' of atoms? As you observe, we must always be able to logically distinguish two atoms from one atom. There is a serious issue of how exactly this is done. Must there be an infinitely small void in between two atoms? Or could two atoms be fully in contact, yet distinguished from one another by the possibility of sliding, i.e. by the fact that one atom had the capacity to move relative to the second atom?

You go on to say, 'Atoms might be an infinite distance from each other, therefore the void must be greater than infinite in size.' At first sight, this might look like a case for applying the principle I referred to earlier, in relation to Hilbert's paradox of the infinite hotel. If you 'add' all the even numbers to all the odd numbers the result is the same 'size' as the original two sets. viz. infinite. However, in the case of spatial infinity, it is not clear what would be meant by saying that A is at an infinite distance from B, which is at an infinite distance from C, where A, B and C are in a straight line (whatever that means in this context). Is C the same distance (i.e. infinite) from A as the distance of B from A?

I'm going to pass on that one. The theory that atoms exist in an infinite void does not require that some atoms are infinitely far apart. I just don't know what it would mean to say that two finite objects A and B are separated by an infinite spatial distance, whereas I think I know what would be meant by saying that the void is of infinite size.

Atoms, or clumps of atoms, might be of infinitesimal size, i.e. be infinitely small. However, it doesn't follow from the assumption that the void is infinite, that there exist atoms or clumps of atoms of infinitesimal size. You might be tempted to argue, 'Compared to the size of the void, any finite object would be infinitesimally small,' but that doesn't follow.

You conclude, 'I am going to be speculative and conclude that logically speaking the assumption [that all that exists is atoms and the void] cannot be accurate because it fails to describe the universe as it actually is.' An atomist would be justified in asking what gives you the right to conclude this? Certainly, experience tells us that there must be more than two atoms and the void. However, that empirical observation does not refute the statement that all that exists is atoms and the void, or require us to qualify that statement in any way.

All the best,

Geoffrey