Wednesday, December 21, 2011

Gorgias 'On What Is Not'

To: Marcus S.
From: Geoffrey Klempner
Subject: Gorgias 'On What Is Not'
Date: 13 July 2006 11:17

Dear Marcus,

Thank you for your email of 29 June, with your fifth and final essay for the Ancient Philosophy program, in response to the question, 'Select an argument from Gorgias' On What is Not and discuss its interpretation and validity.'

You have chosen this passage:
[If what-is] does not have a beginning, it is unlimited, and if it is unlimited it is nowhere. For if it is anywhere, that in which it is is different from it, and so what-is will no longer be unlimited, since it is enclosed in something. For what encloses is larger than what is enclosed, but nothing is larger than what is unlimited, and so what is unlimited is not anywhere. Further, it is not enclosed in itself, either. For "that in which" and "that in it" will be the same, and what-is will become two, place and body (for "that in which is place, and "that in it" is body). But this is absurd, so what-is is not in itself, either. And so, if what-is is eternal, it is unlimited, but if it is unlimited it is nowhere, and if it is nowhere it is not.


You have chosen a particularly interesting argument, which has continued relevance today in the philosophy of space and time, and also to the notion of a universal set or the concept of 'everything'.

Your claim that 'all subjects are really assertions' applies in a Russellian sense, that a proposition containing a singular term like, 'The President of the USA' would be analysed by Russell as a conjunction of two claims, e.g. 'The President of the USA is ill' becomes, 'There is one and only one president of the USA, and that person is ill'.

However, in 'This is a birch tree', the term 'this' arguably does not make an existence claim. 'This' is functioning as a 'logically proper name' in Russell's sense. (A complication is that Russell would only allow names of sense data to be 'logically proper' but we can ignore this for the sake of the present example.)

We know that, for Gorgias, 'there is something which is' is NOT meant in the sense that, e.g. if there is a pen on my desk then it follows that there is something which is. The meaning is clearly intended to apply to a special case of 'what is', namely, 'everything', or 'the whole'. Gorgias is taking as the assumption for reductio, that there is such a thing as 'everything that is', or the 'whole of existence'.

The statement, 'There is something which is, and that thing is not' is a contradiction. However, the statement, 'If there is something which is, then that thing is not' is not a contradiction but is logically equivalent to, 'It is not the case that there is something which is.' We have seen that this is not intended to rule out the existence of my pen, but rather the existence of a thing which we call, 'the whole' or 'everything that exists'. There is no such thing, Gorgias argues, 'as everything'.

In a similar way, in formal logic the statement, while, 'P and not-P' is a contradiction, the statement, 'If P, then not-P' is not a contradiction, but is logically equivalent to, 'not-P'.

In order to grasp the force of the argument it is not necessary to take this in a strictly Parmenidean sense. The argument applies to anyone who believes that it makes sense to talk of 'everything that is'.

If there is such a thing as 'what is', in Gorgiases sense, then where is it? Gorgias argues that to be somewhere is to be in relation to something else. For example, the pen is on my desk. But 'what is' already includes everything, so there is nothing for it to be spatially (or temporally) related to.

Therefore it is 'nowhere'. Therefore, 'it is not'.

Let's take each of these two claims. You offer the suggestion that what is, is over here on my desk, and over there outside my window, and so on. In other words, it is not 'nowhere'. This implies a relational view of space and time, as advocated by Leibniz in his famous dispute with the Newtonian, Samuel Clarke. According to Leibniz, space is not something, 'in itself', it is merely a construct of spatial relations. According to Newton, by contrast, space is something in itself, he calls it the 'sensorium of God'. In which case, the whole of infinite space 'is' somewhere, namely 'in' God (presumably in a non-spatial sense). So we might be tempted to accuse Gorgias of assuming a non-relational (non-Leibnizian) notion of what (spatially or temporally) is. Leibniz would reply, as you do, that it simply does not make sense to ask where 'everything' is, the parts of 'what is' are where they are in relation to other parts of what is.

On second thoughts, isn't this exactly what Gorgias is saying? 'What is' is nowhere, because it doesn't make sense to ask 'where' it is.

I don't agree that we can make anything of the difference between 'nowhere' and 'nowhere in particular'. 'He is going nowhere' and 'he is going nowhere in particular' mean different things, but there is no such thing as being 'nowhere in particular' unless we are talking e.g. about quantum mechanics where (allegedly) an electron exists in a region (as a 'cloud') but nowhere in particular within that region.

What is, is nowhere. Does it follow that what is, is not? No. We have seen that attributing existence to the totality of 'what is' has a special meaning which cannot be explained in terms of what it is for a member of that totality to exist.

However, Gorgias can legitimately point out that the onus is on the defender of a notion of 'everything that is' to explain this unique sense of 'existence'.

It was none other than Russell himself who produced the most powerful argument against the idea of 'everything that is'. It is known as Russell's paradox. Gorgias would have felt vindicated.

The assumption which necessarily underlies the notion of 'everything that is', is the idea that we can form a SET corresponding to any condition. 'Everything that is' is the set which contains everything which exists.

But if that assumption is true, then we should be able to form the following set: 'X belongs in set S if and only if X is not a member of itself'.

First, we may remark that some of the things that exist are objects, and some are sets. Some sets are members of themselves, e.g. the set of abstract objects is itself an abstract object, and some are not, e.g. the set of pens is not a pen.

If X is a set, then X belongs in S only if X is not a member of X. Then we ask the question, what about the case of X=S? Is S a member of itself or not?

If it is, it isn't and if it isn't it is.

Therefore, it logically follows that there is no such set as S.

Therefore we have to reject the assumption that we can form a set corresponding to any condition.

Therefore...

All the best,

Geoffrey