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Philosophical significance of the paradox of the Heap


To: Victoria M.
From: Geoffrey Klempner
Subject: Philosophical significance of the paradox of the Heap
Date: 18 May 2007 11:04

Dear Victoria,

Thank you for your email of 13 May, with your first essay for the Philosophy of Language program, in response to the question, 'How would you explain to a non-philosopher the philosophical significance of the paradox of the Heap?'

This is a good essay which shows a keen awareness of the importance of the problem of vagueness for the philosophy of language. As you say, logic itself, 'one of the key tools to meaning and definitions of philosophical truths' is put into question by the paradox of the Heap.

The key questions are the nature of logic and the notion of truth which is assumed when we define a sound argument as one which never leads from true premisses to a false conclusion, and also (as you again point out) our conception of the relation between the view of the world filtered by our language, and the objective nature of reality.

The logical principle involved is that of 'mathematical induction'. This is arguably the most important principle in maths. Basically, a general proposition of the form, (x)(Ax) or, 'All numbers x have the property A' is proved if you can show:

(1) A is true of the number 1.

(2) If A is true of x, then A is true of x+1.

The principle assumes that we have a domain (in this case the domain of natural numbers) and a way of ordering the domain (in this case by the simple function, 'add 1').

The Heap paradox can be run in both directions, as you show. In fact, these are perfectly symmetrical so far as the logic, and the conclusion is concerned. So let's take the case were we start with one grain.

(1') One grain of sand is not a heap.

(2') If x grains of sand is not sufficient for a heap then x+1 grains of sand is not sufficient for a heap.

Therefore, there is no number of grains sufficient for a heap.

The premisses are true, the conclusion is false. Therefore, it seems, we have to reject the principle of mathematical induction. Or do we?

This would be a catastrophe. With one sweep, virtually all of number theory would be eliminated.

Now, one looks around for alternative solutions. This is the point where we cheerfully hand the problem over to the non-philosopher (or the sceptic about the usefulness of philosophy) and say, 'You try!'

Generally, when non-philosophers respond sceptically to an alleged philosophical paradox, there is in fact a picture in their minds which they are clinging onto, a half-articulated 'theory', which they have never thought to question.

Here's one typical response: 'All you've done is show that there aren't any 'heaps' in reality. 'Heap' is just a rough and ready term which serves a useful purpose, but in principle we could count the number of grains and describe the precise geometrical shape of the so-called 'heap', and that would give all the facts, the literal and unvarnished truth about the object in question.'

What is wrong with that response? One can point out that a large part of our ordinary, everyday vocabulary is vague in one respect or another. In which case, it would seem to follow that in ordinary conversation, we do not succeed in 'describing reality'. When I say, 'There is a heap of sand on the front drive,' I am not saying anything true. In order to say something true, I would have to use terms which had no dimension of vagueness.

'OK,' says our respondent, 'in that case I'm not interested in the 'literal and unvarnished truth' but merely in 'pragmatic truth', saying things that we can generally agree about without getting too fussed about precise details.'

You can point out that this escape from the paradox is purchased at a great cost. So far as the things that we talk about everyday are concerned, there is no 'truth' or 'falsity', but only 'what works', or, maybe only, 'what serves to convince'. All the noble and principled views that we take about the importance of truth crumble into dust. There is no difference between 'truth' and 'lies', only different degrees of lying.

- I'm just imagining how the dialogue might go. I was once at a dinner at my college in Oxford (University College) where there were a number of fellows sitting round the table - History, Law, Philosophy. One History fellow asked one of the Philosophy fellows, 'Are you still working on the topic of proper names? I've never understood why it's such a problem saying what a proper name is!' The Philosophy guy tried and failed. It seemed that they had had this conversation before. This incident has stuck in my mind as a perfect illustration of how difficult it is to get across, even to very intelligent non-philosophers, why a philosophical problem or paradox is perceived as 'important'.

Once you get the philosophical point - whether we are talking about vagueness or proper names or whatever - you realize that every response is a 'theory', even the response which rejects (seemingly rejects) the problem as trivial. There is no way out. Hence the various attempts to deal with the problem of vagueness described in the program (which are outside the scope of this essay). The beauty of paradoxes in philosophy is that they force us to respond, in one way or another. And sometimes none of the responses work and we are left in a state which the Greeks called, 'aporia'.

All the best,