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Philosophical significance of the paradox of the heap


To: Damiano S.
From: Geoffrey Klempner
Subject: Philosophical significance of the paradox of the heap
Date: 19th November 2010 11:14

Dear Damiano,

Thank you for your email of 10 November, 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?'

I found the English to be excellent. I had no problem at all in following your meaning.

It's interesting that you start off with the question, 'Is it possible to construct a logically perfect language?' I think you'd agree that for the non-philosopher this is a rather strange and arcane question. 'Perfect' in what way, exactly? And what is 'logic'? However, a way to motivate this idea would be to point out something that we all recognize, that, somehow, without agreeing on precise definitions of the words we use, we are nevertheless able to convey useful information to one another by means of ordinary language. Yet sometimes we fail, and misunderstandings arise. Who has not experienced the frustration of finding yourself arguing with a friend over a trivial matter, an argument which could so easily have been avoided had one of you expressed themself a bit more clearly.

This motivates the attempt to find better, clearer words to express what we mean. But a logically perfect language? When would we need this, and for what purpose?

I have spent a lot of time thinking about how persons with no previous experience of philosophical thinking can be helped to grasp what philosophers are trying to do, the point of it all. As philosophers, we need to think about this, just to keep ourselves honest. With regard to this particular problem -- the paradox of the Heap -- it is incredibly difficult to motivate learners to think of the paradox as a serious challenge, as something more than merely verbal trickery. They just don't get it. You try explaining it, on someone you know who has never done philosophy before!

As good philosophy students, we learn the importance of logic. We acquire the greatest respect for a logical argument, set out in clear, persuasive steps. The paradox of the Heap is gripping because the steps are so clear, so undeniable -- while the conclusion is so clearly false. How could we get into this mess? Yet, to the non-philosopher, the whole thing is just ridiculous. I think that's a paradox too. The paradox of 'the paradox of the Heap'.

If one was looking for a premise to question, it would be the principle of mathematical induction. As good logicians, we 'see' immediately that what applies to 1 and applies to n+1 if it applies to n, must therefore apply to all the members of a numbered series. You just keep 'doing it', performing the very same action, over and over again for 1, 1+1, 1+1+1, 1+1+1+1 etc. But isn't this begging the question? We know we *can't* just repeat the process endlessly. We know, from experience, that as the numbers rise, so our confidence in applying the predicate in question (e.g. 'non-heap', or 'bald') to the next member in the series falls until we reach a point where the question is just indeterminate. Is it a heap or not a heap? Is the man bald or not bald? We just don't know what to say in this particular case.

Looking at this more closely, and comparing it with mathematical induction as used in arithmetic to prove theorems, it seems to me that there is an important distinction between the mathematical and the empirical case. In the case where we are hypothetically adding grains of sand, one by one, it is *not* true that we are doing the 'same thing' over and over again. Adding one grain to a few scattered grains on the floor is not the same thing as adding one grain to a large number of grains which are beginning to heap up. It's not the same action. That's what someone who had not been 'brainwashed' by a logic course would say.

Moving on to the idea that there is an ideal physical description of the world, at the fundamental level, this does seem easier to motivate. Science is all about precision. In everyday life we express ourselves as precisely as the situation demands (although, as I indicated, sometimes we make a bad judgement call and fail to be sufficiently precise). But science, as you say, aims to give an 'objective' description, it aims to say 'how things are' in themselves and not merely pick out salient features depending on our interests. The problem here is that, in attempting to omit the subjectivity of human perception and interests, we find that we have failed to describe reality completely. We have left a whole lot of stuff out.

What is so interesting about this -- especially for non-philosophers -- is that this isn't anything to do with a 'soul' or Hamlet's 'more things in heaven and earth than are dreamt of in your philosophy'. You can't reduce psychology to physics. Whereas physics might attempt to describe its subject matter with complete precision, psychology can't do this. Donald Davidson accepts this point. He calls this the 'anomalousness of the mental': see his article 'Mental Events'. (David Chalmers goes further, I think, in denying the claim about supervenience: his 'zombie' argument hypothesizes two entities which are physically identical, but one is a zombie while the other has consciousness. The point of his argument is we *don't know* or *can't prove*, on the basis of all we know, e.g. about neurophysiology, that this situation could not arise.)

All the best,