Monday, March 19, 2007

I Get Paid for Doing This

Remember Steve Martin's old stand up routine from the late 1970s? That's OK, I don't remember it very well, either, but I do remember that he used to sing a song while playing his banjo and wearing a fake arrow-through-the-head about how much fun it is to be a stand up comic. The chorus of the song including the line "I can't believe I get paid for doing this." Sometimes I get the same feeling about being a professional philosopher.

In the most recent issue of the International Journal of Philosophical Studies (vol. 15.1, March 2007) there is a short essay called "Hyperheaps" by W. D. Hart. The essay is only two pages long, but it manages to introduce one of the most important concepts in the history of philosophy: a formal definition of a "heap". A "heap", it seems, consists of at least four items: three to form a "base" and a fourth that is stacked on top of them. This is a proposed "solution" to a paradox that is variously called the Sorites Paradox (from the Greek word for a heap) or the Paradox of the Heap. The alleged paradox is that it is hard to tell what constitutes a heap if one is to try to give a formal definition of a heap in the form of an algorithm. You see, if you have only one grain of sand, or one grape, you evidently do not have a "heap" of sand or a "heap" of grapes. Now, suppose you propose the following algorithm: adding one more element to something that is not a heap does not constitute a heap. So adding one more grain of sand, give you two grains, does not make a heap; adding one more grape, giving you two grapes, does not make a heap. But surely if you apply the algorithm enough times, eventually you will get a heap. How could you not? But as the algorithm is defined you will never get a heap, which seems paradoxical.

See how important philosophy can be in the real world?

The purpose of the paradox is not to make you declare that there are no heaps, but to get you to think about the nature of boundary conditions, especially seemingly vague boundary conditions. But now that we have an official definition of "heap" I suppose we won't have much occasion to think about such things any more.

It is a tremendous loss to philosophy and to our culture generally.


Michael said...

Dr Carson,

I think there's plenty left to argue about: "4 items" seems like a pretty arbitrary definition to me. Why couldn't two items form a base?

Your calling yourself a "professional philosopher" amused me. I don't know if you'd be interested, but over on the site where I contribute there's a heated discussion over the legitmacy of this term:

Also relevant to a discussion going on at my site is the abstract you quoted a post or two ago. Would you mind if I lifted it?

--Michael Sullivan

Scott Carson said...


Hart claims that you need to have a base of three in three dimensional space because anything less would be insufficient to support the topmost grain. He gives formalizations for extending his "result" into varying spatial dimensions--I'm supposing its this extendability that's supposed to be the really cool part.

Feel free to lift the abstract--I lifted it myself from the first page of the article. I think it's OK to quote them in full as long as you give full attribution. At any rate if there is a copyright violation I suppose we'll hear about it eventually!

Michael said...

But what if your "grains", whatever they are, are slightly oblong? Why can't I have a "heap" of two lego blocks? Maybe the minimum number of elements wouldn't remain constant across different kinds of heapables?

Now I don't think that two can make a heap; but I'm not too happy with this kind of definition. It just begs for counterexamples.