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pathways (letters)

12 February 1998

Dear George,

Many thanks for your essay, dated 4 February, for units 10-12 of The Possible World Machine, in response to question 5: 'In what way, or ways does the existence and nature of space pose a problem for philosophy?'

The essay was an invitation to pose some philosophical problems concerning space — or at least those you find gripping! Let's see.

Now it would be natural to suppose that part of that task is distinguishing philosophical questions about the nature of space from those that one would expect the physicist to answer. This is, after all, one way of grasping what it is that distinguishes the approach of philosophy from the empirical sciences. — Not so easy as it sounds! A contemporary physicist will tell you that space has been discovered to be finite, not infinite. That raises an obvious philosophical difficulty posed in ancient times: Go to the edge of space and throw a spear: Where does it go? (I try to respond to this problem in the text.)

We need to approach the question in a far more elementary way. When one talks of 'space' people naturally think of empty space ('space for my car') or outer space. You are right to make these points. It is a basic philosophical discovery (again, one made in ancient times) that space is everywhere, and not just where objects are not.

Let's look at this 'abstract', physical or philosophical notion of space. Could there be more than one such space? You suggest one consideration. If space were infinite then there could not be two spaces. 'If it were infinite then it would have to fill all available space, including "other" space.' I wonder about this. The problem is that we have not got a clear idea of what we mean by 'infinite'.

Take the clearest example of infinity, the series of numbers 1,2,3,4... . One of my students, training to be a primary school teacher (she also has a 1st in Philosophy!) was surprised and delighted when the light dawned on one little boy: 'There's no last number!' But now take the series of even numbers: that has 'no last number' either. So it is infinite too!

In formal, mathematical terms, an infinite set is a set that can be 'mapped' onto a 'proper subset' of itself. Take a set of anything you like (numbers, cars, assorted objects) and remove some of its members. The result is a 'proper subset'. The idea of 'mapping', meanwhile, is just the idea of pairing off: setting out a dinner table, the hostess doesn't need to count the total number of knives and forks to know that each knife is paired off with a fork. (Apart from the fruit knife!) Clearly, no finite set can be mapped onto a proper subset of itself. If I set out a table with two columns, 'The cars in this road' and 'The cars in this road minus my Ford Capri' then the items in the two columns cannot be paired off. But that is just what one can do if the items are 'all the natural numbers' and 'all the even numbers' (or 'all multiples of 1,000', 'all fractions' etc.)

So the short answer would seem to be: Yes, there is no objection, on the score of the nature of 'infinity', to there being two or more 'infinite' spaces. If you wanted a picture for this, just imagine a point with lines stretching out to infinity. Give each segment a different colour. Then each segment is an infinite segment of infinite space. (Of course, this is a picture of one space, not of several spaces!)

I'm not saying there isn't a problem with the idea of two or more infinite spaces. Only that the problem isn't to do with infinity. It seems to me that you would have the same problem if space could (as the physicists believe) be finite.

You make a good point about idealism. If what we term 'space' is, in ultimate reality, merely the 'inside' of God's mind, then there is no obstacle to there being two or more spaces cordoned off, just as my fantasies about being a racing driver are cordoned off from my fantasies about being a concert pianist. (The plurality of spaces is in this way analogous to the plurality of possible worlds.) Perhaps even clearer than this, if we leave God out of the picture, if, on the idealist picture, one thinks of the minds of the intelligent creatures who mentally 'inhabit' space, then two sets of such creatures could conceivably inhabit two spatially unrelated 'spaces'.

Of course, to say that there could be several 'spaces' if idealism were true is not much help if you're not in the least bit tempted by idealism!

I also liked your discussion of the contrast between 'knowledge of those things with which we cope every day' and the kind of questioning that leads to science and philosophy. That would be a very useful way of approaching an essay on the nature of philosophy, perhaps taking the philosophical problems concerning the nature of space as a prime example.

Yours sincerely,

Geoffrey Klempner