Question How do we resolve the fact that our finite brains can conceive of mental spaces far more vast than the known physical universe and more numerous than all of the atoms? For example, the total possible state-space of a game of chess is well defined, finite, but much larger than the number of atoms in the universe (http://en.wikipedia.org/wiki/Shannon_number). Obviously, all of these states "exist" in some nebulous sense insofar as the rules of chess describe the boundaries of the possible space, and any particular instance within that space we conceive of is instantly manifest as soon as we think of it. But what is the nature of this existence, since it is equally obvious that the entire state-space can never actually be manifest simultaneously in our universe, as even the idea of a board position requires more than one atom to manifest that mental event? Yet through abstraction, we can casually refer to many such hyper-huge spaces. We can talk of infinite number ranges like the integers, and "bigger" infinite ranges like rational numbers, and yet still "bigger" infinite ranges like the irrationals. In what sense can we say all of these potential huge spaces exist (and they must, since we can so easily instantiate well-formed members of them, at will) yet we don't have even the slightest fraction of sufficent space for them in our known universe?

Accepted:
October 11, 2005

Comments

Consider a small finite set--say, the set of members of your immediate family. You understand this set by knowing a list of all the members of the set, and being familiar with all of them. Now, the point of your question is that that approach clearly won't work for infinite, or even very large finite, sets. My grasp of the state-space for chess, or the collection of all integers, cannot involve being familiar with all the elements of the collection. So what does it involve? You hint at a couple of possibilities in your question: In the case of the state-space for chess, you say that "the rules of chess describe the boundaries" of the state-space. So your understanding of that set consists in knowing the rules that determine what is in the set and what isn't, even though you don't have a complete list of the elements. You also refer to the fact that "we can so easily instantiate well-formed members" of these collections. For example, I can generate as many positive integers as I want by counting, although I could never list all the integers.

One of the reasons math is so powerful is that it is possible to draw conclusions about large, and even infinite, collections of objects based on just knowing the rules for what counts as a member of the collection, or knowing how to generate elements of the collection, even if you don't have a complete list of all the members. For example, by reasoning about an arbitrary integer x, I can determine facts that will be true of every integer I generate by counting--even if I count further than I have ever counted before, thus generating integers I have never considered before.

Now, you also raise the question of the "existence" of these collections. This is a tricky question, on which people disagree. Many mathematicians believe in platonism, which is the view that these objects do exist--not as part of the physical universe, in which, as you observe, there isn't room for them, but as part of some separate universe of abstract objects. This view raises difficult questions--for example, if these objects aren't part of the physical universe, then we can't interact with them using our sensory organs, so how can we ever acquire knowledge about them? How can their existence make any difference to us? You might want to check out the article about platonism in the Stanford Encyclopedia of Philosophy.

There are alternatives to platonism. One alternative is intuitionism. An intuitionist would say that the set of all integers doesn't exist. We have the ability to generate integers by counting, but there is no collection containing all the integers "out there" (somewhere), independent of our counting activities. General statements about integers are not statements about some independently existing infinite collection of integers, but are rather predictions about what will be true of any integer we might generate by counting. Intuitionism was developed by L. E. J. Brouwer--check out the article about him in the Stanford Encyclopedia of Philosohy.

By the way, there's an interersting quote from Voltaire that is relevant to your question. Voltaire said that the infinite "astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads."

To add a word or two to Dan's great response: there is no questionthatmathematics deals with infinite collections, but what those are, whatwe mean when we make claims about them, which claims are correct —these have been hotly disputed issues for thousands of years. (Inthe history of mathematics, concern for these foundational questionshas waxed and waned. There have been times, for instance in theearly part of the twentieth century, when disputes over these issues,were very heated and split the mathematical community. There have beenother times, for instance now, when mathematicians have been lessinterested in these issues — although of course there are alwaysexceptions, like Dan.) The basic question — what does it mean to call aset "infinite"? — is so fundamental that it's simply astounding that wedon't know how to answer it.

Onone way of looking at the matter, what Dan called "platonism", to saythat a set is infinite is simply to have given a measure of its size.To say that a set is infinite is much like saying that it's got 17elements in it: if you counted up the elements in the second set you'dfind there were 17 of them, and if you counted up the elements of aninfinite set you'd find there were infinitely many of them.

But on another of way looking at the matter, this is insane. How can one finishcounting up the elements in an infinite set? Isn't that what "infinite"means, that the process of counting never stops? On this way of lookingat things, to call a set "infinite" is not to describe the size of someactual collection, but rather to mark it off from all finitecollections: finite collections are ones for which the process ofcounting their members eventually stops, while infinite ones arecollections whose elements we can keep on generating without end.

The first conception accepts the existence of the actual infinite:a collection that actually contains infinitely many objects. The secondconception rejects this as unintelligible and talks instead of the potential infinite:to say that a set is infinite is not to make a claim about the size ofan actually existing object but rather to say that each of its elementscan potentially be brought into existence. (The two conceptions will beconfused if you think that an entity that can potentially be broughtinto existence really exists after all — and has the property ofpotential existence attached to it. See here for some comments on a comparable error.)