God gave us the integers...
"God gave us the integers, all else is the work of man". This is the famous quotation from Kronecker, saying that the only numbers we could be sure about were the integers, other types such as irrational numbers and imaginary numbers were inventions, which couldn't really be said to exist.
Most people nowadays would consider Kronecker's ideas to be far too restrictive, and consider other sorts of numbers to be perfectly respectable. I would agree with this in general, but in this article I would like to explore the other direction, and question whether we are perfectly sure about the integers.
Gödel's Incompleteness Theorem
|See my article on the undecidabililty of Fermat's last theorem|
In the 1930's Kurt Gödel proved his celebrated incompleteness theorem, showing that whatever set of axioms you took for number theory, there would always be statements that couldn't be proved one way or the other. for more information People often misinterpret this as saying that there are statements about the integers which cannot be proved, but that isn't what it shows For any conjecture about the integers there will always be the hope of finding a proof. What Gödel's Incompleteness Theorem does say is that the properties of the integers can't be encapsulated in any finite set of axioms.
|Closer to 'real' undecidability are the theorems, related to Gödel's, which show that for certain classes of problem there is no general method of solution|
This is because any such axioms will always have other possible models, which differ from the integers. Given a set of axioms we can find a statement which is true for the true integers, but is not provable from the axioms. However, it is possible to create a consistent model of the axioms in which the statement is false. This is called a nonstandard model of the integers.
But this leads to the question: How do we know about the true integers. Any precise, finite description of the integers will be equivalent to an axiom system, and so won't capture the nature of the integers. But we can't visualise all of the integers at once, some sort of finite description is all we have. I feel that this is something of a paradox.
1, 2, 3 ...
|A note on terminology: The natural numbers are the counting numbers 1, 2, 3 ... and so on (some people would include zero). The integers are the natural numbers together with their negatives and zero, ..., -3, -2, -1, 0, 1, 2, 3,... if I seem to be switching between them then be assured that it doesn't make any difference to the arguments|
Generally we would think of the natural numbers as being 1,2,3 ... , just continued forever. But forever brings a notion of time, and of some physical being doing the counting. However, we don't have sufficient knowledge to be sure that the universe behaves in this way. Suppose that you were an immortal being and counted the days one at a time 1,2,3 .. Can you be sure that there wouldn't come a day when you woke up and found that you had counted all of the integers and had to start again with ω, ω+1, ω+2, ... ( OK, I know that there won't be such things as 'days' for more than a few billion years, but you know what I mean.) There's nothing to say that the universe isn't that way. To give an example that relates more to what people believe, suppose that our universe follows the many-worlds version of quantum theory. You start counting 1,2,3 .. but then a quantum event happens which results in two versions of you, so the sequence of numbers splits to give not only 4, 5, 6 .. but also 4', 5', 6' ... I think that you'll agree that the integers can't split like this. So trying to relate the integers to a physical being counting indefinitely doesn't work. On the other hand, for these example to make any sense, there has to be a notion of the 'true' integers to compare other possibilities with.
If you're a Platonist then there is no problem here - you just think of the integers as existing 'out there' in some sense. But if you're not then I'm not quite sure how you'd deal with it. There is constructivist mathematics, but that doesn't seem to get to the root of the problem. Kronecker was fairly extreme as constructivists go - he seemed to go back to the time of Pythagoras is rejecting irrational numbers such as √2. But even such an extreme constructivist accepted the integers. Rather I think the question goes back to earlier thoughts about the infinite. Is it possible for anything to be actually infinite, or can we only talk of potential infinities? We might think of the natural numbers (or the physical universe for that matter) as having no end, but then not accept that the set of integers can be thought of as a complete entity.
I hope that I've shown that the problems of existence can arise even for these very familiar numbers.