To paraphrase Douglas Adams, “There is a theory which states that if ever anyone discovers the foundations of mathematics to be contradictory, they will instantly disappear and be replaced with something even more bizarre and inexplicable. There is another theory which states that this has already happened”
Imagine you’ve discovered something that will make you famous. Your name will echo through history for decades, maybe centuries. You’ve been working on the details for months. You’re putting together a book on the topic. Then, someone reads a draft of your book, and points out “You’ve made a mistake. Right here.” You look carefully, and they’re right.
What do you do?
If you are Ed Nelson, you say “You’re right, I withdraw my claim”
I don’t know what Ed Nelson is feeling right now, but if I were him, I’d be feeling pretty disappointed. Not to mention somewhat embarrassed. However, the fact that he so willingly, almost offhandedly withdrew such an earth-shattering claim speaks volumes, perhaps – positive volumes – for the kind of person he is.
What happened was this. Mathematicians have a set of axioms, called the Peano Axioms, that describe how the natural numbers work. In plain English, they say
- 0 is a number
- If x is a number, x = x.
- If x and y are numbers, and if x=y, then y=x.
- If x, y and z are numbers, and if x=y and y=z, then x=z.
- if x is a number, and x=y, then y is a number.
- We can count by ones from any natural number.
- You can’t start counting by ones and later reach 0. Remember we are talking about the natural numbers here, so no negative numbers, no fractions or decimals.
- If the next number after x equals the next number after y, then x=y as well.
- If you start with 0, and keep counting forever, you get every (natural) number eventually.
- Suppose something is true for the number 0. Suppose also that if it’s true for x, it’s true for the next number as well. Then it’s true for all natural numbers. Or, in plainer language, if something works the first time, and every time it works, it works the next time, then it always works.
- x+0=0
- x+(y+1) = (x+y)+1.
- x times 0 is 0
- x times (y+1) is x plus x times y, that is, x*(y+1) = x+x*y.
If you understand each of these, they are obviously true of the natural numbers. Note that there are other true statements missing from the list : for example, x + y = y + x. Amazingly, this can actually be proved from the list of axioms above. So can a lot of other things, like “32+42=52“, or “there are infinitely many sets of numbers a,b,c with a2+b2=c2“. We can define the concept of a prime number and prove that every number has a unique factorization into primes.
The question that remains unknown is – suppose we prove something logically from these axioms. Could we perhaps also prove the opposite? For example, if we proved that there were infinitely many prime numbers, could we also prove there wasn’t? In other words, do the Peano axioms contain contradictions??
It seems silly to think so, but it’s easy to write down a set of axioms (or rules) where you can get contradictions. For example, I could write
- if x+1 = y+1, then x equals y
- a does not equal b
- a+1 = b+1
Ed Nelson said he’d found a contradiction in the Peano axioms. He had been writing a book about it. It would have been the pinnacle of his career. He put a preview of the book online. Another (very famous) mathematicin read it, and pointed out some problems in the logic in public on the internet. Ed checked, saw that the problems were real, and retracted his claim.
Just imagine, though, if he had been right. If a contradiction were found, one of the Peano axioms above would have to be thrown away or replaced – which would be hard, because they are all so obviously true.
What would this mean for mathematics as practised? Probably not much. In fact, something similar has happened before. A brilliant mathematician called Cantor wrote down a set of rules for thinking logically about sets. Much later, Bertrand Russell found a contradiction – now called Russell’s paradox. He imagined a set X, defined by this rule :
If S, a set, does not contain S, then X contains S. On the other hand, if S does contain S, X doesn’t contain S.
Then, the burning question is, does X contain itself? To find out, let’s put X in place of S in the rule.
If X, a set, does not contain X, then X contains X. On the other hand, if X does contain X, X doesn’t contain X.
There’s a clear problem here. X can only contain X if it doesn’t. The response of the mathematical world to Russell’s paradox was to invent a new set of axioms for set theory. A set of axioms more complicated than the original, but immune to Russell’s paradox.
If the Peano axiom ever turned out to have contradictions, mathematicians would find some replacement pretty quickly. In fact, there are already some replacements hanging around in the wings – some mathematicians don’t like, for example, to jump to the conclusion that just because something works the first time, and every time it works, it works the next time, that it will always work. “Always” is a long time, after all. They’ve worked out systems of arithmetic that don’t need or use that assumption.
While mathematicians were sorting all this out, the rest of the world (of course) would happily go on using the same old rules of arithmetic as before, hardly noticing that the foundations were shaking.
When Douglas Adams made his famous quote about the Ultimate Answer (and Question) to Life, the Universe and Everything, he was really talking about mathematics.