During our brief intermission last week, commenters chose to revisit the question of whether arithmetic is invented or discovered—a topic we’d discussed here and here. This reminded me that I’ve been meaning to highlight an elementary error that comes up a lot in this kind of discussion.
It is frequently asserted that the facts of arithmetic are either “tautologous” or “true by definition” or “logical consequences of the axioms”. Those are three different assertions, and all of them are false. (This is not a controversial statement.)
The arguments made to support these assertions are not subtly flawed; they are overtly ludicrous. Almost always, they consist of “proof by example”, as in “1+1=2 is true by definition; therefore all the facts of arithmetic are true by definition”. Of course one expects to stumble across this sort of “reasoning” on the Internet, but it’s always jarring to see it coming from people who profess an interest in mathematical logic. (I will refrain from naming the worst offenders.)
So let’s consider a few facts of arithmetic:
- Every number is either odd or even. This is a tautology. It does not follow that every fact of arithmetic is a tautology.
- 1+1=2. This (depending on what you take as your starting point) is true by definition. It does not follow that every fact of arithmetic is true by definition.
- Every number is a product of prime numbers. This is neither a tautology nor a definition. It does, however, follow from the standard Peano axioms. It does not follow that every fact of arithmetic follows from the axioms.
- There is no solution to the equation (x+1)n+3+(y+1)n+3=(z+1)n+3. (This is the famous Fermat’s Last Theorem.) This is not a tautology, it is not a defintion, and I have no idea whether it follows from the axioms. Neither, as far as I know, does anyone else. All we know for sure is that it’s true.
- In Chapter 10 of The Big Questions, I give an example of a fact of arithmetic that is not a tautology, not a definition, and surely does not follow from the axioms. Here is a sketch of a proof that there must be some such facts.
It seems that people are often led astray by thinking that the “facts of arithmetic” consist solely of statements like “seven squared plus one equals five squared times two”. The more logically interesting statements are those that speak not about specific numbers, but about infinite collections of numbers. These are the statements that begin with phrases like “Every number…” or “There is a number such that…” or “There are only two numbers such that….”. When mathematicians speak of the “facts of arithmetic”, they means facts like these:
- Every number is the sum of four squares.
- Every prime number that leaves a remainder of 1 when divided by 4 is a sum of two squares.
- No prime number that leaves a remainder of 3 when divided by 4 is a sum of two squares.
- For every number n, there are infinitely many squares of the form 1+ny2
- Between every number and its double, there is at least one prime.
- 8 and 9 are the only successive numbers that are both powers of primes.
- For any two prime numbers p and q, the equations p=x2+yq and q=x2+yp are either both solvable or both unsolvable, unless p and q both leave remainders of 3 when divided by 4, in which case exactly one of them is solvable.
- If you want to black out enough squares on a tic-tac-toe board to make winning impossible, then, as you pass to boards of higher and higher dimensions, the fraction of squares you must black out gets arbitrarily close to 100%. This is the density Hales Jewett theorem, and I have intentionally glossed over some technicalities in the statement.
Now to the point: First, if you want to tell a story about whether arithmetic is invented or discovered—in other words, if you want to tell a story about where arithmetic comes from—then your story has to account for where the facts of arithmetic come from. If your story says that the facts of arithmetic consist entirely of tautologies, definitions and logical consequences of standard axioms, then your story is wrong. Try again.