At a party last night I was discussing mathematics with a grad student from a different department. I expressed how glad I was to be in mathematics, where things are objective: if I come up with a new result and submit it, it will be judged based on its proofs, and if those are correct, it is unimpeachable. This in contrast to other departments, where, apparently, new ideas are very difficult to get accepted if they vary at all from the status quo, and where they can be rejected for very subjective reasons. My friend pressed me: surely there must be some subjective part to the process, or else why isn’t mathematics automated? And he had a point. In the end, I had to concede that, while the mathematics itself is objective, the interestingness of math is what is subjective. Whether or not the mathematician’s proofs are correct, the referee can suggest the paper is uninteresting. Of course, I’m merely speculating here, since I only really started submitting papers to journals in late 2010 and thankfully no referees have accused them of being uninteresting so far.
Let’s focus a little bit closer on an idea my conversation partner expressed: if mathematics were totally objective, could it be automated? In at least one sense, the answer is a resounding Yes! Start with a system of axioms, be it ZFC, Peano arithmetic, the axioms of a complete ordered field, or the postulates of Euclid. You can easily program a computer to begin listing all provable theorems in the system, by brute force. At one point, I even began work on such a program (starting with Peano arithmetic), but I quickly realized it would take millions of years and more memory than any supercomputer has, before the brute force attack would yield, say, the infinitude of primes.
But who cares? It will take humans millions of years, and more human memory than the current population has, to discover X by our current methods, where X is: the first theorem which the human population will discover, by current methods, after 2 million years. Now, this X might seem more advanced or more arcane than the infinitude of primes, but that is subjective. You can’t even necessarily say that X has (e.g.) a higher information-theoretical complexity, as a sentence, than the statement of the infinite of primes: it might be that X happens to be a very simple statement, which just happens to have a very long, elusive proof. Stay tuned and I’ll let you know in about two million years.
If I were to complete my abandoned Peano Arithmetic program, and start it churning out theorems by brute force, and I start submitting those to journals, should they be accepted? No, but why not? “Not original,” one might suggest, but this is false. Computers work fast, and it won’t take long for my program to churn out more theorems than mankind has ever proved by traditional means. And as time goes on, the fraction of original theorems will probably(*) approach 1. (*I hesitate slightly here. Some theorems might have multiple synonymous articulations. For example, “there are infinitely many primes” is ultimately the same statement as “there are infinitely many primes and 1=1″. But still, I suspect an automated theorem-generator would quickly obtain original results.)
At the end of the day, there seems to be something subjective going on, which lets human mathematicians publish results they found by traditional methods, but doesn’t let a clever-ass like me generate a million easy publications by brute force.
And thank God. Who would ever want to read a bunch of opaque symbols churned out by a machine? The statement “there are infinitely many primes” appeals to us because it uses some vocabulary, like “infinitely many” and “prime”, which is not built in to Peano Arithmetic. If you want to formally write that statement out in raw PA, it might look something like this (shield your eyes):
∀x ∃y ((y>x) & ∃p ∃q ((∃r p.r=y) & (∃r q.r=y) & (∀s ((∃r (s.r=y)) → (s=p or s=q)))))
Why is that any worse than “There are infinitely many primes”, preceded by definitions for those terms? Well, that’s subjective…
One could make the program more sophisticated, and have it make definitions in such a way as to ultimately minimize the sheer complexity of its theorems. But could it come up with those names in a good way? Going about it the most obvious way, it might define a noun “N9524″ to mean what you and I mean by “prime number”, simply because that happens to to be the 9,524th noun that it defines (and in reality, 9,524 is probably a hilarious underestimate of a cosmically infinitesimal nature). That’s assuming its definitions match up with ours at all. More likely, its definitions would have only a loose relationship, if any, to human-made definitions. To get around this, to give these nouns halfway decent names, would probably require full-on, post-singularity, open-the-pod-bay-doors-Hal, Artificial Intelligence. (Or maybe I’m wrong. Many a pessimistic futurist has been humiliated in the past…)
So to sum it all up? Mathematics must necessarily be somewhat subjective as a defense mechanism against trolls who could otherwise flood the journals with true-but-useless flotsam and jetsam. But it is far more objective than any other discipline in the world.