Math is highly creative. Mathematicians forge onward into unknown worlds, artfully shaping and uniting diverse tools in the resolution of their problems. Their progress, however, is regulated by the rigid requirements of logical consistency; unbreakably bound to itself, the discipline progresses forward without risk of retreat or collapse. The logic guiding it, of course, must come from somewhere beyond those who employ it, and the mathematician seems but an agent in the revelation of something much more profound.
Mathematical research is difficult to describe. To call math “discovered” in the Platonic sense – that is, an already-existing “fact of the universe” – is to neglect the role of the mathematician as leader of an expedition: he or she makes very real, and difficult, decisions regarding the path through the unknown which most promisingly portends success. To call it “created”, however – in the sense that math is but a human invention – is to ignore the role of a seemingly supra-human logic in dictating the progress of the field.
I’ll explore the question below using several different approaches.
Is math bound by earthly reality? Early mathematics (much to my chagrin) was largely a mere toolset for application to physics. It’s tempting to believe that math’s early progress, and by consequence its current direction, owe much to the physical sciences. This would, indeed, suggest a view of math as “created”; math is, perhaps, that which generalizes physical phenomena into abstract structures which remain logically consistent with themselves. This view, however, seems beset with problems. Math has an uncanny ability to describe, and even to predict physics, as argued by Eugene Wigner in his paper famously titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” (1) Wigner, in this article, notes a common trend in which crude physical observations lead to generalized, precise mathematical formulations, which in turn, then, miraculously describe unexpectedly vast arrays of further physical phenomena. “It is difficult to avoid the impression that a miracle confronts us here,” suggests Wigner. It’s possible that physics, like math, is bound by an analogous (though perhaps even more elusive) internal logical consistency – and that the mutual consistency forces the two disciplines to march in unison. The deep connection between the two ensures that it couldn’t be any other way.
Or could it? While accepting that the two march in lockstep, we might still imagine a world in which physics, and by consequence math (or is it the other way around?) were entirely different from those which we recognize on our planet (or universe). Math as we know it today, indeed, might represent but one of the many paths the discipline could have taken while still remaining logically consistent with itself.
Still more unpalatable is the possibility that the two could diverge. What if physics were different, yet math remained the same? Perhaps Wigner’s “unreasonable effectiveness” is but an incredible miracle. I find this unlikely. There must be some unifying force.
Shinichi Mochizuki is a Japanese mathematician at Kyoto University. Mochizuki, recently, without fanfare or warning, posted four papers to the “polished mathematical underground” called the ArXiv. (2) These papers were like nothing before them: totaling an unfathomable 512 pages, they were filled with the most deeply incomprehensible mathematics ever seen. The first paper is titled “Inter-Universal Teichmuller Theory I: Construction of Hodge Theaters.” Mochizuki’s papers claim to prove the elusive “abc conjecture” in number theory, concerning the properties of sums of prime numbers. (3)
Mochizuki constructed an entire world. The mathematician, truly, is widely said to have created a new field of math to solve this problem; no living mathematician has succeeded in confirming the validity of the paper and few have tried. Yale graduate student Vesselin Dimitrov pointed out a small error, the error was fixed, and no more was heard.
The conjecture is simple. It’s also almost certainly true – the property has been confirmed computationally for numbers ranging as high as we might reasonably request. But is Mochizuki’s the only proof? Must all possible proofs be identical? Must all proofs concern “Teichmuller Theory” and “Hodge Theaters?”
Probably not. Mochizuki, indeed, seems to have been an inconceivably adroit leader of an expedition. He led an expedition into an entire universe, in fact – one whose unfathomable complexity ensures, I can safely say, that neither you nor I will ever understand it. What does this say about math? Though math explores a world that seems to have come from elsewhere, it’s humans – brilliant, creative humans – at the helm. The conjecture is true. But the manner in which it was explored was entirely created.
Math seems to be held together by an inexorable force that lies deeper than any human has thus far been able to conceive. We, however, seem to be the only ones exploring it. Let us continue, and let the crevasse never end. I don’t think it will. I don’t think it can.
References and further reading:
- Wigner’s “Unreasonable Effectiveness”
- Mochizuki’s papers: I, II, III, IV
- ABC Conjecture. Wikipedia MathOverflow