My good friend Jason Rosenhouse had an interesting discussion a few days ago on whether or not modern mathematics is reliable. The gist is that the deepest of maths are increasingly built on a foundation of theorems and proofs that are so complex that it effectively becomes an act of faith to take them as true and correct. And when further theorems are built on these theorems, how confident can you be in the structure of mathematical knowledge?
It’s an interesting point, and it gets to something I think about from time to time – the limits of mathematical knowledge.
The idea is that humans, when doing math, have an understandable bias towards proofs that are readable and (easily) checkable. The question then is what portion of mathematical ‘truth’ can be expressed in such a way.
As I imagine it, a large part of the problem is, to unfortunately borrow someone else’s language, the idea of irreducible complexity. Consider the simple proposition that successive square numbers differ by successive odd numbers. The difference between 1 and 4 is 3. The difference between 4 and 9 is 5. The difference between 9 and 16 is 7, and so on. One way to prove this would be to compute the difference between all the infinitude of successive square numbers. But this is fastly unfeasible, not to mention silly. Consider instead, computing the difference between (k+1)2 and k2
Thus the difference between k-th square numbers is the k-th odd number. Or something like that. The point is, we were able to reduce the entire scope of the problem to a simple case that encapsulated the complexity of the full set.
Many proofs can be viewed the same way. Reducing the complexity of the problem to a simpler form that encapsulates everything about the larger problem. But this depends on there -being- a reducible form, one that is less complex but still holds all the information about the original problem. What if there weren’t, though? For example, what if there were no easier way to prove the Riemann Hypothesis than to list out ever single zero of the zeta function? There is no reason to think that there are not problems for which any representation would not be as complex as the original problem itself. I will admit that I don’t have any examples offhand, but it strikes me that the Collatz Conjecture or the 196-sequence might fit the bill. And almost certainly anything involving the ridiculous Pi Primes.
As it applies to proofs then, there may not be shorter proofs, easier to understand proofs, better proofs. It may well be that the 1000+ page computer generated monstrosity of the proof of a particular theorem cannot be improved. Or indeed, that may be the simplest of all proofs of that particular theorem.
In many ways, this gets to the heart of what I think Mathematics is. Mathematical proofs are basically explorations of consequence. Starting with a set of axioms, simple definitions and assumptions, it is the burden of mathematicians to tease out mathematical truth. All problems and ideas are expressed in the language of those assumptions and definitions, but in so many ways this is because those problems and ideas exist, albeit hidden and indirect, within the axioms themselves. You can hardly begin to think about and describe lines until they have been defined by Euclid’s Postulates. Proofs then study how truth arises as an interaction of these axioms. The axioms themselves become a reduction or simplification of any given problem – the question is are the interactions that give rise to a specific problem and solution simple enough for us to understand? Or are the mere statements of those axioms themselves the best we can hope for in expressing an idea or solution?
But more importantly, indeed most importantly, when will we know?
I don’t know, and I don’t know if anyone does, but I certainly hope we will continue to push on and on, reaching further into the depths of logic and our understanding until we run face first into this wall. And when we find this wall, if we haven’t already, I know we will explore it and seek to understand it with the full force and rigor with which we encountered it to begin with.
And on a mildly less cerebral note, thinking about what mathematical truths can be expressed in a human readable, and understandable form, there are a finite number of characters, and a finite number of characters you can fit on any one page. And there is certainly a finite upper bound on the number of pages any given mathematician will be willing to read. Factoring that all together, there are only a finite number of math papers that can ever be written such that people will be willing to read and verify them. Assuming that the depths of mathematical knowledge is effectively infinite, it becomes not a worry, but rather a guarantee that some things cannot be proved in a satisfactory manner.
But I don’t see that as any reason to stop trying : )
“There is no reason to think that there are not problems for which any representation would not be as complex as the original problem itself. I will admit that I don’t have any examples offhand”
I’m sure this is a little patronising, but dare I point out the relevence of Godel’s incompleteness theorems here? Its not only the case that we already know that *any* set of axioms we pick will have its limitations, but we have known this for many decades and is still studied by logicians and philosophers today. In short we can actually *prove* that problems this complex must always exist and that we cannot escape from them.
Its also interesting to note your comment that “Mathematical proofs are basically explorations of consequence. Starting with a set of axioms, simple definitions and assumptions, it is the burden of mathematicians to tease out mathematical truth.” is in some sense a very old and now largely ‘debunked’ view. The wittgensteinien stance that the whole of mathematics is a tautology misses the point somehow. An algorithm that solves a problem may turn somthing that a human would struggle to verify into somthing that’s tractable. Then there are questions of beauty: most mathematicians would agree that the deductions made in Euclid’s elements were beautiful, but few would claim that axioms themselves are. Then there’s the ’signposting’ that proofs provide us with: if an idea works once it may work again. Furthermore the fact that certain objects enable us to prove results may make those objects worthey of interest in themselves ie a proof can tell us where to go next.
In short, deductions from axioms genuinly give us results that are more than simply “exist[ing]… within the axioms themselves”.
Sorry to sound critical. This is a very interesting post
Comment by Ben Fairbairn — July 28, 2008 @ 1:04 pm
I’m familiar with Incompleteness, certainly, but what I was trying to say/ask is if the near incomrehensibility of some ultra-high level maths is a result of this? I think that’s maybe making the issue too rigid – I don’t honestly think of maths purely in terms of axioms and the like, but I do wonder if the point has been reached where problems incapable of being reduced to a tractable point? And I don’t really know. I never read enough about Godel to figure out how he could be applied to a specific problem.
And I guess my statement does miss a lot of math, a lot of the beauty of math – and certainly there is beauty in math. If I could revise myself, I’d say that … mathematicians imagine and create, and then determine how the objects of their creation fit and function in the larger known framework. The best proofs I’ve read or written are always the ones relying on an insight or construct that allows access into the problem. And certainly the worst proofs I’ve had the misfortune of experiencing are the ones that start from first principles ^^ But the exploration of consequence remark still holds I think, as it applies to determining how and why the construct of interest interacts with the body of maths, through the framework laid by your assumptions and definitions. The genius of math does not lie in the axioms themselves, but it only makes sense with respect to the axioms themselves. If it were all working from first principles, I’d shoot myself.
This is what I get for being overly simplifying and glib ^^
But then the question remains, I imagine, to what extent is the inherent reducibility due to mathematical insight tied up with the complexity of the problem itself?
Don’t worry about being critical ; ) I’m just glad someone is reading and cares enough to respond!
Comment by Fox — July 28, 2008 @ 2:10 pm
Bertrand Russell is quoted as saying “The greatest challenge to any thinker is stating the problem in a way that will allow a solution.” That seems to perfectly encapsulate what you’re saying.
Comment by Seamus — July 30, 2008 @ 4:23 pm