Alonzo Church, one of the great logicians of the 20th century, published a proof to the Entscheidungsproblem, known as Church’s Theorem:
No single algorithm can determine whether a statement of number theory is a theorem or not.
The statement is fairly simple, I think. You can write a computer program, which, when fed a mathematical statement, would test it arbitrarily to see whether it held in random test cases and then proceeded to attempt to determine its theorem-hood through a variety of preprogrammed “strategies.” It might work in a lot of cases. However, you cannot write a computer program which follows the same procedure every time to determine whether something was a theorem or not, according to Alonzo Church.
The implications of it were, in the late 1930’s, enormous. Church was an important contributor in a great wave of demolishing the illusion of certainty about the physical and mathematical world. Scientific culture was transformed, and with it, eventually, popular culture as well. If you went around saying “There are true statements of mathematics that we can never prove.” 100 years ago, people would you have looked at you funny. If you said that today to someone who had never heard of Godel or Church (there should be plenty) they would probably shrug their shoulders and nod, the implications having already seeped into the greater cultural consciousness.
However, I think that Church’s result is additionally important for mathematics education, and perhaps education in general. A lot of people see mathematics as ‘inaccessible.’ Following initial difficulties, they often give up, huffing about how ‘their mind wasn’t designed to learn mathematics.’ But in a sense, even the most insightful mathematicians are imperfect theorem-provers – they are more like the computer programs described above, throwing a lot of strategies at problems and hoping something works, than demi-gods who work and think on a higher plane of existence. They do what normal people can do, just a whole lot better and faster.
Even though I have personally overcome the initial despair of confusion and lack of understanding, I still have my own shaky self-confidence to deal with. As a mathematics student, I’m nervous about the possibility of having to do original work (I have to discover something no one else has done before AND it’s graded???!!!). Not only that, but from looking at college texts as a high-schooler I noticed a certain diminished familiarity in the relationship between textbook author and mathematics student – there are far fewer examples, and you’re required to do a lot more proving intuitive results. For example, a book on linear programming and convex sets asked me to prove that a line is uniquely determined by two points – I didn’t even know where to begin.
And the problem of widespread thinking that mathematicians are demigods is that mathematical proofs are presented in an extremely clean manner in textbooks – the extraneous work that went into proving them, the incorrect assumptions made on the way, all of that ‘junk’ is cleaned out of the way. It saves space, sure, and obviously it’s not realistic to publish the entire body of work that went into proving an important result over decades. But it certainly would help to gain some insight into this process.
I’ve read parts of How to Solve It, a popular instructional guide on mathematical problem solving by George Polya. It’s a fascinating read, especially because it highlights an important popular misunderstanding of the mathematical community – namely the one highlighted in the previous paragraph: Doing mathematical work and solving difficult problems is a lot more empirical than rational. Mathematicians have to toy with a problem for years on end. And then, the answer doesn’t just appear to you from perfect deductive reasoning. They usually have to take a lot of data, guess an answer that works for those cases that they chose, and then come up with a proof. And how do you work out the details, or direction, of that proof? Guesswork again.
If I have to get from my house to the supermarket, it’s perfectly reasonable that some of the movements that I take on the way in fact physically lead farther away – maybe the front door faces away from the supermarket and I have to walk down to it, so I walk away…this is intuitively obvious. As is the notion that birds-eye-view distance almost never corresponds to actual distances on a map, because if you have points A and B on a map, only rarely does any perfectly straight road connect them.
So extending this analogy to mathematics, going from a problem to a solution is rarely if ever a straight road. To mathematicians, all those detours you have to take may be more easily identifiable than to laymen, but they’re usually not obvious. In any case, no matter how good you are, you’re going to have to try different pathways, take risks, and see what works. The humorous adage “Good judgment comes from experience, which comes from bad judgment” comes to mind – experienced mathematicians have just done a lot more of problem-solving, so they already know the better paths. But that isn’t to say that other people can do it.
Think of rock-climbing. In free-climbing, where one leader has to attach hooks to create a pathway for the rest of the climbers, that leader is pretty much on their own to get up the slope. The path isn’t immediately obvious, but they can always try new things with relative safety – they only fall down to the last hook they attached. That way, they never lose the fundamental work they’ve put in, but there’s always an element of risk in going forward. A lot of times, the leader will have to try roundabouts, tricky sections to get onto an easier part of the slope, or other random experimentation to get to the top. But if it’s a sufficiently difficult slope, they won’t get it on the first try, no matter how experienced they are. And they always have hooks already attached to fall back upon.
I came up with quite a different analogy I like to extend to literary analysis, which is quite different from mathematical problem-solving, but requires some similar heuristics, or solution-searching methods. In essence, literary analysis is looking at literature which uses a variety of techniques, consciously or subconsciously inserted by the author, which communicate purpose and give us a sort of “taste,” be it, again, conscious or subconscious. So literary analysis is a lot like a sugar-cube dissolving in a glass of water. At first, we can sense the presence of an underlying message, but it’s not obvious to us. We look more closely, underlining strange turns of phrase, identifying metaphors and similes, finding little quirks and oddities that seem to make us feel a certain way. We heat up the water, stir it, shake it, crush the sugar cube so it dissolves faster. We propose a possible meaning for the passage, and then check whether the rest of our observations fit that. Maybe they don’t. We go through acceptable meanings, and finally find one which makes sense, and write a fantastic essay. It’s almost like science!
The sugar-cube analogy fits quite well in a variety of contexts. Often, problems like like insoluble plaques which, given a little bit of thought and time, dissolve quite nicely.