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Mathematicians Are Chronically Lost and Confused
An anonymous reader writes "Mathematics Ph.D. student Jeremy Kun has an interesting post about how mathematicians approach doing new work and pushing back the boundaries of human knowledge. He says it's immensely important for mathematicians to be comfortable with extended periods of ignorance when working on a new topic. 'The truth is that mathematicians are chronically lost and confused. It's our natural state of being, and I mean that in a good way. ... This is something that has been bred into me after years of studying mathematics. I know how to say, “Well, I understand nothing about anything,” and then constructively answer the question, “What’s next?” Sometimes the answer is to pinpoint one very basic question I don’t understand and try to tackle that first.' He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ... But more often than not you'll find that by the time you revisit a problem you've literally grown so much (mathematically) that it's trivial. What's much more useful is recording what the deep insights are, and storing them for recollection later.'"
I was the best mathematician in my university math classes. Who knew?
Math should not be taught as a linear process, but as a spiral. Visit the topics at first, so the student can understand why something is important when it is presented rigorously.
All too often I've encountered math teachers who failed to properly explain advanced mathematical concepts because to them it was obvious and trivial.
Gee, thanks.
There are two types of theorems: trivial and unproven.
I suppose you can reduce arithmetic and geometry (both quadrivia) to logic (trivia), but the liberal arts are seven in number for a very good numerological reason.
He then provides some advice for people learning college level math like calculus or linear algebra: 'I suggest you don't worry too much about verifying every claim and doing every exercise. If it takes you more than 5 or 10 minutes to verify a "trivial" claim in the text, then you can accept it and move on. ...
While I agree that one shouldn't waste time questioning every statement you encounter, there's a very ancient and useful tradition in math pedagogy that emphasizes these sorts of things. See, for example, gradually building up geometrical theorems from a few axioms, a la Euclid.
Often, the process of working out complex proofs for yourself is crucial to understanding why things work, not to mention developing and practicing logic skills that are essential in math and elsewhere.
I'm not saying one should waste time trying every exercise or redoing every proof, but some of my greatest insights into the inner workings on math have come from exercises that took me a couple hours to work out or textbook passages I went over a number of times and really dug into how the details worked. If I skipped everything I couldn't do in 5-10 minutes, I doubt I'd ever have developed the more advanced skills and intuitions that would be necessary to see why some results are "trivial."
We must all learn to exist in that exquisitely uncomfortable place where everything we know is always up for reassessment. Otherwise, we miss change, and change is the only constant.
If this is how things stand, then the Philosophy of Mathematics to date is a catastrophic failure. When there is no better methodology than "fumble around in the dark a bit until suddenly you're convinced" then the project of attempting to guide students in understanding maths has done no work at all.
Is this the fault of the philosophers or the mathematicians? I'm inclined to think that the philosophers have at least failed in their advocacy, if not in their actual subject.
Myu:
Some similar effects occur with engineering and programming. For instance:
An engineer is ALWAYS working on something that's broken. That's because, when he gets it fixed, he moves on to the next thing that's broken. (It's like the thing you're searching for always being in the last place you look. It's not Murphy's law, It's becaue, when you find it, you stop looking.)
A good programmer doesn't come to a problem with all he needs to solve it. Instead he comes to it with a big toolbox, SOME domain knowledge, and the skills needed to learn the rest during the project. This will be mostly stuff related to the project, but may include more programming tools as well.
Designing/architecting a program or system is like handling a black bag with the solution inside, in the form of blocks connected by strings. You squeeze it around until you get it into two lumps with very little string running through the thin neck. Then you it into two bags and document all the strings that went through the cut. Repeat unti the bags are small enough to understand easilyj and keep the entire explanation in your head. (In the case of a program that means the code itself fits on a page, with over half of the page being comments.) Then you can open the little bags and grok each one - which by now will be either trivial or maybe embody a single deep concept or "neat hack". (But avoid "neat hacks" if they're not obvious or if something straightforward does the job just fine.)
There is nothing like teaching a topic to force you to learn it.
Rhapsody in Numbers
the constipated mathematician ?
He worked it out with a pencil.
Old School.
Nowadays, he'd work it out ... [*dons sunglasses*] ... digitally.
If it weren't for deadlines, nothing would be late.