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Mathematicians Are Chronically Lost and Confused

Posted by Soulskill on from the dude-where's-my-cartesian-plot dept.

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.'"

6 of 114 comments (clear)

  1. That settles it by immaterial · · Score: 4, Funny

    I was the best mathematician in my university math classes. Who knew?

  2. Failing as a math teacher by Anonymous Coward · · Score: 5, Insightful

    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.

    1. Re:Failing as a math teacher by jones_supa · · Score: 4, Insightful

      I have sometimes thought that the best teacher might be another student who has just a moment ago barely (but still correctly) grasped the concept.

  3. An old mathematicians' joke by Anonymous Coward · · Score: 5, Funny

    There are two types of theorems: trivial and unproven.

  4. Bizarre advice by AthanasiusKircher · · Score: 4, Insightful

    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."

    1. Re:Bizarre advice by vdorie · · Score: 5, Interesting

      I came here to post a similar sentiment. I think it is a terrible idea to just blow ahead every time an assertion is too confusing. Getting the big picture and developing mathematical intuition is great but it doesn't mean that you'll actually be able to do math. For that, practice.

      I actually advise the opposite, to bang your head against a problem over and over until it breaks (the problem, that is). I don't think that we as a society or a species or whatever deal with confusion very well, and tend to take it as some sort of personal deficiency. We've also done a great deal of dumbing math down, so that when someone tries to make the jump from, say, AP calculus to real analysis, minds get blown and souls shattered. It's probably not that mathematicians enjoy crushing students, but rather that higher levels of math are just plain confusing for most people. They're based on abstractions that are pretty far removed from the human experience. None of this is to say that people who are good at math are better somehow, but it usually means that they put in a lot of time. I suspect that a lot of people who are math-phobic would get over it if you locked them in a room with nothing but math books to keep them busy.

      One that is clear, however, is that most mathematicians have no fscking clue what the word "obvious" means. There are some brilliant, dead authors that I would love to punch in the face.