Thread

1. pxagntuvzt · 1791115149 · focus · HN ↗

   Mathematicians lie.

   The three lies (by omission) are:

   1. This solution is how I solved it. If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.

   2. This idea is abstract and has nothing to do with reality. Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)

   3. This is the only and right way to do it. You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).

   This is not what it looks like doing math. It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).

   Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).

   To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.

   1. zmberber · 1791118987 · focus · HN ↗

      i think your points 1 2 3 are true in the sense that this is perceived, and that sometimes, the way things are being taught lead to this. but in my mathematics career, as well as in school, i never had the impression that these sentiments were actually the beliefs of teachers.

      i also think it is wrong to so confidently state in which way mathematics relates to reality. this is a very nuanced philosophical issue.

      also, i don't see how your "counterexample" is one, let alone a well-known one, both things, the way you stated them, are very abstract and not "real world" at all, the way i see it. or are you here still talking about the fact that math gets inspired by things that happen in real life, as opposed to just inventing stuff just out of pure abstract inspiration? no mathematician i have ever met will deny this. and in fact, philosophically, it is an interesting question if it is even possible to separate a human's "purely abstract ideas" from how a human reasons in real life. that goes back to plato.

      i also think one should be careful with these strong assignments of terms like "universal language" and "can describe everything however you want".

      sorry for the rant. i agree with your sentiments about how we teach and how researches find out things.

   2. chrisweekly · 1791127920 · focus · HN ↗

      Interesting points. Regarding intuition, I highly recommend <a href="https:&#x2F;&#x2F;betterexplained.com" rel="nofollow">https:&#x2F;&#x2F;betterexplained.com as a great example of an alternative approach that aligns with your ideas.

Unofficial Hacker News client; not affiliated with Y Combinator.