Habits of highly mathematical people • Hacker News Board

Re: Habits of highly mathematical people

#101

Post by Joof » Sun, Jul 31, 2016, 12:48 AM UTC

d33 wrote: post #17

coliveira wrote: post #13

Earlier quoted context omitted.

True, but if you're not good at school math, I believe there is little hope you can be good at the style of math that mathematicians do...

I'm not sure about this one. I studied CS and was kinda forced to learn university-level maths and I found that it differs both on the pedagogical level (different ways to motivate students, I preferred the ones in the university) and on the logical level, when for the first time in my life I was actually EXPLAINED the terms we operated on. University maths is much more logical than the school voodoo I had.

I didn't quite have this experience in my engineering math courses. Admittedly, the other math courses were much better about this.

Re: Habits of highly mathematical people

#102

Post by Joof » Sun, Jul 31, 2016, 12:48 AM UTC

marcosdumay wrote: post #24

coliveira wrote: post #13

Earlier quoted context omitted.

True, but if you're not good at school math, I believe there is little hope you can be good at the style of math that mathematicians do...

I wouldn't bet on that, they are really different. Some of the most blatantly off-putting characteristics of school math, like mindless repetition, are antagonical to real math.

Orthogonal?

Re: Habits of highly mathematical people

#103

Post by wutbrodo » Sun, Jul 31, 2016, 12:52 AM UTC

gunnihinn wrote: post #88

wutbrodo wrote: post #58

There's one habit in particular that he didn't really touch on that has been one of the most impactful parts of getting my math degree on my thinking. There seems to be a gap between formal definitions and what we feel actual definitions are. Every once in a while, a professor would prove something that was clearly right, but felt like a violation of some unstated implicit part of a definition (for example, that the…

> for example, that the set of all vector spaces is itself a vector space Hmm... under what operations? It's a semigroup under direct sum, and probably something under the tensor product, but I'm having trouble imagining what the scalar multiplication should be. (Or was that a "fictional" example? :)

Sigh, teach me to write a comment quickly without proofreading on HN, on a topic of any complexity. As written, my comment is wrong: I meant to write "vector space functions"[1] but left out a word. What I was getting at was the unintuitiveness that arises from thinking of vector space elements as number-like things (in the sense that we're familiar with them from basic math: numbers or coordinates) to being "anything that meets the definition", including functions. Seriously though, thanks for pointing that out, part of what I still love about the HN community is getting called out for even esoteric incorrectness.

[1] which was my concise but imprecise way of expressing "the set of functions from a set over a field to a vector space over the same one", with + and * defined as you'd expect.

Re: Habits of highly mathematical people

#104

Post by sn9 » Sun, Jul 31, 2016, 12:52 AM UTC

nhaliday wrote: post #87

There's a really great Quora answer along the same lines: https://www.quora.com/What-is-it-like-to-understand-advanced...

This is without question my favorite Quora post of all time.

I find myself rereading it every once in a while.

Re: Habits of highly mathematical people

#105

Post by adrusi » Sun, Jul 31, 2016, 1:37 AM UTC

wutbrodo wrote: post #58

There's one habit in particular that he didn't really touch on that has been one of the most impactful parts of getting my math degree on my thinking. There seems to be a gap between formal definitions and what we feel actual definitions are. Every once in a while, a professor would prove something that was clearly right, but felt like a violation of some unstated implicit part of a definition (for example, that the…

When a conclusion feels wrong despite being derived logically, that usually means that the definitions are flawed. In math the defintions of structures don't have to reflect anything real, it's all about starting with arbitrary rigorously stated definitions and exploring their implications. So when it feels like definitions are wrong in math, that's just a reflection of that "definitional gap".

But the virtue of always siding with the logical-but-unintuitive isn't as valuable outside of mathematics, so it's maybe not the best point to bring up in an article that's about skills math teaches that are useful in "real life". In philosophy, definitions aren't arbitrary and rigorous, they're meant to be reflections of reality. So when a conclusion feels unintuitive, that's a decent argument against the definitions it was derived from. If mathematics teaches people to just pay attention to the definitional gap, that might be useful enough, but isn't that covered by "discussing definitions"?

Re: Habits of highly mathematical people

#106

Post by Waterluvian » Sun, Jul 31, 2016, 2:33 AM UTC

I did poorly in high school. Largely because of untreated ADHD. I got to university and did no math. But really excelled at my studies (STEM related).

Somehow I got into a software engineering position in robotics. And regularly I need not just the logic and rational thought you gain from school, but the practical math skills.

I've been picking up what I need to learn from open courseware like MIT lectures. The thing that really upsets me is that I'm learning it without issue. I was always completely capable of it, but my time in high school convinced me I wasn't. I wish I had been convinced/allowed to take more math in university.

Re: Habits of highly mathematical people

#107

Post by jknoepfler » Sun, Jul 31, 2016, 2:33 AM UTC

joe_the_user wrote: post #65

BariumBlue wrote: post #59

Earlier quoted context omitted.

The "hard analytical disciplines" require understanding the fundamentals of whatever they work with, in order to process those fundamentals together into a useful effect. Getting meaning across via words is fundamentally the same process.

"Getting meaning across via words is fundamentally the same process." -- Here, I think you've distilled the basic assumption of the op and I would claim someone with enough exposure to mathematics will know this claim is wrong. Informal arguments may indeed be carefully and even exactly argued but they nearly always involve implicit emotional appeals, appeals to unstated common assumptions and definitions and so-fort…

Having done graduate work in both computer science and philosophy (and having a healthy respect for both), I agree with this assessment. It's part of what I was trying to articulate. Although I think some of the philosophers (particularly the historians of philosophy) that I've met have been some of the most intelligent, rational, and well-spoken people I've met, I think there's a keen difference between the critical thinking abilities of the computer science undergraduates vs. the philosophy undergraduates that I've seen graduate from a university/college, and I chalk a lot of this up to learning to be wrong very quickly in formal disciplines.

Re: Habits of highly mathematical people

#108

Post by yomritoyj » Sun, Jul 31, 2016, 4:22 AM UTC

wolfgke wrote: post #61

yomritoyj wrote: post #46

Mathematics promotes bad habits too. Example 1: paying too much attention to worst-case scenarios. When creating a mathematical proof you want to make sure that your conclusion holds in all possible scenarios allowed by your assumptions. In real life decision-making worst-case scenarios are really bad, really expensive to guard against but also really rare so often the best thing to do is to ignore them. You may find…

Concerning example 1: There are different concepts, such as average-case scenarios (for example modeled by expected value) or quantiles. Use them if this is what you want to model (but pay attention to the assumptions). Concerning example 2: If you want some 'common sense' property to hold, why don't you specifiy it then?

What you are saying is definitely true within mathematics and its applications. But the original article was about “The […] skills that students of mathematics […] will practice and that will come in handy in their lives outside of mathematics” and in the same spirit I wanted to point out habits of thought developed doing mathematics which have a negative effect when the problem you are facing cannot be fully formalized.

Re: Habits of highly mathematical people

#109

Post by dschiptsov » Sun, Jul 31, 2016, 5:07 AM UTC

Yeah, if we remove every mention of mathematics to make it less "elitist", then one could notice a few nice things, such as justification behind some TTD practices (writing down examples and counter examples to come up with more refined definition), healthy obsession with precise usage of words (plus not mentioned but important - to use as less of them as possible - just enough), and that, basically, the whole process is a heuristic search, so a "wrong" is nothing but merely an empty branch of the search process, not a "failure" - so, just accept it and backtrack. The need of assertions and tests (proofs) is obvious. And the ladder of abstractions is the very same notion of layers of composable abstractions, where each layer is a "language" and building blocks for a layer above, which was popularized by the on SICP and On Lisp books.

This, by the way, is also the answer to that question "do we need to study math for programming". Discipline in the use of ones own mind is what is required.

Re: Habits of highly mathematical people

#110

Post by marcosdumay » Sun, Jul 31, 2016, 1:31 PM UTC

Joof wrote: post #102

marcosdumay wrote: post #24

Earlier quoted context omitted.

I wouldn't bet on that, they are really different. Some of the most blatantly off-putting characteristics of school math, like mindless repetition, are antagonical to real math.

Orthogonal?

Sorry. I meant antagonistic.