Habits of highly mathematical people • Hacker News Board

Re: Habits of highly mathematical people

#71

Post by SatvikBeri » Sat, Jul 30, 2016, 7:08 PM UTC

yummyfajitas wrote: post #55

unfamiliar wrote: post #37

Earlier quoted context omitted.

I strongly disagree with the claim that philosophers fit this description. As a mathematician, it often seems to me that 90% of philosophy is spent arguing about things with extremely loose definitions. As a result, you can argue from the same starting point and come to completely different conclusions (as separate philosophers often do), because the starting point was already self-contradictory for some interpretati…

For example, many of the moral dilemmas that get tossed around as interesting become trivial to answer once you give a mathematically rigorous definition to words like "good", "moral", "utilitarian" or whatever. Yes, but the while the conclusions become trivial, actually coming up with those rigorous definitions is not. I'm not going to provide examples because they are likely to start a flamewar. But if you search f…

Even in Math, coming up with good definitions is often the hard part. For example, modern point-set topology is both extremely powerful and consists of theorems that seem obvious to prove, but that's because the definition of a topology is so well-chosen.

Re: Habits of highly mathematical people

#72

Post by maroonblazer » Sat, Jul 30, 2016, 7:11 PM UTC

jknoepfler wrote: post #56

maroonblazer wrote: post #7

Couldn't this also be titled "Habits of high rational people", with mathematics simply being one application? I'm not a mathematician and was miserable at math in school, but I apply these habits in the business world every day. They help me cut through a lot of crap that comes from other people's sloppy/lazy thinking. >Anyone who has gone through an undergraduate math education has known a person (or been that perso…

I do think there is something unique about training in the 'hard' analytic disciplines (math, computer science, physics, etc.). In particular, one learns to be wrong often, and that the result of one's hard-fought intellectual effort is often objectively incorrect. At the very least, math accelerates the process of discovering this, and so is a very good teacher of intellectual humility. The fact that you jumped to '…

>The fact that you jumped to 'other people's sloppy thinking' as your first example is somewhat telling.

You'll have to take my word for it but I'm under no illusion as to my smartness relative to others. More often than not I'm the dumbest person in the room. I learned the hard way that I have plenty of blind spots and I'm the first person I should be checking for 'sloppy thinking'.

Re: Habits of highly mathematical people

#73

Post by catnaroek » Sat, Jul 30, 2016, 7:11 PM UTC

SatvikBeri wrote: post #71

yummyfajitas wrote: post #55

Earlier quoted context omitted.

For example, many of the moral dilemmas that get tossed around as interesting become trivial to answer once you give a mathematically rigorous definition to words like "good", "moral", "utilitarian" or whatever. Yes, but the while the conclusions become trivial, actually coming up with those rigorous definitions is not. I'm not going to provide examples because they are likely to start a flamewar. But if you search f…

Even in Math, coming up with good definitions is often the hard part. For example, modern point-set topology is both extremely powerful and consists of theorems that seem obvious to prove, but that's because the definition of a topology is so well-chosen.

I like how Conor McBride put it:

> I usually take the presence of any proofs in my code as the cue to question my definitions.

http://stackoverflow.com/a/13241158/46571

Re: Habits of highly mathematical people

#74

Post by SatvikBeri » Sat, Jul 30, 2016, 7:17 PM 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…

My coworkers and I use a related technique all the time, e.g.:

"I think X, Y, and Z are true." "If those are true, then A should also be true. But A seems clearly false." "Hmmm...I agree A is surprising, but not clearly false. What would distinguish a world in which A was true or not?" "Well, if A was true, we'd expect B, otherwise C." "Ok, I agree C seems much more likely than B, so that suggests one of X, Y, or Z is wrong."

Re: Habits of highly mathematical people

#75

Post by Noos » Sat, Jul 30, 2016, 7:22 PM UTC

I'm not sure these traits exist outside of mathematics as much as posited. I would think this happens mostly because mathematics is such an abstract realm that it is possible for mathematicians to cultivate that sort of detached mindset effectively. Once you start getting into topics and subjects that are more personal and unable to be completely abstracted, these traits may be a burden or not exist in practice.

Not many people will take the same approach to their own values, and philosophy shows the difficulties in using the mathematical/logical approach to value systems and human thinking.

Re: Habits of highly mathematical people

#76

Post by bsaul » Sat, Jul 30, 2016, 7:24 PM UTC

his piece about discussing ideas and switching opinions made me remember a comment i made once to someone that said "evolution is just a theory, people need to hear both that and creationism". i said "only evolution is a theory, because it's based on facts and bits of logic, the other's not".

Re: Habits of highly mathematical people

#77

Post by tunesmith » Sat, Jul 30, 2016, 7:30 PM UTC

The parts that jump out to me in the context of being a professional programmer:

1) Definitions == overloading. We regularly run into overloaded terms that mean very different things to different people. Microservices. Technical debt.

2) Being wrong - there is a perceived cost to being wrong, and in many organizations it is unfortunately a reality. When new or young, there is so much emphasis on proving oneself that there is easily a disincentive to opening oneself up to risk of being wrong. After being around a while, it is easier - sometimes I like advancing a position I know to be wrong because I know it will more efficiently generate explicit reasons why, as people protest (although even there I usually make clear I am doing devil's advocate).

3) Abstraction scaling - it's unfortunately so common for programmers and technical people to get pedantic, which frustrates people and wastes time. Sometimes it's driven by insecurity. For instance, needing to be the smartest person in the room, or proving competence in an area that is not directly relevant. And sometimes that insecurity is rational, for instance if part of an organization that encourages insecurity through its systems. But oftentimes it is because of a lack of experience with abstraction, or an inability to recognize what "rung" a discussion is on, or losing one's place on the abstraction tree. Complicating this is that sometimes it is important to drill down when no one else wants to. Recognizing this and keeping track of relevance is one of the hardest things to do when dealing with deep subjects and arguments.

Re: Habits of highly mathematical people

#78

Post by srtjstjsj » Sat, Jul 30, 2016, 7:33 PM UTC

johncolanduoni wrote: post #39

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...

This is not true at all. I know many people in mathematics that are below average when it comes to arithmetic, some to the point of suffering from dyscalcula. I myself am probably in at least the bottom quartile of arithmetic ability, and there have been quite a few historic examples of top mathematicians with a similar "problem", Hilbert being a common example. There's a huge difference between reasoning about abstr…

Hilbert's supposed dyscalcula is a myth..unless you have a source we haven't seen?

It's similar to the Grothendeick story -- a mathematician making a simple mistake -- that get exaggerated to "mathematicians can't do arithmetic." https://en.wikipedia.org/wiki/57_(number)

Another common myth was that Einstein was bad at math, because he (and others) had trouble in school because the material was too easy and he quarreled with teachers, and "failed" tests for early college admissions administered in a foreign language. https://www.washingtonpost.com/news/answer-sheet/wp/2016/02/...

When a mathematician confesses being bad at calculation, they don't mean "couldn't pass high school", they mean compared to their elite peers.

Re: Habits of highly mathematical people

#79

Post by srtjstjsj » Sat, Jul 30, 2016, 7:37 PM UTC

GarrisonPrime wrote: post #54

louprado wrote: post #41

> The most common question students have about mathematics is “when will I ever use this?” The author then goes on to provide 6 semi-abstract reasons as to why abstract reasoning matters. Since this question is most likely to be asked by a child, it is code for "how can math-skills make me money or get me a job." Upon hearing the author's answers, 13 year old me would conclude, "it doesn't, and math is as useless as…

The point of the article isn't to show that the math will be used. The point is that people who ask such a question are missing the point. "Education is what remains after one has forgotten what one has learned..." - Albert Einstein

And the point is that high-horse mathematicians are missing the point when they ignore the relevant lives of their audience.

Re: Habits of highly mathematical people

#80

Post by srtjstjsj » Sat, Jul 30, 2016, 7:42 PM UTC

wolfgke wrote: post #8

javier2 wrote: post #5

7. Being precise. The precision you hold yourself to through a rigorous approach to a problem is also very valuable and is the reason I think calculus is very important.

> The precision you hold yourself to through a rigorous approach to a problem is also very valuable and is the reason I think calculus is very important. What property concerning precision does calculus have that doesn't hold for any topic in mathematics?

Infinity (and infinitesimals) is intuitively east to use but easy to misuse. Arithmetic isn't so easy to misuse.

Students usually see infinity for the first time in calculus (or maybe when working with infinite series)

These are the situations where we commonly see people being confident in their incorrect answers (which is worse than being unconfident and unable to get correct answers)

Difficulty wrangling infinity comes up a lot on Hacker News, even https://hn.algolia.com/?query=infinite%20series&sort=byPopul...