Habits of highly mathematical people • Hacker News Board

Re: Habits of highly mathematical people

#81

Post by pm90 » Sat, Jul 30, 2016, 8:10 PM UTC

I really liked reading this article. The author describes not just the skills that are important for reasoning well (mathematics seems to just be an arena in which this kind of thinking is required), but in the end, also the pitfalls of taking these skills to their logical extremes in social situations. As the author points out, context is very important.

Most of the Senior Engineers I look upto and who I consider as role models seem to know this. They are carefree and jovial in most conversations, but when its time to design a system or drill into root causes of an outage, they are capable of asking (and answering) these types of precise questions. I've learned a LOT from working with them and do hope to be like that in the future.

Re: Habits of highly mathematical people

#82

Post by Falkon1313 » Sat, Jul 30, 2016, 8:11 PM UTC

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…

>It is sad that a student can leave high-school with almost no classes that ever makes math less abstract.

I find it sadder that many can get through high school without any classes that make it more abstract. Too much of it is just mindless repetition of seemingly-useless incantations that have no meaning or apparent purpose (especially if the concepts aren't understood). Where there are word problems, many students are confused and hate them because they haven't really learned how the math abstracts the situation. (Or how to look at a situation and recognize "Hey, I could abstract this with math to get more information!")

I once tutored a student who hated math and after 11th grade was still failing pre-algebra and wouldn't be able to graduate. After a few months, we had covered algebra 1 and 2, geometry and trigonometry, and a brief intro to calculus and statistics. She aced the test, graduated, and went on to become a successful professional in a job that routinely used math. By then she loved it. Problem? It had never been about the joy of discovering new ideas and new ways to use them. It had always just been shuffling numbers around without any concepts or thought processes that could lead to understanding.

My answers are simple:

Why should you learn it? As with learning to read/speak/write, it gives you another way to think and communicate, and exposure to concepts, some of which you may never use, others of which may someday be very valuable. Even if you don't use them everyday, just knowing that they exist and could be helpful (and recognizing when) is valuable.

When will you use this? You may not personally sit around calculating statistics and probability, but as a manager you may recognize that they could help you make important decisions, avoid problems, and increase profits. So you know to hire someone who can do the calculations, and you have a general idea of what to ask them for and how to understand the reports they give you. And to ask questions to get them to explain the business meaning of things affected by confidence levels or standard deviation so you can understand how that affects your decisions. You'll use it to increase knowledge, success, and profit.

Re: Habits of highly mathematical people

#83

Post by Obi_Juan_Kenobi » Sat, Jul 30, 2016, 8:20 PM UTC

ktamura wrote: post #63

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 studied math and computer science. Walked out of a math Ph.D. program before starting one) >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. That's possible. The OP does not claim that the traits he listed apply if and only if you are a mathematician. It'…

My small issue is that he attacks empirical science at its periphery (statistical muckery, mostly going on in politicized debates, pharma, etc.) rather than it's core. My experience in biology is quite similar to what he describes with people more-or-less dispassionately seeking the truth. It's a 'messier' field, so perhaps there's more room for personal pride, but I have my doubts about that. Anyone who holds hypotheses dear is a poor mathematician or biologist, just the same.

Basically, it's a basic vs. applied science critique; basic science is quite brutal, and you will be wrong very often. Usually there's little at stake regarding which conclusion you come to, so long as you're investigating something interesting. Someone trying to create effective medicine is in a very different situation.

> there’s very little fame outside of the immediate group of people you’re talking to

This is very true in most basic science fields. There are about 4 or 5 other labs in the world that are familiar with the details of my sub-field, and this is true for most of my colleagues. Mathematics is certainly not privileged in this sense.

Re: Habits of highly mathematical people

#84

Post by mkagenius » Sat, Jul 30, 2016, 8:21 PM UTC

Kenji wrote: post #43

Mathematicians need logical precision because they work in the realm of things which can be definitively proven or disproven. Not really, by Gödel's incompleteness theorems, they are always working in a realm where some things cannot be definitely proven or disproven.

Yet you disapprove OP's statement ;)

Re: Habits of highly mathematical people

#85

Post by preygel » Sat, Jul 30, 2016, 8:22 PM UTC

Math PhD here. I have also had the chance to work with many bright "analytically-minded" people from other backgrounds while a management consultant.

This article is spot on, and some of the behaviors really do seem more indicative of "mathematical people" -- which I suggest really stands for "those who have done research in a 'mathematical' field." (The key being the mix of cold, hard precision in the idealized proof with the squishy, intuitive, human activity of discovering what is pretty and true -- an aspect usually lost in math education!)

Some examples I found most poignant:

- The article mentions "fluidity with definitions" and illustrates it well with the anecdote about Keith Devlin. This is a skill distinct from pure "analytical reasoning," as it requires comfort with definitions that are at once precise but also open to (frequent) change. The process of forming and changing definitions is creative and imprecise, and falls into what is sometimes called "conceptual reasoning." (A programming analog might be API design.)

- Several of the other points are tools for figuring out what is true, and for precising imprecise statements. For example the need to "teas[e] apart .. assumptions" is only natural when reading papers with Theorems that have very precise conditions .. which do not exactly hold in the case you need! In many other "analytical" contexts pre-conditions are not made as precise and arguments by analogy are considered acceptable provided the conclusion is believed. (A programming analog might be debugging when some implicit pre-conditions or invariants break.)

Re: Habits of highly mathematical people

#86

Post by unfamiliar » Sat, Jul 30, 2016, 8:24 PM UTC

redipiny wrote: post #52

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…

One of the problems of philosophical discourse, imo, is that it seems to be impossible to give mathematically rigorous definitions of concepts like 'good' or 'moral' or even 'knowledge' that somebody won't be able to disagree with by counter example. What you sometimes see among philosophers is that even definitions that are the result of many iterations are still treated more like rules of thumb than precise definit…

> that somebody won't be able to disagree with by counter example

I really don't understand this. You give a counterexample to a theorem; you don't give one to a definition. If someone disagrees with the definition and wants to use a different one, then they are talking about a different thing (even if they want to use the same English word to refer to each of them) and their conclusions can not be compared in any meaningful way.

Imo, you can't "proceed carefully" with a definition that is open to interpretation. If you could do so safely, then you know enough about how people could interpret your definition to form a more rigorous definition.

Re: Habits of highly mathematical people

#87

Post by nhaliday » Sat, Jul 30, 2016, 8:27 PM UTC

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

Re: Habits of highly mathematical people

#88

Post by gunnihinn » Sat, Jul 30, 2016, 8:43 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…

> 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? :)

Re: Habits of highly mathematical people

#89

Post by ewjordan » Sat, Jul 30, 2016, 8:45 PM UTC

sidek wrote: post #21

raldu wrote: post #18

Earlier quoted context omitted.

Nice spot on. The described qualities would also apply to philosophers, logicians, etc. I think a broader term would be, habits of highly "analytical" people.

Philosophers? Using precise definitions? Maybe some philosophers do... But not most.

Oh, they use them. The problem is that each philosopher has a different precise definition that they feel is natural and matches what we "really mean" when we ask deep questions. Since most arguments and conclusions are phrased fuzzily, you end up with the following situation:

Fuzzy premise -> precise logic -> fuzzy conclusion

where each of those arrows leaves so much room for interpretation that you can build literally an entire subfield by arguing back and forth about what the most reasonable mapping from fuzzy to precise and back again might be, even if everyone agrees that the manipulations in between are rigorously correct.

Re: Habits of highly mathematical people

#90

Post by woopwoop » Sat, Jul 30, 2016, 9:00 PM 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? :)

Unless I'm mistaken, The totality of vector spaces is not even a set.