10 ms·

Habits of highly mathematical people

429 points by CarolineW 10y ago

geertj 10y ago
Only six habits, I would have hoped for a seventh one :)
- shoo 10y ago
  in line with "Teasing apart the assumptions underlying an argument", perhaps the definition initially included a seventh habit, but it was found to be implied by the remaining six and discharged
- dev1n 10y ago
  Isn't there an old quip about how poor mathematicians are at arithmetic? ;)
  - mathgeek 10y ago
    I don't recall. How are poor mathematicians at arithmetic? ;)
    - dev1n 10y ago
      lol my bad.
- tlammens 10y ago
  Come on, it's a perfect number!
  - fibo 10y ago
    ahah you can chose among 6 ...or 28 bullet points if you like perfect numbers :)
- Joof 10y ago
  The seventh is left as an exercise to the reader.
- jameshart 10y ago
  7: Schematizing things into, for example, lists.
- wolfgke 10y ago
  If there were seven habit, there would probably be someone who would have hoped for an eighth one.
javier2 10y ago
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.
- wolfgke 10y ago
  > 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?
  - eximius 10y ago
    Indeed. The most precise I ever had to be was in my real analysis course. It seems agreed upon that all professors that teach it will be utterly pendantic about all proofs in that class. Which I agree with as a sort of gateway to graduate mathematics but man was it frustrating haha
  - srtjstjsj 10y ago
    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=byPopularity&prefix=false&page=0&dateRange=all&type=comment https://hn.algolia.com/?query=infinite%20series&sort=byPopul...
    - wolfgke 10y ago
      > Students usually see infinity for the first time in calculus (or maybe when working with infinite series) Students also see groups or R-modules the first time in abstract algebra. So what. > 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.. https://hn.algolia.com/?query=infinite%20series&sort=byPopul.... I rather the reason why "infinity" is misused so often, but, say, groups or R-modules, not so lies rather in the fact that too most math instructors too much to appeal to intuition in calculus, but not in abstract algebra. Thus mathematics should be taught in a much more abstract way where you are not misled by your bad intuition because you simply aren't able to formulate wrong thoughts in the abstract framework (that's why the abstractions and formalism was invented).
maroonblazer 10y ago
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 person) to regularly point out that X statement is not precisely true in the very special case of Y that nobody intended to include as part of the discussion in the first place. It takes a lot of social maturity beyond the bare mathematical discourse to understand when this is appropriate and when it’s just annoying. I don't disagree but would argue that the far more common problem is people - not just mathematicians, mind you - not considering definitions enough which ultimately leads to confusion and/or misunderstandings and consequently additional, unnecessary, cycles spent in discussion about "what do you really mean?"
- vanderZwan 10y ago
  > I'm not a mathematician and was miserable at math in school Maths in school is not really like the maths that mathematicians do.
  - coliveira 10y ago
    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...
    - d33 10y ago
      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.
      - 10y ago
mrcactu5 10y ago
#6 scaling the ladder of abstraction This is one I have a tough time explaining to students who are so focused on the task at hand they don't feel there's time to sit back and reason in generality.
- bjz_ 10y ago
  Came across this Dijkstra quote via https://www.youtube.com/watch?v=GqmsQeSzMdw https://www.youtube.com/watch?v=GqmsQeSzMdw > The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise By taking away all the non-essential things and leaving only on what you need, it allows you to truly understand the structures that you are studying. Definitely rings true when I work in ML/Haskell-style type systems. Easier said than done convincing the students though...
  - catnaroek 10y ago
    > By taking away all the non-essential things and leaving only on what you need, it allows you to truly understand the structures that you are studying. This is the essence of type abstraction and parametricity!
haasn 10y ago
Just reading those 6 bullet points reminds me very, very strongly of the kind of discussions I have with my friends (all of whom are CS students).
auggierose 10y ago
Very good article. I might add that a further pitfall in being "highly mathematical" is when you assume that the person you are communicating with is also "highly mathematical" and therefore has similar habits. That's not true MOST OF THE TIME and can lead therefore to severe misunderstandings.
- unfamiliar 10y ago
  True. It can be very frustrating for people when their "fuzzy-logic" (for lack of a better word coming to mind) is not accepted as a proof and they don't see how it isn't iron-clad. Fuzzy-logic gets us through most of everyday life with minimal mental burden, and so is definitely useful, but it doesn't hold up in an argument where you're trying to prove something. It can also go the other way though; some topics are inherently fuzzy, and trying to impose rigour on them just ends up wasting everyone's time and leaves nobody convinced.
- wutbrodo 10y ago
  IME you get used to this pretty rapidly, and what you're left with KS even better: the ability to translate between the two modes of communication (and expose the lack of rigor in one, which often leads to its proponent discovering errors in their assumptions).
fibo 10y ago
mathematician here, so true
louprado 10y ago
> 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 writing poetry". A child who isn't thinking in this practical way is already academic minded and doesn't need an answer. "When will I ever use this?". It obvious! "In next year's math class". It is sad that a student can leave high-school with almost no classes that ever makes math less abstract. The most beautiful moment in my life was my freshman college physics class. It first filled me with joy and then resent. Because I almost didn't go to college and there was no reason I wouldn't have understood this course at 13. Same goes for finance. So answers should focus on applications like modeling simple Newtonian physics of a satellite in orbit or predicting the RPM when a 4-stroke engine red-lines based on max-stress specs. Or show them useful short cuts like the Rule of 72 and if they are curious show a derivation. Show examples of ratios/percentages, how fractals can simulate a landscape, ray tracing, or examples of the Fibonacci series in nature, etc.
- GarrisonPrime 10y ago
  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
  - srtjstjsj 10y ago
    And the point is that high-horse mathematicians are missing the point when they ignore the relevant lives of their audience.
- Falkon1313 10y ago
  >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.
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Kenji 10y ago
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.
- mkagenius 10y ago
  Yet you disapprove OP's statement ;)
- ktRolster 10y ago
  Your quote didn't say that all things can be disproven or proven. You need to think mathematically!
yomritoyj 10y ago
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 that hard to do psychologically if you spend too much time doing mathematics. Example 2: being too parsimonious. When doing mathematics you have a clear cut objective (your conclusions) and your goal is to achieve it while using the least resources (assumptions). In real life a person thinking mathematically may put in just enough effort to fulfill the formally specified part of their objective while ignoring informal requirements that are assumed to be 'common sense' by non-mathematical people.
- wolfgke 10y ago
  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?
  - yomritoyj 10y ago
    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.
kamaal 10y ago
Only problem is people tend to build whole theories and conclusions out of assumptions, and their assumptions could be wrong. Mathematics only shows you deductions that could be make out of initial axioms and rigid rules. It doesn't mean the assumptions you made while forming axioms or the rules itself are right. Having said that being mathematical increases your chances of arriving at the right results.
- catnaroek 10y ago
  > It doesn't mean the assumptions you made while forming axioms or the rules itself are right. There's no absolute notion of “right” to begin with. There's just “provable” (in a theory) and “true” (in a model of a theory). Furthermore, a statement that's true in one model might not be true in another.
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wutbrodo 10y ago
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 set of all vector spaces is itself a vector space). Despite feeling wrong, I couldn't come up with a particular reason that it wasn't right (short of falling back on some weasel word like "technically that's correct"). Math education forces that gap out into the open, and makes you pay more than lip service by building more and more theorems on top of it. The real world effect is to make you examine whether there's some actual value in this definitional gap (which can then be explicitly stated) or whether it's just fallacy. I know far too many people who are comfortable with stopping at "this doesn't feel correct despite me having no counterargumemt to its 'proof' ". I've never been inclined that way but my math education certainly sharpened my ability to avoid this.
- SatvikBeri 10y ago
  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."
- gunnihinn 10y ago
  > 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? :)
  - woopwoop 10y ago
    Unless I'm mistaken, The totality of vector spaces is not even a set.
mturmon 10y ago
I'd like to propose another habit: the habitual use of certain idioms that are present in mathematical reasoning, in reasoning about other problems. I'm thinking of constructs like - "if and only if", versus mere implication - TFAE, as in, the following are equivalent ways of establishing some property - Defining a relation, say R, between entities, and establishing properties about R, like transitivity. - Defining notations, labels and symbols for certain quantities or relationships within a problem. You typically first learn these idioms when doing proofs or reading lots of math and trying to summarize the ideas in it. Some of them are also aligned with the more abstract side of CS (unsurprisingly). Someone who has had a little experience with these habits can make a real contribution to clear thinking. How many times have you stood at a whiteboard discussion where people repeatedly talk about "the properties that this object inherits from its parent" (or similar) -- but without just inventing a notation for that thing?
Noos 10y ago
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.
bsaul 10y ago
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".
tunesmith 10y ago
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.
pm90 10y ago
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.
preygel 10y ago
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.)
nhaliday 10y ago
There's a really great Quora answer along the same lines: https://www.quora.com/What-is-it-like-to-understand-advanced-mathematics/answers/873950 https://www.quora.com/What-is-it-like-to-understand-advanced...
- sn9 10y ago
  This is without question my favorite Quora post of all time. I find myself rereading it every once in a while.
woopwoop 10y ago
There is an interesting Math Overflow thread called "Mathematical habits of thought and action which would be useful to non-mathematicians". The answers there all interesting (and sometimes conflicting), but my favorite (part of an) answer comes from Terence Tao: >Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two things can be superficially similar but functionally quite distinct). For instance, paying off a credit card at 10% is equivalent (as a first approximation, at least) to investing that money with a guaranteed 10% rate; once one sees this, it becomes obvious why one should be prioritising paying off high-interest credit card debt ahead of other, lower-interest, debt reduction or investments (assuming one has no immediate cash flow or credit issues, of course). Not understanding this type of equivalence can lead to real-world consequences: for instance, in the US there is a substantial political distinction between a tax credit for some group of taxpayers and a government subsidy to those same group of taxpayers, even though they are almost completely equivalent from a mathematical perspective. Conversely, the mistaking of superficial similarity for functional equivalence can lead to quite inaccurate statements, e.g. "Social Security is a Ponzi scheme". [1]http://mathoverflow.net/questions/74707/mathematical-habits-of-thought-and-action-which-would-be-of-use-to-non-mathemati/75022#75022 http://mathoverflow.net/questions/74707/mathematical-habits-...
Guyag 10y ago
'Scaling the ladder of abstraction' applies quite strongly to dealing with a new codebase, and is something I think inexperienced people (myself included) initially struggle with.
HiroshiSan 10y ago
Haha that's funny I was just scrolling through the comments before clicking and thinking "I wonder what Jeremy Kun has to say" and lo and behold...
mfn 10y ago
It might also be that individuals who have these traits self-select into studying math. So we can't easily conclude that studying math leads to the development of these "highly mathematical habits", because the causation could also flow the other way.
Waterluvian 10y ago
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.
dschiptsov 10y ago
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.

429 points by CarolineW 10y ago

geertj 10y ago

Only six habits, I would have hoped for a seventh one :)

shoo 10y ago
in line with "Teasing apart the assumptions underlying an argument", perhaps the definition initially included a seventh habit, but it was found to be implied by the remaining six and discharged
dev1n 10y ago
Isn't there an old quip about how poor mathematicians are at arithmetic? ;)
- mathgeek 10y ago
  I don't recall. How are poor mathematicians at arithmetic? ;)
  - dev1n 10y ago
    lol my bad.
tlammens 10y ago
Come on, it's a perfect number!
- fibo 10y ago
  ahah you can chose among 6 ...or 28 bullet points if you like perfect numbers :)
Joof 10y ago
The seventh is left as an exercise to the reader.
jameshart 10y ago
7: Schematizing things into, for example, lists.
wolfgke 10y ago
If there were seven habit, there would probably be someone who would have hoped for an eighth one.

javier2 10y ago

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.

wolfgke 10y ago
> 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?
- eximius 10y ago
  Indeed. The most precise I ever had to be was in my real analysis course. It seems agreed upon that all professors that teach it will be utterly pendantic about all proofs in that class. Which I agree with as a sort of gateway to graduate mathematics but man was it frustrating haha
- srtjstjsj 10y ago
  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=byPopularity&prefix=false&page=0&dateRange=all&type=comment https://hn.algolia.com/?query=infinite%20series&sort=byPopul...
  - wolfgke 10y ago
    > Students usually see infinity for the first time in calculus (or maybe when working with infinite series) Students also see groups or R-modules the first time in abstract algebra. So what. > 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.. https://hn.algolia.com/?query=infinite%20series&sort=byPopul.... I rather the reason why "infinity" is misused so often, but, say, groups or R-modules, not so lies rather in the fact that too most math instructors too much to appeal to intuition in calculus, but not in abstract algebra. Thus mathematics should be taught in a much more abstract way where you are not misled by your bad intuition because you simply aren't able to formulate wrong thoughts in the abstract framework (that's why the abstractions and formalism was invented).

maroonblazer 10y ago

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 person) to regularly point out that X statement is not precisely true in the very special case of Y that nobody intended to include as part of the discussion in the first place. It takes a lot of social maturity beyond the bare mathematical discourse to understand when this is appropriate and when it’s just annoying. I don't disagree but would argue that the far more common problem is people - not just mathematicians, mind you - not considering definitions enough which ultimately leads to confusion and/or misunderstandings and consequently additional, unnecessary, cycles spent in discussion about "what do you really mean?"

vanderZwan 10y ago
> I'm not a mathematician and was miserable at math in school Maths in school is not really like the maths that mathematicians do.
- coliveira 10y ago
  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...
  - d33 10y ago
    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.
    - 10y ago

mrcactu5 10y ago

#6 scaling the ladder of abstraction This is one I have a tough time explaining to students who are so focused on the task at hand they don't feel there's time to sit back and reason in generality.

bjz_ 10y ago
Came across this Dijkstra quote via https://www.youtube.com/watch?v=GqmsQeSzMdw https://www.youtube.com/watch?v=GqmsQeSzMdw > The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise By taking away all the non-essential things and leaving only on what you need, it allows you to truly understand the structures that you are studying. Definitely rings true when I work in ML/Haskell-style type systems. Easier said than done convincing the students though...
- catnaroek 10y ago
  > By taking away all the non-essential things and leaving only on what you need, it allows you to truly understand the structures that you are studying. This is the essence of type abstraction and parametricity!

haasn 10y ago

Just reading those 6 bullet points reminds me very, very strongly of the kind of discussions I have with my friends (all of whom are CS students).

auggierose 10y ago

Very good article. I might add that a further pitfall in being "highly mathematical" is when you assume that the person you are communicating with is also "highly mathematical" and therefore has similar habits. That's not true MOST OF THE TIME and can lead therefore to severe misunderstandings.

unfamiliar 10y ago
True. It can be very frustrating for people when their "fuzzy-logic" (for lack of a better word coming to mind) is not accepted as a proof and they don't see how it isn't iron-clad. Fuzzy-logic gets us through most of everyday life with minimal mental burden, and so is definitely useful, but it doesn't hold up in an argument where you're trying to prove something. It can also go the other way though; some topics are inherently fuzzy, and trying to impose rigour on them just ends up wasting everyone's time and leaves nobody convinced.
wutbrodo 10y ago
IME you get used to this pretty rapidly, and what you're left with KS even better: the ability to translate between the two modes of communication (and expose the lack of rigor in one, which often leads to its proponent discovering errors in their assumptions).

fibo 10y ago

mathematician here, so true

louprado 10y ago

> 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 writing poetry". A child who isn't thinking in this practical way is already academic minded and doesn't need an answer. "When will I ever use this?". It obvious! "In next year's math class". It is sad that a student can leave high-school with almost no classes that ever makes math less abstract. The most beautiful moment in my life was my freshman college physics class. It first filled me with joy and then resent. Because I almost didn't go to college and there was no reason I wouldn't have understood this course at 13. Same goes for finance. So answers should focus on applications like modeling simple Newtonian physics of a satellite in orbit or predicting the RPM when a 4-stroke engine red-lines based on max-stress specs. Or show them useful short cuts like the Rule of 72 and if they are curious show a derivation. Show examples of ratios/percentages, how fractals can simulate a landscape, ray tracing, or examples of the Fibonacci series in nature, etc.

GarrisonPrime 10y ago
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
- srtjstjsj 10y ago
  And the point is that high-horse mathematicians are missing the point when they ignore the relevant lives of their audience.
Falkon1313 10y ago
>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.

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Kenji 10y ago

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.

mkagenius 10y ago
Yet you disapprove OP's statement ;)
ktRolster 10y ago
Your quote didn't say that all things can be disproven or proven. You need to think mathematically!

yomritoyj 10y ago

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 that hard to do psychologically if you spend too much time doing mathematics. Example 2: being too parsimonious. When doing mathematics you have a clear cut objective (your conclusions) and your goal is to achieve it while using the least resources (assumptions). In real life a person thinking mathematically may put in just enough effort to fulfill the formally specified part of their objective while ignoring informal requirements that are assumed to be 'common sense' by non-mathematical people.

wolfgke 10y ago
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?
- yomritoyj 10y ago
  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.

kamaal 10y ago

Only problem is people tend to build whole theories and conclusions out of assumptions, and their assumptions could be wrong. Mathematics only shows you deductions that could be make out of initial axioms and rigid rules. It doesn't mean the assumptions you made while forming axioms or the rules itself are right. Having said that being mathematical increases your chances of arriving at the right results.

catnaroek 10y ago
> It doesn't mean the assumptions you made while forming axioms or the rules itself are right. There's no absolute notion of “right” to begin with. There's just “provable” (in a theory) and “true” (in a model of a theory). Furthermore, a statement that's true in one model might not be true in another.

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wutbrodo 10y ago

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 set of all vector spaces is itself a vector space). Despite feeling wrong, I couldn't come up with a particular reason that it wasn't right (short of falling back on some weasel word like "technically that's correct"). Math education forces that gap out into the open, and makes you pay more than lip service by building more and more theorems on top of it. The real world effect is to make you examine whether there's some actual value in this definitional gap (which can then be explicitly stated) or whether it's just fallacy. I know far too many people who are comfortable with stopping at "this doesn't feel correct despite me having no counterargumemt to its 'proof' ". I've never been inclined that way but my math education certainly sharpened my ability to avoid this.

SatvikBeri 10y ago
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."
gunnihinn 10y ago
> 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? :)
- woopwoop 10y ago
  Unless I'm mistaken, The totality of vector spaces is not even a set.

mturmon 10y ago

I'd like to propose another habit: the habitual use of certain idioms that are present in mathematical reasoning, in reasoning about other problems. I'm thinking of constructs like - "if and only if", versus mere implication - TFAE, as in, the following are equivalent ways of establishing some property - Defining a relation, say R, between entities, and establishing properties about R, like transitivity. - Defining notations, labels and symbols for certain quantities or relationships within a problem. You typically first learn these idioms when doing proofs or reading lots of math and trying to summarize the ideas in it. Some of them are also aligned with the more abstract side of CS (unsurprisingly). Someone who has had a little experience with these habits can make a real contribution to clear thinking. How many times have you stood at a whiteboard discussion where people repeatedly talk about "the properties that this object inherits from its parent" (or similar) -- but without just inventing a notation for that thing?

Noos 10y ago

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.

bsaul 10y ago

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".

tunesmith 10y ago

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.

pm90 10y ago

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.

preygel 10y ago

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.)

nhaliday 10y ago

There's a really great Quora answer along the same lines: https://www.quora.com/What-is-it-like-to-understand-advanced-mathematics/answers/873950 https://www.quora.com/What-is-it-like-to-understand-advanced...

sn9 10y ago
This is without question my favorite Quora post of all time. I find myself rereading it every once in a while.

woopwoop 10y ago

There is an interesting Math Overflow thread called "Mathematical habits of thought and action which would be useful to non-mathematicians". The answers there all interesting (and sometimes conflicting), but my favorite (part of an) answer comes from Terence Tao: >Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two things can be superficially similar but functionally quite distinct). For instance, paying off a credit card at 10% is equivalent (as a first approximation, at least) to investing that money with a guaranteed 10% rate; once one sees this, it becomes obvious why one should be prioritising paying off high-interest credit card debt ahead of other, lower-interest, debt reduction or investments (assuming one has no immediate cash flow or credit issues, of course). Not understanding this type of equivalence can lead to real-world consequences: for instance, in the US there is a substantial political distinction between a tax credit for some group of taxpayers and a government subsidy to those same group of taxpayers, even though they are almost completely equivalent from a mathematical perspective. Conversely, the mistaking of superficial similarity for functional equivalence can lead to quite inaccurate statements, e.g. "Social Security is a Ponzi scheme". [1]http://mathoverflow.net/questions/74707/mathematical-habits-of-thought-and-action-which-would-be-of-use-to-non-mathemati/75022#75022 http://mathoverflow.net/questions/74707/mathematical-habits-...

Guyag 10y ago

'Scaling the ladder of abstraction' applies quite strongly to dealing with a new codebase, and is something I think inexperienced people (myself included) initially struggle with.

HiroshiSan 10y ago

Haha that's funny I was just scrolling through the comments before clicking and thinking "I wonder what Jeremy Kun has to say" and lo and behold...

mfn 10y ago

It might also be that individuals who have these traits self-select into studying math. So we can't easily conclude that studying math leads to the development of these "highly mathematical habits", because the causation could also flow the other way.

Waterluvian 10y ago

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.

dschiptsov 10y ago

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.