
Habits of highly mathematical people - CarolineW
https://medium.com/@jeremyjkun/habits-of-highly-mathematical-people-b719df12d15e
======
maroonblazer
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?"

~~~
ktamura
(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's certainly possible that you hone these
skills without being mathematically inclined.

I do contest, however, that these habits apply to _all_ "highly" rational
people. I know plenty of people who are rational (or so are they deemed to be
by their friends and colleagues) yet decidedly do not possess some of the
listed habits, especially "scaling the ladder of abstraction" and "Being wrong
often and admitting it".

On "being wrong often and admitting it," based on my experience, it's not that
you become modest as a result of studying math. By studying math, you realize
how prone your mind is to arriving at a faulty, unproven conclusion, and you
learn to be more skeptical of your own thinking and become more at ease with
being "wrong", allowing yourself to let your intellectual guard down.

~~~
aswanson
_On "being wrong often and admitting it," based on my experience, it's not
that you become modest as a result of studying math. By studying math, you
realize how prone your mind is to arriving at a faulty, unproven conclusion,
and you learn to be more skeptical of your own thinking and become more at
ease with being "wrong", allowing yourself to let your intellectual guard
down._ Indeed...math if anything promotes mental hygiene.

------
wutbrodo
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.

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

~~~
gunnihinn
There's one for every cardinality (up to isomorphism). I forget my set theory,
but if the cardinalities form a set, then the totality of vector spaces does
too. If not, we can amuse ourselves by looking at the set of finite-
dimensional vector spaces.

------
yomritoyj
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
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
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.

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

------
preygel
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.)

------
mturmon
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?

------
pm90
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.

------
tunesmith
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.

------
woopwoop
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-...](http://mathoverflow.net/questions/74707/mathematical-habits-of-
thought-and-action-which-would-be-of-use-to-non-mathemati/75022#75022)

------
auggierose
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.

~~~
wutbrodo
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).

------
mrcactu5
#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_
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
> 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!

------
Waterluvian
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
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.

------
haasn
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).

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

~~~
sn9
This is without question my favorite Quora post of all time.

I find myself rereading it every once in a while.

------
Noos
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.

------
HiroshiSan
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...

------
javier2
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
> 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?

~~~
srtjstjsj
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...](https://hn.algolia.com/?query=infinite%20series&sort=byPopularity&prefix=false&page=0&dateRange=all&type=comment)

~~~
wolfgke
> 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).

------
kamaal
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
> 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.

------
Kenji
_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.

~~~
ktRolster
Your quote didn't say that _all_ things can be disproven or proven. You need
to think mathematically!

------
Guyag
'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.

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

~~~
dev1n
Isn't there an old quip about how poor mathematicians are at arithmetic? ;)

~~~
mathgeek
I don't recall. How are poor mathematicians at arithmetic? ;)

~~~
dev1n
lol my bad.

------
bsaul
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".

------
mfn
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.

------
fibo
mathematician here, so true

