
What is it like to understand advanced mathematics? - maverick_iceman
https://www.quora.com/What-is-it-like-to-understand-advanced-mathematics/answers/873950?srid=p6KQ&amp;share=1
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Smaug123
One curious thing I've noticed after completing my master's degree in maths is
that the jargon of maths sometimes lets mathematicians communicate with far
fewer words than non-mathematicians. Irritatingly, I can't remember any off
the top of my head, though I know I've experienced this a few times. Here are
some contrived and ineffective examples, where I put context in [square
brackets]:

\- "To first order, [the intervention turned out badly]" \- "Modulo [this
error, everything is going fine]" \- "[The situation is] symmetric on
interchanging [me and you]" \- "[They are in] the same class under the equiv.
rel. of…"

~~~
SatvikBeri
I think this extends to jargon from scientific fields in general–e.g. "on the
margin" from economics, "failure mode" from engineering, "significant" vs
"substantial" (statistics), "system 1/system 2" (psychology), etc.

~~~
yyhhsj0521
Just curious, what's the difference between significant and substantial?

~~~
bunderbunder
Significant, as in "statistically significant", usually that you've got
sufficient evidence to conclude that your result is not simply due to random
chance.

Substantial means that your result is big enough to have any practical import.
In other words, is there any substance to the result?

For an example of a result that is significant but not substantial, suppose
you find after surveying millions of people that members of demographic group
X score 1/10 point higher than average on an IQ test.

For an example of a result that is substantial but not significant, sales
figures are often so variable that it's impossible to determine with
confidence whether even a 100% jump in revenues is due to a recent ad campaign
or just a random fluke.

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unabst
Bullet point #6 is worth emphasizing.

>biggest misconception that non-mathematicians have about how mathematicians
work is that there is some mysterious mental faculty

Popular perception would have you believe that advanced math is like weight
lifting, where you need the brain muscle to process it. This weight is
measured in IQ, and above a certain measure you are a genius. Einstein is the
champion, and so if "you're no Einstein" then you have a perfect excuse to be
ignorant and incapable of any advanced topics in math.

Of course, the reality as it should be perceived is that any advanced
scientific topic is more like a journey, and that it never truly gets more
difficult. It just becomes more specific and hence technical. We just go
deeper and deeper into the rabbit hole of our choosing. Unfortunately, with
this view, there are no excuses in being bad at anything. It's more dedication
and obsession.

And Einstein himself has rejected any notion of genius being quoted often as
saying, "It's not that I'm so smart, it's just that I stay with problems
longer."

Except, didn't work, because the former group has written the quote off the
words of a genius.

~~~
cdevs
I like your answer because there is something strange about generalizing a
group of people who have dug deep enough down that rabbit hole. Are there
signs of a newbie "person A" to a field that they would have more potential to
always take on a problem in a better way than "person b". What if einsteins
work never involved mathematics and he wrote mysteries about a detective and
from his books we could tell this person understands taking on problems and
how to learn from failing etc,etc..

~~~
unabst
There are more ways of looking at what appears to be obscene amounts of
talent... some suggest the media also has a hand, since they create the
celebrities (monsters). Einstein was definitely celebrated.

But it's odd how popular perception can end up so far fetched. There still
exists this notion that knowing everything and memorizing everything and being
able to compute like a computer are all super powers. If only.

Except, we've been able to write things down forever, the internet is in our
pockets now, and we have supercomputers crunching obscene amounts of data. As
it turns out, none of these are "super" powers. Their capacity is easily
enhanced with simple tools, which we don't even bother to take out most of the
time.

You'd get definitely get better grades if your biological capacity were
higher, since using tools during exams is cheating. But we're starting to
think it's silly basing tests on these things.

------
chestervonwinch
Neat answer; although, the bullet points come off a bit like mathematical
horoscopes (perhaps because of the use of the second-person, "you"). I often
wonder if mathematics would be more popular if the beauty of works by it's
masters were more accessibly appreciable by those without the mastery, as it
is, e.g., in the arts or music. Anyone can look at a painting, see a movie, or
listen to Bach, and at least have an opinion - not true for any random person
flipping through, say, a functional analysis book.

~~~
erdevs
We must remember the frame of reference for these works. The audience.

Most art, and certainly most paintings, movies, and most of Bach's work, is
written _for a general audience_. Works of advanced, pure math however are
written for a very specialized audience, purposefully.

This does not mean, though, that math cannot be written for a general audience
and be understandable or even "beautiful". As an example, Cantor's
diagnolization is understandable by a 10 year old. The profoundness of the
question it asks is also understandable, as is its initially seemingly mind-
bending inscrutability. What's more the solution and the logic of the proof
are perfectly understandable and some of the easiest to understand
explanations literally use a child's method of counting (mapping fingers to
numbers). This is a beautiful work of math. And much math could be written for
broader consumption. It's not done frequently only because that isn't the task
most mathematicians set themselves to.

Moreover, even outside the actual _proof_ , many math results can be
_described_ in understandable and even beautiful ways. And the derivative
_results_ of mathematical discoveries are often easy to describe, understand,
and appreciate as impactful and beautiful.

So, I think this is a function of choice and focus. Perhaps math could indeed
more popular if greater efforts were made here, but mathematicians as a group
on the whole haven't set out to popularize it.

Now, no matter how much effort is applied, math will probably never quite be
as _enjoyable_ to most people as music or movies. But with such efforts math
may be more appreciable to a wider audience, as say paintings or poetry are.

~~~
saghm
Cantor's diagonalization is a great example; I'm definitely not a "math
person", but I was awestruck the first time I learned about it. People had
explained the idea of uncountability to me before, but I never really "got" it
until then. I can't think of a better word to describe its effect on me other
than "beautiful".

------
jordigh
> A theoretical physicist friend likes to say, only partly in jest, that there
> should be books titled "______ for Mathematicians", where _____ is something
> generally believed to be difficult (quantum chemistry, general relativity,
> securities pricing, formal epistemology)

At least one of these does exist:

[https://www.amazon.com/General-Relativity-Mathematicians-
Gra...](https://www.amazon.com/General-Relativity-Mathematicians-Graduate-
Mathematics/dp/038790218X)

Given how the author seems to be most familiar with geometry by the kind of
examples that she or he gives, I'm surrpised that they're not already
familiarw with this book.

------
Koshkin
It would be interesting to know what a book on advanced theoretical physics
would look like if the author assumed that the intended reader already knows
everything there is to know about the mathematics involved. In other words,
what does theoretical physics minus math look like?

~~~
freyr
To some extent, this reminds me of Feynman's Lectures on Physics. The lectures
were originally given to undergraduates at Caltech (so not "advanced
theoretical physics"), but they rely more heavily on intuitive explanations
than mathematical derivations compared to a conventional physics textbook. The
books were highly regarded by physics experts, but were essentially a failure
at their intended purpose of teaching undergrads and preparing them for future
work.

~~~
knite
I've seen many glowing references to Feynman's Lectures on Physics, but this
is the first time I've seen them called a failure. Could you share a bit more
about why they're considered a failure?

~~~
freyr
The lectures are well-regarded by many, but it's been argued that they're
ineffective for teaching undergraduate students the process of solving physics
problems.

Feynman himself was an exceptional mathematician, and used a very mathematical
approach to solve problems. But once arriving at the solution, he identified a
concise intuitive explanation, which he then presented to others. Everyone was
impressed with the brilliant intuition, but it didn't accurately reflect the
more laborious and mechanical approach he used to solve the problems [1]. The
Lectures of Physics are similar: a series intuitive explanations that would be
difficult to discover independently without actually working through the math.

[1] [http://www.stephenwolfram.com/publications/short-talk-
about-...](http://www.stephenwolfram.com/publications/short-talk-about-
richard-feynman/)

------
graycat
> What is it like to understand advanced mathematics?

It's really nice! Get to understand a lot of stuff.

Some of the advanced math is really powerful for applications.

And, can look back at the math saw in physics where wondered if the physics
profs really understood the math and conclude, right, they didn't or at least
not very well.

E.g., recently I saw a physics lecture where they explained more than once
that a quantum mechanics wave function was differentiable and "also
continuous" as if there was some question, doubt, choice, or chance otherwise.
Of COURSE it is continuous! Every differentiable function is continuous!

There's a lot more on why it's nice to understand the advanced math!

~~~
smaddox
> Every differentiable function is continuous!

Under who's definition? The Heaviside step function is discontinuous, but it's
derivative is usually considered to be the Kronecker delta function. Both of
these are used extensively in physics and engineering.

~~~
zodiac
Did you mean Dirac delta instead of Kronecker delta?

~~~
smaddox
Yes, my mistake. Kronecker is used often in quantum mechanics, but when
working in Hilbert space rather than real space.

------
ccvannorman
Wow - This article made me think I identify with and act as a mathematician
much more than I would've thought. I only started programming six years ago,
but the way the author talks about breaking down problems, building frameworks
and tool, and using multiple methods of attacks from an arsenal of knowledge
that builds up over time, and how specific problems are not as interesting as
insights into the general case -- these are all exactly the directions I find
myself moving in as a programmer that just feel 'natural'.

~~~
evincarofautumn
Programming is undoubtedly the largest branch of applied mathematics—although
for some reason we as a field don’t usually think of it as such.

------
j-pb
> your brain can quickly decide if a question is answerable by one of a few
> powerful general purpose "machines" (e.g., continuity arguments, the
> correspondences between geometric and algebraic objects, linear algebra,
> ways to reduce the infinite to the finite through various forms of
> compactness)

To me these tricks all feel like the same trick. "Can I find a program* that
gives a finite representation of a generator or a transformation, from
something I know, to my problem."

* Not as in php but as in lambda calculus.

~~~
auggierose
That's like saying each book ever written is based on the same trick,
generating a finite number of words. True, but trivial.

------
weinerk
Thanks. Great read. Among other things this particularly resonated: You are
easily annoyed by imprecision in talking about the quantitative or logical.
This is mostly because you are trained to quickly think about counterexamples
that make an imprecise claim seem obviously false. On the other hand, you are
very comfortable with intentional imprecision or "hand-waving" in areas you
know, because you know how to fill in the details.

------
erdevs
This is a great answer. It brings into relief many of the thought processes
and experiences which change in tackling questions after studying advanced
mathematics deeply.

However, the answer seems mostly to cover what it feels like to study
tractable problems. Things like reading and understanding others' research or
work, or studying new questions which seem "within reach, or nearly so." This
is representative because this kind of work forms the majority for most
mathematicians, and nearly _all_ of the work for non-professional
mathematicians who have studied math deeply and keep abreast of their fields,
but don't research full-time.

One aspect of the experience of understanding advanced mathematics which
_doesn 't_ seem to be covered thoroughly here is what it feels like to study
truly _intractable_ questions, let alone those questions which you fear may
actually be inscrutable. The simultaneous awe, respect and _consternation_ one
feels when confronting truly difficult questions which you intuitively feel
you just don't have the tools for. The problems which you think you'll need to
discover new tools to even begin breaking down. This answer generally projects
confidence and fluidity in tackling problems. And that's fair, in that this is
the biggest change one undergoes when tackling mathematical questions after
studying advanced mathematics deeply.

For deeper questions-- those which you feel you are very far from being able
to answer-- the experience isn't quite _opposite_ of what is described here,
but it is _different_. When you feel that all of your fluid mappings and
transformations, all of the most powerful tools at the ready in your toolbox,
and all of your simpler analogues are not only not going to solve the problem,
but are unlikely even to lead to a truly deeper _understanding_ of it... when
you're not even sure whether breaking the problem down in a particular way
will be productive or _counterproductive_... when you've wrestled with a
problem for days, weeks, and months and you're still not sure if the
"foundations and frameworks" you've built are even of the right type or in the
right vicinity to solve the problem... then the sense of confidence projected
in this answer falls away. The sense of assured solution-- even assured
_understanding_ \-- and fluidity of movement in problem solving is no more.
You're still confident in your mathematical knowledge and ability, and
certainly you feel differently than you did as a beginner. But it is a
humbling experience.

The process of tackling these questions is not quite so structured as the
impression that might be given by reading this answer. In these situations,
Wiles' analogy of stumbling in the dark through a great mansion for months and
years (which the author also quotes) is closer than the analogy given by the
author of building a house. What even Wiles' analogy doesn't quite capture, at
least in the section quoted here, is the _uncertainty_ of the process of
stumbling through that mansion. At various points you ask, "does this room
even _have_ a light switch? At least one that _I_ can reach?" There is
tremendous backtracking as well, rather than the sense of steady progression
in discovering and understanding room after room. Imagine stumbling through
the dark for weeks or months, finally discovering the lightswitch in a room...
moving to the next room and doing the same... then on again to the next room
to do the same again for yet more weeks and months... then only to discover,
based on what you've seen in these rooms, that you must in fact be in the
wrong hallway. Or the wrong wing. Or even the wrong _mansion_ entirely.

In any case, I think this is a wonderfully insightful answer overall and its
author deserves great credit for being so thorough, so accurate regarding the
great majority of work, and so descriptive and relatable. I just wanted to
enhance the picture painted here, or expand one little corner of it, as it
relates to other types of experiences one is almost sure to have in
understanding advanced mathematics and trying to apply that understanding to
work in the field.

~~~
ScottBurson
Not a mathematician, but I have tilted at the P vs. NP windmill enough to have
a sense of what you're talking about. It is indeed an awe-inspiring
experience. I think I learned something by trying it (beyond the obvious
lesson in humility, heh), but I'd be very hard put to say what that was.

------
curiousgal
>When trying to understand a new thing, you automatically focus on very simple
examples that are easy to think about, and then you leverage intuition about
the examples into more impressive insights.

Sounds like something Feynman would say.

------
haruspex
You get to explain the rules of the game set in one line:

"Pick 3 cards, such that for none of the properties exactly two are the same."

Not very advanced though.

~~~
Chinjut
Just as well, "Pick 3 points comprising a line (in the four dimensional space
over the three element field)"

------
mietek
NB. There is an amazing discussion in the comments following one of the linked
blogposts.

[https://rjlipton.wordpress.com/2009/12/26/mathematical-
embar...](https://rjlipton.wordpress.com/2009/12/26/mathematical-
embarrassments/)

~~~
GFK_of_xmaspast
It devolved into a FLT truther, I wouldn't call that "amazing".

------
cookiengineer
Disclaimer: Contrary opinion that might not be yours

I'm building AIs and intelligent botnets for over 13 years now. One thing I
learned is that I now think humans can't understand mathematics.

No matter how good you think you are, a simple AI algorithm will proof you
wrong. From physics, to algorithms, to simulations, to automation, to analysis
- and in evolutionary AIs also the solution generation for a problem.

Every single scientific mathematical model that occured in my life was in an
"imperfect" state where humans think that our current "model of things" in
form of mathematics is correct while it's totally incomplete and a few days of
AI simulation figured out a better way to do it.

I somehow see mathematics not as a knowledge pool; more like a serialization
format for transferring knowledge. And I think that serialization format often
is too complicated for other humans to be understood and therefore has bias in
interpretation and resulting conclusion errors.

And I personally think that this is a bad thing and a problem that needs to be
fixed desperately.

~~~
jamez1
An algorithm has limits that humans don't, as Godel's incompleteness theorems
prove.

The study of mathematics is communicating some deep logic, but it doesn't end
there as you trivialize. It's not just some poetry that we memorize, but it's
a new insight to the nature of everything.

~~~
mroll
Godels incompleteness theorem says that no formal system can prove or contain
all truthy theorems. It says nothing about humans being immune to this effect,
which it seems to me your comment is implying

------
snaky
"Number, sets and categories or what do mathematicians actually do?" \-
[https://youtu.be/Lrp0M-p5pMU?t=4m59s](https://youtu.be/Lrp0M-p5pMU?t=4m59s)

------
DarkContinent
This is a rather populist answer, but still pretty cool :)

~~~
auvrw
first, i recoil at "advanced" and would prefer "pure". that aside,

> You move easily among ... different ways of representing a problem

is like 3 answers in one. representation has specific meaning.

for me, however, it's not an ideal phrasing: i prefer not to think in terms of
"problems". it can be productive to do so but also messy.

(as if i knew a/b math :p)

------
tunesmith
I don't know anything about architecture, but I know I dislike houses where it
seems like the architect had just recently discovered the "extrude" tool and
was picking all sorts of random things to pull out an extra three feet.

~~~
pimlottc
The McMansion thread is over here:
[https://news.ycombinator.com/item?id=12286724](https://news.ycombinator.com/item?id=12286724)

