What is it like to understand advanced mathematics?
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See.
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.
I've been a physicist and am now a software engineer. I can use the maths based jargon in both those fields, but not vice versa.
That's partly because everyone in a STEM field had to learn at least some maths. But more importantly: maths is about the logical interconnection of things, while the things themselves are substitutable. So it is no suprise that its jargon can be applied in many fields.
Made me remember this. If only the 'upper' fields adopt mathematical naming where applicable :D
I fold everything, once, and then make one trip per handful-of-piles-of-items. Precious seconds saved! Seconds that I can use to have pointless arguments about saving precious seconds with my wife!
>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.
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.
Likewise for avant garde cinema, modern art, poetry, and more!
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.
Infinity MOD X is undefined... How sublime.
At least one of these does exist:
https://www.amazon.com/General-Relativity-Mathematicians-Gra...
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.
On the other hand, perhaps the "math for physics" needs a different emphasis than general math-for-the-sake-of-math. E.g. perhaps for mathematicians certain topics are covered without much detail as they are relatively uninteresting, but a math-for-physics course would put extra emphasis on them because of some applications. Don't know enough to give concrete examples.
Trivial example:
a @ b
b @ c
c @ a
Makes "sense" if @ is = but not if @ is <.
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-...
The page addresses two points: Feynman's fluctuating opinion of his own work, and the drop-off in student attendance (Feynman's opinion is then regurgitated as the third and final point). In fact, Feynman's opinion has no bearing on whether or not LoP is an empirically effective textbook. And regarding student attendance, the page argues against one person's recollection of dwindling attendance with another person's recollection (years after the fact) of steady attendance. Again, no evidence, just two contradictory and possibly flawed memories that have no bearing on whether or not LoP is effective.
The page provides as evidence an undated photograph of Feynman in front of a full classroom. When was this photo taken? Was it on the first day of class, or in the middle of the semester? Was it during a regular class, or a special lecture? We don't know.
Perhaps the LoP approach really is an effective textbook for teaching undergrads (despite Caltech abandoning it, no other nobody else to my knowledge successfully adopting its methodology), but the cited page doesn't provide relevant facts regarding the matter.
I took Caltech's intro physics sequence in the mid-80s -- IIRC my freshman year was when they switched from the Feynman Lectures to Goodstein teaching from his new The Mechanical Universe. (Goodstein was one of the pair with the negative quote above.) I thought at the time that the new textbook wasn't bad, but that Feynman was amazing. IIRC much of why they wanted a new book and video series was to take advantage of 3d computer graphics; I don't remember that making much of a difference to me.
(Added: I also attended a Feynman lecture live, but can't fairly rate him vs. others because I'm hard of hearing and rarely get much from lectures. The lecture hall was indeed packed with non-freshmen, but it didn't seem like any of us frosh skipped it. This isn't very relevant since it was a one-off.)
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!
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.
https://en.wikipedia.org/wiki/Differentiable_function#Differ...
W. Rudin, Principles of Mathematical Analysis.
W. Fleming, Functions of Several Variables.
For the set of real numbers R, a function
f: R --> R
x in R, d in R, function f is differentiable at x with value d provided for h in R
lim_{h --> 0} (f(x + h) - f(x))/h = d
In this case we write
f'(x) = df(x)/dx = d
In that case, in particular,
lim_{h --> 0} (f(x + h) - f(x)) = 0
so that f is continuous at x.
Differentiation is important in physics, e.g., in Newton's second law.
Also this isn't advanced mathematics, this is something most who are interested learns in high-school and I can assure you that most physics professors are aware of this fact.
No, they said what I wrote they said. It was at YouTube, from MIT, from a course in quantum mechanics.
Uh, I do understand this stuff: E.g., given a positive integer n, the set of real numbers R, the usual topology for R^n, a set C a subset of R^n closed in that topology, there exists a function
f: R^n --> R
so that for x in R^n, f(x) is 0 for x in C, f(x) > 0 for x not in C, and f infinitely differentiable. E.g., for n = 1, the result applies for set C a Cantor set or a Cantor set of positive measure. For n > 2, the result applies for a sample path of Brownian motion or the Mandelbrot set.
When I was a grad student, I discovered and proved this result. Later I published the result in JOTA.
Or for example, this famous dialogue between Geoffrey Chew and Arthur Wightman from the 1960s bootstrap fad:
Wightman asked Chew: why assume from the start that the S-matrix was analytic? Why not try to derive it from simpler principles? Chew replied that "everything in physics is smooth". Wightman asked about smooth functions that aren't analytic. Chew thought a moment and replied that there weren't any.
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.
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.
Sounds like something Feynman would say.
"Pick 3 cards, such that for none of the properties exactly two are the same."
Not very advanced though.
https://rjlipton.wordpress.com/2009/12/26/mathematical-embar...
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.
I don't think I understand your claim and I'm rather skeptical of what you seem to be saying, so can you give three examples?
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.
> 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)