>"There is a general feeling in the pure math community that popularizing mathematics is betraying mathematics,”
I don't think this is true at all. My experience with math professors is that they either would welcome more popularity in the field, or they simply don't care.
I DO think that they would be very dissatisfied with the kind of farcical representation of physics that sometimes goes on (looking at you Michio Kaku).
I also think it's very misleading to say mathematics has nothing to do with the real world. The obvious contradiction would be physics, since it's applied math, but I think the quoted person was talking about mathematical research, which is a much fairer point. While it's true that a lot, if not most, of mathematical research has little to offer in terms of applications, but a lot of times "useless" math can still describe real world phenomena, sometimes very beautifully.
You and the author are conflating two different things. The pure math community probably DOES feel that popularizing only applied mathematics is betraying pure mathematics, and they're probably right. We don't really teach proofs outside of highschool geometric proofs and maybe a single class in university. However there's no good reason why we can't popularize both applied AND pure mathematics.
The real tragedy is that pure mathematics is hidden from students for too long and many opt-out of that path in favor of "Actuarial Science" and applied maths.
Real talk: pure maths was way more fun, interesting, and applicable to my life as a software engineer than all the applied math and modelling that I wrote/learned/etc. Pure mathematics expanded my mind unlike any other subject matter.
> The real tragedy is that pure mathematics is hidden from students
Physicists make a similar complaint. "Why would anyone want to study physics, when kids are not shown any interesting physics?"
Chemistry education research characterizes pre-college chemistry education content as incoherent, leaving both students and teachers confused and deeply steeped in misconceptions.
Biology education... starts by deemphasizing it's core organizational principle, and goes downhill from there.
> The real tragedy
Here's another. "It has been more than a decade since science education research surprised us with how badly science education was failing students. But even now, how often are students told? Instead, K-13, students have been left to think their confusion and cluelessness, are strictly their own fault, their own failing. Left to think science is boring. They are not told that these are known defects of the wretched content we're giving them. Fixing the content is hard. But telling them isn't. At what point does our failing to notify them become an ethical failure?"
> pure maths was way more fun, interesting, and applicable to my life [as a software engineer]
It can be fun to create ELI5-ish stories about math in everyday life. Math as recognizing patterns. So for instance, moving one way on a "circular" path, you come back around to the same position. As with boardgames, clocks, parking, etc. Missed doing something at lunch today... perhaps try lunch tomorrow. Missed a parking space and can't back up... perhaps drive around the block. Category theory for kids.
I'm flabbergasted. I just learned more about biology than I have in something like 20 years of state schooling. Genuinely, thank you so much for your comment.
I'd very much appreciate if you could give some sense of what specific highlights or benefits I might expect to gain from an 80 minute time investment?
There's an 18 minute cut from the original lecture, I'm thinking of starting with that.
Dates from a time when books were expensive but time was relatively cheap.
I also find the concept of lectio, meditatio, quaestio (vetted!), and disputationes to be interesting, perhaps an ideal model for the article / discussion online format. The better for having everyone actually listen to, and think over TFA.
I teach mathematics at a community college. The lecture resonated with me and I found it captivating. Here is a synopsis based off of memory:
Masur discovered that there is something called the Force Concepts Quiz. He learned that students of physics across the nation did very poorly on the test.
The test asks basic questions. For instance, a large truck and a small car collide head on. Which vehicle has more force imparted on it? This type of question requires no computations. It's easy if you really understand the concepts and hard otherwise.
He teaches at Harvard and thought his students would do well. They didn't. They did terribly. What he found is that regardless of the quality of the teacher (note he is a very good lecturer), type of class, or size of class students universally did bad. It didn't matter the quality of the institution either.
Conclusion, we are teaching physics wrong. He found that students who could do well on the Force Concepts Quiz could do tedious calculations and mastered that part of the course. Students who could master the calculus computations may or may not be able to pass the Force Concepts Quiz. Hence he decided to concentrate more on concepts and less on calculations. He does not lecture in the classroom.
In my own experience I've found similar things hold in mathematics. For instance, I ask College Algebra II students to give me an example of an equation that has no solution. Most can't do it even though 1=2 is a simple example of such an equation.
Wikipedia says "an equation is a statement of an equality containing one or more variables", but your example lacks a variable.
How about "-1=x²"? If x can be a complex number a solution is "x=i", but what if we only use real numbers? Wikipedia says "There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. A conditional equation is true for only particular values of the variables." Does that imply that without a solution "1=x²" is not actually an equation? The two parts are not equal after all.
If you prefer, 1 + 0 x = 2. We agree that is an equation. It has no solution. Typically in math we don't write 0x in expressions.
Suppose we start with 2x+5 = x - 4 + x. If we solve this by showing all work we'll end up at the end with
5 = -4
At this point students are told this means that there are no solutions. But what they miss is what an equation really means. Anyone who really understands the concept of an equation can easily give an example of an equation with no solution.
An equation is a statement that two expressions are equal. What one tries to find is the solution set. That is, the collection of tuples (if more than one variable is present) that makes the expressions equal when evaluated at the tuple. An equation is really a question. How can these two expressions be evaluated at the same input to produce the same output?
Lastly, We agree that f(x) = 1 and g(x) = 0 are valid functions. Here is the equation 1 = 0 stated in a more sophisticated way. f(x) = g(x) where f and g are defined above.
My personal definition of equation is just "an equation is a statement of equality", no variables required. But if you want to talk about solutions, you usually need variables. (The example "1 = 2" could be saved by saying it has no solutions, while "1 = 1" is solved by the empty assignment of variables.)
The distinction between identity equations and conditional equations is basically one of quantification. (Quantification should really be introduced as soon as you have variables, but is unfortunately often left implicit.)
An identity equation is essentially a statement of the form "for all valid assignments of the variables, ... = ...", where a valid assignment is constrained by conditions like "x is a real number". (Those are also often implicit when they really shouldn't be.) This means that "1=x²" is not an identity equation when x is taken to be real, but it is one when x is constrained to be a root of unity. (This is just another way of saying 1=x², but all identities are like that.)
A conditional equation amounts to "there is a valid assignment of the variables such that ... = ...". Then "1=x²" is a conditional equation that can be made true on the real numbers. The solutions are variable assignments that make it true, in this case 1 and -1. If you constrain x to be an imaginary number, then it has no solution; the equation and the constraint on the variable contradict each other.
If you are taking college level algebra, you should be familiar enough with equations to set up a contradiction like that.
I suspect in context that's not a fair question, because students are probably be expecting a technical definition of no solution - maybe imaginary roots, maybe something else - instead of a much simpler inconsistent statement.
Those two answers live in different conceptual categories, and most math lessons and problems live exclusively in one category and not the other.
This is related to Force Concepts because physics-as-math is taught as a series of algebra problems with a rather distant relationship to lived experience rather than the other way around.
Teaching physics as experience is harder than it looks. When I learned physics the course really tried to emphasise experience, but it did it in a stylised and abstract way with canned "experiments" using special equipment. This actually made it feel more distant and uninvolving, not less.
I suspect to do a better job you have to get out the classroom. E.g. introduce force, momentum, and acceleration to kids on bikes and skateboards, not with wooden trolleys on ramps.
It's a fair question in that a student's inability to give an example of an equation that has no solution indicates that we aren't really teaching students what an equation is/means. If one can't instantly give an example of an equation with no solution then they don't really understand what an equation is. Which means something is wrong with our teaching.
I think you're right that pure math is hidden too long, and with too many "applied" detours along the way. When I was getting my BS in math, I started hearing concepts my junior/senior year that would have done a lot to clear up the motivation and context for lower level classes, even if wouldn't understand them fully until later. Particularly in regards to linear algebra. I did go the Actuarial Science/applied math route in the end, so your comment rings true, but I'm not sure I would characterize it as a bad thing. I wasn't going to be a world-class researcher and the marginal return on a graduate degree in pure math wasn't there.
That said, there was a small feeling of "And now the time for fun and games is over" when I stopped studying pure math. I enjoy my job, but math provides a certain satisfaction that business problems do not.
As someone who ended up majoring in pure mathematics, I always thought it was a shame that theres this wall of arithmetic and calculus between you and the first pure mathematics class.
I suck at those, but I do well with abstractions and proofs. Makes me wonder how many pure mathematicians the field misses out on.
My university kept the pure math courses for "honors" students; I'm still outraged by this to this day. It's very misleading and as someone who didn't do great in the "regular" (i.e. rote repetition/memorization calculus and linear algebra courses), it kept me out until a prof in one of these classes introduced me to real math. At that point it was a much steeper learning curve to switch to the "honors" route but I'm glad I did it.
It does seem strange that you have to typically "pay your dues" with 4-ish semesters of calculus (calc i, ii, ii, diffy-q, and maybe pre-calc) before you take anything else. It's very strange as a gate-keeper to the subject.
There's this joke in applied math departments that the best way to spook a pure mathematician is to whisper in his ear "We've found an use for your theorem..."
I found an application for a theorem that was originally proved by the authors because for its interesting geometrical meaning. One of the authors who I talked to really liked my application.
So in my impression pure mathematician might not be motivated by applications, but if other people find cool applications, they typically love it.
I got a BA from NYU and took a fair amount of grad classes and spent most of my time in CIMS. Firstly I'd probably want to hear what you think of as applied mathematics, because if you're talking about the field of applied mathematics that lies outside of engineering and physics, it gets about as much "popularization" as pure mathematics.
Putting that aside, of the professors who had pure focuses, I'd say more often than not they tried to do their responsibilities as educators and put forth examples to try and make the material more accessible—but that kind of thing seemed to be how they thought on the matter. Actually, the applied mathematicians were very similar. I felt that most of their thoughts were constructive and there wasn't really any sense of comparison with the traitorous physicists or anything.
In the first place, before any popularization of mathematics can occur, it needs to be introduced in a way that doesn't leave all our secondary school students traumatized for their entire education, which goes back to the education system.
Speaking of the lack of education in proofs and popularizing math, do you (or anyone else here) know of good resources to practice and learn more about proofs? I've asked several professors and TAs in my CS program and they say "practice," but they couldn't point to any good tools for self-teaching and I don't really feel confident evaluating the material I've found on my own.
Velleman's "How to Prove It" is a nice textbook on how to develop mathematical proofs. You can apply the techniques to any other field you are interested in, and that is probably the best way to practice: by trying to prove things you already have some intuitive sense about.
Picked up a copy, thank you! And thanks for the suggestion on proving things in area I already have an intuition for, maybe I'll do that in conjunction with reading the book.
We can't popularize pure mathematics because, by its definition, pure mathematics is not related even tenuously to every day life, so popularization requires that the audience do years of idiosyncratic study.
I recognize a piece of background that the article is leaving unstated. A decade ago, the mathematical group E_8 was the subject of a unified theory of physics by an outsider, Antony Lisi, who called it "An Exceptionally Simple Theory of Everything". [1]
This theory never got published (it was supposedly the most downloaded paper on arXiV for a while, though). Peer review was not favorable to it. But the popular press ran with it, especially because of the story they could tell about how a surfer dude was maybe the next Einstein.
I assume that the echoes of this are why a mathematician writing down facts about E_8 got caught up in an explosion of weird publicity about God Particles.
I remember that. IIRC, the draw of the "surfer dude" wasn't just that he was a surfer, but that he lived out of his van, was a mathematician or physicist, and basically just traveled around surfing and doing theoretical math when he wasn't surfing. It's fairly easy for a lot of people to romanticize that.
I don't recall whether I read about him, or watched a Ted talk, or both, but a good portion of the time when I read "unified theory" some recollection of that story comes back. I'm sure it helped that a very dumbed down version could be understood almost purely through visual aids with pretty colors and symmetry. ;)
I met Lisi back in the day. It also helped that he was actually not a crack pot. Certainly not in the classical sense. He understood the math at a level typical of a theoretical physicist in the field, and he engaged the scientific community about his ideas in a deep proper way. The rebuttals and criticism he drew also showed his work was at least taken somewhat seriously and considered.
At the end of the day his theory remained to incomplete. Chirality problems were hand-waved away unconvincingly. It's more a mathematical curiosity than a proper proposal for unification.
You'd be surprised at how many mathematicians are avid hikers. I've heard stories of people flying to Calgary, disappearing in the Rockies, then showing up at UBC for a conference over a month later.
>Even when researchers do want their work shared widely, why don’t we read more about the fuel that makes math grow? “The physicists tell exciting stories,” Vogan said. “In some ways, this is a failure of mathematicians to tell exciting stories.” The physicists also have better names. Black hole and God particle quicken the pulse somewhat more than “irreducible unitary representation.”
Mathematics also has some cute names such as Monstrous Moonshine[1] and Hairy Ball Theorem[2]. I don't understand Monstrous Moonshine but the name had enough poetry to cause stickiness in the brain for later recall. A math problem closer to layman's experience with a cute name might be the "Secretary Problem".[3]
Leaving aside the naming and surveying the "importance" of open math problems... I would guess the P!=NP proof would be an example. Unfortunately, it doesn't have any easy description that the average person would understand. The same goes for all the other Clay[4] Millenium problems.
P versus NP isn't hard to explain if some of the problems are first familiar with the person you're explaining it to.
"It's very easy to check if sudoku is correct, right? You just go line by line, box by box. But isn't it fairly hard to solve?"
You'd have to go more in-depth than that, you're right: there's not a good one or two sentence explanation.
But, explaining that "some things are really easy to check if they are correct, but really hard to solve: what does easy and hard mean? generally, it means it takes either a short time (easy) or a long time (hard)" isn't too bad.
I can't do it in two sentences, but this is approximately the version I teach to first-year non-CS undergrads:
A problem is called "NP" if I can give you a solution to it and you can easily verify that the solution is correct. For example, "find 3 different odd numbers that together add up to ten" (here's a possible solution: 3, 5, 7). What's awesome about this is that there are a lot of problems of that form where we don't know whether or not it's possible to find a solution to the problems before dying of old age -- or before the sun runs out of fuel, even though you could check the answer in a few minutes using pencil and paper. Nobody has ever figured out how to solve them fast, but it's a huge open question of whether they can be solved fast. The P = NP question is whether there exists a fast solution to any problem that you can verify easily if I give you a solution.
And it's not just of theoretical interest: some those problems underlie everything from cryptography (communicating and storing information securely using encryption) to things like optimizing airline routes or the way UPS chooses where to route packages. Amazingly, there's a set of problems that if you find way to solve any one of them fast, you can solve all problems in NP fast. Which means the NSA and everyone else will try to kidnap you to find out the secret. Fortunately, most people who study it don't think that there's a solution. But it's one of the most famous and important problems in computing, and nobody -- of some of the smartest people in the field -- has ever managed to prove it either way.
(Truth in advertising: We actually use subset sum as the example, and walk through it a little, but that's because we're trying to teach it in more detail than just telling the story. And have also taught a handwavy version of asymptotic analysis. If you want to show someone how hard it gets quickly, subset sum is a nice way to do it. The "talking" example above is, of course, an instance of subset sum, but it doesn't explain the general form of the problem.)
I can only think of a few ways in which my definition fails. First, that it doesn't identify that NP only requires polytime verifiability for satisfying answers, not for "no" answers; second, that the use of the informal "you" instead of "a deterministic Turing machine" makes assumptions about the bounds of human reasoning that I assume are correct but don't know for certain; and third, that the ease of verifying certain solutions may mislead a lay-observer into thinking that a problem is in NP when, in fact, it is harder. ("This program exits because it consists only of the single statement exit(0);").
I think the third problem with my definition is the most thorny. But if you take the technically accurate form form of it that doesn't risk confusing easy instances with worst-case behavior - that you can verify in polytime a solution to any instance of the problem - it's technically correct.
But which things did you have in mind, and how would you improve the explanation?
No, that's what NP means. Given a witness to a solution of a problem, verifying that the solution is correct in polynomial time means the problem is in NP.
there are a lot of very interesting math things with cool names.
heres the problem. mathematicians dont build expensive shit. theres no math lhc that cost the equivalent of feeding bangladesh for a century that econ minded folks could whine about.
theres no "mathematicians will create a black hole under geneva that will swallow us all".
if the riemann conjecture is proven, precisely nothing happens. its not even possible that something happens in the tangible world, because of some result in mathematics.
mathematics isnt irrelevant because it doesnt have cool names. its irrelevant because its impossible to make the news.
a biologist can accidentally create a virus that kills us all. obviously science fiction, but its at least potentially possible.
a chemist can invent some kind of explosive that isis could steal and make some building go boom.
physicists can totally make black holes appear and build warp drives and all this weird stuff that is more fiction than science, but spock talked about it and light sabers duh.
mathematics just powers all those things. there are no lasers without math, either, but they arent attributed to mathematics. they are physics stuff.
when the first fusion reactor finally runs profitably, mathematics will have contributed a lot of necessities. but its the billionaire douche who funded the thing who gets to talk about how the engineers built an impressive machine.
mathematics doesnt get any kind of fame outside the scientific world, because its invisible. but the scientists who use all the math stuff know that without mathematics, they'd be almost useless.
I tell people, sometimes, that studying mathematics felt like studying god's (goddess') thoughts. Physics and the like describe this universe - and somehow mathematics reaches something different.
If anything, I wish we had something akin to a mathematics monastery. I'm not talking about academia - that avenue clearly leads to financial enslavement at this point. I'm talking about a place with lots of chalk boards and pretty basic room/board.
What would it contribute to society? How about...people with increased thinking skills and a place to belong? It could also taken on consulting work from everywhere.
It may end up looking like a liberal arts college or the math wing of the NSA - sans secrecy. I'd also want the barriers to entry to be low.
>>I wish we had something akin to a mathematics monastery. I'm not talking about academia - that avenue clearly leads to financial enslavement at this point. I'm talking about a place with lots of chalk boards and pretty basic room/board.
As soon as you wrote that, Princeton's Institute for Advanced Study[1][2] sprang to mind. In 2012-2013 a bunch of mathematics professors got together to organize a research program, from which homotopy type theory[3] (HoTT) was born and a book[4] written to explain their monastic achievement!
If you haven't read "Anathem" by Neal Stephenson, you'd probably enjoy it. It's a science fiction story about basically math monks (there's also a heavy dose of philosophy, though it's pretty much all connected to math in some way).
I second this suggestion. Plow through until it clicks and be prepared to give it a second read-through for an all-new experience. That is if you're slow to catch on, as I was.
> if the riemann conjecture is proven, precisely nothing happens. its not even possible that something happens in the tangible world, because of some result in mathematics.
Unless I'm mistaken, if proved the Riemann Conjecture could provide insights on the distribution of prime numbers or a polynomial-time algorithm for prime factorization, which would enable attacks on ciphers that rely on the difficulty of this problem (e.g. RSA).
This is not assured by the yes/no answer of the conjecture, but the underlying work to reach that.
im not saying that proving the riemann conjecture doesnt do anything for mathematics. it does a whole lot.
but the world itself isnt going to change. i mean the 5 pm news world.
heres the thing. we already kind of assume the riemann conjecture to be true. you can do a lot of things with it. its not PROVEN, and mathematicians will tell you that its not proven, but if it was such a big massive RSA attacks deal, hackers would just "assume" its proven and then calculate those primes. they would very quickly figure out whether the conjecture holds or not, in a practical sense.
i think what you misunderstand is that the riemann conjecture already exists. you can already use the result of the conjecture. the missing link is the proof. but you dont need the proof to apply the result. you can just apply it. maybe some mathematician comes along and disproves it. but the result is very accessible.
so the result cant be what would enable you to perform polynomial time attacks. it must be the proof. but since you dont know what the proof would look like (because it doesnt exist), it cant be in the proof. therefore, i kinda doubt what youre saying.
a conjecture is just a statement by a smart person that lacks the proof. from a mathematical point of view, its not a result. but for everyone else, including serious physicists, conjectures can be used as is to make progress. if the riemann conjecture allowed for primes to be calculated so fast that modern cryptography would suffer, then that would already be happening.
but thats just a tangent. my initial point was that the riemann conjecture isnt going to say anything about the world that a journalist will find interesting enough to construct something reportable out of.
I think they meant that finding a proof could involve discoveries leading up to that proof which could lead to <blah>, not that, once we have a proof, we know it, and then we can deriving things using it as an assumption and be more confident in our conclusions, which could lead to <blah>.
remember that millenium problem solved by perelman? its a problem in topology, but the proof is expressed in terms of elliptic curves which lives in the number theory space, and so far, we havent found a way to express it in terms of topology.
this is pissing topologists off a good bit. that a number theory guy solved their problem.
similar things could happen to the riemann conjecture. proven in unforeseen ways without revealing prime number stuffs. betting on how a particular conjecture is GOING TO be proven sounds far fetched at best.
How does Perelman proof involve elliptic curves? I thought it was about properties of the solutions to the Ricci flow equation, hence more of a difficult problem in PDE. I have no clue where elliptic curves would arise
"Hamilton created a list of possible singularities that could form but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: essentially three-dimensional cylinders made out of spheres stretched out along a line. An ordinary cylinder is made by taking circles stretched along a line. Perelman proved this using something called the "Reduced Volume" which is closely related to an eigenvalue of a certain elliptic equation."
"an eigenvalue of a certain elliptic equation" is probably an elliptic curve. I know very little about number theory.
There are many techniques involved. What I know about the Poincare Conjecture comes from a story a professor told during one of his classes.
As far as I know, there is no direct link between elliptic PDEs and elliptic curves.
Elliptic PDEs are called like that because the coefficient in the second order term look like those of the equation of an ellipse (as opposed to parabolic or hyperbolic PDEs)
Elliptic curves take their name because they were first studied in the context of elliptic functions, which are certain integrals that arise in computing the arc length of an ellipse. Even then, the kind of elliptic curves used by number theorists are generalizations where the algebraic form of the curve is the same, but the field of coefficients is a finite extension of the rationals (while the first appearance was with complex numbers). There is a relation between the two, but in general the study of the curve over the complex field only tells part of the story.
I am not aware of any more direct relations, but it would be fascinating to learn that there is something that relates them!
> which would enable attacks on ciphers that rely on the difficulty of this problem
I don't get this. If such attacks are possible, wouldn't assuming the truth (without proof) of the Riemann Conjecture allow you to develop such attacks?
If you're bored and hanging out with physicists and/or mathematicians or maybe some engineers, perhaps over some beers you can have a fun conversation along these lines:
Here's something uncontroversial: if its in physics, then its in math.
Here's a weird proposal: if its in math, then its in physics at a greater than trivial level or it can easily be proven not to be in physics. Or if its in math, either its obviously not in physics or it is in physics even if we don't know yet.
Discussion: Aside from souls and such your brain used physics to come up with some math thats not in physics so at a very trivial level that non-physics math was simulated in your physics brain, so it is now in physics because a physics simulation of the particles of your brain, however impractical to actually simulation ... naw thats trivial cheat that would include everything. And speaking of easily be proven, a classic example is we know two dimensional geometry fairly well and it doesn't apply to the greater world although it makes an entertaining plot for a book named "Flatland".
So as your buddies gulp down some beers, challenge them to think up a math that is not in physics as kind of a parlor game. It can be fun, especially if you allow extremely extended analogies and have a very relaxed beer fueled attitude toward handwaving arguments.
Some topics like string theory or the multiverse will have to be banned to prevent pointless fighting.
Anyway the relevance to OP and the comment is that in a drunken handwaving sense math that hasn't been found in physics is possibly a failure of the imagination of the physicist, so Riemann seems useless today is an attack on physicists not an attack on mathematicians.
You can just picture some seance between Newton and Maxwell, Yo Mr Physicist I invented all the math you needed centuries before you got around to noticing electromagnetism, like whats your excuse for the delay?
Interesting point. I guess encryption (and indirectly, blockchains) are some of the few closer to "pure" applications of math. But even then, they're treated as black boxes to some extent - not sure math gets all the credit it deserves there, for the average user.
> I would guess the P!=NP proof would be an example. Unfortunately, it doesn't have any easy description that the average person would understand.
"Hard. Really hard. Don't be ridiculous.". Now backtrack 2 small examples on paper. Oh, look, the number of steps here grow, but these grow even faster!
Especially mathematics when explained to a layperson so much depends on the language used. If one manages to keep the rest of the complexity zoo[1] out of the explanation and invests time into very basic examples, a lot things can be explained that otherwise the average person would not understand.
EDIT: Looking for a video I stumbled upon last week that had a very nice, short explanation of P vs NP (which I did not find again), I found this one: Not lay-person simple, but very funny and also provides a cursory glance at what lies beyond P vs NP, i.e. the complexity zoo: https://www.youtube.com/watch?v=YX40hbAHx3s (10 min)
P!=NP is a hint at one great application of mathematics, computer science. CS is closer to math in my opinion than any of the physical scientists. Proofs in information theory have profound effects on IT security, for example.
By far the best math teacher I ever had, Fred Irani, taught math in a way no teacher I'd had before or since did. We didn't use the textbook, and we didn't have homework. We just worked through the problems in class, the way the people who originally figured them out did: by working from what we already knew and making logical inferences.
Learning calculus from him was as much history as it was math, but the math stuck. I really wish more math teachers taught that way.
My favorite math professor would ensure he taught to and beyond the homework. He always had the toughest homework - but he managed to keep it accessible. Then the tests were based on the same sort of questions and material - but easier. Somehow most of my professors at that college got that mixture wrong. They would seemingly get bored with the topics and decide to throw out curveballs everywhere.
I've had profs like that. What really struck me about Irani is that even without the homework, even with only weekly quizzes, it still stuck. It wasn't an AP class, but he encouraged us to write the AP exam at the end of the year and nearly everyone got a 4 or a 5.
I do agree with the notion that profs are more effective when the homework is harder than the exam, however. I had the exact opposite for a fourth-year crypto class. For homework, we'd work through some basic cryptanalysis with lots of hints. Then the test would come around, and two or three of the ten questions would involve breaking a cypher, from scratch. I got about 25% on the first test, which ended up being 65% after he scaled it. Brutal class.
>working from what we already knew and making logical inferences.
The hardest & most valid exam I ever wrote used the same technique. Every single question on the exam was novel. I flipped through each page looking for something familiar, got to the end.
The class required about 30 hours of homework per week, but we were told we could do it while walking, showering, etc.
The second chapter of the book The Mathematical Experience has a hilarious interview between a hypothetical pure mathematician working on "non-Rimenan hypersquares" and a reporter who wants to do a piece on his work (pdf link, starts on page 39: http://www.springer.com/cda/content/document/cda_downloaddoc...). It demonstrates this exact issue.
It is an extraordinary book. I've read and re-read it. Just a chunk from the part you mentioned:
"He finds it difficult to establish meaningful conversation with that large portion of humanity that has never heard of a non-Riemannian hypersquare. This creates grave difficulties for him; there are two colleagues in his department who know something about non-Riemannian hypersquares, but one of them is on sabbatical, and the other is much more interested in non-Eulerian semirings."
Math's 'god particle' may be prime numbers. Though on the surface the concept of prime numbers involves just integers and you can explain it to 8-year olds, there are major unproven conjectures and deep allied mathematics. And now the worlds digital banking system and private messages depends on special properties of prime numbers [factorization].
That's just because we know very little about them. It very well could be the case that factoring semiprimes might be ridiculously easy, but we are just too dumb right now to figure out how to do it quickly (even if P != NP -- we haven't even shown that factorization is NP complete).
Given the very structural nature of the Lie algebra, wonder if the work, or part of it, can be automated and done by computers ... Enlightenment from any experts would be appreciated.
Relatedly, "Mathematics for the non-mathematician" is a great book that talks about the stories of math: history of how certain ideas, techniques, and discoveries happened. Recommended to anyone that wants to learn / review math in a more story-oriented way.
Even basic biological research has this problem. Anyone remember CRISPR? The press ran with all sorts of analogies about laser-like precision to reprogram genomes. Turns out it's not as laser-like as the press made it out to be.
Really the problem is writers and journalists who are not domain experts trying to explain this stuff to laypeople. There is no requirement for story writers to have been mathematicians or biologists when it comes to popularizing so you end up with half-truths and outright lies.
"""
The purpose of this paper is to give a finite algorithm for computing the set of
irreducible unitary representations of a real reductive Lie group G. Before explaining
the nature of the algorithm, it is worth recalling why this is an interesting question.
A serious historical survey would go back at least to the work of Fourier (which can
be understood in terms of the irreducible unitary representations of the circle).
"""
This work is also related to the "Langlands program", which is perhaps the mathematician's version of a grand unified theory of mathematics.
I don't think this is true at all. My experience with math professors is that they either would welcome more popularity in the field, or they simply don't care.
I DO think that they would be very dissatisfied with the kind of farcical representation of physics that sometimes goes on (looking at you Michio Kaku).
I also think it's very misleading to say mathematics has nothing to do with the real world. The obvious contradiction would be physics, since it's applied math, but I think the quoted person was talking about mathematical research, which is a much fairer point. While it's true that a lot, if not most, of mathematical research has little to offer in terms of applications, but a lot of times "useless" math can still describe real world phenomena, sometimes very beautifully.