> If you don't want to cough up the required rigor...

Forgive me, but this sounds absurdly elitist.

Mathematics is not a subject reserved for a select set of chosen elite. It is a very necessary tool for almost every scientific endeavour. And this is a huge problem because people in those fields aren't just studying maths. Case in point, I'm a mechanical engineer. I have to study thermodynamics, materials, structural analysis, mechanics and on and on. In the midst of all this, I needed to learn advanced numerical methods, but was flummoxed because it was taught by a senior professor, to a class of mostly mathematicians, with an inordinately huge amount of effort placed towards proving things etc. and not much thought spared for us non-applied mathematicians.

I once heard maths described as something like "a person sees a category of something that exhibit a certain behaviour and then abstracts away the specifics to be left with a set of "general principles" that let the something exhibit it's behaviour". Thing is, it sounds like you start from the specific and abstract towards the general, but it is taught as a general and I am left to my own devices when dealing with the specific.

All for good reasons I'm sure, but then who is making a case for how we lesser mortals learn it? I was nearly in tears by the end of the course, convinced of my own stupidity and incompetence, because it was as if the class was communicating in their own language, and I (and other aerospace guys) were the village idiots who accidently made it in.

It was only after I took it upon myself later to learn using a lot of youtubing, "maths for idiots" type backtracking that I was able to learn just enough.

>I once heard maths described as something like "a person sees a category of something that exhibit a certain behaviour and then abstracts away the specifics to be left with a set of "general principles" that let the something exhibit it's behaviour". Thing is, it sounds like you start from the specific and abstract towards the general, but it is taught as a general and I am left to my own devices when dealing with the specific.

You made my point far better than I ever could. When I am faced with a general abstraction I have to conceptualize it in terms that I understand (the specific). Yet when the general case is presented, very little is said about the specific so I have to grind for months until I can finally maybe understand what was being meant. All of that could have been prevented had the concept been communicated in a more friendly, yet still mathematical, manner.

>All for good reasons I'm sure, but then who is making a case for how we lesser mortals learn it? I was nearly in tears by the end of the course, convinced of my own stupidity and incompetence, because it was as if the class was communicating in their own language, and I (and other aerospace guys) were the village idiots who accidently made it in.

That's how I felt constantly in college and it is actually part of the the reason I dropped out. It almost felt as if this alienation is intentional. Which makes me somewhat sad!

I see.

What I'm talking about is mathematics for the sake of mathematics. Any training aimed at mathematicians needs to emphasize rigor. Otherwise, there will be a bunch of impotent mathematicians. Not what we want.

I'm not opposed to teaching engineers "just the calculations". But such classes should be organised by their departments, not the mathematicians'. Your guys know better what you need to know.

This has nothing to do with elitism, though. A mathematician needs to know how mathematics works. Most other disciplines do not. I wouldn't claim that engineers are only allowed to earn their degrees if they pass math classes, thats just how colleges organise their course requirements.

As a counter point, I've a PhD in applied mathematics and I believe there's plenty of elitism in the field.

Unless you're a logician following a very specific set of reduction rules that can be verified mechanically, a proof is largely just an argument that something is true or not. Just because something is true doesn't mean the argument for it is strong, nor comprehensible. Part of the elitism within the field is that, at times, the community places pride on being as incomprehensible as possible because we should be smart enough to fill in the "trivial" details. That's morally wrong and it turns people away from the field. Further, it locks out a huge number of people who are not specialists from accessing the material.

Awhile back, I had a coauthor on a paper screaming at me in his office that it was condescending for me to include a full set of calculations for a proof that I authored. We had the room the paper for the result. I, as the author of the proof, couldn't follow the proof without the full set of calculations. We actually walked around the office and most people agreed with my coauthor. Although I ultimately prevailed and kept the calculations, it's an example of the underlying problem. Namely, yes, a large portion of the field is elitist. "I don't need these calculations to understand the result, so why should my readers?"

Math is not so cut and dried that there is precisely the minimum number of statements to describe a problem or give a proof. I very, very strongly believe in rigor. We can be rigorous and not be a dick in how we write or present material.

Thank you!

I am 100% certain that had authors of a lot of papers that I have read included some examples and perhaps even calculation results, I would have understood and perhaps even been able to reason about some of their publications a whole lot better than I currently can.

Just the mere inclusion of a few examples makes a world of difference. The idea that X is clear to everyone and therefor we do not need to exemplify it for the reader is really back-breaking. Especially when you are not coming form a background that includes years of expertise within said domain!

Sure, but math papers are not written for general audience; they're written for mathematicians.

Elucidation that would be useful for the general public is very difficult or impossible to write in such a way that it wouldn't be extraneous distracting fluff for mathematicians. It would also make most papers dozens of times longer.

I'm not exaggerating here: I think you could probably state and prove the results from basic one-variable calculus (assuming as axioms some basic properties of the field of real numbers) in 10-20 pages in such a way that mathematicians would understand it, whereas beginner calculus textbooks are regularly over 1000 pages long. A hypothetical research mathematician who didn't know calculus would find the 20-page version way easier to understand than a slog through Stewart's 1300-page book.

What you want is undergraduate-level textbooks, not research papers. Good textbooks do contain all the stuff you want: examples, calculations, motivation, etc. I can recommend some if you are interested in a particular field.

Some of the point that I was trying to make is that a large number of papers are incomprehensible for people in the field as well and often not for any good reason. Yes, sometimes it's hard to come up with good illustrations or examples, but, frankly, they help everyone. Let me give an example of a good way to do this. To me, Mark Embree is a really good mathematician. He also works in a somewhat esoteric area called pseudospectra, which helps us understand non-normal operators and, ultimately, how to effectively solve differential equations. Look at the following page:

http://www.math.vt.edu/people/embree/EF.html

Mark isn't dumbing down his material, but he is giving context for why it matters as well as graphical descriptions of how some of these ideas work. This is some of why I admire Mark's work. It's not just the technical results, but working to come up with better ways to relate the material.

Right, but we are talking about two different things. I agree with you 100% that mathematicians dickwaving about how hard they can make things for other mathematicians to understand is stupid, and I think quality of exposition for a mathematical audience is very important in math papers.

I strongly suspect the person who originally started this discussion doesn't understand "we are interested in the spectral properties of quasicrystals, as modeled by discrete Schrödinger operators with aperiodic potentials", at all.

(I'm not saying that to be an elitist asshole -- I have an undergraduate degree in math and I don't understand most of that jargon.)

So Mark isn't doing what OP is asking for, and so he isn't a good example to show that what OP is asking for is possible or desirable for mathematicians.

Well, there's a lot going on in the thread, so I'll try to be more specific. The context for much of what I've been writing here, other than my own frustrations, is the OP that stated:

Mathematics is not a subject reserved for a select set of chosen elite. It is a very necessary tool for almost every scientific endeavour. And this is a huge problem because people in those fields aren't just studying maths. Case in point, I'm a mechanical engineer. I have to study thermodynamics, materials, structural analysis, mechanics and on and on. In the midst of all this, I needed to learn advanced numerical methods, but was flummoxed because it was taught by a senior professor, to a class of mostly mathematicians, with an inordinately huge amount of effort placed towards proving things etc. and not much thought spared for us non-applied mathematicians.

This is something that I'm sensitive to and passionate about. Why I think Mark provides a good example for this is that an physicist or engineer may come looking for new techniques to analyze or solve the Schrödinger equation and the linked page provides context for why this may or may not be important. It also provides a visual description to help this explanation. For the technical results, this wasn't necessary. If Mark was just writing for mathematicians this isn't strictly necessary, but, to be clear, it helps someone like me dramatically.

What I'm interpreting from the overall thread, perhaps falsely, is that there's a sense of frustration that results are being made overly opaque. I agree with this and I don't think it's necessary.

I think its necessary to preserve a certain opacity.

I believe that there should probably some effort made to write tutorials on top of mathematical literature that explain how to use and apply at least the important results of mathematics to make it more accessible. But core literature should not be diluted for the purpose of making it more accessible.

That's a perfectly valid point of view. I believe that's not necessary and doesn't dilute the material. That said, I believe what you propose does exclude a sizable group of people from the field, which is elitism. It's a consciousness of belonging to a group of select few that have ground through the material.

> Part of the elitism within the field is that, at times, the community places pride on being as incomprehensible as possible because we should be smart enough to fill in the "trivial" details. That's morally wrong and it turns people away from the field. Further, it locks out a huge number of people who are not specialists from accessing the material.

Saying something is trivial when its not is insecurity. See it for what it is. Has nothing to do with elitism. This is no different from a driver who claims he doesn't need a seatbelt because he is such a good driver.

But even recent PhDs have to understand that our old-ish professors simply deem things trivial that are very difficult for us to understand. They've just gotten used to the material. The mathematician I admire the most, from whom I've initially learned algebraic topology, would react to questions to the material with such sweeping, elegant, simplistic arguments that you would either be mesmerized by the beauty or call it arrogance/elitism. But that's just the level he operates on.

Pretend there was an alien arriving on earth asking "stupid" questions like "whats 2+2", expecting you to explain in detail how you arrived at the answer. If you can't "dumb down" enough to get to where you lose him, are you an elitist because you "pretend" the calculation is "trivial"?

I don't think we need to postulate about aliens to come to a similar scenario. If someone in primary school, an undergraduate, or a practitioner asks me a question and I continue to lose them because I am unwilling to revise my explanation to something suitable for them, then, yes, I am elitist. That's not to say that I can't write something incredibly high level that only specialists in my field will readily understand. Frankly, I don't start my papers by telling the readers that we will be using Zermelo–Fraenkel set theory before going off and proving something about differential equations even though I likely depend on these results in order to get things like the real numbers. However, I don't think that's where the frustration is. I believe the frustration is that there is a dismissive attitude toward others when they don't understand the results. This is elitism. I also think there's a general unwillingness to self-reflect and consider that maybe the problem isn't that everyone else isn't educated enough to understand the results and maybe I'm just a terrible writer.

For me, I'm about a decade out from my PhD and I've been working in my chosen field for almost two decades. As I've matured in my understanding of the material I actually write quite a bit more because I realize more closely that there are lots of little details that are really important and it's highly unlikely that only outside of a select few will pick up on that. Now, that is my choice and others can choose differently, but just because someone masters a field doesn't automatically mean that they can just skip the details.

As an aside, 2+2 isn't that hard to explain. "2" is just our name for 1 + 1. So 2 + 2 is 2 + (1 + 1). By associativity of addition, this is (2 + 1) + 1 or ((1 + 1) + 1) + 1. And "4" is just our name for that last value.

Says you. It took me weeks when I first encountered it to understand the Peano axioms, how the natural numbers were constructed, and how addition is defined recursively given that definition.

what if its brain is binary and it cant comprehend our way of expressing numbers.

>Awhile back, I had a coauthor on a paper screaming at me in his office that it was condescending for me to include a full set of calculations for a proof that I authored. We had the room the paper for the result. I, as the author of the proof, couldn't follow the proof without the full set of calculations. We actually walked around the office and most people agreed with my coauthor.

This is not a math-specific problem. I had the same issue with physicist and engineering department's coauthors.

Do you have any particularly memorable example of a complicated proof which was lacking complicated "trivial" details? My experience (I work in interactive theorem proving) is that many such proofs are subtly wrong.

Off the top of my head, I'd probably say integration by parts in multiple dimensions. Generally, I see people given an outline of a proof in two dimensions for Green's theorem. I'd say it's an outline because the typical qualification is that the proof is for simple regions and that all we have to do is patch these regions together for more complicated areas. However, the details of how that works, in my opinion, are complex. The only full proof that I've ever seen for this result comes from Daniel Stroock's book, "A concise introduction to the theory of integration." He takes a little over 100 pages of background work to get there. This actually leads to my math confession, "I have a PhD in math and can't prove integration by parts in multiple dimensions."

By the way, if anyone has another reference or good proof for proving integration by parts in multiple dimensions, please let me know. I'd love to see it.

As another aside, I'd love to start using interactive theorem proving tools in my field, which tends to be related to things like optimization and differential equations. That said, I've never seen examples for how to prove results for stuff in calculus using interact theory proving tools, so do you know of any? Really, things like the mean value theorem or Taylor series would be great to see.

Most mathematics taught in universities at the undergraduate level is the mathematics for non-mathematicians. The typical calculus, ODE, PDE, vector calculus, complex variable stream taught for engineers and scientist is not at all rigorous, and misses a lot of the interesting details (for mathematicians) in order to speed up and cover more practical results.

It sounds like you ended up in the wrong numerical methods course, which is unfortunate. But there is a lot of the system is actually oriented exactly at what you are asking for.

> When taking a math exam, you can not contest having "misunderstood the question". The question is precise. By saying you "misunderstood it", you acknowledge that you don't understand the topic well enough to pass the exam.

This right here is completely unfair and infuriating.

Great mathematicians always find themselves taking the wrong turn and producing the wrong result(s) because they "misunderstood the question". Granted they learn form their mistake and revise and iterate until they get their theories and proofs right. But if they constantly fail, how could you then claim that they "don't understand the topic well enough". Are you telling me that Tao does not understand prime numbers well enough because he cannot solve for formula X?

This is the 3rd post that I reply to that seem to illustrate my point. You are RIGHT about the nature of mathematical language. But you CANNOT appeal to the general public with that arrogant attitude. You cannot communicate using dense language and expect non-math people to relate or even care. Granted, you might not give a fuck about that. But that is counter productive because if you want more rigor in society you should introduce more people to math rather than having them being put-off by some elitist attitude.

Once again, this is a M.SC. Exam so it should be this way. My issue is with math communication to the public in general, not this pdf specifically!

I don't expect non-math people to relate or even care. If you open a textbook on algebraic topology and you've never studied even the most mundane calculus, you won't understand anything.

Somehow, this infuriates people.

The same people would have no problem admitting that they should probably not read specialized literature on law that requires readers to have gone through 8 years of law school before they can make sense of the content.

Mathematical literature is not meant for casuals. Its impossible to express advanced concepts concisely, in such a way that people who don't know the material can understand them.

This is like asking someone for an explanation on how to walk and then throwing a fit when they reply with "well, place one foot on front of the other in such a way that you dont fall over". You don't expect them to discuss activation levels in skeletal musculature resulting from biochemical processes in the nervous system. But in Math, thats what you want. Clear and concise language that inspires and is fun AND fully describes the problem at the same time.

Not going to happen.

You still don't get it... You are arguing against something I did not say!

Theres no elitism in math. We just require you to write proof in such a way that it is actually a proof, and not just some idea you've had.

In order to do that, you need to study math. Not because we are elitists but because you need to learn how to prove something. You need to learn the tools.

The same way a mason expects you to learn his craft before he takes you seriously. If you don't have a clue about masonry, you can still semi reliably build a brickwall in your backyard. But not a cathedral. Obviously, you dont expect anyone to allow you to work on a cathedral. But you expect mathematicians to take your thoughts on math seriously.

Its like a bootcamp idiot trying to piss all over a big software project. Just because they know how to vomit on a keyboard doesnt mean they get to make decisions.

you STILL dont get it!

gonna build those really important walls instead :D

Posting that opinion in this thread was bound to generate confusion, but I get your point. That's usually called divulgation or public awareness. The problem is that it's often quite hard to do, because most practitioners only understand the concepts in terms of that mathematical language, and it's quite hard to translate them, since you can't easily put yourself in the shoes of someone who doesn't understand that language.

Also, it's not clear that there's any point in doing so; say you can explain a very technical concept to a layman using terms and concepts that don't require two years of study. Then what? What is that person supposed to do with that knowledge, if they lack the mental tools to apply it? It becomes little more than trivia, no?

Still, there are some attempts: http://math.stackexchange.com/questions/275/best-maths-books...

Mr. Tao does, in fact, not understand prime numbers. That's not to say that he is stupid or hasn't done his reading. He probably understands them better than anyone else in the world, but he still doesn't understand them. Otherwise, he wouldn't make any mistakes.

But the literature is not to blame for that. He simply tries to do something that nobody else has ever achieved before. Of course he's going to make mistakes.

The difference between math and writing programs is that in math, there's no compiler. When you go in the wrong direction, nobody is going to tell you about it. But this doesn't mean that any of the proven theorems that he bases his work on are faulty. He can rely on each and every one of them. They are proven to be true.

His issues stem from the fact that he wants to do things that are unexplored. Think columbus discovering america. Its not that his maps are faulty. There simply is no map for the new continent yet.

The rigor in math has nothing to do with an elitist attitude. The italians tried to do handwaving math for a hundred years. 100 years of italian algebraic geometry had to be erased because they began dropping the elitist attitude. They made progress at an amazing pace until eventually, they discovered inconsistencies in their "findings" and had to start over.

> The difference between math and writing programs is that in math, there's no compiler.

Increasingly, there are tools so that there could be, if you want to do math that way.

http://us.metamath.org/index.html

You mispelled Kato. Tao is an analyst not a number theorist. As for the Italian school tge foundational issues were common to all algebraic geometers. Ultimately concepts like generic point pointed the way towards scheme theory, even if not rigorously defined.

Tao definitely knows about prime numbers. Whether mathematical approach he takes, he is relevant to the field.

https://en.wikipedia.org/wiki/Green–Tao_theorem

https://en.wikipedia.org/wiki/Terence_Tao

> You cannot communicate using dense language and expect non-math people to relate or even care.

You seem to be misunderstanding mathematicians' goal. It is not to communicate with the general public. It's to communicate with other mathematicians.

You could argue that the culture of the field should change and should be focused on public exposition (although I'd disagree with you, as I don't think modern math is really interesting or important for more than a small subset of people). But regardless of the outcome of that separate argument, it's currently the case that in practice that is not the goal.

> if you want more rigor in society you should introduce more people to math rather than having them being put-off by some elitist attitude.

Again, "mathematician" is not the same job as "general public mathematics educator". Most professional mathematicians do not care about what you're talking about except in the same vague way everybody else cares about it.

This is a bit like asking me to write my code in such a way that people who don't know C++ could understand it. That might be nice but it isn't the primary goal and I'm not paid to put extra effort into it.

> The things you say are always falsifiable.

is that necessarily absolutely true? i myself would take a deep breath before making such a wide-sweeping claim. first thing that pops into my mind is the axiom of choice and also godel's work. i am sure there are a ton more examples.

> There are no ambiguities in correctly expressed math.

not accurate because define "correctly expressed". ambiguities absolutely exist in mathematics, though people do indeed seek to weed them out.

but i think you are romanticising mathematics too much. it can be messy and confusing because even mathematicians can disagree what the right way to do mathematics is. and once you get into various models and approaches (e.g., set theory vs category theory, classical logic vs intuistic or constructive logic) this is made more apparent. take smooth infinitesimal analysis for example. it completely turns on its head what you know about differentiable functions because it chooses to work in a different logic and model than classical calculus and analysis. the amiguity is clear because which is correct?

people need to relax when bible thumping their mathematical sermons. mathematics is not something etched in stone by the universe. it is very much an art as it is an application of logic and should not be treated as such an elitist subject.

> is that necessarily absolutely true? i myself would take a deep breath before making such a wide-sweeping claim. first thing that pops into my mind is the axiom of choice and also godel's work. i am sure there are a ton more examples.

thats kind of true, but axioms are not "results" of math, they are starting points. they are deliberate. although most mathematicians would argue that our set of axioms is "reasonable", we have just agreed to use them. that is true.

> > The things you say are always falsifiable.

> is that necessarily absolutely true? i myself would take a deep breath before making such a wide-sweeping claim. first thing that pops into my mind is the axiom of choice and also godel's work.

Broadly speaking, if things aren't falsifiable, they're either provably not falsifiable (so everyone knows that debating their truth isn't particularly fruitful, like the case of Choice), or they're precise but not provably unfalsifiable (Gödel might permit this, but we have never yet found a plausible example of such a statement), or they're imprecise (in which case they're recognised as being "not yet rigorous").

He is very comfortable with hurting science.