Mathematical language is precise. Problem 1 of this problem sheet is expressed in precisely as many words and symbols as is required.

You dont need any more language to fully understand the problem. Any less, and the problem would not be correctly specified.

That is the beauty of mathematics. The things you say are always falsifiable.

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. There are no ambiguities in correctly expressed math.

Whatever youre going to write down as your answer for problem 1a) is either correct or its not. No amount of arguing is going to change that. We don't need to write essays and we don't want to. We just apply what's been proven correct to solve new problems.

If math problems were "communicated in more natural ways", they would be reduced to problems people are going to argue and reason about, like they do in a liberal arts class. If you don't want cough up the required rigor, just go do something different.

Mathematics does not make amendments for the rigor impaired. If you want to study something math-y without all the rigorous proofs, physics is what you're looking for.

> 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.

The questions are actually pretty clear. The problem is that you lack a general understanding of mathematical concepts and therefore you don't know what to do with it. But that's fine and that's actually what the exam is for, i.e. only those may pass who understand these things.

>This is probably not a view shared by many here, but if math problems were to be communicated in more natural ways, far more people would be interested in the sciences.

I would argue the opposite. If more problems could be as well posed as mathematical problems, we wouldn't have people bullshitting their way through arguments that are in dire need of some scientifical rigor (see climate science, psychology, and other disciplines that are too complex to isolate phenomena completely). You can't handwave your way through a math exam and you shouldn't be able to do it in other fields of science.

> I would argue the opposite. If more problems could be as well posed as mathematical problems, we wouldn't have people bullshitting their way through arguments that are in dire need of some scientifical rigor (see climate science, psychology, and other disciplines that are too complex to isolate phenomena completely). You can't handwave your way through a math exam and you shouldn't be able to do it in other fields of science.

Amen. I have been saying that for years. But I think you misunderstood my original post.

I am all for 100% scientific rigidity and less bullshit in society. But what you seem to classify as "clear" is not-so-clear to others. It's actually about how condensed the information is in each question. In order to unpack those questions, you need years of experience within that field in order to get anywhere.

I bet you anything that each one of these questions could be posed in a different, yet still mathematical, manner that would make most non-mathy people (such as myself) at least understand the gist of what is being asked. But that is not something mathematicians are interested in. Which is what I was trying to say.

The issue is that each provable discovery in mathematics is named; it has to have a unique name. These discoveries are exact with at least one proof for their validity and for their limitations and assumptions; all that information is tied up in a name, say for example eigenvalues. These proofs are built on top of previous proofs (which of course are named as well), names and phrases built upon each other.

So this ever growing collection of proofs and names is an accretion (or in our moments of grander hubris a pyramid). What you are asking for is the tip of the pyramid (with its wonderful view) without the stones below, in effect what you are asking for is magic, a stone floating in the air with nothing to support it.

>in effect what you are asking for is magic, a stone floating in the air with nothing to support it.

Isn't that how we all learn math? in fact, isn't that how we all learn anything really?

When you are in 2nd grade, you don't start by learning about Riemann hypothesis. You start at the tip of the pyramid and drill downwards into complexity.

I am not asking for something magical here. I just wish that more math was communicated in a simplified manner that would help bring in people from all walks of life into the world of science.

The simple way is to follow the steps, start at the bottom and work your way up. Eventually you will get to the point where it makes sense. The terminology and the understanding is opaque to a neophyte because ideas are non-trivial.

Any sufficiently advanced technology is indistinguishable from magic -- Clarke

> You can't handwave your way through a math exam

This reminds me of a calculus exam a long time ago. One of the problems was "given the ellipsoid define by formula ______ and a certain line, find the point in the line closest to the ellipsoid". The goal for this problem was for us to use Lagrangian multipliers and I hadn't studied that one particular subject. However... They made a mistake writing the exam. The ellipsoid was symmetrical, its intersection with the xy plane was a circle and the line was also in the xy plane. I only needed high school math to successfully handwave it. Lots of facepalms from my TAs.

You've just described the vast majority of my experiences in my EE E&M class. Very hard multi-variate calculus problems, but as soon as you can find the symmetry they get dramatically easier.

The issue is that even a little bit of non-technical language can cause loss of precision. A regular person would love if the technical language was watered down and sprinkled with colorful analogies but this is not often helpful to actually attack the problem. The problems in OP link builds on large pyramid of concepts and abstractions all the way starting from numbers. There is no way out than to walk through this hierarchy of concepts one at a time for many years to finally gain understanding of the problem. Sugar coated explanations might make you feel good but don't actually help work on it.

I understand that and agree 100%. However, what I was trying to say is that for a non-technical person who is trying to understand a concept rather than solving a problem, accuracy is not the main concern. Sometimes you just need to understand the line of reasoning "roughly speaking". and as I said at the end, I understand that this is a set of M.SC. exam question and my issue is not with this pdf specifically.

Take the problem of Integer Factorization, something that I have personally been fascinated by for years. The problem is so ridiculously simple that a 5 year old could understand it. With that said, google for a few wikipedia articles and see the notation and the way it is presented. The non-mathematician does not care about difference of squares or triangular numbers or what.have.you.

It seems to be the case that mathematicians are unable to talk about math without requiring the person they are talking to to have extensive knowledge of the topic. That's IMO why Carl Sagan and Feynman were so great. They could take complex problems and reduce them (read communicate) in a good way that most could understand and reason about without needing to traverse the entirety of the knowledge pyramid. Similarly, that is what Numberphile does so well.

My suggestion would be to slow down when reading math or about math. Let's take Wikipedia's integer factorization for example: https://en.wikipedia.org/wiki/Integer_factorization

First paragraph is:

"In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these integers are further restricted to prime numbers, the process is called prime factorization."

This is a clear and precise statement of the problem. There are some key words that one might need to look up which takes time, but that's what I mean by slowing down.

Be kind and do this for me: Think of yourself as Sara, an average person in society. Put yourself in her shoes. Now click on the following link and scroll down for a bit.

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

Scroll as slowly as you want/can. Tell me, do you think that this is a good way to communicate to Sara why this problem is of importance? or even better, do you think Sara will ever give a shit if all she can see is that?

If you want to promote a more scientifically literate society, you should not try to render every single person into a scientists. Rather, try introducing scientific thinking into people's every day lives. The only way that will happen is if those who know the sciences learn to communicate better with the rest of us and help us get it into our daily lives.

can you see where I am coming from? I am not annoyed by math or my ignorance toward the topic. I am trying my best to combat it. But the problem I often encounter is that those who know/can rarely speak the same language as those who cannot. Khan Academy grew big because he knew how to communicate, more people should be like him imo.

The first sentence is perfectly comprehensible (factorization is decomposing an object into factors which, when multipled together, gives back the original). As for the most of the rest of the article... there's really no reason for the average Sara to care about polynomial factorization at all (unless it's just algebraic manipulation for doing homework).

To promote a scientifically literate society, improve the secondary school curriculum to teach statistics: How does conditional probability work? What do sensitivity and specificity mean? When a poll comes out, what does the margin of error mean? What are some common probability fallacies and how to recognize and avoid them? Etc.

In addition to Khan Academy, also check out the OpenStax textbooks (https://openstax.org/subjects) over the likes of Wikipedia (which is more often a mix of technicalese or a bunch of trivia depending on the subject of the article).

I was with you until you said "there's no reason for the average sarah...unless...homework"

I was the average Sarah, a lot of the people I went to public school with were the below than average sarah and it's the elitist math attitude that's being talked about here that turns kids off from that.

It wasn't until years later, after a career in concept art, then vfx and now programming that I realize..."hey the Fibonacci sequence isn't just some parlor trick for 'math types', it's a thing we can look at to study recursion and integrate in our code to make actual products".

Products that the average sarah uses and maybe even loves and would be supremely interested in learning about but doesn't because she's not a "math person".

I also lament the fact I didn't get into maths and see the beauty of it until years later when it was really too late to get into it at any professional level just because I was always implicitly told I was never meant to be a "math person".

Maybe I'm not, but if we could get more kids into maths, even if they're not geniuses, I think society as a whole and they themselves would greatly benefit from that.

It's not elitist to dismiss polynomial factoring, though, just dismissive. I really can't think of any reason to care about the deeper points of polynomial factorization (anything other than repeated trial division), so maybe it's just ignorance on my part.

Stats? Now there's math you can use and is useful in understanding our world! And yet schools prefer to teach calculus in high school over however much stats you can teach without calculus. No one uses the integration bag of tricks in daily life, but everyone gets lied to with numbers.

a-fucking-men!

Same thing happened to me. I picked up math at 25 after not having done a single math related thing in almost 10 years. I picked it up after realizing that a lot of the things that I do on daily basis are heavily related to concepts such as triangular numbers and other sequences. My life would have been completely different had my teachers communicated math in better ways than simply saying here is an equation, solve!

> Scroll as slowly as you want/can. Tell me, do you think that this is a good way to communicate to Sara why this problem is of importance? or even better, do you think Sara will ever give a shit if all she can see is that?

"In all cases, a product of simpler objects is obtained."

That right there seems like a fairly well-stated explanation of the importance of factorization. As long as you're able to understand that a formula can be composed of objects, which I honestly don't think is that much of an abstraction, the very first paragraph (and in fact the very third sentence, the one that immediately follows two easy-to-understand examples of things that can be factored) tell you that factorization lets you express something in simpler terms.

I think an aspect of the problem is that wikipedia has become a highly technical reference rather than a traditional encyclopedia for a layperson to educate themselves. e.g. Articles often swamp the basic idea with myriad qualifications and connections to other topics.

There's demand for both levels.

I'd also like a wikitorial or wikixtbook... taking a layperson through to solid understanding but I suppose something like Khan is what's needed for that role.

BTW I've found wolfram often better than wikipedia for maths topics.

I agree that there is a demand for both levels. That's why there is also a "Simple English" version of this page. [1]

I agree that discoverability of this feature can be hard, though.

I also miss the fact that this feature is only available to the English language. There could be a "simple article" feature built into the website, I think.

[1] https://simple.wikipedia.org/wiki/Factorization

Ah! Right on all counts. I have come across this feature before... but (eg) looking at the ordinary wiki page for factorization, I'm not seeing a link to it...

So, to access the user-friendly, non-technical, layperson version is simple, all you need is to know how to edit the url... that's wot you call ironic, that is.

EDIT It's not even linked under "see also"... ok, your remark on it being only english was a tip-off... it's counted as the language "simple english", and is available under the language icon (a funny looking "A" on the left... which I would never have guessed was for languages). I can see that'a a very easy hack to add an alternate version of a page, since that's what languages are already... but IMHO, layperson versions of a page (or even for experts, wanting to just get the gist) are an essential part of wikipedia's mission and purpose.

Re: simple versions in other languages: they could use the same hack, and have (eg) "simple french", "simple japanese" etc, but because it's so important, I'd suggest another explicit level, something like "https://simple.en.m.wikipedia.org/wiki/Factorization" (BTW that's also a mobile url, since I'm on a phone... I'd say, "simple" is just as important).

I understand wikipedia reached a level of completion a few years back, and the organization consequentally changed in character. A push for simple versions of everything coukd revitalize it.

what the f... is this sorcery?! this thing exists? O_o

If anything, this should be promoted and slapped on every page as a big green button!

I feel weird posting this link, because I was relatively old when Wikipedia became a thing and still remember what "real" encyclopedias were like.

https://www.britannica.com/topic/algebra/Applications-of-gro...

> Some other fundamental concepts of modern algebra also had their origin in 19th-century work on number theory, particularly in connection with attempts to generalize the theorem of (unique) prime factorization beyond the natural numbers. This theorem asserted that every natural number could be written as a product of its prime factors in a unique way, except perhaps for order (e.g., 24 = 2∙2∙2∙3).

That seems like a pretty good first step to me.

All your examples of the problem you are talking about are from Wikipedia, but you give multiple examples in your posts of explanations of maths done the right way. Feynman, Khan Academy, Project Euler. So I guess some maths related text is written one way and some is written another way?

Sure. But I hope that you can see how little access the average person have to mathematical concepts unless he/she have had years of formal training. Khan academy, Numberphiles and the likes have done wonders for lots of us out here. But it is still not enough and more needs to be done. Especially when talking about concepts that are new or truths that we constantly "take for granted".

But is Khan Academy really so different to formal training? I mean, who is going to understand a Khan Academy video on solutions to 2nd order linear homogeneous differential equations if they have never learned any algebra? There is language in there that is necessary to explain it that they will have had to learn before.

It would be great for there to be more resources for learning maths outside being officially enrolled in formal education, but I don't think this has anything to do with the language used in mathematics being too terse or obtuse. Lots of good undergrad textbooks are no less understandable than a Khan Academy video, try those instead of Wikipedia.

Wikipedia isn't really meant for introductory articles

I am in complete agreement. I've done a fair amount of teaching, which is to say I was employed to communicate information, such that someone else (from a fairly random selection sample) could and would remember it well enough to actually use the information productively.

I have done this so much that I feel poorly abou tleaving the last paragraph as it is, given that I know there is most certainly a way to find a more parsimonious way to express what I am saying, and I think that is the trust of why mathematical language (and the language in wikis, or APIs, for example) is succinct to the point of absurdity.

Everyone is striving for what's called maximum information density, communicating as much data with as few words as possible, and they do so in the jargon of their respective fields of interest. The problem is, those that are not specialized in that field cannot decipher the language well enough to gain anything of use from their efforts at being succinct. Quite the contrary, the field of potential interest becomes even less appealing for having appeared so mentally taxing to parse.

Reddit has this ELI5 idea, which I think stems from Denzel Washington's character in the movie "Philadelphia" (Mr. Washington plays an attorney defending an AIDS patient who has been unduly fired) saying "Explain it to me like I'm five years old", to get people to reduce their use of jargon and get to the heart of the matter.

Needless to say, the point I am making is that it is one of the great intangibles to be able to communicate one's self effectively, and in my experience is only done well when the communicator understands the level of interest and/or understanding of the communicatee, and can adjust their vocabulary "filter" accordingly.

I think math is particularly tough in this regard because of the amount of time that goes into understanding topics that might be complex for an untrained observer (like an Eigen value, for instance, which are critical for a basic understanding of linear regression).

> Everyone is striving for what's called maximum information density, communicating as much data with as few words as possible, and they do so in the jargon of their respective fields of interest. The problem is, those that are not specialized in that field cannot decipher the language well enough to gain anything of use from their efforts at being succinct. Quite the contrary, the field of potential interest becomes even less appealing for having appeared so mentally taxing to parse.

That sums up my frustration perfectly. When talking to the general public, information density is your enemy not your friend. I have been trying to get mathematicians to understand that for years but they almost unanimously actively refuse to do so. I can list the reasons as to why I believe they refuse to do so but one thing is clear in my mind: you don't understand a concept unless you can ELI5.

For an expert to be able to dive into a dick measurement competition with his PhD friends about who can use the most fancy jargon possible is of no use to society. I just wish more people in the sciences could understand that!

> you don't understand a concept unless you can ELI5

Do you have any particular reason to believe that there are no ideas which are simply difficult? For example, I assert that the Monadicity Theorems are difficult and defy easy explanation; and general relativity is difficult and defies easy explanation. It's not that we don't want to explain these things simply: it's that we can't because they just have too many prerequisites.

There are some open problems in exposition (such as forcing, which ought to admit an intuitive explanation but no-one has yet found one), but knowing that we don't yet know how to explain something simply is not the same as refusing to explain something simply.

> This is probably not a view shared by many here, but if math problems were to be communicated in more natural ways, far more people would be interested in the sciences.

To the mathematician, this is the most natural way. Sparse, simple, down to business. Providing extra information can be confusing (if I tell you some random facts, you might be worried that your answers aren't using them at all) and the last thing you want to read on the exam is a page of problem motivation.

I think you are asking for too much out of a post like this. These problems are really constructed just to test you. The techniques you use to solve them will be important ones that they want to see that you have mastered, but the problems themselves are not really that meaningful.

We don't do math in human language for the same reason we don't write computer programs in human language: it's ambiguous and hard to define abstract objects using it. Only abstraction enables us to operate higher level, non-trivial problems.

I just wish Math followed programming best-practices on commenting, variable naming, and including tests and their expected output. Bonus for IDE-like tools that can instantly tell me the type of a symbol or show me the expanded version of an operation.

I agree, maybe because I learned to program way before I got into rigorous math. I really wish the community standardized notation more.

Matlab then?

"We don't do science for the general public. We do it for each other. Good day." -- Renato Dalbecco, complete text of interview with H F Judson

I assume the same applies to mathematics.

I can sympathize with the frustration, but at the same time there's frustration from the other end. It takes a very skilled person who can at once describe something with enough detail to be understood by their peers and also be able to describe the same thing with enough preamble to cover and make clear where any obvious inference level gaps are for a layman. You simply cannot understand factorization (to use the example you've given before) without first understanding several things prior. For general literature, it is better to assume readers have the prerequisites and write for them. When there is confusion people can always do their own study and dive down the definition/link chain themselves. For literature aimed at a beginner of the subject matter, it makes sense to be more verbose and search for other analogies, but you still need to start at some base level and build inferences from there. Start too low, and it takes years, start too high, and it's inaccessible on its own.

I remember some math concepts became a lot clearer to me when I realized how I could express them as computer programs, but it wasn't something any of my math teachers ever pointed out. It would have been helpful if they had. Still, having understood the concepts in the new way, I would rather use their mathematical formulation every time as there is no ambiguity and less room for error. Math is beautiful in that it doesn't really care what your internal representations are, in fact most people probably have widely different internal representations (Feynman's story about his friend who counts by visualizing instead of vocalizing comes to mind: https://www.youtube.com/watch?list=PLE80C041156D48764&v=Cj4y...) so it's better to be precise independent of internal representation than to always burden yourself with some preferred internal representation that you use for metaphors.

I do not agree this is the case. Can you give an example? How would you express these questions "in more natural ways"? (alternatively, feel free to point me to some example with other questions)

What you need is a book on how to solve problems using a strategy, like the first few chapters of 'The Art and Craft of Problem Solving' by Zeitz and 'How to Solve it' by Polya. The Zeitz book specifically refers to the issues in abstract word problems and strategies for understanding them.

'Introduction to Mathematical Reasoning' by Eccles was another great book to explain the meaning of mathematical statements and how to think like a mathematician. If you read some of those, and want to learn more math, start with functional programming in SML since it naturally leads to understanding proof by induction https://existentialtype.wordpress.com/2011/03/21/the-dog-tha...

What do you think could be stated better? I studied math (I.e., I'm the target audience of this) and the problem statements were perfectly clear to me (although I don't remember how to solve most of the stuff)

There really isn't a shortcut to math. I can tell you what to read in order to understand these questions, but it might take you a few years.

> it might take you a few years.

My complaint was not about these questions. These are M.Sc. exam questions and it should take you years of study to undertand and be able to solve them.

My complain was about math communication in general. It shouldn't take me years of study to understand a question on a basic level. I gave the example of prime factorization earlier but you can think of how most math problems are communicated to the general public.

A great example to illustrate my point is Project Euler. The way problems are communicated on PE is vastly different than the way they are communicated by professors, wikipedia or most other outlets. Every problem there starts with an introduction followed by a set of assumptions and a posed problem to be solved. It is far more friendly and inviting of a way to communicate math problems to the general public!

It doesn't take years of study. Those are mostly fairly basic linear algebra and calculus questions. You could learn enough to understand the basics by watching some khan academy videos.

You're assuming a solid understanding of pre-calculus math. It might well take 2+ years for people with no exposure to proofs, or who have a weak foundation in elementary algebra.

I don't know, I was able to understand the initial problem statements, and even a bit of the numbered sub-problems, and I'm a trained classicist pig-ignorant in math.