Some Fundamental Theorems in Mathematics (2019) [pdf]

http://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf

234 91 6 years ago

people.math.harvard.edu

Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.

Konohamaru4• 6 years ago

You cannot teach people--even people at a young age--to enjoy mathematics because mathematics is based on logic and almost everyone except for a few weirdos hates logic. Mr. Spock was designed to point out how unusual people who like logic are, but as a character from a minstrel show, not as a role model....

082349872349872• 6 years ago

Joan Pearce and Peter Sloman of De Forest Research, Inc. were probably unusual people who like logic, having provided the science feedback for TOS.

Some other people certainly found Spock to be a role model in 1968:

https://ca-times.brightspotcdn.com/dims4/default/6870952/214...

bryanrasmussen3• 6 years ago

These design requirements in regards to Spock that you remark on - I can't really find them here https://en.wikipedia.org/wiki/Development_of_Spock and though never a Trekkie I don't think this sounds like something Roddenberry would have considered.

idclip4• 6 years ago

I disagree. Mathematics is a tool to practice our hobbies .. see Feynman‘s biography.

Rot mathematics is where it goes wrong. Meta mathematics, abstract, detached from real life, as alot of the academy with its publish pr perish thinking ... that produces pain. Existential voids filled with an exercise that does not help many to move forward optimally.

saying weirdos hate logic is extremely dismissive to a lot of what I would call the human experience.

The same stuff can be taught with soul and purpose If we actually show children how they can apply mathematics to improve and enhance what they’re actually doing whether it is building castles in Lego, Or designing faster toy car, A better school project volcano, etc

francasso• 6 years ago

Mathematics is as much logic as poetry is grammar. Sadly most people don't know that because they never get exposed to the creative/discovery side of math. Logic is just a tool to check correctness and enable communication.

domnomnom4• 6 years ago

Not true, logic is great. It's just taught in a tortured way and mostly goes by another name.

yesenadam• 6 years ago

almost everyone except for a few weirdos hates logic

Well, all kinds of people seem to love Sudoku puzzles. And they seem to involve nothing but various kinds of logical reasoning, and the pleasure of achieving things with them.

cinntaile2• 6 years ago

(Here I am focusing on math outside of the math taught in pure mathematics programs.)

If anything the focus on math being a tool should be even bigger than it is today if you ask me. Most if not all of the math you get taught in school is math that was invented to solve a problem, it was not created as art. Introducing this historical context would probably help students in understanding why learning math is useful instead of just teaching math without this context as is usually done today. The beauty, in theorems and their proofs, comes in my opinion from understanding the nitty-gritty details. Without understanding the logical steps it's hard, at least for me, to understand where the beauty lies. I cannot appreciate mathematical ingenuity without understanding it. This is a bit of an abstract question, but perhaps you could elaborate on how you see this beauty without understanding?

curiousllama• 6 years ago

Not OP, but I agree with OP, assuming by "people" they mean "people already inclined towards math." Self-evidently, IMO, most students need numeracy much more than they need pure math.

In school, I struggled with memorization & attention to detail. For the life of me, I couldn't quite finish an algebra-heavy problem without writing 3*4=15 or outright skipping a step in a conic rotation or whatever. I'd get dinged every... single... time, and got relatively bad grades.

Shortly later, when I arrived back at higher level math through cs & stats, I could ruin the curve in my applied classes. I'd relied so much on mathematical intuition early on to compensate for a lack of memorization ("what are the steps again? Whatever - I'll just think it through now") that I was quite good at putting pieces together in odd ways to solve data science problems.

Not to mention - with proper context, as you say, all that linear algebra that so annoyed me all the sudden seemed intuitive. "OHHHH so THAT'S why this exists"

I was first driven out of math by a relentless focus on math-as-a-tool, and was able to return at a level where it became more about creating new logic. I'm convinced that I would have thrived in an earlier class that focused on elegance & logic - what makes a good proof? - and I'd have stuck with it.

JackFr• 6 years ago

Replace arithmetic with “math appreciation”? No thanks.

I’m still disappointed by the way music and art are taught at the elementary and middle school levels. Singing, playing an instrument, drawing are all things which can effectively be taught and are so much more useful than useless watered down “art” and “music” I had to take in grade and middle school.

laichzeit02• 6 years ago

I'm convinced that if Book of Proof [1] was taught in high school, most people would (a) understand math, and (b) not hate it.

http://www.people.vcu.edu/~rhammack/BookOfProof/index.html

bryanrasmussen3• 6 years ago

the beginning part on Sets is definitely enjoyable reading. thanks for putting that here for me to find.

jimhefferon• 6 years ago

That's a fine book. But to answer that question surely calls for emperical data?

Generations of teachers of mathematics-- surely not all incompetent-- have found that many people, perhaps most people, do not love math. There are very good teachers (I had a few), and very good ways to teach it (shout out to the Mr Barton Maths podcast), but experience shows that it is a hard go much of the time.

Scarbutt• 6 years ago

Would you classify that book as good introduction to discrete math?

codethief• 6 years ago

This is a good book but I'm not sure it's low-level enough for those people in high school who don't like or haven't learned enough math – he mentions matrices, sin/cos, real numbers, … right in the first chapter!

GaussBonnet2• 6 years ago

The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with MULTICOLORED (whoa!) text. There is no communication between modern mathematicians and grade school teachers in America; they may as well live on separate planets. Math classes in American grade school are little more than busy work meant to weed out people who hate busy work. Infer from that what you will.

dorchadas• 6 years ago

Yes, yes, and absolutely yes! I'm a high school math and physics teacher (who majored in physics, not education, and is now doing self-study in more proof-based mathematics), and so much the issue is in grade school, in my opinion. The teachers there don't appreciate math. And, more than that, they don't understand math. Lots of them don't have a good well-developed number sense, so it's no wonder our kids don't either!

Then the problems compound when they just give kids calculators without the kids understanding why the calculator works, and by the time they get to high school hope is lost because they have no number sense and we're trying to explain variables and such...often divorced from real life. I've found my kids get a much more intuitive sense of slope when I explain it as the rate of change rather than "rise over run" that they're so often taught in middle school. Give them examples, like amusement park tickets (it cost $50 to get there, then $70 per person) and have them incorporate that into a slope-intercept form. It just makes things so much clearer that they often don't get because they don't have the number sense and because many teachers don't understand/appreciate math.

zorked• 6 years ago

So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics?

Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defense against scammers. By people who wanted to blow up their enemies in war and who were moved by hatred and revenge. A recent mathematical construct born out of academia wants to destroy the US dollar and turn the world economy upside down. There are people trying to take over another planet with Mathematics. Others are trying to achieve geopolitical supremacy building artificial intelligence and robots armies.

Mathematics is blood. Mathematics is power.

There's a disconnect between academics and how real humans think if the modern solution to Math education is proofs.

pastrami_panda• 6 years ago

I can't agree more. Speaking as someone who never enjoyed math at school, but now have an avid interest in it - the thing that converted me was building stuff with math. Much like you talk about.

When we learn woodworking we learn some basic tools and how they work, and then we build stuff. In my experience we don't typically get to build stuff with math. At least not the type of stuff that most people find remotely interesting.

I'm now building stuff with math all the time and enjoy it alot. The set of projects to choose from must be diverse though, some math isn't for everyone, just like not all kids will find working on building a baseball bat rewarding.

imvetri• 6 years ago

Floating points are absurd. It helps the scientific community and slows us down to solve interesting frictionless problems. Floating points take too much computation power. The amount of power we spend for floating point math and return on knowledge gained doesn't make sense.

A: Hey, but floating points are main thing in scientific problems.

Me: Ignore scientific problems, it doesn't amount to nothing and scientific problems doesn't bring a smile on face. By the time we solve, we would have lost our sense of self. And that's why it is meaningless.

A: What do you propose? Me: Not proposing anything. Just putting my thoughts out. Chase only whole numbers. Even try to cut out division. It is the root cause of all the problems.

Me: Addition makes sense, Multiplication makes sense. Subtraction doesn't make sense. It is used to check the correctness not for real math. You may call, just like multiplication is repeated addition, division is repeated subtraction. I'll stop right there. Division gives two answers - reminder and a quotient. Both are useless.

IF YOU HAD FUN, I'm glad.

domnomnom4• 6 years ago

We make circular cpus and do everything in base pi! Yes!

agumonkey• 6 years ago

I'd say wrong and wrong :)

respectfully of course.

School removes the pragmatic nature of math thus cannot even teach it as a tool because there's no concrete goal (worded problemes are not in situ enough).

The beauty may evade some souls, they're just not in tune with the subject at that time.

mechaniker2• 6 years ago

I'm not even convinced that schools teach math as a tool. Too few students are aware of the raw power that learning how to think mathematically and use mathematical knowledge as a problem-solving tool gives them. It provides new insight into fields as diverse as visual art and as applicable as physics. The latter of which only became the potent subject it is today after being mathematized.

Forget about about students, there are career programmers out there who only began their careers after a google search on whether they needed math to be a good programmer assured them otherwise. It's so ridiculous to me that I can't help but feel that some shadowy worldwide illuminati-esque force is intentionally sabotaging mathematical education to keep the populace unaware of how much more capable they could be when fully trained in mathematics. (Why would they do this? I don't know, but I think it has something to do with the pyramids.)

Jokes aside, I do wonder what the flaws in the mathematical education system are. Anecdotally, I've found that the truly skilled mathematics students rarely go into teaching careers, which is understandable. One of my favorite mathematics books, How to Solve It by George Polya, outlines a style of teaching that requires the teacher to be as much a confident performer as well as a skilled mathematician. It's worth saying that Polya was Hungarian, growing up at a time that mathematics teachers were expected to have serious postgraduate qualifications. Whilst it would be nice to go back to such rigorous standards, I can't imagine the resulting lack in qualified teachers such a policy would produce...

Koshkin2• 6 years ago

I'm not even convinced that schools teach math as a tool

V.I.Arnold has written an excellent rant about this: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

ur-whale2• 6 years ago

Very, very strongly disagree with you.

First, some of the most beautiful math stemmed from an attempt at solving an actual, practical problem, and the path the "discoverers" of such math followed to get there is often profoundly different from the way it's taught.

Your view, which I've been subjected to by most of my math teachers my entire curriculum is the exact reason why most people hate maths because it feels like a bunch of very complicated and hard to learn stuff that seems to serve no purpose whatsoever.

When you've been learning a bunch of very theoretical stuff for decades and finally find an actual application / use for it, my experience is that:

     - you do *then* see the beauty of it
     - you curse your math teachers and their ancestors to the seventh generation for not having shown you the practical application of the (typically over-generalized) theory he/she was force-feeding you back then.

ivansavz• 6 years ago

Wow I really like this document. As someone who really like mathematics but didn't get to do this as my major in undergrad, I'm missing out on so much, but also there is not enough time to start with undergrad books and do three years of basics... This document with all the clear statements at least allows me to see what is there to learn. Yes it's long, but compare that to dozens of the other books I would have to read to encounter these theorems, it's actually very short!

There is some useful "about this document" info near the end:

The motivation to try such a project came through teaching a course called Math E 320 at the Harvard extension school. This math-multi-disciplinary course is part of the “math for teaching program”, and tries to map out the major parts of mathematics and visit some selected placed on 12 continents. [...] A goal of this project is also to get back up to speed up to the level of a first year grad student (one forgets a lot of things over the years) and maybe pass the quals (with some luck). via http://people.math.harvard.edu/~knill/graphgeometry/papers/f...

jointpdf• 6 years ago

You might enjoy Oliver Knill’s lecture notes on undergrad math topics, which are similarly concise: http://people.math.harvard.edu/~knill/teach/index.html

ngcc_hk2• 6 years ago

Thanks. Same as the poster. But some “horrible” memory did meet me as I did math stat. Within 1/2 hour the lecturer tried to do a proof of central limit theory using an assumption that one can draw sphere to fill up a 3D space. Reading this does frighten me to a great extent at the same time appreciate what humanity have reached for doing such apparent “useless” things. Public goods are hard as it might look useless but unlike private goods it can be consumed and reuse many times. And some may find use of matrix in AI, number theory and geometry in encryption ... all because it is a public good that can be shared.

The trick whoonearth waste their life to create the first and develop later such “useless” things.

augustt• 6 years ago

This is awesome, although a bit in the old sense of the word, since it's scary to see entire courses I've taken distilled into one result that I probably couldn't even attempt to prove now.

aside: wonder why for group theory (51) Lagrange's theorem is only stated for cyclic subgroups, instead of any subgroup.

dan-robertson3• 6 years ago

I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the silly well-behaved ness conditions of the theorems you want apply, and every equation has one solution”.

ilyagr3• 6 years ago

every equation has one solution

So when you see the sun rising, cherish the moment, as it has never happened before and will never happen again.

hnarn2• 6 years ago

In a philosophical manner of speaking, that's true. You never step into the same river twice.

pmiller23• 6 years ago

The funny thing is, besides this one you picked out, pretty much all the rest of these either are continuity assumptions or follow from continuity. I think that's because physicists tend to dismiss discontinuities and singularities as "unphysical," but it does make good intuitive sense.

I suspect that "all equations have one solution" is a lot easier to satisfy when all functions are continuous, or otherwise nice, as well.

dan-robertson3• 6 years ago

The idea isn’t so much that the sun rises only once, but rather that if you were to solve a differential equation for how the sun’s position in the sky varies over time, there would only be one possible solution (provided that you had sufficient initial conditions). The idea is that your equation is supposed to be a good model for reality and in non-quantum physics, only one thing can happen so either your equation has one solution or it is a bad model for reality.

itcrowd• 6 years ago

Do you remember where you read it? It's a nice and funny formulation but a quick attempt at googling didn't bring this up for me.

ellis-bell• 6 years ago

i wonder if it's a folk theorem kind of thing? i heard something similar from one of my physics professors back in undergrad

dan-robertson3• 6 years ago

It’s basically just a joke. I got it from a professor of mine. But like all good jokes, there’s some element of truth in it: most problems in applied maths relate to something physical, and in (non-quantum) physics, something happens and only one thing happens. Therefore your problems are either bad models of reality or they have exactly one solution. It often turns out that even if your methods aren’t technically correct or properly justified, you can still reach that solution.

It is reminiscent of what Euler called “the universality of analysis” (note that over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry. In this case it means roughly the sort of mathematics a physicist or engineer might know well, so non-abstract algebra with integration and differentiation). He could do crazy things like solve combinatorics problems using infinite expansions of series in ways that didn’t really make sense (I think this example is well known but I can’t think what it is) but get to the right answer. A non-Eulerian example would be Dirac delta functions which aren’t functions at all and can be tricky to formalise but also just magically work within the framework of abstract symbol manipulation that is used in applied maths without significant modification (except when e.g. some fundamental property of your model of the universe depends on the value that the Heaviside step function takes at 0)

ur-whale2• 6 years ago

Sounds like many of my physics teachers.

madhadron• 6 years ago

The more precise statement of it is that physics and much of applied math is done in the topos of synthetic differential geometry, where by removing the law of excluded middle from the logic, you can basically make all of this true.

Well, except the Taylor series part. Numerical analysis is perfectly happy to take you to the cleaners even in ultrafinitist settings.

ogogmad2• 6 years ago

in applied maths ... all functions are continuous

I guess that's related to the fact that all computable functions from real numbers to real numbers are continuous. It seems reasonable that the laws of physics should be computable to a large extent otherwise we'd have been able to build a hypercomputer by now.

canjobear• 6 years ago

A step function is not computable?

staycoolboy• 6 years ago

I followed it for three pages, then recognized the next 3-5 pages, and then literally started to laugh at the explosion of exotic names and terms as I crossed page 30. I had no idea mathematics was so ginormous. Humbled.

Micoloth• 6 years ago

Absolutely relatable!

alderz• 6 years ago

In the subject of Artificial Intelligence it says:

Theorem: No AI will bother after hacking its own reward function.

and then goes on:

The picture [263] is that once the AI has ﬁgured out the philosophy of the “Dude” in the Cohen brothers movie Lebowski, also repeated mischiefs does not bother it and it “goes bowling”. Objections are brushed away with “Well, this is your, like, opinion, man”. Two examples of human super intelligent units who have succeeded to hack their own reward function are Alexander Grothendieck or Grigori Perelman.

Grothendieck abandoned the mathematical community after modernizing an entire field and eventually secluded himself in France. He is considered one of the greatest mathematicians https://en.wikipedia.org/wiki/Alexander_Grothendieck#Retirem...

asgard1024• 6 years ago

I think it just happened: https://news.ycombinator.com/item?id=24061653

GPT-3 just wrote an essay arguing that intelligence is doing nothing.

jmount• 6 years ago

That was fantastic. Often things titled "some fundamental" are a short superficial list, but this was some great deep stuff.

mazesc2• 6 years ago

Ctrl+F Gödel "not found".

Also proof theory is completely absent, like Gentzen's cut elimination theorem.

These are "fundamental" theorems of mathematics in the literal sense.

ogogmad2• 6 years ago

Goedel is there. See pages 8 to 9.

mazesc2• 6 years ago

Oops, thanks! Didn't try "oe".

JadeNB2• 6 years ago

It's great to see someone assemble some of their favourite theorems, but, if you have occasion to use these, make sure that you do so with caution.

Aside from some typos ('Perseval' for 'Parseval') and some curiosities (stating on p. 3 that the cardinalities of a set and its power set are different, which is true; but why not say that the latter is larger?), I noticed a wrong statement on p.7: assuming that to 'extend' a monoid to a group is to realise it as a submonoid of a larger monoid that is also a group, this cannot be done in general. (For example, the 'min-monoid' whose objects are the natural numbers, and in which the product of two natural numbers is their minimum, does not embed in a group because it obeys no cancellative law.) What is true is that, for every commutative monoid, there is a universal homomorphism from that monoid to a group, in the sense that all other such homomorphisms factor through it. (In my example, this universal homomorphism is the constant-0 homomorphism to the 0 group.)

generationP• 6 years ago

How useful is this? Not much that these things have in common beyond "results cited a lot".

Can't say that the subjects are chosen very precisely either -- the Fundamental Theorem of Algebra isn't actually a theorem of algebra; Tychonoff's theorem is fundamental only to the set-theoretical part of topology; the Fundamental Theorem of Counting is just a particular case of the "Fubini" interchange-of-summations formula, which I would call the real fundamental theorem of counting. The number of platonic solids is mostly a curiosity from a modern perspective; so are the transcendences of pi and e.

Also, the word "extended" in the Number systems section needs to be taken with some artistic license; the "extension" introduces new symbols (negatives) and new relations that can occasionally render some old elements identical (in the worst case, the whole monoid can collapse to a point). The most famous example of this is what happens if you divide by 0 (= extend the multiplicative monoid of real numbers to a group). This is not something the author should be blamed for; it's the only real error I've spotted at a quick skim, and it's rare for a collection as diverse as this to have this few errors. The real problem is: what's the point of such a survey if pretty much any of the results is given so little time and space that only those who already know it can understand it from the description?

guerrilla• 6 years ago

Can't say that the subjects are chosen very precisely either

The explanation of how they were chosen starts on page 80.

spappal• 6 years ago

Compact summaries are useful when revisiting something that was learnt before. Such a document might be more useful for mathematics than most subjects, since many have studied maths but stopped using it, and those teachings are generally still true and relevant.

The doc would be at least 20 % more useful to me if the pdf had a table of contents. Should be easy to include assuming that it was written with latex. Opinion: when writing a lengthy latex document, the extra 0.5 % of work required to add automated pdf metadata (table of contents, clickable references) has outsized usability effects.

I stumbled upon typos:

* "Basel problem formula": pi should be squared.

* The "more general" statement related to Bayes theorem lacks a right parenthesis.

dang• 6 years ago

Discussed at the time: https://news.ycombinator.com/item?id=19039525

mike00632• 6 years ago

If someone tells you they are fluent in 100 languages then you get a documents like this. The list starts out solid but then gets into areas where the author is clearly less versed.

Some Fundamental Theorems in Mathematics (2019) [pdf]

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