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...
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.
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.)
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”.
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.
Well there are 3 fragments related to this, one likely fake. It is a way to wake you up in his crazy paradox.
There are books and articles and may be one can start with the key phrase What is same river
And the hidden issue is you. If you think it is unique every time frame ... what is same river and same you. Can One same you even step into the River
And if step into it and swim in parallel is the molecules mostly the same ...
Back to What is river? If it is keep on changing as any river that is not changing and moving is not a river, the sameness if you accept then you can step into the same river as many time as you like. But that is wrong per the statement. Hence what is “same river”.
Indeed. But the scientific approach is different -- it's all about finding patterns that do repeat and making predictions.
The original comment's "theorem" is funny, but unless you are doing a homework problem, you better have a good intuition for when to take a second look at a seemingly simple situation.
> the scientific approach is different -- it's all about finding patterns that do repeat and making predictions
Sure, but if you're scientifically looking to answer "where and when will the sun rise" you're only going to collect enough variables to answer that question, and within an acceptable margin of error, right? If you can measure with greater accuracy and collect more data, then you would probably realize that every sunrise is not strictly identical.
We're splitting hairs at this point but I wonder if the comment I replied to that stated that "it has never happened before and will never happen again" can't be argued to be actually true. In a philosophical sense it's more obvious, but in a scientific sense, the more precision you get in your analysis of a sunrise, the more data you would get that differentiates it from other sunrises, no? After all, our solar system isn't closed and constant and there are minor changes not only in smaller factors like weather on Earth, but also larger factors like the orbit of the earth and the drift of the different planetary bodies.
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.
That's mostly true -- however, physicists devote a lot of time to studying phase transitions and characterizing the kinds of discontinuities that are possible etc in that context.
Ah, yes, I forgot about phase transitions. Those are actually interesting discontinuities. It seems to me the uninteresting ones are the ones that have roughly the same behavior all around them (e.g. poles of complex functions are not that interesting).
Would you say that's an accurate characterization?
sorry don't have much more insight here. I think the goal is to generally make any approximation that isn't relevant to the physical behavior one is trying to model. if a pole sitting somewhere isn't related to the phenomenon you're trying to explore, ignore it!
Great point. There are a ton of good models that break down in certain regions (e.g. quantum mechanics and general relativity at large and small physical scales, respectively).
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)
> 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.
In fact, yes! This is counterintuitive if you think of the floats as a model for the reals, but in fact the floats give the wrong intuition, being (a finite subset of) rationals. [1] details this and refers to two related articles (particularly [2], which I find a lighter read). This fact is also remarked on in [3] (by example of the signum function being uncomputable).
For the particular example of a step function, consider the function f which is 1 on the positive reals (>=0) and 0 otherwise. What representation of the real numbers would you choose for the computation of f(0) to terminate in a finite amount of time? It cannot be the binary representation, as those are infinite. Your algorithm, terminating in a finite amount of time, will have checked only a finite number N of binary digits of the argument, and so I can choose x = 0.0{..N..}01 and obtain f(0) = f(x) by this algorithm, which is incorrect. You can choose the Cauchy sequence representation, or the Dedekind cut one, but this problem will persist (and [1] proves this in general).
You can "cheat" by saying that the leading two bits will store the sign of the number: 00 for 0, 01 for positive numbers and 10 for negative ones. But then suddenly arithmetic operations are not computable (see comments on [2])!
If I gave you 0.0000000... as input, you would have to scan forever to find out whether I put in a nonzero digit. Scanning to a partial depth would not result in partial accuracy, because the step function is discontinuous.
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.
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.
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.
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.
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.
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.
I mean, that's mostly a nonsense sentiment, but I really enjoyed the allegory so I will not hold it against you. The widest ranging theorem in mathematics was created by a bunch of greek cultists who believed that the natural numbers were at the heart of existence. The most important mathematical textbook, one that was taught almost exclusively for over a millenia in the west, was steeped in the traditions of said cult. That tradition is still there and has, for the most part, always been there (although, as an extremely applied mathematician, I find it's practioners to be alien).
Plus, tell me: how do you learn proofs? They aren't statements in a textbook that you just learn and recite like poetry. You have to learn to see them, to understand how all those moving parts fit together. When done right, teaching proofs is an exercise in illumination. I find it is more enjoyable all around when performed as a creative, exploratory exercise: give the students the axioms, set them lose on a conjecture.
Of course, for that you need a) Teachers who can serve as guides in the mathematical worlds and b) Students who aren't culturally predisposed to dislike mathematics.
Proofs are only an example of what grade school teachers don't know. They are obviously missing all of the modern applications as well. Whether you want to call those applications "math" or something more specific ("engineering", "cryptography", etc) is a matter of taste I guess. The point still stands that kids are learning hardly anything of theory OR practice in their grade schools, and this is due to the teachers not knowing those things either.
We can talk about why the teachers don't know these things, but I think the real issue is American culture still sees math and science as something for pitiful, nerd-virgin dorks. You can try to sell math as this sexy, dangerous thing you use to win wars, but ultimately you will have to reckon with the image that most Americans have of mathematicians and scientists, which is illustrated in shows like the Big Bang Theory: they're ridiculous laughing stocks who use science as a kind of cope for not being socially successful.
> So the way to get kids to like Math is by forcing them to learn... proofs?
I mean, there are definitely issues with the whole "forcing them to learn" part, but assuming we're given the option to change the content of the math curriculum but not to fundamentally change the education system, then yes, absolutely. You don't need a crazy level of rigor, of course, but I think learning simple proof techniques could make students experience mathematics more like solving puzzles than just completing calculations.
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.
(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?
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.
While I can relate to your complaint that some steps were a bit too simple to include... The point of all those steps was precisely to show that you hadn't memorized the results. At the risk of overly simplifying high school maths (it's hard if it's new) I would say that in high school the math is quite basic. You could pretty much remember the first step, not get most of it in between, and still produce the right answer because you remember the answer format. It's just not feasible to have a individual discussion with every student to see if they really understood the material. Requiring students to include the steps in between is a way to prevent memorization, although maybe not the best way for each student.
It's good that the same thinking served you well later on though!
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.
Based on the username and its recent birth, you created this account just to paste this comment. What does it mean? (I think that you are commenting sarcastically on the way math is taught, but that's just a guess.)
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...
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.
This was downvoted, but I think it's an interesting argument, in this context and in general.
Should we be rooting for things that make supply of labor in our field more plentiful? Wouldn't it be in our self-interest to prevent as many people as possible from learning math and computer science? One could argue it wouldn't be in the interest of humans as a species, but I'm skeptical that the marginal loss to each of us is greater than the gain.
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!
He gets deep into some stuff, so read the parts that interest you. I also highly recommend his book, the problems of philosophy. It's really what I think math is all about.
Yes, definitely. Albeit a very brief introduction. You can probably skip chapter 13 and 14. Although, arguably chapter 14 is the “culmination” of the book, the Cantor-Bernstein-Schroder theorem. It would be fun to do it anyway, since you would fully have the mathematical tools to understand it.
So why do I say it would be a good introduction to discrete math? Well, a lot of the examples and problems are discrete math type problems. Counting, graph theory, etc. Your life will be so much easier.
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.
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.
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.
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....
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.
Logic goes beyond truth tables: it's about being able to see the simplicity and symmetry in objects. That's why it comes from the word /logos/ which means "prevening order" (see Gospel of John).
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.
Did Gene Roddenberry research biographies of logical people such as physicists, number theorists, and other "heavy researchers" (for a lack of a better word) before he wrote or specified the character? If not, then he was very ignorant of the personality type of heavy researchers.
It seems that Spock and Surak were meant to be parodies of stoic philosophers such as Marcus Aurelius. The fact that Surak is listed as a "great mind" alongside Newton and Einstein in the series reflects Roddenberry's high view of stoicism, so perhaps I was too extreme in the judgment of "minstrel character," seeing that he, Surak, and T'Pau were meant to reflect his favorable judgments of Stoicism, but it is still ignorant because
1. Nobody considers the stoics to be legendary thinkers in the real world. Nobody says "Aurelius, Einstein, and Newton."
2. Heavy researchers aren't at all like Spock but he does have a resemblance to people with Aspergers
3. How heavy researchers feel emotions is not at all like stoics. Strongly externally-oriented thinkers feel emotions and see emotional events in a specific way and "like Aurellius" isn't it.
Probably nobody, except Douglas Hofstadter, says "Zeno, Einstein, and Newton" but there I can give one example of a notable thinker who might say that, so probably there are some other people in our very populous planet that say "Aurelius, Einstein, and Newton"
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
But im talking children ... im not against the abstract, i actually infatuated with category theory.
The method, the context id wrong.
Its physics and haskell that made me fall inlove. People like me need context, beyond the aesthetic - math TEACHING is lacking, its not a criticism of math itself.
And also thank you for the balanced reply, i was a bit emotional as i wrote that reply. Ill reread and rethink the parent. Maybe i missed something
I feel like in Today's day and age it's about understanding the fundamentals of a system and using software tools that automate the heavy math to allow for creative problem solving quickly and dynamically, that wouldn't be possible on paper only or through manual computation only. I'd rather give the kid advanced tools to analyze and improve on the structure than force them to learn the already abstracted ulta-fundamental and impractical math that runs those tools...
Look at how he started, and he wasnt super human. I think youre elivating him to a place he isnt in. I honestly think you him less credit by doing so. He was one of us, which makes his feats even more impressive.
Feynmann ENJOYED working on puzzles. He started with radios, and fiddled with electricity... and learned hid math as a kid from bools written for „working men“ (which i read and workd through) .. their language is not what you see in textbooks.
He also invented his own notation (talking pre-uni, not physics notation hes also known for) and says he finally switched because he realized he wanted people to understand him, and partially due to conventions.
He played, he didnt get taught, he explored. He learned math to pursue his passion, not vice versa
xkcd is a webcomic about logic (at least it was in its initial days) and it has gained an extremely passive-aggressive (e.g. explainxkcd) hatedom that makes the Barney the Dinosaur hatedom look outright amiable by comparison.
Why does xkcd have such a strong hatedom? It's not because they love logic, lucidity, and symmetry.
> 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.
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.
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?
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.
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.
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.
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...