Goodness me, this brings back memories having taken the entrance exam back in 2009. I am glad that they have removed the "We do not guarantee the accuracy of the English translation of the original Japanese version of the exam" clause. It should be noted that this is only one "session" of a series of exams that you take in a single day. If I remember correctly there were two computer science sessions as well, followed by a presentation of your thesis in front of the faculty once your exams had been graded. After this you had to wait for a few weeks for the final result.
It should be noted that I don't find these exams to be particularly difficult, the main difficult lies in the fact that the number of potential topics are drawn from the whole University of Tokyo undergraduate curriculum, so if you studied as an undergraduate at a different school or country -- like me -- you are at a significant disadvantage. What I did was to look at ten years or so of exams to get some statistics on what was likely to be on the exam, then locked myself in my room for the whole of January to cover portions that I had not studied before -- like network protocol specifics. In the end, it worked out for me, but I got somewhat lucky in that there were two statistics questions that I could breeze through to cover for my weaker calculus skills given my old school's focus on discrete mathematics.
Problem 1.1 seems to be within the realm of casual curiosity for somebody who hasn't even entered undergraduate school. It's basically "I know what a matrix is, and how a matrix multiplies with a vector".
The general structure of the problems is that they start out with easy questions and get progressively more difficult. If you don't get a single "last question" right, the committee will most likely tear you to pieces. At least this was the common understanding back when I was at Todai.
Fun random fact, back in 2009 Todai had their own "brand" of toilet paper made from recycled paper -- old archived exams that is. I think that I still have a roll stashed away in a box somewhere. Using it did give you a bit of a weird feeling knowing the anguish old students must have gone through.
> Fun random fact, back in 2009 Todai had their own "brand" of toilet paper made from recycled paper -- old archived exams that is. I think that I still have a roll stashed away in a box somewhere. Using it did give you a bit of a weird feeling knowing the anguish old students must have gone through.
And this is admission to Graduate School, so the equivalent of a M.Sc. or Phd in the US, right? (i.e. Is the term graduate school the same or is this entrance to college?)
It seems reasonable to me. Certainly, even if you don't get all of the questions correctly, this exam tells you more about the proficiency of a candidate than a standard graduate school exam like the GRE (in the US) does. Especially if a student can explain some reasoning for incomplete problems.
I am not sure when this would even be desirable. If a student fails a question regarding a definition (simple, memorisation) and succeeds in proving the complexity of a novel algorithm (complex, innovation). Why would I emphasise the former? Besides, there are your grades and the presentation/interview as well -- at least for the PhD.
> And this is admission to Graduate School, so the equivalent of a M.Sc. or Phd in the US, right? (i.e. Is the term graduate school the same or is this entrance to college?)
Yes, that is correct. If I remember correctly, they use the same exam for the MSc and PhD.
Only vaguely related, but I've made it all the way through high school (doing the 2 hardest of three maths subjects) and a software development degree without ever having a lesson on Matrices.
There really seems to be a hole in the system somewhere.
Yeah, matrices are also very useful. Instead of needing the "Tribonacci" equation definition, you can approximate or exactly match a large number of equations and operations with matrices.
Anyway, there are many things missing. I feel that An Introduction to Law and Legal Reasoning should be standard middle-school reading; but alas, civics has not been part of the curriculum in my life time.
I mean, yeah and I never use an array ints since school, but I still work as a developer. However, I'm sure some folks in other areas certainly do.
I'm not sure anymore as I get older that it is really a hole in a negative way. Like yes, it is a hole, but you know, you can't learn everything in your finite time.
> I am glad that they have removed the "We do not guarantee the accuracy of the English translation of the original Japanese version of the exam" clause.
What languages do they run the exam in? Do they accept a lot of foreign students / students for which Japanese proficiency is a barrier for writing the test?
Japanese and English are the only languages for the exams. Unless the curriculum has changed radically I would not recommend doing a MSc without already being proficient in Japanese. It would be far better to enter as an undergraduate on the MEXT fellowship that includes a year of full-time studies learning Japanese before you proceed to enter a university. Only a handful out of hundreds of classes are held in English at a graduate level and I suspect that the situation is even worse as an undergraduate. PhD studies should be fine, most schools have Japanese classes and as long as your supervisor is fluent you will most likely have a good time.
But hey, I am looking for a tenure track position at a decent university back in Japan over the next few years. There is a will to become more international and hopefully I can find a way to contribute towards that goal. Still, we should not fool ourselves and say that things are fine as they currently are -- know what you are getting yourself into.
Note that the linked paper is a mathematics exam that applies to the entire faculty (not just CS). In addition to that exam, one must also take a CS-specific exam. It seems that the Summer and Winter exams are rather different:
Edit: Looking at the exam guide [0], it seems that it's only if you want to sit the exam in the Summer that you must also take the separate maths exam; the Winter CS paper already includes maths questions so there is no separate maths exam.
I just looked at a few of the problems and these are really well designed. From what I've seen, there's always a clear progression and a way to get partial credit. Plus the problems themselves are interesting and non-obvious.
So honestly I think that this is a great exam. The only problem I see is that 5 hours of exam work in a single day is a bit much, even with a one hour break in the middle. It's probably a logistical problem though, with many prospective students being in town just to take this exam...
Whoops, I didn't do too much research. I actually thought I linked to that CS-specific exam. I think these look rather hard! (I'm even a last year Bachelor student in Math/CS.)
I'm glad I'm not the only one who thinks this looks difficult. As someone who studied pure undergraduate CS (the course only included a couple of relatively basic maths units) in the UK, I feel I'd have to do a fair amount of self-study before attempting this exam (even the CS-specific exams are quite tough!).
Why is it that I find these questions always posed in a way that is MEANT TO make me NOT understand them? Granted, math is not my topic. However, I am interested and want to learn.
I don't mean to be a dick, but I just simply find it frustrating that the goal of most mathematicians is to pose some question/conjecture in as few sentences as humanly possible. It really grinds my gears how interesting problems are always abstracted behind confusing vertical specific language. Even when reading a Wikipedia article about some cool problem, you get hit in the face with some abstraction that makes most problems seem far more complicated than they actually are.
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.
Just my 2 cents!
Ps. I understand that this is an M.SC. exam. I am talking about math in general, not this pdf!
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!
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.
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:
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.
>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.
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.
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?
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.
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.
> 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.
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.
> 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").
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
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.
"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.
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.
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.
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.
> 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).
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.
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.
"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.
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.
Some number theory, linear algebra, calculus of variations, combinatorics, etc. From the applied math side of things, it isn't terribly deep. It's actually right about what I would expect from an applied math student who was good at the undergraduate level and ready to study at a competitive university.
For general admissions for any discipline though, I'll agre it's probably a bit rough for non applied math grads. At least it doesn't include any analysis, abstract algebra (groups, rings, fields, etc.), or topology. Which would be murder for non-math majors.
Yep. My Math is undergraduate Electrical Engineering and Graduate Comp Sci. Some of the stuff I'd have to bone up on because I haven't done it in a few decades, but all the problems were familiar and seemed tractable---seemed appropriate for a recent maths undergraduate. (That said, I'd probably bounce hard off of the time limit, and it's been an awful long time since I did any maths where I wasn't allowed to pull a book down off a shelf.)
I missed the "graduate" part of "graduate school" and was temporarily impressed that Japanese high school students would have to learn eigenvalues/vectors. The second question about rotations seemed right out of an AP calculus class, until I saw the differential equations.
> I missed the "graduate" part of "graduate school" and was temporarily impressed that Japanese high school students would have to learn eigenvalues/vectors.
I had the same experience, followed by a "No wonder Sony produced such great products!" moment.
Yeah it looks more like the math component of an engineering exam. If people look at this and think "oh cool, I can do well in this, I will become a mathematician," will be in for a rude shock when they take Grad School Real Analysis 1.
There isn't anything on that exam that didn't get covered in my undergrad Physics program. From the title I thought it was an exam for entering a Mathematics graduate program rather than a Mathematics test for graduate programs, so I was surprised that the questions were so "straightforward".
Yep. I was going to say the same thing, seems like a pretty solid test. Series and linear algebra, calculus, probability sounds like just about everything I saw in my Mechanical Engineering degree.
I believe I could have done all of these questions 12-16 years ago. Shockingly I can't do any of them anymore without refreshing myself for what they are even asking in some cases.
I think that if you have just passed your undergrad degree, these questions are indeed not _that_ hard, because the stuff is still quite fresh in your mind.
If you have left university for some time, chances are you did not do any math in that time inbetween, if you are a software engineer. Software engineering requires logical thinking in a different way than mathematics, so we do not really keep our mathematics skills up to date.
Looking at document, I can remember touching the necessary mathematics to solve this, but not having done any mathematics of the kind makes me too rusty in mathematics to be able to solve it.
With a bit of revision, someone having seen this somewhere in undergrad will probably be able to solve it.
An undergraduate degree in maths, sure. But if your degree is in computer science, I think it's likely you'd never come across any of this during the course. Some universities may teach it, but there are probably plenty that don't.
Any self respecting university that teaches a formal computer science degree would teach this level of mathematics. Mine went beyond and happily forced students to learn introductory quantum mechanics as well, which annoyed the CS students to no end.
Same story here, as computer science students we got a hefty dose of mathematics, and was in some respects close to what students of physics and mathematics got.
For example, when compsci students get Linear Algebra in their second year, they are getting the same lectures as Physics students in their first year. (Same room, same exam).
We also had shared physics courses in the first year.
I believe that physics, mathematics, and computer science should have some shared courses, not just having compsci students take mathematics. For example, for programming, we were a joined group of mathematics,physics and compsci students in the first year.
They are all courses related with logical reasoning anyway, so a compsci student should not hate mathematics and physics that much. Most of the complaints I noticed were the physics students complaining about _programming_ rather than the compsci students complaining about physics.
I agree; my undergraduate course (not in the US) wasn't even called Computer Science, it was "Software Engineering", but we definitively learned enough math to pass that test easily after the second year.
MIT, for example, doesn't require their CS students to take a differential equations class. So you're wrong in that respect. The rest is certainly doable though.
Though if you've taken a course on variational calculus then question 2 is completely routine and you're at a great advantage relative to someone who hasn't taken such a course.
I'd love to know where you went to school and earned an ABET accredited CS degree (assuming US) and didn't take calculus, linear algebra, or have any coursework in probability and statistics.
I studied in the UK. I think our systems are rather different because a lot of that stuff is taught in secondary school, so you're expected to know it before starting uni.
Which is all well and good if you stay on at secondary school (or attend a sixth form college) until you're 18, then go straight on to uni. But that doesn't apply to everyone (some people leave school at 16, or take a break before going to uni). So I can imagine that a lot of CS graduates here in the UK would have studied these aspects of maths quite some time back (and therefore can't remember them very well) or not have studied them at all.
That makes sense, well, that it happened, at least. Typically those topics are covered in US schools before university as well, just they are covered in more depth and rigor again in university and with the applications of those topics to CS as well.
In the US we have this body called ABET which helps define a standard engineering curriculm.
One of my close friends studied in the UK. University of bedfordshire, which was also a computer science degree.
Compared to me (Belgian university), he indeed barely got any mathematics, but I always thought that was just because of the university.
I don't see anyone saying they are "really simple". But they are indeed at the expected level of knowledge for somebody who has had completed few advanced math courses. I did my bachelor in electronics engineering and all these problems look familiar and could probably solve them after refreshing my dusty math skills.
Assuming it is directed towards admission into a Graduate Mathematics / Engineering Program, it looks pretty average to me.
For a student who is in-form (e.g. a fresh graduate or a mature student who has taken 2-3 advanced calculus and algebra+stats courses as refresher), they should be able to perform.
And I was angry at my uni,that my cs curriculum contained almost no maths. After reading this paper I realized that I am glad I didn't have to study this. Some calculus would have been useful though, as I am in a machine learning field now
That's really shocking to hear. When I was in university CS was almost exclusively math. At the time that was the delineation between CS and computer engineering. One was mostly theoretical expressed through equations. The other was just practical application of EE and software development. And the finance college had the MIS program. They were using java.
In my personal experience studying undergraduate CS, apart from the dedicated maths modules, we did use mathematical notation for lambda expressions, regular expressions, finite automata, Turing machines and the like. We didn't cover calculus or probability though (I think you were expected to learn that in secondary school). I vaguely recall that we touched on matrices and eigenvalues/vectors. But essentially, we were only taught the maths that we needed to know for CS. And of course there were units that didn't use much maths at all, such as systems development and human-computer interaction.
Your post is equally surprising given that for Waterloo, our undergrad ECE curriculums covered all this material. Arguably, we covered significantly more math (albeit in Electrical Engineering, but there was significant overlap with Computer) than what the CS students did.
Maybe they teach advanced algebra during graduate studies, so you are not expected to know it at the entrance exams? At least in my experience that was the case -- I've only learned group and field theory while doing my CS master.
Todai has a strong traditional science profile and traditionally science deals mainly with calculus and statistics. This leads to less of an emphasis on algebra for these exams since they are based on the undergraduate curriculum. It is regrettable, especially for those like myself that came from a school that emphasised algebra and theoretical computer science and only taught advanced calculus at a graduate level.
So I never really got too deep into math, once I decided I wasn't going into physics as a career I backed off, but this throws me back into 12th grade AP Physics exams. I'll never forget refusing to take a bad grade on a matrix problem(?) and skipping the next few classes to spend hours trying to do it the long way... I didn't get credit for that question.
That being said, now that I'm on the other side of college and in a steady career, I'm really interested in diving deeper into math. Does anyone have any recommendations on resources for learning from Calculus on? I could probably do with a Calc refresher as well. Thanks!
For a grad school entrance exam this isn't too difficult. It's first and second-year maths stuff for a decent CS degree that doesn't skip out on the basics.
Interesting that this is entirely in English. Is this the English language version for foreign students, or are Japanese masters students expected to be fluent in English?
That's interesting. I know Japanese has multiple scripts. Problems 1 and 3 appear to be in a much simpler script that looks less dense. Problem 2 is using whatever script that looks closer to Chinese and is full of much more complicated characters. Is there a reason for that? Did Japan import whatever concept question 2 is about from China and developed the math for the other questions independently?
As someone with moderate Japanese literacy, I don't really see the distinction between the problems that you're referencing. All problems here are using the same mix of phonetic and logographic characters, which is typical of modern written Japanese.
>Did Japan import whatever concept question 2 is about from China and developed the math for the other questions independently?
Very unlikely. Japanese concepts expressed by the borrowed Chinese logographic script (known locally as kanji) are frequently quite different than the meaning that would be expressed in Chinese.
Additionally, choosing to write a text with relatively dense usage of kanji characters doesn't necessarily imply anything meaningful. For one, the swap between phonetic script and kanji often plays the role that the empty 'space' character plays in western languages. Sometimes the concepts being discussed include words that are homophones/homographs that are difficult to distinguish between in phonetic script, but are easy to distinguish with kanji. For instance, 'hashi' = chopsticks or bridge depending on emphasis. Aside from use of context it's not necessarily clear what you mean if you spell it out, but there are two distinct kanji for either 'chopsticks' or 'bridge' that are usually known, so it's easier to use those to ward off ambiguity. In other cases, choices between use of phonetic scripts or kanji simply comes down to nothing more than convention.
The Japanese language uses three scripts: kanji, hiragana and katakana. Both kanji (the complicated-looking script which is imported from Chinese) and hiragana (the simpler-looking alphabet unique to the Japanese language) are used in all three problems. Katakana — used for (normally English) loan words — appears a few times in the first two problems and look quite simple with mostly sharp, straight lines.
I think this exam paper represents how the Japanese language is normally used, with kanji for the main nouns and verbs, hiragana to modify and conjugate, and the occasional loan word in katakana (for tribonacci, rank, vector and Euler-Lagrange) as necessary. I don't think the second problem has particularly more kanji than the other problems.
Japanese uses three scripts, and each one coincidentally appears more prominently in one of the three problems than in the others:
1. Katakana is a sparse, alphabetic script used to transcribe sounds. Problem 1 looks sparse because much of the space is used to phonetically spell out the word "tribonacci".
2. Kanji is a dense, glyphic script used to express ideas. Problem 2 looks dense because big words like "surface area" and "twice differentiable" are translated to their native equivalents.
3. Hiragana is a sparse, alphabetic script used to add grammatical structure. Problem 3 looks sparse because of the high number of prepositions and auxiliary words like "in", "be", and "of".
I understand that Japanese has 3 fonts, what I am pointing out here is that the second question is using what appears to be the more complicated font or it just happens to be using characters that require many more strokes than in the other two questions.
Compare the average number of strokes per character between question 2 and the other questions to see what I mean.
My question is whether this is due to some historical or cultural reason, or just happens to be coincidence
All three questions use a combination of kanji and hiragana. To me, the average number of strokes per character in all three questions look about the same. The second question does use quite a lot of kanji with lots of strokes, but so do the other questions. If I'm just not seeing it and there are indeed more complicated kanji in the second question, then it's probably mostly a coincidence, although complicated concepts are often represented by complicated kanji.
All three problems descriptions use all three scripts. I think problem 2 uses a heavier font, for no clear reason. The English version is the same. Maybe the problem maker submitted it as bitmaps rather than Latex source.
Not sure about Japan but a very large number of computer scientists also speak English. Due to the youngness of the field, the most important material hasn't been translated from its source language, which is English :)
No they are not. From my experience with Japanese PhD students, most can express themselves reasonably well in written form, but few can speak fluently.
This is really really simple btw, for a grad school entrance exam. Anyone who has successfully studied mathematics for a year should be able to score 100% on this exam.
I guess it's pretty feasible to take all of those in a year, especially if mathematics is your area of study. This wasn't the case in engineering in my experience.
In any case, this wouldn't be so bad with a 4 year degree. Not really any harder than other PhD candidacy exams I've looked at
Oh wait, is it for a PhD program? Then it is pretty easy indeed. Although I would still possibly choke on proving that the surface area equals that formula.
Wow you sure are smart why don't you write an email to the faculty of the highly ranked university that wrote this test to let them know how easy it is? Oh wait, you didn't even realize that this is just one of many tests you take on the examination day and you also must orally defend your undergrad thesis to a panel. But go on, continue telling us how smart you are
I would have been able to do this with no effort 20 years ago, but nowadays I would be totally incapable of solving even one of the problem. Damn I feel the years.
Hah! I gave 3.1 - 3.4 as an assignment to second-year CompSci students literally three weeks ago. Although my version was to determine the values experimentally for n<=8 bits and then estimate values for n up to 64. Bonus marks awarded for a mathematical proof of the pattern (which isn't really all that bad if you apply probability to patterns of two bits).
Solving 3.1 I think the solution is binomial(n1+r-1,r-1) where n1=n-r, but I obtained that formula by calculating the formula for r=1, then r=2, and the relation f(r,n) = sum(f(r-1,n-i),i,0,n) and using maxima with simpsum and factor to obtain a simplified expression whose form suggest the general rule, but I don't see a simpler way.
I'm a CS grad who is fairly strong with maths but never really did learn to write a proper proof. All of these questions I would have been able to "solve", but the proofs would have been very casual. What would be a valid answer it 2.3 where it basically asks you to integrate an expression of F after substituting y'=dx/dy?
"2.3" is a bit ambiguous as a label; do you mean part 3 of question 2, or the part of question 2 in which equation 2.3 occurs?
If you mean part 2 of question 2: my answer would be "dF/dx = del F/del y dy/dx + del F/del y' dy'/dx; but we know del F/del y = d/dx(del F/del y'), so dF/dx = d/dx(del F/del y') dy/dx + del F/del y' dy'/dx, which by the product rule is d/dx(y' del F/del y'). Now integrate both sides."
If you mean part 3 of question 2: just substitute F(y, y') from (2.2) into (2.4).
I think a major problem with mathematical language is lack of discoverability. If I don't understand a specific symbol or notation, there is no way to click "go to definition" or look it up on Google.
Furthermore while mathematicians pride themselves on rigor, I would bet that a lot of proofs require the reader to fill in incomplete steps and/or contain notational mistakes. Just try to imagine what kind of code you would end up with if you did not have access to a compiler and relied only on your logical reasoning for writing correct code. Even if the code was reviewed by other peers, who themselves do not have access to a compiler, they would miss some mistakes because, being familiar with the subject, their brain would fill in the correct meanings.
Well, is f really a function of x and y when only one of them can be chosen? Seems like an abuse of notation that introduces unnecessary special cases. Abstractions (here: a function) should be simple and well-defined, not overloaded by natural language.
Strictly, if one knows in advance that y is a function of x, then one should write f(x, y(x)) to make the dependency clear. This just gets unwieldy, though, as anyone who's ever programmed in Mathematica can tell you. However, what if a priori we only know that f is a function of two variables, and then we decide to evaluate it along a path {(x, y(x)): x in reals}? Under your scheme, we'd have to define a new function g(x) = f(x, y(x)) and reason only about g, which is even more painful, not least because maybe g(x) only depends on x through y and could be written more succinctly as g(y(x)).
The current notation is simply more flexible and easier to use, and it remains unambiguous because we have access to these two operators $\dfrac{\partial}{\partial x}$ and $\dfrac{d}{dx}$.
if a function depends only on multiple variables then those are partial derivatives, if the function depends only on one variable then there is only one derivative d/dx
I get that, but ultimately, why does the notation distinguish between the two classes "function with zero arguments held constant" and "function with n >= 1 arguments held constant"? It seems somewhat of an arbitrary distinction, when in fact the value of n is always clear from the context.
I think that the difference is from historical reasons, anyway when if you introduce in school partial derivatives with functions of only one variable that is confusing because here partial is not partial but global.
Yes, the graduate school paper exam of Todai is not difficult compared to the undergraduate entrance exam for that age. The graduate one only requires the very basics to do research. Requirements are different.
I have not truly figured out the difference between permutations and combinations. I know about permutations having 'order' but when I see real questions I got lost. Could someone help this 'lost' soul?
In problem 2, the Euler Lagrange equation imply that we are looking for the function y(x) such as the surface is minimal, so the solution is a cylinder, ie y(x)=constant=c=2.
The solution to the Euler equation gives the minimal surface with is obtained when y(x) is a catenoid, y=c cosh(x/c) here c satisfies 2=c cosh(1/c). The Euler equation gives the minimal distance when the function F is the length of the arc.
I don't know about other countries, but here in the UK, most universities don't require an entrance exam to study computer science (they go by the grades you leave secondary school / sixth form with).
The only two universities I'm aware of that have entrance exams for CompSci are Oxford and Cambridge (because they have to be different):
It should be noted that I don't find these exams to be particularly difficult, the main difficult lies in the fact that the number of potential topics are drawn from the whole University of Tokyo undergraduate curriculum, so if you studied as an undergraduate at a different school or country -- like me -- you are at a significant disadvantage. What I did was to look at ten years or so of exams to get some statistics on what was likely to be on the exam, then locked myself in my room for the whole of January to cover portions that I had not studied before -- like network protocol specifics. In the end, it worked out for me, but I got somewhat lucky in that there were two statistics questions that I could breeze through to cover for my weaker calculus skills given my old school's focus on discrete mathematics.