University of Tokyo Graduate School Entrance Exam – Mathematics (2016) [pdf]

That's delectably inhumane.

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?)

I could kinda imagine what you say happening in the US, but calling it "maths" instead of "math" implies you're elsewhere.

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.

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.

First paragraph is:

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

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

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.

Somehow, this infuriates people.

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

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

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

Not going to happen.

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.

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

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

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.

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!

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.

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.

> 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").

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.

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.

>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.

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.

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

"del F/del y = d/dx(del F/del y')" is just the Euler-Lagrange equation.