For number theory, there's a few things you could pick up fairly easily, like understanding what 1 and 3 mod 4 primes are, but there's no way you could do the homework in 1 weekend unless you've aced real analysis, and then maybe you could finish 10-20% of the homework (proofs) in a book in a weekend or so.
Many engineers I've met can't even handle using proof by induction to solve a proof.
Rather, writing out any formal proof is monotonous, tedious, lengthy, and prone to minor screw-ups, and to someone with an engineering mindset, not useful in any practical sense.
One thing: How does acing real analysis help with number theory?
Secondly, number theory is not calculus; it requires you to be familiar with college-level math courses (e.g. abstract algebra) before you can even start making sense of it. It's certainly plausible that an engineer might pursue it as a hobby, but it's rather unusual, and certainly not for the faint of hear (it requires major commitment in terms of patience and time).
The same goes true for diff. geometry, although it may be more approachable (e.g. many undegraduate math syllabi have it, while number theory is usually a postgraduate course).
Wait, no, people study those things all the time. They're difficult, but maybe he has unknown prior background in math, or is gifted in that regard.
It's not as though he achieved something that no human ever has before. It's a reasonably routine though difficult task - akin to perhaps completing a marathon.
Sorry, but reading a textbook like a novel is not studying. I've met a fair number of mathematicians but I have yet to see one that can tackle a postgraduate textbook in one sitting (unless they are already intimately familiar with the subject).
A marathon is usually run in about four and a half hours. The best marathon runners in the world (i.e. current record holders) ran one in about 2 hours.
The university that I'm at has about 45-48 hours of classroom instruction per semester; you're expected to spend between two and three times that preparing; that's an assumption of between 135 and 192 hours of work per semester. I find that for my upper level mathematics courses (first year graduate), I spend about 2-3 hours outside of class working on problem sets and reading the material. This is also, as far as I can tell, about how much the other students in my courses are working. My understanding is that this is the expectation across most high tier universities and math programs.
If we assume that the average marathon runner is roughly equivalent to a normal mathematics student in terms of experience, saying that you learned (and completely understood the implications of) an entire semester of mathematics in a day (16-20 hours) is equivalent to claiming you are able to run a marathon in 40 minutes. For the record, that's a rate of about 1 mile every minute and a half.
This is an imperfect metaphor, since math has the same benefits that CS has of being able to absorb material faster the more you already know. On the other hand, my experience (and what most professors I have asked also say) is that most hard math requires a sort of 'processing' time to really start to get it, so the actual case might be worse: I don't think you'd actually learn much math if you just spent 135 consecutive hours (with sleep as required) working on learning math.
Suffice to say: I don't think what he claimed to do is within the realm of reason.
Personal bio, as to why I'm a semi-reputable source: I'm currently both finishing an undergraduate degree in mathematics and a masters in math at highly ranked public university (in general and in math). Most of my time is currently spent learning math, either in class or self taught (for a seminar). The professors I interact with regularly are about as mathematically talented as it is possible to get.
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The above being said:
I would, however, say that someone with a good handle on multivariable calculus and differential equations, along with a really good background in math, could learn a decent amount of applied differential geometry in a weekend. Similarly, you could learn/memorize a lot of the results of number theory in a weekend.
Both of the above could give you the illusion of having learned all of a subject without building a deep understanding of it. Both are also in line with the 'hacker tendency' to be able to stuff a lot of information in your head in a short amount of time. The computer science equivalent would be learning that there exists an algorithm called quicksort that runs in O(nlog(n)) that doesn't work well on sorted data without modification, but without learning how to implement it.
The take away is that it's very easy to fool yourself into thinking you've learned a lot from a math book when you're reading it like a novel. In practice, though, if you aren't doing the exercises (or if they don't exist, working through your own), then you probably aren't actually engaging with the material in any deep and significant way.
I actually think this is in line with many of his other claims: he seems to be a person who seems to believe he has learned everything important about a field in a relatively short amount of research time, without having a deep background in it already, either through reading a single textbook or by spending a few hours/days/weeks researching and reading. I think most of his claims should probably be considered in that light.
Agreed. That doesn't mean he didn't read it and glean a little from it though.
> I actually think this is in line with many of his other claims: he seems to be a person who seems to believe he has learned everything important about a field in a relatively short amount of research time, without having a deep background in it already, either through reading a single textbook or by spending a few hours/days/weeks researching and reading. I think most of his claims should probably be considered in that light.
Agreed, it just smacked of elitism to say that it's not possible to read a textbook without interpretation or instruction. That sort of doctrinaire thinking is just bad policy.
I'd note that that's different from having a rigorous understanding of arbitrary dimension differential geometry, and being able to rigorously show new (if simple/uninteresting) results.
I wasn't trying to imply that you can't read a textbook (or any math text, for that matter) and not learn the material at as deep a level as anyone who's main reference is that text. It's a bit like reading someone's code after very heavy optimization, though: it's easy to miss little parts of how or why the algorithm works, and if you go over it once, without trying possible inputs/etc., then you're likely to miss something.
Instruction and interpretation are like comments in code when you're dealing with specific proofs (they make it easier, but it isn't impossible without them). The thing that is harder (but not impossible) to get without some sort of feedback is a deep understanding of when a proof is rigorous and mathematical aesthetics.
The persons who mention that in order to impress others tend to not be the ones who learn the most.
"That sort of doctrinaire thinking is just bad policy."
Is it impossible? No. Is it likely considering the context? No.
Almost nobody, yes. Not in one sitting.
Anyway, I just read the paragraph with the sentence quoted by geoka9 (it's in the "How I Stopped Eating Food" post), and I call BS on it, too.