Things That Make Mathematics Hard

I received email from a student who had decided to learn calculus by independent study. After looking at a few standard calculus texts, this student decided that it would make more sense to learn from serious upper-division texts on real analysis.

Normally I don't respond to requests for help from students. But in this case, we wound up exchanging quite a bit of mail.


[Konrad Knopp and the Theory of Functions]

>>>I looked at ToF I and II and found that Knopp was presupposing a knowledge
>>>of things like line integrals, which are somewhere in my future.
>> Maybe you need to invent line integrals for yourself.
>Actually, I took a look in my multivariable calculus book; it seems to me
>that, based on the presentation there, the line integral measures, more or
>less, the degree to which a given directed curve is "going with" or "going
>against" the vector field it's embedded in.

Well, that's part of it. But then the size (norm) of the vectors in the vector field is also relevant.

Start by understanding the case when the curve is a straight line and the vector field is constant.

When the curve is not a straight line, think in terms of relativity. A person moving on a bus along the curve actually thinks that he's moving in a straight line and it's the vector field that's turning. The line integral gives a result that agrees with his perception. In other words, we're integrating just as we would if the curve were a straight line, but we're always seeing the vector field as if we were in the curve and we're integrating the extent to which it is pushing us forwards or backwards as we move. For instance, if the vector field is tangent to the curve, the integral sees it as a field that is always pointing straight ahead and the line integral is just the integral of the norm of the vectors.

If the vector field is always perpendicular to the curve, then it's essentially invisible to us on the curve.

It's like we're riding on a roller coaster. At the top and the bottom of curves, the gravitational field is pulling us straight up or straight down (from our perspective in our seats) and therefore doesn't speed us up or slow us down. But whenever the roller coaster car is not horizontal, the gravitational field will be speeding us up or slowing us down to an extent that depends on our angle (given that the gravitational vector is always a constant vector, pointing downwards when viewed from an external perspective).

Now to see how much our speed changes over the whole trip, we need to integrate this acceleration over the whole curve. That's what the line integral does.


James Berlinski, A Tour of the Calculus

>nicely. Yes, there are a lot of things about the book that I like; chiefly,
>Berlinski is one of the few authors capable of simultaneously sticking to
>the formal mathematics while giving one a simple, clear notion of what the
>hell the real idea is behind the formalism.
>All my research so far has only strengthened my perception that mathematics
>is taught in such a way, typically, as to leave out various crucial pieces
>of information which the student is presumably expected to "just know" or to
>quickly pick up as implicit. In particular, there's very little attempt made
>to discuss explicitly questions like "What's the significance of the
>peculiar way that English (German, French, ...) is used in mathematical
>writing?" and "What are some useful techniques I can use to make sense of a
>theorem or proof?" and even "How do I get started with proving a theorem
>myself?" In short, as a rule one receives no explicit training on how to
>-think about- mathematics.

This is what has always interested me. But it does not interest most mathematicians. Even when they are very interested in how to teach mathematics, they're just not capable of analyzing the process of mathematical communication and the things that are taken for granted by this communication.

Furthermore, the academic system gives no rewards for studying these issues. Value is placed on the discovery of new knowledge. Finding better ways of communicating existing knowledge does not count as serious research.

>Another important issue is all-too-often neglected: How do I condense a
>complicated theorem or proof into its essential idea? A proof, for example,
>typically contains one (or a small number) of key insights, the rest being
>just formal machinery. Extracting the fundamental insight contained in the
>proof is crucial for truly understanding its significance, and often the
>significance of the accompanying theorem as well. Not only is this issue,
>as far as I can tell, basically ignored, it's not even generally realized
>that it's important to perform this extraction! I think most of the
>mathematics students out there have no clear idea of the distinction between
>idea and formalism, and less grasp of the fact that the latter is
>principally a tool designed, originally, to serve the former.
>An example: consider the intermediate value theorem, stated thus:
>(1) "Let a function f be continuous on a closed interval [a, b], a < b,
>such that f(a) < f(b). Then for every y such that f(a) < y < f(b) there
>exists a number x in [a, b] with f(x) = y."
>When one understands the idea of the IVT, one can state it directly in
>something like the following way:
>(2) "If f is a continuous function over a closed interval [a, b], then f
>takes on all values between a and b."
>Of course, (2) is less precise than the formalism of (1), but it captures the
>essence of the IVT. I believe it is of much more use to encounter (2) and
>get the idea that's being communicated, after which one can appeal to (1)
>for the formal details. Indeed, I think most students would, if they
>heard (2) first, be curious enough to want to study (1) for themselves to
>figure out how such a concept was expressed in mathematical language.
>Now, some might choose to present (1) first and follow it up immediately
>with (2)--the order is pedagogical matter, and either order of presentation
>could work well in the appropriate circumstances.

Most teachers will present both versions. For some reason, though, it seems to be regarded as almost unprofessional to do the same thing in textbooks.

One of the results of this is that in many cases to a distressing extent many mathematicians themselves don't understand what their theorems really say. This is one of the reasons why teaching a subject is such a good way of learning it. In the process of figuring out how to explain things to students, you have to really understand them yourself.

>This seems to me to be one of the things that makes your teaching
>exceptional; for example, speaking of the chain rule, you write:

>> Well, let's just say that y is changing 5 times as fast as x. And
>> x is changing 3 times as fast as t. Then y is changing 15
>> times as fast as t. That's basically what the chain rule says. The
>> reason the proof is complicated is that one has to deal with certain
>> exceptional cases.

I only came to understand this through a process of teaching calculus dozens of times. It's not that any mathematician would be surprised by what I've said about the chain rule. And yet.... It's one of those things that everybody knows, and yet in a way they don't know that they know it, because they've got so lost in the formalism. I certainly taught the chain rule many many times before it occurred to me that there's a level on which it is intuitively obvious.


[multivariable calculus as prerequisite for analysis]

>> This is basically the same logic as in making calculus a prerequisite
>> for linear algebra. Namely, if a student hasn't made it through
>> multivariable calculus, it is extremely unlikely that he has seen
>> enough mathematical reasoning to be able to cope with real analysis.
>I understand this too, but I think it's important to realize the limitations
>of the approach. It certainly has the potential to turn away students who
>are interested in higher mathematics and who are frustrated by the "things
>being left out" of the "heavy on computation, light on rigor "calculus
>sequence. I'm not convinced that the calculus sequence is better suited to
>introducing students to mathematical reasoning and language than something
>like an introductory course in analysis; in fact, I think the opposite is
>true. Of course, this doesn't mean handing the third- or fourth-year
>real analysis course to first-year students--it means designing a hybrid
>course operating on two levels: the content of analysis on the one hand, used
>as the backdrop for introducing mathematical reasoning and language on the

Well, to repeat, you have to realize that very few mathematicians have much interest in the sorts of things we're talking about. When designing curricula, the thinking is not very profound. It's something like, "Well, we'd better make multi-variable calculus a pre-requisite, because most of the students we have in Calc II are not the sort of students we'd like to see in Real Analysis."

>Of course, analysis is a more or less arbitrary choice, but I think there
>are some things about it that lend well to this sort of thing. It's worthy
>of note that this is kind of the idea behind many a "Discrete Mathematics"
>course being offered today. These discrete math courses are, all in all, a
>good thing, I think; they provide an introduction to many different areas of
>mathematics that otherwise lack such an introduction, and go some way toward
>helping students with some of the basic techniques of mathematics, such as
>induction. Still, these courses have some disadvantages the way they're
>currently designed: they still don't spend much time, usually, on some of
>the important issues mentioned above, and they usually don't "hang together"
>very well.

The problem I have with Discrete Math is that they try and teach a lot of formalism completely out of context, with the motivation that students will need it in later courses. But I don't think it works very well to learn something that seems completely pointless because somebody tells you you'll want to know it later. To some extent, though, you can force it to work in the university system by forcing students to pass tests. But I hate the whole idea of learning things just to pass tests.

If you look at the Discrete Math syllabus on my web page, you'll see that I basically refuse to teach the subject. Students can learn all that stuff about cartesian products and the like very easily when they actually have a reason to know it. Instead, I concentrate on algorithms and make the idea of recursion the unifying idea of the course. (Proof by induction is just recursion applied to truth values rather than to calculations.)

> The diversity of material on the one hand is paid for with
>student confusion on the other. This is one of the reasons why I think
>something like a modified intro to analysis as mentioned above might work
>well: analysis hangs together well, at least at the beginning (I wouldn't
>expect the actual content of the analysis covered in such a course to go
>beyond, say, continuity). This makes it a good backdrop for the real aim of
>the course, namely to address techniques for thinking about and working with
>the language of mathematics.


Defining area for all subsets of R^2.  Integral as area under the curve.

>> There are those, especially in probability theory, who would strongly
>> object to this approach. However I think it's probably the best way to
>> go.
>Can you elaborate on this a little?

Herman Rubin indicated a little of the objection in his articles in sci.math. The idea of the integral really goes way beyond the Riemann integral or even the Lebesgue integral. You can define all sorts of different integrals by starting with different "measures." Even summation --- the sigma notation --- is a type of integral.

Every mathematician learns this, as part of the graduate real analysis course. But I think that only a few mathematicians really need it. The ones who definitely need it, though, are the probability theorists.

You see, in probability theory you have a random variable," which just means a variable whose values are determined in some probabilistic manner. For instance, you look at the thermometer and see what the temperature is. Here in San Francisco is probability of its being over 90 is very small, the probability of its being between 50 and 60 is reasonably large. Or you throw a pair of dice and see what number comes up. The answer you get is a discrete random variable, whereas the currect temperature at a given point and moment is a continuous random variable.

So given points a and b, there is a certain probability that the value of your continuous random variable will be between a and b. Now the random variable is basically a function (my explanation isn't making this very clear) and so what you have is a mapping that associates to this function and a closed interval [a,b] the probability that the value of the random variable lies between a and b. This mapping will be given by an integral with respect to some measure. The "probability" at issue can be identified with this measure.



(And which probably falls more in line with the original development; how
>can anyone expect us to learn how to do mathematics, as opposed to pass a
>class in it, if attention isn't paid to how and why one might be driven
>to create a concept or come up with a result in the course of working on
>a particular problem? I believe this is called "motivation" in many
>books; unfortunately I see more lip service to this concept than books
>which actually strive to present a self-contained development of a
>subject, and the fact that the word is bandied about so much suggests
>that there exists a real alternative--that it might be possible to
>really teach a subject without "motivation."

Not only is it possible, but there are many books that do this. In fact, I may have said before that I think that most mathematics books on the graduate level and beyond could better be referred to as manuals.

In a way, this was the great insight of 20th Century mathematics, starting perhaps with Edward Landau's book The Fundamentals of Analysis which was dramatically innotative in its time in having no discussion whatsoever, nothing but theorems and proofs: the insight was that motivation is mathematically irrelevant. A mathematical theory can be developed on a purely formal level, using only axioms, definitions, and theorems, without any regard for the significance or importance of the theory.

In its way, this insight was an enormous step forward. And yet I think that in the next century, people will look back on the way it has been applied in wholesale fashion as an enormous step backwards. Basically, it has made mathematical knowledge inaccessible to all except the mathematical mandarins, and even mathematicians can understand knowledge only in their own specialty.

> I think this is a
>dangerous disease, in that such a course lacking in "motivation" might
>give one the illusion that one was teaching mathematics while actually
>disabusing students of the notion that mathematics might be a human
>activity, organized and developed by real people working on real problems.

For the most part, it's not. The people developing mathematics are real, to be sure. All people are real. But very few mathematicians have any awareness of how mathematics relates to "real" problems --- which, for a mathematician, would mean problems in physics or engineering or one of the other physical sciences.

>The context of a development in mathematics, or in any other field, is
>critical to understanding the development itself. Many people understand
>this in other disciplines now; physics is a good example. I have no idea
>why the academic mathematics establishment seems to exhibit such
>bullheadedness about this, unless it stems from a misguided notion that
>providing context and motivation somehow detracts from the "purity" of the

I don't know that it's bullheadedness. It's not so much a deliberate decision.

I don't know exactly how it came about myself, although it was happening pretty much during the time when I was an undergraduate. I know that at the time, it was very exciting to realize that everything could be done abstractly, without reference to physical interpretations. And then the nature of the academic world became such that there were no rewards for being able to communicate with people outside one's own field. (It's not only mathematics where this is the case. Academics no longer write for the educated public at large, but now write only for their own peers, and their results are largely published in conference proceedings and profession journals rather than in books sold in bookstores.)


Not Only What Is Said But How It's Said

>I also think you underestimate your communicative abilities with, e.g., the
>logarithm paper. Perhaps the ideas aren't entirely original, but the fact
>is that in general I can read something you've written and come away with a
>solid understanding of what's going on, whereas this is not the case when I
>pick up a random textbook that discusses the same thing and may even use
>exactly the same mathematical ideas. There's a lot to be said for

Well, yes. You see, one of the core principles of mathematical thinking is that if two statements are logically equivalent, then they are interchangeable. But the human mind doesn't work that way, as you and I know. The fact that the human brain can be used for logical thinking at all is an amazing example of using a device for a purpose very different from that for which it was designed.

So if you want to communicate with people, you have to take into consideration the way the mind actually functions. It's not just a matter of what you say, it's a matter of how you say it. But this is contrary to the very training of mathematicians, which is to be able to focus on the underlying logical structure and try to ignore all those other elements of communication which are actually so important to the human brain.

>Right--this is what I encountered working with that set of exercises. I
>now consider the experience to have been sort of a short course in "What an
>axiomatic system is, why you'd want one, and how to approach it." As the
>authors of another book I picked up recently (Topological Spaces: From
>Distance to Neighborhood
, Springer-Verlag UTM, by Gerard Buskes and Arnoud
>van Rooij, ISBN 0-387-94994-1) say,
>"If you are unused to the axiomatic method, you may find it strange that
>formulas like these have to be proved. ... The idea is not to make sure that
>they hold for the system of real numbers you have in mind but that they
>follow from the axioms."
>That is, the point of proving "obvious" theorems like "0x = 0 for all x"
>isn't to demonstrate a seemingly obvious statement but to test the strength
>and validity of your axioms.

Well, no, I don't think that's quite a valid interpretation of what was said. Or, yes, in a way it's valid, but I don't like your words "strength and validity." You need to know that you have enough axioms. And so when you think about "obvious" facts, it's natural to wonder whether these should also be added as additional axioms. So one needs to know whether these facts are consequences of the existing axioms.

To some extent, though, from the point of view of the person who wants to use a mathematical subject as a tool, the whole idea of axiomatic development can seem like an annoying fetish. If one is working with the real numbers, then one needs to know all the basic facts about real numbers. Which of these are axioms and which are theorems is not important.

>(There are many things about the above book that I like very much, from the
>style of presentation to the "Extras" at the end of each chapter on random
>interesting things in mathematics vaguely related to topology. There are
>"Extras" on the Axiom of Choice, Nonstandard Analysis, The Four-Color
>Theorem, Cauchy, Bolzano, The Emergence of the Professional Mathematician,
>Axiom Systems, and Continuous Deformation of Curves (aka homotopy theory).
>The extras contain two to five pages of an overview of each subject, or a
>biography in the case of the mathematicians, followed by references pointing
>to more information. Splendid idea--this is how I learned what little I know
>know about homotopy, probably one of the most lovely mathematical concepts
>I've come across in all my research so far.)

Well, it seemed pretty ugly when I learned about it as a graduate student. Definitely not my favorite part of algebraic topology.

>[applications of mathematics]
>> Sometimes seeing the application makes the fundamental concept a lot
>> clearer. It enables you to look at the mathematical concept in a more
>> intuitive way, in terms of physics or whatever.
>Right--very important sometimes, as I mentioned earlier on, but I think this
>generalizes to "context" in general. In other words, if a particular piece
>of mathematics was developed in the context of physics, then the physics is
>almost certainly going to make the math much easier to understand; but the
>same applies to a piece of mathematics developed in the context of other
>pieces of mathematics. It's a question of what the original "motivation"
>was for the idea, as noted earlier.

I see in large part a sort of crisis in faith among many mathematicians. They don't know why they do the mathematics they do, and they don't see what sort of value it has, despite an intrinsic, almost artistic, interest that can be appreciated only by those actually involved in creating it.

I think that what you're saying is a big part of the reason for this.

I remember how eye opening it was for me hearing a talk by Ralph Abraham, of UC Santa Cruz. Suddenly I understood that the tangent bundle of a manifold, which I had encountered in so many mathematics books, was nothing but a generalization of what physicists call a "phase space." No book on differential geometry had ever pointed that out. And now, suddenly, not only tangent bundles but the whole idea of a fibre bundle became a lot more meaningful to me.

I took a graduate course in differential geometry and we defined the curvature tensor of a differential space and proved a number of theorems about it, and at the end I still had no idea why it was called "curvature" or what it had to do with the concept of curvature I'd learned about in calculus. For that matter, it was not all that easy to see why the subject I was being taught was even called geometry.

But there are a few books out there written with the idea that meaning and interpretation are important. You've found a couple of them. Michael Spivak's books on differential geometry (way over your head, unfortunately) are another.

>> Formal set theory, as in Halmos's book Naive Set Theory (a complete
>> mis-title) can be a trip.
>> As far as the real numbers go, I'm an advocate of taking them for
>> granted. But understanding them thoroughly and rigorously is certainly
>> worthwhile.
>Analysis has been interesting because on the one hand, it seemed like a
>difficult and unforgiving subject (most of mathematics is unforgiving, but
>there's something hard-edged about analysis).

Yes, because there are so many things that seem quite self-evident that are quite difficult to prove. And some other things that seem quite self-evident that turn out not to be true at all. Although you're not really on that level yet.

But in sci.math they were talking about a function which is continuous at every irrational point and discontinuous at every rational point. (Define f(p/q)=1/q if p/q is a fraction in lowest terms. For instance f(3/14)=1/14. And define f(x)=0 if x is irrational.) That's not really over your head, although it's quite demanding, since it can't be seen by thinking on an intuitive level. On the other hand, the theorem that there can't exist a function with the opposite property --- continuous at rational points and discontinuous at irrational points --- is beyond what I could explain to you.

> On the other, there's
>something very graceful about continuous mathematics, such that I quickly
>became fascinated with its foundations. (It all seems to come down to the
>completeness property of the reals, actually--obviously a deceptively
>powerful axiom. Thanks to Berlinski for introducing it in the context of
>Dedekind's original idea.) My experience with analysis has remained
>paradoxical.  :)


Learning How to Find Proofs

>[about evaluating one's own proofs]
>> Yes indeed. It's almost impossible without a teacher. (And often
>> damned difficult with one!)
>I think I'm getting better at this, though; actually, I'm getting fairly
>good at reading proofs, and mathematical books that aren't too far beyond
>my scope of knowledge in general. It's the proof discovery I have trouble

This is almost never taught. I believe that I'm exceptional in teaching it in my undergraduate courses in linear algebra and number theory, but in the graduate courses I simply have to assume that students can learn it on their own.

One learns it by imitation. Sometimes a pattern somehow "makes sense" to a student and after seeing a few proofs using that pattern, he can easily use it to construct proofs of his own. But other times --- such as me in an analysis course --- although a student seems to be able to follow a proof, somehow he never gets the overall pattern, and so he's at a loss when trying to prove another similar theorem.

Mostly the educational system deals with this by saying that the first student is "smart" and the second one is "dumb." I was "smart" in algebra courses, but much less smart in analysis.

> I should say that I'm able to follow a fair number of proofs; some
>of the more complex ones are beyond me, even when I can follow the
>individual steps; again, it's the difference between following the formalism
>vs. grasping the essential idea. Actually, George Exner spends some time on
>this in his book (another Springer UTM: An Accompaniment to Higher
>Mathematics, ISBN 0-387-94617-9) or rather, points it out in the "Where do I
>go from here?" section near the end of the book.
>Exner's book is interesting; as he puts it, reading, discovery, and writing
>of proofs depends on certain techniques as well as a "wild-card called
>talent" that's hard to pin down. The aim of the book is to arm you with
>good techniques in order to "go as far as your talent can take you." It's a
>fairly slim volume, with three chapters ("Examples", "Informal Language and
>Proof", and "Formal Language and Proof") as well as an extra chapter
>consisting of several "Labs". The book goes a good way toward making
>explicit many of the things one is expected to learn implicitly when one
>first encounters higher mathematics. (As it happens, in my specific case, I
>did manage to pick up much of this implicit knowledge more or less by
>osmosis; I attribute this at least in part to my prior experience with other
>"formal systems" like computer languages. I was able to pick up an
>understanding of the language cues used in proofs, for example, by spending
>time looking at lots of books on mathematics. For students without prior
>experience with formal systems, or who don't yet have a knack for picking up
>implicit information "on the fly," I think Exner's book could be quite


The Ability to Read

>> Furthermore, when someone tries to tell you how they came up with a
>> proof or an idea, most often they give you an ex-post facto lie. In
>> retrospect, yes, it's quite obvious that if you think about A, B, and
>> C, and then wonder about X, then you will come up with Gamma. However
>> in actual fact, usually the idea came from some much more clumsy train
>> of thought which the author is no longer able to even remember.
>That certainly holds true with the proofs I've managed to formulate myself.
>With the analysis, I've really gotten myself into hot water, in that I'm
>running smack up against the part of mathematics that I find most difficult,
>namely proving statements or theorems as given by exercises in a book. This
>was, of course, exactly my intention; I wanted to confront my own mind
>head-on, as it were, and this is certainly taking place.
>It's been quite frustrating in a number of ways, but mostly because of this
>simple fact: I know enough, now, about quite a few major fields of
>mathematics to be terribly fascinated and interested in the material, but
>after four months I'm only on page 25 in my analysis book (13 pages of TeX
>proofs in terms of work done). My ability can't keep up with my interest;
>I can read a great deal more mathematics than I can do.

That's very understandable. That sad thing is, though, that most mathematics students and even many mathematicians are not very capable of reading mathematical books.

One of the characteristics of mathematics courses is that there is no assigned reading. The book is regarded as a back-up to what is taught in class. Students would consider it completely unfair if they were held responsible for something in a book which had not been taught in class. This is very different from courses in many other disciplines.


>That said, let me hasten to add that I don't necessarily consider this to be
>a problem. As you may have guessed, I'm not terribly patient by temperament
>when it comes to my own achievements; I'm used to being able to learn things
>quickly--in fact, since you're well-acquainted with NLP, you may remember
>some quotes by the "founding fathers" to the effect that the brain learns
>quickly by nature; it's often much more difficult to get it to learn slowly
>than quickly. Well, I haven't found out how to learn mathematics (formal,
>rigorous mathematics, that is) quickly, and in some respects this is making
>me crazy.

The brain learns one chunk quickly, but it's not good at learning a whole lot of chunks all at once.

> However, I'm willing to accept that formal mathematics may just
>be a fundamentally slow process. Or, perhaps it's only terribly slow when
>one is getting started, and it picks up speed as one continues. Perhaps it
>picks up speed six months through; or twelve months; or five years.

One never becomes really speedy. And it's not just because the material gets harder as one advances. Even in (re-)learning material for undergraduate courses, I found it very slow going. And I am fairly excepional among mathematicians in my ability to learn from books. (This is because I've devoted an enormous amount of effort to reading books, in a fairly wide range of mathematical subjects.)

>Any of these, or any other, answer is perfectly acceptable to me--the
>problem is that I don't know. I don't mind spending six months working on a
>set of exercises if it's really going to make a difference; but I do have
>some concern that, if others of somewhat comparable background and orientation
>can work considerably more quickly than I, either I'm out of my depth or I
>need to revise my methods, or both. Perhaps this is a question you could
>shed a little light on, as someone who's been doing mathematics for quite
>some time.

I think that this is an inevitable consequence of trying to forge your own path. If you want to be sure that you're never wasting your efforts inappropriately, then I think you have to take courses. Even then you're subject to the extremely variable quality of teaching.

>All of this banging my head against the wall hasn't been totally useless,
>though--I actually have learned a lot about the foundations of the real
>number system, and about the axiomatic method and formal mathematics in
>general. I've learned a thing or two about the natural numbers, having
>proved things like the fundamental theorem of arithmetic and even some basic
>facts about continued fractions, and perhaps most importantly I've worked on
>enough exercises to begin to notice some patterns which I hope may be of some
>use in answering the all-important question, the question that just about
>everything else seems to boil down to, when it comes to learning mathematics:
>how does one discover a proof?
>Some of the patterns I've noticed: some statements can be proved "purely
>mechanically." No doubt these results would typically be called trivial;
>they follow more or less directly by, e.g., inserting the definitions of
>the terms in the theorem for the terms themselves. These proofs are
>distinguished by their lack of any particular insight, short of a basic
>understanding of the material; they follow by straightfoward computation.
>Other proofs involve computation combined with insight; for instance, here's
>an exercise I wasn't able to solve:
>"Let a = \sqrt{1 + \sqrt{2}}. Then a is irrational, and there is a
>positive real c satisfying |a - m/n| \geq c/n^4, n,m \geq 1."
>The irrationality of a is obvious, of course.

If a were rational, then a^2 would also be, i.e. 1 + \sqrt 2 would be rational, and it follows that \sqrt 2 would be rational, a contradiction.

I'm not sure that that quite qualifies as "obvious." Most of my undergraduate students would have a quite difficult time with it unless they'd already seen a very similar proof.

As to the rest of the problem, I don't understand it. It seems like there has to be some additional reqirement on c. Or you haven't made clear the order of the quantifiers.

> The existence of c defeated
>me--the key to the proof turned out to be the factoring of
>f(x) = x^4 - 2x^2 - 1 into (x - a)g(x) with
>g(x) = (x + a)(x^2 + \sqrt{2} - 1). The proof was purely computational,
>but not entirely obvious.
>Still other theorems rely almost entirely on some not-immediately-obvious
>insight which, that having been discovered, follow more or less easily.
>These ideas are entirely vague, but they suggest a sort of layered system in
>which one can think of the proof for a theorem as being more or less "close
>to the machinery." I've found the distinctions to be at least marginally
>useful when attempting to classify the type of problem I'm confronted with.
>Here are the first few exercises taken from the section I'm working on, just
>to lend a few concrete examples (and to give you a small taste of Hijab's
>----- begin snippet -----
>1) Fix N \geq 1 and (a_n). Let (a_{N+n}) be the sequence (a_{N+1},
>a_{N+2}, ...). Then a_n \upto L iff a_{N+n} \upto L, and a_n \downto L iff
>a_{N+n} \downto L. Conclude that a_n \to L iff a_{N+n} \to L.

Seems quite routine.

>2) If a_n \to L, then -a_n \to -L.

Definitely routine.

>3) If A \subset R^+ is nonempty and 1/A = {1/x : x in A}, then inf 1/A =
>1/sup A, where 1/\infty is interpreted here as 0. If (a_n) is a sequence
>with positive terms b_n = 1/a_n, then, a_n \to 0 iff b_n \to \infty.

The second sentence is an immediate consequence of the first, given that the terms are positive. (Oops! Perhaps not quite true. I was thinking of a series rather than a sequence.) The first is algebraic manipulation, but it takes a little careful effort.

>4) If a_n \to L and a_{k_n} is a subsequence, then a_{k_n} \to L. If (a_n)
>is monotone and a_{k_n} \to L, then a_n \to L.

The first sentence is totally easy. The second is only slightly harder: an exercise in logic rather than algebraic manipulation.

>5) If a_n \to L and L \neq 0, then a_n \neq 0 for all but finitely many n.

Very easy, but perhaps tricky the first time you see it. Rephrase it: We know that a_n is very close to L for large n. Why should we believe that a_n is positive for large n? (The most useful thing to learn from this exercise: "For sufficiently large" = "for all but finitely many.")

>6) Let a_n = \sqrt{n+1} - \sqrt{n}, n \geq 1. Compute a^*_n, a_{n*}, a^*
>and a_*. Does a_n converge?

a_n "clearly" converges to 0. Proving it might be tricky, though.

>7) Let (a_n) be any sequence, with upper and lower limits a^* and a_*.
>Then (a_n) subconverges to a^* and a_*, that is, there are sequences a_{k_n}
>and a_{j_n} satisfying a_{k_n} \to a^* and a_{j_n} \to a_*.
>----- end snippet -----
>The first two exercises were quite easy, and followed from a basic knowledge
>of the material; the third wasn't too much harder, but required a little
>extra thought, likewise the fourth. The fifth was a one-liner, I suppose to
>see if you were paying attention at all. The sixth required the simple
>trick of converting \sqrt{n+1} - \sqrt{n} to 1/(\sqrt{n+1} + \sqrt{n}).

A non-routine trick unless you've been doing a lot of these recently. I would have tried the binomial theorem, which makes for a much less satisfactory proof.

>seventh seems finally to have entered more difficult territory, as I've been
>stuck on it for a couple of weeks now.
>(If I'm not mistaken, isn't 7 above a version of the Bolzano-Weierstrass
>Theorem, "Every bounded sequence in R has a convergent subsequence"? If
>(a_n) is bounded, then the lower limit a_* and the upper limit a^* are not
>equal to +/- infinity so, if 7 is true, (a_n) actually has two convergent

Hmm.... I don't think that Bolzano-Weierstrass will tell you that these convergent subsequences converge to a^* and a_*. You have to be a little more discriminating in the way you choose the subsequences. Suppose your sequence is 0, 1, -1, 0, 1/2, -1/2, 0, 1/3, -1/3, 0, 1/4, -1/4, ..... Your convergent subsequence might be 0, 0, 0, 0, ... which is not the one you need.

Besides, Bolzano-Weierstrass is too big a cannon for this problem. I think that one of the things that makes this hard to think about is the implicit use of the Axiom of Choice in constructing these sequences.

But also, the problem is too easy. It only requires that you find a subsequence b_n such that for every \epsilon, |b_n - a^*| < \epsilon for sufficiently large n. But it's easier if we make the problem more demanding: let's require that |b_n - a^*| < 1/n for all n. Now for given n, it's obvious why we can choose such an element b_n from among the terms of the original series. But does this automatically give us a subsequence? What can go wrong? Well, there is one thing that might go wrong. We have to think about the definition of a subsequence, as opposed to a mere subset. Then once we realize what can go wrong, it's fairly obvious how to fix it.

>I'm really starting to wish more authors would follow Donald Knuth's
>exercise-rating system as used in The Art of Computer Programming and, I
>think, Concrete Mathematics--it would be nice to know the general difficulty
>level of the problem one is working on.

The first time I taught the graduate algebra course, I rated problems G, PG, R, and X. However it's not that easy, because a problem may be extremely difficult even though the answer seems obvious once you see it.

When I was a graduate student, lots of times, having another student tell me, "Oh that one's really easy" was all the hint I needed to find a proof I'd been seeking several days for in vain.

>Oh, and before I forget, another URL in the spirit of investigating the
>mathematical thought-process:
>Perhaps you've encountered this before. The Seldens make some worthwhile

Yes, that seems good. However I think that their emphasis on relying on academic research in cognitive psychology prevents them from talking about the deeper and more important issues. The psychologists' methodology tends to make them look only at the superficial aspects of situations.