Riemann for Anti-dummies: Introduction and Critique

by Lyle Burkhead

I am now going to link to some pages, the "Riemann for Anti-dummies" pages, which require an apologetic introduction. Instead of going directly to that, I want to talk about something else first. This may appear to be a digression, but it isn't.

To set the stage, I want to introduce the idea that there are different schools of thought in mathematics. This may come as a surprise to many people. When you take calculus, it's just calculus, right?  It's the same everywhere. Calculus is calculus.

No, I'm afraid not.

Vladimir Arnold is one of the giants of contemporary mathematics. If you are not familiar with him, check out some of his books - you are in for a treat. In the 1990s he was invited to serve on the committee that awards the Fields Medal. He declined, for personal reasons, but the fact that he was asked should give you some idea of his stature in the mathematical community. In fact, if Soviet mathematicians had been given the same consideration as Western mathematicians during the Cold War, he might have won a medal himself. He plays in that league. Here is an interview with him, and a lecture he gave:

An Interview with Vladimir Arnold

"On teaching Mathematics" by V.I. Arnold

In the lecture, he said this:

In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences. They first began teaching their ugly scholastic pseudo-mathematics to their students, then to schoolchildren (forgetting Hardy's warning that ugly mathematics has no permanent place under the Sun)...

To the question "what is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!

Another French pupil (quite rational, in my opinion) defined mathematics as follows: "there is a square, but that still has to be proved".

Judging by my teaching experience in France, the university students' idea of mathematics (even of those taught mathematics at the École Normale Supérieure - I feel sorry most of all for these obviously intelligent but deformed kids) is as poor as that of this pupil...

Mentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching. Calculus textbooks by Goursat, Hermite, Picard were recently dumped by the student library of the Universities Paris 6 and 7 (Jussieu) as obsolete and, therefore, harmful (they were only rescued by my intervention).

In other words, this is not happening by accident. It's not just random cultural drift. Mathematics teaching has been taken over by zealots who believe  in what they are doing. They try to impose their belief system on everyone. Mathematical correctness, so to speak, is a force in academia just as much as political correctness. "Incorrect" textbooks may be removed from the library. This is a fact, not a paranoid fantasy. The authors of the pages I am going to link to are part of this intra-mathematical conflict, and their point of view is similar to Arnold's, up to a point.

Arnold describes how mathematics should  be taught:

...these students have never seen a paraboloid and a question on the form of the surface given by the equation xy = z^2 puts the mathematicians studying at ENS into a stupor. Drawing a curve given by parametric equations (like x = t^3 - 3t, y = t^4 - 2t^2) on a plane is a totally impossible problem for students (and, probably, even for most French professors of mathematics).

Beginning with l'Hospital's first textbook on calculus ("calculus for understanding of curved lines") and roughly until Goursat's textbook, the ability to solve such problems was considered to be (along with the knowledge of the times table) a necessary part of the craft of every mathematician.

ENS students who have sat through courses on differential and algebraic geometry (read by respected mathematicians) turned out be acquainted neither with the Riemann surface of an elliptic curve y^2 = x^3 + ax + b nor, in fact, with the topological classification of surfaces (not even mentioning elliptic integrals of first kind and the group property of an elliptic curve, that is, the Euler-Abel addition theorem). They were only taught Hodge structures and Jacobi varieties!

When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old classical calculus course of French type in the Goursat version. He told us that integrals of rational functions along an algebraic curve can be taken if the corresponding Riemann surface is a sphere and, generally speaking, cannot be taken if its genus is higher, and that for the sphericity it is enough to have a sufficiently large number of double points on the curve of a given degree (which forces the curve to be unicursal: it is possible to draw its real points on the projective plane with one stroke of a pen).

These facts capture the imagination so much that (even given without any proofs) they give a better and more correct idea of modern mathematics than whole volumes of the Bourbaki treatise. Indeed, here we find out about the existence of a wonderful connection between things which seem to be completely different: on the one hand, the existence of an explicit expression for the integrals and the topology of the corresponding Riemann surface and, on the other hand, between the number of double points and genus of the corresponding Riemann surface, which also exhibits itself in the real domain as the unicursality.

Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum.

By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems.  (The word "schoolchildren" must be a mistranslation. They had to be high school students. LB)

I remember well what a strong impression the calculus course by Hermite (which does exist in a Russian translation!) made on me in my school years.

Riemann surfaces appeared in it, I think, in one of the first lectures (all the analysis was, of course, complex, as it should be). Asymptotics of integrals were investigated by means of path deformations on Riemann surfaces under the motion of branching points... The "obsolete" course by Hermite of one hundred years ago (probably, now thrown away from student libraries of French universities) was much more modern than those most boring calculus textbooks with which students are nowadays tormented.

So, what about the idea that calculus is calculus?  When you took calculus, was all the analysis complex, as it should be?  Did the professor tell you about Riemann surfaces at the beginning of the course, and put the whole subject into a context that makes sense?  Probably not!

All this is by way of introduction. I am setting the stage for what follows. The pages I am leading up to are written from a similar point of view, and I am quoting Arnold to give this point of view some credibility.

When you read these pages, you find yourself entering a new mathematical world - a wonderland, no less. However, you don't have to read very far before you encounter invective which is not normally heard in polite company, let alone in the mathematical community. The mathematical wonderland is, alas, imbedded in a paranoid belief system. That is why this apologetic introduction is necessary.

The pages I am going to link to are written by associates of a man I will call Mr. LR.

We have established that there are different schools of thought in mathematics. That much is not controversial. However, Mr. LR and his associates go much farther. According to them, mathematics is a war between the saints (Plato, Kepler, Leibniz, Gauss, Riemann, and a few others) and the devils (Aristotle, Descartes, Galileo, Newton, Euler, Cauchy, and their partners in crime). The devils are absolute devils with no redeeming features. Descartes is a Bogomil. Galileo is a slime-ball. Newton is a Satanist. No, I am not making this up. They actually say that. Here are a couple of samples:

"All Aristoteleans are liars. In fact they must lie. For Aristoteleans believe that their minds are empty vessels, indifferent to what is put in them. They project this view of themselves onto the Universe, which, they insist, must conform to their degraded view of man: an empty box devoid of principles, and subject to no cognizable lawfulness." (#52)

"Given that Isaac Newton, by his own admission, was a fraud and a man convinced he had no soul ('Hypothesis non fingo'), it is a cause of some amazement that he was ever held in high esteem." (#53)

On a page called "The Crimes of Klein," this statement occurs:

"Klein's separation of the theoretical and practical is pure evil Bogomilism, in addition to being a fraud."

That is the level on which mathematics is discussed here. Not only that, we are supposed to believe that the "good" mathematicians themselves saw mathematics as a war between good and evil.

For a reality check, let's look at what one of the leading saints said to the leading devil of his time. In 1692 or 3 (the date of the letter is uncertain), Leibniz wrote to Newton as follows:

To the celebrated Isaac Newton: Gottfried Wilhelm Leibniz sends cordial greetings

How great I think the debt owed to you, by our knowledge of mathematics and of all nature, I have acknowledged in public also when occasion offered. You had given an astonishing development to geometry by your series, but when you published your work, Principia, you showed that even what is not subject to the received analysis is an open book to you. I too have tried by the application of convenient symbols, which exhibit differences and sums, to submit that geometry which I call "transcendent" in some sense to analysis, and the attempt did not go badly...

The devil replied to the saint, as follows:

As I did not reply at once on receipt of your letter, it slipped from my hands and was long mislaid among my papers, and I could not lay hands on it until yesterday. This vexed me since I value your friendship very highly and have for many years considered you as one of the leading geometers of this century, as I have also acknowledged on every occasion that offered. For although I do my best to avoid philosophical and mathematical correspondences, I was however afraid that our friendship might be dimished by silence, and at the very moment too when our friend Wallis has inserted into his imminent new edition of his History of Algebra some new points from letters which I once wrote to you by the hand of Mr. Oldenburg, and so has given me a handle to write to you on that question also. For he asked me to reveal a certain double method which I had there concealed by transposed letters. And so I have been compelled to expound as briefly as possible my method of fluxions which I had concealed by this sentence: given an equation involving any number of fluent quantities to find the fluxions, and conversely. I hope indeed that I have written nothing to displease you, and if there is anything that you think deserves censure, please let me know of it by letter, since I value friends more highly than mathematical discoveries...

These letters can be found in Newton, Philosophical Writings, edited by Andrew Janiak, pages 106 - 109. They were written when Newton was about 50, and Leibniz was a few years younger.

In his second paper on the catenary, Leibniz wrote

Finally, since he (Jacques Bernoulli) attempted to imagine the circumstances that led me to these ideas, and which works I had been using to help me, I insist on revealing to him my sources in all honesty. Advanced geometry was a total stranger to me until I met Christian Huygens, in Paris, in 1672, and to whom I publicly acknowledge in this article, as I did in personal letters, I owe the most, after Galileo and Descartes.

Galileo and Descartes!

The Master himself acknowledges that he owes the most to Galileo (the slime-ball) and Descartes (the Bogomil), along with Huygens. He does not mention Cusa, Kepler or Fermat.

For another reality check, let's look at what Riemann, the most heavenly of the mathematical saints, said in his habilitation lecture:

[Section III.3] The progress of recent centuries in understanding the mechanisms of Nature depends almost entirely on the exactness of construction that has become possible through the invention of the analysis of the infinite and through the simple principles discovered by Archimedes, Galileo, and Newton, which modern physics makes use of.

Galileo and Newton!

If you want to verify this, an English translation of this lecture may be found in Geometry from a Differentiable Viewpoint by John McCleary. In Riemann's lecture there are no references to Plato, Nicholas of Cusa, Kepler, or Leibniz. But Riemann does acknowledge a debt to Galileo (the slime-ball) and Newton (the Satanist). Earlier in the lecture he acknowledges the contributions of Lagrange, one of the devils, and he mentions Lagrange in the same breath with Jacobi, one of the saints, without making any distinction between them. In other words - this should be stated as strongly as possible - Riemann himself did not subscribe to Mr. LR's belief system.

Riemann said (Werke, p 507 f, quoted in Klein, Development of Mathematics in the 19th Century, p 233)

My main work concerns a new conception of the known laws of nature - their expression by means of other basic concepts - whereby it became possible to use the experimental data on the reciprocal relations between heat, light, magnetism, and electricity to investigate their relations with each other. I was led to this by studying the works of Newton, Euler, and - from another aspect - Herbart.

Newton again!  And Euler!  But no mention of Cusa, Kepler, or Leibniz.

I have seen no evidence whatever that Riemann thought of himself as part of a tradition founded by Plato, Nicholas of Cusa, and Johannes Kepler, as opposed to an evil tradition exemplified by Aristotle, Galileo, Newton, etc. He did not think in those terms at all.

Neither did Gauss, although Mr. LR and his associates accuse Gauss of "implicitly" agreeing with them. The idea of "good" mathematicians and "evil" mathematicians apparently comes from Kaestner, one of Gauss's teachers. When he was very young, Gauss was pretty impatient with his elders, like any brash 22-year-old genius, and he complained about the "shallowness" of the mathematics of his time, but even at that age I think he would be appalled at the views attributed to him on these pages. He did not take cheap shots at his opponents, and he did not say they were slime-balls and Satanists. I don't see any indication that Gauss considered Newton to be an opponent at all.

Euler is supposed to be one of the most depraved mathematical villains, right up there with Galileo, Descartes, and Newton. In Gamma  by Julian Havil, on the page facing the title page, the following Gauss quotation may be found:

"The study of Euler's works remains the best instruction in the various areas of mathematics and can be replaced by no other."

In Disquisitiones Arithmaticae, Gauss refers to Euler as "this illustrious mathematician" (page 27) and "this shrewdest of men" (page 36), among other such references. Lagrange is referred to as "the illustrious Lagrange." In the Preface, Gauss says:

"Far more [than is owed to Diophantus] is owed to modern authors, of whom those few men of immortal glory P. de Fermat, L Euler, L. Lagrange, A. M. Legendre (and a few others) opened the entrance to the shrine of this divine science and revealed the abundant wealth within it."

On page 410 of Disquisitiones, he says "By a well-known theorem of Newton, from the coefficients of equation W we can find the sum of any powers of the roots a, b, c, etc."  Gauss expresses no distaste or reluctance to use a theorem of the Great Satanist.

On page 61, Gauss says

"We have already mentioned the writings of other geometers concerning the same subjects treated in this section. For those who want a more detailed discussion than brevity has permitted us, we recommend the following treatises of Euler because of the clarity and insight that has placed this great man far ahead of all other commentators..." [after which he lists three papers of Euler]

The vicious, small-minded personal attacks that are so characteristic of the Anti-dummies pages are absent from Disquisitiones Arithmaticae.  Totally absent. No such attacks occur. Period.

When Mr. LR says Galileo was a slime-ball, that tells me more about Mr. LR than it tells me about Galileo. When he says everybody is either absolutely white or absolutely black, he is telling us who he is. He cannot admit any imperfection in himself, so he projects his own faults and weaknesses onto others. I am not using the word "paranoid" loosely, I mean it literally. This is exactly how paranoia works: the paranoid individual projects his own malice onto others. He sees it coming from outside himself.

Why then am I linking to these pages?  Because I keep coming back to them. I have finally found what was missing in my mathematical education. There is a great truth hidden in here, mixed in with the invective. To me, this is like rain after a long drought. Rain mixed with baseball sized hailstones, but still much needed rain.

Mr. LR is right  that the topics discussed here are what is important  in mathematics.

I know a mainstream mathematician - actually a computer scientist who specializes in automatic theorem proving. I won't give his name here, so as not to embarrass him, but he is not an obscure person. He is a professor at a major university, and his name is mentioned in computer science textbooks. He thinks the idea that a nonsensical premise implies any arbitrary statement is one of the most important things in mathematics. For example,  "If x is a member of the empty set, then the moon is a hamburger." This concept of implication is supposed to be almost the essence of mathematics, and it's one of the main things he tries to teach his students. (Not always with success - he complains that many of his students resist what he is telling them. Good for them. There is hope.)  This is a perfect example of the fatuity of (some, not all) contemporary mathematics.

Mr. LR is right that this sort of thing is frivolous, judged as mathematics, and pernicious, judged as philosophy. My old friend the professor recently told me, "Either you're doing math [i.e. doing it his way], or you're gassing." All right, Bob, if you want to put it in either/or terms, then what you're doing isn't math any more than Messiaen's "Eclairs sur l'Au-Dela" is music. What Gauss did is math. Either you are Gaussing, or you are gassing.

The famous professor and I were students together in the 1960s. We met at a NSF course for gifted math students in the summer of 1963, when we were 16, and then we both went to the University of Texas. We were taught the same bullshit. As a student he believed it, and as a teacher he teaches it. I have been trying to get away from it for more than 40 years, but until I discovered the anti-dummies pages, I didn't know how.

I didn't know anything about projective geometry, Dirichlet's Principle, and Riemann surfaces when I was a senior, never mind when I was a freshman. I didn't even know that mathematics has a center, let alone what it is.

I was very, very good at proving theorems. I took a Moore-method advanced calculus class, i.e. a class in which the instructor states definitions and theorems and the students are supposed to prove the theorems, without a textbook. I could run circles around the other students in the class. I proved the Bolzano- Weierstrass Theorem!  Wow!  Nobody else in the class did.  So I thought I was pretty hot stuff. I thought I understood calculus better than anybody.

It soon became painfully clear that I didn't. When it came to setting up and solving differential equations, I was not at the head of the class, to say the least. Some people had a feel for it, an intuition, that I didn't have. As I read the following pages, I finally learned what Vladimir Arnold learned when he was a freshman. I wish I had read the Anti-dummies pages and Hermite's calculus book when I was 16 or 17. The course of my life would have been very different.

As a student, I never came to grips with the concept of curvature, and never understood its importance. It did not occur to me that curvature is a deep concept. That, more than anything, was the gap in my mathematical education. I could prove theorems all day long, and I was also very good at linear algebra, but calculus, after all, is about nonlinear phenomena - "Analyse des infiniment petits pour l'intelligence des lignes courbes" as l'Hospital said, i.e. "analysis of the infinitely small for understanding of curved lines" - and I just didn't get it. I was like the unfortunate students described above by Arnold.

This was also a gap in my philosophical education. Plato put a sign over the entrance of his Academy which said No one ignorant of geometry may enter here. I always thought he meant "If you don't know how to prove theorems, you have no business here." That is true as far as it goes, but that is only part of what Plato meant. Geometry itself is essential to philosophy. They are inseparable. However, in today's universities, they have been separated. It would be inconceivable for a philosophy department to require a geometry course as a prerequisite for Philosophy 101. No one in either department would understand the point of such a requirement. The philosophy department and the math department overlap in only one place: courses in mathematical logic may be offered in both departments. Other than that, the two departments have nothing to do with each other.

There is more in the Riemann for Anti-dummies pages than just mathematics. There is a philosophy which is derived from Plato and Leibniz. That is what keeps drawing me in, even more than the math. This is almost the Pearl of Great Price. It's not quite that, but it's very close.

The tragic thing about this situation is that Mr. LR and his associates are right, or almost right, about a lot of things. They are trying to say things that need to be said - things that desperately need to be said - but they are saying them in such a way as to ensure that almost no one will listen.

It is true that there are different schools of thought in mathematics - different ways of thinking about it, different ways of conceiving it. Gauss and Riemann are among the deepest thinkers of all time, and their work is as important for philosophy as it is for mathematics. Gaussian mathematics is philosophy. As far as that goes, Cauchy's mathematics is philosophy too, and so is Hilbert's mathematics and Bourbaki's mathematics, not to mention Robert S. Boyer's mathematics, but they are radically different philosophies. It is true that most of contemporary philosophy is sterile, and the way to bring it back to life is to go back to the mathematical roots, all the way back to Plato - and yes, the path from here back to Plato does pass through Riemann, Gauss, Leibniz, and Kepler, just as they claim.

It is also true that different ways of conceiving mathematics have different political implications. The general disintegration of Western culture in the 20th century, both in politics and in other areas, is related to the change in philosophy. When Schubert and Chopin gave way to Schoenberg, Stravinsky, etc., it was just like Hermite and Goursat being replaced by today's calculus books. These are two aspects of the same phenomenon, and it's not just random cultural drift. There is a belief system behind it. When your "classical" radio station plays Bach and Messiaen on the same program, or when MTV plays rock and hip-hop on the same program, this is not an accident. They believe  in what they are doing. That belief system has to be identified and challenged. It's a shame that this line of thought has not been pursued on a higher level of discourse.

The following pages contain the only attempt I have ever seen to explain and illustrate (with animations) the Gauss/Riemann point of view, and to put it into a philosophical context. At the beginning of the 21st century, this is the closest thing we have to mathematical philosophy. I hope someone will use this as a starting point and do it better, but for the moment it's all we have.

I know it can be done better.

When you read these pages, you can wear earplugs to shut out the gangsta-rap soundtrack, and just lose yourself in the mathematics. You may give up in exasperation and put the whole thing aside, as I did, more than once - but if you are like me, you will keep coming back. There really is a wonderland in there, and they keep teasing you with glimpses of it, just enough to keep you coming back for more.

And now, on to the math!

The best place to begin is a paper about Gauss: How Gauss Determined the Orbit of Ceres.

This is by far the most polished of their essays. This is an 88 page PDF file with some nontrivial mathematics. Even V. I. Arnold would find something to chew on here, especially if he read the first eight chapters and then tried to figure out how Gauss solved the problem before reading further (assuming he read it when he was very young and didn't already know the answer). The signal to noise ratio is high. It's almost all math, and hardly any polemics. If all their papers were like this, my apologetic introduction would not be necessary. A brief excerpt:

Characteristically, Gauss’s method used no trial-and-error at all. Without making any assumptions on the particular form of the orbit, and using only three well chosen observations, Gauss was able to construct a good first approximation to the orbit immediately, and then perfect it without further observations to a high precision, making possible the rediscovery of Piazzi’s object.

To accomplish this, Gauss treated the set of observations (including the times as well as the apparent positions) as being the equivalent of a set of harmonic intervals...

Now, compare the orbital arc between P1 and P2 with the straight line joining P1 and P2. Together they bound a tiny, virtually infinitesimal area. Evidently, the unique characteristic of the particular elliptical orbit must be reflected somehow in the specific manner in which that arc differs from the line, as reflected in that “infinitesimal” area.

Finally, add a third point, P3, and consider the curvilinear triangles corresponding to each of the three pairs (P1, P2), (P2, P3), and (P1, P3), together with the corresponding rectilinear triangles and “infinitesimal” areas which compose them. The harmonic mutual relations among these and the corresponding time intervals lie at the heart of Gauss’s method, which is exactly the opposite of “linearity in the small.”

... Skipping ahead to the end -

Readers may have noticed that Gauss made no use at all of “the calculus,” nor of anything else normally regarded as “advanced mathematics,” in the formal sense. Everything we did, we could express in terms of Classical synthetic geometry, the favorite language of Plato’s Academy. Yet Gauss’s solution for Ceres embodied something startlingly new, something far more advanced in substance, than any of his predecessors had developed.

Gauss’s method is completely elementary, and yet highly “advanced,” at the same time. How is that possible?

Far from being a geometry of fixed axioms, such as Euclid’s, Platonic synthetic geometry is a medium of metaphor — a medium akin to, and inseparable from the well-tempered system of musical composition. So, Gauss uses Classical synthetic geometry to elaborate a concept of physical geometry, which is axiomatically “anti-Euclidean.” A contradiction? Not if we read geometry in the same way we ought to listen to music: the axioms and theorems do not lie in the notes, but in the thinking process “behind the notes.”

Through a gross failure of our culture and educational system, it has become commonplace practice to impose upon the domain of synthetic geometry, the false, groundless assumption of simple continuity. It were hard to imagine any proposition, that is so massively refuted by the scientific evidence! And yet, if we probe into the minds of most people — including, if we are honest, among ourselves — we shall nearly always discover an area of fanatically irrational belief in simple continuity and, what is essentially the same thing, linearity in the small.

Is it true, that the adducible, net change in direction of a physical process over any given interval of space-time, becomes smaller and smaller, as we go from macroscopic scale lengths, down to ever smaller intervals of action?

Well, in fact, exactly the opposite is true! As we pursue the investigation of any physical process into smaller and smaller scale-lengths, we invariably encounter an increasing density and frequency of abrupt changes in the direction and character of the motion associated with the process. Rather than becoming simpler in the small, the process appears ever more complicated, and its discontinuous character becomes ever more pronounced. Our Universe seems to be a very hairy creature indeed: a “discontinuum,” in which — so it appears — the part is more complex than the whole.

"Platonic synthetic geometry is a medium of metaphor — a medium akin to, and inseparable from the well-tempered system of musical composition"  — that is the kind of amazing statement one finds in these pages. That's what I mean when I say there is a radically different way of looking at mathematics. This is as far as you can get from "If x is a member of the empty set, then the moon is a hamburger."

The rest of the series can be found here: Riemann for Anti-dummies: Table of Contents. Unfortunately the signal to noise ratio is much lower than the Ceres essay. However, this may not be such a bad thing. When you don't fully trust the authors of the pages you are reading, it forces you to read slowly and question everything, trying to separate the wheat from the chaff. That's good. It keeps you on your toes.

There are some new articles that are not listed in the Table of Contents:

#65. On the 375th Anniversary of Kepler's Passing

#66. Gauss's Arithmetic-Geometric Mean

#67. The View from the Top

#68. An Insider's Guide to the Universe

One way to read the Anti-dummies pages is to start at the beginning and read them sequentially. The problem with this approach is that the series gets off to a slow start. There is an old saying in mathematics, "The best way to learn a subject is to teach it." Apparently Bruce Director, the primary author of these pages, is learning as he goes. The more recent pages are better, and they cover the same material as the earlier ones. The Anti-dummies pages are very repetitive. The same figures are reproduced again and again, and the same points are made in almost the same words. The whole thing could be consolidated to a fraction of its length. I'm not saying the first few lectures should not be read - some of them are important and contain material not repeated later - but I would recommend starting in the middle, with #27. First, however, let's check in with Gauss himself:

Gauss's dissertation (1799)

This is where he proves the Fundamental Theorem of Algebra. He begins with a long critique of previous attempts to prove the theorem. His own proof begins on page 15.  Felix Klein, in Development of Mathematics in the 19th Century, page 24, says: "Even in these early years, Gauss chose his manner of presentation and expression with great care; he disguised his deepest ideas to keep them for himself. In his thesis he carefully avoided any mention of imaginary quantities, although an awareness of them clearly underlies his whole train of thought; he spoke only of resolving a polynomial into real factors of one or two. Indeed, he operated on an xy-plane without ever indicating that it was a question of the geometrical interpretation of complex numbers x + iy."

Gauss, Contributions to The Theory of Algebraic Equations (1849)

Fifty years later, Gauss published another proof along the same lines as his 1799 thesis. This time he explicitly used complex numbers.

Jason Ross, Gauss's Geometrical Approach to Algebra

A short paper which explains what Gauss is doing. Now we have finally arrived at the Anti-dummies pages:

#27. The Fundamental Theorem: Gauss's 'Declaration of Independence'

"Looking back on his dissertation 50 years later, Gauss said [in the "Contributions to the Theory" article just mentioned], 'The demonstration is presented using expressions borrowed from the geometry of position, for in this way, the greatest acuity and simplicity is obtained. Fundamentally, the essential content of the entire argument belongs to a higher domain, independent from space, in which abstract general concepts of magnitudes are investigated as combinations of magnitudes connected by continuity, a domain, which, at present, is poorly developed, and in which one cannot move without the use of language borrowed from spatial images.'

"In essence, Gauss was defending, and extending, a principle, that goes back to Plato, in which only physical action, not arbitrary assumptions, defines our notion of magnitude.

"A physical concept of magnitude was already fully developed by those circles associated with Plato, expressed most explicitly in the Meno, Theatetus, and Timaeus dialogues. Plato and his circle demonstrated this concept, pedagogically, through the paradoxes that arise when considering the uniqueness of the five regular solids, and the related problems of doubling a line, square, and cube. As Plato emphasized, each species of action generated a different species of magnitude. He denoted such magnitudes by the Greek term, 'dunamais', a term akin to Leibniz' use of the word 'kraft', translated into English as 'power'. That is, a linear magnitude has the 'power' to double a line, while only a magnitude of a different species has the 'power' to double the square, and a still different species has the 'power' to double a cube. In Riemann's language, these magnitudes are called, respectively, simply, doubly, and triply extended. Plato's circle emphasized that magnitudes of lesser extension lacked the capacity to generate magnitudes of higher extension, creating, conceptually, a succession of 'higher powers'...

"By the time Gauss left Goettingen, he had already developed a concept of the physical reality of the square roots of negative numbers, which he called complex numbers. Adopting the method of Plato's cave metaphor, Gauss understood his complex numbers to be shadows reflecting a complex of physical action, action acting on action. This complex action reflected a power greater than the triply-extended action that characterizes the manifold of visible space. It was Gauss' unique contribution, to devise a metaphor, from which to represent these higher forms of physical action, so those actions could be represented, by their reflections, in the visible domain."

#28. Bringing the invisible to the surface - a continuation of the previous article.

"The issue for Gauss, as for Gottfried Leibniz, was to find a general principle, that characterized what had become known as 'algebraic' magnitudes. These magnitudes, associated initially, with the extension of lines, squares, and cubes, all fell under Plato's concept of 'dunamais', or 'powers'.

"Leibniz had shown that while the domain of all 'algebraic' magnitudes consisted of a succession of higher powers, the entire algebraic domain was itself dominated by a domain of a still higher power, that Leibniz called 'transcendental'. The relationship of the lower domain of algebraic magnitudes, to the higher non-algebraic domain of transcendental magnitudes, is reflected in what Jacob Bernoulli discovered about the equiangular spiral. (See Figure 1.)

"Leibniz and Johann Bernoulli (Jakob's brother) subsequently demonstrated that his higher, transcendental domain exists not as a purely geometric principle, but originates from the physical action of a hanging chain, whose geometric shape Christaan Huygens called a catenary. (See Figure 2.) Thus, the physical universe itself demonstrates that the 'algebraic' magnitudes associated with extension are not generated by extension. Rather, the algebraic magnitudes are generated from a physical principle that exists, beyond simple extension, in the higher, transcendental, domain.

"Gauss, in his proofs of the fundamental theorem of algebra, showed that even though this transcendental physical principle was outside the visible domain, it nevertheless cast a shadow that could be made visible in what Gauss called the complex domain."

#49. The hidden history of the complex domain

"When Kepler discovered the elliptical nature of the planetary orbits, he uncovered a paradox whose solution would require the development of an entirely new way of thinking, and he called on future generations to develop it. This "Kepler Problem", as it has since become known, was not merely a mathematical lacuna, but reflected the ontological paradox indicated by Nicholas of Cusa in "On Learned Ignorance" and other locations. Kepler's demand provoked Leibniz to develop the infinitesimal calculus, which revealed a new manifestation of that same paradox. This led Leibniz to indicate that the solution existed in a higher, yet to be discovered, domain of the imagination. Reflecting on these developments, the young Carl F. Gauss discovered that what both Kepler and Leibniz had sought. He called it the complex domain.

"The above sketch is the true history of the origin of the discovery of the complex domain. It was known to Gauss's immediate collaborators and followers, but today it lies hidden, even to the relatively best scientific thinkers. What has been substituted is the myth that complex numbers arise as "impossible" solutions to formal algebraic equations - a myth whose malignancy has infected today's popular thinking far beyond the domain of pure mathematics."

#21. It is principles, not numbers, that count

"The first principle of generation of numbers, to which Gauss points, is the generation of numbers by the juxtaposition of three cycles. While this concept was introduced in a new form in the Disquisitiones Arithmeticae, by the concept of congruence with respect to a modulus, the principle underlying it is perhaps the earliest, and most elementary concept of number. In this case, no number exists on its own. Rather, all numbers exist as players, whose parts (i.e. their roles in the play) are a function of their relationship to one another and a One, which Gauss called a modulus. Thus, all numbers are ordered according to the characteristics of the modulus. Those characteristics are themselves determined by an underlying generating principle, which will become more clear below. The so-called "natural", counting numbers are only the special case of numbers ordered with respect to the modulus 1."

#22. Your education was not merely incompetent

"Thus when thinking about numbers from the bottom up, as formed by adding 1 to 1 to 1, the prime numbers are mysterious and arise from an unknown. However, when thought about from the top down, the prime numbers are that from which all numbers are made. The question that Gauss and Riemann contemplated was, "what principle generates prime numbers". This led to the investigation, not of the numbers, but of the manifolds in which those numbers were generated."

#34. Power and curvature

"Yet read backwards, Euclid's Elements begin to reveal a completely different comprehension of the universe. The Elements end where they should begin - with the construction of the five regular (Platonic) solids from the characteristic of spherical action. This investigation leads to the discovery of magnitudes of different powers, as exhibited in the problem of doubling the line, square, and cube. The relationships among these powers give rise to the proportions called the arithmetic, geometric, and harmonic means, and to the prime numbers and the relationships among them. Only then do the investigations concern the reflection of these relationships in a plane. Only at the end, should we arrive at the point, line, surface, and solid. Seen in this way, these objects are concepts arising from a higher principle - the action that produced the five regular solids from a sphere - not as objects created by arbitrary decree from below, in the form of axioms, definitions, and postulates."

To understand wave phenomena at the quantum level - to understand them in a way that they are not understood now - we have to have a fundamentally different point of view. The ideas of Plato, Kepler, Leibniz, Gauss, and Riemann are going to be essential - and yes, the ideas of Mr. LR and his associates are also going to be essential. They have put everything together in a way that has never been done before. More power to them. -Pun fully intended.

Mr. LR - Lyndon LaRouche, if you haven't figured it out yet - is an eccentric genius whose eccentricity has taken him one toke over the line, but he is still a genius. As I said on another page, it is very, very stupid to refuse to learn from "Jewish physics." The same principle applies to LaRouche's mathematics. It doesn't matter who he is. The math stands on its own merits. The math, after all, comes from Gauss, who was anything but eccentric. Gauss was just a genius, period - as solid as they come.

I don't mean to imply that Vladimir Arnold and Lyndon LaRouche have anything to do with each other. If they were aware of each other at all, LaRouche and his associates would probably say Arnold is an evil Newtonian, and he could retort that they are amateurs who have contributed nothing to mathematics. As mathematicians they are not even comparable. Arnold knew more math when he was 23 than LaRouche will ever know. However, this is beside the point. LaRouche isn't a mathematician at all - as he himself said, he is "not a physicist or mathematician as such." He is a philosopher who wants to reunite mathematics and philosophy. Comparing LaRouche with Arnold is like comparing Plato with Archimedes. They don't live in the same space and don't really compete with each other.

They both say there is something wrong with mathematics as it is currently taught, and they say that for similar reasons. They have arrived at the same conclusion by different paths, and they are both right. I want to combine them. I don't care if Arnold is an evil Newtonian, and the LaRouchites are amateurs. As far as I am concerned, those are nonissues. As for whether they are really doing the same kind of mathematics, that remains to be determined.

I am going to stop here, for the time being. This is a new page, written in the spring of 2006. I will have more to say about this subject later. As I said above, I know it can be done better - both the math and the philosophy.

One question I want to explore is this: how could these ideas be applied to programming?  How could a programming language express metaphorical and harmonic relationships?  We have procedural, object-oriented, and functional languages - could we have metaphorical languages?  Could that be done within the von Neumann scheme of things? 

How could a program express the idea that numbers are generated by actions?  "Gauss understood his complex numbers to be shadows reflecting a complex of physical action, action acting on action..."

Or functions acting on functions.  But the thing is, in all existing computers, arithmetic is built in. Of course it would be possible to construct a function that would "generate" numbers, and that would be an instructive exercise, but there would be something artificial about it. You would just be "generating" what you started with. Numbers are already there, in high level languages, in assembly language... all the way down to the ALU.  In a von Neumann computer, it is not the case that "all numbers exist as players, whose parts (i.e. roles) are a function of their relationship to one another and a One, which Gauss called a modulus."  Binary numbers are primary, and everything else is constructed from them. If you are going to do it the other way around, "from the top down," where do you start?  What is the "top"?  What kind of computer architecture - and what kind of physical device - would you have to have to do it the other way around?

A Gaussian computer would truly turn everything inside out.

Note added in January, 2009.

It has been two and a half years since I wrote this page. In the last few weeks I have been editing it a little, but the edits do not fully reflect the change in my views. I have learned more about LaRouche, and I take him more seriously now. The more I explore his websites, the easier it is to ignore his conspiracy theories and his megalomania. His philosophical insights make all that stuff fade into the background. He is certainly a more important thinker than people who get a lot of shelf space in bookstores, such as Nietzsche, Russell, and Heidegger. Please note that I am putting him in the philosophy section, not the economics section. In the future, he will be remembered as the most important philosopher of his generation. His Triple Curve will be forgotten. He started as a Marxist political organizer. He has come a long way since then. Forty years ago he was helping workers organize strikes against their employers; now he is planting seeds that will grow into a new Renaissance. He has changed a lot over the years, but his self-image has not kept pace with the changes in his life. He still thinks of himself as the Great Leader who is going to take over and solve all the world's problems. As far as I am concerned, this does not detract from the validity of his ideas.

Some people accuse LaRouche of appropriating the ideas of others without giving them credit. The truth is just the opposite: he gives too much credit to others and not enough to himself. He asserts that there is a thread running through Plato, Cusa, Kepler, Fermat, Leibniz, Gauss, and Riemann. It was not there, at least not consciously there, until he pointed it out. It is his idea, not theirs. Get any of Leibniz's books and look for Cusa and Kepler in the index. Read anything by Gauss and see how many references you find to Kepler and Leibniz. Please do not tell me that Gauss refrained from mentioning Leibniz because he was afraid of the "British fascists." That's bullshit. He did not mention Leibniz because he did not think of himself as belonging to a school of thought founded by Leibniz. It was LaRouche's idea to connect Gauss and Riemann with Leibniz and the others. Good for him.

Dynamis - a journal with cutting edge articles. This comment from one of the editorials is the best short summary of where LaRouche is coming from:

Lyndon LaRouche has stated that, the most important work of the LYM is their Florentine BelCanto voice training, through daily vocal warm-up exercises and a study of the emergence of the Pythagorean Comma in, particularly, Bach’s “Jesu, Meine Freude” motet, and Mozart’s short piece “Ave Verum Corpus.”

Beautiful animations of number theory - in the menu on the left, click on bi-quadratics. This is what LaRouche's students do in his youth movement (besides singing). This is where the renaissance starts, with music and visual number theory. No one else has been able to motivate students to do what needs to be done. LaRouche, with his manichean belief system, does motivate them. This is a mysterious fact, but a fact.

Harmonia Mensurarum - "Another interesting and 'harmonious' result found by Cotes concerns the points of intersection between a pencil of straight lines and an algebraic curve. Recall that an algebraic curve is defined by a function f(x,y) = 0 where f is a polynomial in the variables x and y. This means that f(x,y) is the sum of terms of the form c_m,n x^myⁿ. The largest value of m+n for any term with non-zero coefficient is called the degree of f. An algebraic curve of the first degree is a straight line, whereas the algebraic curves of degree 2 are the conics, i.e., circles, ellipses, hyperbolas, parabolas. We can have algebraic curves of any degree, including cubics, quartics, etc. Given an arbitrary algebraic curve of degree n, and an arbitrary point O on the plane, suppose a straight line through the point O intersects the curve at n points, labeled P₁, P₂, ... P_n , and let r_j denote the distance from O to the point P_j. Now mark the point Q on the line at the distance r_q from O, where r_q is the harmonic mean of r₁, r₂, ..., r_n. If we repeat this process for many different lines through O (each intersecting the curve at n points), Cotes stated that the constructed "Q" points will all lie on a straight line. ... This theorem is actually quite easy to prove, provided we accept the fundamental theorem of algebra, which assures us that a line always intersects an algebraic curve of degree n in exactly n points (not necessarily distinct), with the understanding that the points of intersection may have complex coefficients."

Exercises: (1) Why is it called the "harmonic" mean?

(2)  Finish the proof as indicated, i.e. show that Cotes's Theorem follows from the fundamental theorem of algebra.

(3) Can you use a Gaussian argument to prove Cotes's Theorem directly, i.e. prove it without assuming the fundamental theorem of algebra?

---

Now, I am going to back off to a more elementary level. Geometry, especially projective geometry, is the starting point.

Euclid's Elements - This is a link to an online edition with Java applets by D. E. Joyce. The Riemann for anti-dummies pages presuppose some background in classical mathematics. Simple things like Proposition VI.30 are usually omitted from today's dumbed-down high schools, not to mention the more advanced material.

I am recommending Euclid for the content, not the axiomatic approach to the subject. Some scholars have suggested that the silly definitions of point, line, etc. appear to have been tacked on later by someone else. I hope they are right.

Radu Vero, Understanding Perspective - The title is misleading. This book is about art, not math. There are no theorems here. Understanding Perspective is a series of exercises to develop one's geometric imagination. The point is, that can be done - i.e. visualization is a skill that can be learned. The author says:

But the neglected necessary training deals with the mental process we call "vision in space." This is the ability to project images on a mental screen prior to any attempt at transferring them to paper. For the untrained student, the mental images are blurred and lacking in precision, whether they are the details of a person's face, a color resulting from the combination of two or more different tones, the shape of a building, or even a simple straight line. The general mistake of untrained students is to try to work directly on paper without a precise visualization of the image, thus losing the three-dimensional values of the construction lines. The trained student will project a definite image on paper, and his construction lines will make sense in space.

The second and more important ability to be trained is the manipulation of mental images. Whether he is visualizing points, lines, solids, or colors, the student will need to modify, move, compose, replace, translate, rotate, intersect, and build them in accordance with a precise sense of space in order to achieve a desired result. Without this ability it is impossible to imagine lines running toward the horizon, planes changing position, solids being raised one on top of the other, and, more generally, perspectives of the same complex spatial structures seen from various points of view. The student will need the ability to play with shapes in time, to imagine them in motion, and to visualize perspective four-dimensionally. Only when this mental training has been completed does the student gain the full ability to visualize.

The inability to visualize is one of the main things that holds people back in mathematics. If you read Understanding Perspective and do the exercises the way they were intended to be done, it works. Of course the same thing can be done with any geometry book, or with the Riemann for Anti-dummies pages. You can invent your own exercises. That's how mathematicians have trained themselves ever since the time of Archimedes and Euclid. The quality of your mathematical work will be a function of the exercises you invent for yourself.

One thing Vero does not emphasize enough is that when you manipulate mental images, it is essential to be able to adjust the speed. This too can be learned with practice. Seeing shapes move in slow motion is more difficult and more important than speeding them up, but you have to be able to do both.

Another problem with Vero is that he begins with imagining shapes. If you try to imagine something you have never really looked at, you are going to imagine it wrong. Instead of trying to imagine a parallelepiped from various points of view, the way to begin is to get a box, look at it, and draw it from various angles. Draw what you see. Then draw it again, without looking at it, and keep working on this until you get it right. Do the same thing with a cylindrical object. Then you are ready for Vero's exercises.

C. Stanley Ogilvy, Excursions in Geometry

The Anti-dummies pages presuppose a background in geometry that few students have. Ogilvy's little book gives you the essential facts about harmonic division and projective geometry.

Brannen, Esplen, and Gray, Geometry

This is much longer and more comprehensive than Ogilvy.

Farin and Hansford, The Geometry Toolbox for Graphics and Modeling - From the Preface:

The Geometry Toolbox approaches linear algebra from a geometric viewpoint and geometry from an algorithmic viewpoint. Every matrix or vector equation has an underlying geometric meaning, and we focus on this geometric meaning rather than on plug-and-chug exercises in matrix arithmetic.

Evar D. Nering, Linear Algebra and Matrix Theory (first edition). There are two things you need to know about linear algebra. One is the geometric significance of linear transformations, which is discussed in Farin & Hansford. The other is dual spaces. Nering covers this very well, and his book was one of my favorite books as a student. I have not seen the second edition.  The price is ridiculous. Used copies of the first edition are still available, cheap. This book is written in such a way that you can easily segue into infinite-dimensional linear spaces. Of course there are a lot of linear algebra books out there, and I have not read all of them. There may be others that would work just as well as this one.

Victor Gutenmacher and N.B. Vasilyev, Lines and Curves: A Practical Geometry Handbook - This is a book of problems. From the publisher's description:

"One of the key strengths of the text is its reinterpretation of geometry in the context of motion, whereby curves are realized as trajectories of moving points instead of as stationary configurations in the plane. This novel approach, rooted in physics and kinematics, yields unusually intuitive and straightforward proofs of many otherwise difficult results."

Kenji Ueno, Koji Shiga, and Shigeyuki Morita, A Mathematical Gift, vol. II

This is a three volume set, of which the second is the most relevant for our purposes. The selection of topics is amazingly close to the Anti-dummies pages. This is the only book I know of that discusses elliptic functions and lemniscates (at an elementary level), the arithmetic-geometric mean, and the Poncelet Theorem. The main shortcoming of this marvellous little book is the lack of problems. You have to invent your own.

Felix Klein, Development of Mathematics in the 19th Century - This book (which is available online as a Google book) gives a fascinating overview of the mathematics of Gauss, Riemann, and their contemporaries. Klein said something about Gauss which ties in with what I said about making up your own exercises:

A natural interest, I might even say a certain childlike curiosity, first led the boy to mathematical questions, independently of any outside influence. Indeed, it was simply the art of calculating with numbers that first attracted him. He calculated continually, with overpowering industry and tireless perseverance. By this incessant exercise in manipulating numbers (for example, calculating decimals to an unbelievable number of places) he acquired not only the astounding virtuosity in computational technique that marked him throughout his life, but also an immense memory stock of definite numerical values, and thereby an appreciation and overview of the realm of numbers such as probably no one, before or after him, has possessed. Aside from arithmetic he was occupied with numerical operations on infinite series. From his activity with numbers, and thus in an active, "experimental" way, he arrived quite early at a knowledge of their general relations and laws. This method of work was not so rare in the 18th century - for example, with Euler - but stands in sharp contrast with the practice of today's mathematicians.

One of the earliest topics to arouse Gauss's appetite for discovery is the arithmetic-geometric mean. To unite, as it were, the two means m' = (m + n)/2 and n' = sqrt(nm), he continued their method of construction:

m'' = (m' + n')/2 ,  n'' = sqrt(n'm').

He remarked - of course computing with definite numbers, for example m=1, n=sqrt(2) - that this procedure converges to a value that he determined to many decimal places. At this time, of course, Gauss did not suspect the importance this fact would have in the theory of elliptic functions. Here we encounter a strange, certainly not accidental phenomenon. All these early intellectual games, devised solely for his own pleasure, were first steps towards a great goal that became conscious only later. It is part of the anticipatory wisdom of genius to place the pick-axe precisely on the rock vein where the gold mine lies concealed, and to do this even in the half-playful first testings of its power, unconscious of its deeper meaning.

Gauss set out huge tables: of prime numbers, of quadratic residues and non-residues, and of the fractions 1/p for p=1 to p=1000 with their decimal expansions carried out to a complete period, and therefore sometimes to several hundred places! With this last table Gauss tried to determine the dependence of the period on the denominator p. What researcher of today would be likely to enter upon this strange path in search of a new theorem?  But for Gauss it was precisely this path, followed with such unheard of energy - he himself maintained that he differed from other men only in his diligence - that led to his goal. Thus, like Euler before him, he discovered the law of quadratic reciprocity, the theorema aureum, in a numerical-inductive way.

Now, returning to our alternate reality, the Anti-dummies pages:

Pierre Beaudry, The Metaphor of Perspective

Kepler, The Harmony of the World, book 1 - background for #65 - spherical triangles, Napier's pentagramma mirificum, etc. You have to click on the arrow at the bottom of each page to continue through the series.

Anti-dummies #65 - This is a long article which discusses the same complete quadrilateral, cross-ratio, and point at infinity that Beaudry is concerned with, and relates all this to the rest of the Gaussian mathematical universe. We get our first glimpse of the center:

But as Desargues' construction demonstrates, the "infinite" in the complete quadrilateral, is, as in Kepler's projective conics, a single point of change, which maintains an harmonic relationship to the finite parts, just as any "finite" point does.

The complete quadrilateral also illustrates an early expression of what would later be called by Riemann, "Dirichlet's Principle". There is a single connected relationship among the position of points A, B, C, D, and the angles and lengths of the sides and diagonals of the quadrilateral. This relationship is an effect of the harmonic principle reflected by the invariance of the cross-ratio. It is this harmonic principle that is primary. The positions of the visible objects are a function of that harmonic principle.

The harmonic relationship exists, even if one of the points appears to be infinitely far away, because the point at infinity is not outside the process, but within it. What appears to be infinite is merely a point of change within an otherwise non-infinite self-bounded manifold.

THE MISSING BOOK - At this point there should be a differential equations book that picks up where the previous books left off, and introduces calculus. That book has yet to be written. There are some excellent d.e. books available, including several by Vladimir Arnold himself, but of course they all assume that the reader already knows calculus.

One time I was looking through the calculus books in a library. I found one by Lipman Bers. He made this statement in the preface:

"Calculus is the art of setting up and solving differential equations; this is how it originated, and this is what it is about."

Unfortunately he did not follow up on this striking idea. His book is not noticeably different from all the others. But what if someone did write a book using that idea as the organizing principle?  Better yet, why not just write a differential equations book in which the concepts and techniques of calculus are introduced when they are needed?  Why have a separate calculus book at all?

In Advanced Calculus, Buck says "The notion of the differential of a function does not appear in its true light in the theory of functions of one variable; one must draw the subtle distinction between a number c and the 1-by-1 matrix [c]."  Likewise, the fundamental theorem of calculus does not appear in its true light until you see it as a special case of Green's theorem, Gauss's theorem, and Stokes's theorem - and all these theorems have to be seen in the context of differential geometry. In Visual Complex Analysis, Needham says "Taylor series and Fourier series of real functions are merely two different ways of viewing complex power series."  In other words, elementary calculus is not a self-contained subject. You are not going to understand what's going on in calculus until you study it in a more general context. And, as Arnold would insist, you are not going to understand it without a strong background in geometry.

The more I ponder this, the more I am inclined to think that the traditional calculus course is a fossil. It is time to completely redesign it, and maybe just discard it. People who write calculus books just keep repeating the same cliches that have been accumulating for more than 300 years. It is time to start over. There should be an Advanced Geometry course in which curves, surfaces, tangents, evolutes, and areas are explained - including how curves, surfaces, and areas behave under transformations - and then that course should pass the baton to a course in differential equations, where calculus proper is introduced. This would be followed by courses in vector analysis (with and without differential forms), complex analysis, Fourier analysis, partial differential equations, classical mechanics, differential geometry, and algebraic geometry.

I am constantly reviewing everything I have learned about mathematics and building up a multivolume textbook in my mind. The calculus book that I have in my mind now is very different from the one I read in high school (Apostol). As I get farther into mathematics, my mental calculus book gets closer and closer to The Missing Book. That's what this page is really about: how to write The Book.

The Ten Essential Skills - from a handout that comes with the basic Differential Equations course at MIT. (They may change the URL from year to year, but you can use the search box on any page of the MIT math site to find it.) "You should strive for personal mastery over the following skills. These are the skills that other courses at MIT will expect you to have when you finish 18.03. This list of skills is widely disseminated among the faculty teaching courses listing 18.03 as a prerequisite. You must become proficient at them to prepare yourself for those courses." This is what calculus is for. You can start with this list of skills and work back.  For each one, ask "Which calculus topics and techniques are used here?" That gives you the outline of the basic calculus course.

The lecture notes and applets for the MIT course can be found here.

For each of the skills, you can also ask: How would Newton do this?  How would Leibniz do it?  How would Gauss do it?  Some of the MIT skills involve concepts that were not known in those days, so the point of the question is, if Newton, Leibniz, and Gauss were faced with these problems, how would they proceed, using their typical ways of solving problems?  Is there any essential difference in their approaches?  We have already seen examples of Gauss at work. Leibniz's methods are illustrated in his catenary papers and many of the RFAD pages, particularly #7, #8, #9, #10, and #59. I should warn you that #59 is a long page which contains animaged gif's. It uses a lot of memory and takes a long time to load (you may have to close other programs to get it to load at all), but it's worth the wait. If you think Leibnizean calculus is just the dy/dx notation, you will be amazed. This is not your daddy's calculus book.

If you segue from #59 to The Poetry of Logarithms, and apply Leibnizean calculus to the logarithmic spiral, it gets even better. This is where you really get a chance to make up your own exercises.

What about Newton?  Obviously, Satan's mathematics is not discussed in the RFAD pages.  In Huygens and Barrow, Newton and Hooke, Vladimir Arnold says "Integration had already been encountered with Archimedes, and differentiation with Pascal and Fermat; the connection between these operations was known to Barrow. What did Newton do in analysis?  What was his main mathematical discovery?  Newton invented Taylor series, the main instrument of analysis... Newton correctly assumed that all the calculations in analysis need to be carried out not with repeated differentiations, but by means of expansions in power series."  So Newton would use power series, plus the geometric techniques presented in the Brackenridge book, to solve the MIT problems. That is the general answer, but I want to go into the details, for Newton and for the others too.

"These studies [on power series] stand in the same relation to algebra as the studies of decimal fractions to ordinary arithmetic." - Newton

J. Bruce Brackenridge, The Key to Newton's Dynamics - Here is another take on Newton. This book is an explication of the most important parts of Newton's Principia. The same thing that was said above about Gauss in the Ceres essay can be carried over to Newton, mutatis mutandis, thus:  Readers of this book may notice that Newton made no use at all of “the calculus,” nor of anything else normally regarded as “advanced mathematics,” in his Principia.  Everything he did was expressed in terms of classical synthetic geometry, the favorite language of Plato’s Academy. Yet Newton’s theory of dynamics embodied something startlingly new, something far more advanced in substance, than any of his predecessors had developed.

But the thing is, Taylor series are not mentioned - not even listed in the index. I don't know what to make of this.

Other questions present themselves - if MIT could be converted to the Anti-dummies point of view, what difference would it make?  How would the d.e. course change?  What would "Differential Equations for Anti-dummies" look like?  What about "Ten Essential Skills for Anti-dummies"?  And the most important question: With a d.e. course based on harmonic relationships, would MIT graduates be more creative scientists and engineers?

Arnold would say that there is no missing book - the old books were just fine. Unfortunately, when I found a copy of Goursat at the library, it turned out to be not quite as he described it.

Edouard Goursat, A Course in Mathematical Analysis - Arnold says his professor "conscientiously retold the old classical calculus course of French type in the Goursat version. He told us that integrals of rational functions along an algebraic curve can be taken if the corresponding Riemann surface is a sphere and, generally speaking, cannot be taken if its genus is higher..." Goursat does not discuss Riemann surfaces. Professor Tumarkin must have added comments of his own about Riemann surfaces.

Goursat does have a brief section on unicursal curves, where he says "It is shown in treatises on Analytic Geometry that every unicursal curve of degree n has (n - 1)(n - 2)/2 double points, and, conversely, that every curve of degree n which has this number of double points is unicursal." Goursat's discussion of double points and unicursality is in Chapter V, more than 200 pages into the book. This subject is just mentioned in passing. It is not the starting point, nor is it the framework for the whole exposition.

Goursat's book may be more interesting than today's calculus books, but he does not present the subject the way Arnold says (or implies) he does. Not only that, putting Goursat up against an elementary calculus book is not a fair comparison. If you compare Goursat with Buck's Advanced Calculus, it's not obvious who wins.

Leibniz, The Early Mathematical Manuscripts of Leibniz, with an introduction and notes by J.M. Child.

I am adding this book later. I was not aware of it when I wrote the rest of this page. It is a Dover reprint, published in 2005, of the book originally published in 1920. Strangely, it is not listed on Amazon. There is also an edition published by Kessinger Publishing, LLC, and Powell's lists a third edition by Merchant books. I assume the other two are the same as the Dover book.

Unfortunately, Child was mainly concerned with the old argument about priority. Such things are a waste of time, to say the least. However, as long as we are on this subject, I cannot help noting that in his account of the matter ("Historia et Origo Calculi Differentialis," reprinted here), Leibniz does not mention Nicolas of Cusa, and he mentions Kepler only in passing. Not only that, his principle of continuity (page 147) contradicts what Cusa said (De Docta Ignorantia, III, page 115).

Leaving aside the question of who influenced whom, it is fascinating to watch Leibniz struggle with his problems and gradually invent the concepts, methods, and notation for solving them. This is not the missing book, but there is much to be learned here.

Vladimir Arnold, Ordinary Differential Equations - In the Preface, the author says:

In selecting the subject matter of this book, I have attempted to confine myself to the irreducible minimum of absolutely essential material. The course is dominated by two central ideas and their ramifications: The theorem on rectifiability of a vector field (equivalent to the usual theorems on existence, uniqueness, and differentiability of solutions) and the theory of one-parameter groups of linear transformations (i.e., the theory of linear autonomous systems).

Many of the topics dealt with here are treated in a way drastically different from that traditionally encountered. At every point I have tried to emphasize the geometric and qualitative aspect of the phenomena under consideration. In keeping with this policy, the book is full of figures but contains no formulas of any particular complexity. On the other hand, it presents a whole congeries of fundamental concepts (like phase space and phase flows, smooth manifolds and tangent bundles, vector fields and one-parameter groups of diffeomorphisms) which remain the the shadows in the traditional coordinate-based approach.

This book makes heavy demands on the reader. You have to slow down and ponder each figure until you get a vivid image in your mind's eye, and keep looking at it until you see how it relates to the concept or equation in question. Try to imagine being Arnold, as he draws this figure to express what he sees in his mind's eye. This is certainly a mind-opening experience, but it's not easy. As in the discussion above, I would like to know how to write a calculus book by starting with this book and working back. If Arnold expanded the first chapter and explained what an integral is, instead of assuming that the reader already knows, what would his explanation look like?  In other words, if you are using diffeomorphisms and rectification of vector fields to set up and solve differential equations, how does integration emerge in that context?

Incidentally, it is interesting to note that in spite of the statement quoted above - "Newton invented Taylor series, the main instrument of analysis" - Arnold hardly mentions Taylor series in his d.e. book. They do appear briefly in Section 14, but they play a minor role, and they are not even listed in the index. Taylor series are "the main instrument of analysis," but they are barely mentioned in "the irreducible minimum of absolutely essential material."  Hmmm....

The central concepts of the Anti-dummies pages, such as harmonic relationships and Dirichlet's Principle, do not appear in Arnold's book. If you wrote a d.e. book organized around those concepts, that would be Differential Equations for Anti-dummies.

Two questions present themselves: How do you combine these points of view, and why hasn't this already been done?  It's obvious why LaRouche and his group have not done it. They have not used Arnold's ideas in their work because they never got that far into mathematics. But in the other direction, it's not obvious. Why doesn't Arnold incorporate Gauss's and Riemann's point of view into his work?  He is certainly aware of Gaussian mathematics. For example, in Geometrical Methods in the Theory of Ordinary Differential Equations, he says (page 144)

The averaging method has been used to determine the evolution of planetary orbits under the influence of the mutual perturbation of planets since the time of Lagrange and Laplace. Gauss formulated it in the following way: to determine evolution, one has to smear the mass of each planet over the orbit in proportion to the time spent in every part of the orbit and replace the attaction of planets by the attraction of the rings thus obtained.

The paragraph just quoted is in the introduction to Chapter 4, Perturbation Theory. Gauss is only mentioned in passing. As I read Arnold's discussion of differential equations, and compare it with the Anti-dummies pages, I see a strange duality, like one of those optical illusions that flips back and forth. If you look at it one way, Arnold seems shallow compared with the Anti-dummies pages, but if you look at it the other way, it's just the reverse. If Arnold read RFAD he might say most of it is obvious, even Mickey Mouse (or whatever the equivalent Russian expression would be), and I can understand, sort of, why he would say that. But if you look at it from the other side, there is something missing in his chapter on Perturbation Theory. Something vital  is missing. His discussion is flat and lifeless compared with How Gauss Determined the Orbit of Ceres. I think someone who aspires to be the Leonardo or Edison or Tesla of femtochemistry has to find some way to combine Arnold with the nexus of ideas that starts with Plato and goes through Kepler to Gauss and Riemann.

Jan Koenderink, Solid Shape - Don't be misled by the description on the MIT press site. This book is not about artificial intelligence, it's about developing your mathematical imagination. This book does for differential geometry what Understanding Perspective does for projective geometry. The author says:

It seems that these [skills] are hardly being taught at all: the ability to use intuitive, heuristic tools to shape the problem, if necessary to redefine it. Although you may be able to prove theorems given as "problems" or do sums that have been formulated carefully by the teacher or author, you discover to your chagrin that you are in no position to produce something off track by yourself. The theory "doesn't get you anywhere." In this text I try to concentrate on this most elusive part and - I hope - to do something about it.

The value of "visualization" of geometric objects can hardly be overrated and should be exercised and developed continuously. The reader should not rest until each subject developed in this text stands out vigorously clear in his "mind's eye."

Unlike Understanding Perspective, this really is a math book. It covers differential forms, intrinsic and extrinsic curvature, the Gauss map, moving frames, and much else. The author has a unique point of view that I don't know how to describe briefly. Here are some typical Koenderink comments:

What is a dispersed cloud at one resolution looks like a coherent blob at another. Many different shapes at a high level of resolution can lead to essentially identical shapes at a low level of resolution...

In specifying the surface area of a person's skin, do you want to count the area subtended by the inside of the pores? For some applications yes, for others no. This is by no means a mathematical  problem.  It is a decision you make on purely practical and operational grounds... The notion of "surface area" is devoid of any meaning unless you specify the resolution at which it is to be assessed. The same goes ipso facto for the notion of "arc length."

Here's more:

In Euclidean space you have a simple mechanism to switch back and forth between forms and vectors, something unthinkable in affine geometry. It is called "the metric" when used as a geometric object, or "the metric tensor" in coordinate language, the "gauge figure" if you want to stress the pictorial geometric content, and - finally - the "sharp operator" if you want to stress its operational character. Although superficially it may seem that you meet a lot of novel entities here, they really all boil down to the same notion. In fact heuristically they are but different manifestations of the same entity.

And still more (this is about Gauss's Theorema Egregium, p. 218):

Useful heuristic: The total curvature of an area measures the net turn of the tangent plane about the normal for a full circumnavigation of that area.

You end up with two totally different interpretations of the Gaussian curvature. On the one hand there is the extrinsic interpretation of K as the spread of surface normals per unit surface area. On the other hand there is the intrinsic interpretation of K as a density whose integral over an area yields the holonomy angle for a circumnavigation of that area.

One side of the Janus face reveals how the surface is embedded in three-space; the other side can be seen by the intelligent ant that only knows the surface. Small wonder that Gauss got excited about this! If you really want to understand surfaces in three-space, the miracle should somehow be transformed into a basic fact of life - a truth you hold to be self-evident ... Such can only happen through familiarity: playing around with surfaces in the real, in the abstract, and in the computer.

Koenderink uses resolution dependence to neatly sidestep the issue of "linearity in the small."

As of June 2008, Solid Shape is back in print. Amazingly, MIT press called me the other day, with regard to my order of two years ago. That's service! 

Tristan Needham, Visual Complex Analysis - Needham has written a unique, pathbreaking book. Visual Complex Analysis  is not just another textbook, it is a major mathematical event. However, while this is a huge step in the right direction, it is only one step. Needham doesn't venture very far out of the box. He still has one foot planted firmly in the mathematical world of Euler and Cauchy - and yes, that really is different from the world of Kepler, Leibniz, and Gauss. The essential ideas of Riemann for Anti-dummies will not be found here - especially in the first chapter, where they are most needed. This book is a supplement to RFAD, not a substitute. Nevertheless it is a great book, and I don't want to overemphasize the fact that it is not quite perfect. If you have taken a course in complex analysis using another textbook, you might want to read chapters 4 and 5 of Needham, to see what you are missing. See  is the operative word here.

I took a complex analysis course using Churchill's book. I did the homework assignments and made a B on the final, but at the end of the course I had no idea what the subject is about. This was not entirely Churchill's fault, of course, nor was it entirely the professor's fault. Some students do understand what the subject is about, no matter how poorly it is presented. A student who has it in him to be a great mathematician doesn't need to have everything explained step by step. Just give him a few hints to get him started, and he can take it from there. Alas, I didn't have it in me. But I don't think the fault was entirely mine, either. I was not cut out to be Abel or Ramanujan, but with better instruction I could have been a better mathematician than I was. If the professor had given us some of the Anti-dummies material to get us started, and then used Visual Complex Analysis as the main textbook, I would have learned something. Like most of the courses I took, my complex analysis course was designed to hide mathematics rather than to reveal it. To quote the title of one of the RFAD pages, "Your education was not merely incompetent."

In the Preface, Needham says

My book will no doubt be flawed in many ways of which I am not yet aware, but there is one "sin" that I have intentionally committed, and for which I shall not repent: many of the arguments are not rigourous, at least as they stand. This is a serious crime if one believes that our mathematical theories are merely elaborate mental constructs, precariously hoisted aloft. Then rigour becomes the nerve-racking balancing act that prevents the entire structure from crashing down around us. But suppose one believes, as I do, that our mathematical theories are attempting to capture aspects of a robust Platonic world that is not of our making. I would then contend that an initial lack of rigour is a small price to pay if it allows the reader to see into this world more directly and pleasurably than would otherwise be possible.

The "robust Platonic world" is still there, always there, for anybody who wants to seek it out.

--

Coming back to our starting point, Arnold said in his lecture

ENS students who have sat through courses on differential and algebraic geometry (read by respected mathematicians) turned out be acquainted neither with the Riemann surface of an elliptic curve y^2 = x^3 + ax + b nor, in fact, with the topological classification of surfaces (not even mentioning elliptic integrals of first kind and the group property of an elliptic curve, that is, the Euler-Abel addition theorem). They were only taught Hodge structures and Jacobi varieties!

When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old classical calculus course of French type in the Goursat version. He told us that integrals of rational functions along an algebraic curve can be taken if the corresponding Riemann surface is a sphere and, generally speaking, cannot be taken if its genus is higher, and that for the sphericity it is enough to have a sufficiently large number of double points on the curve of a given degree (which forces the curve to be unicursal: it is possible to draw its real points on the projective plane with one stroke of a pen).

I have not found any book which discusses this exact point, but there are many books which at least explain what is involved and lead you up to it. The idea of the genus of a Riemann surface goes back to Riemann's paper on Abelian Functions. In modern books on algebraic geometry, the Riemann-Roch theorem is the centerpiece, so we need to look at algebraic geometry.

Dan Pedoe, Geometry: A Comprehensive Course - This book gives you most of the geometric background you need for algebraic geometry. In fact the last chapter of the book is called "Prelude to Algebraic Geometry." Here is a typically nonrandom Pedoe comment:

We have already encountered a quadric surface in S₃ as the locus of lines which intersect three mututally skew lines. We now discuss this surface once more, and a corresponding construct in S₂, the conic, from another point of view, that of projective generation. This leads to a much deeper insight into the nature of conics and quadrics, which appear in some books on mathematics as unadorned loci given by second-degree equations, to be studied because two is the next higher degree after one.

C. G. Gibson, Elementary Geometry of Algebraic Curves - This book is just what the title implies. As far as it goes, it is a good book. I have two problems with it. The first problem is that it leaves you hanging at the end. In the last chapter, Gibson says "Thus lines have genus 0, whilst non-singular conics, cubics and quartics have genera 0, 1, and 3 respectively. The genus is the single most important number associated to a non-singular algebraic curve. For curves over the complex field, it turns out that the genus can be defined solely in terms of the topology of the curve, a topic we will not pursue here."  Well, we are on page 242, only four pages from the end of the book - since we have gotten this far, surely you could add one more section giving us some idea what that means?  Alas, no. The author decided in advance to avoid topology, so he can't go there.

The second and more serious problem is that Gibson's book does not really show you what a complex curve is. Real curves are easy enough to draw - if you make the effort - but a complex curve is a completely different kind of thing. Complex "numbers" are not the same as integers and real numbers, which are used for counting and measuring quantities and distances. A "complex number" is a measurement of spiral action.  What does a line of actions look like?  or a plane of actions?  And then what about a curve of actions, or a curved surface of actions?  What is the difference between a straight line of actions and a curved line of actions?  This subject should be worked out in great detail, not passed over as if it's obvious. We need a whole new way of looking at "complex numbers." After 200 years we are still using Argand diagrams, and that just isn't going to cut it. What I want to do is use computer graphics to show what cannot be shown on paper. That is the main contribution I can make to this discussion.

Gibson says, tantalizingly,

...that decision [to avoid topology] precluded the possibility of developing one of the great historical ideas of the subject, that complex curves can be viewed as real surfaces. (page xv)

This is the first instance of a general principle, namely that curves in C² can be considered as real surfaces in R⁴; in the single complex relation f(x,y)=0 defining an algebraic curve in C² we can take real and imaginary parts to produce two polynomials in the variables x₁, x₂, y₁, y₂, defining (in principle) a surface in R⁴. (page 32)

The fact that a complex curve is a Riemann surface is more or less what the subject is about.

Phillip A. Griffiths, Introduction to Algebraic Curves - Griffiths takes this as his starting point in Chapter 1:

There exists a close relationship between the study of compact Riemann surfaces and that of complex algebraic curves: any irreducible algebraic curve admits a parametric representation and the domain of definition of this representation is a compact Riemann surface... On the other hand, any compact Riemann surface can be represented by an algebraic curve. From this we see that the study of compact Riemann surfaces and that of plane algebraic curves are in fact the same thing.

I still say no one has made this really clear, I mean as something you actually *see*. I am learning Blender scripting with Python. It should be possible to make a movie with one diagram fading into another, and then merging with an adjacent one, or connecting to others on different layers.

Imagine the plane rotating and expanding. If the plane rotates by a certain angle and expands by a certain number, the pair (angle, expansion factor) defines a complex "number". There should be a better name for it. It is the rotational action that is primary, not the a+bi concept. Instead of numbers we could call them spins, where it is understood that a spin is inherently a two-dimensional quantity, requiring two numbers to specify it. When the plane does not just move through one step, but continually rotates and expands (or contracts), this movement is a complex function.

If the plane rotates back and forth in an oscillating movement, without expanding or contracting, this is the sine or cosine function, depending on the phase. You can adjust and combine these functions and generate a vast array of functions - most functions that are of physical interest. Of course this is well known. The point is that you can see it happening.
You can see the basic spins combining and converging to other functions. You can see this in your mind's eye, without a computer, but doing it with computer graphics would open up a whole new realm.

Miles Reid, Undergraduate Algebraic Geometry. In the Appendix to Chapter 1, "Curves and their genus," Reid says

Commercial break. Complex curves (=compact Riemann surfaces) appear across a whole spectrum of math problems, from Diophantine arithmetic through complex function theory and low dimensional topology to differential equations of math physics. So go out and buy a complex curve today.

To a quite extraordinary degree, the properties of a curve are determined by its genus, and more particularly by the trichotomy g=0, g=1, or g=>2. Some of the more striking aspects of this are described in the table on the following page, and I give a brief discussion; this ought to be in the background culture of every mathematician.

Michael Monastyrsky, Riemann, Topology, and Physics - The first part of this book is a biography of Riemann and an overview of his mathematical work. Is is surprisingly difficult to find anything about Abelian functions. I have four complex analysis books (Needham, Nevanlinna & Paatero, Flanagan, Marsden). "Abelian functions" does not appear in the index in any of them. Monastyrsky explains what they are. The context for this is the problem of inverting an elliptic integral. In the following, R is a rational function, in other words a function of the form g/h where g and h are polynomials. He says

Abel's theorem, in essence, allows us to reduce the study of integrals of arbitrary algebraic functions - Abelian integrals - to several special cases (depending on the type of singularities). For example, we can pick out integrals for which logarithmic terms are absent, and the like. We see that the problem of inversion in the general case requires the study of multivalued functions on an arbitrary Riemann surface. Abel himself did not pose the question of inversion in the general case.

The honor of solving this problem belongs to Riemann. He began his investigation with the construction of a general theory of multivalued analytic functions on a Riemann surface. His basic method was already contained in his doctoral disseration. He shows that, first, it is possible with the help of a system of cuts to convert a multiply-connected domain into a simply-connected one and, second, having defined the behavior of a function and given a passage across the cuts (jumps), to reduce the problem to the study of single-valued functions with a given type of singularities.

Riemann succeded in showing that the existence of multivalued functions with a given type of singularities depends on the topology of the Riemann surfaces. Consider, for example, the Abelian integral:
w = integral from 0 to z of R(u,z)dz.
The integrand is called an Abelian differential and the function R(u,z) is an Abelian function.

One can pose the following question: do there exist Abelian functions which do not approach infinity and which are different from a constant on a given Riemann surface? (Corresponding integrals are called integrals of the first kind.) That such functions ought to be multivalued follows from the theorem of Joseph Liouville, which states that any single-valued analytic function without singularities (of zeroes and poles) on a closed Riemann surface is constant. Riemann obtained the following result: on a surface of genus g, there exist g linearly independent integrals of the first kind. (It follows from this, for example, that on a sphere there are no regular Abelian functions.)

Riemann solved an analogous problem for Abelian functions which approach infinity at a finite number of points. Here the problem is significantly more complicated...

Riemann succeeded in obtaining a most important result - Riemann's inequality. The number of linearly independent meromorphic functions with poles of the order r, which is not greater than n_k in m distinct points P_k, (k=1,...m) is not less than Sigma n_k -g + 1 (where g is the genus of the surface). In 1864, a student of Riemann, Gustav Roch, who died at an early age (1839-1866), succeeded in strengthening this result. It turned out that

r = Sigma n_k - g + 1.

This is the renowned Riemann-Roch theorem. At the present time, numerous multi-dimensional generalizations of the Riemann-Roch theorem play an important role in various branches of algebraic geometry, analysis, and topology.

So that's what Abelian functions are, and what the Riemann-Roch theorem is, as these things are understood in mainstream mathematics.  I am sure this is all perfectly clear... Not.

Some people are going to feel like the renowned Riemann-Roch theorem is an anticlimax. Now we know something about the number of linearly independent meromorphic functions with poles of the order r.  So what?  What is the point?  Is this why Riemann is so famous? 
It seems like we are still in the bunch of random topics school of mathematics. It's like I said about Arnold's discussion of perterbation theory. Something vital is missing.

Rick Miranda, Algebraic Curves and Riemann Surfaces - an Amazon reviewer nailed it. Mathwonk says

Let me observe for beginners that these ideas were introduced by Riemann in the analytic context and only later translated into algebraic language. There is a reason no one thought of these ideas algebraically before Riemann. The concepts are inherently analytic and topological. Hence it is almost impossible to understand how their algebraic versions were thought of unless you learn the analytic versions first. Hence people starting from Walker or Hartshorne, or even Shafarevich are handicapped by trying to understand the motivation for algebraic concepts which were introduced to mimic analytic ones that are not mentioned. How are you going to appreciate the genus of a curve if you think it is the smallest possible integer g such that for every divisor D of degree d on the curve, we have l(D) is greater than or equal to 1-g + deg(D), where l(D) is the vector dimension of the space of rational functions f with divisor greater than or equal to -D? This is the kind of unilluminating definition given in purely algebraic treatments of the subject. Riemann explains it as the number of handles in a surface which is topologically equivalent to a sphere with a finite number of handles. I.e. for a sphere it is zero, and for a doughnut it is one, etc...

To find what is missing here, let's look at the Anti-dummies discussion of Abelian functions, To What End Do We Study Riemann's Investigation of Abelian Functions?  Bruce Director says:

As Leibniz showed in the initial development of the infinitesimal calculus, the relationships among the discrete objects of the visible domain are determined by the continuous, but differential, action of universal principles. Riemann extended Leibniz’s notion, hypothesizing that when a physical process is the effect of a multiplicity of universal principles, those principles can be thought of as constituting a multiply-extended continuous manifold. But these universal principles are not sensible. Therefore, Riemann said [in his Habilitationsvortrag], “there are in common life only such infrequent occasions to form concepts whose modes of determination form a continuous manifold, that the positions of objects of sense, and the colors, are probably the only simple notions whose modes of determination form a multiply-extended manifold. More frequent occasion for the birth and development of these notions is first found in higher mathematics.”

In particular, Riemann notes, such concepts of multiply-extended continuous manifolds arose in connection with his investigation of Abelian functions. To further our understanding of these manifolds, Riemann developed his method of representing them as Riemann’s surfaces.

These Riemann’s surfaces, like the Pythagorean sphere, Archytas’s construction, or Gauss’s surfaces, are not visible objects. They are the metaphorical images of thought-objects that express the relationships of a multiplicity of insensible universal principles which are acting together to produce a physical effect.

From this standpoint, Riemann was able to demonstrate that the essential characteristic of a multiply-extended continuous manifold is determined by the density of singularities that that manifold could sustain. This density of singularities, therefore, indicates the physical “power” or “potential” of the manifold. Just as Gauss expressed the power of an algebraic equation by a geometrical characteristic, Riemann showed that the power of a species of functions can be expressed by the geometrical characteristics of the Riemann’s surface. These species characteristics, Riemann emphasized, are the primary determinant of potential action in a physical manifold while other factors, such as variations among different functions of the same species, are of relatively little importance.

To better understand this concept, it is pedagogically most efficient to work through a series of animations illustrating Riemann’s surfaces, associated with manifolds of action of increasing density and power. Each type of Riemann’s surface is illustrated through an ordered series of animations, comprised of groups of triple-pairings showing the various characteristics of the surface from three different viewpoints: flat, 3D and stereographic projection. Each pair of animations shows an animated trajectory within a continuous manifold of physical action, and the effect on that trajectory of a complex function. To grasp the concept of the Riemann’s surface, the viewer must form a single idea of the effect of the function from the three pairs of animations. The function is not depicted directly in any of the animations. What is depicted is merely the effect. {The function is the concept, formed in the mind, of the characteristics of the principle which has the power to produce the depicted effect}.

However, as Riemann stressed, the change in density of singularities expressed by a Riemann’s surface depends, not on the particular function, but only on the {species} of function (i.e. algebraic, transcendental, elliptical, hyper-elliptical, etc.). Therefore, what is important to know is not the characteristics of any particular function, but the characteristics of a species of functions, and even more importantly, the characteristic of change from one species of function to a higher species.

This is illustrated in the animations by grouping an entire set of tripled-pairs into an ordered series illustrating the characteristic of an entire species. That characteristic is not directly depicted visibly in the animations. Rather it is evoked as a concept (thought-object) when the mind recognizes the total potential of that species to produce a change in the action. {It is this cognitive act, not the visible form, that constitutes the concept of the complex function}.

In this way, the entire set of animations walks the viewer through a succession of cognitive leaps. First, the leap required to put a pair of animations together. Then the leap required to put a tripled-pair into a unified idea. Then, the leap from one tripled pair to the next within a species. Then, the leap from species to species. Thus, it is not any single animation that presents the idea, but the cognitive change that is evoked as the viewer moves from one pair, to the triplet, from triplet to triplet, and from species to species. The animations themselves are not the idea. They are the vehicles by which the idea is evoked in the mind of the viewer. The viewer can then reflect on the succession of thought-objects that have been evoked through an investigation of the succession of animations. Reflecting back on the entire succession, the viewer can then form a single idea–a mental animation–of the different qualities of change embedded in the process.

It is this {change of change} that forms an initial idea of Riemann’s concept of Abelian functions.

Let's back up and repeat something for emphasis:

... Riemann was able to demonstrate that the essential characteristic of a multiply-extended continuous manifold is determined by the density of singularities that that manifold could sustain. This density of singularities, therefore, indicates the physical “power” or “potential” of the manifold.

Now it starts to become clear.  If you go to the page and watch the animations... aha! 
So that's what it's about!

The next thing that is needed here is a problem set to guide the student through this, so he can construct his own animations. That will be in The Book.

The next step beyond that is to connect this with the physical world. How can the "power" of a manifold be resolved into cause and effect relationships that an engineer can use?  How do you get leverage on it?  How would Archimedes, or Leonardo da Vinci, or Michael Faraday use the density of singularities of a manifold to design motors and other mechanisms?

J.A. Shercliff says (Vector Fields, page 37, slightly paraphrased)

Although in general the circulation of v around a closed loop is not zero, nevertheless there are situations where the circulation is zero around all loops, and then the field v is described as conservative. In the case where v is a force field like E or gravity, it is easy to see why the question whether the circulation does or does not vanish is of great interest to the engineer.

When v is conservative, it is possible to extract limited work from the field by making a finite excursion (not forming a loop) but it is not possible to do this indefinitely on a profit-making basis by repeatedly traversing the same closed loop. It is the fact that under some circumstances the electric field E is not conservative which puts the electrical power engineer into business, while in contrast the would-be gravity power engineer goes bankrupt!

So, is the Riemannian manifold power engineer in business, or not?

Felix Klein, On Riemann's Theory of Algebraic Functions and Their Integrals - This short, profound book by one of the great mathematical criminals sheds light on many of the anti-dummies pages.  (This statement would give Mr. LaRouche a heart attack.)  Klein also sheds light on the matter of algebraic curves and Riemann surfaces. This book can be read immediately after, or concurrently with, Visual Complex Analysis.

Monastyrsky, in the book mentioned above, says

There exists a remarkable interpretation of the theory of functions on Riemann surfaces, which owes its origin to Helmholtz and which explains why physicists had such confidence in the validity of Riemann's results. One can explain the theory of analytic functions of Riemann surfaces as a problem of physics. We will show that the theory of a stationary two-dimensional ideal incompressible fluid on a surface as a whole leads to the theory of analytic functions.

That is what Klein does. He uses "streamings" on surfaces to illustrate Riemann's theory. Monastyrsky summarizes all this in a couple of pages and comments:

We have obtained a remarkable result. All the singularities of the analytic function f(z) can be described in terms of the flow of a fluid with a certain number of sources, sinks, vortices, etc. Sommerfeld called this whole circle of ideas "physical mathematics":  "Here it is not mathematics serving the interests and problems of physics, but rather physics inspiring and governing mathematical ideas."

Monastyrsky also credits Klein's book (or paper, since it was originally published as a paper) with making an essential new contribution to the theory:

One must note that Riemann limited himself only to the representation of Riemann surfaces as a collection of sheets with a definite rule of gluing. The fact that a Riemann surface of an algebraic function... is topologically equivalent to an arbitrary orientable closed two-dimensional surface was clarified later, basically thanks to Felix Klein's paper of 1881/1882, "On Riemann's theory of algebraic functions and their integrals."

From Lemniscate to Langlands - You have to get pretty far into the anti-dummies pages to see the relevance of this. The lemniscate is first mentioned in #25 and figures prominently in later pages such as #49,  #51,  #54,  etc. The lemniscate also comes up in chapter 2 of McKean & Moll, when they are explaining what elliptic integrals are and where they come from.

At this point I want to mention a few other books that belong here. I will say more about them as I continue to rewrite this page.

V. Alekseev, The Abel Theorem in Problems - mentioned near the top of this page, in Arnold's lecture. This is Arnold's short cut to the center. Instead of trying to describe it, I am just giving a link to Amazon, where you can use the Look Inside feature to look at the table of contents and browse a bit. As the title indicates, this is a book of problems.

Henry McKean and Victor Moll, Elliptic Curves As Serge Lang said, elliptic functions have been at the center of mathematics since the middle of the 19th century. This is where geometry, analysis, and number theory all come together, and this is also where RFAD and mainstream mathematics start to come together. When you read this book, you know you have gotten out of the "bunch of random topics" school of mathematics, once and for all.

Francis J. Flanagan, Complex Variables: Harmonic and Analytic Functions. Flanagan starts with two chapters about calculus in the plane and harmonic functions in the plane, and then, having established the context, he segues into complex analysis. This is a highly original approach. It makes sense to do it that way.

Nevanlinna and Paatero, Complex Analysis. The names may not be household words, but the authors are heavyweights. This book covers topics such as elliptic integrals which are not found in Needham. This is the place to get the background for McKean & Moll, and for Riemann's article about Abelian functions.

Harold M. Edwards, Fermat's Last Theorem - A Genetic Introduction to Algebraic Number Theory. If you have tried to read Gauss's Disquisitiones and found it daunting, this book will help to prepare the way. This is for people who really love math. Remember what Klein said about Gauss doing lots and lots of calculations - that is what Edwards expects you to do. If you are the kind of person who can happily spend entire days filling page after page with calculations, you will have fun with this.

John McCleary, Geometry from a Differentiable Viewpoint. The title, of course, is a take-off on Milnor's Topology from the Differentiable Viewpoint. This is a very good introduction to advanced geometry (differential, not algebraic). The last chapter makes the transition from manifolds in R³ to abstract manifolds. The book leads up to Riemann's Habilitationsvortrag, which is included at the end.

Theodore Frankel, The Geometry of Physics.  Physics from the point of view of modern differential geometry and topology. This is a beautifully written and beautifully produced book. It is a pleasure to read. Well done, Dr. Frankel.

John Stillwell, Mathematics and its History.

One of the disappointments experienced by most mathematics students is that they never get a course in mathematics. They get courses in calculus, algebra, topology, and so on, but the division of labor in teaching seems to prevent these different topics from being combined into a whole. In fact, some of the most important and natural questions are stifled because they fall on the wrong side of topic boundary lines. Algebraists do not discuss the fundamental theorem of algebra because "that's analysis" and analysts do not discuss Riemann surfaces because "that's topology," for example. Thus if students are to feel they really know mathematics by the time they graduate, there is a need to unify the subject.

Jet Nestruev, Smooth Manifolds and Observables. Reversal of containment is one of my main themes, so this a book after my own heart. From the Preface:

Since there are many excellent textbooks in manifold theory, the first question that should be answered is, Why another book on manifolds?

The main reason is that the good old differential calculus is actually a particular case of a much more general construction, which may be described as the differential calculus over commutative algebras. And this calculus, in its entirety, is just the consequence of properties of arithmetical operations. This fact, remarkable in itself, has numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles. Our book explains in detail why the differential calculus on manifolds is simply an aspect of commutative algebra.

In the standard approach to smooth manifold theory, the subject is developed along the following lines. First one defines the notion of smooth manifold, say M. Then one defines the algebra F_M of smooth functions on M, and so on. In this book this sequence is reversed: We begin with a certain commutative R-algebra F, and then define the manifold M = M_F as the R-spectrum of this algebra.

... what is really new in this book is the motivation of the algebraic approach to smooth manifolds. It is based on the fundamental notion of observable, which comes from physics. It is this notion that creates an intuitively clear environment for the introduction of the main definitions and constructions. The concepts of state of a physical system and measuring device endow the very abstract notions of point of the spectrum and element of the algebra F_M with very tangible physical meanings...

--snip--

In the framework of this approach, smooth (i.e. differentiable) manifolds appear as R-spectra of a certain class of R-algebras (the latter are therefore called smooth), and their elements turn out to be the smooth functions defined on the corresponding spectra. Here the R-spectrum of some R-algebra A is the set of all its unital homomorphisms into the R-algebra R, i.e., the set that is "visible" by means of this algebra. Thus smooth manifolds are "worlds" whose observation can be carried out by means of smooth algebras.

Marcel Berger, A Panoramic View of Riemannian Geometry - Here is a snippet about Arnold:

Is the global theory of plane curves just beginning?... Very recently, Vladimir I. Arnol'd started a revolution when studying plane curves, hammering out a general frame to encompass these results... The curve has to be considered together with its tangent lines, so that the object to study is the set of all oriented tangent directions to the Euclidean plane, an object of dimension three (not two) and inside it the curve consisting of the tangents of a given plane curve. The three dimensional space has the topology of the inside of a torus... Deformations of curves will then be interpreted as wave fronts in a geometrical optics language, following flows given inside that torus by a more or less canonical vector field...

"Very recently" was 1994/95, when Arnold was almost sixty years old. So much for the notion that mathematicians are incapable of creative thought when they are over 40. It is possible to keep on reorganizing your mind at the most fundamental level at any age, and if you take care of yourself it isn't really that much harder in middle age, even late middle age. If you spend your life reading the best books you can find - if you spend decades continually trying to see the world steadily and see it whole - and you reach the age of 65 without getting senile, you can see a panoramic view of science that is not available at any earlier age. It's incredibly beautiful.

Felix qui potuit rerum cognoscere causas.

Most of the books mentioned here are part of mainstream mathematics. Students who read Riemann for Anti-dummies think of themselves as a mathematical counterculture, or even mathematical revolutionaries. There is an element of truth in this, but it is only a partial truth. Riemann is mainstream, and so is Gauss, perhaps to a lesser extent. The farther you get into advanced mathematics, the closer you get to Gauss and Riemann, especially Riemann.  Plato, Kepler and Leibniz are another matter. There is something revolutionary about RFAD, but it is important to understand exactly what is and is not revolutionary about it. Of all the great mathematicians of the past, Riemann is by far the most influential today, so it doesn't make much sense to use him as the poster child for the revolution.

They really ought to call it "Plato and Leibniz for Anti-dummies," or maybe "Kepler and Gauss for Anti-dummies."

The mathematics I am reviewing on this page is only part of mainstream mathematics, which also includes an enormous amount of "ugly scholastic pseudo-mathematics," as Arnold says. Gauss's theorems may be mainstream, but his way of thinking isn't. Most libraries don't even have a copy of Riemann's collected works, and if they do have it, it's hardly ever checked out. Nevertheless, at the highest level, the very best mathematicians do seek out the classics and read them. See the Lemniscate to Langlands link above. Abel said:  "Study the masters, not the pupils."  In each generation, some people still do that.

There are still some gaps that need to be filled in before this page is complete. I need to say something about Fourier analysis and functional analysis. The main thing I need to do is to go back and take a more fine-grained approach. Instead of a list of books, I need a list of topics, and a set of exercises, leading up to "Differential equations for Anti-dummies" and the calculus book that could be derived from it. First of all I need to make my own animations of the catenary problem, the lemniscate, and the complete quadrilateral. Then I need to apply those ideas to some specific differential equations. How do you get from the catenary to the heat equation? or from Gaussian logarithms to the wave equation?

They never say anything about that. They go on and on about the catenary, as if that is the only problem that ever comes up. Even with the catenary, they keep reprinting the same diagram over and over, and it appears that they are just repeating things by rote. I have read most of the anti-dummies pages, and I have seen no indication that they ever use these ideas to solve actual problems.

Coming back to the optical illusion that flips back and forth: It may be true that mainstream mathematics is flat and lifeless, and something vital is missing; but something is missing on the other side too. Mainstream mathematicians solve problems and prove theorems. As far as I can tell, the LaRouche school of mathematics does not do either of those things.

One of my oldest friends is an engineer who designs airplanes. He does not use LaRouche math. He uses evil Newtonian math. His planes fly.

Are there any engineers who design airplanes (or anything) with LaRouche math?  Do the planes fly?  Inquiring minds want to know.

As far as that goes, the same question can be asked about Arnold's mathematics. Do engineers actually use his abstract approach?  Maybe they do in Russia, but Russia is not known for its excellence in engineering. I think most practicing engineers would rather get their math from old-fashioned books like Granville's calculus book and the o.d.e. book by Tenenbaum and Pollard. Those books reflect how engineers actually solve problems. I have the 1911 edition of Granville's book. Mainstream modern mathematicians despise it. Arnold would like it, or at least he would like some things about it, since it is sort of an American version of Goursat. For example, Granville tells you about double points of algebraic curves, a topic unknown in more recent calculus books. A lot of engineers keep a copy of Granville handy for reference. It's their secret weapon.

Looking over the previous paragraph, it occurs to me that there is a contradiction, and it goes all the way back to the top of the page. Arnold likes the old books, the same books engineers like, but he himself writes "modern" books that are unintelligible to engineers. They don't want to hear about one-parameter groups of diffeomorphisms.

I can imagine a situation where an engineering student, having taken a d.e. course using Arnold's book, is asked  "What is XY?"  and he replies "YX, because vector fields are rectifiable." (To complete this joke, I need to think up something specific for XY and YX; the idea is that it would be as absurd as "3+2, because addition is commutative.")

When I started writing this page, I was not aware of Ordinary Differential Equations by Morris Tenenbaum and Harry Pollard. I just discovered it recently (2011), and I still have not really read it; I have just looked through it. It is pretty close to what I had in mind - the authors take differential equations as the starting point and introduce calculus as needed. Not only that - what they say about calculus makes sense. I wish I had read this book in high school, instead of the books I did read.

No, wait a minute. In my senior year, I read Calculus by Apostol, Introduction to Matrix Algebra by the School Mathematics Study Group (SMSG), Introduction to Inequalities by Beckenbach and Bellman, Introduction to Modern Algebra and Matrix Theory by Schreir and Sperner, Foundations of Analysis by Landau, and What is Mathematics by Courant and Robbins. The SMSG matrix book and the inequalities book were excellent.  Schreir & Sperner was a waste of time. I was not ready for it and didn't know what to make of it. Landau was of doubtful value, and Courant & Robbins was a disaster.

What I meant to say was, I wish I had read Tenenbaum & Pollard instead of Apostol. But that is still not quite right. Apostol was not the wrong calculus book, it's just that I read it in the wrong way.  I thought mathematics was all about definitions and "foundations", and calculus was all about limits. I think I got that idea from Courant & Robbins. I thought I had to understand the foundations of calculus before I could do anything else. I refused to even work problems, if the statement of the problem appeared to be nonsense. I am an intellectually fastidious person, and I did not want to soil myself with any kind of superstition. If an idea does not make sense, I will not let it enter my mind. I was the class atheist starting in the 9th grade. No hocus pocus for me. I thought differentials and infinitesimals were on the same level as astrology. I would not touch them.

When I say differentials, I am not talking about differential forms. I mean expressions like "Choose a differential volume dV contained between two planes perpendicular to the x axis and a distance dx apart," where dV and dx are supposed to be "infinitesimal".

There was more to it than that. It was not just infinitesimals that bothered me. In fact I only encountered infinitesimals in books about physics and engineering. Calculus books did not use infinitesimals in an objectionable way. Apostol did not mention them, as far as I recall. Even Granville is careful to explain that there are no infinitely small quantities. What bothers me about calculus books is that they define dx to be one thing, and then use it as if it is something else. dy/dx is defined as an inseparable unit in which the parts to not have meaning by themselves. Fair enough. But then they blithely go on and say "multiply both sides of the equation by dx and integrate." Well excuse me, but if dx has no meaning by itself, then you can't do that. If you multiply your phone number by your birth date, what do you get?  Nothing, those things can't be multiplied and divided.

In 1968, Spivak's Calculus appeared. Finally, a calculus book with no differentials!  He only talks about derivatives. Instead of dy/dx, it is always f' in Spivak's world. But there were two problems with this. After reading Spivak, I still had no clue about differential equations. It occurred to me that there is no such thing as a book about "Derivative Equations." Apparently people who work in this subject find that differentials are the natural way to think about it. The second problem was that I discovered that the masters themselves used differentials. At the end of his book, Spivak has suggestions for further reading. He says when you are ready to start reading original papers, the ideal place to begin is Abel's paper in which he proved the binomial theorem in the general case. So, I sent off for Abel's Oeuvres Completes, which only cost $22.50 for the two volume set in those days. - I still have the books right here in front of me, and I just looked on the flyleaf to check the price. When the books arrived, I discovered to my dismay that Abel's papers are full of dy/dx.

By that time I had spent enough time in the library to know that all the old books were full of differentials. So I started to wonder what dx meant to Abel and his peers. If it was good enough for them, why can't I use it?  Were there no valid math books prior to 1968, when Spivak's book appeared?  If infinitesimals are on the same level as astrology, then why do planes fly?  Engineers have been using infinitesimal arguments to arrive at correct answers for more than 200 years. I should be able to do it too. Applied mathematicians have been saying "multiply both sides of the equation by dx and integrate" for a very long time, and they always get correct answers. The operation of multiplying by dx must mean something. It must make sense geometrically. What they say about it does not make sense, but that is a different question. It should be possible to look at it, i.e. look at the multiplication operation, in such a way that the meaning is apparent. As I continue to rewrite this page, one of the things I want to do is make animations of "multiply both sides of the equation by dx" so the meaning is apparent. When Gauss and Abel used differentials, they must have had an image in mind, and it should be possible to reconstruct that image and demonstrate it with computer graphics.

Actually I think the "infinitesimal" concept might be salvaged. A differential like dV is not an infinitely small number. It is not a number at all. It is just a shape without size. For anyone who is familiar with topology, that concept should not be difficult.

I have totally gotten over my aversion to applied math. I have the good fortune to live near one of the last remaining bookstores that has a collection of old technical books. I recently found Analytic Mechanics by Faires and Chambers, originally published in 1934, revised in 1952. It is about mechanical engineering, and it is full of infinitesimal arguments. This is where the example above came from - "Choose a differential volume dV", etc. This is the kind of book that I thought was beneath contempt when I was a student. Now I think it is as beautiful in its own way as The Geometry of Physics. This is how calculus is used by people who actually use it to design mechanisms.

Coming back to my high school books, and Landau's book specifically: the very idea of foundations is a misnomer. Mathematics does not have or need foundations. Set theory does not support the rest of mathematics. It was tacked on as an afterthought. The fundamental theorem of algebra is no more secure now than it was in the 19th century. Set theory does not make it any more secure than it already was. On the other hand, I would not be able to write this paragraph, or this page, if I had not started trying to understand everything philosophically in high school. Reading Landau's book would have been a valid thing to do, if I had done something else as well.

One time Mozart described how he composed music: "First I see it, then I hear it, then I write it down." I skipped the first step. You have to see it. Even at the time I knew I was missing something, and I have been trying to go back and do it right ever since. If I had  R_E_A_D Tenenbaum & Pollard instead of READING Apostol, it would not have made much difference.
If I had read Tenenbaum & Pollard AND solved most of the problems AND drawn lots of direction fields and integral curves, that would have gotten me off to a good start, but I could have done that with Apostol. What Apostol says about dy/dx does not make sense, but I should have brushed that aside and plunged ahead. If I had spent my senior year working problems, i.e. exploring mathematics instead of talking about it, I would have learned something.

When you solve enough problems, eventually you get a Gestalt of how it works. dy/dx may not make sense philosophically until many years later. Don't worry about it. Just keep on drawing direction fields and integral curves and surfaces until you see what's going on. It does make sense if you persist.

Unfortunately, nobody told me this when I was 17.

If you want to do Gaussian mathematics, you have to do it the way Gauss did it. There is no short cut. Mozart didn't just "see" the music, he also spent many hours practicing the piano.

Think infinitesimal!

With a d.e. course based on harmonic relationships, would MIT graduates be more creative scientists and engineers?  Maybe, maybe not. That remains to be determined. The idea here is to think about atoms, plasmas, and cells the way Leonardo would have thought about them.  I don't really care if we use LaRouche math or evil Newtonian math. Whatever it takes.

In The Metaphor of Perspective, Pierre Beaudry said

And since the Jacobin terror had destroyed the laboratories and guillotined the scientists (such as Lavoisier), there was no better and more necessary idea than to establish a curriculum based on geometric discoveries, as the catalyst that would lead to the discovery of the creative process of the human mind, and give France the scientists, the engineers, the metallurgists, the chemists, and so forth, that the nation-state needed so desperately. And so began the real French Revolution when, in 1794, Robespierre was defeated by Carnot, and the Committee of Public Safety passed a resolution for the creation of the Ecole Polytechnique and the Ecole des Arts et Métiers [Arts and Trades]. As Poncelet, one of the very first student brigade leaders would later express it,

We do not intend to teach you a method and a process for each art, but instead what is the principle common to all of the arts ... with the purpose of making inventors out of you, inventing new machines and new processes.

The LaRouche school seems to have lost sight of this, but as far as I am concerned this is still what it's about. Most of the inventions I have in mind are at the atomic scale, and the "density of singularities" concept definitely comes into play at that level.

Looking over what I have written so far, I think what I am trying to say is that mathematics is like a hologram. When you cut off a little piece of a hologram, the image is still there in that little piece, it just isn't as fine-grained as it was in the original hologram. You lose the details, but the shape is there. Likewise, mathematics is "all there" fairly early on, in the first two or three years of college. You don't have to wait until graduate school to see the "center." If you start out in the right way, you can see it almost at the beginning. In fact if you don't see it at the beginning, you aren't going to see it in graduate school, either. You may get a Ph.D., but if you are in the "bunch of random topics" school of mathematics, you are just going to know more and more random topics.

Gauss saw mathematics whole when he was about 19, and spent the rest of his life filling in the details. Newton saw it too, when he was about 21.

The RFAD pages are not a substitute for the original works of Gauss, Abel, and Riemann. As Harold Edwards said in another of his books, reading textbooks instead of original papers is like bringing a sack lunch to a banquet.

Abel on Analysis - Papers on abelian and elliptic functions and the theory of series - an English translation (finally!) of some of Abel's papers, published by the Kendrick Press.

Bernhard Riemann, Collected Papers - also in English, also by the Kendrick Press (bless them!)

In LaRouche's universe, Aristotle is supposed to be the ur-villain, so before closing, I want to mention something René Thom said about Aristotle. In his youth, Thom was one of the outstanding mathematicians of the mid-20th century (Fields Medal in 1958 for his work in topology), and then in later life he moved on to other things. He published Structural Stability and Morphogenesis in 1977, and Semiophysics: A Sketch in 1988. The second part (chapters 6, 7, and 8) of Semiophysics is based on his study of Aristotle. He looked at Aristotle with fresh eyes and saw something no one else had ever seen, something only a topologist could see. In the Foreword, he says

"It was only quite recently, and almost by chance, that I discovered the work of Aristotle. It was fascinating reading, almost from the start. I knew of course that the hylomorphic schema - of which I make use in catastrophe formalism - originated in the Stagirite's work. But I was unaware of the essential fact that Aristotle had attempted in his Physics to construct a world theory based not on numbers but on continuity. He had thus (at least partly) realized something I have always dreamed of doing - the development of a Mathematics of the continuous, which would take the notion of the continuum as point of departure, without (if possible) any evocation of the intrinsic generativity of numbers."

I also want to mention this very sad page:

New Bretton Woods: Russia's Role in a Recovery by Lyndon H. LaRouche, Jr.

This article appeared in the September 5, 2008 issue of Executive Intelligence Review. Leaving aside the main content of the article, the important thing for my purposes is the introduction:

A fraudulent representation of the Franklin Roosevelt Bretton Woods system was recently launched at Modena, Italy, by a pair of seasoned turncoats, Jonathan Tennenbaum and Paolo Raimondi. The targets of their fully intended fraud included both important Russian scientists and notable Italian political figures. That pair of hoaxsters, who had gone over to the proverbial "other side" during recent years, represented a small, London-oriented circle of hoaxsters which have put themselves out for sale in search of hire and fame to be supplied by British Euro-oligarchical intelligence circles. The method by which that pair of hoaxsters perpetrated their fraud on the Russian and other guests, was passing themselves off, flagrantly, by representing themselves as being currently associated with me.

Jonathan Tennenbaum is co-author of the essay mentioned above, How Gauss Determined the Orbit of Ceres. He has been associated with LaRouche for at least 25 years. If you read the various LaRouche websites - 21st Century Science and Technology, the Schiller Institute, the Executive Intelligence Review, Fidelio, Dynamis - his name comes up many times; for example, Eurasian Infrastructure and the Noösphere and The Isotope Economy.

Now he is a "seasoned turncoat" who has committed "fully intended fraud", and so forth. In the best Marxist tradition, he has been declared a nonperson. The poisonous atmosphere of the LaRouche organization comes through clearly here.

Near the end of the upper part of this page, I said that Lyndon LaRouche will be remembered as the most important philosopher of his generation. Well then, why not join him?  This is why.

If anyone reading this page would like to study Gaussian math without joining the LaRouche organization, let me hear from you. I can tell from the log files that this page has been read, or at least visited, thousands of times since its inception (I am writing these last paragraphs in November of 2011, so the page has been here for more than five years). If all those people were connected, and working together, we could accomplish a lot more. I would particularly like to hear from anyone who is doing computer graphics along the lines I have suggested above.

The great schools of the past - Plato's Academy, the Ecole Polytechnique and the Ecole des Arts et Métiers, and the 19th century German universities - can be revived in our time.

my address:

math.to.lyle-----at-----recursor.net

my home page

"I keep the subject constantly before me and wait until the first dawnings open little by little into the full light." - Isaac Newton