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 1990's 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.
You can click here to jump farther down the page and skip my critique of Mr. LR and his associates.
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.
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 founded by Aristotle and Galileo. 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 19-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.
Some of Gauss's praise of Euler may contain an element of sarcasm, since he points out that "this shrewdest of men" failed to see things that were obvious (to Gauss, at least), but it's always good natured sarcasm. It's friendly ribbing, delivered with a twinkle in his eye. I think most of his praise was sincere. His view was that Euler was indeed a man of immortal glory, but maybe not always as shrewd as he could have been. The vicious 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.
The belief system one finds on these pages is also paranoid in another sense: everything fits. All the dots are connected. One of the hallmarks of a paranoid belief system is that there are no loose ends. Everything is accounted for, everybody has a role to play, good or evil, and everything that happens is part of some grand conspiracy. As my uncle Pete used to say, never trust anybody who thinks he has all his shingles nailed down.
According to Mr. LR, mathematical and philosophical errors have political ramifications. Maybe so. Well then, what is at stake here? What is he driving at? What kind of politics does he advocate? Two of his heroes are Franklin Delano Roosevelt and Martin Luther King, Jr. The mountain has labored and given birth to a mouse. If Mr. LR had a new and inspiring vision of human society, I might be willing to make allowances for a certain amount of paranoia, but FDR and MLK? Did we come all the way out here for this?
Mr. LR is also a drug warrior. He used to publish a magazine called "War on Drugs." Darryl Gates, who was police chief of Los Angeles at the time, said anybody who smokes marijuana should be taken out and shot. A few years later he backed off and said the statement was hyperbole. Mr. LR is not given to half measures, and if he ever came to power (which he won't), I think he would actually do it. He would do the same thing to pot smokers that Mao did to the prostitutes in China. Just kill us and get it over with.
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 1960's. 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 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, and I could run circles around the other students in the class. One time the teacher, who was just finishing his dissertation, said "Mr. Burkhead knows more about advanced calculus than anyone in this room," leaving open the question of whether he included himself in that statement. If he was just referring to the other students, the statement would be obvious and pointless. So naturally I thought that I understood calculus.
As the years passed, it 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 - "calculus for understanding of curved lines" as l'Hospital said - 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. 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, 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.
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 with these:
#27. The Fundamental Theorem: Gauss's 'Declaration of Independence'
"Looking back on his dissertation 50 years later, Gauss said, '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."
"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."
This page (i.e. the page you are reading, not the one I just linked to) relates to my page about coherent energy from the subatomic domain. 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.
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 want to come down too hard on LaRouche. His views on economics make a lot of sense. The same things Gauss said about Kaestner can also be applied to LaRouche: sometimes he may be a silly old man, but he is also an extraordinary man. His horizons are wider than just about anybody's, and I respect that. We all have our flaws. I think his heart is basically in the right place. I realize that he probably would not say the same about me.
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.
My own view is that mathematics does not have or need an ultimate foundation, either at the bottom or at the top. We start in medias res and work our way down towards set theory and up towards higher abstractions, or higher manifolds, "powers," metaphors - however you want to frame it. Mathematics is not a crystalline structure, it is an ongoing dialectical activity. Metaphors lead beyond themselves to new metaphors.
My premise here is that there must be some young mathematicans who are intrigued by Arnold's lecture and the anti-dummies pages, and who want to learn more about this kind of mathematics.
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, he's 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.
When I was a student, I was just taught a bunch of random topics. We were not supposed to ask "What is the point of all this?" There was no point to it. There was no goal that gradually came into better focus as we approached it. The idea that mathematics fits together into a coherent whole was not on anybody's radar screen. The 1960's were a mathematical dark age. Some professors thought mathematics is just a game in which you make up arbitrary axioms and see what follows from them. This is what led, a few decades later, to the classrooms described by Arnold, where if you ask a pupil what 2 + 3 is, he says 3 + 2, because addition is commutative.
I think things have improved a bit since the 1960's. There are mathematicians who think mathematics is about something, and they may be in the ascendance now. Nevertheless the view that math is a game with no content still has powerful advocates. If you want to be a real mathematician, you can't depend on anybody to give you an education. You have to take matters into your own hands. I have a goal now. I am still a long way from getting it in focus, but at least it is finally coming into view.
The rest of this page is just beginning to take shape. So far it is just a list of links and books, with some comments by me. The list has been changing from day to day. The idea of making it into a "curriculum" only occurred to me yesterday (6/25/06), and I realize it may be a bit premature to call it that. A lot of work remains to be done. I look forward to it. Eventually this page will tie in with the rest of the site. There will be a philosophy curriculum, too. I have had writer's block for some time now. This site has been in the doldrums, probably because I have been trying to write about unpleasant subjects that I don't really want to think about. Writing this page has finally broken the block. When I write about what I want to write about, it's effortless.
So, here goes:
First, some transitional material. This is intended for readers who have actually read the links given above. Doing the following exercises will nail down your understanding.
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 cm,n xmyn. 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, ... Pn , and let rj denote the distance from O to the point Pj. Now mark the point Q on the line at the distance rq from O, where rq is the harmonic mean of r1, r2, ..., rn. 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.
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 the main thing 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. The discussion of the cross-ratio beginning on page 39 is essential for just about everything that follows below.
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.
Victor Gutenmacher and N.B. Vasilyev, Lines and Curves: A Practical Geometry Handbook - 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."
Most of the problems in this book involve motion, so they could be recast as differential equations - but no calculus is used in this book! Of all the books mentioned here, I think this one is my favorite. When I read it, the lights come on. I see bright, vivid mathematical images.
Kenji Ueno, Koji Shiga, and Shigeyuki Morita, A Mathematical Gift - This is a three volume set, of which vol. 2 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), arithmetic and geometric means, and the Poncelet Theorem. The main shortcoming of this marvellous little book is the lack of problems. However, it leads naturally to the next book, which is a set of problems.
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 Search Inside feature to look at the table of contents and browse a bit.
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 differential geometry, advanced calculus, complex analysis, 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. "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 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.
I.R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces - The title is meant to be a takeoff on the title of the original Bernouli/l'Hopital calculus book, Analyse des infiniment petits pour l'intelligence des lignes courbes. This book is very much in the spirit of Riemann for Anti-dummies. Fermat, Huygens, and Leibniz would love it. It also connects RFAD with Arnold, who is mentioned several times. For example:
"We shall prove that the evolute of a plane curve does not have any points of inflection. However, as L'Hopital (or was it Jean Bernouli?) first remarked, there is nothing to stop one from swinging a pendulum from a curve with an inflection. The resulting family of non-regular evolutes has an intimate relationship with the group of symmetries of an icosahedron - a deep and mysterious fact only recently noted by the Russian school of singularity theorists under the leadership of V.I. Arnol'd."
Even though the title of this book imitates the title of the original calculus book, it is not a calculus book. The author assumes that the reader already knows calculus. He only discusses differentiation. The rest of calculus is not considered here. He does not go into integration, the fundamental theorem, and how to set up differential equations. Nevertheless this book is a step in the right direction, and it could be morphed into Calculus for Anti-dummies.
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:
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."
Unfortunately, Solid Shape is an extremely rare and hard to find book, even more than Understanding Perspective. I found it in a library. When I tried to order it, it was always out of stock. For a long time I couldn't even find a used copy to buy. I finally found one in February of 2007. Used copies are almost impossible to find, because anybody who owns the book doesn't want to part with it. Everyone who is interested should try to order it directly from MIT Press. They won't fill your order, but that's not the point. The idea is to get their attention. If they get a steady stream of inquiries about Solid Shape, maybe they will decide there is enough interest to justify bringing out a new edition.
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. 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. More precisely, I should say it's possible to continue doing this in middle age if you have been doing it all along.
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 typical 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.
Hans Schwerdtfeger, Geometry of Complex Numbers - More background for what follows. Our old friend the cross-ratio appears once again:
In the geometrical discussion of paragraphs 1-4, the complex number notation provided a convenient alternative for the cartesian coordinate method. From now on the complex numbers will have a deeper significance. The notion of cross ratio has its origin in real projective geometry where it is defined for a set of four points on a straight line, for instance on the x-axis. This line (as well as any other line) may be considered as carrier of the field of all real numbers, which, for the purposes of real projective geometry, has been been completed by one point at infinity. For a set of four non-collinear points in the plane of projective geometry the cross ratio is not defined. It is just the association of the points of the plane with the complex numbers and the completion of the plane by the one point at infinity, which enables us to extend the definition of the cross ratio to sets of four points in the completed plane. This suggests another interpretation of the completed z-plane, viz. as the complex projective straight line.
Coming back once again to Arnold's statement, "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..." This reminds me of something in Applied Differential Geometry by William Burke. Burke says "Vector fields that do not commute are called anholonomic. If two transformations commute, then the system would never leave a 2-surface. This obvious result is called the Frobenius Theorem." An astute Amazon reviewer picked up on this and said "Now after reading about the Frobenius Theorem elsewhere, few people would call it 'obvious.' Nonetheless, when you read Burke, you will agree." A real mathematician is someone who can not only see curves and surfaces in his mind's eye, but can also see the obviousness of deep theorems, and can describe what he sees in such a way that the reader or student can also see it. This is a rare thing. I am reminded of what some people say about Rembrandt's paintings - after looking at them, you see the world differently. You see the way Rembrandt saw.
So our goal here is to see the obviousness of the connection between integrals of rational functions along algebraic curves, and the genus of the corresponding Riemann surface. This takes us into deep water. Unfortunately this subject has not yet found its William Burke, or its Rembrandt. I am not really satisfied with the next two books, and this part of the page may change eventually, but for the moment this is what I have come up with.
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. To pursue that idea you have to move on to the next book.
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.
Frances Kirwan, Complex Algebraic Curves - Kirwan finishes the story Gibson started. You can get a detailed table of contents here. But the thing is, this is a senior level book which presupposes quite a bit of mathematical sophistication. I don't know of any shorter way to reach this point. I am still pondering the question of how this material could be presented to freshmen, without all the machinery Kirwan brings to bear on it.
I agree that it should be presented, however vaguely, at the beginning, to get one's education off to a good start. And then at the end of each school year, if not more often, the student should ask: How close am I to the center? What gaps have I filled in, and what gaps remain to be filled in? I am still asking those questions.
Michael Monastyrsky, Riemann, Topology, and Physics - The first part of this book is a biography of Riemann and an overview of his mathematical work. The discussion of Abelian integrals and Riemann surfaces is as clear and intuitive as possible, assuming the reader has the necessary background. However, Arnold says his professor told him about this when he didn't have the background. The question still remains - how do you explain this to someone who is just now encountering the concept of an integral, and has no idea what "integrals of rational functions along an algebraic curve" might be?
Putting it another way: if you are teaching calculus to beginners, how do you introduce the concept of an integral so that Arnold's statement is intelligible to them? If you tell them that an integral is, by definition, the area under a graph, then the question about integrals of rational functions along an algebraic curve just doesn't compile. The area under a graph is not "along" anything. But if you start with differential equations, and let integration emerge from that context, there is some hope of making this clear.
Of all the things Riemann did, the "Riemann integral" is the most famous - and it is the least important.
It is interesting to juxtapose the books I have listed here with the Anti-dummies discussion of Abelian functions: The Dramatic Power of Abelian Functions (#54) and To What End Do We Study Riemann's Investigation of Abelian Functions? (#61). Like #59, these are long pages with animated gif's. Unless your computer has a lot of memory, you may not be able to run any other programs while these pages are loading. When you do get them loaded... "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." The RFAD pages are not talking about quite the same thing Arnold had in mind, but that's ok. There is more than one way for Abelian functions to capture one's imagination. There is more than one way to approach the center.
The authors of the next book mention Kirwan as one of several books that would be useful to read in conjuction with their book.
Henry McKean and Victor Moll, Elliptic Curves --- Drumroll, trumpets --- We finally arrive at Gaussian mathematics. 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. McKean and Moll casually refer to Gauss as if he is part of the contemporary mathematical scene. Which he is! Eventually I want to have a more detailed discussion of this book, but that will have to wait. Returning to Riemann -
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, and on Arnold's enigmatic statement that I was discussing above. 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, 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.
Marcel Berger, A Panoramic View of Riemannian Geometry - Gauss, Riemann, and V.I. Arnold, all in the same book. 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.
For some reason Marcel Berger does not mention Lyndon LaRouche in his book about Riemannian Geometry. Must be part of the oligarchical plot.
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.
Euclid's Elements - The Riemann for anti-dummies pages presuppose some background in classical mathematics. Books 9, 10, and 13 of Euclid are particularly important. This is a link to an online edition with Java applets by D. E. Joyce.
Euclid's Elements - a new Green Lion edition, much more readable than the Dover edition. It should be read backwards, of course :)
Applonius, Conics - Another Green Lion book. For our purposes here, Applonius is even more important than Euclid.
The Green Lion Press - They also publish new editions of other scientific classics. Here is what one of their contributors says about Faraday. RFAD enthusiasts should recognize some familiar themes:
"What then is Faraday's concept of science? It is evident that he is not merely an experimentalist, cleverly providing data for theoreticians to work up into theories. On the contrary, Faraday had one of the most fertile and insistent of speculative minds; in a certain sense, he was constantly producing new hypotheses and his mind was constantly reasoning from them. The result of this, reported in the thousands of paragraphs of the Experimental Researches and the still more numerous paragraphs of the Diary, is not theory, but a vast weaving and unweaving of powers, a process of discovery and identification, a great, highly unified formulary for the production and classification of effects."
IEEE International Conference on Plasma Science -- Continuing Faraday's work... maybe. Judge for yourself. This is not mainstream. Proceed at your own risk :)
Carl Gauss, Disquisitiones Arithmaticae - The Anti-dummies pages will get you started, but there is no substitute for going to the source. The only way to learn Gauss's point of view is to read what he wrote in his own words, not filtered by anybody. An English translation by Arthur A. Clarke appeared in 1966, and is still in print, published by Yale. Springer published a new edition in 1986, in which a few errors were corrected. I went ahead and splurged on the Springer edition, but if you are on a budget, it's better to read the Yale edition than not to read the book at all. When you read any math book, you have to be on the lookout for typos and slips of the pen. The fact that a few errors exist should not stop you from reading the book.
I never understood the importance of number theory until recently. I was first exposed to it in high school - I mentioned earlier that I went to a NSF course for (more or less) "gifted" math students, and number theory was one of the things we studied. Like everything in those days, the NSF course belonged to the "bunch of random topics" school of mathematics. From that course, and from a couple of books that I read, I got the impression that number theory was a frivolous subject, on the level of magic squares, and I retained that opinion until I started reading the RFAD pages and then the Disquisitiones. Now I am finally beginning, but only just beginning, to see how number theory fits into the mathematical universe. I am also beginning, but only just beginning, to appreciate who Gauss was.
When I read the Disquisitiones, I try to imagine being Gauss, as he wrote the words I am reading. I try to join my mind with his, and let his thoughts shape my thoughts, like a morphogenetic field. Mozart wrote a letter in which he described how he composed music: "First I see it, then I hear it, then I write it down." Gauss must have composed his mathematics the same way. What did Gauss see in his mind's eye, before he wrote it down?
What is needed is a book and/or website called Visual Number Theory, in which the phenomena of number theory are shown emerging - from where? - as Platonic "shadows."
We have hardly scratched the surface of what can be done with computer graphics. When full-fledged nanotechnology and femtotechnology appear, it will be possible to create the "Gaussian computer" that I was speculating about earlier.
Carl Gauss, Theory of Motion of the Heavenly Bodies Moving about the Sun in Conic Sections The question I have is, if Gauss had been a young man in the 1920's, how would atomic physics be different? Not to take anything away from Pauli, Dirac, etc. - if Gauss had been born in 1900, he would not have complained about the shallowness of his contemporaries! - but he would have approached the subject from a different point of view. If he wrote a book called Theory of Motion of Electrons Moving about the Nucleus in Conic Sections, what would it look like?
That question leads back to the coherent energy page.
As of March, 2007, I think this page is just about complete, to the extent that it's just a list of books. I can think of many other books that might be included, and I may add a few more, but that's not important. Now I need to go back and take a more fine-grained approach. Instead of a list of books, I need a list of topics, or a list of exercises, along the lines of Understanding Perspective, leading up to "Differential equations for Anti-dummies" and the calculus book that could be derived from it. I already know what Arnold has to say about differential equations, so I'm going to be coming at this from the RFAD side of things, not the Arnold side. 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?
I also need to pause and rethink what I am doing. I'm not going to spend the rest of my life writing math books. Writing websites and books is not an end in itself, it's a means to an end. As I said above, the main question is, With a d.e. course based on harmonic relationships, would MIT graduates be more creative scientists and engineers? The "curriculum," after all, is basically for my own use. The idea here is to think about atoms and plasmas the way Leonardo would have thought about them.
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 year or so 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.
That's one reason why I like "Riemann for Anti-dummies" so much. It gives you a holographic view of mathematics. They keep going back to their starting point and extending the discussion to more and more advanced material. This is totally different from the way mathematics is usually presented. Even the best conventional books, such as Visual Complex Analysis, are not a substitute for the RFAD pages.
The hologram metaphor does not tell the whole story, however. It misses the self-reflective aspect of mathematics. Math is kind of like Lisp, but it's "curved Lisp," or "harmonic Lisp" - I'm straining for something that's not quite within my reach yet. As I said, there will eventually be a philosophy curriculum - philosophy for geeks - "No one ignorant of namespaces may enter here" - and this page will be connected to the rest of the site.
my home page
my address:
math.to.lyle-----at-----recursor-----dot----- net
"I keep the subject constantly before me and wait until the first dawnings open little by little into the full light." - Isaac Newton