Exchange :: Self Study

The Self-Study of Mathematics

Author(s): R. Earle Harris
Released under Creative Commons-3.0-SA (as modified in Guidelines)

Since the early 21st century, serious mathematical works, from the late 18th century to the present day, have proliferated in PDF format on the Internet. The opportunities for self-education in mathematics are now amazing. But while mathematicians may know what they want and what is useful, many people interested in mathematics find the choices mind-numbing. The Open Mathematics Depository was founded in 2017 as a selective library of these works. This text is an introduction to the methods and materials for self-study of mathematics and assumes only that the reader is interested in learning (and has a sense of humor).

Since June 1, 2017, 78 people have read this exchange.

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Guidelines

This is an open exchange and an experiment in collaborative writing. Comments are welcome. You are also welcome to join the exchange. Then you can freely contribute directly to the work. Posts are straight HTML, using header-tag-only posts, H3 through H5, for organization. Please either fit your posts into the existing Mind, Methods and Materials H3 sections or add new H3 sections at the end. Technical instructions are on the post-form page. More guidelines will be added as necessary. While, as original author, I reserve the right to control access to the exchange and to edit the contents at any time, the intention and hope is for free and mutual collaboration.

Legal Notice: The Creative Commons license above is hereby modified in the following way. Anyone is free to commercially publish these contents. But no one may publish this work commercially without first contacting the authors through the project email. If you are not an author on the author list above, you are entitled to no more than 15% of any monies coming from such an action. The authors must be included in any agreement and are entitled to their percentages calculated, using the Unix wc tool, as the word count of their posts as a percentage of the word count of the entire text minus the word count of any posts by authors not on the author list. In order to be considered an author, you must contact me through the project email and have your name added to the author list. And you must notify the exchange if your email changes. Failure to reply at any time will remove you from the author list and from your rights to the contents. You are welcome to author posts without being added to the list. But you must be on the list before any commercial agreement is made in order to participate in any profits. Posts by authors not on the list are considered the property of the exchange until their author claims authorship as above. Finally, all comments become the absolute property of this exchange.

The above modifications are an attempt to combine open licensing with economic justice. (And writing as if you are getting paid by the word is not permitted.)

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Mind

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The Short Mind

If you are reading this, you are interested in mathematics. And there is no conceivable reason for you not to study mathematics. I want to assure you that, no matter what else you have been told or believe, your mind is perfectly adequate for the serious study of mathematics.

But if you are like most people, this mind which is more than sufficient is not the mind you are used to. Let me explain. Every demonstrable spiritual discipline recognizes two minds or states of consciousness, if you will. We'll avoid ideologies and call them the short mind and the long mind. Let's talk about the short mind first.

The short mind is short in attention span, in outlook, and in wisdom. Everyone experiences this mind. It's the one that rattles on and on, all day long if you let it, like marbles in a commercial drier. Everything that is unpleasant or limiting or discordant is a product of this short mind. It's like a radio that comes on when you wake up and keeps you up with its fretting noise when you try to sleep.

And this mind, which people have told you is limited and of which you have believed the same thing, is not even yours. Except for regional variations and a handful of personal preferences, your short mind is the same as everyone else's. It's the mind we all get for free, the lowest common denominator of current human consciousness. And it is totally lacking in originality and individuality.

This unoriginal mind is the source of all selfishness, egotism, and human will. And it has never accomplished anything good. You can look at your own life and see that everything you have done that is worthwhile has come from another state of mind. It has come from a mind which is independent and creative and individual. And unlike the short mind, this real mind is yours and yours alone.

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Equality of Mind

The short mind is quick to make comparisons of greater or lesser intelligence. And many people let themselves be limited by such false judgements. A habit of this mind is to call someone a "genius" and thereby place a great distance between what another has achieved and what one believes oneself able to achieve. But there are no geniuses.

Let us briefly consider Isaac Newton. By birth, he was supposed to be a farmer. But because he was clever in creating mechanical solutions in public school, a well-to-do uncle procured a place for him in Cambridge. Up to this point, he showed no particular talent for mathematics.

At that time, professors lectured irregularly; wealthy students hired tutors; and beyond that, you were on your own. The first book Newton purchased for study continually referred to Euclid. So he bought Euclid and, like you will be if you look at it, was surprised by how simple it was. So he began to study mathematics. Eventually, he was known to work at his desk for nineteen or twenty hours a day.

Given this diligent effort, it is not surprising that he, like Leibniz, plucked the low-hanging fruit of the calculus. For a generation, calculus had been staring mathematicians in the face. Newton, in this case, basically put two and two (actually, curvature and quadrature) together. But Newton could not explain why his calculus worked and was mocked when he tried. Further, he was completely wrong about the nature of light. And his notion of centripetal force, while it works, appears to be the first epicycle upon the laws of motion, for which gravity remains a mystery (unless I missed an update).

People of real moment who are called "geniuses" to their face, resent it. It deprecates the real sacrifice of their effort. And natural talent only goes so far. It might get you a doctoral degree. But there are professors who, having reached the limits of natural talent, have learned almost nothing beyond that limit -- because they never learned how to learn. Such men and women are by no means rare. And less rare are those who reach their limit before getting their "pedigree" and turn away to work at something far easier.

Do you believe for a moment that if you pursue your study with honesty and diligence until it consumes five-sixths of your every day, that you will not surpass the majority of those who make a lesser effort? If you but knew "the sublimity of your hope, the infinite capacity of your being, the grandeur of your outlook," you would turn away from any idea of your intelligence being limited and get on with your work. Carl Friedrich Gauss would tell you the same thing. He said, "If others would but reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries."

Your individual mind is the equal of anyone else's -- because any two infinities (Cantor notwithstanding) are equal.

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Out of School Mind

This section is intended for both those in or recently in school and for those long out of school. If you are one of the former, get it into your head that the self-study of anything has nothing in common with school.

Your parents will not be asking to see your grades. Mostly because you won't be getting any. Your are entirely responsible for your own progress. You will have to make your own time for study and work at your own rate. And when you have questions, there will be no one there for you to turn to.

Well, actually, there will be. A great many of the enduring texts in mathematics were written by excellent mathematicians who cared about making real mathematicians of students. In this class, I would put Augustus De Morgan, Isaac Todhunter, Richard Courant, E.A. Maxwell, just to name a few off the top of my head. Benjamin Peirce. Morris Kline. These are the people to turn to.

For those who have long been out of school, it is never too late to do what you love. One of my secret gardens is taijiquan. In traditional China, not so long ago, it was not unusual for a man in his sixties to take up taijiquan and in his seventies to be a formidable fighter. There is nothing to prevent the same from being true of the study of mathematics.

As Gertrude Stein pointed out: except when we are infants or senile, we are always and forever the same young men and women. So, in truth, all who take up mathematics are at the same point. And the mind which realizes mathematics exists outside the claims of youth and age.

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The Long Mind

Anyone who was expecting a mystical, Zen, or otherwise airy-fairy definition of the long mind will be disappointed. The long mind is the long-term, joyful, diligent, honest, and selfless cultivation of demonstrable excellence. Its object is realization, a step beyond understanding.

If I give you two one-digit numbers, you already realize, without effort, their sum, differences, product, fractions, and results of division. Not so far from this is the realization, given the postulates of a finite geometry, of how many lines, points, and planes there must be. And so on. Without end.

If you are not diligent, you will never get there. If you aren't honest, you'll have nothing to show for your effort. If you are not selfless, you will be bound by all the limitations of the self. And if you aren't joyful, you are on the wrong bus. Go do something else. The right application of the long mind results in joy.

Here's another taijiquan metaphor, this time for realization. For five years or so, I had only faint intimations that doing two longforms a day was getting me anywhere. In my sixth year, descending an ice-covered staircase, I slipped. All I actually did was think, "This is gonna hurt in the morning" as a scene from Hellboy flashed through my head.

But instantaneously, I found myself rooted on one leg on an ice-covered step, with the muscles down my right thigh ripped apart. I had begun the practical realization of my practice. And this allowed me to expand my practice to doing taijiquan on the ice. Which is really cool. And so on, two longforms a day, without end.

I don't think I have to say anything more about the long mind. You have always had one and you know it. Just use it.

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Note to the Fairer Sex

Whoa. I may be spending too much time with treatises of mathematics written before women gained universal suffrage. The point I wanted to make was that everything I write is intended for every individual interested in mathematics. Mind is not limited by gender. Reject any suggestion of same.

Of course, men in mathematics are about as sexist as men in the general population. So roughly nine out of ten of them will be wankers, in this respect, if somewhat timid and unsocialized ones.

But don't let that stop you, darlin'. If you've got the cojones, get out there and beat them at their own game. That would be the game of mathematics. Not the game of sexism, reverse or otherwise. Let us all try to raise the level of social discourse. Yes? Jolly good.

And remember, if any of these gentlemen should cross the unwritten boundaries of courtesy, you can remind them, physically if necessary, with bended knee, that their cojones are horribly vulnerable in a way in which yours are not.

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Fame and Gain

The big picture is this: The established, senior mathematicians of the establishment, unable to understand or jealous of Abel and Galois, were responsible for those two young men's early deaths. (See Paul's scriptural remark on "pricks.") The establishment hasn't changed. If you are in fact brilliant, watch where you kick.

The small picture is this: If you are serious and develop your own interests, you will occasionally make discoveries which are not in your book. Check your work. Then look in a newer, bigger book. If your discovery can't be found this way, it is still probably something older than George Washington. But you can check.

Write it up very simply without the proof. Just put down the assumptions and the conclusion. Then take or send it to a university. Not a college or an (Oh, Your God) community college. Find a friendly professor and say: "I have a question. But where I live there is no one who can answer me. If I assume [your assumptions here], is this [conclusion here] true?" Hold out your simple paper if necessary. Make sure you do not appear to expect excessive praise for your immense talent.

In the small picture, you may even impress the professor with an old, well-known result. This would happen if your proof (which you also have with you) shows elegance. Most people's proofs (certainly mine) look like they were carved on a cave wall with rocks, mentally speaking. They may be correct. But any real mathematician could think of ten shorter ways of getting there.

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Dangers of Mathematics

Death or Something Very Like It

Abraham de Moivre, of De Moivre's Theorem fame, told his friends that he needed to sleep about fifteen minutes longer each day. And that's what he did. He slept 8 hours, 8 and 1/4, ... , 17, 17 and 1/4, ... , right up to 23 and 3/4 hours. So he stayed up fifteen minutes and went back to sleep. And then he died. Or at least, has yet to wake up. There's a moral here somewhere. But I couldn't begin to guess what it is.

I do find it interesting that the series {1 +1.25 + 1.5 + 1.75 + ...}, with terms greater or equal to 1, converges and quite rapidly at that. It calls some of my fundamental beliefs about divergence into question.

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Methods

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Leave Nothing Undone

You will naturally ask yourself, "How much should I do?" You will have your own interests. Laying the foundations for these, lay them thoroughly without worrying about how long it takes. Make sure you lay all the foundations for what you wish to do. Since about 1950, there has been a rush to Calculus in the schools. And pure geometry, algebra, and trigonometry have suffered. Look in late 19th or early 20th century texts of these topics. Even if you have a bachelors in mathematics, what you see should make you feel sketchy in these foundations. The contents of these books are not somehow out of date. They've just been glossed over in our time. How will you see wider relations if the objects of those relations are not in your mind?

Foundations laid, you will start working on your real interest. Do everything which you come to understand as bearing on your interest and leave nothing undone. But this also means to do everything proportionately. You will have a growing sense of how something relates to your interest. Give it as much time and interest as it deserves. This can vary from making sure you write it down completely in your notes for later reference to marking your place in a current text and working all the way through another one before you come back. As anyone who has contributed to mathematics would tell you, time has no bearing on the study of mathematics. Whatever it takes to achieve realization is what you must do.

Okay. But how much should you do?

Think about it this way. A mathematics text averages 400 pages. If you did five pages a day, five days a week, the average text would take you four months. Two sessions a day, two months for one text or two books in four months.

The main thing is to train yourself in diligence using the long mind. Don't count days and pages. But set an amount of work that you will actually do and do it a set amount of days a week. As Archimedes teaches us, you can empty the ocean with a teaspoon. If you work really fast. For a very long time. And ignore evaporation. And have a place to put the spoonfuls. What was he thinking?

His mathematical point actually works with human endeavor, though. Start with what you can and will do. And the books start passing by like mileposts on a highway roadtrip. Where you are going about sixty-six feet a day. Maybe you are pushing the car. With your wheelchair. Or your four-year-old is pushing the car. I don't know. Clearly, metaphors don't mix well with mathematics.

But if you express diligence in this way, you will make real progress. And the books start stacking up behind you as your realization grows.

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Reading the Text

These are excerpts from an essay by Isaac Todhunter.

I say, then, that the student should read his author with the most sustained attention, in order to discover the meaning of every sentence. If the book is well written, it will endure and repay this close examination: the text ought to be fairly intelligible, even without illustrative examples. Often, far too often, a reader hurries over the text without any sincere and vigorous effort to understand it; and rushes to some example to clear up what ought not to have been obscure, if it had been adequately considered. The habit of scrupulously investigating the text seems to me important on several grounds. The close scrutiny of language is a very valuable exercise both for studious and practical life. In the higher departments of mathematics the habit is indispensable; in the long investigations which occur there it would be impossible to interpose illustrative examples at every stage, the student must therefore encounter and master, sentence by sentence, an extensive and complicated argument.

I suppose the student, then, to read his author with close and sustained attention. Of course it must happen that in some cases the author is not understood, or is very imperfectly understood; and the question is what is to be done. After giving a reasonable amount of attention to the passage, let the student pass on, reserving the obscurity for future efforts. If the text-book has been well arranged, in separate independent chapters, it will be generally found that if a few difficult passages in one chapter are left unconquered, still some progress can be made in the subsequent chapters. After a time the student, having left behind him several points not cleared up, will find that he is no longer proceeding with satisfaction to himself; he must then turn back and begin again at the beginning. It will commonly happen that in the revision of the work some of the former difficulties will disappear, and the student will be able to carry his reading beyond the point at which he formerly turned back. The process should be repeated until the whole work is mastered, or at least such parts as may be pointed out for a first course.

It might be conjectured perhaps that this advice is likely to be abused by the student in such a manner as to lead him to give up a difficulty after only a very faint attempt to overcome it; but practically I think that the danger is but slight. The natural tendency of solitary students, I believe, is not to hurry away prematurely from a hard passage, but to hang far too long over it; the just pride that does not like to acknowledge defeat, and the strong will that cannot endure to be thwarted, both urge to a continuance of effort even when success seems hopeless. It is only by experience we gain the conviction that when the mind is thoroughly fatigued it has neither power to continue with advantage its course in an assigned direction, nor elasticity to strike out a new path; but that, on the other hand, after being withdrawn for a time from the pursuit, it may return and gain the desired end.

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How to Solve Problems

Buy Polya's "How to Solve It." Beg, borrow, or steal a copy. You want an actual book, a codex with paper pages (you remember those, right?), not a PDF. The entire book is useful. It's a kind of pattern language for solutions. And Polya's heuristic, outlined in the opening pages, is essential.

However, I recommend skipping the problems in the back. The first one is cute. But wrong. If you can do spherical geometry and assume a left turn is actually ninety degrees, how far is the bear from his starting point in decimal or English distances?

The Peter, Paul, and Mary (or whoever the driver was) problem is also wrong. The answer, which to be fair is from university exams and probably not Polya's, is algebraically elegant, clever -- and wrong. I mean, on a simple example used to test the equation, awfully wrong. Not even hand-grenade close. There's an essay in Todhunter's "Conflict of Studies" that talks about these kinds of clever exam questions.

If you want the correct answer, there is a section, early in De Morgan's "The Study and Difficulties of Mathematics," where a similar problem is permuted to show the possible meanings of negative numbers. His method was dead on.

But then, De Morgan was interested in the meaning of mathematics. Formalism came after his time. I think we would all do well to recall Wittgenstein's remark that there are two kinds of meaningless propositions: contradictions and tautologies.

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Problems? We don't need no stinking problems.

Some time back, I was reading about the self-study of mathematics. And I was surprised to find this quote, by Isaac Todhunter:

I come now to a very important point, namely the solution of examples. I hear repeatedly from solitary students that although they believe they understand what they read in an ordinary Cambridge book of good reputation, yet they find themselves completely baffled by the exercises and problems, to their very great discouragement. The answer which I give to such complaints is substantially this: do not trouble yourselves with the examples, make it your main business, I might almost say your exclusive business, to understand the text of your author.

But here is Augustus De Morgan saying the same thing:

[P]roblems as are usually given in the treatises on algebra rarely occur in the applications of mathematics. The process is a useful exercise of ingenuity, but no student need give a great deal of time to it. Above all, let no one suppose, because he finds himself unable to reduce to equations the conundrums with which such books are usually filled, that, therefore, he is not made for the study of mathematics, and should give it up. His future progress depends in no degree upon the facility with which he discovers the equations of problems; we mean, as far as power of comprehending the subsequent sciences is concerned. He may never, perhaps, make any considerable step for himself, but, without doing this, he may derive all the benefits which the study of mathematics can afford, and even apply them extensively. There is nothing which discourages beginners more than the difficulty of reducing problems to equations, and yet, as respects its utility, if there be anything in the elements which may be dispensed with, it is this. We do not wish to depreciate its utility as an exercise for the mind, or to hinder all from attempting to conquer the difficulties which present themselves; but to remind every one that, if he can read and understand all that is set before him, the essential benefit derived from mathematical studies will be gained, even though he should never make one step for himself in the solution of any problem.

I will simply suggest that you look at their textbooks and see what use these two authors made of problems. And then you can draw your own conclusions as to what they mean by the above recommendations.

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Partial Answer to Last Exercise

You can check your conclusion regarding the use of problems with this remark by Todhunter:

I may remark here incidentally that when private students have not time or have not inclination to work steadily through a collection of some examples, but yet desire some exercise of the kind, I always recommend them to follow some rigid rule, such as that of taking the first third of the set, or of taking every third example. If a student allows himself an arbitrary license of selection he naturally chooses those examples which for some reason appear most attractive to him; and very probably he thus takes those which really he least requires.

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Extra Reading

Let's say you are working your way through Calculus towards Vector Calculus. You have time for more study. But not enough for slogging your way through something in earnest. Read something. Graduate students in mathematics take reading courses. The professor points you at books in your field. You read some books, write some papers.

A few suggestions on what to read:

You could read another, but differently conceived, Calculus text. You're probably already studying something relatively recent. Read Lamb's Calculus. Or De Morgan's. Peirce's. It will give you a broader perspective of what you are doing.

Or make a reconnaisance of the next text on your list. Read a Vector Calculus or other advanced Calculus text. This will teach you to push through more difficult reading. You won't be able to follow it all this way. But it will prepare your mind for the future.

Or go slightly aside from your path. Read a beginning Real Analysis or Complex Analysis text. This will deepen your understanding of what you are doing. And it will give you an idea of where your path is taking you, as you go into more pure and abstract math.

Or go backwards. Pick something you already know but wish you were better at. Then find something thick like Chrystal's Algebra and just read it, making pertinent notes as you go.

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Rough Index

Studying mathematics, I end up with a lot of notes. Especially with PDFs, where it's a hassle to search backwards in the file for something specific, I take more notes than I ever did at university. I have about six spiral notebooks sitting next to me right now. But they aren't much help if I can't find things in them when I need them. And if I'm going to do more than just collect notes, I need to gather notes by topic from different notebooks and bring them all together elsewhere.

For quite a while my notebooks just got further and further out of control. Finally, I spent a week or so figuring out what to do and came up with a system of rough indexing. The current version of this ongoing process is this:

When I sit down to study for the day, I begin by reading over my notes from yesterday. I have overcome the impulse to skim them and actually read them carefully. This way I catch my egregious mistakes and make sure I put enough down to make sense of it all.

As I read, I'm numbering the pages up in the corner. Then as I finish a page or a section, I add to my index in the back of the notebook. The index starts on the back page and works inwards. When the notes hit the index, new notebook.

I'm not trying to do a professional index as I do this. I index the topics of the notes. But I index them through my eyes with my interests rather than the author's. And I have topics that I'm interested in specifically. These are things I want to develop further. One topic like this is infinite series, so I index any page that has any kind of infinite series, under "series" in the index.

That way, when I have sufficient material to synthesize and develop my understanding with, I can find all the mentions of infinite series and copy them to another notebook. Copying all that again makes me think it through again. I can imagine doing this cycle one more time when I am working on something more seriously, putting the cooked-down notes from the second notebook into a third.

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Materials

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Hey. Aren't these books too old?

Nope.

If you are already into your graduate specialization, you can judge matters for yourself. Otherwise, you can rest assured that not much below that has changed since 1920. And books newer than 1920 are easy to find. For Calculus and down, not much more has changed since 1850.

Okay, sure. You might get a Calculus text old enough not to distinguish between Cauchy's older pointwise convergence of limits and Weierstrass' slightly newer continuous convergence. But that's only before 1900. And you won't appreciate the difference anyway, if you ever do, until you make it to Real Analysis. And if you do make it that far, you have to know both convergences and some others anyway.

The books are not too old.

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Where to Begin

Some suggestions for those who find the infinite choices mind-numbing.

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Whitehead's Suggestions
- OR - decline in public education since 1920.

Let's say you've gotten no further than Calculus, if that far. Where to begin? Alfred North Whitehead had his ideas for how 12- to 14-year-olds should prepare for university at 16:

Passing to the serious treatises on the subject to be read after this preliminary course [he means his "Introduction to Mathematics"], the following may be mentioned: Cremonaís "Pure [Projective] Geometry" (English Translation, Clarendon Press, Oxford), Hobsonís "Treatise on Trigonometry," Chrystalís "Treatise on Algebra" (2 volumes), Salmonís "Conic Sections," Lambís "Differential Calculus," and some book on Differential Equations.

Go take a look. I'll wait here.

(Pause)

You're back. And you look like someone just kicked your honey-buns across the kitchen. So that was where your great-grandfather would start on this journey. If you're up to it, I have some suggestions on what you might study first which Whitehead would assume you would already know at 12.

He would assume you know the first six books of Euclid. So either spend a couple months doing 5 pages and a few problems a day or work through Maxwell's "Deductive Geometry." And before you hit the Calculus, I'd do Green's "Series and Sequences." And probably Maxwell's "Coordinate Geometry and Tensors" before the Salmon. (You could skip the tensors on the first pass.) Then you should be good.

Here's your hat and coat, old boy. Do write me when you arrive at Oxford.

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A More Sane Approach

The Whitehead Diet is a bit over the top. What does he think you will do when you get to university? Teach? For the less spectacularly talented self-studier, you might think about De Morgan's Element Series.

Really, he starts with "The Study and Difficulties of Mathematics." This is like Whitehead's "Introduction to Mathematics." Reading Whitehead, I noted some of his suggestions but took no notes. Reading De Morgan's intro-work, I took (looking at them now ...) 16 pages of notes and fell in love with Augustus De Morgan.

In his "Elements of Arithmetic," he works through all the algebra most people get in high school and ends with as much combinatorics as I got in undergraduate probability. One of the problems was: Given a round table of eight chairs, how many place settings are there where no one sits next to the same people twice? This for twelve-year-olds.

"Elements of Algebra" for the older child, works up to more combinatorics, convergent and divergent series, the derivation and proof of the Binomial Theorem, and three different contexts from which arise Pascal's Triangle.

"Elements of Trigonometry" derives sin(A+B) in three lines of algebra using a simple geometric diagram of two stacked angles and Euclid's Book VI proportions, passes through exponential series and creation of exponential trig functions and their derivatives, and finishes with the products of infinite series.

Also, in his works to prepare students for university, he wrote "The Connection of Magnitude and Number" explaining Euclid V, "First Notions of Logic," "Trigonometry and Double Algebra," and "Elementary Calculus."

De Morgan loved to teach and loved his university students. So he treated them all like adults. If you like to be treated like an adult, De Morgan is a good place to start.

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Todhunter's Beginnings
A few words may be devoted to the order in which the various branches of mathematics should be taken. After Algebra should follow Plane Trigonometry, and then Plane Coordinate Geometry, The latter subject however has received such enormous extensions in recent years, that it is necessary to address a caution to beginners to prevent them from giving an exorbitant amount of time to developments which however elegant cannot be considered as of essential importance. The elaborate discussion of the modern methods of abridged notation should be reserved at least until a reasonable acquaintance has been made with the Differential and Integral Calculus, and Analytical Mechanics. Great care is necessary in commencing the study of Mechanics; the subject appears to be extremely interesting to most persons, but it is fatally easy to acquire incorrect notions and phrases which will afterwards cause serious trouble. Many ordinary popular works, which are readily accessible, are very deficient in clearness and accuracy of expression; and the beginner cannot be too strongly recommended to be cautious in the selection of his guide in this subject.

--Isaac Todhunter

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An Armed Reconnaisance

Another approach to self-study is to pick up more or less where you left off. Let's say you didn't make it much further than the Calculus and you don't remember much of that.

Go pick out a Calculus text that you think you can make it all the way through. Some of the more basic ones are Thompson's "Calculus Made Easy," De Morgan's "Elementary Calculus" and Maxwell's four-volume Calculus. Just browse what you can find and pick one that seems sympathetic.

Then I suggest you read every page and do every problem. This is the only context where that is really useful. You are nailing down exactly where you are in mathematics. So you note what you don't know at all, what you aren't comfortable with, and what you know a bit of but are sketchy on.

When you finish your armed recon, you look at what you don't know and pick out a series of texts that will fill out the missing foundations. Because you are making a long-minded effort, you don't really care how big a hole in your foundations you have to fill. You just settle in and fill it.

Eventually, you are ready to move forward. And by the time you're done, you have a much better sense of where you want to go.

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For the Weekend Hobbyist

Not everyone who is interested in serious mathematics wants to spend hours a day working at it. Like Fermat, the simple relaxation of an evening with a glass of wine and a bit of algebraic exploration is all they are looking for. A place for these people to start is with the volumes of Moscow Press' "Little Mathematics Library."

In the Soviet Union, as with Revolutionary France, serious mathematicians were expected to contribute to the education of the masses. So in the Little Math Library, Soviet mathematicians contributed small texts for highschool students like "Complex Numbers and Conformal Mappings" and "Goedel's Incompleteness Theorem." And several other somewhat less intimidating titles.

But all of these texts are non-trivial even if they are generally strong on demonstration and weak on proof. They are for beginners. Which is one more example of the failings of our current public education...

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Or start at the beginning

Start with Euclid. Everyone should study Euclid. But don't take my word for it. Newton said he wished he had studied it more seriously before he studied algebra. Then he did all those proofs in the Principia in pure geometry. Leibniz said that pure geometry opened a new world for him. De Morgan considered Euclid irreplaceable. I could go on and on like this because every serious mathematician who studied Euclid praises him.

Euclid was no geometric genius, though. Almost none of "Euclid" is Euclid's. Euclid was a harmonizer on the order of Raphael (the artist, not the turtle). Euclid took all the geometry that existed up until his time and combined it. He created a unified form of exposition. And he organized it with a brilliant sense of granularity. Four-hundred-and-seventy-six grains. Less than a thousand objects in all of Euclid including axioms and postulates.

Adrian Marie Legendre, who was no slouch, wrote a substitute for Euclid that was used in Europe for a time. If you look at it, you see that he removed Euclid's minor constructions and lemmas, so that only the meat remained. And then every proof goes on twice as long as Euclid's do because Legendre has to make up for what he tossed out.

Newton, as a teenager, was surprised at Euclid's simplicity. And you will be, too. At five pages and a few problems a day, you can work through Todhunter's Euclid in three months (82 days, to be exact). You will effortlessly learn how to do direct and indirect proofs. You will internalize all those little relations of triangles and circles, lines and arcs. Most importantly, you learn how to think mathematically. With so few objects of such fine granularity, you find yourself naturally able to hold them in your mind and consider them at length. Euclid is a joy and a pleasure which deserves to be savored at length.

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Another thing you learn from Euclid is the scale of mathematics. It is common to feel as if each new mathematic is an immense dark cloud of unknowing full of mysterious and half-discerned objects. The earlier you do Euclid, the sooner you will get over this.

Every mathematic is no more than what you have at each step in aquiring its realization. If you are no further than Euclid I.1, there is no mystery to any of the text's problems at that point. All you have is a line which for its endpoints is the radius of two circles. For any problem at that point, the solution can require no more than than these tools. And this continues to be true at every step.

One learns to distinguish between actual tools and minor accessories. The former are useful in solutions. The latter are just Euclid covering his geometric ass every step of the way. You pay attention to the tools. And then, given a problem, you ask which tool fits. Triangles and the mention of two equal lines or angles has you reaching for the two isosceles theorems. The glimpse of similar triangles calls up I.4, 8, and 26. Once you hit the parallelograms, every triangle becomes a half-finished parallelogram.

You also learn to think more generally in terms of solutions. Let's say I give you a triangle with a line connecting the midpoints of its sides and ask you to prove that line is half the base. This is a particular problem. Turn that triangle into a parallelogram and the more general question becomes: prove that a line parallel to the sides of a parallelogram passing through the intersection of its diagonals bisects the parallelogram. This sounds, to the ignorant ear, like a much bigger problem. Actually, it's much, much easier. You've just tripled what you have to work with.

And while it won't help you answer your own personal questions in mathematics, you soon realize that problems in Euclid (and every other textbook) refer only to the most recent section. If you just finished, I.47, the Pythagorean Theorem, look at that tool first and at what you can add to the problem to bring this most recent tool to bear.

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Two Possible Games

When working through Euclid, or any other text, there are two games you can play when solving the problems. In the first case, you work out the problems as you come to them. You only use the tools, as in constructions and propositions, which you have at that point for your solutions. This is the "traditional" or "pure" game. But this is not how mathematics is actually done. Let me explain.

The second game is to work through a given amount of the text every day. Then you turn to the problems and spend a certain amount of time solving however many you can. In this way, the problems lag behind the reading. And you allow yourself to use everything you have read in order to solve the problems.

Looking in my scratch-book, I see a problem at III.17 that I solved with III.36. I remember beating my head against proving, using Euclidean algebra, that a rectangle equaled a square. It didn't work. Then I realized that my diagram reminded me of another diagram which turned out to be in III.36. So I added a subsidiary circle to my problem and used III.36 to solve it.

I'm writing this today because I was working on polars in Euclidean geometry, a locus having to do with a point and related tangents. The proof of the locus was the algebra I had used on the old problem which today solved exactly what I had needed. It simply went one step further than I had. Call me slow. But as you can see, I learned what was needed in the end.

The point of this little essay is that the second game is how mathematics is actually played. One does not discover for one's self EVERY step of every proof. One uses every prior result that comes to hand. Elegance and laziness go hand in hand, as long as the former governs the latter.

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The Joys and Sorrows of PDFs

PDFs are a mixed blessing. The good news is that two hundred years of mathematical texts are yours for the downloading. A wealthy collector couldn't have put together the collection I now carry on the SD card of my eight-inch tablet.

Without wishing to detract from this amazing abundance on the good side, the bad news is rather extensive. Let me forewarn you of a few things.

For some reason, early scanners of texts (Google, Microsoft, and others) thought they should scan in full color. Some of them apparently used HD. These books would look good on the big screen. But more than a few will crash your mobile device. A real culprit in too-big PDFs is the Indian government. Bless them for scanning their entire library system. Curse them for all the PDFs over 100 megabytes which, amazingly, when you look inside, have alternating big pages and little pages. I can't explain or guess how it was done. Or why they continue to this day with the big-little-big-little. Just go look on archive.org. I think the problem with the corporations and India is that the people involved are not exactly bookish types. They read only Facebook posts, graphic novels, and technical books with too much white space. And the rest of us suffer for it.

The other problem with putting all this unnecessary data in the file, is load time. If you download a PDF, open it, and start scrolling, it looks like a blank book. Just jump to page 20 and wait for it. If it takes too long to load, find a different scan of the book. Note that black-and-white PDFs created from colored ones are bigger than the color original and load at least as slowly. Check file size before downloading.

Then there are the technical limitations. Adobe Reader is different on every platform and worst on Android where it turns some color-scanned books into encaustic art. It's pretty but not helpful. Non-Adobe readers vary from quite good, to passable, to completely sucky. You're on your own with finding a livable solution. But once found, you're good to go.

The PDF, when everything is otherwise shiny, has its own limitations. You cannot flip through the pages, jump back and forth, check the index, or anything else at anywhere near the speed of an actual book. Take lots of notes. More or less condense the book into your notebook. Math texts all have section numbers. Put all those into your notes so that, when referenced (the reason the author numbered it in the first place), you can find it. And after each study session, note the PDF page you are on. Then, if you do have to scan back to find something, you know within five or ten pages where it will be.

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Idiocy Preserved

Only idiots write in books. I've owned thousands of books in my lifetime, mostly used ones. And only one, a copy of "Zen Flesh, Zen Bones," had something intelligent written in it. Two somethings in blue ink. I wish I knew the writer. I still have the book.

An unintended consequence of scanning millions of library books into PDFs is the preservation of generations of idiocy. Plenty of mathematics texts have been written in by generations of these idiots. So far, in my experience, many of them have undertaken to "correct" the text. Not surprisingly, they are often wrong.

Heaven help you if you trust these people more than you trust the author. If they're not obviously correcting a printer's typo, determining what the author wrote under someone's black-squibbed scribble is a neverending exercise for the reader.

Related to this are the common errata pages. Publishers have apparently been poor proofreaders for the past 200 years. When a new edition was published, errata were listed, often on the final page, if not in the frontmatter. Look for these and print them out, if you can.

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