/The-Math-Group

THE MATH GROUP

The-Math-Group

Curriculum for group 1 and 2 Welcome to the The-Math-Group wiki!

Group 1

Curriculum:

  • Read Lang’s book first. When you get to Algebra 2 type stuff, use Gelfand side by side with Lang. So, read lang first, do his exercises, then do Gelfandd’s exercises to make sure you can apply it.

  • When you get to the Trig stuff in Lang’s book, either use Axler’s precalc book or another trig or precalc book side by side to make sure you can apply the material.

  • Crack open Rudin’s book next. It will review the algebra 2 and precalc stuff in the very begining, and introduce set theory. You can work through Rudin to learn Limits (stuff that’s sometimes taught in pre calc). It’s also a key concept in calc and, in actuality, it’s important in high school math too, especially when you think about inverse functions, domains, codomains (which range is a subset of, you’ll learn more about this later) and continuity.

  • Make sure you get friends or family who know what they’re doing to test you to make sure that you know what you think you know.

  • Finish Rudin side by side in Calc class.

  • Take multivariable calculus at a college during a summer session (like stanford or berkeley or harvard, and surprisingly UC Santa Cruz summer math classes are open to anyone, affordable, and are surprisingly more proof based than standard or try to take a class online through an online extension school like Harvard’s extension school or UC online (UC Santa Cruz’s Math 23 is surprisingly rigorous (uses Marsden and Tromba) and way better than a standard calculus class).

All textbooks are provided @ the homepage

Group 2

Curriculum:

  • Go online and read the short paper “The Three Crises in Mathematics: Logicism, Intuitionism and Formalism” by Ernst Snapper

  • Learn formal logic with: Introduction to Formal Logic by Peter Smith

  • Go online and read the Scientific American article “Dispute over Infinity Divides Mathematicians” by Natalie Wolchover (also in Quanta Magazine)

  • Learn Set theory with Karel Hrbacek and Thomas Jech. Introduction to Set Theory. Don’t get too bogged down with this, just enjoy the read and move on when you feel ready. Go back to it when the situation arises that you need it to move forward with math.

  • Read a little bit about (don’t read it all the way through; just enjoy it until you get tired of it; go back to it as you work through math and see how it all fits together) Category Theory with Lawvere, Conceptual mathematics: a first introduction to categories, 2nd Edition, 2009

  • Go online to the CSUSM Spring 2009 Math 378 course website by Prof. Aitken and download all of the class lecture notes (Ch. 0 - 10). Save them before they’re taken down, and work through these excellent notes as if they were a textbook. Learn it all as if you were given the Sports Alimak in Back to the Future series in 1985; it’s literally that good. This is the most important step in this entire list; if you do nothing else, at least do this.

  • Get a copy of Russell’s Principles of Mathematics. Books Like the books on Set and Category thoery, read it, but not like your life depended on it. If you mastered Smith’s book on formal logic, you will master this book too, and it will help you clarify things that seem like magic in mathematics. But know, Russell’s work isn’t the end all be all.

  • Learn about non-classical logic. Question the law of excluded middle; think for yourself — does it make sense to you? Do you believe physical reality follows this rule? This is a big moment. It will decide whether you want to be a intuitionist or a formalist (built on top of the logicist framework). All of formal mathematics from this point on, including calculus, is built on the idea that the law of excluded middle is right. In fact, even the books by Smith and Prof. Aitken, as well as all of Set Theory assume this notion. Maybe just let this question simmer in the back of your mind and continue to read about more mathematics. Don’t forget that it’s still a valid philosophical question.

  • Read Principles of Mathematical Analysis by Walter Rudin; accompany this read with lecture notes and free online midterm exams from Stanford’s Math 19, 20 and 21 and Harvard’s Math 1a, 1b and 112. Just google the course websites and use what you can find. You should realize that Prof. Aitken’s lecture notes should make this transition seamless. After all, his notes could could well be called “Analysis of the Natural Numbers, Arithmetic and Algebra.” He even covers some real and complex number stuff, so when you see Rudin, you should be in a very, very solid position to blow this material out of the water. Do it. When you have, congratulations, you’ve probably surpassed the majority of college graduates understanding of mathematics. But don’t stop here, you need to understand more than just two dimensional mathematics after all.

  • Read A First Course in Topology by James Munkres. This should be tons of review by this point. You should recognize things from set theory, real analysis and logic popping up everywhere. This should be an easy A, and it comes in handy as you move up to more than two dimensions.

  • Now you have options. You can now start Abstract Algebra if you wish (I’d recomend this), or multivariable mathematics with linear algebra (this is how things get extended beyond two dimensions, making use of everything you learned up to this point; it will act as if your limbs were extended further. In order for this learning experience to truly feel this way, you need to do Abstract Algebra first. So I’m just going to say, do Abstract Algebra now. Go ahead and read Topics in Abstract Algebra by Herstein first. Follow it up with Algebra, second edition by Artin. Artin is more hand wavy, but covers more material, so I think it makes a lot more sense to go in this order. Both are excellent books.

  • Read Linear Algebra, Vector Calculus and Differential Forms, 5th edition, by Hubbard and Hubbard. Much like Rudin should have flowed seamlessly from Aitken, Hubbard and Hubbard should role off the tongue like butter to you now. You should easily grasp this material, and you should learn it because it’s important in real life. Accompany your reading with lecture videos of Math 3500/3510 by Shifrin on youtube (excellent lectures of an honors class covering multivariable math). Supplement Hubbard and Hubbard with either: 1) Linear Algebra by Levendosky, 2) Vector Calculus by Marsden and Tromba or 3) Multivariable Mathematics: Linear Algebra, Multivariable Calculus and Manifolds by Shifrin. It’s hard to say which is better. Don’t waste money buying all 3. Personally, I’d probably buy Shifrin based on his lecture videos, and also because Marsden and Tromba is on Scribed online. You can pickup Levendosky cheaply on amazon ($30 or less). If that sounds like a super good deal, buy it. It’s excellent.

  • Go ahead and solidify your linear algebra because it’s really important from now on. Have Linear Algebra done Right by Axler at hand and take Berkeley’s Math 110 midterm and final exams before opening Axler’s book (you can find them online easily enough). If they are easy for you, just scan the table of contents of Axler and read anything that sounds unfamiliar; skip the rest unless you want to read it. If you want to do the HW from 110 as well, then pick up a copy of Linear Algebra by Friedberg, Insel and Spence (optional). Don’t spend too much time on this. Just make sure you have Hubbard and Hubbard, and Math 3500/3510 down really well (youtube). Glaze through Axler to patch up anything not covered in Abstract Algebra and multivariable mathematics.

  • Now you have gotten to the point where you can go online and buy any math book that interests you, and you should be able to just learn it with ease. Explore whatever you want. Algebraic topology, differential geometry, differential topology, complex analysis, physics, cryptology, computer science, statistics, anything at all. My advice, try to learn about multilinear algebra and tensors in depth. I don’t know why, but multivariable math textbooks don’t teach it, in fact the only school that I know of that teaches it is Stanford in their Math 52h class. The sky is the limit man! Have fun!

Readings:

Only listing readings up to Topology

Pdfs are also available for download @ the homepage

Resources:

Most importantly, http://thebestpageintheuniverse.net/c.cgi?u=math