Reading Challenge: The Road to Reality

The Road to Reality. A complete guide to the laws of the universe.

Prior to my more serious interests in Mathematics I challenged myself to read Goedel, Escher, Bach. Not an easy task if you aren't familiar with musical theory at all. It took me about six months to complete the book. Of course I was in doubt if I really understood it all ( I didn't, not sure if I do know, even after M381 Mathematical Logic, but math grows on you ). Anyway, I have been told that The Road to Reality by Roger Penrose is a book of similar importance as GEB. With it's 1100+ pages it's a challenge alright.

I intend to read it while commuting. Who says commute time can't be made productive? Clearly this is a self-motivating post ;-), but an announcement of a log as well. Sort of a summary of the book in parts. More later. I completed one chapter sofar or 25 pages which is less than 3%.

Proof: Trivial

#mathematics #books #krantz

Have you ever come across something like: "This course has no prerequisites except a certain level of mathematical maturity." To me this sounds just as awful as: "It is easy to see that..." or "Proof: trivial." What is mathematical maturity anyway? As far as I know, the concept of maturity is only used in relation to mathematics. Doesn't it simply means knowing a LOT about mathematics? Anyway, If I would have to describe my own mathematical development then I would not use the words mature or maturity. I would probably say that "I am learning how little I know and how little I will ever know". It is as though if I set one step towards my goal, my goal takes two steps back. I keep walking and learning but I will clearly never reach that final goal. You are never done in mathematics.

Krantz (left) Lederman (right )

Stephen G. Krantz wrote a book about mathematical maturity called "A Mathematician comes of Age.". Sol Lederman interviewed Krantz in his series 'Wild about Math'. Krantz has a website too and I happened to found that he left a copy of his book on it: here ( PDF ). There may be a zillion reasons why he left it there so let's not speculate about it. Get the book while you still can and read it if you are interested in the concept of mathematical maturity.

Link to A Mathematician Comes of Age on Amazon.

Fearless Symmetry 6/23: Equations and varieties

I read the sixth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 6: Equations and varieties

Logic of Equality

An equation is a statement, or assertion, that one thing is identical to another. In mathematics we replace is by = and use symbols that stand for the terms.

History of equations

Long before algebra as we know it, ancient peoples were working with equations.
A triangle whose three sides have lengths 3, 4, and 5 is a right triangle which is an example of a Diophantic equation because the unknowns are restricted to integers. Around the late 1500s Descartes added the connection between algebra and geometry now known as analytic geometry. Descartes, as a philosopher believed that the physical universe was governed entirely by the laws of geometry. Newton ( and Leibniz ) discovered that this wasn't true, they had to invent calculus to solve their scientific problems mathematically.

Z-Equations

A rational number is any number that can be expressed as the ratio of two integers. Real numbers are rational iff it is a terminating decimal or a repeating decimal. The set of all rational numbers is usually denoted as Q. We will deal mostly with equations where all the constants are integers. Or equations "defined over the integers". A Z-Equation is an equality of polynomials with integer coefficients. One of the main problems in number theory is finding and understanding all solutions of Z-equations.

Varieties

Fix the attention on a particular Z-equation. Write S(Z) for the set of all integral solutions of that equation, S(Q) for the set of all rational solutions of it, and so on. We call S an "algebraic variety". The variety S defined by a Z-equation ( or a system of Z-equations ) is the function that assigns to any number system the set of solutions S(A) of the equation or system of the equations.

For example define the Variety S as x^2 + Y^2 = 1
Then
S(Z) = {{1,0),(0,1),(-1,0),(0,-1)}.
S(Q) = {t in Q | 1-t^2 / 1+t^2, 2t/1+t^2}.

We can reformulate Fermat's Last Theorem using varieties as follows.
For any positive integer n, let V_n be the variety defined by x^n + y_n = z^n. Then if n > 2, V_n(Z) contains only solutions where one or more of the variables is 0.

Systems of equations

The system
x^2+y^2=1
x>0
is valid and has solutions, but it does NOT define an algebraic variety because inequalities are not defined in C nor in any of the finite fields.

Take the system
x^2+y^2+z^2=w
w^4=1
x+y=z
then S(R) is the ellipse x^2+y^2+x y = 1/2.

Finding roots of polynomials

The easiest general class of varieties to look at would be those defined by a single Z-equation in a single variable, for instance, x^3 + x - 2 = 0. The study of this type of variety is dominated by the concept of the Galois group. ( More in 8 and 13 ). If f(x) is a polynomial the roots of f(x) are the numbers c such that f(c)=0.

Are There General Methods for Finding Solutions to Systems of Polynomial Equations?

On a purely number-theoretical level, leaving philosophy and logic behind, we also have the famous theorem of Abel and Ruffini: Unlike quadratic polynomials, for which we can use the quadratic formula, for polynomials f (x) of degree 5 or greater, there is no formula involving just addition, subtraction, multiplication, division, and nth roots (n = 2, 3, 4, . . .) that can solve f (x) = 0 in general.

Deeper understanding is desirable

The amazing discovery of Galois is that there is more structure to S(A). As we shall see, S(A) is not just a set; it is the basis for defining a representation of a certain group, called the Galois group. We will look at another series of very interesting and very important, though not so very simple, Z-varieties: elliptic curves. These two kinds of varieties will give us some of our main examples to help us understand Galois groups and their representations.

To be continued with 7. Quadratic reciprocity

Fearless Symmetry 5/23: Complex Numbers

I read the fifth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 5: Complex Numbers

Well, I think it is safe to assume that readers of this blog know the complex numbers. This chapter is in fact about a subset of the Complex Numbers called the Algebraic Numbers. In FS they use $\mathbf{Q}^{Alg}$ as notation, whereas I have seen mostly the notation $A$ for the Algebraic Numbers.

Every algebraic number can be expressed as the root of of a polynomial equation with integer coefficients. So $\pi$ is not a member of $\mathbf{Q}^{Alg}$, but $\sqrt{2}$ is because $\sqrt{2}$ is a solution of $x^2 - 2 = 0$.

Visualisation of the (countable) field of algebraic numbers in the complex plane.
( From Wikipedia )

To be continued with 6. Equations and varieties

Fearless Symmetry 4/24: Modular Arithmetic

I read the fourth chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 4: Modular Arithmetic

Chapter 4 is all about modular arithmetic.

Considering the goal of the book somewhere fields have to be introduced and in this chapter we find the first definition of a field.

Definition: A field is a number system where we can divide by anything nonzero.

Anything more precise would scare off the laymen casual reader for who the book is intended. I kind of like the definition myself. '... where you can divide anything by nonzero'.

Modular arithmetic is introduced as clock arithmetic of course with examples like: "Today is Tue. What day is it in 25 days?" or "The analog clock shows 8. What time will it show in 33 hours?"

Also, the extremely important concept of an equivalence relation is defined. There is much more about modular arithmetic in the book, of course.

To be continued with 5. Complex Numbers

Fearless Symmetry 3/23: Permutations

I read the third chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 3: Permutations

In chapter 3 the concept of a permutation is explained and how they form groups.

Definition: A permutation is a one-to-one map from a set to itself.

Example 1:
Given the set {1,2} the possible one-to-one maps ( permutations ) are:
1->1, 2->2 and
1->2, 2->1.

Example 2:
For sets of three elements there are 6 = 3! possible permutations.

If we put all the permutations of a set in a set by itself and add the composition of permutations as the operation then this set becomes a group. Permutation groups are among the most important objects in group theory because every finite group is a subgroup of some permutation group.

There are two possible ways of notation when it comes to permutations. Let's consider the set {1,2,3,4} which has 24 possible permutations.

The permutation 1->1, 2->2, 3->4 and 4->3 can be written as [1 2 4 3] and also as (1)(2)(3 4) or short (3 4) this is the so called cycle notation. Thus (1 2 3) and [2 3 1] represent the same permutation. Clearly the cycle notation is more efficient, especially when considering permutations of large sets.

The composition of permutations means permuting one after the other. Unfortunately in some books it is done from left to right, in others from right to left.

Exercise 1:
Show that (ab)(cde)*(ae)(bc)(d)=(ac)(bde).

( I would solve it as follows: )
Right hand side:
A B C D E
_ D _ E B Apply (bde)
C D A E B Apply (ac)

Left hand side:
A B C D E
_ _ _ D _ Apply (d)
_ C B D _ Apply (bc)
E C B D A Apply (ae)

E C B D A
C D _ E _ Apply (cde)
C D A E B Apply (ab)

And it shows that LHS = RHS

Sets of permution form a group under composition because:
- composition leads to a new permutation ( closure )
- the neutral element is the do-nothing permutation, i.e. (1)(2)(3).
- every permutation has an inverse because it can be permuted back to the original positions.
- composition of permutations is associative.

Note that the composition of permutations is NOT ( always ) commutative.

To be continued with 4. Modular Arithmetic

Fearless Symmetry 2/23: Groups

I read the second chapter of Fearless Symmetry.

Part 1: Algebraic Preliminaries

Chapter 2: Groups

Definition: A group G is a set with a composition defined on pairs of elements, as long as three axioms hold true:
1. For any three elements x,y,z in G: x*(y*z) = x*(y*z)
2. G contains an element e such that for all x in G: x*e = e*x = x.
3. For any element x in G, there is an element y in G such that x*y=e.

Example 1:
The group of rotations of the sphere in R3: SO(3) or the Special Orthogonal Group in 3 dimensions. -The set G is the collection of all rotational symmetries of the sphere, i.e. if we rotate the sphere by any angle, the sphere doesn't noticeably change. The group property basically means that if we rotate the sphere over any angle A, after this over an angle B, it is the same if we would have rotated it in one go, but over some different angle. Also any rotation has an inverse: rotating it over the opposite angle. This makes the rotations a group. SO(3) is in fact a Lie group because these rotations can be done arbitrary small which is not the case when considering the symmetry group of for example a cube. Lie groups capture the concept of "continuous symmetries".

For me personally, this is the time to review chapters 1,2 and 3 of Naive Lie Theory by John Stillwell, Springer 2008. There you will find that the ( 4 dimensional ) quaternions are intimately related to the group SO(3) and that the quaternions can be expressed as 'complex 2-dimensional rotations' or complex 2 by 2 matrices. - This explains why quaternions are frequently used in 3D-(game)-programming.

To be continued with 3. Permutations

Fearless Symmetries - Reviews

In the eye of the general public the protypical math student suffers a range of personality disorders, i.e. autism, Asperger of course. When they suffer from a Narcissistic Personality Disorder as well, they merely look dumb and arrogant, the complete opposite of what they are trying to achieve.

Perhaps the most accessible introduction for a "very naive layperson" is Ash and Gross's Fearless Symmetry.

by Anonymous on Mathematics StackExchange

Anyway, Ash, one of the authors of Fearless Symmetry listed several reviews of the book on his page here.

Fearless Symmetry 1/23: Representations

I read the first chapter of Fearless Symmetry. See: Reading ( conflicts time management ) for the history on this topic.

Part 1: Algebraic Preliminaries

Chapter 1: Representations

The goal of the book is 'Mod p linear representations of Galois groups' and how these representations help to clarify the general problem of solving systems of polynomial equations with integer coefficients.

It is very important to know that a mathematical definition can redefine a commonly word used elsewhere. ( A simple group is not 'simple' but complex. A tree is a graph, a 'tree' in the forest is not. ) Sometimes an object is defined by listing its properties and following that a proof is given of the existence of such an object.

Definition: A set is a collection of things which are the elements of the set.

Definition: A function f: A-> B from a set A to a set B is a rule that assigns to each element in A an element of B.

Definition: A morphism is a function from A to B that "captures at least part of the essential nature" of the set A in its image in B. ( Clearly "captures at least part of the essential nature" needs to be refined later. )

Definition: A representation is a morphism from a source object to a standard target object.

Example 1: Take A,B and the fact that B represents A. A may be a citizen, B her state rep and X the legal fact that B represents A by voting in the legislature on her behalf. A may be a(n abstract) group, B a group of matrices, and X a morphism from A to B. The ultimate in abstraction is representing A,B as dots and X as an arrow from A to B.

Example 2: In the context of counting, given any two finite sets A and B, a morphism is a one-to-one correspondence from A to B. A representation in this case is a morphism from a given finite set to one of the sets {1}, {1,2}, {1,2,3} and so on. So a flock of three sheep has the set {1,2,3} as its target.

To be continued with 2. Groups.

Combinatorica

The following function in the Mathematica Combinatorica package ShowGraphArray[Partition[FiniteGraphs, 4]] produces the following graphic.

I found it in the book 'Computational Discrete Mathematics, Combinatorics and Graph Theory with Mathematica, by Sriram Pemmaraju and Steven Skiena, Cambridge 2003.' It's not the typical Proposition / Proof mathematics book but it does give you access to about 450 -practical- freely available ( if you have Mathematica ) tools to extend your problem solving toolkit.

Reading ( conflicts time management )

When I was in high school I ( and others of my generation ) considered reading a waste of time. Teachers took every effort to convince us of the opposite.

There is only one way you can effectively change yourself and that is through reading.

Some of the best memories I have are somehow related to reading. Good times! I have read quite a few time management books. These books are primarily written for people who want to do better in life, i.e. change themselves into more productive people. In none of these books I have found the advice: "Read more". Strange. Anyway, I managed to undo the habit of reading. And that is bad.

In "The New Student Hat" Hubbard explains why people don't read, he talks about the strange sensations people experience when they sit down and attempt to read a book.

Amazing reactions occur when conscious effort is made to do this. Dullness, perception trouble, fogginess, sleep and even pains, emotions and convulsions can occur when one knowingly sets out to BE THERE AND COMFORTABLY PERCEIVE with the various parts of a subject.
These reactions discharge and vanish as one perseveres ( continues ) and at last, sometimes soon, sometimes after a long while, once can be there and perceive the component.

When I sit down with a book ( or device ) I get these sensations which only go away when I CAN OCCUPY MY MIND ON A (MATH-)PROBLEM. I know exactly what Hubbard means, I have to -simply sit there and read-, confront it.

Many years ago Goedel, Escher, Bach helped me to through a reading-barrier. This time I chose "Fearless Symmetry, Exposing the Hidden Patterns of Numbers, by Avner Ash and Robert Gross published by Princeton 2006.

I'll keep you posted about my struggle to become a reader again.

The book will never die.

A lot has been written about real paper books versus ebooks. Both have their distinct advantages and disadvantages. I have to admit that I read mostly ebooks. They are cheaper and easier to get, store and carry. Mathematics ebooks often have the PDF format. So you need a reader that can handle PDFs. If you use your PC or laptop than a PDF reader is all you need. Most people know Adobe Reader but there are much better programs than Adobe around. With free readers, just like Adobe. My PDF reader of choice is Foxit Reader 5. What I particularly like about Foxit is that it is lightweight, i.e. loads and acts fast. Foxit uses tabbed reading, like internet browsers. If you were in the middle of five books, close Foxit, the program nicely remembers which books you were reading and on what page you left. Most of all, I like the feature that I can highlight what I read. It is almost as if I was reading in a paper book.

Link:
- Foxit Reader 5.

Hypercomputation

Only recently lightspeed as the ultimate limit of speed got challenged. It seems they have been challenging the Turing Machine for a while too.

If you start with studying mathematics you are only three or four centuries behind on contemporary mathematics. That's quite a lot of catching up to do. - Some fields started their development in the previous century though. Like mathematical logic, a field I have been studying this year, and have written about so now and then in this blog. Part of mathematical logic is the theory of computation which showed us what can be computed and what can't. That what can be computed is what can be computed on a Turing Machine, period. - That idea is challenged however in the theory of hypercomputation. A new field in mathematics which is trying to go beyond the limits of the Turing Machine.

A book with an overview of the theory is the following.

The Limits of Mathematics ( or: a Lisp interpreter in Mathematica )

( ... ) mathematics because it is an extremely difficult road to traverse. The terrain is extremely demanding. The amount of work and concentration required to build the foundation necessary to continue extending the framework is immense. ( ... ) - David Andrews

Mathematics, as if you have never seen a skyscraper and are traversing the streets of Manhattan. With that mindset, you can only think that people -walk- to the 60th floor... Anyway, feeling overwhelmed by the sheer size and complexity of maths is not going to help. Only people willing to teach, without ulterior selfish motives, can help. One can write a book about mathematics to impress peers, as a way to meet publication quotas or to -teach-. Like the book The Limits of Mathematics does for example. It is a clear taste of the best mathematics has to offer, an invitation to go on to the next level.

The table of contents says it all:
- Randomness in arithmetic and the decline and fall of reductionism in pure mathematics
- Elegant LISP programs
- An invitation to algorithmic information theory
- The limits of mathematics
- Appendix. LISP interpreter in Mathematica

The appendix contains the source code of a Lisp interpreter coded in Mathematica. I love that. But the book starts with a clear description of the massive changes taking place in ( the thinking about ) mathematics during the first half of the twentieth century, from Hilbert to Turing.

Study Tip - 3 ( mathematics does not have to be difficult )

Being a programmer who studies math (, a math studying programmer, whatever ) gives me access to the literature of the foundations of computer science. Now and then I browse the libraries. I thought to have a found a book that fits my level, so I read the preface to check if I was right. The author Alan Parkes made the following interesting statement.

For Students : I wrote this book partly because when I studied this material as part of my own Computing degree, I had to work really hard to understand the material, a situation which arose not because the material is too difficult, but because it was not well presented ...

from Preface of A Concise Introduction to Languages and Machines by Alan P. Parkes, Springer 2008.

I have seen video of a lecture on MSRI where the researcher / professor held 'a talk' ( there is a subtle difference between 'a talk', a presentation and a lecture, 'a talk' is a euphemism for a not well prepared presentation ) about a subject from scribbled notes. Most of the lecture he stood with his back to the audience mumbling while scribbling math. End of lecture. Goodbye. It was expected of the audience to write up notes from his mumblings and scribbles. - Folk like that, probably based on a brilliant thought they had decades ago, get the opportunity to write books too. Once in the hands of ( then ) students like Alan Parkes they make the learning experience troublesome to say the least.

Well written, illustrated mathematics followed by examples, more examples, examples of exercises, exercises with written solutions and so forth make studying mathematics easy. ( Think of an Open University booklet, a good one ). Open University booklets are written by course teams for students and not by professors who write to impress... who actually?

Tip 4: Change books

If you have difficulty with a ( mathematics ) subject then find another book about the subject. Check how the other author explains the topic. Try a third, fourth if necessary. This method is guaranteed to work. When you are close to revision for an exam it stimulates to read through the material written by someone else. Different exercises, and so on.

Finally: authors who write for students are read, respected and remembered.

Reading fiction about mathematics

This is a lighthearted, enjoyable novel about a bumbling but likable mathematics professor at an unnamed university who believes that he has discovered a solution to a famous mathematical puzzle known as "Beauregard's Wild Number Problem."

A novel set in the world of contemporary mathematics: The Wild Numbers by Philibert Schogt. Doesn't that sound great? I mean actually having the time to leisurely read novels, stroll through the city, drink coffee somewhere. Haven't read it yet, but I will, someday, I must really. :-)

Mathematics made difficult

I found an interesting book.

It's hard to describe the book or assign it to a category. Let me give you two quotes from the book.

Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige.

All numbers are interesting, since the first uninteresting number would be interesting.

If you like mathematics and you are ready for some light reading while giving the impression you are reading really hard stuff then this book might be for you.

Mathematics made difficult
A handbook for the perplexed
Carl E. Linderholm
Out of print - but download-able ;-)
( Unbelievable it is out of print in the age of Kindle, iPad and what have you. )

Michael Schneider

Twelvealone made a comment about the Egyptian math video. The mathematician / presenter is Michael Schneider. He wrote a book called 'Constructing the Universe'. If you have seen the movie Pi, you'll recognize many similarities between Schneider's and Maximillian Cohen's views about mathematics and the physical reality. Apple Computer employs Evangelists. The mission of a Mac Evangelist is simply to disseminate the beauty of an Apple computer. Unlike teachers who teach from a position of superiority, even if they only want to help their ( engineering ) students to get through Calculus, Michael Schneider does not teach: he disseminates. He invites you to investigate the beauty of mathematics, for yourself. One of my favorite ( alternative ) radio stations is the Swedish Red Ice Radio. It turns out they interviewed him last year about his book, mathematics and so much more.



Parts 2, and following at the Red Ice Radio channel.

Mathematica programming: an advanced introduction

I don't know what people who go to watch a performance of the New York ballet actually experience. In the Black Swan Thomas Leroy ( played by Vincent Cassel ) says:

To beauty! - Thomas Leroy

I can't agree more. Beauty, it is probably the best word to describe what happens when we get touched by a piece of mathematics, or even a line of Mathematica code.

I found a free, Creative Commons licensed, book about Mathematica programming. I must emphasize that it is a book about programming the core Mathematica language. It is possible to solve and communicate about mathematical problems without actually programming Mathematica but there will come a time when your problems require ( some ) programming. Although Mathematica ships with extensive on-line documentation several ( excellent ) books have been written about the subject most of which I have reviewed in this blog. Among them, and not reviewed yet, is Mathematica programming: an advanced introduction.

An ideal reader for it would be a person who has some Mathematica experience as a user, needs to write programs more substantial than a few one-liners, and wants to understand the logic of the language and ways to program idiomatically, minimize programming effort and maximize program's efficiency. - Leonid Shifrin

I don't know why people write 'free' books. In general I am not in favor of free books simply because the author spent ( a lot of ) time creating the book and if it is of value to a reader some form of exchange needs to take place. Free does not mean 'of less quality' although it has to be said that an author of a free book is not helped by an editor. I can only say that I like Leonid Shifrin's Mathematica book and that I suggest you have a look at it. It can be read online, or downloaded as a PDF at his site http://www.mathprogramming-intro.org/

Linear Algebra Thoroughly Explained

M373 ( Optimization ) renewed my interest in Linear Algebra which always has been one of my favorite branches in mathematics. Unfortunately I never got any further with it then the standard course up to Eigenvectors. Benedict Gross said in one of his lectures "You can't learn too much Linear Algebra." I suppose that he meant that any investment in learning more Linear Algebra always pays off. Quite a few books on advanced linear algebra have been published, understandably most of them by algebraists which doesn't make these books very accessible, let alone of any practical value. I recently discovered a book called Linear Algebra Thoroughly Explained by Milan Vujicic and published by Springer in 2008. From the foreword: "There are a zillion books on linear algebra, yet this one finds its own unique place among them." The book introduces Complex Inner-product Vector Spaces, new methods for solving linear systems, Dual Spaces, Tensor Products and a lot more.

Linear Algebra Thoroughly Explained @ Springer.