Mathematics

[Raphael]

Math is central to the understanding of the natural sciences. So I have started this thread for mathematics and statistics.

[rivka]

Raphael wrote: "Math is central to the understanding of the natural sciences."

Agreed!

And I recommend Sylvia Nasar's wonderful book about John Nash, A Beautiful Mind. While less about math than it is about a specific mathematician, it does include a bit of Nash's game theory. And unlike the movie based on the book, it gets game theory right! (The movie is fun and worth watching, but only loosely based on the book. The book OTOH, is clearly very well researched and meticulously detailed.)

[Raphael]

For starters I recommend What is Mathematics by Richard Courant/ Herbert Robbins. It takes you through a lot of mathematics and really stretches the imagination and your cognitive abilities. Many people find mathematics boring and dull but this book opens ones eyes to the vast world that is mathematics.
My only warning is that it is not an easy read. be prepared to really think!

What Is Mathematics?: An Elementary Approach to Ideas and Methods

[Raphael]

the next book I will recommend is e:the story of a number. Not only does this book give an excellent history of this irrational transcendental number. the number e is really first introduced in high school mathematics but one doesn't really start understanding the deep mystery behind this special number till calculus.
in addition to the history of this number this book also discusses where the number appears in the world around us through other explorations in mathematics, music, physics and finance.

e: the Story of a Number

[Raphael]

This book is the reason I wanted to pursue mathematics professionally when I was in high school.
It is the story of one of the most famous unsolved problems in mathematics.
Questions about the distribution of prime numbers have been around since the ancient Greeks.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

[Raphael]

Another book I strongly recommend is How Not to Be Wrong by Ellenberg. This was a great read about mathematics all over popular culture. Mostly drawing examples from statistics in popular culture he thoroughly explains how to analyze "facts" and how to appropriately draw conclusions from those "facts". I recommend this book for everyone that wants to understand current events in a new light.
I would also add there is a section in this book on the bible codes. Although the author seems to reject the divinity of these codes I dont agree with his conclusions at all. The appearance of codes in Tanach is complicated and there is a vast knowledge of these codes. There really hasnt been that much statistical research towards this yet except for the sole research paper that he sites as proof for his opinion. One will also notice in the book that the author openly admits how much we still dont understand about the nature of statistics. really fundamental open questions which are still not answered to this day. (this is a scary notion since statistics is relied upon heavily in so many ways in our modern world) that being said, my personal belief is that the bible codes are still a very real thing but I appreciate his analysis of the subject. This is still an active research area at the forefront of our holy Tanach and modern mathematics.
(i bet the majority of people dont even realize that math can be applied to the phenomenon of the bible codes!)

How Not to Be Wrong: The Power of Mathematical Thinking

[rivka]

I say this with some hesitation, because this is not the place to have an argument. However, I cannot allow some of those statements to pass unchallenged.

My father is one of the mathematicians who has quite thoroughly debunked the statistical claims of the bible codes, so I am quite familiar with the subject. He (and I) quite firmly believe that the Torah is divine. However, the codes are not proof of that. They have no statistical significance.

If you don't believe me, I can show you a wonderful set of codes for Chanukah, with short skips and lots of related words. They are found in the Hebrew translation of War & Peace.

[Raphael]

small world isnt it? well Rivka in all honesty, the reason I said so is because I dont see the research. If you have the citations to these research papers I would be very interested. the paper cited in this particular book was from the 90's. the paper by, Dror Bar-Natan, Maya Bar-Hillel, Gil Kalai and Brendan McKay. I do understand that paper and accept the findings. Just to add, the paper really was ingenious and a great landmark paper. However this was a far cry from really showing that all codes are bogus. I am well aware of the codes that can be found in various books but I dont believe that all codes are of equal "merit". I can read the original literature with no problem.

I would also add again that statistical methods are still in their infancy to say the least. I am not a statistician I am doing things in differential equations but have enough of a background in probability theory that I can understand the literature.

also to add Rivka, there's a problem with your reasoning above. to find something random about chanukah in war and peace is one thing. The codes in the Torah are not about random things in random places. there are codes in very specific places about very specific RELATED events. Thats a far cry from finding codes in war and peace about chanukah. pretty sure war and peace isnt about chanukah at all.

[rivka]

Raphael wrote: "I would also add again that statistical methods are still in their infancy to say the least."

I would not agree with that statement at all.

Since my father has not published anything new in this field in over 10 years (having said what he had to say), I suspect that you have seen his work. I see no utility in rehashing arguments as his proxy.

I do not feel you have represented what I said accurately, but again, I see no point in having that debate here.

Perhaps we can just agree to disagree on this subject, and allow this thread to return to a discussion of any additional book recommendations.

[Raphael]

Yes, I do know that must be the paper you are referring to because that is where they take War and Peace as their example. Again, I don't dispute the findings of that paper. I dispute the claim that they really have debunked all possible codes. a big part of the paper was based on the "wobble" of the representation of different rabbis names. it was really an ingenious answer. However as I stated above this is a far cry from proving that all codes are bogus. the claim that your father was debunking was a very specific claim by another paper that was published some 5 years prior I believe. Again I repeat I do not dispute his findings at all. it was ingenious and very clever.
I dont think its wrong to talk about this for its talked about in the book I cited and I dont think Kressel would be opposed to good heated discussion.
anyway, looking forward to more book recommendations from you rivka. ~kol tov

[Raphael]

Another book I strongly recommend is Chaos. It is a great read about the unpredictability of models in science. It is important that one distinguish between different kinds of unpredictability. Classical Mechanics, originally pioneered by the work of Sir Isaac Newton, is basically the notion that everything is dictated by equations. these equations can in theory be solved and the whole future of anything can be determined.
There were two breakthroughs that shattered this simplistic view of the universe. The first is the development of the Quantum Mechanics/Quantum Field Theory that was thoroughly developed throughout the 20th century and is the current accepted fundamental theory of our universe today. this theory is a theory of probabilities or chance. Dr. Albert Einstein fiercely opposed this theory till his death and is famously quoted, "God does not play dice". However to the best of our knowledge today God does indeed play with dice.
The second theory that toppled the Newtonian philosophy in a different way is the subject of this book, Chaos. Chaos is still a very new and exciting active field of research and is believed to be at the heart of many phenomenon such as turbulence. Although the systems themselves are deterministic in nature it does not give them so much predictive power. in popular culture they coined the term "butterfly effect". When there is a small change, even the flap of a butterflies wing, in a deterministic system, like the weather, it changes the future in a dramatic unpredictable way.

Chaos: When the present determines the future, but the approximate present does not approximately determine the future. - Lorenz

Chaos: The Making of a New Science

[Kressel]

I don't mind a healthy debate, but if Rivka doesn't want to, then please drop it. I think I'm going to check out How Not to Be Wrong: The Power of Mathematical Thinking .

Here's one for parents of very little kids:

by Sam Ed Brown