An Understanding of our fundamental limitations is among the most important contributions of science and of mathematics. There are quite a few cases where things that seemed possible and had been pursued for centuries in fact turned out to be fundamentally impossible. Ancient geometers thought that any two geometric lengths are commensurable, namely, measurable by the same common unit. However, for a right triangle with equal legs, the leg and the hypotenuse are incommensurable. In modern language (based on the Pythagorean theorem), this is the assertion that the square root of two is not a rational number. This was a big surprise in 600 BCE in ancient Greece (the story is that this discovery, attributed to a Pythagorean named Hippasus, perplexed Pythagoras to such an extent that he let Hippasus drown). Two centuries later, Euclid devoted the tenth book of his work The Elements to irrational quantities. The irrationality of the square root of 2 is an important landmark in mathematics. Similarly, the starting point of modern algebra can be traced back to another impossibility result. Algebraists found formulas for solving equations of degrees two, three, and four. Abel and Galois proved Continue reading
This post collects some brief philosophical thoughts about mathematics that appeared as part of my paper “Combinatorics with a geometric flavor: some examples,” from the proceedings of the conference “Vision in Mathematics, towards 2000.” I added two small items (the first and fifth).
1. Mathematical truths – theorems.
“There are infinitely many primes;” “The three angles of a triangle add up to 180 degrees;” “A continuous real function defined in a closed interval attains there its maximum;” “A non-constant polynomial over the complex numbers has a solution;” If you substitute a matrix A in its characteristic polynomial you get zero;” “A simply connected closed 3-dimensional manifold is homeomorphic to a sphere.”
These truths appear very different from truths in other areas of life. This sharp difference is the secret to some of the successes of mathematics and explains also its limitation.
What makes a mathematical theorem important, deep, or central?
2. Proofs, more proofs, “proofs from the book” and computer proofs
Science has a dual role: exploring and explaining. In mathematics, unlike other sciences, mathematical proofs are used as the basic tool for both tasks: to explore mathematical facts and to explain them.
The meaning of a mathematical proof is quite stable. It seems unharmed by the “foundation crisis” and the incompleteness results in the beginning of the 20th century, and unaffected by the recent notions of randomized and interactive proofs in theoretical computer science. Still, long and complicated proofs,
as well as computerized proofs, raise questions about the nature of mathematical explanations.
Proofs are gradually becoming intolerably difficult. This may suggest that soon our days of successfully tackling a large percentage of the problems we pose are over. Also, this may reflect the small incentives to simplify.
Be that as it may, we cannot be satisfied without repeatedly finding new connections and new proofs, and we should not give up hope to find simple and illuminating proofs that can be presented in the classroom. Continue reading
Correlation and Cooperation
In our spring school devoted to Arrow’s economics, Menahem Yaari gave a talk entitled “correlation and cooperation.” It was about games as a model of people’s behavior, and Yaari made the following points:
It is an empirical fact that people (players in a game) act in a correlated way,
It is unscientific not to take this into account (although this is not taken into account in game theory and economics).
The prisoner’s dilemma
A basic example in game theory (which also played a central part in Yaari’s lecture) is the Prisoner’s dilemma. Let’s talk about this example a little, before getting to Yaari’s claims. Continue reading
1. Here is a quote from Karl Popper’s paper “Science, Problems, Aims, Responsibilities” about Francis Bacon: “According to Bacon, nature, like God, was present in all things, from the greatest to the least. And it was the aim or the task of the new science of nature to determine the nature of all things, or, as he sometimes said, the essence of all things. This was possible because the book of nature was an open book. All that was needed was to approach the Goddess of Nature with a pure mind, free of prejudices, and she would readily yield her secrets. Give me a couple of years free from other duties, Bacon somewhat unguardedly exclaimed in a moment of enthusiasm and I shall complete the task – the task of copying the whole Book of Nature, and of writing the new science.”
2. Here is a tale from Arundhati Roy’s book “The God of Small Things”. In the book Margaret Kochamma tells the following joke to Chacko in Oxford, England, where the two meet: A man had twin sons… Pete and Stuart. Pete was an Optimist and Stuart was a Pessimist… On their thirteenth birthday their father gave Stuart – the Pessimist – an expensive watch, a carpentry set and a bicycle… And Pete’s – the Optimist’s – room he filled with horse dung… When Stuart opened his present he grumbled all morning. Continue reading
Many people do not regard mathematics as a science since it does not directly probe our physical reality; some mathematicians even like to think about mathematics as being closer to art, music or literature. But is there really a big difference between exploring the physical reality and exploring the logical/mathematical reality?
It is perhaps too early to have an open discussion thread on this blog but let me try anyway. What do you think? Is mathematics a science?