Category Archives: Philosophy

Randomness in Nature II

In a previous post we presented a MO question by Liza about randomness:

 What is the explanation of the apparent randomness of high-level phenomena in nature?

1. Is it accepted that these phenomena are not really random, meaning that given enough information one could predict it? If so isn’t that the case for all random phenomena?

2. If there is true randomness and the outcome cannot be predicted – what is the origin of that randomness? (is it a result of the randomness in the micro world – quantum phenomena etc…)

Before I give the floor to the commentators, I would like to mention a conference on this topic that took place in Jerusalem a year ago. The title was “The Probable and the Improbable: The Meaning and Role of Probability in Physics” and the conference was in honor of Itamar Pitowsky. Let me also mention that  the Wikipedia article on randomness is also a good resource.

Here are some of the answers offered here to Liza’s question.

Qiaochu Yuan

One way to think about what it means to say that a physical process is “random” is to say that there is no algorithm which predicts its behavior precisely which runs significantly faster than the process itself. Morally I think this should be true of many “high-level” phenomena. Continue reading

Some Philosophy of Science

The Bayesian approach to the philosophy of science was developed in the first half of the twentieth century. Karl Popper and Thomas Kuhn are twentieth-century philosophers of science who later proposed alternative approaches.

It will be convenient to start with the Bayesian approach since we already talked about probability and Thomas Bayes in this post. The Bayesian approach (mainly associated with Ramsey and Savage) can be regarded as a verification-based philosophy of science; it is based on different scientists gradually updating, according to new empirical evidence, their (different) prior (subjective) probabilities of scientific explanations and theories, until the cumulative evidence is strong enough to reach a common conclusion.

One difficulty with the Bayesian approach is that in cases of disagreement, there are also disagreements on the interpretation of the evidence.

Bayesian view does not give a way to test a scientific theory but rather to update our beliefs in the theory given new evidence. In practice, scientific theories primarily explain existing observations. For example, the main motivation of Newtonian mechanics and the main support for its validity was the explanation of Kepler’s laws. Kepler’s laws concerning the elliptic orbits of planets around the sun were discovered seventy years before they were explained by Newtonian mechanics.



              Karl Popper                          Thomas Kuhn

 Popper is famous for basing philosophy of science on the notion of falsification. According to Popper, the mark of a theory as scientific is falsifiability: the possibility to empirically refute the theory – in principle. This is in contrast with other approaches that can be viewed as basing philosophy of science on confirmation or verification. Famously, two principal examples of non-scientific theories according to Popper are the Marxist theory of capital and Freudian psychoanalysis.  

If the Bayesian approach, like approaches based on verification, suggests that the optimal way for a scientific theory to proceed is by making safe conjectures which may lead to small incremental progress, Popper’s approach suggests making bold and risky conjectures. One concern about practical implication of the Popperian approach is the fact that bold conjectures and theories that pass the falsifiability test are of little value if they are absurd or simply false to begin with.

Critics assert that neither Popper’s theory nor earlier approaches based on verification give a proper description of how science is practiced. Also, they have limited normative value regarding how science ought to be practiced. It is especially difficult to use the insights from philosophy of science for scientific theories under development.

Thomas Kuhn is famous for his notions of paradigm shifts and scientific revolutions. According to Kuhn, science is normally carried out inside a certain paradigm that is shared by a community of scientists, and it is furthermore characterized by “paradigm shifts,” which occur when the current paradigm is no longer capable of explaining the new evidence.  Kuhn referred to the process of switching from the common paradigm to a new one as a “scientific revolution.” An important example of a scientific revolution analyzed by Kuhn is the shift from Newtonian mechanics to Einstein’s theory of relativity. Continue reading


organic chaos

What is the correct picture of our world? Are noise and errors part of the essence of matters, and the beautiful perfect patterns we see around us, as well as the notions of information and computation, are just derived concepts in a noisy world? Or do noise and errors just express our imperfect perception of otherwise perfect laws of nature? Talking about an inherently noisy reality may well reflect a better understanding across various scales and areas.

Fundamental Impossibilities


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

About Mathematics

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

The Prisoner’s Dilemma, Sympathy, and Yaari’s Challenge

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

Optimism – two quotes

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