Why is Mathematics Possible: Tim Gowers’s Take on the Matter

lnc math

In a previous post I mentioned the question of why is mathematics possible. Among the interesting comments to the post, here is a comment by Tim Gowers:

“Maybe the following would be a way of rephrasing your question. We know that undecidability results don’t show that mathematics is impossible, since we are interested in a tiny fraction of mathematical statements, and in practice only in a tiny fraction of possible proofs (roughly speaking, the comprehensible ones). But why is it that these two classes match up so well? Why is it that nice mathematical statements so often have proofs that are of the kind that we are able to discover?

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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

Randomness in Nature

Here is an excellent question asked by Liza on “Mathoverflow“.

What is the explanation of the apparent randomness of high-level phenomena in nature? For example the distribution of females vs. males in a population (I am referring to randomness in terms of the unpredictability and not in the sense of it necessarily having to be evenly distributed).

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…)

Where can I find resources about the subject?

Some answers and links can be found below the question in MO. (The question was closed after a few hours.) More answers and further discussion are welcome here.

Related posts:  noiseThomas Bayes and probability, four derandomization problems, some philosophy of science

And here is a related post on probability by Peter Cameron relating to the question “what is probability”.