# Test Your Intuition (11): Is it Rational to Insure a Toaster

Here is a question from last year’s exam in the course “Basic Ideas of Mathematics”:

You buy a toaster for 200 NIS (\$50) and you are offered one year of insurance for 24 NIS (\$6).

a) Is it worth it if the probability that damage covered by the insurance will occur during the first year is 10%? (We assume that without insurance, such damage makes the toaster a “total loss”.)

b) Is it worth it if the probability that the toaster will be damaged is unknown?

As an additional test of your imagination, can you think of reasons why buying the toaster insurance would be rational?

# Itamar Pitowsky: Probability in Physics, Where does it Come From?

I came across a videotaped lecture by Itamar Pitowsky given at PITP some years ago on the question of probability in physics that we discussed in two earlier posts on randomness in nature (I, II). There are links below to the presentation slides, and to  a video of the lecture.

A little over a week ago on Thursday, Itamar,  Oron Shagrir, and I sat at our little CS cafeteria and discussed this very same issue.  What does probability mean? Does it just represent human uncertainty? Is it just an emerging mathematical concept which is convenient for modeling? Do matters change when we move from classical to quantum mechanics? When we move to quantum physics the notion of probability itself changes for sure, but is there a change in the interpretation of what probability is?  A few people passed by and listened, and it felt like this was a direct continuation of conversations we had while we (Itamar and I; Oron is much younger) were students in the early 70s. This was our last meeting and Itamar’s deep voice and good smile are still with me.

In spite of his illness of many years Itamar looked in good shape. A day later, on Friday, he met with a graduate student working on connections between philosophy and computer science.  Yet another exciting new frontier. Last Wednesday Itamar passed away from sudden complications related to his illness.

Itamar was a great guy; he was great in science and great in the humanities, and he had an immense human wisdom and a modest, level-headed way of expressing it. I will greatly miss him.

Here is a link to a Condolence page for Itamar Pitowsky

 Probability in physics: where does it come from?

## Itamar Pitowsky

### Dept. of Philosophy, The Hebrew University of Jerusalem

The application of probability theory to physics began in the 19th century with Maxwell’s and Boltzmann’s explanation of the properties of gases in terms of the motion of their constituent molecules. Now the term probability is not a part of the (classical) theory of particle motion; so what does it mean, and where does it come from? Boltzmann thought to reduce the meaning of probability in physics to that of relative frequency. Thus, eg., we never find a container of gas in normal circumstances (equilibrium) with all of its molecules on the right hand side. Now, suppose we could prove this from the principles of mechanics- that a dynamical system with a huge number of particles almost never gets into a state with all its particles on one side. Then, to say that such an event has a vanishing probability would simply mean (and not only imply) that it is very rare.I shall explain Boltzmann’s program and assumptions in some detail, and why, in spite of its intuitive appeal, it ultimately fails. We shall also discuss why quantum mechanics with its “built in” concept of probability does not help much, and review some alternatives, as time permits.

Additional resources for this talk: video.

(Here is the original link to the PIPS lecture) My post entitled Amazing possibilities  about various fundamental limitations stated by many great minds that turned out to be wrong, was largely based on examples provided by Itamar.

# Noise Stability and Threshold Circuits

The purpose of this post is to describe an old conjecture (or guesses, see this post) by Itai Benjamini, Oded Schramm and myself (taken from this paper) on noise stability of threshold functions. I will start by formulating the conjectures and a little later I will explain further the notions I am using.

## The conjectures

Conjecture 1:  Let $f$ be a monotone Boolean function described by  monotone threshold circuits of size M and depth D. Then $f$ is  stable to (1/t)-noise where $t=(\log M)^{100D}$.

Conjecture 2:   Let $f$ be a monotone Boolean function described by  a threshold circuits of size M and depth D. Then $f$ is  stable to (1/t)-noise where $t=(\log M)^{100D}$.

The constant 100 in the exponent is, of course, negotiable. In fact, replacing $100D$ with any  function of $D$ will be sufficient for the applications we have in mind. The best we can hope for is that the conjectures are true if  $t$ behaves like  $t=(\log M)^{D-1}$.

Conjecture 1 is plausible but it looks difficult. The stronger Conjecture 2 while tempting is quite reckless. Note that the conjectures differ “only” in the location of the word “monotone”. Continue reading

# 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.

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.

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

# Midrasha News

Our Midrasha is going very very well. There are many great talks, mostly very clear and helpful. Various different directions which interlace very nicely. Some moving new mathematical breakthroughs; very few fresh from the oven. Tomorrow is the last day.

Update: I will try to describe some highlights an links to the presentations, related papers, and pictures at a later post.

# Four Derandomization Problems

Polymath4 is devoted to a question about derandomization: To find a deterministic polynomial time algorithm for finding a k-digit prime.  So I (belatedly) devote this post to derandomization and, in particular, the following four problems.

1) Find a deterministic algorithm for primality

2) Find a combinatorial characterization (and a deterministic algorithm) for generically 3-rigid graphs

3) Find an explicit construction for Ramsey graphs

4) Find a deterministic method for polling

(Remark: I was very slow in writing this post, and I had to give up elaborating the details on some very nice topics in order to finish it.)