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?

The Beauty of Mathematics

This semester I am teaching an introductory course in mathematics for students in other departments.  I taught a similar course last year entitled “Basic Ideas in Mathematics,” and this year, following a suggestion of my wife, I changed the name to “The Beauty of Mathematics”. Another change is that starting this year we have a general program in the university, called “Cornerstones“, (initiated by the rector,) whose purpose is to widen the education we offer to our students, and this course is part of the new program.

Talking about beauty rather than about basic ideas, combined with the new Cornerstone program have led many  more students to enlist to the course this year, and subsequently the lectures will use computer presentations.

Of course, the challenge has become  harder. I truly think mathematics is beautiful, but trying to convey its beautiful facets has never been easy. Also, I do not want to sweep under the rug the difficulty of mathematics, and the students will have to learn some basic mathematical skills and some abstract mathematics.   Ideas and suggestions are most welcome.

What do you regard as a great example of the beauty of mathematics?

There will be  is a separate page devoted to the course and the slides for the first lecture (in Hebrew) are linked there (and here). The first lecture was devoted to “numbers.”

Muhammad ibn Mūsā al-Khwārizmī

(From the Wikipedia article on zero.) In 976 Muhammad ibn Musa al-Khwarizmi, in his Keys of the Sciences, remarked that if, in a calculation, no number appears in the place of tens, a little circle should be used “to keep the rows.”

Nerves of Convex Sets – A Recent Result by Martin Tancer

Martin Tancer recently found a very beautiful proof that finite projective planes can’t be represented by convex sets in any fixed dimension. This was asked in the paper entitled “Transversal numbers for hypergraphs arising in geometry” by Noga Alon, Gil Kalai, Jiri Matousek and Roy Meshulam some years ago. I am thankfull to Jirka Matousek for telling me about this development.

Here’s a very rough sketch of the argument:  Suppose that (P,{\cal L}) is a finite projective plane with n points, and suppose that for every line L \in P  there is a convex set C_L in R^d such that a subcollection of the C_L has a common point iff the corresponding lines L have a common point in the projective plane. Continue reading

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.

For more information about Itamar Pitowsky, visit his web site. See his presentation slides.

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

The Möbius Undershirt

“Look at this brand new undershirt,” my wife said. “I am shaking it and shaking it but still I have this twist.  Can you see what to do?” 

I gave the undershirt a good shake. And another one. And one more. And then it struck me. It was a Möbius undershirt!

What a rare case in which mathematics can come to the rescue in domestic matters

“There is no way in the world this twist can be undone,” I said. “This is a mathematical fact! It is a Möbius undershirt!” My wife listened carefully to my firm statement.

I started to day-dream about the bright future of this rare Möbius undershirt: I will show her to my colleagues!, I will display her in public lectures, and even let some selected graduate students hold her. However, Continue reading

Anat Lotan: Who is Gina II, My Own Shocking Revelation

Who’s Gina? (Part 2): My Own Shocking Revelation

By: Anat Lotan

It was one of those typically hot Israeli end-of-August days; a scorching summer morning, where you have to convince yourself that the cool breezes of autumn are just around the corner.

 To escape the unbearable humidity of the coastal area, I decided to treat myself to a visit to Jerusalem. My cause for excitement was twofold: not only would I enjoy the sites and atmosphere of this beautiful city, but this impromptu visit would also be the perfect opportunity to meet with dear Prof. Gil Kalai, the mastermind behind these very pages, a man who boldly followed Gina’s adventures through cyberspace, and who cleverly pieced them together into a highly entertaining book. 

Little did I know, however, that this would not be just another visit to Jerusalem, or that this seemingly typical summer day would turn out to be anything but typical. Indeed, it was to be a memorable day of SHOCKING REVELATIONS!

 Shortly after arriving at the Hebrew University’s scenic Givat Ram campus, I made my way to Prof. Kalai’s office. After we exchanged pleasantries and discussed our respective summers, our conversation naturally turned to Gina’s Adventures. Gil had recently added the book to his blog, and he shared with me some of the most notable and amusing comments that had been posted in reaction to the book.

Laughing, Gil told me that one of the bloggers had even “accused” him of being Gina. This comment prompted me to ask if the “real” Gina had, in fact, attempted to contact Gil – it only seemed natural that she would have something to say to the man who had so blatantly “outed” her.

 I’m not sure I recall Gil’s precise response to my question. Whatever it was Continue reading

Anat Lotan: Who is Gina I

 
Several people asked me to explain who is Gina, the hero of my book “Gina says:  Adventures in the Blogosphere String War.” There was a chapter written by Anat Lotan about who Gina is.  And in view of the comments Anat wrote a new chapter about Gina.  Here are two posts with the two chapters.     

 

 Who is Gina?

As seen by Anat Lotan    

 

Perhaps it’s time to say a few words about our fearless Master of Ceremonies in cyberspace – Gina. 
  

35 years of age, Gina is of Greek and Polish descent.  Born in the quaint island of Crete, she currently resides in the USA, in quiet and somewhat uneventful Wichita, Kansas. Gina has a B.Sc in Mathematics (from the University of Athens, with Honors), and a Master’s Degree in Psychology (from the University of Florence, with Honors).   

Currently in-between jobs (her last job was working with underprivileged children), she has a lot of free time on her hands, which gives her ample opportunities to roam the blogosphere.    

Forever the proud Grecian, Gina is the happy owner of Papa, her beloved pet tomcat, named after “that dear man”, Christos Dimitriou Papakyriakopoulos, Continue reading

A Discrepancy Problem for Planar Configurations

Yaacov Kupitz and Micha A. Perles asked:

What is the smallest number C such that for every configuration of n points in the plane there is a line containing two or more points from the configuration for which the difference between the number of points on the two sides of the line is at most C?

We will refer to the conjecture that C is bounded as Kupitz-Perles conjecture. It was first conjectured that C=1, but Noga Alon gave an example with C=2. It is not known if C is bounded and, in fact, no example with C>2 is known.

Alon’s example

Kupitz himself proved that C \le n/3, Alon proved that C \le K\sqrt n, Perles showed that C \le K\log n, and Rom Pinchasi showed that C \le K\log \log n. This is the best known upper bound.  (K is a constant.) Pinchasi’s result asserts something a little stronger: whenever you have n  points in the plane, not all on a line, there is a line containing two or more of the points such that in each open half plane there are at least n/2-K\log \log n points. 

The proof uses the method of allowable sequences developed by Eli Goodman and Ricky Pollack. Another famous application of this method is a theorem of Ugnar asserting that 2n points in the plane which are not on the same line determine at least 2n directions. Continue reading