Chomskian Linguistics

Here is another little chapterette from my book. It follows a chapter based on discussions that followed a post by David Corfield from n-Category Cafe. There, the following thought was raised: Is there something analogous to Chomsy’s theory of language’s structure and language acquisition when it comes to mathematics. One interesting aspects is trying to understand “dyscalculia” which is a term describing children’s learning disabilities in mathematics.  Continue reading

(Eran Nevo) The g-Conjecture III: Algebraic Shifting

This is the third in a series of posts by Eran Nevo on the g-conjecture. Eran’s first post was devoted to the combinatorics of the g-conjecture and was followed by a further post by me on the origin of the g-conjecture. Eran’s second post was about the commutative-algebra content of the conjecture. It described the Cohen-Macaulay property (which is largely understood and known to hold for simplicial spheres) and the Lefshetz property which is known for simplicial polytopes and is wide open for simplicial spheres.

The g-conjecture and algebraic shifting

Squeezed spheres

Back to the question from last time, Steinitz showed that

any simplicial 2-sphere is the boundary of a convex 3-polytope.

However, in higher dimension

there are many more simplicial spheres than simplicial polytopes,

on a fixed large number of vertices. Continue reading

Answer to Test Your Intuition (9)

Two experimental results of 10/100 and 15/100 are not equivalent to one experiment with outcomes 3/200.

(Here is a link to the original post.)

One way to see it is to think about 100 experiments. The outcomes under the null hypothesis will be 100 numbers (more or less) uniformly distributed in [0,1]. So the product is extremely tiny.

What we have to compute is the probability that the product of two random numbers uniformly distributed in [0,1] is smaller or equal 0.015. This probability is much larger than 0.015.

Here is a useful approximation (I thank Brendan McKay for reminding me): if we have n independent values in U(0,1)  then the prob of product < X is

X \sum_{i=0}^{n-1} ( (-1)^i (log X)^i/i!.

In this case  0.015 * ( 1 – log(0.015) ) = 0.078

So the outcomes of the two experiments do not show a significant support for the theory.

The theory of hypothesis testing in statistics is quite fascinating, and of course, it became a principal tool in science and  led to major scientific revolutions. One interesting aspect is the similarity between the notion of statistical proof which is important all over science and the notion of interactive proof in computer science. Unlike mathematical proofs, statistical proof are based on following certain protocols and standing alone if you cannot guarantee that the protocol was followed the proof has little value.