TITLE: Math vs Probability vs Statistics
QUESTION [8 upvotes]: For a certain job interview, I gave myself a 6 in SQL and 8 in Statistics.  I love math and probability but I always found significance testing and confidence intervals rather dry.  
What is the difference between math, probability and statistics again?  This is a soft question, but it should have a clear answer.

REPLY [5 votes]: Probability is a branch of mathematics.  Period.  (Nowadays, that is... in Poisson's day, it was a branch of Physics....but that is what makes this question so interesting...)
So what is mathematics.  Mathematics is the branch of logic which deals with sets, and hence, with any or all of what we usually call `mathematical objects', e.g., numbers, transformations, equivalences of categories, points, generic points, etc.  
There is a famous quotation, which André Weil repeats, «Qui dit mathematiques, dit preuves.»
You are not doing mathematics unless you define your terms and state your axioms and come up with a proof of your assertion.  (Well, formulating a conjecture counts, at least if you have a lot of numerical evidence for it, i.e., have proved many special cases.)
Now what is statistics.  Statistics can be part science or part math or both.  Mathematical statistics is a branch of mathematics which is founded on probability theory.  Its practitioners do proofs just like any probabilist or other mathematician.  (As stated by the great Boris Levit.)  But Statistics can also be the analysis of data, in which case it is a branch of science which merely uses mathematical results like any other branch of science does, but which requires scientific intuition and experimental results to compensate for lack of precise definitions, lack of proofs, etc.  For this reason, many statisticians have the intellectual traits which scientists have and mathematicians almost always lack these days.  Sir Ronald Fisher had mathematical ability but preferred to hide it, because he had so much more....
Last, as to the curious way the field of probability has evolved.  In its beginnings, it was really thought of as a branch of Physics or Natural Science.  (See Leo Corry on HIlbert's Sixth Problem,  http://www.icm2006.org/proceedings/Vol_III/contents/ICM_Vol_3_82.pdf.)  When Poincaré began his career, he occupied chairs of physics, of probability, and of rational mechanics, at different points.  He did not consider himself only a mathematician (and, didn't he discover the Theory of Relativity?) but this was already beginning to seem a little strange to other physicists.  Still, this was the last gasp of the tradition that The Calculus of Probabilities was part of Physics, just like Mechanics.  It was very hard for people to think of it as part of mathematics since for millenia, mathematics had been defined as that which dealt with numbers or points.  Only after both numbers and points had been finally unified, via the axiomatic method and the use of set theory, could some mathematicians begin to think of the possiblity of a purely Mathematical Probability which would be axiomatised and founded on logic plus set theory: Frechet and Wiener worked this out and Kolmogoroff offered the definitive formulation.  But even now we then have inherited a curious question, focussed on by Littlewood in his math club talk, The Dilemma of Probability, (see http://books.google.ca/books?id=MjVgeT7Laf8C&pg=PA72&lpg=PA72&dq=Littlewood+dilemma+of+probability+%22inverted+commas%22&source=bl&ots=ou2am-HEHZ&sig=baPTMGuK9VYXOQ4CFi8OkG4gSh8&hl=es&sa=X&ei=qbGOUqqPIafP2wXF54GwCg&ved=0CC8Q6AEwAQ#v=onepage&q=Littlewood%20dilemma%20of%20probability%20%22inverted%20commas%22&f=false), published in his Mathematician's Miscellany, that the meaning of «physical probability» is not explained by the mathematical theory of probability, nor by its axioms, and remains somewhat mysterious in spite of the fact that the mathematical theory of probability works very well when applied to the real world applications.