Benacerraf: What Numbers Could Not Be

 Without commentary, here's a link to Benacerraf's original paper and a cut-paste job from the Stanford Encyclopedia of Philosophy.

A resource for formal logic

Peter Smith's Logic Matters has plenty of useful material for the student of logic.

Dehydrated Elephants

Not Even Wrong noted the recent passing of mathematician Robert Hermann 1931-2020.

In that blog post, Peter Woit wrote:

Being ahead of your time and mainly writing expository books is unfortunately not necessarily good for a successful academic career. Looking through his writings this afternoon, I ran across a long section of this book from 1980, entitled “Reflections” (pages 1-82). I strongly recommend reading this for Hermann’s own take on his career and the problems faced by anyone trying to do what he was doing (the situation has not improved since then).

The reference is to Robert Hermann's book, "Cartanian Geometry, Nonlinear Waves, and Control Theory, Part 2", which you can read in Google Books. This is where I first encountered dehydrated elephants.  To quote:

On Graduate School

For a graduate student it can be very difficult. It’s very easy to get discouraged if you have no real understanding that you’re going to be stuck for a very long time.

I could very easily have imagined myself getting discouraged and dropping out of graduate school. That could have easily happened to me under different circumstances.

Graduate school is a different environment and then you start to wonder—“Am I really good enough to do research-level math?” It’s hard, in fact, to tell. You see all these people who are doing great things around you and then you think, well, maybe you’re not cut out for this. So I think it’s important [long pause] to have someone who believes in you.

from an interview with Akshay Venkatesh.

Akshay Venkatesh in the news.

mathpages.com

Stumbled across this yesterday - http://mathpages.com/ - looks like a lot of interesting reading - mostly mathematics, and some physics.  I came across this via a trail that began at a classic - The Art of Unix Programming.

Lincoln and Euclid

Blatantly copied from: http://the-american-catholic.com/2012/08/16/lincoln-and-euclid/. 
I learned of the Lincoln's love of Euclid from Stepanov's recorded lectures; and the link above was a good written source of what Stepanov mentioned.

Abraham Lincoln was not an especially well-read man, but what he read he retained, thought about and frequently used.  One author he was fond of was the Greek mathematician Euclid.  His law partner Billy Herndon relates how Lincoln studied Euclid’s Elements:

He studied and nearly mastered the Six-books of Euclid (geometry) since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for Euclid, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.

Lincoln wrote about why he decided to study Euclid:

In the course of my law reading I constantly came upon the word “demonstrate”. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof? I consulted Webster’s Dictionary. They told of ‘certain proof,’ ‘proof beyond the possibility of doubt’; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man.

At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.

In the fourth Lincoln Douglas debate Lincoln used Euclid to illustrate a point:

If you have ever studied geometry, you remember that by a course of reasoning, Euclid proves that all the angles in a triangle are equal to two right angles. Euclid has shown you how to work it out. Now, if you undertake to disprove that proposition, and to show that it is erroneous, would you prove it to be false by calling Euclid a liar?

In a speech in Columbus, Ohio in 1860, Euclid came up again:

There are two ways of establishing a proposition. One is by trying to demonstrate it upon reason, and the other is, to show that great men in former times have thought so and so, and thus to pass it by the weight of pure authority. Now, if Judge Douglas will demonstrate somehow that this is popular sovereignty,—the right of one man to make a slave of another, without any right in that other, or anyone else to object,—demonstrate it as Euclid demonstrated propositions,—there is no objection. But when he comes forward, seeking to carry a principle by bringing it to the authority of men who themselves utterly repudiate that principle, I ask that he shall not be permitted to do it.

However Lincoln did not merely cite Euclid in speeches, but used him in his private thoughts about slavery.  In an unpublished note from 1854 on slavery:

If A. can prove, however conclusively, that he may, of right, enslave B. — why may not B. snatch the same argument, and prove equally, that he may enslave A?– You say A. is white, and B. is black. It is color, then; the lighter, having the right to enslave the darker? Take care. By this rule, you are to be slave to the first man you meet, with a fairer skin than your own. You do not mean color exactly?–You mean the whites are intellectually the superiors of the blacks, and, therefore have the right to enslave them? Take care again. By this rule, you are to be slave to the first man you meet, with an intellect superior to your own. But, say you, it is a question of interest; and, if you can make it your interest, you have the right to enslave another. Very well. And if he can make it his interest, he has the right to enslave you.

Lincoln throughout his life was fascinated by logic and mathematics.  In considering him as a thinker, it is always best to keep this in mind when looking at his thought processes as reflected in his writings and his speeches.

Binge watching mathematics

If you are into binge-watching, and are of a certain bent of mind,  I recommend Professor Alexander Stepanov's Four Algorithmic Journeys (actually only 3 were made).  Be sure to go in order.

1. Spoils of the Egyptians
2. Heirs of Pythagoras
3. Successors of Peano
4. Epilogue

The collection of Stepanov's books, papers, class notes, and source code, covering generic programming and other topics.

In these talks, Stepanov traces the history of some simple, foundational mathematical ideas and their value in computer science.   In the process he also demonstrates that the love of mathematics for its own sake is good for the soul.

November 10, 2015 - likely a big day for computer science

As R.J. Lipton narrates:

László Babai is one of the world experts on complexity theory, especially related to groups and graphs. He also recently won the 2015 ACM Knuth Prize, for which we congratulate him.
Today we wish to discuss a new result that he has announced that will place graph isomorphism almost in polynomial time.

More exactly László shows that Graph Isomorphism is in Quasipolynomial Time: that is time of the form

for some constant . Polynomial time is the case when , but any is a huge improvement over the previous best result.

Luca Trevisan already has made a post on this result, and Scott Aaronson likewise. Luca further promises to be in Chicago next Tuesday when László gives his talk on the result......