We started the week off talking about paradigm shifts. Paradigm shift, according to Wikipedia, was a term that was first used by Thomas Kuhn in his book The Structure of Scientific Revolutions in 1962 to characterize a foundational transformation in the dominant theory of a science. Since its introduction the phrase has had to go through one of itself to arrive at its current meaning of a over-arching change in any area within the realm of homosapiens. In class quite a few paradigm shifts were discussed from the Calculus of Leibniz and Newton to the training regimes of the Williams sisters, but there is one that I feel is just as crucial and that is the invention of computational theory by Alan Turing.

One interesting way to look at the idea of a paradigm shift is through the lens of the Singularity that Venor Vinge was kind enough to give us. The term singularity is one that mathematicians are very comfortable as they are a term for where some mathematical object happens to not be defined. Say we are talking about a function that is not defined at zero, then the function tends to behave oddly near this singularity. The same can be said for matter near black holes, which are called gravitational singularities. It was from these ideas that Vinge came up with the term Singularity, in this case referring to some point in the future when technology will stop increasing in speed at an algebraic level and start progressing at essentially infinite speed; usually this refers to AI or self-replicating machines. The reason that I feel this lens could be useful to look at paradigm shifts through is because of something that Singularity Science Fiction, the Singularity does have it own sub-genre of Science Fiction literature, author Cory Doctorow once said, that for all essential purposes the Singularity is the point at which human beings that were raised under the conditions caused by the Singularity are incapable of meaningfully communication with those born before the event. He went on to posit that human beings have already gone through multiple points of Singularity, the greatest of which would be the invention of spoken language. After which it is quite clear that meaningful communication with those who do not have spoken language by those who do would be, for any practical purposes, impossible.

While I do not believe that any of these paradigm shifts qualify as full Singularity events, I do appreciate the problems that those who learned mathematics after we had the calculus would have communicating the mathematics of, say a thrown projectile to those who came before the calculus was know. It is in this way that I wish to discuss Alan Turing and the beginning of computing.

Computers for a long time did not refer to the machines that make the soft white noise hum that constantly surrounds us wherever we go, or who’s flickering LED’s let us know that the world is continuing to function. No, computers for a long time referred to the human beings who computed the values of arithmetic operations. Whether they did it on paper, with an abacus, or some early predecessor of the digital computing, such as Leibniz’s Stepped Reckoner, the computers were people. It was Alan Turing, 1912-1954, who changed this paradigm on the 28th of May, 1936 with the publication of one of the most important papers of all time, “On Computable Numbers, with an Application to the Entscheidungsproblem”. The paper was a response to another on the list of all-time important work, that of Kurt Godel and his incompleteness theorem. Turing introduced the idea of Turing machines, and through manipulations of these incredibly powerful and simple abstract machines showed that any computation that can be done can be done using a Turing machine.

This basic meaning of this, also known as the Church-Turing Thesis, is that instead of having to build machines, or write an algorithm that is computable by hand, all one has to do to show that any computation is possible is show that it can be done using a Turing Machine. This was not the only important thing in that paper though, since it also contained a proof that the Halting Problem is undecidable proving there is no solution to the Entscheidungsproblem; not the first such proof but by far the most intuitive and easy to understand. It is this intuitive nature of Turing’s work that is so important. All of the theoretical work of people like Godel, Church, and Turing was fine and dandy, but in order for it to have a real impact on human being’s lives, set theoretic work such as this is quite often the most abstract and hardest to apply, some application was going to have to occur. This application turned out to first be the analog computing devices, such as those used by Bletchly Park to crack the Nazi Enigma code in WWII, and then the digital computer, upon which I am currently writing this sentence. The type of people who tend to build things like computers from the ground up, engineers primarily, are usually quite different from a mathematician who spends his life at a blackboard abstracting the world away. Therefore the equivalent definitions of Godel and Church which are nearly impossible to get a good handle on without decades of training, which is an unlikely time investment for the engineer who just wants to build a machine, but, this is the the most important component of Turing’s work, the Turing Machine would be easy for the engineer to grasp at an intuitive level and begin to apply.

We are now nearly 74 years past the submission of Turing’s paper and our entire world is run by the physical manifestations of his machines. I try to imagine going back 80 years and discussing research with a mathematician of the time. No matter what problem we decided that we were going to research for I would inevitably just say that we should Google it as an initial step, then use things like JSTOR or MathSciNet to find more scholarly results. The mathematician of the 1920s would look at me as if I were a loony and then I would quickly become very happy I did not go farther back in time as if I had I probably would have been burnt at the stake as a warlock for such insane sounding statements. I would then have to join this mathematician at some library, walking through the stuffed shelves looking for some forgotten tome, and, sparing that, writing with a quill on paper to some mathematician elsewhere asking for their help in procuring the material that we needed to continue. The differences caused by this 80 year gap are nearly impossible for me to gain any purchase on, and I think would make it very hard for two people of the eras to work together. This is not simply conjecture though, as I have had the experience of dealing with those who refuse to adapt to the computer age and I find it very hard to find any sort of shared ground upon which we can share a conversation since computers are such an integral part of my life and being. It is really this inability to communicate with those before, this mini-Singularity, that really characterizes this paradigm shift.

I am not fully satisfied just talking about paradigm shifts that have already happened though, I want to speak a bit about one that I feel is coming in mathematics and that harnesses that which is so powerful, crowds. Crowds have become the poster child of internet, and more specifically how to Get Things Done using the internet. Want to make a repository of all the world knowledge, well crowds can do that and have on Wikipedia. Want to have an algorithm that gives out meaningful search results, Google leveraged the knowledge of crowds to do this using Page Rank. Want to prove a mathematical theorem, well this is the question that Field’s medalist Timothy Gowers asked on a pot to his blog, gowers.wordpress.com, on the 27th of January last year. In the post “Is massively collaborative mathematics possible?” Gowers outlined a problem, Density Hales-Jewett, and a method, using the comment sections on his blog, and then let the crowd go. A couple of months later he had his answer, which was: YES! The first polymath paper, “A new proof of the density Hales-Jewett theorem, is available over at cs.cmu.edu.

This is a can of worms that will not be able to be closed; in fact fellow Field’s medalist Terrance Tao jumped on board with his own polymath projects, as has Gil Kalai and Gowers himself has a couple new ones running at this time with many more on the horizon. I truly believe that this is not simply something new, it is something great which will lead to some incredible, interesting, and unexpected results. In fact it was even listed by the New York Times as on the Big Ideas of 2009. While it leverages many of the same people that already did mathematical research, professors, graduate students, and industry mathematicians, the real breakthrough is the way that this method of research can bring in the mathematically inclined amateur who up till this point had no easy path into mathematical research. These people bring such a different perspective, one that has not been tainted by the methods and biases present in high level mathematics education, that they will be able to see and attack problems in ways that the current mathematical elite would never dream of using. I will go on record right now that with in a decade a major result will be proven using a version of the polymath method and that within a few decades the students coming into their own in the field of mathematics will wonder how work was ever done before their predecessors started using the wisdom of the crowd.

Other than the obvious intelligence of deciding to use crowds to help prove theorems, the shrewdest thing that Gowers did was realizing that mathematics is not a lonely pursuit. There are of course very foundational examples that would prove me wrong on this, most recently the Poincare Conjecture proof by noted recluse Grigori Perelman, but if one looks through the mathematical publications in any journal it is an incredibly rare event to find one that does not have at least two authors. By understanding that most work is collaborative and the importance of an active community within mathematics, and the strengthening of this community by working on theorems together may have an even stronger impact than the polymath results themselves, Gowers was able to bring together many people who would not have worked together under other circumstances and knit them together into a functioning mathematics producing machine. I am just glad that I will be able to look back and say I remember when this started.