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. Read the rest of this entry »

I am here with another entry from my weekly write-up of topics talked about in my History of Mathematics class. This one is a bit longer:

We started this week with a reading of a few section from chapter two of our book, The History of Mathematics by Roger Cooke, specifically those dealing with the mathematical history of India and the Maya. The mathematical history of India is itself, removed from any external context such as the over-all study of the history of mathematics, incredibly interesting.

One of the oldest cultures in the world the history of Indian mathematics reaches, of course, well into BCE. As is common with mathematics of that time, the math seems to be mostly concerned with geometry and artimetics. In fact, according to Cooke, sometime between 800 to 500 BCE the Sulva Sutras, who’s root words come from measure and cord, a collection of mathematically based verses were inserted into the Vedas. These verses, and the idea that the content probably springing from the maintenance of  altars, are intimately tied to a conversation that we had in class on Tuesday: The importance that culture and religion have on studies, and mathematics in particular.

Professor Bhatnagar brought up the semester he spent at the University of Nizwa in Oman, and the perspective the students there brought to their education, specifically that they came into classes expecting to be able to memorize their way through instead of learning basic concepts and then extrapolating from there to solve here to fore unseen problems. Professor Bhatnagar then posited that there was a good reason for this and someone else from the class spoke up that it could have something to do with the practice of memorizing large section of the Qur’an for recitation, a hypothesis that was quickly seconded by many in the class and was agreed with by our Professor. Of course it does simply end there because, as our Professor quite rightly pointed out, it is also a great honor to be one chosen to do the recitation and because of that the students were not only well practiced in memorization but have a large respect for the method.

There is no reason to stop the speculation on the effect that religion and culture have on mathematics there though, let me spend a second talking about mathematics in the United States. As we spoke about on Thursday after the USA declared its independence from the Untied Kingdom way back in 1774 it was not only in governing that we decided to break away from the British model. We also changed our education system quite a but as well, so much so in fact that there is very little in common with the two systems now only 236 years since independence. The United States university system tends to function on the idea of: If more than one person wants to study it, it is probably worth studying as opposed to a more track based system such as that in the United Kingdom. While I can not say I agree completely with this idea, I am proud to say that I am a product of a system that does, for some reason that eludes even my radically liberalized mind, offer underwater basket weaving as a for-credit course in more than one university.

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