Realities

For this reading the dichotomy of mathematical reality versus physical reality continued to distinguish itself to me as I read (annoyingly almost like peanut butter stuck to the roof of my mouth). Hardy reiterated and supported this distinction using several different ideas, most notably real math versus applied math.

The distraught Hardy laments that this mathematical reality is "useless," questioning if the world without the efforts of such as Abel, Riemann, and Poincare "would have been as happy a place without them."Is the mathematical playground, another reality for geniuses like Hardy to play in, only there because it is held up and supported by the applied math of the masses? Or are the masses alive and living to serve and sustain the real and serious mathematical entrepreneurship of mathematical gods like Hardy?

It seems for me that real mathematics (terminology from Hardy) cannot be done without the time and conveniences applied mathematics lends. But for some reason it seems wrong for me to say physical reality is more real than mathematical reality (side thought: which is more real?). Physical reality is complicated, has uncertainty, and causes irrational emotion. Mathematical reality gives humans, though a selected few, the ability to live in this ideal world, objective from human spin or interpretation according to Hardy.

We may live in a physical reality, but maybe our existence is in a more ideal mathematical reality. It is sort of like an example from non-Euclidean geometry versus Euclidean; a straight line on the surface of the earth may seem straight right in front of you, but over a long distance it curves.

C.P. Snow & Ramanujan

The author of the foreword to Hardy's A Mathematician's Apology is C.P. Snow, a very important figure in his own right. As we begin discussion of Hardy, be ready to answer the question, "Who was C.P. Snow?"

And perhaps the story of Ramanujan seems slightly familiar to you, but you can't remember where you heard it before. It may be that you saw Good Will Hunting:

Mathematicians are People, Too

 

               The Apology and Hardy’s own “Ramanunjan” piece seem to portray a dichotomous picture of the manner in which mathematicians (and, to a greater but less specific extent, creative artists in general) come into being. We have a stark contrast between two individuals: Hardy himself and “The Indian Mathematician” whom, from nearly unrecognizable backgrounds, play a part in the further development of mathematics as a whole. And finally, we are given Ellenburg’s article, which I think was a very interesting choice made by our professors to pair with these two. What I want to talk about, then,  is both the explicit and implicit relationships given between creative work, humanity, and our own mortality – how each is influenced by the other, and create a unique experiences for each individual.

               I think it is obvious that these pieces illustrate that neither man’s accomplishments can be separated from the men themselves, both in how they are perceived today and how they were perceived at the time. Ramanunjan’s story is a good illustration of this. The Apology mentions that the first documents that Hardy received for review from Ramanunjan had already been ignored by two other esteemed mathematicians, and Hardy mentions that Englishmen have difficulty in “relating” to an Indian. Hardy himself dismisses these documents at first. I think that this is more indicative of the social climate of the time period than anything mathematical. Coming from almost no educational background, he obviously would have been dismissed had it not been for Hardy’s sponsorship. The foreword to the Apology mentions that Hardy once (erroneously, in his estimation) wrote that had Ramanunjan had a better education, he would have been less than Ramanunjan. Both Hardy and the author think that this statement is false; however, we are all by-products of the manner in which we were raised, and mathematicians are no exception. Rather, I think that we should embrace this statement as one of wholehearted truth: had Ramanunjan been indoctrinated within the colonial education system of India under England, I highly doubt the world would have received the same product. Hardy himself, a Victorian scholar who dwelt in the Olympus of academia in England, should credit his upbringing as much as his own skills in mathematics for his success.

               As Ramanunjan died young, it is hard to apply Ellenburg’s article to him. Applying it to Hardy, however, is a very easy thing to do, at least on the surface. In an examination of his story, it would seem that yes, mathematics is a young man’s game; and yet, one need only look at his life experiences to see what other factors may have led to his creative end. Having lived through the First World War, Hardy (like many others) believed that nothing could be as bad as that war had been. I think that having to live through the Second World War caused Hardy, troubles at Cambridge, and a certain loneliness that he must have felt did more to dampen his creative drive than age. I am convinced that age has nothing to do with creative drive in any subject matter (save, perhaps, the actual deterioration of the brain). At least, in Hardy’s case, there are just too many other factors to assuredly state that his age had anything to do with possible future accomplishments. It is more important, I think, to look at how a person thinks and feels about themselves and their past work; if they believe they’ve done the very best that they can, then what possible drive can they have to go even farther?

Prime Years for Mathematicians

          In Apology and the Ramanujan readings, we learned a lot about the lives of the mathematicians Hardy and Ramanujan, respectively.  While this information presented the lives of the mathematicians not strictly in terms of their work, the thing I took away from the readings was highlighted by the third article.  Is math really a young person’s game?

          The Ellenberg article assumes that everyone knows mathematical discoveries are the work of the young and accomplished is a moment of epiphany.  I would tend to agree with the second part of that assumption, but I never even considered the first part.  I always thought mathematicians spent years of their life working in secrecy and locked up in a room, much like Wiles, whittling away at some new theorem, and presented their findings when they were around 40 years old.  But Hardy says “the years between eighteen and twenty-five are the critical years in a mathematician’s career.”  So hypothetically, I would have to come up with something brilliant either before I finished my undergraduate math major or graduate school for math.  Personally, I think that’s an unrealistic expectation for math students.  But Hardy’s statement was mentioned about 70 years ago.  More recently, the Ellenberg article says that “today one doesn't find mathematicians who revolutionize their field – even once – before the age of 22,” and that young probably means under 50.  The only reason the earliest “prime-time age” has been pushed back is because there is more math to learn.  

          I don't think we can really say when someone is past their potential as an academic.  To me, that's like saying that person can no longer create unique connections within their subject area, just because they are past a certain age.  I know the age range is not definitive, but it still bothers me that society has constructed an age range for when someone is going to be the most brilliant.  Are the math prime years just something people extrapolated from the information about past brilliant mathematicians?  Do those prime years exist in other fields or even at all?  We label prime years in athletics; is it fair to label the prime years in academics?  

Noneuclidean pancakes

http://www.gocomics.com/foxtrot/2012/10/14

Contradicting Contradictions

These two articles by Gray were really a great insight into how mathematics steps out from behind the shadows of science and philosophy. As math began to emerge you can almost see the growing pains that it goes through, as it tries to find its place in this world. The two biggest pains that math tries to tackle are: how can we quantify the world, if we even can, and  how do we deal with the complexities of contradictions that seems to be shaking our foundations? It was interesting to see that when it came to quantifying the world, math quickly ran into walls. 

The first reading really reminds of the class a couple weeks ago when we looked up a copy of Newton's Principalia to see how much it was worth. It really makes you wonder how can we put price on things like that? How do we determine how much a piece of artwork is worth, is it based on the technical difficulty, on the virtue of who made it, or do we put more weight into how the artwork makes us feel? This can then be extended to trying to figure out how much a human life is worth in the case of accidents where someone gets killed? How can insurance companies put a monetary value on a human life, how can you quantify the value of a soul? It is these types of questions that mathematicians quickly realize they couldn't answer, and that they were also questions they had no reason to answer.

 It was then interesting to see how mathematics try then to formalize itself into what it is today, as a set of rules and symbols. That as mathematics came out from behind the shadows it had to restructure itself so that it could hold up to scrutiny as it went beyond this world, and transverse into the realm of hypothetical. It was interesting to see how as the more formalized and "modern" math got, the more it took on the shape of philosophy. It would ask questions that you would not normally think to ask, then it would set out to find the answers. To me mathematics are one in the same, in that both aim to try to unravel the mysteries of the universe. The only difference is that mathematics deals with the quantitative questions while philosophy tackles the qualitative answers. That is how I view them, but what do you guys think? Is math and philosophy two sides of the same coin working in unison to unravel the mysterious complexities of the universe, or are they mortal enemies in an eternal struggle for dominance over each other till the end of time? 

At the end of class on Tuesday, Dr. Crist suggested that I volunteer for these readings due to the philosophic implications discussed, probably hoping that, being a philosophy major, I would be well equipped to find and display said implications for discussion.  To be honest, in one article I was frustrated by my limited knowledge of Set Theory and in the other I couldn’t help but recoil at the seeming dismissal of philosophy by many of the thinkers discussed.  Most notable among these men was Veronese, who is portrayed with the opinion that “If, [Veronese] suggested, philosophy is intended to be the highest level of research into truth, then mathematics does so much better than philosophy ‘being not only the most ideal but also the most positive of the sciences, because it is the oldest and the most precise expression of the truth.’”  Ouch.  Certainly our kind author should offer some sort of solace for the wounded philosopher: “But is every cardinal of the form of a natural number? … More food for the philosophers.”  As if they’re animals at a petting zoo. 

Anyway, putting that behind me, there are some serious philosophic implications to each of the readings that deserve discussion, the first being the comparison of shapes in mathematics to the idea of Plato’s forms.  Plato’s forms are ideal beings (in that they exist somewhere in reality) that act as a standard for other beings in the real world to “participate in,” functioning much like a concept.  An example is the idea of Beauty.  A thing can be beautiful, but it is never as beautiful as Beauty, and it is only beautiful because it “participates in” Beauty.  Other examples of the forms are Goodness, Hotness, and pretty much anything else you could imagine.  Contrast these forms with mathematical shapes and numbers.  Perfect squares and circles don’t exist in our world, but it seems plausible that they exist within reality.  This notion, considering a conceptualization of something as a validation for its existence, can be traced all the way back to Parmenides and his ontology.  So I guess the following questions arise: 1. Do the forms exist? 2. Do shapes and numbers exist? 3. If we assume the existence of one, must we assume the existence of the other?

According to Hermann Weyl, math is not true because it is logically consistent, but it is true because it exists within Physics.  That is to say, we can use math to express and interpret the world around us.  Suppose mathematics did not accurately describe the real world in tandem with Physics, but was still just as logically consistent as it is today.  Imagine if Newton never connected Calculus and Physics, but with a disjunction infinitely wider going all the way back to Euclid’s geometry.  Would math still be true?  Or do you need a grounding in physical reality for truth?