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.
I find the distinction between real and applied mathematics that Hardy explains to be very interesting. I've never really seen math as being divided into separate categories, it had always been that annoying class that I had to take. But Hardy makes the distinction that applied mathematics tries to express physical truth from a mathematical framework whereas real math expresses truths that are independent of the physical world. If this actually is the case, if there actually are very distinct categories of mathematics than to me, "real" mathematics has no true value, there's no cross over from the mathematicians playground into the real world. Sure, it could serve possibly as an escape for mathaholics, a place for certain people to get their math fix. But for the rest of us it serves no practical purposes. I feel like applied math, although may seem more "boring" than real math, is much more "real" than what Hardy gives it credit. There are actual applications to this sort of math that the majority of the populous can see and touch, not just the select few mathematicians out there.
ReplyDeleteI like the thought of real an applied mathematics. Physics does deal with uncertainty and more real world phenomenon, both classically and "quantumly," but most of its foundations are based in math. The differentiation of real math and applied math may be just a restriction on our human selves. We can't know the entire universe, at least not without bending a few physical rules first, and this restricts our scope of applying real math to the world. It could be entirely possible that all of the real mathematics that is known, and to be known, actually manifests itself in our universe but we can't see its implications on our scale.
ReplyDeleteIn response to Hardy saying that he seems to be devoid of any profound implications and genius breakthroughs, it seems to me that this is an exercise in humility. He could not know, for one on can, whether or not in the future some genius will spring forth after reading his book. To apply real math into a sociological perspective, he changed the conditions for how people view mathematics and mathematicians by writing said book. He also helped to educate other mathematicians thus creating an attractor in that field. Hardy has already set into motion small changes that perturb the perception of mathematics and has irrevocably changed that course of mathematics history. Though this is probably only at a small scale now and probably be drowned out by other socioeconomic norms.
Jason, I find your side thought of "which is more real?" as pertaining to mathematics versus physical reality very interesting. Also, when you say that our existence may be in "a more ideal mathematical reality," it makes me think of an age-old ontological question: Is reality what is in my head or is it what is in the world around me? Descartes's famous "I think therefore I am" claims that only two things are absolutely knowable; the first is that "I am a thinking thing" and the second is the existence of God, neither of which have much to do with physical reality. Similarly, since the time of Kant, epistemology has been dominated by the idea that we represent the world in our mind with systems of symbols, and those symbols are what we treat as real, an opinion I tend to agree with.
ReplyDeleteThis, combined with Hardy's elitism ("since most people can do nothing at all well") and especially his exclusionist view of mathematics, makes me think that perhaps men like Hardy simply use different symbol systems to represent truth. It's a common notion that people use images and language in order to think; these are the building blocks in our symbol system, likely chosen due to their accessibility and easy application to the real world. Perhaps the greatest mathematicians find math and number to be more accessible and easier to apply, so they substitute or integrate that as their way of thinking and come to the conclusion that their system of thought is truth. From their perspective, this way of thinking is natural and the rest of the world is lacking something when it comes to math, but wouldn't the greatest composers say the same of the world when it comes to music? Or, as Hardy seems to prefer, when it comes to poets and poetry?
I guess I just dislike Hardy's elitism, but that is to be expected of most top scholars from that time period in England. Like we discussed in class, they saw strict and rigid separations between men in the world, whether it was class, race, or intellect; it was integrated into their way of thought.
Zach, Your epistemic argument (or account of the argument) that we represent the world with symbols within our mind is a very interesting notion. Yet, I cannot help but wonder what you exactly mean by symbols. After all, I do not think in literal characters, and doubt many people do, so I must wonder if you are using the term a bit more broadly to mean any representation (such as sound or imagery) as a symbol?
DeleteNow my reaction to Hardy's claims about the real and applied mathematical world seem paradoxical to me. At one point he claims his arguments about the veracity of euclidean geometry would be invariant of the given physical perturbations to the environment. But I really wonder if this is true? Can we truly know if a logical system is fully independent of the fundamental nature of the world? Could it be possible that we find such things true, not because they are true independent of our current physical state but rather because of it? The surface of the planet on our length scale after all IS locally euclidean. In fact all of space-time is thought to be this way... and if I recall correctly ALL differential manifolds, a very mathematical idea, are locally 'flat' about a point on the surface regardless of the overall curvature present. Therefore, is it possible we only find such things true because they are inherently true, or at least potentially realizable physical states?
I also found this idea of real and applied mathematics to be a very interesting topic. To be honest, the math one needs to solve problems which aren't applied to everyday life, is very confusing to me. I love math, don't get me wrong, but when I stopped understanding it, up to calc 3, it didn't make sense to me to take anymore. I got my usefulness of it, and find the rest 'trivial' as Hardy calls it. I would have to disagree with Hardy though, that ballistics and aerodynamics are not real. I actually think the applied math for war is real.
ReplyDeleteI...umm..hmmm.... hahah okay see cuz I wanted to direct my comment at your... well, I guess it would be kind of the general question that most of these posts appear (to me) to beat around and that is your aforementioned dichotomy between mathematical reality and physical reality.
ReplyDeleteAnd initially I was just going to throw down my opinion but... I don't know after reading all these posts I feel like either everyone is misinterpreting the point or else I am completely mistaken...but I'll lay it out anyway and I hope it doesn't sound too combative because I think you brought up some very good points.
Anyway regarding your third paragraph I quite disagree with.. well I guess I disagree with the train of thought in which you took this topic; and by that I mean that I don't think it is even at all an issue of which is more real or whether "the masses" are there to serve and sustain the growth of mathematics... in that regard I find your questions surprisingly difficult to answer because I don't agree with them.
Because from my point of view (which, from an excerpt that I remember reading, appears consistent with Hardy's), mathematical and physical realities are both completely real to the same extent, and if I had to summarize their relationship in one phrase, I'd call it a guidance system of mutual checks and balances.
Let me explain: as I understand it, the true passion of the mathematician lies in theory/"mathematical reality" - as they imagine all the possibilities of what could be and, out of curiosity and (above all) fun, they spend their time deriving seemingly random curious relations among numbers almost just for the hell of it and the result is a beautifully creative concept of their imagination. But then reality kicks in and they have to start obeying its rules, judging whether their new concept is worth anything in the laws of the real world/"physical reality".
But where else could physical reality come from if not having stemmed in the domain of the imaginative and potentially irrational?
And beyond that, what if reality proves that relation wrong? Does that make it completely useless? Of course not.
Basically my point is that mathematical reality and the theories of numbers and their relations, though not technically practically useful to the human population as a whole, are nonetheless absolutely necessary for the progress of applied mathematics as whole - take modern particle physics, for example, where such entities as the Higgs boson were predicted by pure math decades before experimental physicists were able to even remotely observe it.
Along those lines and my "definitions" above, I turn to disagree with your thought in the third paragraph that physical reality is complicated with uncertainty and "irrational emotion" while the mathematical reality supplies the objective view. I believe, in fact, that it is almost the exact opposite - it is the mathematical realities that reside in the realm of the "irrationally emotional" imagination, full of generalities and uncertainties that can only be clarified as "fact" through the check system of observation in physical reality, where there is far less human spin on the subject.
And again, I could have misinterpreted the whole topic here but hopefully that is not the case.