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Mathematics is the language of nature

SunnySingh's picture
Projects: 
I should preface this post with some warnings. First and foremost, it’ll probably be rather verbose. Second, I anticipate catching much flak for this; I have discussed these opinions with many others in the past and have always received strange looks. Finally, it's 2:30 am and though this is all clear in my head, it may come out as pure incoherent blabber. Last night I watched the movie Pi for probably the 108th time. Yeah, I know it’s just a Hollywood movie sprinkled with inaccuracies…but my love for math, physics, and computer science was born when I saw this movie. It opened up my mind to ideas I would never have fathomed in my wildest dreams. There are redeeming qualities in this movie-—many of which are related to our study of emergence. For those who have not seen it, I highly recommend doing so. Keep in mind that it IS a movie and not completely factual. Anyway, I’d like to post and comment on some of the more illuminating quotes/scenes which pertain to emergence. Before I delve into the quotes, here’s a brief synopsis for those of you not interested in seeing the movie: “Maximilian Cohen is on the verge of the most important discovery of his life. For the past ten years, he has been attempting to decode the numerical pattern between the ultimate system of ordered chaos—the stock market…” You can read more about it here. Max: “12:45, restate my assumptions: 1. Mathematics is the language of nature. 2. Everything around us can be represented and understood through numbers. 3. If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature. Evidence: the cycling of disease epidemics, the wax and wane of caribou populations, sunspot cycles, the rise and fall of the Nile. So what about the stock market? A universe of numbers that represents the global economy. Millions of human hands at work, billions of minds—a vast network, screaming with life. An organism. A natural organism. My hypothesis: within the stock market there is a pattern as well…right in front of me. Hiding behind the numbers…Always has been.” It was this quote that first slapped me across the face and made me consider the possibility that there is much more to mathematics than just long division and counting how many apples I have left after I eat two of them. We see that mathematics is everywhere around us. Mathematics is the conductor in the musical that we call reality. Everything we see and understand is governed by mathematics and physics—well, I should just say mathematics since the language used for physics IS mathematics. Some may say that mathematics is the product of man over thousands of years. I agree with this only to an extent. I agree that our current interpretation and understanding of mathematics was created by man—namely, the alphanumeric system and logical reasoning that we now utilize to understand our world. However, much like Pythagoras, I believe that the universe is made of numbers. When I say ‘numbers’ I don’t necessarily mean the conventional alphanumeric system that initially comes to mind—because, as I have claimed, this was created by man. I believe that even without our current foundations of mathematics/mathematical thought, there would be some ‘underlying math’ governing the universe. I can see where some may find this claim to be outrageous since, in a philosophical sense, how could something exist when it transcends us to such a degree that we cannot even contemplate it? (I don’t know how to articulate that coherently…my apologies). The degree to which we can understand our universe, in my opinion, is directly proportional to the overall understanding of math that mankind currently has under its belt. What I’m trying to get at is this: how do we know that our understanding is complete? How do we know that there isn’t more math out there that we haven’t discovered? What if that math transcends human thought? This leads me to the bane of my existence: the idea of ‘randomness’. The word random is thrown around so flagrantly in many circles. I could write an algorithm to spit out numbers, conceal the code and show someone just the output. Some may say that they are seeing a random number generator. Yet, there is an underlying order to it—namely, the algorithm producing it. I am not convinced that there is such a thing as randomness. I’m beating a dead horse with this, but I still hold onto the belief that if we perceive something as ‘random’, it is in fact orderly…however, we (mankind, not our class) are just not intellectually mature enough to see what’s truly going on. Should we define ‘randomness’ as something that may have order, but is beyond human thought? I don’t know. These thoughts are exemplified in Pi. For example, Max talks about his mentor, Sol, who spent his life searching for a pattern in pi: Max: “Sol died a little when he stopped research on pi. I wasn’t just a stroke. He stopped caring. How could he stop when he was so close to seeing pi for what it really is? How could you stop believing that there is a pattern, an ordered shape behind those numbers when you are so close? We see the simplicity of the circle, we see the maddening complexity of the endless string of numbers: 3.14 of into infinity…” I don’t know of any true research into patterns in the distribution of the digits of pi, but I would be highly interested in learning more about it. Regardless, the question Max poses here is what we’ve been asking ourselves in emergence. How can something seemingly so simple be complex? How can it NOT have an order behind it? I’d like to argue that there is order, but I don’t have any proof. Maybe it could be that my understanding of ‘randomness’ is primitive—which would render this blog both embarrassing and moot/flawed. I’d like to think that these thoughts are insightful and are not a product of a limited understanding. Sol argues with Max about how the universe does not take on order and that it is actually chaotic. Max replies with some interesting insight before they both completely snap: Sol: “The ancient Japanese considered the Go board to be a microcosm of the universe. Although when it is empty it appears to be simple and ordered, the possibilities of gameplay are endless. They say no two Go games have ever been alike. Just like snowflakes. So, the Go board actually represents an extremely complex and chaotic universe. And that is the truth of our world, Max. It can’t be easily summed up with math. There is NO simple pattern.” Max: “But as the Go game progresses, the possibilities become smaller and smaller…The board DOES take on order. So then ALL the moves are predictable.” Sol: “So? So?” Max: “So maybe, even though we’re not sophisticated enough to be aware of it, there IS a pattern—an order…underlying every Go game. Maybe that pattern is like the pattern I the stock market…the Torah…” Sol: “This is insanity, Max!” Max: “Or maybe it’s genius!” Until I’m truly convinced, which I doubt will ever happen, I will always side with Max on this. Am I insane? Am I putting too much faith in this movie? Have I been sucked into the romantic and quixotic ideas it conveys? Does this make all of my arguments biased and clouded? Moreover, I don't know if it's necessarily a good or bad thing that I dissected the movie in such detail...Anyway, I’m sure I’ll smack my forehead out of embarrassment when I fully recover from my stupor in the morning. Every time I watch this movie, I pick up on something new that has real-word manifestations. It makes sense in my head, and I felt compelled to share these thoughts with the class. Hopefully what I said is illuminating and not pure drivel. There is much, much more I want to say...but I really NEED sleep. Let the discussion begin!

Comments

DavidRosen's picture

I would argue that true randomness is actually inherently simple. For example, if you fill a CA with "random" noise, and then zoom out a little bit, you will get a uniform shade of gray representing the average cell color of the noise. This, to me, is why it does not really matter if the behavior of microscopic particles are nondeterministic or not. If you look at a slightly higher level, these larger patterns act in a deterministic way. The digits of pi are definitely not random; considering that there are formulae that return the nth digit of pi where n is any real number. I am no mathematician, but my problem with that movie's portrayal of this kind of research is that patterns in the digits of pi are meaningless, because like any PSEUDO-random number, or large enough body of information, it is possible to find any conceivable pattern if you look hard enough. In order to find any interesting or useful patterns, you would need to look at the formula for calculating pi and reduce it somehow, and we already know that it is irreducible in terms of our current mathematical rules. If anyone thought that we had discovered *all* math already, I think a lot of mathematicians would be very unhappy. My impression was that the field only exists because there is a lot of math we have not discovered, and are constantly discovering. If Max is so interested in finding formulae that govern how people behave, he should become an economist and learn about equations of supply and demand, trade relationships, and those kinds of things, that actually *do* have closed-form relationships to some extent.
Kathy Maffei's picture

I'd argue that this was a useful post, Sunny. Sure, films can oversimplify, and theatrical devices can detract from the subject in the attempt to make the movie entertaining, but sometimes it's also important to make it entertaining to get people to consider something! (and since I haven't seen this - yet - I can't even begin to judge) In short, there were some really interesting concepts you brought up. I also feel comfortable with the concept of a mathematical universe. It's funny, ever since I was a child I've had a habit of judging music by two properties: emotional and mathematical - I think of good pieces as being mathematically satisfying. I haven't yet met anyone (except my brother) who didn't think this odd. I'm less disturbed by the idea of randomness, though. Like I said before, I think randomness might give us the philosophical wiggle-room for free will in what might otherwise appear to be a deterministic system. But then again, I'm not disturbed by the idea that the universe is deterministic (as it sounds like you're suggesting). I was satisfied with Doug's comment on this in class (please forgive the paraphrase): does it really matter, since we can't tell the difference?
Kathy Maffei's picture

One more thing: a philosophy student once tried to tell me that math did not exit until there was language - that language had to come first. I strongly disagreed (without any study in the area, mind you) - it seems to me that someone without the development of language skills could assess quantities, consider the addition or subtraction of quantities, etc without the benefit of number characters. Some answers might be found in literature about feral children or about deaf children - I haven't looked into it yet.
Kathy Maffei's picture

Oddly, this article just happened to be in the Scientific American links. I guess that's some decent support for math irregardless of language!
jrohwer's picture

Sunny: you should read the article on Omega by Chaitin. I think it does a good job of explaining randomness and is very relevant to your belief that there is order underlying apparent randomness. What I got from it was that randomness is contextual. Within a givin context (a "theory" or "math"--set of rules, I guess), something is random if it is irreducible--that is, if there exists no algorithm for it that consists of less information than the actual random entity. So, if information can be compressed, it is not random. If it can't be, it is random. Chaitin argues that there cannot ever be a Theory of Everything in math because of the existence of such numbers as Omega-the probability that a program will or will not halt (embodying the indeterminability of the halting problem). I think this is really another way of saying what you said when you suggested that we as humans are somehow incapable of grasping the mathematical explanations for phenomena we perceive as random. For example, Chaitin talks about a scenario that Leibniz talked about in which a sheet of paper is splattered with ink drops by shaking a pen. They are random, he says, because although an algorithm could be written to produce a curve through all the dots (thus "predicting" their locations deterministically), no algorithm could accomplish this simply enough that it consisted of less information than just a list of the positions of the dots. The way I see this as relating to your post is that within the context of just page and ink spots, yes the ink spots are randomly distributed. But in a larger context in which one considers the shaking of the pen, its initial placement relative to the page, the viscosity, density, and surface tension of the ink, gravity, air resistance as those ink drops fall, etc., the outcome can be (mostly?) predicted with physical laws whose algorithms are relatively simple. Of course, you would bring in a lot of new information to make these predictions (so the change within the smaller context--from blank paper to paper with dots--would probably still be simpler to describe just with a list of the locations of the dots), but the change in the larger context, from before-shaking-pen to after-shaking-pen, might be reducible (i.e. not random). Maybe, like the [paper] vs. the [paper + pen + ink + person shaking pen, etc], the context in which we exist is "transcended" (to use your word) by some other context(s) in which phenomena that appear hopelessly random to us can be reduced/compressed/efficiently predicted.
LauraKasakoff's picture

My bias, I’m a math major. I love math. Part of the reason I love math is its universality, and my belief that everything really can be explained through mathematics. I have been warned before of the pitfalls of a naïve reduction argument, but I still can’t help believing that every discipline, everything physical or mental can be understood through mathematics. The other reason I love math is Square One TV, an outstanding 80’s public television educational program., but that’s another story... A really fun story. Pi = The Movie Pi is an awesome movie. In fact it is the only movie ever to cause me physical pain. Be warned if you are going to watch it, you will get a headache. Pi = The Number Sunny, I believe you are right in your hopes that we may be able to understand much more about pi (and other mathematical concepts we currently classify as “random”). As of now, mathematicians do not understand why the base 10 decimal expansion of pi is random. A big focus of Pi research is dedicated to trying to prove that pi is a normal number. The article Pi à la Mode gives a nice overview of pi research and normality. My personal hope is that there are no human limitations to mathematical thought, i.e. there is nothing in the world of mathematics that is beyond human comprehension. Some things that we have trouble understanding might not be limitations of human epistemology but may perhaps just be byproducts conventional barriers of thinking. For example, it is important to remember that we are used to thinking of numbers in the familiar base 10 decimal form. There are many other alternatives. We can look at pi in other base systems such as binary or as a continued fraction. Perhaps randomness is just a human construct, a word we use we haven’t discovered the right way to explain things. I don’t believe it will be impossible in every case to understand random objects. The pi story is far from over, and new ways of looking at pi can lead to lots of new information that can nicely chisel away at the randomness… The Debate: What is Math? What are its Limits? Does mathematics exist independent of us, or is it simply a human construct. Is the work of mathematicians to uncover new theorems and solutions or to invent them? In either case what are the limits of mathematics and what are our limits of understanding? This debate directly relates to the question of whether or not all of the world can be expressed by mathematical laws. If (as I like to believe) math is built into the world around us, Chaitin’s omega is a scary black hole of incomprehensibility. But for whatever reason, it doesn’t scare me. Intuitively, it makes sense that if math could account for everything in the universe, it would also have account for chaotic things. Is it possible that chaotic/ random objects could be understood through simpler rules? Sounds like a job for an emergent miracle! Perhaps it will be an emergent tool that will eventually allow us to understand the randomness behind omega!!! I really believe that math is inherent in the universe AND that there is no frontier outside the reach of human intellect. Here’s hoping.
Doug Blank's picture

Does your definition of "math" include computation? I guess I make a distinction from the topics that mathematics has studied (e.g., relationships between abstractions, and how to manipulate them) and computation. Otherwise, computer science is really just mathematics.