I was watching a video on the Hubble space telescope (indeed very geeky) and there were a few very interesting thoughts that came to my mind. Its not that I was not aware of this line of reasoning but maybe I never developed it to any appreciable extent. The more I think about it, the more astonishing it gets. Let me explain.
The first thing that struck me in the video (90 minutes documentary) was the immense forces at play on a universal scale, the almost incomprehensible extent of the universe and the unfathomable distances and time scales involved. We are all very aware of this aspect of the universe. The next thing that caught my attention was the immense cosmic dance giving rise to supremely exotic phenomenon occurring almost with a mundane regularity in the universe. From the devilish grasp of Black holes to the concept of cataclysmic Supernovae and immense energies of the Quasars to galactic collisions, nature plays the game of destruction and beauty at a level we can hardly comprehend and she plays it with the virtuosity of a Horowitz gently stroking the keys of a grand Piano. But these were not the things that impressed me the most about this video. It was something else.
Einstein once said that the most surprising thing about nature is that it's comprehensible. And if you think about it, its rather disconcerting and very astonishing. You see, nature is not obligated to make sense. The fact that a few equations on a piece of paper can accurately describe phenomenon as weird as gravitational lensing and stellar implosion is nothing less than startling. I do not have much idea about Quantum theory but I have read a bit about Einstein's gravitation and all I can say about it is that its a triumph of human intelligence. I do not want this to be a geeky post so I will go straight to the essence of it all. The only assumption in Einstein's theory is the constance of the speed of light. Its hardly a theory of physics. Its pure mathematics. Its just a geometrical statement. And whats seriously weird is that the final equation was found by a guess since there are infinite other equally correct such equations. Einstein's equation just happens to describe the universe with a scary accuracy.
I find it strange that nature and mathematics are such close bedfellows. Why is it all so simple and so logical ? Why does nature dance to the tunes of purely mathematical laws and relations ? I am not sure if I am communicating this idea well enough. You see, mathematics is a very rigid discipline in which if a+b=c then there is no way a+b=d unless c=d. If, on the other hand, we represent two physical quantities by a and b and then try to find a+b, nature is not obligated to give us c as an answer, but it does. An example would be the conservation of energy. Saying that energy is conserved in a physical system and that 2+2=4 (always) in a mathematical system have a deep connection because we have chosen to describe nature via mathematics. But in the end they are two very distinct entities. The fact that we never see a violation of conservation of energy and that we never find that 2+2=5 somehow signifies a deep inter-dependence of the most basic natural laws and the most fundamental mathematical tenets. And this thinking rests on the sole fact that physical reality and mathematics form two ends of a very interesting spectrum. While physical reality is the ultimate truth which does not depend upon anything else for its sustenance, mathematics is the sole discipline which does not seek to explain anything and which does not depend upon any other science. Everything in between including physics, chemistry, biology either serve to explain the physical reality or emanate from mathematics or both. I find it interesting that the two fields which are just not obligated to be connected end up getting so closely tied together. Which makes me think that if there is such a thing as an ultimate truth, an ultimate reality, the only way it will be found would be in the abstract dance of purely mathematical symbols. And when you think about it, you would wonder if its all too obvious that within the infinite relationships between purely mathematical concepts, there would be one relation that would be the statement of the ultimate truth. Its just that humanity is just not intelligent enough to zero in on it, as yet.
Addendum: Well, thinking a bit more upon the topic, I have realized a rather grim possibility. If we take it that nature and mathematics are closely tied to each other and that all natural laws, howsoever deep, are ultimately expressible mathematically, we will soon reach a dead-end. A brilliant mind, with the name of Kurt Godel, gave a landmark theorem called the Godel incompleteness theorem which proves that a mathematical system cannot be both complete and self-consistent. In other words, a mathematical system that seeks to explain everything must necessarily be inconsistent and vice-versa. I wonder if it has ramifications in our understanding of reality.
The first thing that struck me in the video (90 minutes documentary) was the immense forces at play on a universal scale, the almost incomprehensible extent of the universe and the unfathomable distances and time scales involved. We are all very aware of this aspect of the universe. The next thing that caught my attention was the immense cosmic dance giving rise to supremely exotic phenomenon occurring almost with a mundane regularity in the universe. From the devilish grasp of Black holes to the concept of cataclysmic Supernovae and immense energies of the Quasars to galactic collisions, nature plays the game of destruction and beauty at a level we can hardly comprehend and she plays it with the virtuosity of a Horowitz gently stroking the keys of a grand Piano. But these were not the things that impressed me the most about this video. It was something else.
Einstein once said that the most surprising thing about nature is that it's comprehensible. And if you think about it, its rather disconcerting and very astonishing. You see, nature is not obligated to make sense. The fact that a few equations on a piece of paper can accurately describe phenomenon as weird as gravitational lensing and stellar implosion is nothing less than startling. I do not have much idea about Quantum theory but I have read a bit about Einstein's gravitation and all I can say about it is that its a triumph of human intelligence. I do not want this to be a geeky post so I will go straight to the essence of it all. The only assumption in Einstein's theory is the constance of the speed of light. Its hardly a theory of physics. Its pure mathematics. Its just a geometrical statement. And whats seriously weird is that the final equation was found by a guess since there are infinite other equally correct such equations. Einstein's equation just happens to describe the universe with a scary accuracy.
I find it strange that nature and mathematics are such close bedfellows. Why is it all so simple and so logical ? Why does nature dance to the tunes of purely mathematical laws and relations ? I am not sure if I am communicating this idea well enough. You see, mathematics is a very rigid discipline in which if a+b=c then there is no way a+b=d unless c=d. If, on the other hand, we represent two physical quantities by a and b and then try to find a+b, nature is not obligated to give us c as an answer, but it does. An example would be the conservation of energy. Saying that energy is conserved in a physical system and that 2+2=4 (always) in a mathematical system have a deep connection because we have chosen to describe nature via mathematics. But in the end they are two very distinct entities. The fact that we never see a violation of conservation of energy and that we never find that 2+2=5 somehow signifies a deep inter-dependence of the most basic natural laws and the most fundamental mathematical tenets. And this thinking rests on the sole fact that physical reality and mathematics form two ends of a very interesting spectrum. While physical reality is the ultimate truth which does not depend upon anything else for its sustenance, mathematics is the sole discipline which does not seek to explain anything and which does not depend upon any other science. Everything in between including physics, chemistry, biology either serve to explain the physical reality or emanate from mathematics or both. I find it interesting that the two fields which are just not obligated to be connected end up getting so closely tied together. Which makes me think that if there is such a thing as an ultimate truth, an ultimate reality, the only way it will be found would be in the abstract dance of purely mathematical symbols. And when you think about it, you would wonder if its all too obvious that within the infinite relationships between purely mathematical concepts, there would be one relation that would be the statement of the ultimate truth. Its just that humanity is just not intelligent enough to zero in on it, as yet.
Addendum: Well, thinking a bit more upon the topic, I have realized a rather grim possibility. If we take it that nature and mathematics are closely tied to each other and that all natural laws, howsoever deep, are ultimately expressible mathematically, we will soon reach a dead-end. A brilliant mind, with the name of Kurt Godel, gave a landmark theorem called the Godel incompleteness theorem which proves that a mathematical system cannot be both complete and self-consistent. In other words, a mathematical system that seeks to explain everything must necessarily be inconsistent and vice-versa. I wonder if it has ramifications in our understanding of reality.
4 comments:
I've been thinking for a while and can't seem to come up with anything convinceing. But I am too tempted to take a shot-
Mathematical thought (logic/connectedness/building comlex forms out of a set of fundamental forms) is probably the most fundamental mechanism by which human beings think and express themselves. We are at the same time products of evolution and Nature.
Mathematics is just the "language" in which humans express their "discovery" of the "Nature" that led to and is ingrained in their existence. Hence the connection.
Ok. Let me combine your previous post to this one.
There is a direct mapping of the needs of a human being to mathematics, in particular mathematical tools. Both of them keep on evolving. A researcher few decades back would be thrilled at using some of the then sophisticated mathematical tools (say Expectation Maximization). He would have his own reasons to claim that the tools were needed to his end. His peers might frown upon claiming that it was not and the same end could have been accomplished using less sophisticated tools. But its highly probable that those tools are now a necessity. No one thinks of them as a luxury. Just like having a TV was thought of by many people as an unneeded-sophistication few decades back, but now its not given a thought.
One thing that froths up is the notion of choice. I as a researcher in 1980 could have managed without expectation maximization. However, it can only be my decision to claim it as necessary or unnecessary. I as a consumer in 1980 (assuming I was born much before that) could have managed without a TV. However, it can only be my decision to claim it as necessary or unnecessary.
Another interesting analogy stem from the basis on which the choice was made. A researcher may use expectation maximization to only make his research accepted in a journal, as expectation maximization is cool thing to use. The other end is that he actually felt that using expectation maximization will help him optimize some of his costs. Both are permissible needs. Similarly for TV.
Btw if you have been wondering why expectation maximization? I felt the need to use it ;)
@Vadrevu: I would agree with you for the most part. The only thing I would want to add is that although mathematics is the language humans express Nature in, the aim of mathematics is never that. In fact, its the most abstract of languages which concerns solely with its own existence. Its the other fields like physics, statistics etc. which apply mathematics to nature. So the connection cannot be explained by saying that one explains the other. Rather we have to keep in mind that these two fields are not connected at all and mathematics was never invented to explain nature and its never intended to do it even.
@Rasia: Mathematics is not a luxury in the explanation of nature. Although some researchers might find it cool to use expectation maximization, I am sure there are deeper understandings to some phenomenon which depends strongly on the concept of expectation maximization. I think that nature remains hidden to us because the necessary mathematics has not been invented/discovered till now. Like tensor algebra and its role in explaining curved geometries of higher dimensions which directly led to relativity. I am talking more about fundamental science. But I see what you are saying. For most part its a choice but for true advancements in natural understanding, mathematics is not a luxury.
As the protagonist despairs in '1984', "Freedom is the freedom to say that two plus two make four. If that is granted, all else follows!" As Orwell shows us, math can probably show the truth, but we shouldn't be blindfolded! I disagree with your statement about math being the only way, as to the reason, 'general principle' is the only resort to ignoramus like me!
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