I found after reading these articles wondering what truly inspires us to pursue the advancement of personal and public knowledge of ourselves, abstract ideas and manifested realities. Hilbert appears to suggest that much of the advancement of mathematics has historically been spurred by a handful of seemingly innocent looking questions, such as the curve of least time (a cycloid or brachistochrone curve for those of you who are familiar with the calculus of variations or classical mechanics) or Fermat's last theorem. Indeed, it seems to be the case that such problems, while having little theoretical implications in and of themselves, tug at our understanding by being deceptively simple yet have no obvious resolution. These questions seemingly mock the intellect by their very simplistic nature. How well do we really understand the concepts behind the question? For the curve of least time, the problem really relates to how one finds minima or maxima functions given a set of constraints using calculus. What is amazing about these questions is not the solutions themselves, but rather the winding paths that one may take in investigating the problem. Often, along the way, a person may discover a myriad of phenomenon that they would have otherwise been ignorant to. Insofar as these difficult challenges promote the growth of intellectual understanding of a field, such as mathematics as a whole, they drive the fields evolution and our own knowledge.
On the other hand however, one must consider how the individual actors that make up an intellectual group come to the realization of new knowledge. In Poincare's discussion on the matter, he attributes much of his realizations to the subconscious, an assertion that prima facie I found discomfiting. He later goes on to elaborate that one still requires conscious effort to get the "atoms" of intellectual knowledge that may later form the combinations necessary to come to a sudden realization of a profound truth. This analogy to the subconscious formation of new connections between ideas and problems made me wonder about the fundamental nature of truths. Specifically, Poincare appears to be asserting that knowledge or truth comes in 'atomic' packets (not literal atoms) through which one can construct new propositions via underlaying connections between these atomic truths and the new problems a proposition may seek to resolve. If truth is truly atomist, then one may cut away, in principle, at any given proposition and determine whether or not its constituent parts are members of the set of atomic truths. If it is the case that none of the atomic constituents are true, does it follow that the statement may be definitively declared false? What about if some are and some are not members of the set of atomic truths? Note here, that one must make the distinction between whether the truth of a proposition and the justification for believing whether or not a proposition is true. One may also ask if a proposition is constituted of all truths, and is valid (i.e. does not violate standard propositional logic) does it necessarily follow that the conclusion is true?
To perform a major redirect, the necessity of creativity in the construction of new knowledge is also an interesting consideration in the personal construction of new knowledge. It definately takes a different kind of person to understand deep connections between various abstract concepts such as non-euclidean geometry and functional theory, as in the case of Poincare. Does a through understanding of existing knowledge lead one to draw new connections? Alone, it seems, that such knowledge is necessary but not sufficient for drawing fundamental connections between new problems and existing abstract knowledge, but the connective step is not guaranteed. The construction of the bridge between existing knowledge to new problems is certainly not a straightforward process. I cannot even estimate the number of times I've begun to construct a solution of a 'new' problem using existing framework provided to me in a text, only to find that I had little clue as to where to begin the construction. When I come to impasses such as this I've found that noodling with an example or two often highlights a mechanism that I had previously not seen, and proves a promising route to the solution. But even this type of problem solving fails regularly, what then? To quote a quote: "We sleep, we dream and we write it down." It does appear that poincare is correct in his analysis that many answers jolt their presence into existence. A spark ignites some insight, often something "obvious" when written down but difficult to initially come up with, which is critical to the desired end result. Where does this spark come from? What is the nature of sudden insights? Why does it often appear, after reflection, that such insights are 'obvious'? Is it the case that the mind has inherent barriers that it needs to break down in order to allow itself to come to a conclusion or is the conclusion happenstance?
I would like to respond by positing something... probably very controversial with I feel not nearly enough proof to state as fact, but it is a philosophy which I have nonetheless come to believe is true of all persons and situations - it's actually very reminiscent of Eastern philosophies.....
ReplyDeleteAs in all walks of life, when it comes to the realm of thought, there exists a dualism. We think with logic and we think with emotion, and because logic is the only one that is "universally" and logically provable, we tend to raise it above the latter; yet I find that, more often than not, properly guided "emotional thought" reaches the very same conclusions, and often with a much greater variety of other knowledge learned along the way. Of course, this is a very hard concept to understand and impossible to explain in logical terms, because at its very root, “emotional thought” is fluid, chaotic, and never confined to an absolute answer. Indeed it flies in the face of the absolutist discipline of logic such that the two are practical opposites, and so if you don't understand what I mean, just try and bear with me.
As with all dualisms, though, neither of these modes of thought by itself can ever truly capture the full truth that comes when one acts as what Poincare referred to as the inventor. In keeping with Eastern-esque philosophy, the best understandings of any situation can only come from the coalition of logical deduction and emotional induction; one defines relations through what is known, the other through what could be.
THIS is what I believe Poincare was talking about with his dissertation on the realizations of the subconscious.
Think about it - let's use all prior knowledge of mathematical relations as the "logic" and, say, Poincare's aesthetic sensibilities as the "emotion". If a person only approaches math and the discovery of the unknown with such pure, base logic, whatever could they possibly accomplish? Using only what is already known and has already been done, how likely is it that you will stumble upon something new if you do not introduce any new ideas? It's possible, yes, but highly unlikely.
So, too, if you were to approach a mathematical problem armed solely with the emotional sensibility of aestheticism (i.e. the belief that the answer should be beautiful in its simplicity)... well quite honestly I wouldn't be surprised if you were laughed out of any serious institution.
I think Poincare was describing just that; after spending inordinate amounts of time trying to logically determine the solution to a problem [long enough that your efforts have dwindled down to simple plug and chug operations - not because your mind is decaying but simply because you are seemingly out of alternatives], you are so steeped in the one-sided extreme of logical thought that you no longer have any sense of how it all fits together. Enter the emotional sensibility. When your mind takes a break or you fall asleep and your subconscious takes over, you begin once more to access your emotional thought processes that see the aesthetic beauty of math and indeed of the universe and how it all fits together so perfectly and you begin thinking well why shouldn't there be a solution that makes the whole situation preposterously simple? The emotional component of the mind - the one that irrationally predicts - is now beginning to meet the logical component - the one that empirically proves - and only then can you accurately move forward and discover the truth of invention.
This is my answer to, well, just about all of your questions, as they seem to me to revolve around the same main point. I have tried to propose this explanation/"proof" as seamlessly as possible in the small space that I have blog it, but I am positive there are things that some people [assuming anyone bothers to read all that] will take opposition to, so I gladly welcome any and all feedback :)