Why I Like Mathematics

Mathematics is rather like philosophy’s Siamese twin. They both share a far reaching ubiquity, as these studies can be used to describe elements in such diverse areas of inquiry as biology, physics, sociology, or spirituality, although the last of these may only be touched by mathematics in a metaphorical sense, although I find this quite valuable. However, despite their like levels of applicability, mathematics and philosophy also have strong elements of abstractions within them.

Mathematics differs from philosophy only by virtue of its rigor and well-defined terms. Whereas philosophy is understandably and necessarily vague in its use of some terms, such as “beauty” and “good” mathematics builds up its definitions from initially meaningless symbols, so every term can be explained in terms of other definitions, all the way back to the axioms. It is a gargantuan system of deduction built by us for our own use—mathematicians generally avoid making proofs and definitions longer and more complicated when they could be shorter and simpler. And yet, despite the most rigorous efforts to make such a vast chain of deductive truths understandable and manageable, many mathematical truths still manage to come as a complete surprise; therein lies the true beauty of mathematics.

As far as I have observed in my relatively few years in this lifetime, many non-mathematicians seems to view mathematics and symbolic logic as being predictable, mechanical, and unquestionable—set in stone. What I find particularly distressing is the tendency of many philosophers and scientists making assumptions in their arguments based on this view. How many times have I seen a militant atheist talking about the “unquestionable mathematical truths of logic”? In reality, mathematics and logic defies all expectations—rather than being straightforward and predictable, it is a frenetic merry-go-round of unexpected connections, and seeming coincidences run-amuck. Intuition is defied at every turn, as the impossible is proven and common sense is abandon.

How can such unpredictability occur in a system so rigid? Do the following. Add Contintue this process for all of eternity. When you arrive at the end of forever, take the mysterious number that eternity has produced, and divide it by six. Take the square root of the resulting number. What is it? It’s But why? It can be proved beyond a reasonable doubt that it must be , that it cannot be anything else but . And yet, despite the level of certainty we can have about this claim, it is rather difficult to shake of the feeling that it is still a rather bizarre coincidence.

I like mathematics in much the same way that I like chess. Now, I’m terrible at chess; truly abysmal if I am to be honest with myself. I stare at the board, and do not know where I should move my piece. Finally, I feel a glimmer of understanding—I make a connection, and thus a move. Of course, two or three turns after my epiphany has passed, and I have lost my Queen. Not only have I lost my Queen, but it is completely obvious to me why I lost my Queen, and what mistake I made. No matter how firmly I declare that I shall not make such a mistake again (after all, I understand my mistake well!) the same thing happens game after game. Mathematics is like a chess game, save for the fact that there is no winning or losing—only rules and strategy. Breaking a rule results in a contradiction, equivalent to moving a rook diagonally. Like chess, mathematics follows definite rules, and yet can still take you by complete surprise.

Quantum Mechanics, Chaos Theory, and More

I’m on another one of my math/science sprees, where for a short measure of time my learning efforts are more concentrated toward disciplines of a rigorous mathematical nature (although rigor does indeed succumb to the constraints of the practical where science is concerned). Due to some fascinating things that I have begun to learn in astronomy class, and some things that I have investigated on my own time, I am becoming increasingly interested in the wild and wonderful realm of knowledge where advanced math and science intersect. I like science of a mathematical nature (in harmony with my natural tendency to favor the general and abstract over the specific and concrete), and have finally found some areas of scientific inquiry where math is applied beyond mundane uses of multiplication and addition. Quantum physics deals with wave functions, partial differential equations, and even lie algebras, a topic from modern algebra. Meanwhile, chaos theory busies itself with recursive functions and fractals. All brilliant, all fascinating!

Chaos theory is a real treat. Theoretically speaking, it deals with deterministic processes (math has yet to produce the truly random!) However, even in deterministic processes, the systems can be so complex and touchy that it is impossible to predict specific phenomena beyond a certain level of accuracy. Consider the weather—it is so dependent on minute fluctuations in air currents and slight changes in temperature that we might never be able to know for certain ahead of time whether or not it will rain. Chaos theory instead looks for patterns that are threaded through the chaos, patterns that make it possible to discover astonishing and bizarre facts that arise out of seemingly meaningless data.

For instance, some things, such as stock market fluctuations, were shown by Mandelbrot to exhibit fractal behavior! That is, the fluctuations of a day to day basis were rough copies of the fluctuations on a month to month basis. Another interesting relationship between fractals and chaotic systems is shown in bifurcation diagrams. Depending on the growth rate of an animal population, it might stabilize, or oscillate between two, three, or more values. Or, it might descend into “chaos”. Also, chaos theory deals with things called “attractors”, states to which chaotic systems tend to if they get close enough.

I’m also beginning to understand a bit more about the strange and wonderful world of quantum mechanics. We’ve encountered quantum mechanics multiple times in astronomy class, and I think that I may finally be beginning to attain a glimmer of understanding as to what Heisenberg’s uncertainty principle really means. It’s confusing, because it’s often explained with metaphors that make it sound as if the uncertainty principal is merely a facet of our tendency toward human error, rather than a property of quantum mechanics itself. But it seems that it really is a property of physics, at least insofar as it is described by current quantum mechanical theory.

Two things helped me understand—the double slit experiment, and the properties of wave addition. It is all very complex, and I am still struggling to wrap my mind around it, but it seems that practically speaking, measurement is inseparable from something being restricted, an action that effects the properties of the thing measured. For instance, if a particle travels through a small slit, it’s momentum at the point of the slit will be quite uncertain. Total momentum will be known, but its maximum direction, or its value in any particular direction, remains unknown. Thus, you don’t know where the particle will hit a sheet on the opposite side of the slit.