Flames of Math

The history of mathematics is one of the longest histories of the world. Since the early days of humanity, formal inquiries were given proper names. Though the findings of Pythagoras (and the Pythagorean bravado of Plato) are still closer to mythology, the post-Aristotle time seems to entirely sit in treatise and document. Probably, this heap of data has something to do with the development of mathematics as such. Still, it is much more reachable from the philosophical angle, including both philosophy of math and philosophy of science in general. Once we need a historical allusion to support an abstract discourse, there it is, the illustrative mathematical fact. An encyclopedia of method, that's what we've got.

However, philosophy does not come to mere cognition theory. An exaggerated care for the techniques of knowledge mining is a good way for somebody to distract the public attention away from much more important problems. Accordingly, too much scientism in philosophy (as well as its formal inverse, phrase mongering) is nothing but a disguise, a showy trick. Real wisdom may look like science; but it does not scorn neither a vivid picture nor a piece of (non-abusive) exhortation. In any case, philosophy is to be primarily concerned with human deeds, examining their worldview and the view of the world's future.

To attenuate the traditional pretentiousness of mathematical philosophizing, let us commence with a flaming metaphor, tracing an analogy between the history of mathematics and the history of conquering fire. The both stories are of approximately the same size.

Primitive mathematical knowledge was mainly implemented in the daily practices of counting, measurement, distinguishing forms. It was preserved in a practical manner, passed from the tutor to an apprentice, from one generation to the next. Rare independent ploys were immediately incorporated to an appropriate craft chain; the chains ramified and replicated, so that the overall level of mathematical competence grew millennium be millennium, then century by century, then age by age. The history of fire shows exactly the same: we cannot yet produce fire, but we can keep it and pass to the others. The both techniques were insistently improved, which might keep it burning for a few thousand years.

Later, people learned to produce fire by friction. This liberated them from the need of centering the culture on a fireplace; a new hearth could be built anywhere at any urge. Fire became more reliable and more secure. As a result, the whole bulk of technologies related to fire conservation and transfer gives way to the progressive technologies of fire production. For this end, people invented ingenious aids and appliances, designed new surfaces with efficient friction control, used steadily flammable substrates, and so on. The hot motivation accelerates the development of industry in general, which even more simplifies fire management. Eventually, we come to a real masterpiece, the climax of the method, an ordinary match.

Similarly, in mathematics, the first clumsy substantiations get gradually polished up to becoming a well-regulated deductive scheme sometimes teachable even to robots. The numerous reformulations of the axiomatic carcass of mathematical theories provided us with a selection of elegant patterns to refine any new theory starting from just a few basic considerations.

Well, friction is not the only way to do fire. There is a parallel industry based on spark production at the impact of one solid against another. Here, too, one finds a lot of progress-boosting opportunities and comes to the modern lighter as an acme of the trade. Of course, the two branches interact, speeding up each other. Virtually, they use the same principle: we need to introduce a heated body in an easily flammable medium and to lead the combustion to a well-controllable mode. Which is the best? What is primary? A void question, since it is the practical considerations that rule. We use what better suites our needs. In the same manner, the agitated discussions about the foundations of mathematics do not much influence a working mathematician, who may resort to one approach or another to enhance the overall feasibility. Just aim at a stationary burning, that all.

In this way, the humanity approaches a new epoch in its fiery history. We learn that open fire is a sort of plasma, and the flames could be perfectly tamed when confined to a controlled gas flow or even put inside a sealed discharge tubes. As a first step, we develop a novel ignition technique, electrically incited discharge in the air or a special gas mixture. We no longer depend on the errant ways of tiny sparks; now, this is a full-fledged electric arc. We proceed with using electric lighters to kindle open fire; still, the principal direction of technological development is shifting elsewhere. The same trends could be observed in modern mathematics as well: the focus shifts from derivation and calculation to modeling. The mathematical fire is now playing in computer hardware, in algorithms and networking.

Don't stop at that. We know about yet another (exotic) way of flame production, namely, by concentrating light from the sun. This method never occupied a significant place in the common trade, since the good lenses it needs came on the crest of the other ignition technologies; solar lighters have remained a children's toy or a playful rite. On the other hand, the sun is here now, and hidden in the clouds the next moment; one cannot put in in the pocket like a box of matches or a gas lighter. There are regions where you need to wait for a few months to admire the sun over the horizon. So, can we get anywhere in these lines? Yes, we can. At least, we arrive at a big dream. Indeed, the Sun is the virtual source of all flames on Earth. Why not dare to light many artificial suns? After all, the Sun is a mere dwarf star in just one of the myriads of galaxies, while we are humans, and we can endeavor more than that, we'll light new worlds.

Probably, some of the mathematical trifles that do not yet attract any academic interest keep the light of a mathematical dream burning within us, making us long for mysterious reigns beyond the formal universe of the present. It is much later that we'll get really acquainted with them. For the time being, please, no disparaging remarks about philosophy! Yes, it is out of any rigor. But here, we are not entirely tied up by formal decency, and we can play and dream.