“Mathematics is the queen of sciences and arithmetic the queen of mathematics” Carl Friedrich Gauss
What exactly is mathematics? What is the core function of mathematics? Did mathematics precede philosophy and science? How do these concepts relate to each other? While we don’t exactly have all the answers, this article will attempt to elucidate some of the concepts. Just treat it as though we are thinking aloud – that helps a ton in intellectualizing ourselves to mathematical fulfillment, a process you may love to undergo if you have been suffering through the mathematics syllabus in your respective countries.
How did it come about?
Mathematics is mainly based on logic. In its earlier stages of formation, it was usually married with philosophy, as the ancient Greeks used mathematics on top of philosophy to explain the scientific phenomena around us. Using draw the often quoted dichotomies (they are not really as polarized as they are; which our article would attest to) of rationalization versus empiricism, mathematics is rather empirical. Since it arguably preceded our understandings of science and the scientific method, it was always used as that scaffold for logic based on our comprehensions of abstract thought (to be explained later). Thus, to complete Gauss’ definition, philosophy would have been the king of sciences. Philosophy tirelessly draws from scientific discoveries fresh strength, material for broad generalizations, while to the sciences it imparts the world-view and methodological impulses of its universal principles.
Many general guiding ideas that lie at the foundation of modern science were first enunciated by the perceptive force of philosophical thought. One example is the idea of the atomic structure of things voiced by Democritus, whose cutting of watermelons led him to question if one could eventually break down the object to its indivisible parts. Certain conjectures about natural selection were made in ancient times by the philosopher Lucretius and later by the French thinker Diderot. What he anticipated became a scientific fact two centuries later. And it was with the help of the logical reasoning that gave birth to the Cartesian reflex and the philosopher’s proposition on the conservation of motion in the universe.
Sadly, as the sciences progressed, the public began having the notion that the sciences could stand apart from philosophy, that the scientist should actually avoid philosophizing, the latter often being understood as groundless and generally vague theorizing. The specific sciences cannot and should not break their connections with true philosophy – and mathematics the key node to binding the reasoning together.
How did we learn to pick it?
Gauss could not have put it better by choosing to distinguish arithmetic by explaining it as the queen of mathematics. Yet, how did humans really piece it all together?
The reality of the matter is that most humans did not really approach mathematics from any objective viewpoint. In fact, the philosopher and mathematician Bertrand Russell had this to say of his fellow brethren: “They don’t know what they are doing.” (His judgment of philosophers was even harsher: A philosopher in his eyes is a blind man in a dark room looking for a black cat that isn’t there). That is true in the sense that most mathematicians do not actually ask what is and what exactly are they doing.
In a classroom setting, such questions tend to be left unattended. Students are usually mechanically expected to learn to pick up mathematical concepts, drawing upon some sort of innate rationalizing capacities they have as children (think Heuristics and pattern solving questions) and solve them.
Surprisingly, humans have by and large shown that ability to pick up these skills. As mentioned by Israeli Academic Ron Aharoni, “Abstract thought is the secret of man’s domination of his environment. “ To him, he believes that the power of abstractions lies in the fact that they enable us to cope efficiently with the world. Abstraction means discovering regularity, and regularity means generality. Generality saves thought; in other words, they save effort. They enable going beyond the boundaries of the “here and now” – something discovered here and now can be used in another place and another time. If 3 pencils and 3 pencils equal 5 pencils, the same can be said of apples, even of another day. This one time effort is permanent and applies universally. The mathematician George Polya thus had this to say of mathematics: “Mathematics is being lazy. It’s letting the principles do the work for you.”
These abstractions are what humans are able to relate. And that is one component of mathematics. You take the reasoning from certain logical problems and attempt them onto a different platform and concept. While some of the questions may not be exactly practical, it sheds light on other problems which similar principles will appear. Through this, mathematics promotes basic habits of thought, such as the ability to reach logical conclusions. These are some of the most significant assets that schooling can provide. If abstractions in general are useful, then all the more so are mathematics, as mathematics this takes abstractions to the limit. Therefore, it is not surprising that mathematics is so useful and practical.
The idea of taking humans taking abstractions to the limit is wonderfully explained by Eugene Wigner, who defines a great mathematician as one who “fully, almost ruthlessly, exploits the domains of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: Certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection that it seems to possess. ”
Of course, to think that mathematics only allows humans to relate abstractions would mean for a much tunneled take on the subject of mathematics itself. From being forged out of physical phenomena, mathematics has since been able to be appreciated as a language based on arithmetic, and further allowed the domains of reasoning to be challenged and subsequently affirmed.
Pure mathematics has since been viewed as a language: its symbols thereafter carry meaning and can be combined into expressions in well defined ways to carry more meaning. This power and reach of mathematics allowed for us to channel the understandings of sciences and rationalizations to create concepts, which were explained based on such mathematical language. And such a rationalizing process taken to create concepts and explain conjectures seemed like a path that suddenly opens before you in a dark forest. Before that the thicket may have seemed unpenetratable, but once it’s open several different paths open in your way. Andrew Wiles, the English mathematician that solved Fermat’s conjecture used another analogy, which was somewhat similar to mine: A good concept is like the switch of an electric light, that you find your way in a darkened castle. When you turn on the light, you know what’s in your room where you are. Afterwards, when you go to the next room, you will have to feel your way again, looking for another switch.
Before that, it is important to note the abilities that a pure mathematician should process, as explained by writer Gaurav Tiwari. To him, a pure mathematician must be able to guess what kinds of mathematical structures can be built – is this conjecture provable? And are worth building? Secondly, to walk along mathematical structures that others have built is the mean ability, and much more. In my opinion, what he should have explained further is the fact that these improvements in mathematical structures should move tangentially to science – most importantly physics.
As things stand today, an increasing number of academics are finding the need to draw in mathematical reasoning to explaining its phenomena – such as biology and political science. The use of sophisticated mathematical models have allowed for the branching of computer science, handling of big data and various other statistical models used in finance.
In our next article on the same subject we would be talking about beauty and aesthetics in mathematics.
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