Recently, one can observe a growing tendency toward a rupture between the scientific and the philosophical worldview. Even major scientists sometimes express the opinion that philosophy is useless for modern science, forgetting that the scientific approach itself was originally proposed by philosophy. In his book A Brief History of Time, Stephen Hawking writes that in the modern world physicists have advanced so far in their understanding of the nature of space and time that philosophy, which traditionally dealt with these questions, has become irrelevant. Another stereotypical view is that mathematics is an exclusively strict logical discipline and develops in a strictly sequential manner. Our task is to show that this is not the case, and to show that mathematics and philosophy have always moved together.
Philosophy is often defined as the history of ideas. Mathematics has always been part of philosophy. It is believed that Pythagoras introduced the very concept of âphilosophy,â and the foundation of Pythagorasâ own philosophy was the idea that numbers are the measure of all things. Plato regarded mathematics as the only true science because of its clear determinacy, since it operates with the immutable and motionless world of ideas. An important part of Aristotleâs metaphysics also consists of abstractions that have always been associated with mathematics: the whole and its parts, sets, wholes, and continuous magnitudes.
Major mathematical breakthroughs, at their foundation, carried above all a philosophical meaning. Already in arithmetic we encounter such a philosophical concept as infinity. Aristotle, in his Physics, noted that although natural counting in mathematics correctly reflects the physical counting of objects, in the physical world the concept of infinity loses its meaning. The concept of a point, which has no size, is also a difficult philosophical problem. Although it has no physical analogue, this basic mathematical concept nevertheless correctly describes the physical world in the language of geometry.
Stefan Banach proposed the following aphorism: âA mathematician is someone who can find analogies between statements; a better mathematician is someone who can find analogies between proofs; a stronger mathematician is someone who notices analogies between theories; but one can also imagine someone who sees analogies between analogies.â In fact, this is precisely the point where mathematics merges with philosophy, because the search for âanalogies between analogiesâ is essentially an epistemological â that is, philosophical â question.
The scholastics tried to derive the essence of the infinitely small through the famous problem of determining the number of angels on the point of a needle: something without size, yet really existing.
Descartes, through the idea of the mechanistic nature of the world, fixed it within his coordinate system and postulated the absoluteness and fundamentality of space. Leibniz, by contrast, considered space only in the context of the interaction of objects, as something that has no meaning outside their movement.
Newtonâs ideas concerning the beginnings of differential calculus were apparently influenced by Masonic conceptions of God â God as the Great Architect who created the mechanism of the universe â combining, on the one hand, the mechanistic nature of the world, and on the other, the infinity of God himself, which implies the infinity and mechanistic structure of space at all levels. To demonstrate this connection more clearly, one may recall that according to scholastic beliefs the spiritual could be correlated with the infinitely small. This means that Newtonâs fluxions may perhaps have been something that allowed him to mathematically formalize the spiritual principle permeating space. This hypothesis is also supported by the fact that Newton himself attached greater importance to his theological works than to physics and mathematics. In some of his theological writings, Newton discussed the idea that the motion of celestial bodies could be the result of the intervention of God or other supernatural forces.
As a counterpoint to Newtonâs ideas, one can refer to Democritus, who preceded him and denied the existence of gods. Matter, in his understanding, was not infinitely divisible, but ended in atoms moving chaotically. Perhaps, had Newton held views of physics similar to those of Democritus â views closer to modern ones â he would not have proposed and developed the ideas of differential calculus and Newtonian mechanics.
The philosophy of modernity brought the concept of determinism to an absolute. A materialist view of the world absolutizes the idea of the hidden and regards the complexity of mathematics merely as a consequence of the weakness of the human brain, placing obviousness at the center of everything.
For a long time it was believed that mathematics rests solely on proofs. Philosophers of mathematics in the past, of course, did not deny the role of creativity, but they tended to attribute it to the imperfection of the human brain and of human nature. They tried to move away from it. Bertrand Russell even hoped to reconstruct the entire edifice of mathematics starting from two simple logical axioms. But with the arrival of Gödel and his incompleteness theorem, it became clear that this approach was deeply mistaken. It turned out that such algebraic systems, in which axiomatics exists, can arise only from other algebraic systems. Bertrand Russellâs goal was to derive all of mathematics from only two self-evident logical axioms. However, upon reaching its limit, this idea collapsed.
Already Russellâs student Wittgenstein proposed a philosophy in which, relying on Russellâs idea that the world described through infinite logical constructions is strictly hermetic, mathematics says nothing at all, since all its statements are identities. In essence, Wittgenstein did in philosophy what Gödel did in mathematics with his incompleteness theorem. The attempt at a logical proof of the absence of the hidden only illuminated the problem of the supra-ontological.
Mathematics is not fundamentally different from other forms of human activity. Heidegger said that in recent times our thinking has become too subordinated to language, ignoring another mode of thinking: vision. The mathematician Vavilov writes: âOne cannot know mathematics, but one can be a mathematician. To be a mathematician means, first of all, to see, to possess a kind of super-vision that allows one to look through walls and over barriers â Ma chi ha gli occhi nella fronte e nella mente.â
Thus we see that being a mathematician and using mathematics are fundamentally different things. To be a mathematician means to understand mathematics as vision, whereas to use mathematics means to understand mathematics as language.
Mathematical axioms are self-evident only at first glance. Already Euclidean mathematics introduces into its foundation such a complex philosophical concept as infinity. Mathematics owes its depth and its power above all to the depth of its axiomatics. The essence of mathematics is not proof. Proofs are the language of mathematics, but not mathematics itself. True mathematics appears where a new vision and a new axiomatics appear.
At present there is a certain divergence between the philosophical and the mathematical view of the world. Kant claimed that arithmetic generates time, since temporal relations can be expressed by numbers, and that geometry generates space, since spatial relations can be expressed by geometric figures and laws. More modern views of the world suggest that neither number nor vector is fundamental; rather, symmetry and action are more fundamental. Arithmetic and geometry exist because transformations of mathematical quantities are symmetrical, and the basic level is the algebra of transformations. In this sense, arithmetic and geometry are equivalent if, in terms of group theory, they are isomorphic.
In his lecture âMathematical Proof: Yesterday, Today, Tomorrow,â Vavilov argued the following: in new mathematical works, what matters is not the absence of mistakes, but the presence of new ideas. He gives as an example the proofs of topologists, which are based on the perception of images illustrating the topologistâs thought. These proofs are inaccessible to non-specialists and may appear insignificant to an algebraist, yet they are nevertheless accepted within the community of topology specialists. He also cites the proofs of Gaussâs theorems, in which mistakes were found only seventy years later, when mathematics had already moved forward.
In reality, the proofs that we will not write do not need to be written, because nobody will read them. Proof consists in believing something oneself â believing it strongly enough to convince others, with a clear conscience. What matters are ideas which, after developing over time, will compress everything into trivial proofs, just as Euclidean geometry has been compressed. At present, for a professional mathematician, none of the theorems of Euclidean geometry presents any problem, because it is a purely mechanical task that, with the help of a Gröbner basis, can be reduced to computer computation. All school geometry can be reduced to computations with polynomials.
