Mathematics and Beauty

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In his book Úhelný kámen evropské vzdělanosti a moci (The Cornerstone of European Scholarship and Power) (Vopěnka 2001, 53), Petr Vopěnka expressed his belief that it was not the desire for truth, but rather awe and beauty, which led ancient philosophers to an interest in theoretical knowledge. Knowledge would be acquired regardless, by the common utilization of objects or by word of mouth,1 but an interest in knowledge for its own sake is what separates a barbaric approach to knowledge to that of the ancient philosophers. A barbarian (whatever interpretation of this word we choose to use) does not observe the world for the world itself. He is only interested in survival, in fulfilling his goals, desires and needs; and knowledge is only one of the means to achieve these ends. That is why he does not hesitate to use objects for purposes other than those which they were made for – he does not find it necessary to learn why a thing is as it is, how it is linked to its history, being, etc. The only relevant aspect of it is whether it meets his needs. Ancient thinkers, however, observed things for different reasons. They were interested in the things themselves, in their nature, laws, being. A pre-eminent example of this is the concept of theory, which is the centrepiece of the Greek understanding of the meaning and nature of knowledge.

The term theory means “viewing”, “beholding”. In his On the Soul, Aristotle adds that it always pertains to observation, speculation, but also contemplation – the highest form of philosophical activity (Aristotle 1996). However, theory is not an observation focused on finding a secondary purpose or personal gain. On the contrary, it is characterized by an interest in a thing for the thing itself. We study it not so we can better manipulate it or to achieve certain goals, but because the thing and its being interests us.2 To be able to study it well, however, as Martin Heidegger (Heidegger 1995, 165) points out, we need sufficient time. Theoretical observations are therefore characterized by us having the time and desire to observe, view, behold, imagine, speculate and test things both hypothetically and in reality – to occupy ourselves with them, to get to know them better. The value of such activity dwells in the finding itself, not in any further use of it. That is what theory and the Greek notion of knowledge have in common with beauty.

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Mathematics is among the oldest disciplines of human knowledge and has developed independently everywhere where there is evidence of human culture.3 It not only includes simple calculations, but also the investigation of the laws of quantity, structure, space and change and in its observations leans on both visual thinking (geometry) and abstract thinking. Mathematicians (gr. μαθηματικός (mathematikós) from μάθημα (máthēma), “knowledge”) – scholars – were thus people who not only knew something, but (according to Aristotle’s claim from the Posterior Analytics) also knew the “why”. Their knowledge did not come purely from a need for solutions to real issues, but from the results of theoretical exercises and a purely theoretical interest in the object of observation and the relationships between individual objects. That may be the reason why ancient scholars often eventually found the objects they were interested in outside of the common material world. Pythagoreans understood numbers as divine entities and did not see geometric objects as bound to the inventory of the material-burdened world. A geometrical triangle is not one that you trace with a finger in the sand, nor is it the one carved in stone. The triangle of which we can speak is the one in the ideal – geometrical world. The worlds of geometry and algebra cannot be plumbed by one’s own eyes. The line you can see is never perfectly straight, infinite and single-dimensional. What you perceive is just a material, imperfect simulacrum. “True” geometrically straight lines only exist in the world of ideas.

No surprise then, that Platonic philosophy is so well received among mathematicians. Its essence is the world of ideas. Truth, if it is to be, in fact, true, needs to apply to something eternal, infinite, immutable, something that can be known without fear that in a few moments that knowledge will no longer be true (Démuth 2013b). That is something the material world simply cannot offer. Entering the world of truths is therefore entering the spaces “beyond the world”4 – it is the art of finding the signs and archetypes of something eternal and divine in this world – of something, which ←13 | 14→is true. The ability to see the eternal contents is not always entirely natural and innate. It is often the case that a person needs to be initiated to allow them to see otherworldly objects.5

One of the classical examples of the initiation of a candidate to knowledge of science is education. As Petr Vopěnka (Vopěnka 2001, 34) suggests, teaching mathematics is about demonstrating facts and their evidence. A teacher can only demonstrate using examples, by means of which he reveals the given knowledge, but the pupil must see the knowledge by himself in the examples. It is therefore not irrelevant which roads are taken to knowledge, which roads do our thoughts travel. Sometimes, in demonstrating the mathematical world, the road of thought itself – the method of thinking – is beautiful, because by understanding it the pupil attains a tool that can reveal other – new (hitherto unknown) content. On such a road, one may encounter many remarkable, unexpected and alluring objects, or not – the road only being a tedious sequence of necessary steps. Sometimes, there are multiple roads to the result,6 and the choice of some of them may be more confusing than enlightening for a candidate. In such cases, a student does not perceive the laws, nor does he fully understand how he got to where he is and what it is that he perceives. He is but the silent witness to a spectacle he does not comprehend. If a candidate does not see the revelation of truth, he only perceives the demonstrated examples and does not understand. In contrast, if he understands why, the route his mind must follow and where his teacher is taking him, he will understand not only the laws and the beauty of the road, but will also see the ideas revealed in their own world and understand them. The Evidence of these relationships is often clear and self-evident and therefore do not need any further proof.7 Some basic proofs are entirely natural and do not need any hints. Some only appear after we are shown, or after the road we need to take to see them is suggested to us.