How do you explain the deep interaction between philosophers and mathematicians throughout history?
In the history of human thought development, there has been constant interaction between philosophical and mathematical aspects. Many intellectual movements have sought their support, inspiration, and even their model in the style and way of mathematics. What could be the reasons for this approach?
Philosophers try to understand and unravel the many enigmas that the real world proposes to them. But the reality is too chaotic to try to address it as it is. The world of mathematics is intended to simplify a few critical aspects of the real world.
It is a partial sketch of the world, human-made for us. It is natural that the philosophers of all time, more or less consciously, in their inability to penetrate the tangle of reality, have rightly considered mathematics a precious first field of operations on their way to more prosperous areas of life. The Pythagoreans’ attitude was transmitted with their influential and peculiar style by Plato and taken up several times throughout the centuries to the present day. This mood of thought makes those ancient philosophers, so “contemporary,” appear before our eyes. It would be more accurate to say that the contemporary style of thinking preserves quite faithfully many of the features of early Pythagoreanism.
But other exciting aspects of mathematics naturally attract the philosophers. The internal dynamics of mathematical thought, the logic of its structure make it a reliable model of reflection that arouses the consensus of all. Philosophers interested in clarifying human knowledge’s mysteries have seen mathematical thought as an ideal field of work to test their hypotheses and theories. In mathematics, general aspects of knowledge appear detached from other components, which makes its study simpler.
Even more recently, psychologists, concerned about aspects related to the study of human creativity, those who study artificial intelligence, have also turned to mathematics because of its paradigmatic, exemplary nature in such elements.
So far, we have seen some of the reasons for the philosophers’ approach to mathematics. But the mathematicians also have them very powerfully to approach philosophy. Ever since the Pythagoreans, mathematicians have been interested in what their activity means, asking themselves endless disturbing questions.
Where and how do mathematical structures arise? Is there already mathematics in things? Are mathematical systems only in the human mind? What is mind-world interaction like so that mathematics can emerge from it? How does the external world seem to adapt to mental structures developed as if by their dynamism, without practical intentionality? How to explain the mystery of the unreasonable effectiveness of mathematics?
The most profound element of mathematical thinking is, undoubtedly, the major challenge we have faced since the beginning of its existence: the lordship of infinite processes of thought. Mathematics did not mean more than a tautology if it were not for the presence of various types of infinite processes. How do you explain the possibility of such procedures? What does mathematical infinity mean to the structure of the human mind?
In the initial opening of the mind to intellectual knowledge, being in its infinity is present as a horizon or a condition of the possibility of concrete knowledge. In this horizon, the concrete knowledge must stand out, and that horizon makes any other knowledge possible. We do not consider it as an object. It is the bottom of our cognitive vision, and if it were not there, there would be nothing knowable.
The mind is, by its very nature, open to this horizon. It is something constitutive of their way of being. Being limited stands out in it precisely in a negative way and shows its limits.
Therefore, it is not surprising that the interplay with infinity is the excellent source of fruitfulness in mathematical thought but at the same time the cause of the deepest frustrations in those who have thought at some point of having it grasped between their fingers. The most fruitful moments in the history of mathematics have taken place precisely in the instants of mathematical audacity towards a new type of understanding of infinity: Pythagoreans, the discovery of the irrational, Zeno, infinitesimal calculus, mastery of infinite processes of passage to the limit, series, integral, Cauchy, Weierstrass, Cantor’s set theory, Gödel’s theorem, non-Cantorian set theories, etc. The infinite has slipped from our hands after each attempt.
Among the great mathematicians who have been concerned with the most profound aspects of mathematics, to the point of leaving a considerable mark on the thought of humanity, a few stand out with whose views we remain enlightened today.
To Pythagoras and his followers, we owe one of the profound characteristics of Western thought: the persuasion that the universe is intelligible, apprehensible by human reason, and more specifically, regarding the physical world.
Although Plato is primarily responsible for the effective transmission of the Pythagorean spirit without striking contributions to the development of mathematics, at the same time, he is the paradigm of the respectful and sensitive philosopher to the world of ideas. However, this veneration for the idea led him and many others, influenced by him, to disdain reality’s observation.
Descartes, Pascal, Leibniz were brilliant mathematicians whose philosophical contributions are marked with the stamp of mathematical style, each with its peculiar air. Among the closest philosopher mathematicians in time, Cantor and Poincaré should be pointed out, and already in our century, Hilbert, Russell, Whitehead, Wittgenstein, Weyl, Gödel, and many others…