Thomas Enda Conlon

The ancient Greeks gave us medicine and mathematics as departments of  specialised knowledge. To this day people trust their doctors and they trust their mathematicians.  They of course feel the need of doctors more often and acutely than they feel the need for mathematicians, but their trust is more safely reposed in mathematics than medicine. In the history of mathematics there is no equivalent of grave faced medical men confidently recommending the application of more leeches, or muscular surgeons wiping their knives on dirty coats or white coated psychiatrists enthusing about the benefits of pre-fontal lobotomies.  Calculation works. A measure of how “scientific” our civilisation is the degree to which we conduct our affairs in the light of calculations rather than solely on the basis of tradition and accumulated experience.  The medieval cathedrals are probably the high point of achievement relying solely on techniques acquired by the slow accumulation of practical experience across the generations.  However no amount of tradition or ancestral wisdom will guide a rocket to land on Mars within metres of a target set before launch. Only vast amounts of calculation will enable this sort of achievement to pass.  Medicine has never been short of confident, not to say over confident, explanations of what underlies the efficacy recommended treatments even when such treatments have not proved to be very effective at all. By contrast mathematicians have been diffident and perplexed as to what the subject matter of their study actually is and why it has been so superbly effective. 

These central perplexities have attracted three major attempts at resolution respectively associated with the names of Plato, Aristotle and Kant. Plato would like us to believe that mathematical objects, like numbers and triangles, exist, alongside the forms of justice and beauty, as immaterial entities in a transcendental realm of being.  The application to the material world arises from the imprinting by the Demiurge of these ideal objects on the material world. Thus the imperfect triangles one draws in the sand stand in much the same relationship to their ideal versions as the behaviour of a just man stands in relation to the form of Justice. The ability of men and women to reason and gain knowledge about immaterial entities stems from a dim recollection of a prior incorporeal existence when we contemplated these entities more directly.  At first sight it might seem that Catholic Christianity might make its peace with Platonism and, in fact, for several centuries it appeared to do so. However Platonism lends too much comfort to dualism which more insidiously undermines classical orthodoxy than simple atheism or materialism, in much the same sense that a forger threatens the integrity of a currency more than a thief. With a renewed emphasis on core anti-dualistic doctrines particularly the Resurrection of the Body,  Transsubstantiation and the Real presence, the high Middle Ages saw a firm and decisive abandonment of any attempt at a profound reconciliation of Platonism and Catholicism.

Kant’s view, which had enormous traction in the 18^th and 19^th centuries can be seen as a radical anthropomorphising of a panentheistic view held by von Guericke and more famously by Newton. This view, radically dissenting not just from Aristotle but also from Leibniz and Descartes, that space and time were indeed objective but immaterial and were respectively infinite and eternal. This position was theologically underwritten. In Newton’s celebrated phrase Space and Time were the sensorium Dei – the aspects of the Divine nature by which God contemplated and sustained His material creation.  Essentially Kant’s mathematical philosophy is the variant of Newton’s view obtained by replacing the Divine nature by human nature or equivalently “sensorium Dei” by “sensorium omnis hominis venientis in hunc mundum”. According to Kant we see things in space and time because that is how our faculties of perception work. Our intuitions of time and space reflect, not a state of affairs in the objective world, but the structure of our brains. In the well-worn comparison, we are like a man who is convinced everything is blue but only because he is wearing blue tinted glasses. 

In a nutshell Kant’s philosophy of mathematics is that the foundational ideas of the subject represented (as it seemed to him) by geometry and the counting out of natural numbers arise respectively from our intuitions of space and of the steady elapse of time.  Mathematical truths are, in the last analysis, not truths about an objective world, be it material or immaterial, but about the structure of our own minds. Catholic orthodoxy is uneasy about Kant essentially because the tendency of his philosophy is to downgrade the great gift of Reason, one of the marks of human, as against animal, nature into a sort of sophisticated navel gazing. Mathematical reasoning is ultimately merely introspective while the external objective world eludes true understanding. More generally the scientific world is ambivalent. On the one hand Kant is the iconic philosopher of the Enlightenment when (supposedly) our civilisation began to rebuild itself on new humanist foundations. On the other hand most mathematicians today embrace the philosophy that makes the grandest claim for their science, to wit Platonism, and natural scientists are every bit as convinced as Aquinas that the senses are a reliable, undistorting conduit between the intellect and the external world. 

 Aristotle seems at first sight a very unlikely figure to have been embraced by Catholic orthodoxy. Reacting strongly against the ontological liberalism, not to say profligacy, of Plato’s academy. His philosophy is ontologically parsimonious. He rejects Plato’s idea of a transcendental realm of being populated by the Forms of Justice and Beauty as well as the mathematical ideal geometric figures, units and arithmoi which Plato held to be the true subject matter of mathematics. Whatever exists, exists in the category of substance. Objectively there are only things which are often just collections of things and human minds to think about them. Generality is achieved not  by postulating ideals which actual individuals somehow resemble but simply by noting the human capacity to advert only to relevant aspects of things. The state can look at you or me and see a citizen, the people we work with a colleague, the people who likes us a friend but the mathematician will see each of us merely as a unit – adverting only to “that by which we call something a one” (as Euclid, an adherent of Aristotle puts it) and see the state, our work colleagues and our circle of friends merely as “arithmoi” adverting to them only as “multitudes of units”.

Similarly geometric figures are just ordinary things – fishing rods, bracelets, necklaces, orchestral triangles etc. adverted to in a geometric way. There is no ideal triangle mystically hovering over the quotidian individual instances of triangularity nor any number “4” hovering  over this bridge  party, that string quartet, or any other concrete foursome and somehow encapsulating  the quarticity of the individual instances.  In developments stemming from the 19^th century  Aristotle’s “units” and “arithmoi” , now more commonly referred to as “elements” and “sets” have become a foundational idea for all mathematical thought. How this came to pass, why it was delayed to the 19^th century, and why the legacy of the Greek theory of arithmoi became in the 17^th century “the natural numbers” rather than a more general set theory is a fascinating topic in the history of mathematics.