Gian - Carlo RotaForeword
There are two erroneous extremes one might fall into in regard to the philosophy of mathematics. In the one, which we could call naive objectivism, mathematical objects, such as the triangle, the regular solids, the various numbers, a proof, or abelian groups, are taken as simply existent apart from the work of mathematicians. They exist whether we discover them or not. In the other, which we could call naive subjectivism or psychologism, mathematical items are taken to be mental constructs of mathematicians, with no more objectivity than the feelings someone might have had when he thought about a rectangle or worked out a proof. The truth, as usual, lies in the middle. Mathematical items are indeed objective. There are mathematical objects, facts, and valid proofs that transcend the thinking of any individual. However, if we, as philosophers, wish to discuss the objectivity of such items, we must also examine the thinkers, the mathematicians, for whom they are objective, for whom they are facts. The philosophy of mathematics carries out its work by focusing on the correlation between mathematical things and mathematicians.
It is in this correlation, this "in between," that Gian-Carlo Rota has developed his own highly original and programmatic philosophy of mathematics. He draws especially but not exclusively on the phenomenological tradition. The point of Husserl's phenomenology, which was further developed by Heidegger, is that things do appear to us, and we in our consciousness are directed toward them: we are not locked in an isolated consciousness, nor are things mere ciphers that are essentially hidden from us. Rather, the mind finds its fulfillment in the presentation of things, and things are enhanced by the truth they display to us. The subjective is not "merely" subjective but presents objectivity to itself. Husserl's doctrine of the "intentionality" of consciousness breaks through the Cartesian straightjacket that has held so much of modern thought captive.
Another principle in phenomenology is the fact that there are different regions of being, different "eidetic domains," as Rota calls them, and each has its own way of being given to us. Each region calls for a correlative form of thinking that lets the things in it manifest themselves: there are, for example, material objects, living things, human beings, emotional facts, social conventions, economic relationships, and political things, and there are also mathematical items, the domain that Rota has especially explored.
One of the phenomena that he develops in this book is that of evidence. Most people think that in mathematics truth is reached by proof, specifically by deriving theorems from axioms. Rota shows that such proof is only secondary and derivative. It is not the primary instance of truth. More basic than proof is evidence, which is the selfpresentation of a given mathematical object or fact. We can know that some things are true, and we can even know that they must be true, before we have found axiomatic proofs to manifest that truth. We know more than we can prove. Rota shows that axiomatic derivations are ways in which we present mathematical objects and facts, ways in which we try to convey the evidence of the thing in question, ways in which we bring out the possibilities or virtualities of mathematical things. Proofs are valuable not because they bring us assurance that the theorem is in fact true, but because they show the power of the theorem: how the theorem can present itself and hence what it can be.
Evidencing could not occur, of course, except "between" the mathematical object and the mathematician as its dative of presentation, and yet it would be quite incorrect to see evidence as "just" psychological. Rota correctly observes that the depsychologizing of evidence is one of the great achievements of phenomenology. To appeal to terminology used in another philosophical tradition, evidence is the introduction of a fact into the space of reasons, into the domain of logical involvements. Does not such an introduction belong to the space into which it enters? Is not evidence a rational act, indeed, the rational act of the highest order? What is given in evidence is just as logical and rigorous as what is derived within logical space. Mathematical facts are objective but they are achieved and even "owned" by someone in a sense of ownership that is sometimes recognized by adding a person's name to a theorem or conjecture. Rolle's theorem is so named not because of a psychological event but because of an intellectual display, an evidence, a logical event, that took place for someone at a certain time. Once having occurred to him, it can occur again to others at other places and times, and the theorem can be owned by them as well. Intellectual property is not lost when it is given away. Furthermore, we have to be prepared and disposed to let mathematical evidences occur to us: we must live the life of mathematics if we are to see mathematical things.
I think that one of the most valuable moves in this book is Rota's identification of evidence and the Kantian synthetic a priori. He observes that "all understanding is synthetic a priori; there is not and there cannot be any other kind." When we achieve evidence, we see something we had not seen before (hence, synthetic), and yet we see that it is necessary, that it could not have been otherwise (hence, a priori). Rota's observation sheds light on Kant's theory of judgment, on Husserl's concept of evidence, and on the nature of human understanding.
Another theme developed by Rota is that of "Fundierung." He shows that throughout our experience we encounter things that exist only as founded upon other things: a checkmate is founded upon moving certain pieces of chess, which in turn are founded upon certain pieces of wood or plastic. An insult is founded upon certain words being spoken, an act of generosity is founded upon something's being handed over. In perception, for example, the evidence that occurs to us goes beyond the physical impact on our sensory organs even though it is founded upon it; what we see is far more than meets the eye. Rota gives striking examples to bring out this relationship of founding, which he takes as a logical relationship, containing all the force of logical necessity. His point is strongly antireductionist. Reductionism is the inclination to see as "real" only the foundation, the substrate of things (the piece of wood in chess, the physical exchange in a social phenomenon, and especially the brain as founding the mind) and to deny the true existence of that which is founded. Rota's arguments against reductionism, along with his colorful examples, are a marvelous philosophical therapy for the debilitating illness of reductionism that so pervades our culture and our educational systems, leading us to deny things we all know to be true, such as the reality of choice, of intelligence, of emotive insight, and spiritual understanding. He shows that ontological reductionism and the prejudice for axiomatic systems are both escapes from reality, attempts to substitute something automatic, manageable, and packaged, something coercive, in place of the human situation, which we all acknowledge by the way we live, even as we deny it in our theories.
Rota calls for a widened mathematics that will incorporate such phenomena as evidence and "Fundierung," as well as anticipation, identification, concealment, surprise, and other forms of presentation that operate in our experience and thinking but have not been given an appropriate logical symbolism and articulation. Such phenomena have either not been recognized at all, or they have been relegated to the merely psychological. What has been formulated in logic and mathematics so far have been grammatical operators. It is an exciting and stimulating suggestion to say that various forms of presentation might also be formalized. Rota makes his proposal for a new mathematics in his treatment of artificial intelligence and computer science. These fields, which try to work with intelligent operations wider than those of standard formal logic, have shown, by their failures as well as by their partial successes, that a much richer and more flexible notion of logic is called for. The logic Rota anticipates will not displace the rational animal, the dative of manifestation, but it will bring the power of formalization and mathematics to areas scarcely recognized until now.
Rota's fascinating and sympathetic sketches of persons and places in twentieth-century mathematics should also be seen as part of his study of the correlation between mathematical truth and mathematicians. He sheds light on mathematics by showing the human setting in which it arises. His exhortations to mathematicians to become involved in the service of other disciplines is another point in his recognition of the human face of mathematics. He calls for a presentation of mathematics that uses intuitive, illuminating examples, and for texts with "a discursive, example-rich flow," as opposed to the rigid style that turns the reader into a "code-cracker." The imaginative example is essential to the achievement of mathematical evidence.
Rota makes use of other authors, but never as a mere commentator. He uses authors the way they would most want to be used, as vehicles for getting to the issues themselves. He is like a musician who listens to Mozart and then writes his own music himself. His instinct for mathematical evidence has made him especially alert to philosophical truth. Mathematics and philosophy were blended, after all, in some of the very first philosophers, the Pythagoreans. Their thinking, along with that of all the presocratics, was given a human twist by Socrates, who turned from nature to the human things. Gian-Carlo Rota makes an analogous turn, complementing objective mathematics by showing how it is a human achievement, an intelligent action accomplished by men. His writings have much of Socrates' irony and wit, and the occasional barb is also socratic, meant to illuminate and to sting the reader into looking at things afresh. In these essays, mathematics is restored to its context in being and in human life.
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