The Mathematical Experience by Philip J. Davis
My rating: 5 of 5 stars

Coming to Conclusions

The Mathematical Experience is just about as close to a phenomenology of mathematics as we’re ever likely to have. It is not about theories or principles, or arcane arguments and proofs but about what the authors do, and what others have recounted before them did. It is not even an explanation since many of the ‘greats’ and great events in mathematics are themselves unexplainable. The contents may constitute a kind of philosophy but if so it is a philosophy of educated tolerance and respect rather than normative or dogmatic.

For me the real value of the book is its insights into what ‘mathematical method’ (and by extension all of scientific method) is about. There are many who insist that method is some attribute of rational thought which produces objective, robust, and incontrovertible facts about the world. That this is what any science does is at least questionable. But that these results are the consequence of following some fixed procedure is categorically wrong. There is no such thing as scientific method. Or more precisely, the criteria of what constitutes good science change, sometimes slowly sometimes almost overnight. Method is something learned right along with the results produced by method. Method cannot be distinguished from a more general culture:

“What was in Archimedes' head was different from what was in Newton's head and this, in turn, differed from what was in Gauss's head. It is not just a matter of ‘more,’ that Gauss knew more mathematics than Newton who, in turn, knew more than Archimedes. It is also a matter of ‘different.’ The current state of knowledge is woven into a network of different motivations and aspirations, different interpretations and potentialities.”

Yes, for the edification of all the fundamentalists in the world (and there are many more of these than the religious sort) scientific knowledge is relative. Even in mathematics, the queen of the sciences, truth is a variable feast. You may think that the internal angles of a triangle have always summed to 180 degrees no matter where you are in the universe. But of course such a conclusion is warranted only is a universe with certain defined characteristics. It turns out that the universe that Euclid described isn’t definitive at all as demonstrated by the Bernard Riemann* and other 19th century mathematicians who simply dropped Euclid’s presumption that parallel lines never intersect. Using different presumptions triangles can have more than 180 degrees internally.

With this in mind, it’s important to note that Euclid’s geometry is not a less accurate or less general description of the world than Riemann’s. Neither geometry is a description of anything other than itself. And each is an entirely different and incompatible world. The fact that one may be transformed into the other does not mitigate their differences. Just that as in physics, Newton’s theory of gravitational forces is not an approximation of Einstein’s space-time, so the ‘principles’ of different geometries are not ‘generalisations’ but very different mathematical life forms with the equivalent of distinct atomic structures. They are relative because they start with different theoretical premises. They are relative by definition. Only when they are used practically does a criterion emerge to judge one against the other.

That criterion is really what is meant by scientific method. When one is conducting research, when results are reviewed with one’s colleagues, when a paper is being prepared for submission to a journal, and when the referees decide it is or is not worthy of publication, the criterion of what constitutes good science prevails. If anything it is this process which has the historic right to be called scientific method. It includes the acculturation of individuals into an instinctive mode of thought which varies by discipline. It also establishes a sort of intellectual hierarchy in which senior members have the final say about the work of more junior members. As in any other such hierarchy, it’s control over recognition, advancement and professional status is almost absolute.

But here’s the paradox: no one knows what that criterion is. Or rather more precisely, there is no agreement among practitioners about what the criterion should be. As the authors note about their own work: “We find that our judgment of what is valuable in mathematics is based on our notion of the nature and purpose of mathematics itself.” And there are widely different views about both the philosophy and the pragmatic usefulness of mathematical thought. So mathematicians, like all scientists, adopt a live and let live attitude. The consequence is that the criteria of what constitutes good mathematics is left purposely vague.

Even the criterion of what constitutes mathematics tout court is indeterminate except as what is produced by those who are accepted as part of the mathematics profession. “The definition of mathematics changes. Each generation and each thoughtful mathematician within a generation formulates a definition according to his lights,” the authors say. This too is part of scientific method, the continuous reconsideration of what the term ‘science,’ that is to say, ‘reason’ actually means. Thus, “In the final analysis, there can be no formalization of what is right and how we know it right, what is accepted, and what the mechanism for acceptance is.”

What must be called the essential ‘tolerance’ of science (despite its hierarchical structure) is captured well in the Preface: “Mathematics, like theology and all free creations of the Mind, obeys the inexorable laws of the imaginary.“ Reason, mathematical or otherwise, guarantees nothing. At best Reason is a continuing conversation among a changing group of people. The conversation is always flawed but there is hope in its continuation:

“There is work, then, which is wrong, is acknowledged to be wrong and which, at some later date may be set to rights. There is work which is dismissed without examination. There is work which is so obscure that it is difficult to and is perforce ignored. Some of it may emerge later. There is work which may be of great importance such as Cantor's set theory-which is heterodox, and as a result, is ignored or boycotted. There is also work, perhaps the bulk of the mathematical output, which is admittedly correct, but which in the long run is ignored, for lack of or because the main streams of mathematics did not choose to pass that way.”

The American mathematician, C.S. Peirce is cited by the authors as defining "mathematics as the science of making necessary conclusions.“ this seems to me an apt summary of the book, keeping in mind that Peirce’s understanding of necessity was always relative to some changing end or purpose.


* Reimann also provided the mathematical foundation for the integral calculus developed by Leibniz and Newton, which had had no proof of its validity since the 17th century except its usefulness. So much for mathematical rigour. On a related topic, remember the very first thing you learned in Geometry, that the circumference of a circle is equal to its diameter multiplied by π? A cast iron Mathematical Truth, right? But only in Euclid’s world. No one in this world has ever, nor could ever, verify this truth empirically. Π is a transcendental number whose mathematical existence once again was only proven in the 19th century (by Joseph Liouville) although it had been in use for over two millennia. Nevertheless, π does not exist in our world at all. By definition, no measurement of π will ever be entirely accurate. No matter what level of precision we can achieve, there are an infinite number of additional levels that will foil our attempts to know what it really is. Something like God perhaps?.

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