Is God a mathematician? (Or is he a computer programmer or a software engineer?) Is mathematics invented or discovered? Why is mathematics so useful in physics? And why isn't it at all useful in biology? These are stimulating fundamental questions, and Mario Livio obviously had fun collecting the material for this book. Livio gives no definite answers, but the the book is entertaining and uplifting, with many beautiful quotations from distinguished thinkers and many photographs of the title pages of books that are inspiring masterpieces of the scientific literature of previous centuries.
Clearly, as Bertrand Russell says at the end of the book, there are no definite answers; everyone must try to answer these questions to their own satisfaction. Here are some of the answers I've come up with. And I'll now give some of my own favorite quotations, not ones that are in Livio.
On why mathematics works so well in physics, I have two answers. First of all, math and physics co-evolved. Secondly, if you believe in the multiverse, there would not be any life in this universe marveling at the mathematical nature of reality if our universe did not have enough structure for life to evolve within it. Chaotic universes have no observers.
On whether mathematics is invented or discovered, I have an ambiguous position. I believe in the Platonic world of ideas, so you might guess that I will answer "discovered." However, our knowledge of this perfect, unchanging, eternal Platonic world is far from perfect, unchanging and eternal. I believe that our theories are necessarily invented, that we do not have direct perception of the Platonic world of ideas. And here are quotations from Bertrand Russell, Kurt Gödel and Albert Einstein which also suggest that the answer to whether mathematics is invented or discovered is "invented:"
My object in this paper is to explain in what sense a comparatively obscure and difficult proposition may be said to be a premise for a comparatively obvious proposition, to consider how premises in this sense may be discovered, and to emphasize the close analogy between the methods of pure mathematics and the methods of the sciences of observation. [Russell (1907)]
The analogy between mathematics and a natural science is enlarged upon by Russell also in another respect (in one of his earlier writings). He compares the axioms of logic and mathematics with the laws of nature and logical evidence with sense perception, so that the axioms need not necessarily be evident in themselves, but rather their justification lies (exactly as in physics) in the fact that they make it possible for these "sense perceptions" to be deduced; which of course would not exclude that they also have a kind of intrinsic plausibility similar to that in physics. I think that (provided "evidence" is understood in a sufficiently strict sense) this view has been largely justified by subsequent developments, and it is to be expected that it will be still more so in the future. It has turned out that (under the assumption that modern mathematics is consistent) the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic, i.e., the domain of the kind of elementary indisputable evidence that may be most fittingly compared with sense perception. Furthermore it seems likely that for deciding certain questions of abstract set theory and even for certain related questions of the theory of real numbers new axioms based on some hitherto unknown idea will be necessary. Perhaps also the apparently unsurmountable difficulties which some other mathematical problems have been presenting for many years are due to the fact that the necessary axioms have not yet been found. Of course, under these circumstances mathematics may lose a good deal of its "absolute certainty;" but, under the influence of the modern criticism of the foundations, this has already happened to a large extent. [Gödel (1944)]
[T]he concepts which arise in our thought and in our linguistic expressions are all—when viewed logically—the free creations of thought which can not inductively be gained from sense-experiences. ... Thus, for example, the series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences. But there is no way in which this concept could be made to grow, as it were, directly out of sense experiences. [Einstein (1944)]
Kant, thoroughly convinced of the indispensability of certain concepts, took them—just as they are selected—to be the necessary premises of any kind of thinking and distinguished them from concepts of empirical origin. I am convinced, however, that this distinction is erroneous or, at any rate, that it does not do justice to the problem in a natural way. All concepts, even those closest to experience, are from the point of view of logic freely chosen posits... [Einstein (1949)]
These are all quotations that I believe support what Imre Lakatos termed a quasi-empirical view of mathematics, which is the idea that although mathematics and physics are different, perhaps they are not as different as most people think. In particular, mathematical quasi-empiricism denies the primacy of the axiomatic method, and maintains that although the axiomatic method may be of expository value, mathematicians actually discover new concepts and principles by trying to unify and organize their mathematical experiences, much as empirical scientists do. For more on this see any of my books, or Tymoczko (1998) and Borwein et al. (2003, 2004, 2007).
Enough of my own opinions. Returning to Livio, what his book shows well, is that even though metaphysics is currently out of fashion in academic philosophy, it is alive and well in the scientific community. Just look at Leonard Susskind's and Max Tegmark's idea that the physical laws of this universe are not particularly interesting, because they are just, as it were, our "street address" in the multiverse/landscape of all possible physical universes with all possible laws of nature. You cannot get more metaphysical than that, but Tegmark is a cosmologist and astrophysicist, not a professor of philosophy, and Susskind is a string theorist.
Mario Livio is to be congratulated for keeping such high level intellectual ambitions alive in spite of a hostile contemporary zeitgeist, and for packaging his book in such a fashion that it is acceptable to commercial publishers while still being of interest to philosophers and theologians.
Gregory Chaitin
IBM T. J. Watson Research Center
http://www.umcs.maine.edu/~chaitin
Kurt Gödel (1944), Russell's Mathematical Logic, in Paul Arthur Schilpp, The Philosophy of Bertrand Russell, La Salle, IL: Open Court, 125-153.
Albert Einstein (1944), Remarks on Bertrand Russell's Theory of Knowledge, in Paul Arthur Schilpp, The Philosophy of Bertrand Russell, La Salle, IL: Open Court, 279-291.
Albert Einstein (1949), Autobiographical Notes, (Edited and translated by Paul Arthur Schilpp), La Salle, IL: Open Court, 1979.
Thomas Tymoczko (1998), New Directions in the Philosophy of Mathematics, (Revised and expanded paperback edition), Princeton, NJ: Princeton University Press.
Jonathan Borwein et al. (2003, 2004, 2007), Mathematics by Experiment, Experimentation in Mathematics, Experimental Mathematics in Action, Natick, MA: A K Peters.
April 2009