Abstract It is argued that to a greater or less extent, all mathematical knowledge is empirical.
Although I have never thought of myself as a philosopher, Harvey Friedman has told me that I am “an extreme Platonist”. Well, extremism in defense of truth may be no vice, but I do feel the need to defend myself from that description.
When one thinks of Platonism in mathematics, one naturally thinks of Godel. In a letter to Gotthard Gunther in 1954, he wrote:
When I say that one can .. .develop a theory of classes as objectively existing entities, I do indeed mean by that existence in the sense of ontological metaphysics, by which, however,
I do not want to say that abstract entities are present in nature. They seem rather to form a second plane of reality, which confronts us just as objectively and independently of our thinking as nature.1
If indeed that’s extreme Platonism, it’s not what I believe. I don’t find myself confronted by such a “second plane of reality”.
In his Gibbs lecture of 1951, Godel made it clear that he rejected any mechanistic account of mind, claiming (with no citations) that
1See , vol IV, pp. 502-505.
M. Davis (B)
M. Davis (B)
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
© Springer International Publishing Switzerland 2016 349
E.G. Omodeo and A. Policriti (eds.), Martin Davis on Computability,
Computational Logic, and Mathematical Foundations,
Outstanding Contributions to Logic 10, DOI 10.1007/978-3-319-41842-1_14
.. .some of the leading men in brain and nerve physiology .. .very decidedly deny the possibility of a purely mechanistic explanation of psychical and nervous processes.
In a 1974 letter evidently meant to help comfort Abraham Robinson who was dying of cancer, he was even more emphatic:
The assertion that our ego consists of protein molecules seems to me one of the most ridiculous ever made.
Alas, I’m stuck with precisely this ridiculous belief. Although I wouldn’t mind at all having the transcendental mind Godel suggests, I’m aware of no evidence that our mental activity is anything but the work of our physical brains.
In his Gibbs lecture Godel suggests another possibility:
If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. The fact is that in mathematics we still have the same attitude today that in former times one had toward all science, namely we try to derive everything by cogent proofs from the definitions (that is, in ontological terminology, from the essences of things). Perhaps this method, if it claims monopoly, is as wrong in mathematics as it was in physics.
I will claim that mathematicians have been using inductive methods, appropriately understood, all along. There is a simplistic view that induction simply means the acceptance of a general proposition on the basis of its having been verified in a large number of cases, so that for example we should regard the Riemann Hypothesis as having been established on the basis of the numerical evidence that has been obtained. But this is unacceptable: no matter how much computation has been carried out, it will have verified only an infinitesimal portion of the infinitude of the cases that need to be considered. But inductive methods (even those used in physics) need to be understood in a much more comprehensive sense.