[Cohen:] ... Even if the formalist position is adopted, in actual thinking about mathematics one can have no intuition unless one assumes that models exist and that the structures are real.[Kanamori:] Cohen then returned to the bedrock of number theory and gave as an example the twin primes conjecture as beyond the reach of proof. “Is it not very likely that, simply as a random set of numbers, the primes do satisfy the hypothesis, but there is no logical law that implies this?” [But weak twin primes has been proved!]
So, let me say that I will ascribe to Skolem a view, not explicitly stated by him, that there is a reality to mathematics, but axioms cannot describe it. Indeed one goes further and says that there is no reason to think that any axiom system can adequately describe it.
Cohen and Set Theory (Kanamori)
Skolem and pessimism about proof in mathematics (Cohen)
The discovery of forcing (Cohen)
Cohen on discovering Godel's Incompleteness Theorem as a graduate student. (From My interaction with Kurt Godel, reprinted in Cohen's Set Theory and the Continuum Hypothesis.)
... I still had a feeling of skepticism about Godel's work, but skepticism mixed with awe and admiration.What is my attitude toward foundational work in mathematics, logic and set theory? the nature of proof and rigor? See this earlier post on the relation between physics and mathematics (GC = Gregory Chaitin).
I can say my feeling was roughly this: How can someone thinking about logic in almost philosophical terms discover a result that had implications for Diophantine equations? ... I closed the book and tried to rediscover the proof, which I still feel is the best way to understand things. I totally capitulated. The Incompleteness Theorem was true, and Godel was far superior to me in understanding the nature of mathematics.
Although the proof was basically simple, when stripped to its essentials I felt that its discoverer was above me and other mere mortals in his ability to understand what mathematics -- and even human thought, for that matter -- really was. From that moment on, my regard for Godel was so high that I almost felt it would be beyond my wildest dreams to meet him and discover for myself how he thought about mathematics and the fount from which his deep intuition flowed. I could imagine myself as a clever mathematician solving difficult problems, but how could I emulate a result of the magnitude of the Incompleteness Theorem? There it stood, in splendid isolation and majesty, not allowing any kind of completion or addition because it answered the basic questions with such finality.
... Let's recall David Deutsch's 1982 statement:
The reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic, cannot be found in mathematics or logic. The reason is that the laws of physics "happen" to permit the existence of physical models for the operations of arithmetic such as addition, subtraction and multiplication.
Does this apply to mathematics too? ...
GC: But mathematicians shouldn't think they can replace physicists: There's a beautiful little 1943 book on Experiment and Theory in Physics by Max Born where he decries the view that mathematics can enable us to discover how the world works by pure thought, without substantial input from experiment.
CC: What about set theory? Does this have anything to do with physics?
GC: I think so. I think it's reasonable to demand that set theory has to apply to our universe. In my opinion it's a fantasy to talk about infinities or Cantorian cardinals that are larger than what you have in your physical universe. And what's our universe actually like?
a finite universe?
discrete but infinite universe (ℵ0)?
universe with continuity and real numbers (ℵ1)?
universe with higher-order cardinals (≥ ℵ2)?
Does it really make sense to postulate higher-order infinities than you have in your physical universe? Does it make sense to believe in real numbers if our world is actually discrete? Does it make sense to believe in the set {0, 1, 2, ...} of all natural numbers if our world is really finite?
CC: Of course, we may never know if our universe is finite or not. And we may never know if at the bottom level the physical universe is discrete or continuous...
GC: Amazingly enough, Cris, there is some evidence that the world may be discrete, and even, in a way, two-dimensional. There's something called the holographic principle, and something else called the Bekenstein bound. These ideas come from trying to understand black holes using thermodynamics. The tentative conclusion is that any physical system only contains a finite number of bits of information, which in fact grows as the surface area of the physical system, not as the volume of the system as you might expect, whence the term ``holographic.'' ...
CC: We seem to have concluded that mathematics depends on physics, haven't we? But mathematics is the main tool to understand physics. Don't we have some kind of circularity?
GC: Yeah, that sounds very bad! But if math is actually, as Imre Lakatos termed it, quasi-empirical, then that's exactly what you'd expect. And as you know Cris, for years I've been arguing that information-theoretic incompleteness results inevitably push us in the direction of a quasi-empirical view of math, one in which math and physics are different, but maybe not as different as most people think. As Vladimir Arnold provocatively puts it, math and physics are the same, except that in math the experiments are a lot cheaper!
CC: In a sense the relationship between mathematics and physics looks similar to the relationship between meta-mathematics and mathematics. The incompleteness theorem puts a limit on what we can do in axiomatic mathematics, but its proof is built using a substantial amount of mathematics!
GC: What do you mean, Cris?
CC: Because mathematics is incomplete, but incompleteness is proved within mathematics, meta-mathematics is itself incomplete, so we have a kind of unending uncertainty in mathematics. This seems to be replicated in physics as well: Our understanding of physics comes through mathematics, but mathematics is as certain (or uncertain) as physics, because it depends on the physical laws of the universe where mathematics is done, so again we seem to have unending uncertainty. Furthermore, because physics is uncertain, you can derive a new form of uncertainty principle for mathematics itself...
GC: Well, I don't believe in absolute truth, in total certainty. Maybe it exists in the Platonic world of ideas, or in the mind of God---I guess that's why I became a mathematician---but I don't think it exists down here on Earth where we are. Ultimately, I think that that's what incompleteness forces us to do, to accept a spectrum, a continuum, of possible truth values, not just black and white absolute truth.
In other words, I think incompleteness means that we have to also accept heuristic proofs, the kinds of proofs that George Pólya liked, arguments that are rather convincing even if they are not totally rigorous, the kinds of proofs that physicists like. Jonathan Borwein and David Bailey talk a lot about the advantages of that kind of approach in their two-volume work on experimental mathematics. Sometimes the evidence is pretty convincing even if it's not a conventional proof. For example, if two real numbers calculated for thousands of digits look exactly alike...
CC: It's true, Greg, that even now, a century after Gödel's birth, incompleteness remains controversial. I just discovered two recent essays by important mathematicians, Paul Cohen and Jack Schwartz.* Have you seen these essays?
*P. J. Cohen, ``Skolem and pessimism about proof in mathematics,'' Phil. Trans. R. Soc. A (2005) 363, 2407-2418; J. T. Schwartz, ``Do the integers exist? The unknowability of arithmetic consistency,'' Comm. Pure & Appl. Math. (2005) LVIII, 1280-1286.
GC: No.
CC: Listen to what Cohen has to say:
``I believe that the vast majority of statements about the integers are totally and permanently beyond proof in any reasonable system.''
And according to Schwartz,
``truly comprehensive search for an inconsistency in any set of axioms is impossible.''
GC: Well, my current model of mathematics is that it's a living organism that develops and evolves, forever. That's a long way from the traditional Platonic view that mathematical truth is perfect, static and eternal.
CC: What about Einstein's famous statement that
``Insofar as mathematical theorems refer to reality, they are not sure, and insofar as they are sure, they do not refer to reality.''
Still valid?
GC: Or, slightly misquoting Pablo Picasso, theories are lies that help us to see the truth!
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