What is mathematics?

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Recently, I read in a certain forum the idea that mathematics should be identified with ZFS. Another idea discussed in the same forum is the idea that mathematics is a closed "thing". IMHO, both ideas are problematic. First of all, ZFS is one formalization of set theory, yet it is not sure whether sets are the most fundamental entity. Indeed, Mac Lane put forth the idea that categories are more fundamental than sets. Thus, it is not sure whether there is an "atom", in the sense of Democritus's atomic theory, in mathematics. The closeness of mathematics is one more absurd idea that seems to stem from finitism, which, in turn, has its root to intuitionism. Although, intuitionism was instrumental in development of certain ideas, especially in computer science, still I believe its ideas are not generally applicable. As a matter of fact, I am convinced that hypercomputation and intuitionism refute each other. Back to the closeness of mathematics. I had always the idea that, for example, complex numbers are a "proof" that mathematics is expandable. Are there any limits? No, of course not because if there are linits, then, in the end, mathematics is closed!

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A comment on a comment

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Recently, "someone" posted a review of my book on hypercomputation on the Brains site. The reviewer argues that This area is full of problems, some of which are physico/mathematical and some of which are conceptual. Unfortunately, from a quick sample, Syropolous's book does not avoid common mistakes and confusions—some of which I've been trying to correct in my own work. First of all, I have to admit that back in 2004 I read a paper by this reviewer, but at that time I did not considered it interesting. However, after I read this review of my book, I read The Physical Church-Turing Thesis: Modest or Bold? , the reviewer's latest manuscript, in order to see what he meant by common mistakes and confusions and how he was trying to solve them. In this manuscript, the reviewer puts forth a number of criteria that every machine has to satisfy in order to be ``useful for physical computing.'' These criteria are: Readable Inputs and Outputs Process-Independ
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Is the Universe a Quantum Computer?

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 The idea that the Universe (whatever we may mean by this) is a some sort of a computer is a recurring theme. Recently, Nature published an article by David L. Chandler that is entitled Could the Universe be a giant quantum computer? This article is actually a tribute to Edward Fredkin, who passed away last June. The main contribution of Fredkin to the philosophy of science is his "digital philosophy" that advocates the idea that the universe is a computer and everything in our universe is discrete. I would not like to argue against the validity of these ideas (I have done so in Hypercomputation: Computing Beyond the Church-Turing Barrier and in Demystifying Computation: A Hands-On Introduction ) . However, I would like to point out that if the Universe is not a computing device, in general, then it cannot be a quantum computer, in particular. Also, what I did not like with Chandler's article is that it takes it for granted that the Universe is a computer. He did not e
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Turing Machine Simulation

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 Hypercomputation is about machines that transcend the computational power of the Turing machine. However, I think it is quite instructive to really understand the power of the Turing machine. Therefore, I think play with a Turing machine simulator is quite instructive. The Turing machine simulator by Martín Ugarte is a very interesting and powerful simulator that I recommend to everyone who wants to play with Turing machines.
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