Please try to understand that because computer science was known first it doesn’t necessarily mean it is the most fundamental natural science, it surely means it is more intuitive than physics(physics is more abstract) hence greeks learn algorithms before physics. Understanding the basics of nature would be very painful without the knowledge of physics. An algorithm can be for anything like baking a cake which is not science, by mathematical induction your statement is false.

Yes, if a physically realizable machine could solve the halting problem, this would disprove the strong Church-Turing thesis. Of course, we would quickly replace it by something else, but it would have profound consequences, I believe.

You keep writing “Church thesis”, so maybe you are not clear on the fact that I address specifically the strong Church-Turing thesis which is as follows:

“The universe is equivalent to a Turing machine.”

However, this is a misuse of the term computable function, and probably arises from confusion about what the Church thesis says. In fact, these issues have been studied in theoretical computer science: one can look up Turing jumps or Arithmetic heirarchy for more.

Further, there are computionally complete systems of total maps. For example, a reflexive object in a Cartesian closed category.

The Church thesis is not a mathematical statement about computer science, but more an insight about the relationship between the system in which one is working, and what that system can express.

Furthermore, cannot it also be said that an understanding of patterns and algorithms themselves, as they possibly exist in our perceived reality, is an investigation of physics itself, that it necessitates a platform to actually perform its operations? And cannot it also be taken one step further back, i.e., questioning the very validity and ability of our own reason to accurately discern and defend the truth that we may or may not discover through an investigation of our physical reality? In this circumstance, does not philosophy, then, become an even more rudimentary discipline than physics or computer science?

The predictions cited and stated to be based off the C-T thesis rest on assumptions, and hence do not necessarily declare absolute, positive truths about the universe. John Searle’s Chinese room argument counters the idea that conscious understanding and original, free direction are shown in a computerized agent even of supreme sophistication. The argument can be reversed back on itself, however, i.e., on the human mind that claims it as original thought, and thus philosophy, once more, prevents epistemological certainty (even preventing the “prevention” of epistemological certainty, and so on, and so on, ad infinitum). And even if a computerized agent can duplicate any existing physical process, it necessitates a physical platform and physical terms first before it can execute its operations.

These are my thoughts, not necessarily my opinions. I look forward to someone rebutting or bolstering this commentary. To the original author: thank you for this post!

In addition, one can easily argue that “Logic is the most fundamental natural science” using your argument.

http://web.ncf.ca/andre/publications/methodology_of_design.pdf

Poincare quipped about the scientific method “man proposes, nature disposes”. I don’t think it’s very different with software, except that “nature” is replaced by “design specifications”.

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(1) Where does this come from? I mean, “How do you know this?”
(2) Why is it relevant? And who said that TMs must have a blank tape?

If the universe is infinite (even just in Turing Machine-ish sense), it isn’t a TM, since the tape won’t be blank anywhere. But if people are finite, then everything they are capable of can be performed with a TM. Church-Turing thesis should be about the nature of human ability, not the nature of reality (of course, the latter can only be meaningfully conceptualized through the lens of the former).

Yeah, they are utter garbage (not just wrong, but thoroughly and embarrassingly wrong with no redeeming features). I went to a talk of his once and spent some time arguing with him afterwards. Basically, he starts with Hilbert’s 10th problem, about the solvability of Diophantine equations. This is known not to be solvable by Turing machines. He then writes down a Hamiltonian encoding this problem, so that finding a solution of the original equation is equivalent to finding the ground state of his quantum mechanical system. This is easy. He argues that of course you can find the ground state using a quantum computer, which is therefore more powerful than a Turing machine.

This assertion is provably wrong in all previously studied models of quantum computers, but he claims these models are unnecessarily restrictive and that in reality a quantum computer could do this. He ends up spending a lot of time arguing for this position, but it’s all just silly. The fundamental computational issue here is that the solutions to the Diophantine equation may involve enormous numbers of digits (growing not just quickly but faster than any computable function). Even aside from the ludicrous difficulty of engineering a physical system that can handle this level of precision, you never know when you have searched the solution space thoroughly enough. If you could really cool your system to absolute zero, you would get the true ground state, but you can’t actually reach absolute zero, and you never know whether you’ve come close enough to say there’s definitely no lower-energy state.

So what it comes down to is this. If you can create and manipulate an exact Hamiltonian of a certain form (on an infinite-dimensional Hilbert space), and if you can find a guaranteed way to bring it into its ground state, then you can use this to do things no Turing machine can do. In my experience, no expert is surprised or impressed by this, since it’s an easy deduction from grossly unphysical hypotheses. However, Kieu has a lot invested in this idea and has spent the last 10+ years arguing for it. It’s kind of depressing.

@Gustavo The analysis of the performance of real-world implementations is tricky, but not always so tricky that nothing can be said about their asymptotic running time.

OTOH, it seems like we’ve accepted that the asymptotic behavior of PCs is modeled by Turing Machines… although we’ve seen in practice that this is an idealization, and PC performance is in fact slightly worse than that of Random-Access Machines.

Did you take a look at the paper I linked? I haven’t had a chance to read it fully yet, but if you’d like to discuss further, I’d be happy to take this over to e-mail. It’s a really interesting topic that I’d like to learn more about.