The most important machine that was never built

I think your hang-up is one of infinity. Just because we must deal the reality of the finite does not mean our computers are not _capable_. For example, the Turing Machine in our minds is able to run infinitely, but the computer plugged in to my wall cannot because it is dependent on me paying the power bill.

Any Turing Machine we will have made into silicon is able to run any Turing Machine we can formulate (at this current moment in time) except only up to a finite point. But we can forever increase that finite point of measure! For example, our computers now may be able to calculate up to N. The next year, we may be able to do N+10000. The following year, N+10000000, and on and on. This is a limit of our reality, but not of the Turing Machine, which our machines must be because they are capable of being extended all the way to infinity, even if we take them there step by step, by adding 1 at a time.

Again, if this were a problem of infinity, then we would have limited the domain and solved the issue already. But we can design very simple, very finite Turing Machine programs for which the Halting Problem cannot be solved:

    while (true) {
        sleep(1);
    }

edit: you were right -- we are talking past each other. The "finite" in "finite state machine" is not useful here. Any infinite Turing Machine which we have made into silicon is still an infinite Turing Machine, just one we have inherently limited due to our realities of only having so much time or room or resources or whatever else. But a FSM is not one we limited artificially, it is a different class entirely and a class we can completely make into silicon without any sort of loss. Again, if this were a problem of "infinity" then we would have limited the domain by our own choice and solved it, which we have done for machines of a lower complexity, but _not_ for Turing Machines.

The Von Neumann architecture is just a Turing Machine, simply a physical manifestation of one, a physical abstraction. My computer has a “tape” which is 64GB wide because I have 64GB of RAM.

If we had infinite memory and infinite symbols (instead of only 2^64), we could build a universal Turing Machine. But that doesn’t mean our physical computers aren’t Turing Machines executing Turing-complete programs that just happen to be in the space of programs executable in a finite time and within the limitations of our current physical realities.

I’m not sure who told you that Turing never intended for the machine to be built. He knew that he could not build a machine with infinite tape, as we both know too. But when he built the machine to crack the Enigma Machine, he did this _after_ writing his paper, “On Computable Numbers.” So, he was in effect making a Turing Machine (but he called them a-machines).

We literally didn’t have computers before Turing formalized a way to execute them, instruction by instruction, and in detail. Once we had some kind of formal method to do this, physical machines could start being built.

Edit: My comments have not shown misunderstanding. There seems to be this consensus that our mathematics here came _after_ our machines. But this is wrong and the implications are tremendous. Our computers we have really are machines which execute other machines. What is the only class of machine that can do that? The Turing Machine.

Edit2: I suppose this is an argument of semantics. I didn’t realize. Yes, perhaps our computers are “Turing-equivalent”. I still think the “tape” and “symbol” and all of those things are merely mental variables, which is why I called these things a Form earlier. We built a physical tape, it just took the form of RAM and has much detail that electrical engineers can talk about. We also built symbols, we just called them instructions and placed them into a CPU. All of our physical devices have come from theoretical findings, not vice-versa.

https://en.m.wikipedia.org/wiki/The_Treachery_of_Images