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Oct 5 '21 at 21:22 comment added cmaster - reinstate monica @JKusin Then you not only have to throw out things like numbers, turing machines and truth itself, but also things like friendship, ownership, stocks, good/evil, power and freedom. All of these only exist because people agree that they exist, yet they shape our very lifes. Likewise, you can't build a car without crunching some numbers, and turing machines are required to answer questions like the halting problem. I understand that it may be difficult to accept the existence of abstract things, but the consequences of their existence are just as profound as the chair I sit upon.
Oct 5 '21 at 19:58 comment added J Kusin @cmaster-reinstatemonica I guess I'm not ready to accept purely abstract objects exist in an ontology.
Oct 5 '21 at 18:47 comment added cmaster - reinstate monica @JKusin Well, the turing machine (which is used to define the notion what computation is in mathematics) is just as abstract an entity as the natural numbers. It is fully specified by the finite state machine that controls its operation. And the state of the turing machine is fully described by the infinite sequence of symbols on its memory, as well as its current control status. All of these are defined by their behavior, only, so there is no need for a physical machine in order for the (infinite) turing machine to exist. The turing machine is mathematical, not physical.
Oct 5 '21 at 16:57 comment added J Kusin @cmaster-reinstatemonica I probably admit within the realm of mathematics n+1 is always defined and there is no max n. But the halting problem speaks of computation, runs, and forever. And it is linked to the incompleteness theorems. These words say something about the world, not just math in some sense. That crossover is hard for me to understand.
Oct 5 '21 at 15:04 comment added Mr.Mindor @JKusin would you be more comfortable with this phrasing?: "the halting problem is... whether the program will finish running." Here 'run forever' just means the opposite of 'finish running'. It is not a promise that the universe is unending. If the universe happens to end, and the program is still running, when the universe happens to end, the program still did not finish, it was interrupted.
Oct 5 '21 at 9:25 comment added cmaster - reinstate monica @JKusin Let's take an analog to "run forever": When I say that there are infinitely many natural numbers, I mean that whenever I have any natural number n, I can always find a greater number n+1. This number is just as real as the number n. It doesn't matter how big n is, or whether there is enough paper in the universe to write it down. Similarly with computation: Given the state of a machine that has computed n steps of an algorithm, the state of the machine after n+1 steps is well defined, and it does not matter whether there's enough time in the universe to reach step n+1.
Oct 5 '21 at 4:02 history edited Just Some Old Man CC BY-SA 4.0
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Oct 5 '21 at 3:15 comment added Just Some Old Man That does not mean the problem depends on time or includes time as a parameter. The “run forever” part is a physical implication resulting from the fact that something cannot be determined, thus cannot be determined no matter how much time you are allotted. It’s an unnecessary cane for the intuition to lean on for a lay-person’s understanding.
Oct 5 '21 at 2:40 comment added J Kusin Do you have any advice about how to regard (wiki): "the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to "run forever". What trick/interpretation do you use to make sense of "run forever"? Just ignore it and call it poor phrasing? I'm fine with doing that if that's the consensus. Runs forever to me means something about the world. Ignoring it makes me gravitate toward fictionalism I must say.
Oct 5 '21 at 2:10 history answered Just Some Old Man CC BY-SA 4.0