I am interested in the effective calculability and solution of problems. You seem to be interested in sort of illusory and abstract calculability of them. As a consequence I fear we will never converge to an agreement.

You put forth an argument based on incompleteness and computability theory, neither of which deal with effective calculability and solution of problems. Regardless, this isn’t a question of our respective interests, it’s a question of whether your claims wrt computability are true or false.

Computability is a well-defined mathematical concept, so it’s not subject to opinion. At least one of us is simply wrong.

This doesn’t mean that our discussion has been unuseful and I thank you for your active participation.

I thank you too for your graciousness.

You seem to believe that any problem finitely defined on a finite number of objects can be resolved by mean of a finite series of instructions or operations (computation or algorithm). Really I don’t understand what your believe is based on, considered the huge range the concept of “problem” covers. This is even more unbelievable for me given you have rightly stated that “virtually all problems are non-computable” (by the way this perfectly agrees with the general thrust of my OP).

If you think that tomorrow’s winning lottery number is non-computable, you misunderstand what computability is all about.

How you can claim that tomorrow’s winning lottery number is obtainable by mean of a series of instructions is beyond me. TMs cannot know the future.

I am interested in the effective calculability and solution of problems. You seem to be interested in sort of illusory and abstract calculability of them. As a consequence I fear we will never converge to an agreement.

Anyway these kinds of situation are typical when an evolutionist (you) and an IDer (me) discuss: the former inclines to oversimplify and reduce things while the latter inclines to see the things from an engineering resolutive viewpoint.

This doesn’t mean that our discussion has been unuseful and I thank you for your active participation.

By mean of your method of the hardwired values there is no incomputable problem.

That’s incorrect. You can’t hardwire answers to the Halting problem because a TM is a finite automaton and the Halting problem has an infinite domain. And there are an uncountably infinite number of problems that are Turing equivalent to the Halting problem.

In contrast, the number of computable problems is countably infinite. Which means that virtually all problems are non-computable.

Also the problem to know the future outcomes of the lottery become computable.

Absolutely. If you think that tomorrow’s winning lottery number is non-computable, you misunderstand what computability is all about.

You seem to believe that all finite problems are computable. I provided a finite problem but you didn’t compute it.

But I explained why there is a TM that computes it.

By mean of your method of the hardwired values there is no incomputable problem. Also the problem to know the future outcomes of the lottery become computable. You provide us a TM with the hardwired answers. Your method is too good to be true.

Computable functions are only a sub-set of functions/problems. You seem to believe that all finite problems are computable. I provided a finite problem but you didn’t compute it.

The problem is indeed to know if a TM can compute the answers from data different from them and without having them hardwired inside itself.

Actually, no. The definition of computability does not disqualify TMs with hardwired answers. You can search any computing theory text and you will find no definition of computability that matches your understanding of the term. Nor will you find any examples of non-computability that are finite-domain functions.

The TM determines the answer by looking it up in the table, which is incorporated in the TM. The question of how the table got populated with the correct answers — i.e. how the TM was made — is irrelevant to the computability issue.

Sorry but disagree. It is a tautology to say that if I insert the answers in a TM then the TM outputs them. The problem is indeed to know if a TM can compute the answers from data different from them and without having them hardwired inside itself. This is relevant to the computability issue.

The problem is not there on that finite lookup table about which we agree, the problem is that any single row of it, i.e. the single attempt to know mechanically a single value of f (say y2=f(x2)) necessarily fails.

The TM determines the answer by looking it up in the table, which is incorporated in the TM. The question of how the table got populated with the correct answers — i.e. how the TM was made — is irrelevant to the computability issue. The only relevant question is whether some TM, out of the space of all possible TMs, implements the function.

Before a string on a computer screen we cannot a priori know by mean of a computation if it was written by a guy hitting on a keyboard or by a TM.

That’s like saying that a TM can’t say whether a given shirt came from Macy’s or JCPenney. But of course a TM can do this. A TM can contain any information whatsoever, as long as it’s finite. If the information contained in the history of the universe is finite, as Dembski argues, then a TM can “know” everything there is to know about the physical history of the universe.

Okay, so the argument is not the string itself, but rather information about a certain physical instantiation of the string. No problem. If the function is well-defined and the domain is finite, f(x) can be implemented with a lookup table. This is not a controversial statement. If we can’t agree on that, then we mean different things by the term “computability” and we have no foundation for a discussion.

I don’t think we are really on a different page about computability. Perhaps the misunderstanding is the following. Given for f(x) a domain {x1, x2, … xn} and the codomain {1, 0} the lookup table has n rows: y1=f(x1); y2=f(x2); … yn=f(xn).

The problem is not there on that finite lookup table about which we agree, the problem is that any single row of it, i.e. the single attempt to know mechanically a single value of f (say y2=f(x2)) necessarily fails. It is this failure that makes me say that f(x) is incomputable, not the lookup table per se.

Before a string on a computer screen we cannot a priori know by mean of a computation if it was written by a guy hitting on a keyboard or by a TM. It is this impossibility that my function describes.

My function f(x) needs as argument a single specific string written somewhere.

Okay, so the argument is not the string itself, but rather information about a certain physical instantiation of the string. No problem. If the function is well-defined and the domain is finite, f(x) can be implemented with a lookup table. This is not a controversial statement. If we can’t agree on that, then we mean different things by the term “computability” and we have no foundation for a discussion.

Does f(”Hello”) have a unique value?

My function f(x) needs as argument a single specific string written somewhere. Only in internet there are 382 millions “hello”. What of them do you specify? If you don’t specify the particular “hello” f(x) is not valid because x is not univocal. If x is univocal f(x) has a unique value.