"The central result of logicians in the 20th century was that, in the end, it will always be necessary to extend your axioms—things you just assume to be true without proving them—if you are to extend your idea of truth."
I'm in total agreement with this.
On a philosophical note, I even hold that it's actually possible to create a sound mathematical formalism that is based directly on this principle.
I personally hold that our current mathematical formalism is fundamentally "wrong". And this mainly has to do with Cantor's "empty set" theory.
IMHO, that was a very wrong turn for mathematics to take.
Henri Poincare, once stated, "Cantor's Set Theory is a disease from which the mathematical community will someday recover from".
I agree with Poincare. Cantor's Set Theory based on an empty set is a bad idea, and we should have never gone down that road. It's an unnecessary path to take. In fact, there are far better paths to take. We will not advance properly in our mathematics until we go back and correct this wrong turn.
The above is purely polemical assertion. No actual argument is given that Cantor's direction was "wrong". Moreover, to be clear, (1) The notion of an empty set (or empty class) does not originate with Cantor. (2) With merely a bit of labor, Cantor's developments can still be essentially expressed without resource to the notion of an empty set (all that is really required is that there is at least one object (not necessarily called a "set") that has no members). (3) Poincare's quote on Cantor needs to be taken in much greater context than in isolation. (4) To my knowledge, Poincare was not objecting to the notion of an empty set.
Interesting, I have seen the same idea on this forum somewhere, don't remember where exactly - doesn't matter... though, I just want to say that Cantor's Set Theory makes sense in biology and other sciences, so far (correct me if I am wrong...),
Speaking about the article, it's a really interesting point of view, but wrongly presented - the Artificial Intelligence works on well defined algorithms - and the problem seems to appear when there is an unexpected event that might occur; there is no algorithm to react for an unplanned factor, and it's normal - that's the definition of an unexpected thing, to be unexpected; I think not only by applying the Artificial Intelligence we have problems with that factors, we as humans very often have the same problems, if we are not appropriately prepared (algorithms again...)
Also, it was tried to draw a parallel between two truths, the "1+1 = " truth, and our understanding of a truth(an excerpt from the Declaration of Independence...), but we should keep in mind that the latter truth is quite vaguely defined(broadly, there are more than one criteria needed in order to define this truth) - so, if we would attempt to define it in clear terms, this truth could be interpreted by a machine, wouldn't it?
We should not overestimate the capabilities of a machine that does not think and just executes whatever we have instructed it to do...the rest is just blabla..
I was a programmer ............................
Computers do not think...we think...they execute....If.... Then...... instructions, a programmer writes in the programming code, is not considered as a decision...it is just another instruction that the machine can execute...
Computers cannot think the way we do nor they will do in the future...alll what they will do is to execute the code we instructed them to do....
If someone believes that in the future a computer can think and be able to prove a theorem of Geometry is an outopia...................
Usually, 'validity' is defined in terms of 'truth'. An argument is valid if and only if it is of a form that ensures that if the premises are true then the conclusion is true. In other words, validity is the notion of preserving truth through arguments. I don't know why one would consider it a mistake to include the notion of truth in logic.
Quite a defeatest and sad failure of imagination.
Done, infact this is rather trivial, finding a proof for a given conjecture is another matter completely.If someone believes that in the future a computer can think and be able to prove a theorem of Geometry is an outopia...................
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Of course, as we know that the argument is of a certain form. But we know that that form provides only valid arguments by having already shown that that form is truth preserving (indeed, as 'valid' is itself defined as 'truth preserving'). Again, I don't see what advantage there is in eschewing the notion of truth nor how one would explain the notion of validity without referring to the notion of truth.
That is basically the if-thenism philosophy of mathematics; and it seems to me to be fine to a large extent. However, it does not itself preclude that there is benefit in having a mathematical notion of truth. And when you say "we" you are not speaking for the great majority of mathematicians; I think that you will find that the great majority of mathematicians do have a strong notion that mathematics is a matter of truth. Now, you may wish to argue that mathematicians SHOULD eschew a notion of truth; but so far you have not given an argument for that. Instead, you have merely asserted it. Unless you provide an argument, it is merely your dogma. Moreover:
(1) There is the matter of the truth of "if a then b" itself.
(2) You skipped my point. I'll say it again: we know that certain argument forms provide only valid arguments by having already shown that these forms are truth preserving (indeed, as 'valid' is itself defined as 'truth preserving').
(3) There is a notion of mathematical truth that may be regarded as even more fundamental than 'if then' statements. I refer to finitistic artihmetic, which is associated with primitive truths regarding the matching of finite sequences of symbols. Such statements as
"The pattern
111000
matches the pattern
111000."
and
"The pattern
111000
does not match the pattern
101010."
are primitive truths that don't need to be drawn out as "if then" statements.
(4) And the if-then view of mathematics does not preclude a notion of mathematical truth. As I understand, usually, if-thenism regards mathematics as assertions that certain statements follow from certain axioms. However, the axioms themselves do not all have to be if-then statements, and we may still regard questions as to the truth of the axioms themselves, both those that are and those that are not if-then statements.
(5) The benefits of having a mathematical notion of truth include (1) that we have a fuller understanding of the subject and (2) that the subject of mathematical truth has shown to be a rich and interesting subject of mathematics itself. Again, merely asserting that a notion of truth should be eschewed is, unless given rational argument, merely dogma.
This is my view too. I know very little about coding myself but I do think that computers will never be capable of independent thought and are doomed to just following instructions depending on whether something is true or false.
What exactly has been "done"?