"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.
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...................
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...................
(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
matches the pattern
does not match the pattern
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.