Recent content by VazScep

My position, from the beginning, is that the question of whether mathematics is invented or discovered is naive. Yes, we cannot choose whether a theorem follows from axioms or accepted assumptions, and in this sense, we can discover whether a given proposition is a theorem. But to say this means...

I edited this into my earlier reply and you may have missed it: An early example of possible religious involvement in mathematical development comes from Eratosthenes. He said that the problem of doubling of the cube arose because an oracle had said that `to get rid of a plague they must...

No. Similarly, my reasons for being fascinated by the Incompleteness Theorems are not intrinsically mathematical. How much do the contingent value judgements and inherited societal beliefs of mathematicians affect the direction mathematical research takes? As another example, a number of 19th...

It doesn't follow, not trivially anyway, though a number of people have tried to develop philosophical arguments against Strong AI via the Incompleteness Theorems. They generally haven't convinced the logicians or the AI community, though.

It would make the people feel it is an important mathematical result. For a more concrete example, according to Eratosthenes, the problem of doubling the cube arose because an oracle had said that `to get rid of a plague they must construct an altar double of the existing one'. Right. There is a...

Both FLT and 2+3+5=10 are theorems of PA. Mathematicians do not simply discover the theorems of PA. They select interesting results from them. In this case, we would discard 2+3+5=10 because it is extremely trivial. In a society which associates spiritual connotations to these four numbers...

Because the way we react to certain results and the decision as to which mathematical problems we wish to pursue is going to affect what mathematics gets done. If you program a computer to churn out the theorems of Peano Arithemetic one by one, it is certainly not behaving anything like a...

I'm not suggesting Fermat invented his theorem. But why is the theorem that the first three primes sum to ten not listed as one of his theorems, or as the theorem of any other mathematician? Very few of the infinity of possible theorems in number theory are ever mentioned. What makes us single...

However, Cincinnatus only said that mathematical objects can be defined syntactically, not that number theoretic truth could be so defined. So the second-order Peano axioms, which are categorical (they have the natural numbers as their unique model up to isomorphism), could be said to...

Sorry, but I'm going to have another go at this, since I would like to be able to explain this stuff without making people more confused. A first-order theory (according to one definition, at least) is a set of wffs closed under logical consequence. Truth and falsehood are not defined in terms...

This is not what undecidable means. As I said earlier in the thread, truth and provability should not be confused. If a sentence is undecidable it means it is unprovable, but it will still be either true or false for a given interpretation. For instance, the undecidable Gödel sentences in...

Okay. In that case, the negation will define the complement of the first set, which will have two elements. I'm not sure what this is a concrete example of though.

Are you asking this to clarify my last post? What do you mean by a set defined by a formula, or a set defined by the negation of that formula?

Generally, a set is decidable if there is an algorithm to determine whether an object belongs to that set or not. A theory is a set of formulas, so we can say that a theory is decidable if there is an algorithm to determine whether a formula belongs to that theory or not. We can also say that a...

Just to correct something Hurkyl wrote earlier, truth is not defined with respect to any set of axioms or any theory. It is defined with respect to an interpretation of a formal language. For instance, suppose the given formal language consists of the predicate symbol p and the constant symbol...