4 ms·

Yes, that's the story that working mathematicians use day to day, but glosses over the relevant question in this context - why do we believe in one set of axiom

by shasta 16y ago

Yes, that's the story that working mathematicians use day to day, but glosses over the relevant question in this context - why do we believe in one set of axioms instead of another? Do you think it's possible that we'll find a factorization of 17 tomorrow, and so will just need to refine our axioms?

Groxx 16y ago
If you redefine 17 to be something not defined by our current set of axioms, sure. Which you're free to do. As to our current set of axioms, they are a useful subset of possible axioms which appear to describe the real world with phenomenal accuracy, even being used by scientists to predict things far in advance of testable evidence (ie: quantum mechanics), and are internally consistent. They're used because they work, and because what we have built off is consistent with our uses. Where there are gaps, other more controversial, potentially less reliable, potentially inconsistent axioms are considered. And, thanks to Gödel, we know we will always have gaps. Factorization of an integer is not one of the gaps. That's an entity described within the bounds of the main set of axioms in use, and factorization is within those bounds as well. So no, I do not think it is possible that 17 as it is being used in this context will ever be factored into anything but 1 and 17.
- shasta 16y ago
  I'm not redefining anything. I'm asking whether you accept it to be possible that one day a flaw will be found in arithmetic, leading to a contradiction. An example might be proving that 17 is prime on the one hand, and then finding a factorization of it on the other. The fact that arithmetic is widely used throughout the world without contradiction does provide some evidence that it's consistent, but other famous theories (e.g. Newtonian physics) have been found to be not quite right. Maybe the flaws in arithmetic only show up when really huge numbers are involved. Stated another way, how do you know that arithmetic is "internally consistent"? Why do you judge it to not be among the "potentially inconsistent axioms" of stronger systems? If it's just due to our empirical success with numbers to date, then you should admit there to be some small chance that we've just not discovered the problems, to date. Or, you can prove arithmetic correct in some stronger system, but then how do you check the consistency of that system? At some point, you have to rely on your intuition about what the axioms are supposed to be about.

by shasta 16y ago

Groxx 16y ago

If you redefine 17 to be something not defined by our current set of axioms, sure. Which you're free to do. As to our current set of axioms, they are a useful subset of possible axioms which appear to describe the real world with phenomenal accuracy, even being used by scientists to predict things far in advance of testable evidence (ie: quantum mechanics), and are internally consistent. They're used because they work, and because what we have built off is consistent with our uses. Where there are gaps, other more controversial, potentially less reliable, potentially inconsistent axioms are considered. And, thanks to Gödel, we know we will always have gaps. Factorization of an integer is not one of the gaps. That's an entity described within the bounds of the main set of axioms in use, and factorization is within those bounds as well. So no, I do not think it is possible that 17 as it is being used in this context will ever be factored into anything but 1 and 17.

shasta 16y ago
I'm not redefining anything. I'm asking whether you accept it to be possible that one day a flaw will be found in arithmetic, leading to a contradiction. An example might be proving that 17 is prime on the one hand, and then finding a factorization of it on the other. The fact that arithmetic is widely used throughout the world without contradiction does provide some evidence that it's consistent, but other famous theories (e.g. Newtonian physics) have been found to be not quite right. Maybe the flaws in arithmetic only show up when really huge numbers are involved. Stated another way, how do you know that arithmetic is "internally consistent"? Why do you judge it to not be among the "potentially inconsistent axioms" of stronger systems? If it's just due to our empirical success with numbers to date, then you should admit there to be some small chance that we've just not discovered the problems, to date. Or, you can prove arithmetic correct in some stronger system, but then how do you check the consistency of that system? At some point, you have to rely on your intuition about what the axioms are supposed to be about.