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Equality is actually quite easy to axiomatise in most logics, here in my favourite logic: 1) x = x 2) x = y => P[x] => P[y] In my opinion, there is a mathema

Equality is actually quite easy to axiomatise in most logics, here in my favourite logic: 1) x = x 2) x = y => P[x] => P[y] In my opinion, there is a mathematical reality, which is shared by everyone, even by those who don't believe in it :-) For example, a logical system exists in that reality, and you can either derive a theorem in that system or not in this reality. I don't think it is possible that there is a third possibility. I don't think it is possible that I have a different reality from you in that respect. This reality is not socially constructed, it just is. Intuitionism will tell you that because you don't know if a certain theorem is derivable, it is in some sort of hybrid state until we know for sure via a intuitionistic proof or a counter example. I think that is bullocks. Either there is a proof or not. Either there is a counter example, or not. Beyond that, extending this mathematical reality, there is a wider, not as easily accessible reality. We can try to understand that reality by modelling it via certain assumptions, and then applying our mathematical reality to those assumptions. I believe the mathematical conclusions we draw from this will be real to the extent that the assumptions are true; but of course you cannot ever be sure about those assumptions, and so you cannot be sure about the conclusions. But if you notice that your conclusions do not hold, you need to challenge your assumptions, not your mathematics.

prmph 4y ago
Can you put into words the symbolic notation you have in your comment? I think I understand pretty well what you mean, but for the avoidance of doubt, explain what the notation means, and then I will indicate all the assumptions on which it is relying.
- auggierose 4y ago
  It would be somewhat lengthy to explain its meaning exactly here. You can read about its exact meaning and its context here: https://doi.org/10.47757/pal.2 https://doi.org/10.47757/pal.2 In short what it usually means (the exact meaning depends on the model under consideration) is that there is a binary operation "=", such that "x = x" is a theorem, that is evaluating "x = x" will evaluate to "true" for any object x in the mathematical universe. Furthermore, for any unary proper operator "P", and any two objects x and y of the mathematical universe, the expression "(x = y) => (P[x] => P[y])" will also evaluate to "true". Here "=>" is another binary operation called implication, which has some special properties outlined in the link. P[x] denotes the application of the operator P to the object x. Edit: Oh, forgot to add the third axiom for equality (it is actually more an axiom about "true", but uses equality): 3) A => (A = true) What this means is that for any object A of the mathematical universe, if you evaluate "A => (A = true)", you obtain the value "true".

prmph 4y ago

Can you put into words the symbolic notation you have in your comment? I think I understand pretty well what you mean, but for the avoidance of doubt, explain what the notation means, and then I will indicate all the assumptions on which it is relying.

auggierose 4y ago
It would be somewhat lengthy to explain its meaning exactly here. You can read about its exact meaning and its context here: https://doi.org/10.47757/pal.2 https://doi.org/10.47757/pal.2 In short what it usually means (the exact meaning depends on the model under consideration) is that there is a binary operation "=", such that "x = x" is a theorem, that is evaluating "x = x" will evaluate to "true" for any object x in the mathematical universe. Furthermore, for any unary proper operator "P", and any two objects x and y of the mathematical universe, the expression "(x = y) => (P[x] => P[y])" will also evaluate to "true". Here "=>" is another binary operation called implication, which has some special properties outlined in the link. P[x] denotes the application of the operator P to the object x. Edit: Oh, forgot to add the third axiom for equality (it is actually more an axiom about "true", but uses equality): 3) A => (A = true) What this means is that for any object A of the mathematical universe, if you evaluate "A => (A = true)", you obtain the value "true".