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> The axioms don't have to be true. You can still talk about their implications in absolute terms. No, you can't. Two mathematicians using different rules of i

> The axioms don't have to be true. You can still talk about their implications in absolute terms. No, you can't. Two mathematicians using different rules of inference (say, those of classical vs. intuitionistic logic) will arrive at different theorems, even if they begin from the same axioms. > "Assuming a=0 implies a=0" is absolutely true. Assuming that “a = 0” and implication are both expressible in our formal system, it would indeed be very weird (though not a priori impossible) if “a = 0” didn't imply “a = 0”. But I have no problem imagining a formal system where “a = 0” isn't expressible (e.g., because equality is inexpressible), or implication isn't expressible (e.g., geometric logic), or implication has strange properties to someone only familiar with the classical and/or intuitionistic interpretation of “implication” (e.g., linear logic).

plutooo 11y ago
You can't have the same axioms with different rules of inference. The rules of inference are axioms. By the axioms being the same, you mean the ink-shapes making up the symbols on a piece of paper being the same. Not the actual meaning behind them. Edit: Since you added more content to this reply later on, let me respond. In this case I was talking about specifically a formal system where a=0 makes sense and is expressible, but to keep it short and concise I didn't choose to write out all the details. So the rebuttal is moot.
- catnaroek 11y ago
  Axioms and rules of inference are fundamentally different: (0) An axiom is an internal statement to a mathematical theory that is assumed to be true. That is, inside of a mathematical theory, you don't need to prove that its axioms hold. However, if you want to construct a model of a mathematical theory, you need to prove externally that the axioms hold. In return, you get the theory's theorems (suitably interpreted) for free. (1) A rule of inference exists in an external metatheory, where the original mathematical theory we were studying is treated as a syntactic object (also known as object language), in very much the same way a compiler treats the program being compiled as a (possibly annotated) syntax tree. A rule of inference defines a class of valid syntactic transformations, but doesn't concern itself with the impact of these transformations have on the meaning of the phases in the object language. It is perfectly sensible to consider the effect of changing the rules of inference, on an axiomatic system.
  - plutooo 11y ago
    If you have different rules, you have a different object and the meaning of the axiom is different. Even if it is written using the same symbols. To continue the compiler analogy.. Just because the ASCII sequence "int c=0;" means different things in C and Java, doesn't imply "int c=0;" is meaningless when specifically talking about only C.
    - catnaroek 11y ago
      The problem with your analogy is that C and Java have completely different abstract syntaxes. A better analogy would be taking the syntax of an existing programming language, and completely changing its meaning. For instance, consider the effect of making Racket use lazy evaluation (Lazy Racket), or making Haskell use strict evaluation (the upcoming -XStrict pragma). Strict and lazy languages validate different sets of equational laws (and hence compiler optimizations!), neither of which is a subset of the other, so this is perhaps a more interesting example than switching between intuitionistic and classical logic.
      - plutooo 11y ago
        But I'm saying if you evaluate the axiom symbols two different ways, it's two different axioms.
        catnaroek 11y ago
        What you're saying is more or less equivalent to “two C implementations targeting different [architectures / operating systems / whatever] are actually implementations of two different programming languages”.
        plutooo 11y ago
        No, I'm saying the meaning behind statements are different. On some architectures, an int is 16-bit, on others 32-bit. Anyway, this analogy was pushed too far a long time ago.

plutooo 11y ago

You can't have the same axioms with different rules of inference. The rules of inference are axioms. By the axioms being the same, you mean the ink-shapes making up the symbols on a piece of paper being the same. Not the actual meaning behind them. Edit: Since you added more content to this reply later on, let me respond. In this case I was talking about specifically a formal system where a=0 makes sense and is expressible, but to keep it short and concise I didn't choose to write out all the details. So the rebuttal is moot.

catnaroek 11y ago
Axioms and rules of inference are fundamentally different: (0) An axiom is an internal statement to a mathematical theory that is assumed to be true. That is, inside of a mathematical theory, you don't need to prove that its axioms hold. However, if you want to construct a model of a mathematical theory, you need to prove externally that the axioms hold. In return, you get the theory's theorems (suitably interpreted) for free. (1) A rule of inference exists in an external metatheory, where the original mathematical theory we were studying is treated as a syntactic object (also known as object language), in very much the same way a compiler treats the program being compiled as a (possibly annotated) syntax tree. A rule of inference defines a class of valid syntactic transformations, but doesn't concern itself with the impact of these transformations have on the meaning of the phases in the object language. It is perfectly sensible to consider the effect of changing the rules of inference, on an axiomatic system.
- plutooo 11y ago
  If you have different rules, you have a different object and the meaning of the axiom is different. Even if it is written using the same symbols. To continue the compiler analogy.. Just because the ASCII sequence "int c=0;" means different things in C and Java, doesn't imply "int c=0;" is meaningless when specifically talking about only C.
  - catnaroek 11y ago
    The problem with your analogy is that C and Java have completely different abstract syntaxes. A better analogy would be taking the syntax of an existing programming language, and completely changing its meaning. For instance, consider the effect of making Racket use lazy evaluation (Lazy Racket), or making Haskell use strict evaluation (the upcoming -XStrict pragma). Strict and lazy languages validate different sets of equational laws (and hence compiler optimizations!), neither of which is a subset of the other, so this is perhaps a more interesting example than switching between intuitionistic and classical logic.
    - plutooo 11y ago
      But I'm saying if you evaluate the axiom symbols two different ways, it's two different axioms.
      - catnaroek 11y ago
        What you're saying is more or less equivalent to “two C implementations targeting different [architectures / operating systems / whatever] are actually implementations of two different programming languages”.
        plutooo 11y ago
        No, I'm saying the meaning behind statements are different. On some architectures, an int is 16-bit, on others 32-bit. Anyway, this analogy was pushed too far a long time ago.