11 ms·

"Formal mathematics has no concept of absolute truth" This is false. The axioms don't have to be true. You can still talk about their implications in absolute

by plutooo 11y ago

"Formal mathematics has no concept of absolute truth" This is false. The axioms don't have to be true. You can still talk about their implications in absolute terms. "Assuming a=0 implies a=0" is absolutely true. Regardless of whether a actually is 0.

catnaroek 11y ago
> 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.
bachback 11y ago
"The axioms don't have to be true." No, not at all. Axioms are better thought of as universally accepted truths which everything else depends on. https://en.wikipedia.org/wiki/Logical_atomism https://en.wikipedia.org/wiki/Logical_atomism
- thomasahle 11y ago
  "Axioms are better thought of as universally accepted truths" Not really universally accepted. You just have to specify which set of axioms you are using. Different problems may call for different axioms. Like if you want to prove theorems about the security of some computer system, you'll define a set of axioms that allow proving interesting results for that purpose. If you want to prove theorems about sets, you'll want to specify what set axioms you are using. You may even try to specify interesting minimal sets of axioms that make your theorem true. There is no reason to use the same axioms for everything, or to assume they are True in some teological sense.
- golergka 11y ago
  Acioms are only viewed as true because they are defined as such. It's obvious when applying math to real world. If you're talking about objects on Earth surface, for example, at first Euclid's geometry will get you good results, because objects you're working with can be assumed to fulfill it's axioms. However, when your scale gets bigger, you'll have apply a more complicated geometry apparatus. Easiest analogy about axioms is interface in software engineering: you don't care what the object really is, but as long as it shows certain properties, you can prove theorems about it. Which will be true for any objects with these properties, and as true as exactly these properties are fulfilled by the real life object.
SeanDav 11y ago
> "Assuming a=0 implies a=0" is absolutely true. This is not correct. There is no such thing as absolute truth. Something can only be true within a set of previously agreed constraints and rules. For example, I can simply imagine a scenario where a=0 implies a=0 is considered to be false, because I define it to be so.
- plutooo 11y ago
  "There is no such thing as absolute truth." Isn't that an absolute truth?
  - erichocean 11y ago
    Although I know you meant that jokingly, presumably the op's statement is only true within the parameters of a world where "there is no such thing as absolute truth", therefore, the statement isn't absolutely true, it's just true within the confines of his own (subjective) world view. The actual world, may, in fact, have absolute truth.
meshr 11y ago
As it was mentioned earlier we can’t think about “absolute truth" in terms of math because math is appliance science. It is like a language. The root science is physics and only it owns “absolute truth", the real axioms and implication rules.

by plutooo 11y ago

catnaroek 11y ago

> 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.

bachback 11y ago

"The axioms don't have to be true." No, not at all. Axioms are better thought of as universally accepted truths which everything else depends on. https://en.wikipedia.org/wiki/Logical_atomism https://en.wikipedia.org/wiki/Logical_atomism

thomasahle 11y ago
"Axioms are better thought of as universally accepted truths" Not really universally accepted. You just have to specify which set of axioms you are using. Different problems may call for different axioms. Like if you want to prove theorems about the security of some computer system, you'll define a set of axioms that allow proving interesting results for that purpose. If you want to prove theorems about sets, you'll want to specify what set axioms you are using. You may even try to specify interesting minimal sets of axioms that make your theorem true. There is no reason to use the same axioms for everything, or to assume they are True in some teological sense.
golergka 11y ago
Acioms are only viewed as true because they are defined as such. It's obvious when applying math to real world. If you're talking about objects on Earth surface, for example, at first Euclid's geometry will get you good results, because objects you're working with can be assumed to fulfill it's axioms. However, when your scale gets bigger, you'll have apply a more complicated geometry apparatus. Easiest analogy about axioms is interface in software engineering: you don't care what the object really is, but as long as it shows certain properties, you can prove theorems about it. Which will be true for any objects with these properties, and as true as exactly these properties are fulfilled by the real life object.

SeanDav 11y ago

> "Assuming a=0 implies a=0" is absolutely true. This is not correct. There is no such thing as absolute truth. Something can only be true within a set of previously agreed constraints and rules. For example, I can simply imagine a scenario where a=0 implies a=0 is considered to be false, because I define it to be so.

plutooo 11y ago
"There is no such thing as absolute truth." Isn't that an absolute truth?
- erichocean 11y ago
  Although I know you meant that jokingly, presumably the op's statement is only true within the parameters of a world where "there is no such thing as absolute truth", therefore, the statement isn't absolutely true, it's just true within the confines of his own (subjective) world view. The actual world, may, in fact, have absolute truth.

meshr 11y ago

As it was mentioned earlier we can’t think about “absolute truth" in terms of math because math is appliance science. It is like a language. The root science is physics and only it owns “absolute truth", the real axioms and implication rules.