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This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or whit

This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an axiom reflects reality or not is out of scope, which is the whole point of their invention. As a philosopher you're welcome to debate them, and as a pragmatist you're welcome to choose them (and be compelled to accept their consequent theorems). > Because it is easy for mathematicians to ignore Gödel’s proof. What lurks in their heart of hearts is a commitment to absolute truth, and a universal formal axiomatic theory for all of mathematics. In one sense it's disheartening that we cannot prove or disprove every proposition in every axiomatic system. This is an unreasonable expectation as Gödel proved, but it doesn't invalidate the theorems we have proven and disproven. It also doesn't invalidate all of the programs we have written, even though there are non-terminating programs and uncomputable numbers.

conceit 11y ago
You are being philosophic asserting that there would be a reality to reflect, in a rather colorful language, while axioms are real enough to not need to pull a strawman from beyond them.
- conceit 11y ago
  > Formal mathematics has no concept of absolute truth That's like an absolute truth in formal mathematics then, I suppose.
_RPM 11y ago
This reminds me of my Calc professor. When I asked about applying a theory. I asked, "So, when applying this, these values would indicate the expression is true?" He said, we don't concern ourselves with truth in Mathematics.
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.
gizi 11y ago
> Formal mathematics has no concept of absolute truth ... Well, it does. If you drag the propositional calculus into the fray, "true" is arbitrarily but probably most elegantly defined as following: true(x) = x or not x False is defined like this: false(x) = x and not x These definitions are pretty much arbitrary. The lambda calculus does this: true(a,b)=a false(a,b)=b It also works absolutely fine. > It's just concerned with axioms and theorems (and their proofs) In the general case, theorems are indeed "provable" or "unprovable" (not "true" or "false"). However, if a theorem has exclusively been derived from propositional calculus, it can also be "true". Gödel's incompleteness is exactly about statements that are "true but unprovable". In terms of propositional calculus, the theorem is "true" but since you can show that the theorem can never be derived from the axioms, it is also "unprovable".
- zvrba 11y ago
  Well, it does not. Mathematics is a symbolic system invented by humans, but that system tells you nothing about the state of the world. Used correctly, it can be used to describe the world, but it can also be used to describe phenomena that have not been physically observed. E.g., currently accepted theories allow for existence of wormholes, but we do not know whether the actual physical laws of the universe ("absolute truth") do. So, no, there is no "absolute truth" in mathematics. There are only valid derivations, but taking them as the "absolute truth" is like mistaking the pointing finger for the moon.
  - Houshalter 11y ago
    Yes there are absolute truths. 2+2 will not equal 5 in any universe. The laws of physics could be radically different, but I have 2 apples, and you give me two more, I will not suddenly have 5 apples. Math can tell you an awful lot about the state of the world. That's why it's useful.
bachback 11y ago
Have you followed Gödel's proof in detail? I know Fields medalists who like to speculate about Gödel, but haven't read Russell and Whiteheads Principia Mathematica (PM). Gödel's system is PM, but PM is quite an impossible read. Most of this discussed has been discussed over 30 years between 1900-1930. By the way Turing also builds on PM in his on computable numbers paper. PM is like a compiler before there were compilers. Today it would look more like metamath.org or https://github.com/vladimirias/Foundations https://github.com/vladimirias/Foundations > Formal mathematics has no concept of absolute truth; this is left for philosophy. No, it is called logic which is the intersection of both. And most mathematicians haven't done any logic whatsoever. They avoid it, because it means you actually have to think about things. You have to think about why greek symbols have special meanings. And as computer scientists/hackers know, computer code is an alternative medium. I remember when I asked a math professor about the "=" sign and how things can be equal at all, since if they are either the same (A=A is trivial), it is trivial, or if they are different they are not the same (A=B is false). About 10 years later I discovered this is why physics has the notion of symmetries, which is a statement of the form A - B = 0. Two things are equal iff their transformation leads to the origin. But I should have known this all along, because in computer programs "=" can stand for various kinds of operations. It is just that the language of mathematics in general is so inaccurate that one gets lost on the way.
- asQuirreL 11y ago
  I'm a bit dubious about your sweeping generalisation that "most mathematicians haven't don't any logic". I also agree with GP that logic also does not deal with absolute truth. Proofs in formal logic may be applied to models, but they only provide truths modulo the assumptions made in those models. As for your remark on equivalence. Your original description is one of syntactic identity, a sort of free equivalence. But any relation that is reflexive, transitive and symmetric can be considered an equivalence (indeed, there are infinitely many equivalence relations over the integers). Also note that this is not the same as how "=" is used in computer programs where for the most part it is asymmetric (a=b is rarely semantically equivalent to b=a).
- __mbm__ 11y ago
  This is the crankiest comment I've seen on HN in a while. You "know Fields medalists"? Right. You think that Pricipia Mathematica is the source of all truth in mathematics? Wrong. Most mathematicians working on foundational questions start out by learning Zermelo-Fraenkel set theory, which is well-understood, and avoids several difficulties that Russel had. While few people have read PM, it remains important because it goes through the hard task showing that high-level mathematics (calculus, &c.) can have rigorous, first-principle proofs. "PM is like a compiler before there were compilers." What? "I asked a math professor about the '=' sign and how things can be equal at all (...)" OMGWTFNO. I've seen comments like this before. Typically, it creates debate around a non-issue by sowing confusion and never nailing down precisely what we are talking about (hence the need for formal methods—it helps us call bullshit on comments like this). "Two things are equal iff their transformation leads to the origin. [A - B = 0]". Wrong. There are equality relations that do not require the existence of a zero or the concept of addition or subtraction.

conceit 11y ago

You are being philosophic asserting that there would be a reality to reflect, in a rather colorful language, while axioms are real enough to not need to pull a strawman from beyond them.

conceit 11y ago
> Formal mathematics has no concept of absolute truth That's like an absolute truth in formal mathematics then, I suppose.

_RPM 11y ago

This reminds me of my Calc professor. When I asked about applying a theory. I asked, "So, when applying this, these values would indicate the expression is true?" He said, we don't concern ourselves with truth in Mathematics.

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.

gizi 11y ago

> Formal mathematics has no concept of absolute truth ... Well, it does. If you drag the propositional calculus into the fray, "true" is arbitrarily but probably most elegantly defined as following: true(x) = x or not x False is defined like this: false(x) = x and not x These definitions are pretty much arbitrary. The lambda calculus does this: true(a,b)=a false(a,b)=b It also works absolutely fine. > It's just concerned with axioms and theorems (and their proofs) In the general case, theorems are indeed "provable" or "unprovable" (not "true" or "false"). However, if a theorem has exclusively been derived from propositional calculus, it can also be "true". Gödel's incompleteness is exactly about statements that are "true but unprovable". In terms of propositional calculus, the theorem is "true" but since you can show that the theorem can never be derived from the axioms, it is also "unprovable".

zvrba 11y ago
Well, it does not. Mathematics is a symbolic system invented by humans, but that system tells you nothing about the state of the world. Used correctly, it can be used to describe the world, but it can also be used to describe phenomena that have not been physically observed. E.g., currently accepted theories allow for existence of wormholes, but we do not know whether the actual physical laws of the universe ("absolute truth") do. So, no, there is no "absolute truth" in mathematics. There are only valid derivations, but taking them as the "absolute truth" is like mistaking the pointing finger for the moon.
- Houshalter 11y ago
  Yes there are absolute truths. 2+2 will not equal 5 in any universe. The laws of physics could be radically different, but I have 2 apples, and you give me two more, I will not suddenly have 5 apples. Math can tell you an awful lot about the state of the world. That's why it's useful.

bachback 11y ago

Have you followed Gödel's proof in detail? I know Fields medalists who like to speculate about Gödel, but haven't read Russell and Whiteheads Principia Mathematica (PM). Gödel's system is PM, but PM is quite an impossible read. Most of this discussed has been discussed over 30 years between 1900-1930. By the way Turing also builds on PM in his on computable numbers paper. PM is like a compiler before there were compilers. Today it would look more like metamath.org or https://github.com/vladimirias/Foundations https://github.com/vladimirias/Foundations > Formal mathematics has no concept of absolute truth; this is left for philosophy. No, it is called logic which is the intersection of both. And most mathematicians haven't done any logic whatsoever. They avoid it, because it means you actually have to think about things. You have to think about why greek symbols have special meanings. And as computer scientists/hackers know, computer code is an alternative medium. I remember when I asked a math professor about the "=" sign and how things can be equal at all, since if they are either the same (A=A is trivial), it is trivial, or if they are different they are not the same (A=B is false). About 10 years later I discovered this is why physics has the notion of symmetries, which is a statement of the form A - B = 0. Two things are equal iff their transformation leads to the origin. But I should have known this all along, because in computer programs "=" can stand for various kinds of operations. It is just that the language of mathematics in general is so inaccurate that one gets lost on the way.

asQuirreL 11y ago
I'm a bit dubious about your sweeping generalisation that "most mathematicians haven't don't any logic". I also agree with GP that logic also does not deal with absolute truth. Proofs in formal logic may be applied to models, but they only provide truths modulo the assumptions made in those models. As for your remark on equivalence. Your original description is one of syntactic identity, a sort of free equivalence. But any relation that is reflexive, transitive and symmetric can be considered an equivalence (indeed, there are infinitely many equivalence relations over the integers). Also note that this is not the same as how "=" is used in computer programs where for the most part it is asymmetric (a=b is rarely semantically equivalent to b=a).
__mbm__ 11y ago
This is the crankiest comment I've seen on HN in a while. You "know Fields medalists"? Right. You think that Pricipia Mathematica is the source of all truth in mathematics? Wrong. Most mathematicians working on foundational questions start out by learning Zermelo-Fraenkel set theory, which is well-understood, and avoids several difficulties that Russel had. While few people have read PM, it remains important because it goes through the hard task showing that high-level mathematics (calculus, &c.) can have rigorous, first-principle proofs. "PM is like a compiler before there were compilers." What? "I asked a math professor about the '=' sign and how things can be equal at all (...)" OMGWTFNO. I've seen comments like this before. Typically, it creates debate around a non-issue by sowing confusion and never nailing down precisely what we are talking about (hence the need for formal methods—it helps us call bullshit on comments like this). "Two things are equal iff their transformation leads to the origin. [A - B = 0]". Wrong. There are equality relations that do not require the existence of a zero or the concept of addition or subtraction.