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"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.

> 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).

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.

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.

    Axioms and rules of inference are fundamentally different:

Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences.

As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen's sequent calculus [2] which has only one axiom (A |- A), everything else being a rule of inference. Most presentations of FOL are between these extremes.

[1] A mathematical introduction to logic, by H. Enderton

[2] Untersuchungen über das logische Schliessen I, by G. Gentzen

> Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences.

Yes, but the derivations themselves will change.

> As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen's sequent calculus [2] which has only one axiom (A |- A), everything else being a rule of inference. Most presentations of FOL are between these extremes.

Yep. I'm aware of the phenomenon that a single mathematical object of type T (say, infinity-categories) may admit multiple presentations by objects of type T' (say, model categories). But, just because two objects of type T' present the same object of type T (e.g., two Quillen-equivalent categories), it doesn't mean that they are equal in all respects (e.g., the category of simplicial sets is much nicer than the category of topological spaces).

Exactly. What are axioms and what are rules of inference ends up being a distinction without a difference.

I'm not sure I would go that far.

There is a difference, but it's not clear quite what it is.

For example, it becomes progressively harder to get nice (with cut elimination and finite rule schemata) sequent calculi for richer axiomatic systems. For example I have not come across a nice (in the above sense) sequent style formalisation of ZFC set theory. Have you?

No, but why is that a problem?

It's an indication, a hint, that rules and axioms are not always completely exchangable.

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.

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.

But I'm saying if you evaluate the axiom symbols two different ways, it's two different axioms.

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”.

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.

Ah, lovely! You just arrived exactly where I was trying to get.

> I'm saying the meaning behind statements are different.

Yes, exactly! And, just like a single C program can have two different meanings under two different implementations, the same axiomatic system can have two different meanings when deducing its consequences (proving theorems) using different rules of inference.

Not necessarily. It's very possible for one system to embed inside the other. In this case you can often talk about exactly the same axiom/theorem/whatever within different inferential systems.

An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic. That said, we can use flattening to embed classical logic inside of intuitionistic logic and then recover a whole family of propositions of which AC is one and also see why it has a difficult time holding in IPL.

> An interesting example is the Axiom of Choice which is, naively encoded, a theorem of intuitionistic logic.

Naïvely encoded in Martin-Löf type theory, that is.

Still, "axiom of choice is a theorem in intuitionistic logic" is absolutely true whether or not you accept intuitionistic logic.

But thas is not Mathematics, they call it "metamathematics" for something.

This seems to be a technical objection that dodges the meat of the parent's comment. In what sense is it true that a given theorem follows from a given set of axioms and a given choice of inference rules?

A “theorem” is by definition a statement in a mathematical theory that has a proof. In what sense is this true? Well, that depends on what “true” means in your model... :-p

I agree with parent comment. At the heart of your disagreement is that you seem to use definition from logic of what axioms are, while your opponents use "definitions" from metamathematics.

Umm. I've been of the understanding that we never change the inference rules, Can you point to some branch of mathematics that involves a change of inference rules?

At my level of understanding, it seems a reasonable characterization of intuitionistic logic to say that it operates with a more restricted set of inference rules. It was the parent who made the stronger claim about differences here, and they speak a bit more to the topic in nearby comments.

> "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.

"There is no such thing as absolute truth."

Isn't that an absolute truth?

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.

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.

"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

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.

"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.

> 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".

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.

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.

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

> 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.

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.

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).

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.

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.

> Formal mathematics has no concept of absolute truth

That's like an absolute truth in formal mathematics then, I suppose.