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.
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).
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.
(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:
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
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).
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?
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.
Anyway, this analogy was pushed too far a long time ago.
> 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.
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.
Naïvely encoded in Martin-Löf type theory, that is.
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.
Isn't that an absolute truth?
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
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.
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".
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.
Math can tell you an awful lot about the state of the world. That's why it's useful.
> 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.
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 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).
> 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.