Absolute Truth in Mathematics

Question

Often in philosophical discussions, the concept of absolute truth will be proposed in a metaphysical manner that supposes supreme authority and the absence of exceptions to rules regardless of context. Naturally, human beings might be tempted to resist the idea of anything in isolation and resort to the comfort of relative truth. Mathematics is notorious for its colloquial consensus as being factual when a proof is demonstrated for any assertion as simple as 1+1=2, which may include a self-evident intuition as evidence.

Is mathematics a sufficient example in the context of philosophy to assert the existence of absolute truth?

Answer 1

According to the formalist view on mathematics, I think the answer should be no - mathematics is not an example of absolute truth according to the definition you've given.

This is how mathematics works, loosely speaking:

• we choose a set of statements ("axioms") and encode them as formulas - sequences of mathematical symbols

• we produce new statements ("theorems", "lemmas", etc) by combining or rewriting those formulas in accordance with the laws of deduction (logic)

A "proof" is a chain of such deductive rewritings. This is only a process of blind manipulation of symbols. It makes no claims about the state of affairs in the physical world, and is by no means of "supreme authority". Remember, we started from unproved axioms, so in essence our proof is nothing but a glorified if/then clause - if the axioms are true, then this theorem is true.

Note that the process of finding proofs could be extremely challenging and is often aided by intuition, however once built, the proof itself does not rely on intuition at all.

When you say 1+1=2 is "true", you really mean it can be derived from a certain axiomatic system, for instance Peano arithmetic. But in another axiomatic system (or using a different choice of symbols) it might be provable that 1+1=0, for instance Boolean algebra.

What I did not explain above is where we get the rules of deduction from and whether they constitute absolute truth. Formalism evades these questions by keeping the rules of deduction out of its realm. In other words, it avoids mixing language and meta-language. But how do we study that meta-language (logic)? By using a formalism...

Answer 2

There are different kinds of mathematical truth, and depending on what you would want in a notion of absolute truth, some of these may fit - or not. My personal stance would be that absolute truth is not a fruitful concept, and probably not worth thinking about too much.

The agreement about truth amongst mathematicians

A defining feature of mathematics as a field is that mathematicians can generally reach consensus about the truth of a particular mathematical statement, by providing a purtative proof, critizing its weak points, shoring them up, etc.

This process can require clarification/abstraction: If the a priori assumed axiom systems of the involved mathematicians differ, they may have to move from "X is true" to "Axiom system Y proves X". If a statement written by a mathematician using classical logic is read literally by a constructivist, it will often be false - but constructivists know how to read classical mathematics in a way that retains truth.

Even on the "fringes" of mathematical foundations, say the ultrafinitists would typically attack results as being meaningless, rather than being false.

We thus see evidence of a notion of truth that trancends individual view points or perspectives, in manner that most fields of philosophy are not privy to. This also differs from the one in the natural sciences, where experiments can provide an arbiter of truth.

On ontology

The notion of truth described above is - on purpose - divorced from any assumptions about the ontological status of mathematical objects. One can refuse to entertain this question at all (which essentially leads to formalism, and the unresolved questions of why we are playing this game, and why math works so well to describe nature).

The stance that most readily leads to a notion of absolute truth would be platonism in the strict, "one mathematical universal", form. This does not mean that any mathematics not done in the "one and only correct system" is invalid - since we can translate between the various systems, we can recover (almost) all other systems as a fragment of the chosen true reality.

A stance that seems widespread amongst mathematicians working on foundations is that of mathematical relativism (described very nicely by Andrej Bauer here). We just accept a plethora of mathematical universes, and study those we find most fascinating. Depending on what is meant by absolute truth, this might be incompatible with such things.

Conclusion

Mathematics offers far more absolute truth than other areas, but asking for perfectly absolute truth probably doesnt make much sense.

Answer 3

First of all, what do we mean by absolute truth? Absolute truth is an ontological assertion. It asserts that there is no fact, now or in the future that will occur to any kind of intelligence that would contradict any part of what is being called absolutely true.

We tend to believe something like this: Statements in math are true or false. And if we can't prove them true, they just aren't true. They may not be provably false, but they are not necessarily true.

If that is true, that truth in mathematics is established via rigorous proof, and that its statements are themselves ultimately clear, and unambiguously either true or false, we might presume that we can never grow math into something that will have internal contradictions. It becomes a candidate for absolute truth.

But then we have Russel's paradox. Given the ordinary intuition of sets and containment, it just can't be either true or false. So we adjusted our intuition of sets into a rigid formalism to accommodate this problem. And we now assume there will never be another such problem. But on what basis do we decide that? Now that we have seen one exception to the first statement above, how do we know there will not be dozens more, slowly turning up over time?

And then we have Godel's theorem. Until we proved it, we predicted that only provable statements in mathematics would be true. But we know that that is not the case. There are necessarily true statements that we cannot possibly prove. So we have an exception to the rest of the statement.

So what is left of our intuition about the reliability of math? Only the experience of its extraordinary power. And there are explanations for that other than access to absolute truth, which have been explored elsewhere on this site.

Answer 4

The best mathematics can do, is prove that its axioms, theories, and "laws" form a consistent system - most of the time. This is a "long way off" from being a "sufficient example to assert the existence of absolute truth"!

Answer 5

There are no absolute truths in mathematics. As with any subject, there are numerous perspectives and unique and informative insights that any one theory may provide. As for absolute truth contained in and in regards to the validity of individual math theories themselves, mathematicians are more concerned with making sure that their theories are logically consistent (especially after when taking Godel's insight on the nature of arithmetic and by large-- all of math into account). Demonstrating proofs to be true is how mathematicians show the logical consistency behind their developed theories.

Answer 6

Truth & relevance are two different but related things.

For example it is true that 120000007+1=120000008 but no one cares about this particular result, to most people at most times that this is true is irrelevant.

Why is this different from the truth and relevance say of 1+1=2. Because this latter statement is the one by which one can be inspired to think through a general theory of addition in a way the first statement is unlikely to inspire.

Mathematical truth can also be relative. For whilst 1+1=2 in the ring Z, it equals zero in the ring Z/2Z. One might answer this by saying that the ring Z was understood implicitly...