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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?

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    We have had to alter basic assumptions about the nature of sets and self-reference in order to remove paradoxes. So was the math absolutely true even when it was inconsistent? If not, why would we assume it is absolutely true now? – jobermark Jan 27 '18 at 1:15
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    This is a good question, but one of the issues that you'll get with answers and comments is that the answer to the question depends on which school of the philosophy of mathematics is true. The answer is yes if platonism is correct, no if formalism is correct, and so on. Nobody knows which school is correct but most people believe that one is. You're probably going to find a lot of questions and answers that claim yes or no without explicitly stating anything about the metaphilosophical assumptions they're making because they feel like the answer is self evident, e.g. the comment above. – Not_Here Jan 27 '18 at 18:22
  • @Not_Here If it helps, I will state my position -- Intuitionism. By the standards of Platonism set theory is inconsistent. By the standards of formalism it was wrong and is now right. In all three sets of assumptions the question remains a question. So I did not simply fail to mention my position. I made a genuine argument The answer to the question cannot be 'yes'. – jobermark Jan 27 '18 at 22:13
  • @jobermark "so I did not simply fail to mention my position" no you did, and that is precisely the point that I was making. You didn't mention your position because you're convinced of the arguments that bear it's truth and you don't see how any other position in the philosophy of mathematics could be correct. Do you agree with the statement "if mathematical platonism is correct, then there is a notion of absolute truth in mathematics?" If you agree with that statement then you agree with what I'm saying. – Not_Here Jan 27 '18 at 22:32
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    @jobermark You find it irrelevant so you simply failed to mention it, I don't understand where the disconnect is. Anyway, you're obviously incredibly upset by the fact that I pointed out that the caution I was giving the OP was present in your comment, but all of this goes beyond the scope of comments as well. My point remains and has been perfectly exemplified by how vehemently you're arguing that your point of view is correct and all others are incorrect. I was cautioning the OP about the answer's they're going to (did) receive, not telling you that you can't post whatever you want. – Not_Here Jan 28 '18 at 3:06
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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...

  • I really don't think that formalism is dominant in modern mathematics. In fact, I don't think I've ever heard a mathematician "professing" formalism, and I've been to a fair share of math conferences with discussions of foundations. – Arno Jan 27 '18 at 20:04
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    @Arno I removed from my answer the unnecessary claim that formalism is dominant. I'd love to read about alternative views on the foundations of mathematics and whether they treat it as "absolute truth". – ngn Jan 27 '18 at 20:19
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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.

  • As a professional mathematical logician, I struggle to match your description of mathematics with what I perceive. Are the different "math theories" you mean the different frameworks (constructive vs classical logic, ZFC vs ZF + AD, etc)? If so, then there is still the general consensus over what the theorems are. – Arno Jan 27 '18 at 20:07
  • @Arno As a professional mathematical logician, you should have heard of intuitionism, which pretty much backs this position. We agree on what the theorems are because we as humans share an intuitive basis. That does not mean that those shared intuitions are in any way absolutely true and will not occasionally bifurcate or contradict one another. – jobermark Jan 27 '18 at 22:25
  • @Arno. And you should be aware of ongoing parts of your own domain, like the theory of large cardinals, which continue to exist outside any containing set theory, and therefore lie in the world that has no solution for Russel's paradox. In a domain with an unresolved paradox, there are no absolute truths, because you can simply deduce their opposite from the paradox. – jobermark Jan 27 '18 at 22:32
  • To ground this position in an established context, consider plato.stanford.edu/entries/fictionalism-mathematics which gives a careful approach to the idea that math has only productive consistency, and no truth. – jobermark Jan 27 '18 at 22:34
  • @jobermark I see intuitionism as a particular stance regarding the ontological status of mathematical concepts. This answer is making statements about the practical reality of mathematics that dont match my own perspectives. This two things are not closely connected. – Arno Jan 28 '18 at 0:05
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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.

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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.

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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"!

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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...

  • I don't think that different rings constitute places where mathematical truths are relative. They are just places where we choose to use similar symbols to mean different things. – jobermark Jan 29 '18 at 1:24
  • @Jobermark: sure, thats why I said 'understood implicitly' in the final sentence. – Mozibur Ullah Jan 29 '18 at 1:45
  • There are just much more genuine examples of relative truth in mathematics: e.g. the Axiom of Choice vs Determinacy of Infinite Games. Given that context, this example seems misleading. It is an example of confused equivocation, and not of relative truth. – jobermark Jan 29 '18 at 1:46
  • @jobermark:Maybe, but it is more easily understood; its possible to argue your example falls into the same family. My example looks like equivocation but then again the axiom of choice is similar - its simply that the situation is so much more complex that it's harder to decide. – Mozibur Ullah Jan 29 '18 at 1:50
  • @jobermark: I'm a pluralist at heart; I don't think that mathematics founded upon one set of axioms - foundationalism - is the best way of looking at mathematics. – Mozibur Ullah Jan 29 '18 at 1:55

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