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Assume that a contradiction in mathematics is discovered, say '0=1'. Then, by the principle of explosion from classical logic (by the rules of which, arguably, the world adheres as well) we can derive the truth (and falsity!) of any contingent (but patently false) proposition about the world.

For example:

0=1, therefore gravity repels, not attracts.

0=1, therefore gravity attracts, not repels.

What are we to make of this? Should we conclude that because it looks like there are no contradictions in the world and the world is not falling to pieces, mathematics itself must be consistent?

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  • There's not just "one" mathematics to be consistent or inconsistent, though. Dec 19, 2023 at 2:44
  • 1
    It is not clear to me what you are asking. Do you believe that the objects of the world are mathematical entities? Do the elementary particles of the standard model compute their evolution in time? It seems a bit of a stretch. We use mathematics to approximate the physical world, but to suggest that the two are identical seems implausible.
    – nwr
    Dec 19, 2023 at 3:04
  • @nwr I have paraphrased the question. Let me know if I need to elaborate.
    – Alex
    Dec 19, 2023 at 3:11
  • But can't one help but use classical logic while reasoning about quantum phenomena?
    – Alex
    Dec 19, 2023 at 3:41
  • Bridges will not fall down if math is found to be inconsistent.
    – user4894
    Dec 19, 2023 at 3:52

6 Answers 6

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Not all those who assert contingent propositions believe that the consequence relation is classical (in relevant logic, for example, the consequence relation is paraconsistent, and so there's no proof of everything from a contradiction.)

But yes, if you assert any proposition whatsoever, and if you believe that consequence behaves classically, then you will want to believe that there are no true contradictions lest you become a trivialist! So you are right in some sense, but this is in no way unique to contingent propositions, nor does it apply universally (e.g. it doesn't apply to paraconsistent logicians that assert contingent propositions).

There are other counterexamples too. For example, in the philosophy of logic, not everyone is a logic monist. Some pluralists believe that various systems of logic are good for one purpose and bad for another. So they might use classical logic in certain instances and abandon it in the cases where contradictions arise; in fact, adaptive logic is precisely that, compromised of both an upper and lower limit logic. The upper limit logic is classical, and you always get to use the ULL until you find a contradiction, in which case only inferences in the lower limit logic (the LLL) are allowed, and the LLL is often substantially weaker than classical logic: it is most commonly a paraconsistent system, so once again no derivation of trivialism is allowed.

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  • Thanks for an interesting answer. Here is a proof that does not use the consequence relation.
    – Alex
    Dec 19, 2023 at 4:08
  • I think you might be slightly confused. All logics have a consequence relation, usually both syntactical consequence (e.g. a proof) or semantic consequence (i.e. all the structures that model the premises model the conclusion). A proof in a system simply shows that the conclusion (in this case, Q or any arbitrary sentence) is a consequence of the premises, and of course this includes the Lewisian proof of explosion! But no proof "uses" the consequence relation in the internal sense of it occurring in the proof, since ⊢ is a metalogical symbol we use to talk about a logic.
    – kuro
    Dec 19, 2023 at 4:14
  • Oh, I thought you were talking about the object theory consequence relation.
    – Alex
    Dec 19, 2023 at 4:17
  • Right, so the consequence relation does not per se exist in the object language (in the sense that there is no wff where it occurs), but you could view every proof within the object language to be a proof that the consequence relation holds between the premises and the conclusion. You might also be referring to the material conditional, which, while not itself the consequence relation, is intimately related to it (see the deduction theorem: in fact, this is how we justify the rule of inference →I!) And if you are indeed referring to →, then yes, you don't need → to prove explosion.
    – kuro
    Dec 20, 2023 at 7:06
  • Recall that relevant logicians believe consequence itself (⊢) behaves non-classically precisely because they don't admit the rules of inference that classical logicians accept, so there is no indeed no proof of explosion in relevant logics, and this is independent of whether → occurs in the proof (although relevant logicians also strongly disagree with the classical interpretation of → as the material conditional, see relevant implication)
    – kuro
    Dec 20, 2023 at 7:09
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If we are some sort of mathematical realist and then distinguish our theories of mathematics from the "reality itself," then the reality itself will always be consistent anyway (supposing that reality does "obey" classical logic), so there is no causal dependence of the world's stability on a contingent consistency of mathematical "reality itself." There is no danger of a logical explosion as a physically real event (the closest risk is of a Big Rip, arguably).

Now, as for some of our theories of mathematics, these have been found inconsistent in various ways at various times. The stability of the physical world is not testimony on behalf of our theories, though, then. It might be an empirical claim to say that, "ZFC will not be proven inconsistent in 1000 years" (Woodin, IIRC), but falsifying this claim would not mean that the world would suddenly fall apart, would it?

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In my opinion, sort of. My position on the philosophy of maths is a formalist one, that is, I believe mathematical truths are essentially constructed and evaluated from a set of rules, or axioms. However, despite the constructed nature of mathematics, the way it functions can still demonstrate truths about reality. For example: any logical or mathematical system consistent with its axioms will effectively have:

  1. Reflexivity of equality; that is, each side of an equality will be of the same value; A=A. Even if in a logical system A+A=AAA, and this logic is consistently applied, that is equivalent to the '+' operand representing the appending of another A to the string; even if a system appears to not retain value on each side of an equality, it could be equivalently rephrased to do so.
  2. Non-contradiction. If a logical or mathematical system is consistent with its axioms, it will follow the law of non-contradiction. This essentially follows from 1, as if the law of non-contradiction isn't true then A+A=AA would also be true. As you implied, non-contradiction is essentially built into the concept of consistency itself.

So, even if a contradiction, 0=1, was discovered, as long as this principle was consistently applied throughout the rest of maths, it would be acceptable; otherwise, that system is not in accordance with reality. Given that maths is constructed, we can discard contradictory systems in favour of better ones rather than re-evaluating the rest of reality.

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The case you describe already happened: Bertrand Russell’s antinomy in set theory from 1901, see Russell's paradox.

After Cantor’s invention of set theory it took a while, until mathematicians learned how to deal with the new concept. Russell’s antinomy showed that one cannot build new sets by just stipulating arbitrary properties for their elements.

Russell’s discovery was a shock for mathematics, but it did not destabilize the physical world. It did not imply a contradiction in the world, the world was "not falling into pieces".

Instead one had to limit the mathematical construction to form new sets.

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  • I don’t believe that any formal system, some foundational theory like ZFC, can really be mathematics. Mathematics is part of the world.
    – Alex
    Dec 19, 2023 at 8:50
  • @Alex Can you support your "believe" with some arguments? If mathematics is part of the world, what do you think is not(!) part of the world?
    – Jo Wehler
    Dec 19, 2023 at 9:00
  • I can certainly say that things like mathematical objects are part of this world. No further comment (unfortunately).
    – Alex
    Dec 19, 2023 at 9:29
  • @Alex Please note that this platform is about philosophy. A philosophical discussion requires arguments, it is not just stating claims or beliefs.
    – Jo Wehler
    Dec 19, 2023 at 9:59
  • @Alex What even is a mathematical Object, to you? Is the set of all Natural Numbers a mathematical Object? What about Powersets? If so, you have a hierarchy of increasingly large infinities, are they all also part of the natural world? Dec 19, 2023 at 14:56
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There are no great metaphysical insights into this question.

First of all, contradictions aren't discovered. They're constructed, and a good explanation of this is Constructive Mathematics (SEP). You give me an example of a contradiction in mathematical logic (Russell's paradox is a good example), and I'll show you how it is constructed, and then how it is obviated through another construction. (Russell himself devised simple type theory). Mathematical statements in this view are nothing more than constructions.

Secondly, given the construction of a contradiction, there is no great implications to reality or thought given your ability to produce systems that admit all manner of true, but meaningless inferences. That's because reason requires relevance for such inferences to be meaningful and useful. In fact, there are now decently complex relevance logics (SEP) that govern such issues. From the SEP:

In addition, relevance logicians have had qualms about certain inferences that classical logic makes valid. For example, consider the classically valid inference... The moon is made of green cheese. Therefore, either it is raining in Ecuador now or it is not... Again here there seems to be a failure of relevance. The conclusion seems to have nothing to do with the premise. Relevance logicians have attempted to construct logics that reject theses and arguments that commit “fallacies of relevance”.

So, while your question is meaningless in a technical sense, it betrays a certain perspective of logic that somehow the rules of the game manufacture meaning, when in reality, meaning dictates the rules of the game. Games that produce meaningless results are simply trivial. This is because the language of logic is governed by the broader nature of the Wittgensteinian language-game. Languages, including those of mathematical logic, are ways about sharing experience and bringing about change in the world through speech acts. There is no great mystery when nonsense is uttered.

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Materialist dialiectics answer this!

From this perspective, math is derived from nature and not the other way around. Ontologically this means that 2km+2km will always be 4km no matter what happens in human minds.

So while at first glance a formal contradiction like 2km=3km are evidently not found in reality, dialectical contradictions do exist in the world. To see them we just have to introduce the dimension of time: A bridge might collapse in 20years not because then 0=1but because the material have eroded with time and shrinked in size therefore making the standing of the bridge materially imposssible and forcing the bridge to fall.

The point is that whatever exists right now is always changing and maybe the external conditions and internal development might in some point in the future form a contradiction (like a bridge shorter than its needed) from which reality needs and does inevitably escape to keep consistency.

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