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I'm thinking every logical derivation as something like all the derivations in the Principle of Explosion - really everything.

It could just be a helpful interpretation, not trying to get super deep into ontological commitments. I guess I really do mean there is a "world" where 1=2, or 1=2 is helpful to some situation (I really prefer the latter, it seems to avoid a lot of ontological commitments as to what 1=2 is even saying, and what exists).

In a way I guess it would help reinforce the empirical stance/side of knowledge (at least for me temporarily) like Quine did about set theory and physics, i.e. set theory is part of our greatest overall empirical theory, and could still be abstract/platonic.

I don't think its paraconistent or dialetheism. If it's pluralism, does it have a name? Explosive pluralism? Or have I missed some obvious and trivial step, that of course we think all derivations "exist" as the backdrop to (our) logical and mathematical knowledge.

*Another misstep I may have made is I just don't know how to define logic and math. (Then, is there a modern conception of them which still makes this wondering possible?)

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I genuinely think I found the proper term for this - really full blooded platonism: "[e]very mathematical theory—consistent and inconsistent alike— truly describes some part of the mathematical realm". [1] Really everything, without being trivial.

My initial question was admittedly muddled by adding in whether or not and on what grounds we should be ontologically committed to platonic objects. I tried to cast the widest net to find this term by selecting from realist and anti-realist commitments.

RFPB doesn't seem like a trick, it seems like a genuine answer to philosophical problems, and I'm surprised (it seems) most shy away from such a large platonic heaven.

[1] https://entailments.net/papers/fbplatonism.pdf

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    The popularity of Hamkins' multiverse standpoint in set theory indicates that RFPB is gaining a lot of traction in the philosophy-of-mathematics community. Even on the level of the forcing multiverse, Hamkins (with Benedict Loewe IIRC) found a pretty exact match in modal syntax for their concept. Quantified modal propositional logic (i.e. where we quantify over axioms rather than/in addition to universes) might be a way to have RFPB without direct/substantial ontological commitments (depends on how egregious one feels abstract propositions are compared to abstract universes).
    – Kristian Berry
    Dec 24, 2022 at 13:05
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    @KristianBerry I find it interesting the growing traction is coming from the realism side. The fictional universe seems perpetually at least as large as the mathematical and logical universe. RFBP seems like the first theory that could unite them. What better unification than realists and fictionalists fully exploring their domains to find they are the same, coming to agreement only at the terminus of their endeavors, and separation and even hostility in the meanwhile.
    – J Kusin
    Dec 24, 2022 at 17:19
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Is there stance that every logical and mathematical derivation exists/is contructable but we only care about a proper subset?

This is not a stance. This is the situation. For mathematicians as well as for anyone else.

I ignore mathematical logic here as totally irrelevant.

So, suppose you have A and B as a system of two independent axioms. The only (moderately) interesting conclusions that you can logically infer are A ∧ B, A ∨ B, B ∧ A and B ∨ A, and then nothing else interesting will come out of that.

You could also infer A ∧ A, A ∨ A, B ∧ B and B ∨ B, but this is not interesting.

You could also infer things like A ∧ (A ∨ B) etc., but this is also of no interest.

However, independent axioms are themselves not interesting to mathematicians. What mathematicians are only interested in are systems of non-independent axioms.

As a basic case, suppose only two axioms, A → B and A, where A and B are however independent from each other.

From A → B and A, we can infer A ∧ (A → B), and then from A ∧ (A → B) we can infer B, by modus ponens.

And this is it. There is nothing else interesting to infer.

You could also infer A ∨ (A → B), B ∧ (A → B) and B ∨ (A → B), for example, but these are not interesting.

This situation is typical of mathematics.

This explains why there is only a limited number of theorems mathematicians will ever be interested in proving.

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  • I thought many axioms were independent of ZFC but mathematicians find them interesting? It seems like there is interesting independence. I still don't understand how classical and int mathematicians for example aren't independent. Or classical and paraconsistent. How is "A ∧ A, A ∨ A, B ∧ B and B ∨ B, but this is not interesting." NOT interesting? I would find it tremendously interesting to find situations where A is useful and another where B is useful, or even necessary. (I didn't downvote)
    – J Kusin
    Sep 23, 2022 at 16:15
  • @JKusin "It seems like there is interesting independence." Like what? 2. "how classical and int mathematicians for example aren't independent" What does that mean? 3. "Or classical and paraconsistent" How does that relate to my answer? 4. "many axioms were independent of ZFC but mathematicians find them interesting?" My answer is about systems of axioms, which is the only aspect which is relevant.
    – Speakpigeon
    Sep 23, 2022 at 16:39
  • 2. I mean't are independent, Classical and Intuitionist mathematics. They have different logical rules/axioms. 4. Similar answer: ZF + AoC, ZF -AOC. How are those not different axiom systems and not independent? plato.stanford.edu/entries/set-theory-constructive/… Says INT math rejects at least two axioms of ZFC, Foundation and Choice. "It seems like there is interesting independence." it is exactly that two different axiom systems can produce relevant mathematics I find interesting.
    – J Kusin
    Sep 23, 2022 at 16:48
  • @JKusin "ZF + AoC, ZF -AOC. How are those not different axiom systems?" My answer is about axioms that are part of one system of axioms, not about axioms from different systems. You are not going to infer anything interesting from axioms that are independent of each other.
    – Speakpigeon
    Sep 23, 2022 at 17:10
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    @JKusin "all internally consistent axiom systems." There is also a countable infinity of axioms systems, at least if we only consider axioms that have a finite number of logical operations and relations and if we consider finite systems of axioms. Beside, for any theory, the set of theorems which would be even moderately interesting is not only finite but also extremely limited.
    – Speakpigeon
    Sep 25, 2022 at 17:46

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