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Some considerations:

  • A related question is whether math is finite, and has been asked here before. Unlike science, math does not seem to be finite. As a dear friend of mine once retorted to me, "You just stop finding patterns at some point?" Mathematics contains the patterns of many a universe, including this one.
  • I see two ways the question could appear to be possible. One, whereby certain axioms can be regarded as 'equations of everything' due to their nature of permitting deduction of all other patterns. Or two, at some point after sufficient mathematical development, results and conclusions are integrated via synthesis into a singular equation of patterns. I wonder then, what would a last equation of math be like?
  • Unfortunately, the first approach does not seem to work. Strongly referring to the ramifications of Gödel's incompleteness theorems:

"...modern logic has had the tools ... to show that Russell was right to claim that the axioms of Principia Mathematica, and the Three Laws of Thought, are no more tautologous and no more self-evidently true than other truth-functional tautologies. All truth-functional tautologies are true for every combination of truth values for their constituent simple statements; they are all on an equal footing, for they are all demonstrably tautologous by the truth-table method. This important fact shows, however, that self-evidence—whether real or illusory or a cognitive function of meaning understanding—is completely unnecessary in logic. What really matters in a system of logic ... is that it be expressively complete (i.e., that every truth function is constructible with the symbols of the system) and that it be deductively complete in that every valid argument can be proved valid by means of its rules" —Introduction to Logic 15e, Irving Copi et al.

  • Unfortunately, the second approach does not seem to work either. As mentioned in the first point, if patterns of math are infinite, there can never be a satisfactory synthesis.
  • Thus then: Is all math doomed to never fit together? Or is this its very nature?
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  • · The uniformity principle is brought to mind. · The two approaches may be seen as the first premises and the last integrative conclusions of a very long endeavor (though, not necessarily all in the right order initially). · I am referring to all patterns, not just 'useful' ones (would useful math not inherently inevitably refer to this universe and its science?)
    – Xeon
    Aug 16 at 21:06
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    Why "equations"? Modern math deals with far more complex objects than that, structures, categories, formal languages, etc. The current "theory of everything" is ZFC set theory, pretty much any mathematical construction or argument can be expressed in ZFC, including "infinite patterns" that nobody has thought up yet. By the way, science is not "finite" either. String theory, the candidate "theory of everything", has "infinite patterns" that we only scratched the surface of, but the same is already true of Newtonian gravity as the many-body problems show.
    – Conifold
    Aug 16 at 22:41
  • Indeed @Conifold! While very much a stretch, to make an attempt to elaborate on part of that: say Geometry connects via the Langlands program with Number Theory, for which numbers can be generated using the Peano Axioms, which is possible via Set Theory, which rests upon First Order Logic, which are via the ZFC system.
    – Xeon
    Aug 16 at 23:31
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    Your above stretch sounds already contains logical circularity, ie, "which is possible via Set Theory, which rests upon First Order Logic, which are via the ZFC system". Also apart from grounding its own consistency PA doesn't need set theory, in fact the creation of 19th century PA was prior to (ZFC) set theories of the 20th century... Aug 17 at 1:50
  • @DoubleKnot I stand corrected, thank you ;)
    – Xeon
    Aug 17 at 2:45

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Once upon a time, set theorists began to tinker with, and then put some of their weird hopes in, things called nontrivial elementary embeddings. They're nontrivial because they're not "the identity": think of them as some function j(x), which just goes to x for many an input, but for critical points goes to something greater. Penelope Maddy describes the commencement of the procedure:

[Let us] imagine ourselves in an Ord-long process, generating an R at each stage, one for each ordinal. Suppose we step several ranks at a time, so that by step a, we are already to Rya, for some y > a. We keep careful track of the structures at each stage by making copious notations on a clipboard, one scoresheet for every stage; we note down every detail of the structure we have just generated, along with every detail of the process that got us there...

Note that in the quoted passage, "R" refers to "ranks," which are similar to ordinals but which, in set theories without the axiom of choice/global well-ordering, play a parallel, not identical, role (see the Wikipedia entry on Scott's trick for some relevant details). Now, the ranks could be models of any strength, from the "smallest" model L (the constructible universe) up to the vague image of V in general. So it occurred to the practitioners of embedding theory to ask if it was possible to nontrivially embed V into itself.

For better or worse, things didn't work out so great. A certain Kenneth Kunen proved that trying to embed V into itself would lead to a violation of the Axiom of Choice (though see also analysis of how to abrogate the foundation axiom, in order to allow for the problematic embeddings, and more, to go through). I can't recall where I read it, I think it was in the work of a certain Kanamori, but the nonexistence of the embedding was given as the positive claim that j(V) does not equal V. (I would've expected the opposite, but I don't actually understand the inner workings of embedding theory.) So one good candidate for "the ultimate equation" seems to have been more or less ruled out; even if we readmitted the equation for some reason, it would be that weird reasoning that we'd be adverting to for our "ultimacy," yet just the same, then, we'd prompt ourselves to go on to yet another "ultimate candidate." At least, the zone above embedding theory is unknown; the theory tries to be all-encompassing by definition, and yet then we see that it breaks down (in fact, not just for V-into-itself, but Va+2 into itself, for some limit ordinal a, and then embeddings of any higher (but not absolute) rank as such).

Some cases of category theory or type theory might strive for cosmic answers; if there is no set of all sets (or set of all other sets?), yet might there be a category of all (other) categories or a type of all (other) types? Or there's Corazza embedding theory, which involves applying category-theoretic techniques to the cosmic questions of set theory. Corazza's own "solution" to the problem of embedding V into itself is to abridge the axiom scheme of replacement, so that the specific stage of V's self-embedding as is implicated in the failure of the choice axiom, is ruled out. This has the dire side-effect of making V cofinal with the zeroth aleph, however, whereas cf(V) = V is widely considered to be more "likely."

So now could cf(V) = V be an "equation of everything"? By itself, it is not rich enough to teach us all that we wish to know in mathematics. Nevertheless, it might be thought of as expressing the theme of "everything," here: of the finality (or transfinality) proper, not only the cofinality, of the numerical cosmos.

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  • I too was thinking along the lines of a set of all sets, or a set of all supposed useful mathematical patterns. Or perhaps, even the notion of infinity itself. Alas, I also came to your conclusion that such formulations by themselves are "not rich enough to teach us all that we wish to know in mathematics." Excellent point
    – Xeon
    Aug 17 at 4:20
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    @Xeon, I just recalled that the anti-principle is sometimes stated as, "Absolute infinity is unknowable (in its complete estate)," by virtue of which they suppose what they call a reflection principle, that says any locally specifiable property had by V is had by some set-sized object (below V as a proper class), or even arbitrarily many such objects. A cosmic recipe, yet one in which the cosmos generated is more an infinite mass of questions than answers? Aug 17 at 11:14

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