My understanding of Benacerraf's identification problem breaks this problem into two subquestions. There is the format issue (the one that leads to the possible "junk theorems") and then there is the "first number" issue, i.e. whether to start from 0, 1, or maybe (with Brouwer?) 2, or maybe (with Christian philosophers like Cornelius van Til, who thought that the Trinity conceptually resolved the one-over-many problem) 3, or...

Now, in the Tao Te Ching there is a passage that reads:

The Tao gives birth to One. One gives birth to Two. Two gives birth to Three. Three gives birth to all things.

I read an article about Plato's Form of the Good that interpreted this as some kind of Form of Unity, something that seems to make Plato into a neo-Platonist before the fact, so to speak. That there could be a concept of unity that was not reducible to the number 1 was also indicated to me by the following passage from Kant's first Critique:

But the conception of conjunction includes, besides the conception of the manifold and of the synthesis of it, that of the unity of it also. Conjunction is the representation of the synthetical unity of the manifold. This idea of unity, therefore, cannot arise out of that of conjunction; much rather does that idea, by combining itself with the representation of the manifold, render the conception of conjunction possible. This unity, which a priori precedes all conceptions of conjunction, is not the category of unity (SS 6); for all the categories are based upon logical functions of judgement, and in these functions we already have conjunction, and consequently unity of given conceptions. It is therefore evident that the category of unity presupposes conjunction. We must therefore look still higher for this unity (as qualitative, SS 8), in that, namely, which contains the ground of the unity of diverse conceptions in judgements, the ground, consequently, of the possibility of the existence of the understanding, even in regard to its logical use.

Although while writing my essay about "deontic set theory," I had started out with an intro that tried to assimilate Plato to the tradition of philosophers who have tried to ground ethics in mathematics (as the first of these philosophers, even), it occurred to me a little while later that something very much else seems to have been going on: that Plato thought that mathematics could be grounded in ethics instead.

This would be cashed out in terms of saying that Plato and the Lao Tzu persona who wrote the Tao Te Ching, can be thought of as implicitly believing in a specific response to the second part of the identification problem for the natural numbers. The Tao itself as the "first number" for Lao Tzu is often mapped to emptiness and hence the number zero; but Plato is mapped to the number 1 as the "first number"; and yet the way in which the Tao gives birth to the One, or in which the Form of the Good transcreates the other Forms, makes it seem, to me, as if, for Plato and Lao Tzu, neither zero nor the One are the first numbers, but something else is.

Aside from being rather anachronistic (though in my defense, I would say that adapting the Tao to the empty set, or the Form of the Good to abstract monadicity, are already anachronistic interpretive maneuvers), how fair is this interpretation?

Some theoretical serendipity was involved on my end, here: for various reasons I had already foregone having zero be the first natural number in deontic set theory, but I was leery of having 1 play the role instead. I also wanted 1 to still be a set, though. So I axiomatized the following: ∃xy((xy) & ∀z((z ~= x) → (z ~∈ y)) & ∀z((z ~= y) → (x ~∈ z)); in other words, there exists an x that is (a) only an element of some y and (b) the only element of y. Let's call this u (with an eye towards its eventual name in English). To round things out, ∀x(x ~∈ u), i.e. u has no elements. So far, then, u might be a set, but empty. So finally add in ∀x(u ~∋ ~x), so u is not a set of no elements, either, so is not a set at all. Presto, u is an ur-element, and in fact here is called the ur-element, and the set with which it is so essentially correlated, is the number 1. So I had the template for the Tao/Form of the Good as the "first number" in place, and all that remained was to code u for a unique Boolean-styled value in the logic of my system, which was J as the fundamental justification value for the Boolean-style justification logic (as one level of deontic (tran)set theory).

The exciting immediate consequence was an albeit extremely slight resolution of the formatting side of Benacerraf's implementation question. In deontic (tran)set theory, since u is only an element of 1, it is not an element of 2, and hence a formatting of the natural numbers that produced a junk theorem according to which u is an element of 2, is ruled out. So even if we were to commence with von Neumann format, we could not do so until after we had the number 2 in Zermelo format instead. That is, we would have u, {u}, {{u}}, and only then, perhaps, {{u}, {{u}}}. "Sadly," this does mean that 2 is no longer the first ordered pair in the system, but 3 is. Perhaps that fact is grounds for revising the implementation further, i.e. to get a system in which either 2 is still an ordered pair somehow (I don't see how) or 3 also is not the first ordered pair, but maybe 4 is. At any rate, the broader attempt in the system, to address the identification/implementation problem, is to assign a strictly infinitesimal positive justification value to at least the elements of one countably infinite set of Benacerraf-theoretic notation styles (for the natural numbers in set theory), maybe the set of so-called "eventually writable ordinals" or at least some other large countable ordinal series of that "kind." Then deontic (tran)set theory's own proper implementation is supposed to sum over all those justification values. (Having said that those are "strictly" infinitesimal in scale, I mean to have said that I was using Conway's surreal infinitesimals, here. So let's say Zermelo format has n/ω-much justification, von Neumann format has m/ω-much justification, etc. Then "deontic format" has a justification value that is (n + m + ...)/ω. But n + m + ... = ω, after all, so the "deontic format" has a justification value of ω/ω = 1 (surreally), and we are adapting the unit interval's alethological role to the deontic scene and saying that 0 = unjustification and 1 = completely sufficient justification (though we allow x < 0 and > 1, and speak also of antijustification and hyperjustification). I am suspicious of this derivation (it's almost an exact copy of a major part of the derivation of CH in the system), but otherwise it seems like a promising way to paint the deontic picture.)

  • 1
    Amusing symbol for foot note!
    – J D
    Feb 7, 2022 at 15:03
  • 1
    Neither 0 nor 1... I think that's the whole point of the Tao... that 0 and 1 come into existence simultaneously in virtue of each other, and that neither is primary, but rather the conjunction of 0 and 1 or rather the dichotomy between nothing and something as a pair is, to use mathematical language, a generator of numeracy. This is also true in a modern psychological context where subitization is a psychological process that differentiates a small set of objects from the background. Quine's ontological criterion of existential declaration is implicitly a carrier of this idea too.
    – J D
    Feb 7, 2022 at 15:07
  • I do have J (for u) as somehow both a variable and a constant in my theory, and variables are traced back to disjunctive functions (on the sentential or predicate level, it doesn't matter) to a great extent, ultimately (in the imperative prelogic). So having this be the disjunction over 0 and 1 seems quite apropos, too. However, I remain deeply troubled by the implications for the theory of the number grid. Having zero be the origin seems inescapable; but here either u is "on top of" the origin, or absolutely perpendicular to it? Or u as J is the grid itself, in a way? I don't know. Feb 7, 2022 at 15:36
  • From a mathematical point of view there is a first number, i.e. the number with no predecessor. To call it zero, one or pluto is immaterial. But if we start counting from two what do we mean? that there is a number that is "prior" to it? And thus, two is the successor of that number (whatever it is)? If so, two has not the unique property of being the number without predecessor... Feb 8, 2022 at 9:49
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    Having said that, we can "measure" here the difference between mathematics: the field where we can state problems in a precise way and (try to) answer them in an understandable way, from philosophy, where the problems are murky and answers are always debatable :-) Feb 8, 2022 at 9:50

1 Answer 1


Let me tell you of some mathematics relevant to these questions.

In 1999 Pavicic and Megill demonstrated that neither Boolean models for classical propositional logic nor orthomodular models for "quantum logic" (the algebraic form devised by Birkhoff and von Neumann) are faithful to their syntax. On both cases, what stands for propositions must be "conditioned" using a map into a hexagonal order. In 2006, Schechter formulated a hexagon interpretation for classical propositional logic. The faithful model is non-distribitive.

In 1979 Assmus and Salwach described non-isomorphic finite geometries related to 16-sets called (16,6,2) designs. The representable design relates a 4x4 array of cells to the lines of the complete graph on 6 vertices. One cell in the array must be viewed as a "point at infinity" because the graph K_6 has only 15 lines. The representable design is called a Kummer configuration. The Kummer surfaces spoken of in quantum gravity speculation and quantum cryptography all have an exceptional 16-set arranged as a Kummer configuration.

In 1985 Simon initiated the field of topological stereochemistry by proving that molecules comparable to the bipartite graph K_3,3 and the complete graph K_5 have mirror images in 3-space. Both of these graphs are provably non-planar. The K_3,3 graph is a subgraph of K_6.

In the last 10 years, Flapan and others have been studying graphs in 3-space. The K_3,3 graph falls into a class of graphs they call Moebius ladders. In contrast with a normal ladder, a Moebius ladder has only one "siderail". In contrast with a Moebius surface, a Moebius ladder has only rungs connecting two points across from each other on the siderail. Traversing a siderail is like counting the vertices of a 2n regular polygon. Opposite vertices correspond to opposite ends of Moebius ladder rungs.

Kant's criticism of Leibniz' principle of the identity of indiscernibles had been based on attributing numerical difference to external, spatial intuition. Topology is the mathematical discipline concerned with separability.

In 2019 Soltan's work on the Kuratowski 14 set problem had been translated. The actual "closure" to only 14 sets occurs because the alternate application of closure and complementation leads to cycles of 4 sets in either direction (starting with closure versus starting with complementation). What remains are 6 central sets which never repeat.

Quantum mechanics organizes its "particles" in a scheme called "the eightfold way." The representation of isospin leads to the use of arranging related "particles" into hexagonal arrays.

It has been determined in neuroscience that a form of grid cell which has a hexagonal firing pattern is implicated in "concept formation" in so far as such can be controlled in an experimental protocol.

A cube is a polyhedron that can tile 3-space. For any cube with two tetrahedra inscribed into opposite corners, a hyperplane separating the tetrahedra intersects the cube so that a hexagon is formed within the hyperplane.

The vertices of the cube and the hexagon form a 14 set.

There is a class of finite geometric structures called Steiner Quadruple Systems. The SQS on 8 symbols forms 14 blocks.

One can label the vertices of a cube with three abstract shapes of two colors so that it's vertices can be mapped to an 8-vertex de Bruijn graph having 16 edges.

So, one can relate the Kummer configuration to the de Briujn graph edges. The 6-set of hexagon edges vanish.

In contrast, oneness is related to finiteness.

Famously, 0.999... = 1

The proof depends on a subtraction. The situation is easier to understand with binary decimals. With exception for the uniform elements consisting of all zeroes every binary decimal which is eventually constant is paired with a similar one that is also eventually constant with the opposite symbol. Everyone simply claims that these denote the same "real number."

Anyone ever see a "real number"?

The claim is a claim of substitutability relying upon a topological characteristic called compactness. Propositional logic has a compactness theorem. Its models can be compared with infinite strings of '0's and '1's. One understands the "individuation" of an infinite set via compactness over finite subsets. Mathematics which does not assume completed infinities from the outset does not always have decidable equality.

The simplistic understanding of finiteness versus infinitude requires "twoness" if finiteness is to be meaningful.

Physics requires the Pauli exclusion principle to explain "space." The "two" Boolean values have their significance relative to the Stone representation theorem and structures based upon order called ultrafilters.

Kant contrasted the numerical difference of space with the oneness of internal temporality at any given moment. If you look up the 3-dimensional projection of the 16-vertex tesseract, you will see that it is a tetrahedron with its 4 vertices attached to a "point at infinity." With proper labeling, one can index the 4x4 array of the Kummer configuration to have a point at infinity for labeling the lines of the K_6 graph.

The individuality of this point at infinity ought to be understood on modern terms with respect to the principle of indiscernibilty of non-existents from negative free logic. In turn, Kant discusses the individuation of "the sum-total of possible predicates" in CPR in the section called "the pure ideal of reason." It is an individual that cannot fall under a concept.

When "time words" are analyzed, the simple temporal predictions are comparable with modal logics. The modal logic S4 has 14 modalities. The Hasse diagram for these modalities has the same shape as the inclusion relations associated with the Kuratowski 14 set problem.

Full circle.

But, science says there are multiverses and many worlds and quantum weirdness

In 1999 Pavicic and Megill showed that naive logic is delusional.

Moebius ladders are cool, by the way.

  • I appreciate the information, but am I appreciating it correctly, here? I gather that, in terms of your list of facts, either (a) there is no "first number," or (b) 14 is the "first number," or (c) some form of infinity might be the "first number." Are any of these interpretations faithful to your intent? Feb 7, 2022 at 21:37

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