The Atlas

In Pythagorean and Neoplatonic number theory, the One: the source the whole series of numbers comes out of, treated as its origin rather than as one number among the others.

The first pair. In the Neoplatonic scheme the monad cannot generate numbers at all until a dyad has appeared from it — a claim Eustratius of Nicaea used to argue that the Son and the Spirit must be perfectly matched to each other.

The Pythagorean figure of ten dots arranged in four rows — 1 + 2 + 3 + 4 = 10 — taken as the emblem of completeness. It is worth knowing because it cuts against a popular argument: in the tradition that produced it, the completion-numbers are four and ten, not three.

How many blanks a relation has. “Loves” has two — ___ loves ___ — and “gives” has three: ___ gives ___ to ___. Arity counts the places, never the things filling them, and that distinction decides a very old theological argument.

A relation that nothing can bear to itself. Nothing is taller than itself, and on Augustine's account nothing begets itself. Irreflexivity is the one property that forces a relation's places to be filled by different individuals — which is why it, and not the number of places, turns out to be load-bearing.

Peirce's three-place identity relation: a is identical with a is identical with a. It has three places that cannot be reduced to two, and exactly one individual fills all three — which makes it the standing counterexample to any argument that moves from “the structure has three places” to “there are three of them”.

The Eastern doctrine that the Father alone is unoriginate and is the single source of the other two, neither of whom originates anything. Drawn as a diagram it is a root with direct children and no chains — which fixes how deep the structure can go, and says nothing at all about how wide.

The discipline of saying what God is not, on the ground that no positive description reaches Him. In Eastern theology it is a rule rather than a mood, and it has a consequence people rarely notice: the rule that forbids over-describing God also forbids certain arguments about Him.

The leading realist account of what an argument-place is: a universal has three places exactly when it contains three slots. Its own axioms allow one object to occupy more than one slot at the same time — so the theory built to take places seriously says, as an axiom, that counting places does not count occupants.

A symmetry of a structure: a way of swapping its parts around that leaves every relation in it exactly as it was. A structure with a non-trivial automorphism contains positions that nothing inside the structure can tell apart.

The view in philosophy of mathematics that numbers simply are positions in a structure — there is nothing more to the number 2 than the place it occupies. It is the purest available version of “the slots are the individuals”, and it has been in trouble for twenty-five years over exactly that.

Leibniz's principle that if two things share absolutely every property, they are not two things but one. Max Black's thought experiment of a universe containing nothing but two exactly similar spheres looked like a refutation of it.

Quine's idea that two things can be told apart not by any property either has on its own, but by a relation neither can have to itself. Black's two spheres are two because each is two miles from another sphere and no sphere is two miles from itself.

Richard of St Victor's term for love shared by two people toward a third. His argument is that private self-love is not charity at all, that mutual love between two is real but unshared, and that love only becomes shared when two are joined in loving a third.

Aquinas's name for the ordering of a thing to itself under two different descriptions — which he classifies as not a real relation at all. It is, thirteen centuries early, exactly the case Peirce's teridentity makes formal.

The principle that in God everything is one except where relations stand opposed to one another. It is how Aquinas gets three persons out of four real relations, and it is the formula the Council of Florence used to license the filioque.

The Greek word Christian theology settled on for what English calls a divine “person” — one of the three who are the one God. It is contrasted with ousia, the single nature or essence all three share.

The doctrine that God is not made of parts of any kind — not physical parts, and not a bundle of separable attributes either. It does real work in arguments about the Trinity: if there is no substrate for a mere aspect to sit in, then any real distinction in God has to be someone rather than something.

Peter Geach's proposal that “the same” always means “the same something” — so two things could be the same being without being the same person. It is often reached for as a way of stating the Trinity without contradiction.

The view that Father, Son and Spirit are three modes or presentations of a single person rather than three distinct persons. The early Church rejected it, and it matters here because collapse into modalism is the direction any purely structural argument for the Trinity falls in when it fails.

The clause “and from the Son”, added in the Latin West to the creed's statement that the Holy Spirit proceeds from the Father. Its addition is one of the standing disagreements between the Orthodox East and the Catholic West, and several arguments in this Atlas were built to attack or defend it.

Peirce's claim, since given formal proofs, that three-place relations cannot be built out of one- and two-place ones, while relations with four or more places can always be built out of three-place ones. Three really is a stopping point — for the shapes of relations.

Jukka Keränen's 2001 result that structures usually contain positions nothing inside the structure can distinguish, and that every account of identity available to the structuralist then makes those positions the same thing — so the theory ends up saying that 1 = −1 in the integers under addition.

Peirce's distinction between three-place relations that genuinely mediate — sign, object, interpreter — and ones that only look three-placed. Specialists concede that no one has supplied a test for telling them apart, which is why the distinction cannot be used to disqualify teridentity.

The last great systematic Neoplatonist. His Elements of Theology argues that every effect stays in its cause, comes out from it, and turns back toward it — three stages, and the turning-back closes the circuit so no fourth stage is needed.

Byzantine bishop and commentator on Aristotle, a pupil of John Italos. Against the Latin picture of the Trinity as a line running Father to Son to Spirit, he set an isosceles triangle: the Father at the apex, the Son and the Spirit perfectly level at the two base corners.

Twelfth-century Byzantine bishop who wrote a point-by-point Refutation of Proclus's Elements of Theology. His objection to Eustratius — that the Son and the Spirit are not from the Father in the same way, one begotten and one proceeding — turns out, in modern terms, to be exactly the right repair.

Prior of the abbey of Saint-Victor in Paris and author of De Trinitate. He built the strongest argument anyone has for exactly three divine persons, and he built it out of the nature of love rather than out of counting anything.

Dominican friar and author of the Summa Theologiae. On the question of how many divine persons there are, he is the sharpest witness against slot-counting: he holds that there are four real relations in God and three persons, and explains the mismatch.

American logician, mathematician and founder of pragmatism, who worked out the algebra of relations and argued that three-place relations are irreducible. The same body of work supplies both the best formal support for a triadic argument and its cleanest counterexample.

British philosopher of religion, for many years at Oxford, and a founder of the modern analytic case for theism. His argument that perfect love requires exactly three divine persons is a descendant of Richard of St Victor's, and has been criticised in print for stopping at three without a reason.

Russian émigré theologian in Paris, and the most widely read modern Eastern Orthodox systematician. On the question here he is decisive and unembarrassed: the Trinity, he writes, “can be deduced from no principle nor explained by any sufficient reason, for there are neither principles nor causes anterior to the Trinity”.

The last of the Greek Fathers and author of the first full Eastern dogmatic summary. He supplies the sentence that closes off a whole line of argument: we have learned that generation and procession differ, and the nature of that difference we in no wise understand.

Patriarch of Constantinople and author of the Mystagogy of the Holy Spirit, the founding Eastern case against the filioque. It contains the most on-point patristic discussion of a fourth divine person anywhere — as the absurdity a reductio ends at, which means he treats “there is no fourth” as a premise nobody disputes rather than as something to establish.

North African bishop whose De Trinitate shaped every later Western account of the Trinity. On the question here he supplies a flat axiom and no argument for it: nothing whatever generates its own existence.

The first council to gather bishops from across the Christian world, convened by the emperor Constantine. It produced the creed that states the Son is “of one essence” with the Father — the vocabulary every later argument about the Trinity has to work in.

The council that attempted to reunite the Latin and Greek churches. Its 1442 bull Cantate Domino states the formula that in God everything is one except where relations stand opposed — the principle Aquinas had used, now made conciliar.

The late-antique development of Plato begun by Plotinus, on which everything flows out from a single ultimate source and returns to it. Its vocabulary — monad, procession, reversion — was absorbed by Christian theology and still shapes how the Trinity gets argued about.

Christian theology done with the tools and standards of analytic philosophy: explicit premises, formal argument, and the expectation that a doctrine can be stated precisely enough to be tested. The name comes from a 2009 collection of essays; the practice is older.

An ancient handbook of the religious meaning of the numbers one through ten, wrongly attributed to Iamblichus. It is the number theory Eustratius of Nicaea drew on — and it treats four and ten, not three, as the numbers where things are complete.