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To understand the title, we should first distinguish between lowercase-p platonism and capital-P Platonism: The former platonism refers to realism as an answer to the problem of universals, the latter Platonism refers to membership in the specific institution which Plato himself founded (the Academy, a school of philosophy at Athens). Plato is ergo a Platonist by definition, having founded the school at Athens. Since this essay deals with the problem of universals, we should also define this matter. The problem of universals was first framed in Plato's Parmenides, with an aporia which has endured for over two millennia. It is probably the oldest unresolved philosophical problem in history. The dialogue takes place during the festival known as the Great Panathenaea, in the deme Keramis of Athens, the site of a (fictional?) meeting between three famous philosophers of three different ages: Socrates (probably ~20), Zeno (~40), and Parmenides (~65). Parmenides was an established Italian philosopher, recorded with prestige in every ancient biography of the pre-Socratic philosophers. Zeno was Parmenides' student at his school in Italy. The Parmenides begins with Zeno reciting one of his many lost philosophical treatises (which remains anonymous in the dialogue). Socrates probes him on an argument advanced therein: an argument which, “maintain[s], in opposition to everything that is commonly said, that things are not many (127e).” This was the characteristic thrust of that Italian school of philosophy: that what exists is one and not many.

Socrates produces a counterargument for Zeno, using a metaphysical principle which Socrates had already invented in prior dialogues like Phaedo¹: The principle of Separation (chorismos) between universals and particulars, that forms exist “themselves by themselves,” (Phaedo 100b, Parmenides 129a; ipsa pro se or auto kath' auto). In fact, there is considerable difficulty in translating the Greek phrase auto kath' auto, since neither the Latin preposition pro nor the English preposition by resemble the exact meaning of the Greek preposition kata (here contracted to kath'). Given the relevant context and grammar, kata can be interpreted as any of the English prepositions “in,” “for,” or “according to”². It is therefore difficult to interpret Socrates' principle of Separation in modern English or in modern logic. So I go for recourse to Dr. Gregory Vlastos (1907-91), certainly the last great scholar of Classical philosophy and Plato to have been born in Constantinople. In 1954, Vlastos' Third man argument in the Parmenides was published, an essay concerning those, “two passages in the Parmenides purporting to prove that the Theory of Forms involves an infinite regress, which came to be dubbed within Plato's lifetime the 'Third Man' Argument (319).” The “Third Man” argument in fact refers to Aristotle's reconstruction of the same one advanced in the Parmenides, with different terms but the same form—a reconstruction which has been lost outside of secondary testimony³. In fact, the original Platonic version of this argument was advanced by Parmenides in response to Socrates' argument against Zeno, and it uses as a premise the same Separation principle which Socrates assumed in his argument. When academics refer to the “Theory of Forms” or especially the “High Theory of Form” or the “Early Theory of Form,” they generally mean a theory of form which adheres to the sort of Separation principle highlighted in the Parmenides.

Now, here is Vlastos' reconstruction of that argument which Parmenides told to Socrates:

(A1) If a number of things, $a,b,c$, are all $F$, there must be a single Form, $F$-ness, in virtue of which we apprehend $a,b,c$, as all $F$.
(A2) If $a,b,c$, and $F$-ness are all $F$, there must be another Form, $F_1$-ness, in virtue of which we apprehend $a,b,c$, and $F$-ness as all $F$.
(Vlastos 320-321)

That is his paraphrase of Parmenides' original argument, but he elaborates on it by expanding (A1) into two smaller premises:

(A3) Any Form can be predicated of itself. Largeness is itself large. $F$-ness is itself $F$.
(A4) If anything has a certain character, it cannot be identical with the Form in virtue of which we apprehend that character. If $x$ is $F$, $x$ cannot be identical with $F$-ness.
(ibid 324-325)

(A3) and (A4) clearly entail (A2), which demands an infinite regress of $F_x$s in order to explain any $F$. This is a basic inconsistency between two principles: Self-predication (A3) and Separation (A4). I hope the modern notation of this argument reveals itself fairly obviously: First, particular $F$s are $F$ in virtue of a form of $F$: $$ \begin{aligned} &\forall x \left(F(x) \to \exists F_\ast \left[ F_\ast \gt_F x \right] \right) \\&\text{where } x \gt_F y \text{ reads: } F(y) \textit{ in virtue of } x \end{aligned} $$ Second, the form of $F$ is itself $F$: $$ \forall F_\ast \left( \left( F_\ast \gt_F x \right) \to F(F_\ast) \right) $$ and finally, no $F$ is $F$ in virtue of itself: $$ \forall F_\ast \left( \exists x \left[ F_\ast \gt_F x \right] \to F_\ast \neq x \right) \text{.} $$ Altogether these three premises yield one relevant conclusion: Each self-predicating form $F$ demands a separate form $F_1$ to ground the former's $F$-ness: $$ F(F_\ast) \to \exists F_\ast^1 \left( F_\ast^1 \gt_F F_\ast \right) $$ Ergo $F_1$ naturally demands an $F_2$ and so on $$ F_\ast^1 \lt_F F_\ast^2 \lt_F F_\ast^3 \lt_F \cdots \text{.} $$ This is the infinite regress which Parmenides spoke to Socrates at Athens, still a valid argument nearly 2,500 years later. The Parmenides (and the TMA in particular) receives so much attention because of its lasting relevance to the problem of universals—the forms mentioned in the TMA are themselves universals! The problem of universals concerns our attitude towards these forms: Dr. Michael J. Loux delineates two such attitudes in the Routledge Encyclopedia of Philosophy, Platonism (or realism) and nominalism⁴. It is often said that nominalists deny the existence of universals, but this is beyond the pale. Instead, nominalists characteristically affirm something which is easily confused with the denial of universals. Dr. Loux writes, ‘Nominalists like Abelard and Ockham insisted that everything that exists is a particular (Nominalism, REP).’ Since universals are contrasted against particulars according to the principle of Separation, it might seem that nominalists deny the existence of universals. But Loux continues, ‘They argued that talk of universals is talk about certain linguistic expressions... and they attempted to provide an account of [the view] that universals are to be identified with them (ibid.).’ Clearly nonexistents can not be identified with anything existing (like ‘linguistic expressions’), so Nominalist can not deny the existence of universals. To remain coherent, Nominalist must deny the principle of Separation, since if universals were separate from their particulars they would be non identical to their particulars; but, as Loux wrote, nominalists identify universals with particular expressions in human language. This is the heart of the problem of universals: Nominalist must deny the principle of Separation in order that he can identify universals with particulars, whereas Platonist must affirm the principle of Separation de facto. The first platonist in recorded history would therefore be Socrates, since he appears to draw on no prior recorded sources when describing the principle of Separation in the dialogues. Who, then, was the first nominalist? I never detailed Socrates' original argument because it was so decisively damaged by Parmenides' several critiques. However, it is unclear whether the principle of Separation is under attack from those critiques. Although the TMA intends to prove that the principles of Separation and Self-Predication are insoluble, nobody is certain whether Parmenides—let alone Plato—conclusively rejects either of those two principles. Parmenides advances several destructive arguments in his eponymous dialogue, most of which end in aporia (aporously, impassably, without any obvious takeaway). Plato does not make Parmenides' personal attitudes toward the axioms explicit, so we can neither prove nor rule out Parmenides as the first nominalist. Although various secondary opinions could be extracted from the Platonists who wrote after Plato's death, we should weigh the fact that, in Plato's own writings, at least one argument can be reconstructed which attacks platonism (the principle of Separation). On the contrary, there are no proofs in his corpus for the principle of Separation. I can decisively conclude neither that Plato was a platonist nor that he was a nominalist, but he did advance the first argument in history which concludes with the characteristic premise of nominalism, viz. the principle of Separation's negation.