As discussed in the analogy of the divided line in The Republic (509d–511e): mathematical knowledge does not achieve the height of knowledge about ideas that are given existence by the Good itself, like the Idea of Justice.

Yet how can we know which ideas derive their existence from the Good? Why not also numbers (and geometric figures, etc.)?

Aside from this, why should the source of an idea make such a difference regarding the perfection of knowledge?

What Plato has to explain: agreement about numbers (calculations) is basically unequivocal, while matters like justice vary enormously and seem dependent on culture.

In this context, he speaks of the ignorance of the common people, who won't ever reach the level to know those highest ideas. But how can this seriously be the only argument?

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    It has something to do with morality here. Does anyone fluent in number theory necessarily become a good person? Of course not. But if someone masters the idea of justice certainly she becomes a good person... Jul 26, 2022 at 1:34
  • I was recently reading through A New History of Western Philosophy. If I recall correctly, they asserted therein that though numbers are a non-physical mental abstraction, yet they permit of multiple instances. Thus you can have an equation like "2 + 2" where each quantity "2" is distinct. There are no duplicate Forms. Aug 25, 2022 at 1:08

2 Answers 2


There is a longstanding read-back of neo-Platonist monadism into Plato's own Form of the Good (though see also William Altman, "Why Plato's Idea of the Good Is Not the One"). Now for Plato (either way), the One leads to the Two, the Two to the Three, and so on forever. There is also a Large Form/Form of the Large, which lends itself to the ordering on the Numbers, presumably (though for where Plato speaks of the Good being greater in glory and might than the other Forms, one wonders how It would be so if Its glory and might did not participate in the Form of the Large? whence is It not the Largest Number of all instead?).

Anyway, the great difference between the Forms and the Earthly exemplars is that the Forms are perfect self-participants. Translated: they are otherwise as general to their terms as possible, not saturated with the differentiae that adulterate the essences of the lower exemplars. Auto to dikaiosune is not just on account of the differentiae that would also shape it in other directions. If there were only one possible exemplar of a Form, i.e. that Form Itself, the reason for distinguishing between general and particular truths would disappear. (C.f. why the most jealous of all possible deities is not held to be "divided into" a general essence and a particular expression of that essence.)

But though the Two-in-itself might be construed as a Form of the Two (or of Duality, or whatever), for elsewhere Plato writes of Forms of anything and everything (over any and every descriptive term), just the same, it is the multiplicity that is their essence that makes the Numbers into an ambient source for the possibility of imperfection. If there had been no Numbers, nothing would have been able to differentiate itself from a pure self-exemplar so as to make counting pluralities of exemplars possible. Conceivably even the other Forms would never have been parted at all from that of the Good.

Whether Plato thought strictly as such, or rather recognized such apparent implications of some claims he floats (if not outright makes), I don't know, but I suspect not quite. Still, he has no clear-eyed appreciation for the problem of evil's full scale in his own transtheology (if he was ever even so dogmatic as to think of himself as trying to advance a settled transtheology!), which problem infects his known discourse on Forms, participation, self-exemplification, demiurges and world-souls, and so on. That is, it seems as if Plato probably has to locate the source of evil, as participatory imperfection, in the Forms somehow, yet then the Forms with this unsavory role in reality then get depreciated along the line of knowledge in The Republic (as interfering with perfect knowledge, as united knowledge, by virtue(!) of their multiplicity, which divides the intrinsic Form of Knowledge itself no less?).


Look at this part 511C-D which I take from Elaine Landry https://youtu.be/IqMWVNHg5Gc?t=372 because it has the best translation I can find imo:

Landry gives Glaucon's shocked reply to hearing philosophy is clearer than mathematics:

"I understand, though not adequately - You see, in my opinion, you are speaking of an enormous task, you want to distinguish the part of what is and what is intelligible, the part looked at by the science of dialectical discussion, as clearer than the part looked at by the so-called sciences - those for which hypothesis are first principles ... and although those who look at the latter part are forced to do so by means of thought rather than sense perception, still, because they do not go back to a genuine first principle in consider it, but proceed from hypothesis, you do not think that they have true understanding of them, even though...they are intelligible, and you seem to me to call the state of mind of the geometer - and the others of that sort - thought but not understanding; thought being intermediate between belief and understanding."

Hypotheses are the base level in mathematics, and that is less secure than what actually is.

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