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I'm reading "The Routledge Guidebook to Plato's Republic" (3rd edition) and Plato's "Republic".

In the Guidebook (chapter 7, pg 158, the section on Knowledge and Opinion), the 12th fundamental premise in the Republic's argument is:

  1. Every level of understanding requires a corresponding level of reality in the object of understanding.

Now, in the guidebook on pg. 160 (chapter 7)

According to (12) it might seem that the distribution of prime numbers can only be a matter of knowledge, never of opinion. Obviously that is false.

Is the distribution of prime numbers not fully understood and is that the reason why the quote says that one can only have an opinion? The guidebook seems to say that a mathematician cannot have this knowledge. Why?

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  • The asymptotic distribution of prime numbers is given by the Prime Number Theorem. A proof of the Riemann Hypothesis would complete the picture and the resulting theory would qualify as knowledge by any reasonable mathematical standard.
    – nwr
    May 12, 2019 at 16:52
  • @NickR: I don't think you can "complete the picture" for something as open-ended as the distribution of primes. Even with Riemann, you still have error terms, they're just a lot smaller than they otherwise would be.
    – Kevin
    May 12, 2019 at 17:11
  • @Kevin You're right! It would be the "best possible" bound on the error.
    – nwr
    May 12, 2019 at 17:24
  • It does not say "one can only have an opinion", it says that there is room for opinion, in addition to some knowledge. Some questions about the distribution of primes may well be undecidable even with "full understanding", like the continuum hypothesis is. Some believe it, others don't.
    – Conifold
    May 12, 2019 at 23:56

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I think the OP misunderstood the guidebook.

The OP said

According to (12) it might seem that the distribution of prime numbers can only be a matter of knowledge, never of opinion. Obviously that is false.

Is the distribution of prime numbers not fully understood and is that the reason why the quote says that one can only have an opinion? The guidebook seems to say that a mathematician cannot have this knowledge. Why?

To declare false the statement: knowledge of primes "can only be a matter of knowledge, never of opinion", implies that it is possible to have an opinion about prime numbers but it does not deny that knowledge of primes is possible.

In the Republic, 506D Plato says "opinions without knowledge are shameful and ugly things" this suggests that you can have an opinion, and that opinion is independent of the knowledge (or lack of) that you have regarding the subject.

This is consistent with the Meno where Socrates says

"The man who does not know has within himself true opinions about the things that he does not know..." (85C)

The questions Socrates posed to the boy about geometry led Socrates to conclude that:

"these opinions have now just been stirred up like a dream, but if he were repeatedly asked these same questions in various ways, you know that in the end his knowledge about these things would be as accurate as anyone's" (85C)

This suggests that one can have a true opinion without knowledge. Again, knowledge and opinion are separate.

Applying this to the Republic and the discussion of prime numbers, it seems that a person can have knowledge of prime numbers but he/she can also have opinions about prime numbers, and those opinions are separate from his/her knowledge. This reading of Plato is consistent with the statement made in the Guidebook.

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