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He is not a participant in his own dialogue.
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He is the author, but not the participant.
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We don't know precisely what Plato thought, but we are helped,
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at least, by a kind of intellectual autobiography
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that he wrote and that we still have,
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in what is conventionally referred to as The Seventh Letter.
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Plato wrote a series of letters that we have.
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People have argued over the authenticity of them,
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although I think by now it is established that they are his.
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In the most famous of these letters, the lengthy seventh one,
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he gives us, again, something of an autobiography and
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tells us a little bit about why he came to write this book.