Or, Infinite Smoothness: From The Discrete Complaints of Individuality to The Ironic Continuum of Small Rules
1. I think it's in book 2 or 3 of The Republic that Plato makes a rhetorical analogy that is engrained in the way we argue and complain about ourselves and our relations to society. He asks to transfer the argument about whether a person's happiness depends on power and money rather than virtue (that is, knowledge of the good) to a much larger body that can be dissected objectively, namely, the body of society. The move from examining the personal body and its object of happiness (what we might now call psychology) to the social body and its aim of good government, may seem quite weak, for how can we trust in an answer to a problem which is at a level different from the problem itself. The problem is more particular and local than the solution which is general and indeed, allegedly universal.
2. This move has bothered me for 42 years. Now, I think I have an answer to the quandary both at the substantive and procedural levels. And both technically can be viewed from what we call a category theory. We can also immediately understand Plato's overall strategy as a movement from a local composition in algebra to a general conception of geometry. (There is an apocryphal saying that the signage above Plato's Academy said something to the effect, "Let no one without Geometry enter [this space]."
3. I would also temper this interpretation with a couple caveats. First, Plato though genius of the first degree did not have the devices necessary to found category theory. Second, when I call out the term of art, category theory, I mean in the first and foremost instance, the conceptualisation of certain data about objects, arrows (morphisms), the associativity axiom [(a*b)*c = a*(b*c)], and the identity axiom [a*I = a = 1*a]. In its most austere form and substance, namely, in diagrams or sketches ("esquisses") very very complex structures that would not be very comprehensible in a grammatical English are very evidently laid out like Dr. Seuss drawings. One can for example re-wire quantum mechanical expressions rather naturally in category theory and understand how so-complex financial contracts work in an infinitely contingent world.
4. Coming back to the Platonic move from small (individual) to big (society), we recall Socrates' reason for this move was because he was trapped. The two nephews of Plato, x and y, [i can't remember their names but will look them up later--maybe Glaucon and Adimentaus?] had argued so eloquently and convincingly that at the PERSONAL level only power and money counted for anything in a real person's real happiness. This was an argument based on personal observation ("what is evidently and evidentially true"). So, Platonic philosophy based whole heartedly on one's personal knowledge and education could hardly win against someone whose argument was based on personally observed facts.
5. Socrates had to go a level up. Plato through Socrates was making an argument about the SPACE around the individual, the interconnections, the networks, the various specialised objects ("specialists are better for the good of the whole society than the inefficient workings of generalists"), and how ultimately, this led to an analysis of governance, from benevolent dictator to timocracy, oligarchy, democracy and tyrant. All of this was rounded by the question asked of Socrates, "And what if none of what you have imagined comes about?" Socrates' answer again is of a SPACE, "Well then, we have not wasted time, because forever and a day, people will now know where they should be heading." In other words, Socrates had literally given future generations a map to travel, with signposts and monuments. Now, just as no one on the tube in London can say that the tube maps are what the tubes and tube lines are, we can't say that Plato's Republic describes exactly what is front of us. Much depends on our understanding of how to make the correspondences between the map and our own presence or location. We use the map as a guide to help make sure we steer ourselves to a particular destination.
6. The more general meaning and thus, philosophical import of the move from individual psyche to the social space is in the form of generalisation. Note that the rules defining the linkages in the local space, i.e., the relations between individuals, which in a strict sense might be defined as "rules" establish a structure which is the definition of the entire space in which any and all individuals exist. This movement from particular individual mind to the structures which link us together may appear to be a type of algebraic expression of composition, that is, combinations of bilateral transactions. But the point is that these groups defined by rules when applied generally literally create a geometry of social space. It is a space that is topologically continuous.
7. As we say in the old lingo, "from the many to the one." We can now understand this as a movement from the particular algebraic rules that appear discrete in form but are actually, when taken together, define the infinite smoothness of the skin of reality. The slogan here is, "small rules define big space."
8. As a meditation one might take the slogan to its limit in breath, balance and humour.
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