Saturday, September 15, 2007

Continuity, Unity and Infinity

I’m back in Glasgow wondering, what are continua? Things that extend smoothly, is the obvious answer, but what is extension? And while things presumably extend smoothly if they contain no gaps that contain something else (whence they are unities), what if the space through which they extend is discrete? So what makes intuited space a continuum (so to speak)? Whilst we think of it as extended because it contains many things like our own body (e.g. many users of our own language), so might something more discrete (e.g. a finite matrix). So presumably another criterion is that those parts of a continuum that are not points must all be endlessly subdivisible (whence continua have infinitely many parts). But that is insufficient, as the rational number line is not thereby excluded, from being the shape of space itself (even though it does not contain the diagonal of the unit square)... (To be continued:)

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