The Owl of Minerva Tweets at Dusk

It seems that I never feel so old as when I try to use twitter. I’ll be turning thirty-four on Tuesday, but returning to twitter after the better part of a year makes me feel like a man out of time, as if I’d gone to sleep and woken up in another decade. It seems that the twitter client I have on my phone won’t handle either the new expanded character limit, or the new threading mechanism, and storify is apparently no more.

So much for micro-blogging then.

Here’s tonights stream of twitter thoughts, compiled into a reasonably coherent sequence. It provides a glimpse of the bigger picture work I’ve been doing in philosophy of logic and mathematics over the last few years, which finally seems to be coalescing to the point at which I can be a bit aphoristic about it. I’ve taken the liberty of inserting a few links to make explicit what I’m referring to.

Time for a couple more philosophical thoughts.

Here’s one inspired by Fabio Gironi: the error of classical Platonism is conceiving the ideal/real difference from the side of the real.

Consider Ladyman and Ross’s ontic structural realism (OSR). It can’t articulate the difference between mathematical and physical structure.

This is essentially isomorphic to the old platonic problem of the relation between universals and particulars. In fact, it compresses it.

The problem for earlier platonists was the tendency to treat universals as analogous to particulars. Hence the worry about ‘where’ they are.

But stripping away the surplus universals (the forms of men, dogs, mud, etc.) and restricting to the mathematical focuses the problem.

So, instead of asking ‘How are the mathematical and the physical related, physically?’ we might ask ‘How are they related, mathematically?’

And this, it turns out, is a much more interesting question.

From this perspective, we can begin to see how traditional ways of conceiving this opposition reflect its intrinsic logical duality.

We might see that being/becoming and intelligible/sensible align with algebra/coalgebra, proof/refutation, even construction/interaction.

A series of dualities that become suspiciously attached to that between intuitionistic/co-intuitionistic logic.

But equally, we might begin to see how these traditional conceptions get things exactly wrong. Take Neoplatonism, for instance.

The opposition between one and many, or unity/multiplicity, is another cardinal theme in presocratic thought that is filtered through Plato.

It is aligned with being/becoming and intelligible/sensible by Plato, but not too tightly. Plotinus though, identifies them in extremis.

Here’s the thing: there are many mathematical objects. Quite famously in fact. The fuckers keep multiplying.

Equally, physical things have a tendency to interact, entwine, and compose all these interesting higher order systems.

So, we might want to say that unity/multiplicity is inflected differently across the mathematical/physical divide.

However, we really shouldn’t treat the identity/distinctness relation on one side as a model for the other. This is what Plotinus does.

The One, and, following Augustine, God, becomes the only true unity, of which all else is reflection, to greater or lesser degrees.

The barrier between the two sides of reality collapses, or rather, is pushed down to the level of pure matter, or brute difference.

But notice how this is not like Anaximander’s apeiron. Not a material unity from which individuals must be cut as if from cloth.

That’s all pretty fucking gnomic, I grant you.

Here’s the logic: in the last instance, we need reasons why, e.g., numbers are identical, but reasons why processes are distinct.

It’s when the logical duality is raised to the level of predicate calculus that the real structure becomes apparent: identity differs.

For anyone still following the train of thought, notice this: the intrinsic logic of algebra is equational, that of coalgebra is modal.

Why? Because identity is understood either in terms of isomorphism, or in terms of bisimilarity, suitably packaged in either case.

The packages are type systems, and the logical duality is expressed by the Leibniz laws, and their asymmetry in each case.

Post-Kripkean analytic metaphysics is mostly the construction of ever more elaborate model-theoretic epicycles to ignore this logical fact.

Lewis basically had it with counterpart relations: algebraic identity can’t cope with modal variation. Didn’t stop him from trying though.

The moral of the story: exclusively teaching FOL and set theory produces a weird sort of Dunning-Krueger effect.

It let’s you talk about mathematical and empirical objects as if they were all just objects. This is handy, but it’s a syntactic device.

This is why analytic semantics tend either to work well for mathematical or empirical language, but never quite both.

You can even trace this to the conflict between Frege’s two most famous papers: ‘The Thought’, and ‘On Sense and Reference’.

What do you want: truth functions and extensions, or identity across variation? Pick one.

This makes more sense in topos theory: subobject classifiers equate predicates as truth functions with predicates as extensions.

But what this means is you’re stuck in a context whose logic is intrinsically intuitionistic. It’s Boolean at best (or, in fact, worst).

The distinction between sense and reference, and thus de re/de dicto behaviour, are highly restricted here (to intension/extension).

How do we find our way out of this Fregean bind? Hint: take the path Russell foreclosed to us. Abandon foundation. Seek regresses and loops.

If this is still too gnomic: check out non-well founded set theory, or hypersets. Take your first step down the coalgebraic rabbit hole.

Maybe check out this excellent paper by David Corfield: ‘Understanding the infinite II: Coalgebra‘

Interestingly fact for Badiouians: he defines Events in B&E using self-membership. They are trans-being because that’s impossible in ZF+FA.

That’s no longer impossible in non-well founded set theory (e.g., Aczel’s ZF+AFA). Now isn’t that interesting?

It’s almost as if he appealed to some ineffable outside by stipulating something outside the formalism, not realising it could be encompassed within it.

Right, time to see if I can retrospectively thread these tweets now I’m at a laptop. The owl of Minerva tweets at dusk, I suppose.

That’s enough for now.

If you’re wondering how I’m doing, the answer is ‘surprisingly well’. I’m still struggling with chronic pain, but it’s a lot more manageable than it was, and I’ll actually be doing some lecturing this year in the philosophy department at Newcastle (now ranked 6th in the country by the telegraph, apparently). That’s, what, eight years from doctorate to lecture theatre? Could be worse I suppose.