Materialism and Cantor’s higher infinities

Let’s take a short break from all things economic.

Previously we introduced the idea of material constraints that any kind of economic formation must face. By “material constraints” we intend to point to structures of reality that cannot be easily altered and yet profoundly affect what is, and what is not, possible. Those structures may be initially hidden to us, and require extensive scientific work to uncover. That work often involves highly abstract thought that deliberately assumes a counterfactual world quite different to the empirical world we (currently) find ourselves in.

For example, the economist Luigi Pasinetti often builds economic theories in which specific economic institutions, such as markets and rules of distribution, are entirely absent (see reference in last post). Pasinetti identifies properties of economic production that are invariant across all kinds of economies (and therefore invariant across transitions in modes of production). In a sense, he’s identified a deeper structure of economic reality, which initially we may not have been aware of, or perhaps conflated with properties of the capitalist economy we actually live in.

So we shouldn’t confuse the materialist, or scientific, attitude with crass empiricism (that simply takes empirical reality as a set of given facts, and dismisses talk of reality of deep and hidden structure in reality). Materialism of course includes Samuel Johnson kicking the stone, but there’s more to it than that. Sometimes we need to creatively imagine new hidden mechanisms before we start sifting empirical phenomena for evidence.

The materialist attitude, therefore, includes an inductive moment, where we creatively introduce new ideas to a problem domain, and where those ideas are not fully justified by the empirical data. But the materialist attitude also has a sceptical and pragmatic side. We expect scientific theories to matter, in the sense of making some kind of practical difference in our lives, some kind of material consequence.

So let’s examine a particularly famous, and difficult, case for the materalist attitude: Cantor’s higher infinities.

In 1874 the mathematician Georg Cantor published a paper that claimed to prove the existence of an infinite hierarchy of infinities, each more vast than the infinity before it, stretching out forever like some vast alien landscape. Cantor had achieved the seemingly impossible feat of counting beyond infinity.

Of course, if true, this is astonishing intellectual achievement.

However, the existence of higher infinities is, on the face of it, absurd, for one very simple reason: infinity, by definition, is bigger than anything, and therefore there cannot be anything bigger than it. And we cannot practically count up to infinity. So what’s the use of postulating abstract structures that we cannot possibly construct?

Cantor’s reasoning is highly abstract. Can we trust it? And, even if we can, does it matter? Does it make any practical difference to our lives? Are there material consequences? Perhaps Cantor’s higher infinities identify deep, hidden structures that we’ve yet to fully interact with or┬ánotice in our empirical stream of experience. Or perhaps Cantor’s higher infinities are like the medieval proofs for the existence of God? We might grant that Cantor’s argument has some kind of logical necessity, but the premises just don’t connect to the reality we actually live in.

Here’s a 30 minutes talk I gave in Oxford UK, in November 2016, that deals with this subject. I refer to a handout in the talk, which I reproduce below.

Audio of talk on materialism and Cantor’s higher infinities

The fun part: you too will be able to count beyond infinity if you follow the audio and keep the handout in front of you. One of the surprising aspects of Cantor’s proof of higher infinities is its elementary nature. With just a small amount of effort, anyone can understand it.

In the talk I give my (personal) conclusion on the status of higher infinities. I point out that, surprisingly, Cantor’s theory does entail some empirical predictions. But currently there’s no evidence to support those predictions (and quite a lot of evidence that suggests material reality prevents them).

But reality has a way of surprising us. We shouldn’t dismiss the creative moment. So the jury is still out. The materialist attitude also includes the cheerful acceptance of ignorance. Sometimes we simply don’t yet know!

Also, Cantor’s ideas raise questions about what rules of logical reasoning we are willing to accept. So they raise very profound and foundational issues about the identity of thought and being.


 

(For those interested in materialist constraints on computation (which I touch upon in the latter part of the talk) I can recommend:

Computation and its Limits by Cockshott, Mackenzie and Michaelson.

which gives a broad overview of this area. Disclaimer: Cockshott and Michaelson are co-authors on Classical Econophysics.)

 

infinityhandout-0

infinityhandout-1

 

 

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