Augenstein's exploration in Links between physics and set theory mentions Ulam relating complexity and regularity:

There are several sources for appreciating Ulam’s ideas and interests. A collection of his papers in Beyer et al. [80]... discusses the issue of whether one might expect meaningful undecidable statements in physics (Ulam’s answer, yes), and the notion that if there are physical structures which increase in complexity indefinitely (see the earlier comment re Hertz), the set-theory axiom of regularity would not hold. This phenomenon has been rediscovered several times; see Scheibe [57].

What kind of complexity is he talking about?

The earlier comment re Hertz is "a general observation" made by Ulam:

The behavior of a truly infinite system is not necessarily obtainable as a limit of the behavior of a finite approximating configuration.

A subsequent quotation summarizes "Ulam's central concern" in his own words as follows:

Is there a true infinity of structures going down into smaller and smaller dimensions? It is not a precise problem, or recognized as such. In physics there has always been an atomistic or a field point of view. If there is a field, then points are mathematical points and they are all the same. But another possibility is a very strange structure of successive stages, each stage different. The topology or the scene on which they exist, that is, space and time themselves, need not be the uniform, smooth Euclidean topology. The miracle is that physics would not be possible if protons and electrons were not very much the same. If this similarity or identity of subsets of the universe did not exist, there would be no physics. It may be that in reality for phenomena in the small and involving high energy, there may be an underlying true infinity that does not allow for similarities. It may be that at the present stage of evolution of the universe a sufficient number of identical situations has not yet been produced. If this is so, then physics will become fundamentally more complicated.

Motivation: section 5 of the SEP article on teleological arguments for God's existence reads:

That question is: why do design arguments remain so durable if empirical evidence is inferentially ambiguous, the arguments logically controversial, and the conclusions vociferously disputed? One possibility is that they really are better arguments than most philosophical critics concede. Another possibility is that design intuitions do not rest upon inferences at all.

Now, take Aristotelian final causality. Modulo the temporal order, this can arguably be reformulated as backwards causation, and it has sometimes seemed to me that an intent/plan might be thought of as a projection of the mind, into the future, to cause the actions that realize a plan. Or, better, some part of our minds is already in the future, and is backwards-causing our bodies, in the past, to act towards our goals. I know that will strike some readers as foolish, magical, etc. but I won't be responsive to such remarks if they were made, here. My point is that an infinite backwards series of causes would not appear to be well-founded, i.e. would conflict with the axiom of regularity. Yet then if there is some level of physical complexity that would conflict with the same axiom, might this pertain to where design intuitions come from, ultimately? We would see a certain complex and think (or feel), "It's too complex to be unintentional," and perhaps what is happening is that we are intuitively/subconsciously applying a concept of final causality modulo a parafounded temporal regress.

I'm not saying this for or against design intuitions, but if anything, I am trying to work out a version of an intelligent-design conjecture that would in some manner not come across as "unscientific" (or whatever along that line). Although I am wary of trying to defend the relative justifiability of theological statements, I'm still committed to Rawls' ideal of overlapping consensus enough to prefer looking for common ground with ID theory (rather than just complain about ID theorists) so far as it might prove available. At the least, the above picture could allow us to intellectually sympathize with ID theorists on a general/abstract level, even if we ourselves don't have design intuitions (I myself don't know that I have such intuitions almost ever, but then I'm not even so confident that I have a stable concept of intentions/plans anyway).

But again, all that is contingent on what Augenstein/Ulam is talking about when bringing up physical complexity. So for now, that's my question: what is the kind and degree of complexity to which they are referring? I might try finding Augenstein's own source per his citation, of course, but I daresay that I would approach it with some dread (Ulam came up with measurable cardinals, after all, and even after years of earnest, even obsessive, large-cardinal studies, I am still mostly at a loss as to the internal characterization of those things).

  • Did you follow the Scheibe reference? I presume to Erhard Scheibe's work..
    – CriglCragl
    Aug 6, 2023 at 17:49
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    @CriglCragl I'll see if I can find an open-source version of it. Aug 6, 2023 at 18:25
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    Scheibe mentions "complexity" in a single place, when he quotes Bridgman on quantum theory:"The mathematical structure... has an infinitely greater complexity than the physical structure with which it deals. In our elementary and classical theories we have become used to discarding perhaps one-half of the results of the mathematics... But here we retain only an infinitesimal part of the mathematical results, and except for a few isolated singular points relegate the entire mathematical structure to a ghostly domain with no physical relevance." It seems they are using it purely colloquially.
    – Conifold
    Aug 6, 2023 at 21:00
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    I could not access Beyer's collection, but Ulam's quote I added to the post, presumably, gives the gist. At smaller and smaller scales we may encounter "subsets" without "similarity or identity" to those at larger scales, and there is a "true infinity" of dissimilar scales. So familiar physical sets are not "well-founded" (no atomism) and physics becomes "fundamentally more complicated". Unlike your temporal/causal, Ulam has spatial non-"well-foundedness" in mind. The use of "subsets" and "well-foundedness" seems to be metaphorical, and "complexity" is in the infinity of dissimilar primitives.
    – Conifold
    Aug 6, 2023 at 22:31
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    Just here to lob my favorite numberphile quote, “randomness is patterns on patterns on patterns…”. Seems to present similar ideas without being so darn domain specific
    – J Kusin
    Aug 7, 2023 at 3:21


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