This bridge between mathematics and physics can be well explained in the thought of Thomas Aquinas. Mathematical objects have a a foundation in extramental reality, but their notions are completed by an act of the mind. Physical objects have an immediate foundation in reality (e.g. 'stone' in this stone), and mathematical objects have a remote foundation in reality (e.g. 'line' in a 'stone' of this stone). Mathematical objects are "mental elaborations remotely based on real quantity but proximately on the mind's constructive activity" (Maurer, 55).
See Armand Maurer, "Thomists and Thomas Aquinas on the Foundation of Mathematics," Review of Metaphysics 47 (1993): 43-61.
One thing that really pops out to me is that there's no inherently rational reason to choose rationality, and this lack-of-reason suffers infinite ingress. This is something people often quip about, but not necessarily good or bad. It means, rather, that one must choose to climb up from a position of non-rationality into a position of rationality, and from there a self-supporting rationale materializes. In a similar way, mathematics is like a house with many doorways but no foundation, where every floor is supported by another ceiling and there is no ultimate truth.
In the extremely-precise refutation of anticipated objections (what Hackernews would call "replying to strawmen and steelmen") the author doesn't make a summary refutation: Any statement about the physical world can either be operationalized into a logically-precise (and thus mathematically-precise) statement, or is meaningless word salad designed to attempt an end-run around any kind of logical or structural analysis. This might be an extreme position, but I see that most typical statements about reality are actually of the former case, and people are just lazy in their precision most of the time. But we can still deduce things logically from what people say, can't we!
The author doesn't know about probability sheaves and other relatively-recent advantages in foundational probability. (An interesting intro is )
The author doesn't know about, or disregards, the de Broglie/Bohm QM model, which is an interesting and IMO much less-philosophically-loaded model than Copenhagen. They do mention that Everett also gets around Copenhagen's problems, but Everett also comes with baggage (although HN seems to really like many-worlds theories, so maybe this is okay!)
I greatly appreciate that Chapter 4 continues the grand philosophical tradition of, having achieved some sort of rational basis, immediately applying it to reach silly conclusions.
There are a couple foundations for mathematics. The traditional one is set theory, and there are some newfangled ones based on type theory.
"Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice"
"a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics"
Imagine standing on a vast desert. You have, floating above you, the HYPERCUBE. You could choose to climb up into the HYPERCUBE, but it has no exits.
In a serious, non-trivial sense, mathematics is the pattern science. If you choose to believe that reality has underlying patterns, then mathematics is unavoidable.
To respond to the grandparent, we can put sets on categories, types on categories, categories on sets, logic on sets, logic on categories, etc. Who cares which foundations you pick, when they all can build from each other?
Not to be nitpicky, but why? I feel like this sentence was generated by a Markov chain. "Hologically," I couldn't find a definition anywhere. I presume he means wholly?
There seems to be a huge asymmetry between our universe or reality or whatever you want to call it on the one hand and mathematics on the other hand; the set - or is it a class? - of all mathematical structures seems to be enormously richer than reality. On the one hand it seems quite unproblematic to think that there is a set of mathematical structures that are useful to describe reality, on the other hand it seems quite problematic that all mathematical structures are embodied in reality.
In what sense does a 79 dimensional vector space over GF(43) exist? Sure, you can take a huge pile of stones, arrange them in groups forming sets of stones, combine those sets to build natural numbers, and work your way through layers upon layers of abstraction to end up with an arrangement of stones that, under the right interpretation, represents whatever mathematical structure you had in mind. But that seems a pretty wired way to justify that a 79 dimensional vector space over GF(43) really exists, I would be much more inclined to say that you managed to emulate it with what reality offers you.
I also would bet that there are mathematical structures that are impossible to emulate in this sense in our universe. And if the position is that there is another universe separate from our universe which is a 79 dimensional vector space over GF(43) or which allows emulating it in a more meaningful way than can be done in our universe, that seems not very appealing either. You are postulating a rather strange collection of universes without any obvious benefit. And it seems like you also get all the problems of Platonism - how do we know about and work with this vector space if it is in a separate universe.
As a lesser point I find the idea repelling because it seems incredibly convoluted, at least in the way presented here. For a - at least at first glimpse - simple idea like all mathematical objects exist, there are a lot of technicalities and complications just to state what you really mean without becoming inconsistent and failing right from the start. So yes, I am really unable to see any merits here. I probably also did not do a good job explaining my position, I don't like this comment at all. But I hope anyone reading this will at least be able to get an idea of what I mean and want to say even if the comment is rather rough and not without flaws.