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I had already independently come to the conclusions in the introduction, so the rest of the paper served as an exploration of an alternative path. One thing th
by myWindoonn 8y ago
I had already independently come to the conclusions in the introduction, so the rest of the paper served as an exploration of an alternative path.
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 [0])
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.
[0] https://www.youtube.com/watch?v=IMGoluu1mgc https://www.youtube.com/watch?v=IMGoluu1mgc
- justinpombrio 8y ago> 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. There are a couple foundations for mathematics. The traditional one is set theory[1], and there are some newfangled ones based on type theory[2]. [1] https://en.wikipedia.org/wiki/Set_theory https://en.wikipedia.org/wiki/Set_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" [2] https://en.wikipedia.org/wiki/Type_theory https://en.wikipedia.org/wiki/Type_theory "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"
- sewercake 8y agoI'm speaking out of my expertise here, but I think he's referring to Godel's incompleteness theorum/the halting problem, when he says 'no basis'.
- bloaf 8y agoIt could also be the admission that you can't beat pure skepticism. That is, you can't actually beat Descartes' deceiving demon; you need to start any line of reasoning with an assumption that your reasoning is somewhat sound. When you're starting out to build a foundation, you have no basis for making that assumption.
- justinpombrio 8y agoWe're up to three completely different interpretations of "like a house with many doorways but no foundation, where every floor is supported by another ceiling and there is no ultimate truth". My takeaway: be wary of analogies; they will be misunderstood.
- myWindoonn 8y agoYou grokked it. 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?