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In what "circumstances" are manifolds and homeomorphisms useful?
If you were writing software for something intended to traverse the Interplanetary transfer network then you would probably use charts and atlases and transition functions, and you would study (symplectic) manifolds and homeomorphisms in order to understand those more-applied concepts.
If an otherwise useful theorem assumes that the manifold is orientable, then you need to show that your practical manifold is orientable before you can use it - and if it turns out not to be orientable, then you can't use it at all. If instead you had an analogous theorem that applied to all manifolds, then you could use it immediately.
The last sentence is not obvious at all. The goal of mathematics is not to be correct a lot. The goal of mathematics is to promote human understanding. Strong axioms help with that by simplifying reasoning.
If you assume A and derive B you have not proven B but rather A implies B. If you can instead assume a weaker axiom Aprime, and still derive B, then you have proven Aprime implies B, which is stronger because it will be applicable in more circumstances.
Your view is not mainstream.
I agree with this statement - and yet you did not contradict my statement that second order logic is also not part of mainstream mathematics.
A topologist might care about manifolds or homeomorphisms - they do not care about foundations of mathematics - and it is not the case that only one foundation of mathematics can support topology. The weaker foundation is preferable.
EY is talking from a position of faith that infinite model theory and second-order logic are good and reasonable things.
I think this is a fallacy of gray. Mathematicians have been using infinite model theory and second-order logic for a while, now; this is strong evidence that they are good and reasonable.
Edit: Link formatting, sorry. I wish there was a way to preview comments before submitting....
Second-order logic is not part of standard, mainstream mathematics. It is part of a field that you might call mathematical logic or "foundations of mathematics". Foundations of a building are relevant to the strength of a building, so the name implies that foundations of mathematics are relevant to the strength of mainstream mathematics. A more accurate analogy would be the relationship between physics and philosophy of physics - discoveries in epistemology and philosophy of science are more often driven by physics than the other way around, and the field "philosophy of physics" is a backwater by comparison.
As is probably evident, I think the good, solid mathematical logic is intuitionist and constructive and higher-order and based on proof theory first and model theory only second. It is easy to analogize from their names to a straight line between first-order, second-order, and higher-order logic, but in fact they're not in a straight line at all. First-order logic is mainstream mathematics, second-order logic is mathematical logic flavored with faith in the reality of infinite models and set theory, and higher-order logic is mathematical logic that is (usually) constructive and proof-theoretic and built with an awareness of computer science.
In response to
Logical Pinpointing
EY is talking from a position of faith that infinite model theory and second-order logic are good and reasonable things.
It is possible to instead start from a position of doubt that the infinite model theory and second order logic are good and reasonable things (based on my memory of having studied in college whether model theory and second order logic can be formalized within Zermelo-Frankel set theory, and what the first-order-ness of Zermelo-Frankel has to do with it.).
We might be fine with a proof-theoretic approach, which starts with the same ideas "zero is a number", "the successor of a number is a different number", but then goes to a proof-theoretic rule of induction something like "I'd be happy to say 'All numbers have such-and-such property' if there were a proof that zero has that property and another also proof that if a number has that property, then its successor also has that property."
We don't need to talk about models at all - in particular we don't need to talk about infinite models.
Second-order arithmetic is sufficient to get what EY wants (a nice pretty model universe) but I have two objections. First it is too strong - often the first sufficient hammer that you find in mathematics is rarely the one you should end up using. Second, the goal of a nice pretty model universe presumes a stance of faith in (infinite) model theory, but the infinite model theory is not formalized. If you do formalize it then your formalization will have alternative "undesired" interpretations (by Lowenheim-Skolem).
I remember reading that article, and not being impressed. He lumps all the sexist talking points into one essay, and therefore it ends up looking like one big strawperson. He may have good points, but unfortunately his own essay undermines them.
My understanding is that the essay's effect is via the horror a reader feels at the alternate-world presented in the essay. It opens the reader's eyes somewhat to the degree that sexism is embedded in everyday grammar and idiom. My understanding is that it is not a persuasive essay in the usual sense.
Please elaborate.
There's nothing about the tables that was not explained in the previous installment of this series; click the links if you're still confused. I came to this knowing nothing about that type of notation, but the tables told me even more than the bubble diagrams--and here's the secret. Looking at the table tells you next to nothing. It's only when you think about the situations that the probabilities quantify, then they make sense. Although, as an additional step, he could have explained each of the situations in sentence form in a paragraph, but probably felt the table spoke for itself.
The second table, for instance, (if I am interpreting correctly) can be paraphrased as:
I believe that my partner loves me, and that the universe knows it, and I can get this answer from the universe. I would also know that if my partner didn't love me, because the universe would know it and I would hear that. It's probably one of those two. Of course it could be that I don't hear the universe, or the universe is lying to me, or that the universe doesn't magically pick up our thoughts (how unromantic!), but I really don't believe that to be true, I only admit that it's possible. I am rational, after all.
I agree that if you don't look at the numbers, but at the surrounding text, you get the sense that the numbers could be paraphrased in that way.
So does h, labeled "I hear universe" mean "I hear the universe tell me something at all", or "I hear the universe tell me that they love me" or "I hear the universe tell me what it knows, which (tacitly according to the meaning of knows) is accurate"?
I thought it meant "I have a sensation as if the universe were telling me that they love me", but the highest probability scenarios are p&u&h, and -p&u&h, which would suggest that regardless of their love, I'm likely to experience a sensation as if the universe were telling me that they love me. That seems reasonable from a skeptical viewpoint, but not from a believer's viewpoint.
In response to
Causal Diagrams and Causal Models
I think it would be valuable if someone pointed out that a third party watching, without controlling, a scientist's controlled study is in pretty much the same situation as the three-column exercise/weight/internet use situation - they have instead exercise/weight/control group.
This "observe the results of a scientist's controlled study" thought experiment motivates and provides hope that one can sometimes derive causation from observation, where the current story arc makes a sortof magical leap.
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There's a difference between assuming that a manifold is orientable and assuming something about set theory. The phase space is, of course, only approximately a manifold. On a very small level it's - well, something we're not very sure of. But all the math you'll be doing is an approximation of reality.
So some big macroscopic feature like orientability would be a problem to assume. Orientability corresponds to something in physical reality, and something that clearly matters for your calculation.
The axiom of choice or whatever set-theoretic assumption corresponds to nothing in physical reality. It doesn't matter if the theorems you are using are right for the situation, because they are obviously all wrong, because they are about symplectic dynamics on a manifold, and physics isn't actually symplectic dynamics on a manifold! The only thing that matters is how easily you can find a good-enough approximation to reality. More foundational assumptions make this easier, and do not impede one's approximation of reality.
Note that physicists frequently make arguments that are just plain unambiguously wrong from a mathematical perspective.
I understand your point - it's akin to the Box quote "all models are wrong but some are useful" - when choosing among (false) models, choose the most useful one. However, it is not the case that stronger assumptions are more useful - of course stronger assumptions make the task of proving easier, but the task as a whole includes both proving and also building a system based on the theorems proven.
My primary point is that EY is implying that second-order logic is necessary to work with the integers. People work with the integers without using second-order logic all the time. If he said that he is only introducing second-order logic for convenience in proving and there are certainly other ways of doing it, and that some people (intuitionists and finitists) think that introducing second-order logic is a dubious move, I'd be happy.
My other claim that second-order logic is unphysical and that its unphysicality probably does ripple out to make practical tasks more difficult, is a secondary one. I'm happy to agree that this secondary claim is not mainstream.