An Interesting Challenge to Physicalism, 2022

Standard disclaimer, I'm not an expert, just an amateur who enjoys learning about and sharing this stuff.
Arguing metaphysics with Mark P Xu Neyer. [link]. His most recent response to me looks like it would require me to write about three different essays!  I'll sort of summarize and link to further reading for now. Especially since further conversation might reveal things that I can't anticipate, so I can't answer it all right now, even if I had the time and motivation.
Might be best to go in this order:
# Objective beliefs, and objective values
# Ontology of numbers on physicalism
# What about Max Tegmark, etc.?
== Beliefs and Values ==
(a lot of the following discussion of beliefs and values can be accepted whether or not someone accepts physicalism)(after all, this is the place I'm starting, it makes sense that it should potentially be a kind of common ground we can share)
Of course to choose some belief as “better” requires some evaluation.  That means some values, or desires.   Normally people assume that truth is valuable, but you might ask: "is it really?" But notice what this is really asking: "is it true that truth is valuable?". So you can't really escape it, even by questioning it. See my musings on what is most important (figuring out "what to value" is itself kind of the most valuable thing), as well as the equivalence of figuring out what is true and figuring out what to do. As well, my notes on truth, and whether we can use something other than truth (like, in Mark's comment, entrails or whatever). Putting this together, I think we have at least a bit of a foundation for evaluating differetn beliefs to see which is a "better choice": based on truth.
Questioning our values is one way to avoid "smuggling" them in. But I also think that, to an extent, what we like is an immediately observable fact. Like seeing what color something is with our own eyes. So questioning is only necessary to be sure our "eyes" are not deceiving us (which they sometimes do). We don't have to start from nothing.
There's also plenty of writing by people other than me that is relevant to this subject:
* Richard Carrier's writings:
**Epistemological Endgame digs down to the root of this:
*** "We are thus faced with an ultimate choice: K or ~K? Which principle do I follow? I can try them both out right now, and immediately see that following K leads to correct predictions and the satisfaction of my desires and the fulfillment of my plans, while following ~K does much poorly in all three respects."
** his more recent "The Objective Value Cascade" also seems quite relevant. This time going down to a kind of root of "should we value anything at all?"
** his "Epilogue to the Sam Harris Moral Facts Contest":
*** "Even at the level of choosing not just actions, but values. An “ideal you” would choose a set of values that might differ from the values you have now, and that is a statement of fact about how a machine (your brain) will operate in a given factual condition (being rational and possessed of true beliefs)."
*** "In the most reductive sense, all moral propositions, of the form “you ought to do A,” are predictive hypotheses about how your brain would function in an ideal condition. If your brain was put in a state wherein it made no logical error (no step of reasoning in its computing of what to believe, value, or do was fallacious) and had all requisite information (no relevantly false beliefs, and all relevant true beliefs), then it would in fact do A. And in telling someone they ought to do A, we are really saying they are acting illogically or ignorantly if they don’t; that even they themselves would recommend they do A, if they were more informed and logical."
** his deep dive into the "Argument from Reason" discusses how all of this reasoning stuff can be physical, rather than supernatural. (this might be more relevant later, when discussing "how numbers can be physical").
* Eliezer Yudkowsky's “The simple truth" is one I come back to a lot. It's an allegory for the technology of truth/belief. And it even outlines how it can work as a physical system. It's very enjoyable to read, but the ending of the story always feels a bit mean to me. [oh dear, the character's name is Mark! So in this case it's especially awkward]
** it is very much talking about stuff like Mark's "entrails" comment: "I try several methods: I toss the divination sticks of my tribe; I train my psychic powers to locate sheep through clairvoyance; [...]"
Just in case there is a question about how values are physical: it's easy enough to think of physical cause and effect. Having a particular value can cause you to do something different than if you didn't have that particular value. I don't know if these come down to individual neurons, or structures of them, or what, but some physical explanation seems possible, and quite likely to be found. (beliefs can also be described and explained in a similar way)
* 
== Ontology of numbers on physicalism ==
Personally I find it a bit difficult to see the difficulty of explaining numbers on physicalism. Some of our most well-known physical technologies, such as computers and calculators, seem to handle numbers even better than they handle other things. It's within reach for me or you to learn how these objects physically do what they do. Even at the stage of counting things, we have computers capable of pattern recognition, identifying objects, and counting how many of them there are (see Object Counting in computer vision, for example at this link). It's not always as easy to see what the human brain is up to, but we can both imagine how it might work, and see some neuroscience that looks a lot like what I would have expected. While writing this I use a hand-held measuring tool, like a physical ruler, as an analogy. Then I was going to say we would expect to have something similar in our heads. Then I searched on google for neuroscience of counting, and my idea looks a lot like what this link claims. Here's another link on some neuroscience of counting.
Googling Richard Carrier and the Ontology of numbers will also bring up numerous results:
*How My Philosophy Would Solve the Unsolved Problems
* Plantinga’s ‘Two Dozen or So’ Arguments for God: The Onto-Metaphysical Arguments
* How Can Morals Be Both Invented and True?, also referenced in Eight Philosophical Questions We’ll Never Solve?
*Defining Naturalism
*The Ontology of Logic [also goes over mathematics]
My attempt to define numbers years ago:
* A label selected from a series of unique labels that are in a fixed order, with a beginning.  For a given set of items, each item receives a unique label given out in the same order as the list of labels is defined, beginning with the beginning label.  A set itself can receive a label that is the same as the final label applied to the items in the set. Then you just have to invent names for those labels.  The beginning we call "one", and so on.
== What about Max Tegmark, etc.? ==
I’m not very familiar with Tegmark’s views.  It appears that most of it will work a lot like physicalism.  I’ll trust your perspective to find any unanswered questions.
While Tegmark's views appear very similar to physicalism, the emphasis is on "reality is mathematical" or "reality is a mathematical structure", and I confess I don't know what that means. First step would be to find out what a mathematical structure is. But I confess the statements seem wrong on an entirely different level. They almost appear grammatically incorrect to me. Another thought that comes to mind is that it seems a bit like saying that "reality is scientific", or that "the nature of the universe is that it is a scientific theory". It just doesn't sound right at all. Sounds mixed up like that to me.
From my discussion of numbers, it should at least be clear that all of reality is not the same as this part of our brains that measures quantities and keeps track of them and stuff. No more than all of reality is a ruler in your hand. We do our best to create a "map" in our heads that matches reality [that map is our beliefs, theories, etc.]. And it seems to me that the word "mathematics" really refers to this tool kind of thing.
On his website, he replies to stuff like this:
* Q: Aren't you conflating the description with the described when saying that our physical reality *is* mathematical rather than just being *described* by math? 
** A: [...] Our language for describing the planet Neptune (which we obviously invent - we invented a different word for it in Swedish) is of course distinct from the planet itself. Similarly, we humans invent the language of mathematics (the symbols, our human names for the symbols, etc), but it's important not to confuse this language with the structure of mathematics. For example, as mentioned above, any civilization interested in Platonic solids would discover that there are precisely 5 of them (the tetrahedron, cube, octahedron, dodecahedron and icosahedron). Whereas they're free to invent whatever names they want for them, they're *not* free to invent a 6th one - it simply doesn't exist. [...] The possibility that I explore in the book is that one of the structures of mathematics (which we can discover but not invent) corresponds to the physical world (which we also discover rather than invent).
He's right that the language of math is different from the structure of it. Just like our name for a ruler" is different from a ruler in our hands. And even with numbers and math in our own heads: the part of the brain actually doing the math stuff (like counting, subtracting, multiplying, etc.) is also not the same part that has names for all of those numbers and operations and so on. But it's still a thing in there, and when we say "mathematics", we are talking about this kind of thing. Just like with a ruler. The ruler might not be the same as our language to describe the ruler, but the ruler is still not the same as the stuff we are measuring. So there's three things. The language, the measuring device, and the thing we are measuring.
At that last part, he says that "one of the structures of mathematics (which we can discover but not invent) corresponds to the physical world", and this sounds much more comprehensible. The measuring device (a mathematical structure) can be compared to the universe, and maybe we see that they match. That's what I'd say. But based on his other statements, maybe when he says "corresponds", he means something else, and is once again mixing things up.
== A reply ==
he has replied below. i tried pasting my response in a comment, but it didn't preserve bullet points, so it's easier to put it here:
I tried to finish my reply a few times, here’s what i’ve written so far.  I could write more, but i’d take too long to finish writing it.  I already took too long!
* My reply
** BEGINNING 
*** It's not that i don't think you can explain numbers physically.  I understand that this is doable, 
**** Huh.  That’s a bit unexpected.  I thought if you’re asking, then you don’t think it’s possible. But I guess “anything is possible”, to an extent, so we’d still have to evaluate which of those possibilities to choose, and this is partly what you’re asking about.
*** You're thinking inside a specific frame [...] In order to disagree with what Tegmark is saying you should perhaps try to get to a place where you can steel-man it. 
**** You’re right, an outsider test of faith is the only way.  So, could you steel man it?  Or at least summarize it comprehensibly?  Or, is there a good specific place I should go to read what he’s written?  You know more about how to steel man it than I do, presumably.
=== WHAT IS TEGMARK PROPOSING? ===
* For now, Tegmark still either
** just sounds like physicalism
** just sounds like word salad
* Seems to me he is proposing something that is scarcely distinguishable from physicalism.  He is proposing a kind of monism, and it behaves in mathematical ways.  I don’t know what part of that could be different from physicalism.
* You [and I think others] seem to propose he is talking about some kind of dualism [with frequent comparisons to Platonism], almost like there is the physical world, and then some other mathematical world of some kind. This is less comprehensible to me than christian dualism, because I can at least conceive of a big ghost magically interfering in the world through miracles and such (“animals exist because god magically made them!”).  I’m not sure what the mathematical world is supposed to be doing, how it's supposed to be related to the physical, or how this is supposed to explain anything.
=== THIS PARAGRAPH: ===
* “But it has problems to me. For example, when the greeks proved that the square root of two was not a rational number, they didn't know, at that time, how to translate proofs and theories into machines. If I undersatnd the physicalist interpretation of the proof, it seems to demand that I believe that what greeks had really done, was not to understand mathematical reality - which exists outside of physics, and can be represented by physical systems -, but instead to figure out a specific property of a specific kind of machine (namely, the results of a proof validating computer when run on an input consisting of a step-by-step proof, coupled with the valid axioms and production rules of number theory) - even thought they could not conceive of, or build, this machine?”
** That is not my physicalist interpretation [see also: The Ontology of Logic by Richard Carrier, it has a section on mathematics that is relevant here].  I’ll cover three points:  first, “the machine”, second "understanding", third “specific things being represented by something else”, fourth “discovering mathematical truths”:
*** 1)  I think there is this tendency [even seemingly in Carrier’s writings, perhaps?] to think that we need to talk about a bunch of other hypothetical things, like hypothetical infinite cats, or “a machine they could not conceive of or build”.  By hypothesis (and as an explanation), the machine exists:  the physical world (with everything in it, such as human brains).  Nothing else is needed.  The Greeks physically came up with the proof.  They were the machine (perhaps along with any physical tools they actually used, like writing on paper and measuring devices and abacus or whatever).  This is the nature of mathematics, according to physicalism.  I suppose you could go on to say “humans don’t really know how to build a human brain, nor do they fully “conceive” of it in all its necessary functional detail” and sure i’d agree with that.  But (by hypothesis) it’s literally right there, just waiting to be understood (and already, apparently successfully, being used, just as most people use computers without really knowing how to build one or how it works).  It’s not like a hypothetical infinite cats or hypothetical star trek computer, which we know we don’t have access to (or a modern day computer, which the Greeks did not have access to). The Greeks had no problem making proofs because they had everything that is required to do that.  And note:  I have no idea how a non-physical “mathematical reality” is supposed to be helpful here. 
*** 2) You say "even thought they could not conceive of, or build, this machine?" which sounds to me like you might also be concerned about whether they understood any of it. So I guess I should also mention about the understanding: doing the proof was ALSO their investigation/analysis to reach understanding. Just as doing the proof was the same as conceiving/building the machine, in a way. Or, you could say, they reached an understanding/construction of a possible algorithm (or the results of a possible set of operations), which is like a virtual machine that can be run on a more generalist/versatile machine (like a brain or computer).
*** 3) Abstraction is the whole point (and is physically possible).  When we learn a mathematical proof, we are not only learning how one specific machine works.  The entire meaning of abstraction is that we recognize some similarities between different things (despite their differences).  All that requires is for the things themselves to exist, and our ability to recognize similarities between them (and, perhaps, to generate categorizations and such).  And it was always “a possibility” that recognition could be performed physically (and now this has been fairly convincingly proven with recent object recognition technology).  
**** An analogy here could be something like testing a scale model of an airfoil in a wind tunnel.  Then you can apply that knowledge to airplane wings, wind turbines, boat propellers, and various other similarly shaped machines.  Mathematics is like a kind of little machine or device (that we can build and run  in our heads if we need to) that behaves a certain way, and we see how other things can behave similarly.  And then, of course, we have language that can communicate about all of this in detail.
**** Or, for an example that doesn’t seem so much like math:  mixing colors.  We know these colors, when mixed, turn into this other color.  It doesn’t matter what physical materials you use, whether fruit juice or oil paint or blood or whatever.  The abstract knowledge about  color mixing can be applied to each case.  (Similarly for the additive colors, and using different light sources and colored glass or whatever else).  This is abstract knowledge.
*** 4)  can facts about math be discovered?  If it’s kind of a tool we made (especially an abstract conceptual tool), what is there to discover?  Well, the answer is a few things.  First, we never know everything about even our own tools we make.  With math, we’re usually just given [or create] some general rules and operations, so that we can calculate results on a case-by-case basis.  All this rule stuff isn’t exactly a requirement for us to work with numbers, right?  We could just memorize the whole thing (like people do with times tables).  You’d be able to use it for all practical purposes.  But memorizing tons of stuff is harder than just learning a few simple rules, and being able to build the rest from there.  
*** Interestingly, our hand computation of math has little to do with the quantities themselves, and more to do with manipulating our digital representations of them.  We start and end with well-defined quantities, but our rules to get between them are not necessarily connected, and alternative methods of multiplication have been developed, for example.
**** Why do rules work?  Because it is a predictable structure (or:  you get a predictable structure when you follow the “straight lines” of the computation rules you learned).  Other human inventions can be predictable, as can physical systems that humans did not make.  Part of the trick here is:  if you repeat something, it repeats.  So we create a digit system where we repeatedly cycle through the digits every ten numbers.  But we could have made different digits after ten, and switch them at unequal intervals, so you could switch after ten, but then have a new set until twenty three, and then switch again at twenty eight, and so on.  But instead we made something predictable, because that’s easy to use.  
**** but there are infinite combinations of them, so we can’t really “know” it all ahead of time.  But (again) this is also true for many other inventions.  People can spend all day testing how a microwave affects various household objects.
=== REPRESENTATION ===
* “Yes, you can understand numbers and proofs and mathematical logic as being mere expressions of physical reality." 
** Hmm, “expressions”?  Not sure what you mean.
* "But you can ALSO understand physical reality as being a mere expression of a mathematical realty.”
** Can i?  What would “a mathematical realty” be, and how could “physical reality” be “a mere expression of” it?  What does that mean?
* “to understand mathematical reality - which exists outside of physics, and can be represented by physical systems”
** Represented?  Like what?   Representation usually means information compression.  So a simpler model “represents” a more complex thing that is similar in some way.  Is Tegmark proposing that there is an extra, more complex reality beyond the physical?
*** So, is it a reversed idea of what “representation” means?
*** I think most people would say “5” represents the number of apples, they would NOT say that the apples represent “5”.  This is backwards in the many-to-one nature of representation [a stick-man can represent this man or that man, but men don’t “represent the stick man”, nor do real people “represent” our abstract knowledge about people]. Unless you mean using apples to represent something else, like sheep [see my link to "the simple truth", above]. .....i'll have to re-think what you're trying to say here. but, again, by hypothesis if you aren't using apples or pebbles (or a calculator or an abacus) to count sheep, you're using the physical system in your head. No idea what Tegmark's view is supposed to do here.
=== CHOOSING BETWEEN SOMEWHAT EQUIVALENT METAPHYSICS? ===
* “This last statement seems very hard for you to grasp, i think because you believe physicalism so strongly that it's difficult for you to avoid asserting the conclusion. You aren't merely saying "the physicalist perspective is a valid outermost conceptual frame" - you seem to be saying it is the only valid outter-most frame. And this strikes me as contradictory, because if there are multiple outermost frames which have the same predictive capacity, how should we select one over the other? And that is where i'm getting at with the point about values”
** No, i think i grasp other metaphysics like idealism fine enough, i think.  But I contend idealism does NOT have the same predictive capacity.  See my wiki page on it [or see this specific section of my page on Bernardo Kastrup, for example][similarly, think i understand christian dualsim just fine, i suppose, but dualism does raise questions] Tegmark's view, on the other hand, is difficult for me to make sense of (as i said before).
** I have linked to various things about how to choose one belief over another.
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