I'd like to know people who agree with me that mental models of people can often be people. Consider contacting me if that's you.
Since multiple answers here mention equality being undefinable in first-order logic, I want to say that that's only true if there are an infinite number of constants/functions/predicates in the language. Since set theory can be formalized using only a single predicate, it is possible to define equality this way:
(Where x and y can be the same variable, but z must be a different variable from them both)
By simply replacing every instance of the formula "x=y" with this definition, set theory can be formalized in first-order logic without equality.
So what part of a mathematical universe do you find distasteful?
the idea that “2” exists as an abstract idea apart from any physical model
It's this one.
Okay, but if actual infinities are allowed, then what defines small in the “made up of small parts”? Like, would tiny ghosts be okay because they’re “small”?
Given that you're asking this question, I still haven't been clear enough. I'll try to explain it one last time. This time I'll talk about Conway's Game of Life and AI. The argument will carry over straightforwardly to physics and humans. (I know that Conway's Game of life is made up of discrete cells, but I won't be using that fact in the following argument.)
Suppose there is a Game of Life board which has an initial state which will simulate an AI. Hopefully it is inarguable that the AI's behavior is entirely determined by the cell states and GoL rules.
Now suppose that as the game board evolves, the AI discovers Peano Arithmetic, derives "2 + 2 = 4", and observes that this corresponds to what happens when it puts 2 apples in a bag that already contains 2 apples (there are apple-like things in the AI's simulation). The fact that the AI derives "2 + 2 = 4", and the fact that it observes a correspondence between this and the apples, has to be entirely determined by the rules of the Game of Life and the initial state.
In case this seems too simple and obvious so far and you're wondering if you're missing something, you're probably not missing anything, this is meant to be simple and obvious.
If the AI notices how deep and intricate math is, how its many branches seem to be greatly interconnected with each other, and postulates that math is unreasonably effective. This also has to be caused entirely by the initial state and rules of the Game of Life. And if the Game of Life board is made up of sets embedded inside some model of set theory, or if it's not embedded in anything and is just the only thing in all of existence, in either case nothing changes about the AI's observations or actions and nothing ought to change about its predictions!
And if the existence or non-existence of something changes nothing about what it will observe, then using its existence to "explain" any of its observations is a contradiction in terms. This means that even its observation of the unreasonable effectiveness of math cannot be explained by the existence of a mathematical universe outside of the Game of Life board.
Connecting this back to what I was saying before, the "small parts" here are the cells of the Game of Life. You'll note that it doesn't matter if we replace the Game of Life by some other similar game where the board is a continuum. It also doesn't even matter if the act of translating statements about the AI into statements about the board is uncomputable. All that matters is that the AI's behavior is entirely determined by the "small parts".
You might have noticed a loophole in this argument, in that even though the existence of math cannot change anything past the initial board state, if the board was embedded inside a model of set theory, then it would be that model which determined the initial state and rules. However, since the existence of math is compatible with every consistent set of rules and literally every initial board state, knowing this would also give no predictive power to the AI.
At best the AI could try to argue that being embedded inside a mathematical universe explains why the Game of Life rules are consistent. But then it would still be a mystery why the mathematical universe itself follows consistent rules, so in the end the AI would be left with just as many questions as it started with.
My view is compatible with the existence of actual infinities within the physical universe. One potential source of infinity is, as you say, the possibility of infinite subdivision of spacetime. Another is the possibility that spacetime is unboundedly large. I don't have strong opinions one way or another on if these possibilities are true or not.
The assumption is that everything is made up of small physical parts. I do not assume or believe that it's easy to predict the large physical systems from those small physical parts. But I do assume that the behavior of the large physical systems is determined solely from their smaller parts.
The tautology is that any explanation about large-scale behavior that invokes the existence of things other than the small physical parts must be wrong, because those other things cannot have any effect on what happens. Note that this does not mean that we need to describe everything in terms of quantum physics. But it does mean that a proper explanation must only invoke abstractions that we in principle would be able to break down into statements about physics, if we had arbitrary time and memory to work out the reduction. (Now I've used the word reduction again, because I can't think of a better word, but hopefully what I mean is clear.)
This rules out many common beliefs, including the platonic existence of math separately from physics, since the platonic existence of math cannot have any effect on why math works in the physical world. It does not rule out using math, since every known instance of math, being encoded in human brains / computers, must in principle be convertible into a statement about the physical world.
I completely agree that reasoning about worlds that do not exist reaches meaningful conclusions, though my view classifies that as a physical fact (since we produce a description of that nonexistent world inside our brains, and this description is itself physical).
it becomes apparent that if our physical world wasn’t real in a similar sense, literally nothing about anything would change as a result.
It seems to me like if every possible world is equally not real, then expecting a pink elephant to appear next to me after I submit this post seems just as justified as any other expectation, because there are possible worlds where it happens, and ones where it doesn't. But I have high confidence that no pink elephant will appear, and this is not because I care more about worlds where pink elephants don't appear, but because nothing like that has ever happened before, so my priors that it will happen are low.
For this reason I don't think I agree that nothing would change if the physical world wasn't real in a similar sense as hypothetical ones.
Reasoning being real and the thing it reasons about being real are different things.
I do agree with this, but I am very confused about what your position is. In your sibling comment you said this:
Possibly the fact that I perceive the argument about reality of physics as both irrelevant and incorrect (the latter being a point I didn’t bring up) caused this mistake in misperceiving something relevant to it as not relevant to anything.
The existence of physics is a premise in my reasoning, which I justify (but cannot prove) by using the observation that humanity has used this knowledge to accomplish incredible things. But you seem to base your reasoning on very different starting premises, and I don't understand what they are, so it's hard to get at the heart of the disagreement.
Edit: I understand that using observation of the physical world to justify that it exists is a bit circular. However, I think that premises based on things that everyone has to at least act like they believe is the weakest possible sort of premise one can have. I assume you also must at least act like the physical world is real, otherwise you would not be alive to talk to me.
Okay, let's forget the stuff about the "I", you're right that it's not relevant here.
For existence in the sense that physics exists, I don’t see how it’s relevant for reasoning, but I do see how it’s relevant to decision making
Okay, I think my view actually has some interesting things to say about this. Since reasoning takes place in a physical brain, reasoning about things that don't exist can be seen as a form of physical experiment, where your brain builds a description which has properties which we assume the thing that doesn't exist would have if it existed. I will reuse my example from my previous post to explain what I mean by this:
To be more clear about what I mean by mathematical descriptions “sharing properties” with the thing it describes, we can take as example the real numbers again. The real numbers have a property called the least upper bound property, which says that every nonempty collection of real numbers which is bounded above has a least upper bound. In mathematics, if I assume that a variable x is assigned to a nonempty set of real numbers which is bounded above, I can assume a variable y which points to its least upper bound. That I can do this is a very useful property that my description of the reals shares with the real numbers, but not with the rational numbers or the computable real numbers.
So my view would say that reasoning is not fundamentally different from running experiments. Experiments seem to me to be in a gray area with respect to this reasoning/decision-making dichotomy, since you have to make decisions to perform experiments.
I don't say in this post that everything can be deduced from bottom up reasoning.
That's also true in first-order logic with equality, since nothing except convention stops us from considering models where multiple objects are equal according to the equality predicate. The choice to exclude models which include duplicate objects is just a side-condition used to filter out inconvenient models when studying semantics. We can include such a side-condition when considering the semantics of set theory without equality, too, so it doesn't seem fair to me to single it out as being uniquely incapable of defining equality.
(In fact, I'd argue that this also applies to second-order logic. Second-order logic can be given Henkin semantics, which have all the same idiosyncracies as first-order semantics. Using these semantics, we can get models of second-order logic with duplicate objects, just like first-order logic. I'd argue that standard second-order semantics are more or less just using a side-condition to filter out models which have missing subsets. But if we wanted to we could include similar side-conditions when considering models of first-order set theory, too, so it doesn't seem fair to me to to say second-order logic can define equality while first-order logic can't.)