And how exactly would you define the word “circle” other than {X \in R² : |x|=r}?

(In other words, if a geometric locus of points in a plane equidistant to a certain point exists, but circles don’t, the two are different; what is then the latter?)

I have also (disappointingly/validatingly) thought of this and then read Tegmark. (It's even more disappointing/validating than that, though, since as well as Tegmark, you appear to have invented Syntacticism. You even have all my arguments, like subverting the simulation hypothesis and talking about 'closure'). However, I have one more thing to add, which may answer the problem of regularity. That one thing is what I call the 'causality manifold': Obviously by simulating a universe we have no causal effect upon it (if we are assuming the mathematical universe hypothesis); but it has a causal effect upon us, because it defines the results of our computation. I explore this theme somewhat in The Apparent Reality of Physics, a footnote to which mentions the problem of consistency when you have a closed loop of universes, and its putative solvability by loop unfolding / closure. Considering the ensemble of mathematical structures with the natural topology, we see that locally it's either a graph or a manifold (almost everywhere), and it has this flow defined by the causal relations (with the flow in the opposite direction to simulation), which we can consider as being a flow of subjective probability (with some equilibrium state). Of course it contains both regular and irregular universes (hereonin RUs and IUs), because adding a delta function to a differential equation gives you, simply, a different DE (well, that's 'morally' why it's true; it's more complicated in practice because not all mathematical structures are DEs; but any continuous mathematical structure can be continuously corrupted). IUs typically cannot simulate RUs, because any simulation is going to keep hitting the delta functions and being corrupted; RUs, on the other hand, can simulate both other RUs and IUs (a cosmic ray can turn your RU simulation into an IU simulation). Consequently, subjective probability flows from {IUs} to {RUs} much more strongly than the other way, so the equilibrium has most subjective probability on RUs. Thus, anthropics and cake for everyone :)

I should add that I haven't yet been able to mathematically formalise the above argument, because I haven't yet worked out the correct definitions/characterisation of the 'causality manifold' (which is, incidentally, not a manifold), and it's possible that the small probability of an IU simulating a RU screws things up, and that we should (perhaps) expect to find ourselves in a Universe with some (say) Poisson-distributed degree of irregularity. Or something like that. But, at least it does allow for a mathematical universe in which anthropic experience can actually be given a probability distribution.

since this-Omega simulates counterfactual-me and counterfactual-Omega simulates this-me

Does syntacticism work if you know Omega likes simulating poor you, and each simulated rich you is counterbalanced by many simulated poor yous? Or only in special cases like you mentioned?

Under my syntacticist cosmology, which is a kind of Tegmarkian/Almondian crossover (with measure flowing along the seemingly 'backward' causal relations), the answer becomes trivially "yes, give Omega the $100" because counterfactual-me exists. In fact, since this-Omega simulates counterfactual-me and counterfactual-Omega simulates this-me, the (backwards) flow of measure ensures that the subjective probabilities of finding myself in real-me and counterfactual-me must be fairly close together; consequently this remains my decision even in the Almondian variety. The purer and more elegant version of syntacticism doesn't place a measure on the Tegmark-space at all, but that makes it difficult to explain the regularity of our universe - without a probability distribution on Tegmark-space, you can't even mathematically approach anthropics. However, in that version counterfactual-me 'exists to the same extent that I do', and so again the answer is trivially "give Omega the $100".

Counterfactual problems can be solved in general by taking one's utilitarian summation over all of syntax-space rather than merely one's own Universe/hubble bubble/Everett branch. The outstanding problem is whether syntax-space should have a measure and if so what its nature is (and whether this measure can be computed).

I don’t quite get what you mean, then. If the various “pieces of math” describe no more and no less than exactly the rocks and trees and leptons, how can one distinguish between the two?

Would you say the math of “x^2 + y^2 = r^2” exists but circles don’t?

But note that there are also patterns of light which we would interpret as "the wrong answer".

I did note that, maybe not explicitly but it isn't really something that anyone would expect another person not to consider.

isn't it a bit odd that whenever we build a calculator that outputs "5" for 2+2, it turns out to have something we would consider to be a wiring fault (so that it is not implementing arithmetic)?

It doesn't seem odd at all, we have an expectation of the calculator, and if it fails to fulfill that expectation then we start to doubt that it is, in fact, what we thought it was (a working calculator). This refocuses the issue on us and the mechanics of how we compress information; we expected information 'X' at time t, but instead received 'Y' and decide that something is wrong with out model (and then aim to fix it by figuring out if it is indeed a wiring problem or a bit-flip or a bug in the programming of the calculator or some electromagnetic interference).

Can you point to a machine (or an idealised abstract algorithm, for that matter) which a reasonable human would agree implements arithmetic, but which disagrees with us on whether 2+2 equals 4?

No. But why is this? Because if (a) [a reasonable human would agree implements arithmetic] and (b) [which disagrees with us on whether 2+2 equals 4] both hold, then (c) [The human decides she ve was mistaken and needs to fix the machine]. If the human can alter the machine so as to make it agree with 2+2 = 4, then and only then will the human feel justified in asserting that it implements arithmetic.

The implementation is decidedly correct only if it demonstrates itself to be correct. Only if it fulfills our expectations of it. With a calculator, we are looking for something that allows us to extend our ability to infer things about the world. If I know that a car has a mass of 1000 kilograms and a speed of 200 kilometers for hour, then I can determine whether it will be able to topple a wall given that I have some number that encoded the amount of force it can withstand. I compute the output and compare it to the data for the wall.

Because, if arithmetic is implementation-dependent, you should be able to do so.

I tend to think it depends on a human-like brain that has been trained to interpret '2', '+' and '4' in a certain way, so I don't readily agree with your claim here.

Yes! (So long as we define computation as "abstract manipulation-rules on syntactic tokens", and don't make any condition about the computation's having been implemented on any substrate.)

I'll look over it, but given what you say here I'm not confident that it won't be an attempt at a resurrection of Platonism.

"Manipulations of symbols according to formal rules are the ontological basis"

I understand "symbols" to be a cognitive shorthand for our brains representation of structures in reality. I don't understand the meaning of the word "symbols" in the abstract, without a brain to interpret them with and map them onto reality.

"existence" being merely how the algorithm feels from inside.

This doesn't really explain anything to me, it just sounds like wisdom.

As I understand it ec429’s intuition goes a bit like this:

Take P1, a program that serially computes the digits in the decimal expansion of π. Even if it’s the first time in the history of the universe that that program is run, it doesn’t feel like the person who ran the program (or the computer itself) created that sequence of digits. It feels like that sequence “always existed” (in fact, it feels like it “exists” regardless of running the program, or the existence of the Universe and the time flow it contains), and running the program just led to discovering its precise shape.(#)

Now take P2, a program that computes (deterministically) a simulation of, say, a human observer in a universe locally similar(##) to ours, but perhaps slightly different( ###) to remove indexing uncertainty. Applying intuition directly to P2, it feels that the simulation isn’t a real world, and whatever the observer inside feels and thinks (including about “existence”) is kind of “fake”; i.e., it feels like we’re creating it, and it wouldn’t exist if we didn’t run the program.

But there is actually no obvious difference from P1: the exact results of what happens inside P2, including the feelings and thoughts of the observer, are predetermined, and are exclusively the consequence of a series of symbolic manipulations or “equation solving” of the exact same kind as those that “generate” the decimals of π.

So either: 1) we are “creating” the sequence of decimals of π whenever we (first? or every time?) compute it, and if so we would also “create” the simulated world when we run P2, or 2) the sequence of digits in the expansion of π “exists” indifferently of us (and even our universe), and we merely discover (or embody) it when we compute it, and if so the simulated world of P2 also “exists” indifferently of us, and we simply discover (or embody) it when we execute P2.

I think ec429 “sides” with the first intuition, and you tend more towards the second. I just noticed I am confused.

(I kind of give a bit more weight to the first intuition, since P2 has a lot more going on to confuse my intuitions. But still, there’s no obvious reason why intuitions of my brain about abstract things like the existence of a particular sequence of numbers might match anything “real”.)

(#: This intuition is not necessarily universal, it’s just what I think is at the source ec429’s post.)

(##: For example, a completely deterministic program that uses 10^5 bit numbers to simulate all particles in a kilometer-wide radius copy of our world around, say, you at some point while reading this post, with a ridiculously high-quality pseudo-random number generator used to select a single Everett “slice”, and with a simple boundary chosen such that conditions inside the bubble remain livable for a few hours. This (or something very like it, I didn’t think too long about the exponents) is probably implementable with Jupiter-brain-class technology in our universe even with non-augumented-human–written software, not necessarily in “real-time”, and it’s hard to argue that the observer wouldn’t be really a human, at least while the simulation is running.)

(###: E.g., a red cat walks teleports inside the bubble when it didn’t in the “real” world. For extra fun, imagine that the simulated human thinks about what it means to exist while this happens.)

If that is so, then how come others tend to reach the same truth? In the same way that there is something outside me that produces my experimental results (The Simple Truth), so there is something outside me that causes it to be the case that, when I (or any other cognitive agent) implements this particular algorithm, this particular result results.

People have very similar brains and I'd bet that all of the ideas of people that are cognitively available to you shared a similar cultural experience (at least in terms of what intellectual capital was/is available to them).

Viewing mathematics as something that is at least partially a reflection of the way that humans tend to compress information, it seems like you could argue that there is an awful lot of stuff to unpack when you say "2+2 = 4 is true outside of implementation" as well as the term "cognitive agent".

What is clear to me is that when we set up a physical system (such as a Von Neumann machine, or a human who has been 'set up' by being educated and then asked a certain question) in a certain way, some part of the future state of that system is (say with 99.999% likelihood) recognizable to us as output (perhaps certain patterns of light resonate with us as "the correct answer", perhaps some phonemes register in our cochlea and we store them in our working memory and compare them with the 'expected' phonemes). There appears to be an underlying regularity, but it isn't clear to me what the true reduction looks like! Is the computation the 'bottom level'? Do we aim to rephrase mathematics in terms of some algorithms that are capable of producing it? Are we then to take computation as "more fundamental" than physics?

Does this make sense?

Your conclusion on sheep is a physical state in your mind, generated by physical processes. But the sheep still exist outside of your mind.

Restating my claim in terms of sheep: The identification of a sheep is a state change within a context of evaluation that implements sheep recognition. So a sheep exists in that context.

Physical reality however does not recognize sheep; it recognizes and responds to physical reality stuff. Sheep don't exist within physical reality.

"Sheep" is at a different meta-level than the chain of physical inference that led to that classification.

That "truth" in the map doesn't imply truth in the territory, I accept. That there is no truth in the territory, I vehemently reject.

"Truth" is at a different meta-level than the chain of physical inference that lead to that classification. There is no requirement that "truth" is in the set of stuff that has meaning within the territory.

When you look at the statement 2+2=4 you think some form of "hey, that's true". When I look at the statement, I also think some form of "hey, that's true". We can then talk and both come to our own unique conclusion that the other person agrees with us. This process does not require a metaphysical arithmetic; it only requires a common context.

For example we both have a proximal existence within the physical universe, we have a communication channel, we both understand English, and we both understand basic arithmetic. These types of common contexts allow us make some very practical and reasonable assumptions about what the other person means.

Common contexts allow us to agree on the consequences of arithmetic.

The short summary is that meaning/existence is formed by contexts of evaluation, and common contexts allow us to communicate. These processes explain your observations and operate entirely within the physical universe. The concept of metaphysical existence is not needed.