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David_Allen comments on Strong substrate independence: a thing that goes wrong in my mind when exposed to philosophy - Less Wrong
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No. This is a wrong idea that seems to have been accepted on this site.
Arithmetic is an abstraction. It is a useful way to carve meaning out of our perceptions. It exists in systems implementing that abstraction, such as our mind. It has no meaning or existence beyond that.
This becomes evident when you consider perspective. What perspective are you adopting when you say "two bottlecaps are being combined with two other bottlecaps to make four bottlecaps"? You have adopted the perspective of somebody who can see, identify, and count the bottle caps in some particular area. You are in fact modeling this perspective in your mind. In this case 2+2=4 isn't in the arrangement of bottle caps, it is in your mind.
I don't know if this is agreement, but the way I have thought about it for a while now is that mathematics, including arithmetic, is more like a game than anything else. Rules like the Peano Postulates exist simply because those are rules that we think are appropriate to the game - they handle cases like accumulating bottlecaps in an elegant fashion - not because they have a seperate, Platonic reality.
If squirrel A puts a peanut in an empty nook in a tree, and then later puts another nut there, and does that twice more, and nothing removes nuts from that nook, and later a squirrel B comes along and eats all the nuts in that nook, my expectation is that squirrel B will eat four nuts.
That expectation depends on certain facts about the world, and among those facts is something that can be expressed as "1+1+1+1=4," which I would label a statement of mathematics. The squirrels don't necessarily have access to that fact, and certainly don't have access to that expression of it.
What is your expectation of how many nuts B eats? Do you agree that "1+1+1+1=4" expresses something that's related in some way to that expectation? If so, do you agree that that's a statement of mathematics?
If the thing "1+1+1+1=4" expresses exists only in minds, whose mind does it exist in, in this case? Squirrel A? Squirrel B? Somewhere else? What would be different if it existed somewhere else?
The statement "1+1+1+1=4" exists in your mind. Your mind generated this expression as a result of contemplating your observations (from the story). This statement is a abstraction of your observations, specified in the terms of arithmetic. Your observations were formed based on your sensory input combined with your prior experience. Your sensory input depended on your relative context, your body's physical capabilities, and the physical laws of the universe.
The statement is only related to the physical reality of the nuts through a chain of inference. The statement does not represent anything about the nuts directly; it only represents something about the state of your mind.
Even the identification that nuts are individual items that can be counted is an abstraction that you hold in your mind. If you consider the nuts from the perspective of a single photon, the nut abstraction vanishes. With a single photon's perspective we can't tell a nut from a bear, much less count the number of nuts.
Ah, I see what you mean now.
Agreed that the notion that arithmetic primitives relate to a relationship between nuts is unintelligible from a perspective that does not allow for nuts, or objects in general, or relationships among objects, in the first place.
And, yes, the existence of objects and relationships among them is an accepted idea on this site, which makes that perspective pretty much incompatible with most discussion here.
I am not arguing for the non-existence of objects and relationships among them. Actually, the nature of relationships is key to my arguments.
I am arguing against the idea that arithmetic has an existence outside of any meaningful context. I am arguing that arithmetic is an abstraction that only exists in the contexts of its actual implementations. Arithmetic isn't occurring when sheep wander or squirrels store nuts. Arithmetic occurs when we interpret our observations of those circumstances.
There exists a relationship between how many nuts squirrel B eats, and how many times squirrel A deposited a nut in the tree.
That relationship does not depend on my observations.
"1+1+1+1=4" is a statement of arithmetic that expresses one aspect of that relationship; specifically, the aspect of it related to counting.
"1+1+1=3" is a different statement of arithmetic that expresses the same aspect of a different relationship, one that could be implemented in a different story, and likely was.
"1000+1000+1000=3000" is yet another statement of arithmetic that expresses the same aspect of a different relationship, one that has probably never been implemented in terms of nuts and squirrels, although in principle it could be.
"1+1+1=4" expresses the same aspect of yet another relationship, one which probably has never been implemented that way, and which probably can't be.
And there are other kinds of relationships, implementable and otherwise, which can be expressed by other kinds of statements of mathematics.
None of those relationships depend on my observations, either. And you say that none of those relationships are arithmetic relationships, precisely because they don't involve us interpreting our observations.
For convenience, let's call them X, instead. You aren't denying the existence of X, merely asserting that X isn't arithmetic.
Well, OK. I'm not sure what I would expect to experience differently if those relationships were or weren't arithmetic, so I don't know how to evaluate the truth or falsehood of that statement.
But I will say that if that's true, then arithmetic isn't very interesting, except perhaps linguistically. Sure, maybe arithmetic only occurs in minds, or in human minds, or in English-speaking minds. I can't see why I ought to care much about that.
The interesting thing is X.
Thanks for sticking with this, I am trying to hone my arguments on this topic and you are helping.
Yes it does.
You are implying that there is some sense of reality that is independent of how we think about it. I agree with that. But your statement adopts a "human mind" centric interpretation which makes it false.
For example, from the perspective of the universe at the level of quarks, the reality within the story's space-time is unchanged by our later observations of the written story. It is independent of our observations.
However, the relationship that you identified has no meaning from the quark perspective. We wouldn't know if a squirrel ate a nut or if a nut ate a squirrel. At that level, there are no concepts for squirrels and nuts -- or counting; those are higher level abstractions.
The relationship you identified is real and it has meaning; but that meaning is found within the context of your mind and does not describe some intrinsic property of the universe, it describes an interpretation of your observations.
Here is why you should care:
Here at LW we are working toward rationality. We want to improve the correspondence between our map and the territory. We want to know what the truth is and how to carve reality and its joints. We want to make ourselves immune to obvious fallacies such as the mind projection fallacy.
My claim is that the context principle -- that all meaning is context dependent -- is essential to understanding existence, truth and knowledge; it provides traction for solving problems and toward achieving our goals.
Consider a particular system, S1, of a squirrel eating a nut.
S1 can be described in a lot of different ways. The way I just described it is, I agree with you, a human-mind-centric description.
But I could also, equally accurately, describe it as a particular configuration, C1, of cells. Or a particular configuration, A1, of atoms. Or a particular configuration, Q1, of quarks.
Those aren't particularly human-mind-centric descriptions, but they nevertheless describe the same system. Q1 is, in fact, a description of a squirrel eating a nut, even though there's no way I could tell from analyzing Q1 whether it describes a squirrel eating a nut, or a nut eating a squirrel, or a bushel of oranges.
That I am using a human-level description to refer to it does not make it somehow an exclusively human-level as opposed to quark-level system, any more than the fact that I'm using an English-language description to refer to it makes it an English-language-level system.
And Q1continues to be a quark-level description of a system comprising a squirrel eating a nut even if nobody observes it.
Essentially you are saying that Q1=S1. This is certainly not true.
Clearly Q1 and S1 are related. If we could vanish a large contiguous chunk of Q1, we might see a chunk of squirrel disappear in S1; so they have some time-space context in common.
But Q1 describes a system of quarks and S1 describes a system of a squirrel and a nut. They are represented in different "languages"; to compare them you must convert them to a common "language". The relationship between Q1 and S1 is this process of language conversion -- it is the layered process of interactions and interpretations that result in S1, for some context that includes Q1.
The process that generates S1 -- in part from observations ultimately derived from Q1 -- includes the recognition of squirrels and nuts; and that part of the process occurs within the human mind.
No. In general you are not guaranteed "equally accurate" descriptions when you convert from one language to another, from one perspective to another, from one domain abstraction to another. For example the fraction 1/9 is exact, but its decimal representation limited to three decimal places, 0.111, is only approximate.
I addressed this above. Q1 is a system of quarks that is part of the context that led to S1, it is not S1.
For the purpose of efficient communication mixing perspectives in this way is generally fine. To answer certain questions on existence and meaning -- for example to identify if arithmetic has an existence that is independent of humans and our artifacts -- we need to be more careful.
You seem to be failing to attend here to the difference between descriptions and the systems they describe.
I'm not saying Q1=S1. That's a category error; Q1 is a description of S1. The map is not the territory.
I am saying that Q1 and "a squirrel eating a nut" are two different descriptions of the same system, and that although "a squirrel eating a nut" depends on a human mind to generate it, the system it describes (which Q1 also describes) does not depend on a human mind to generate it.
Agreed that there are gains and losses in going from one form of representation to another. But the claim "'a squirrel eating a nut' is a description of that system over there" is just as accurate as the claim "Q1 is a description of that system over there." So I stand by the statement that I can as accurately make one claim as the other.
Why does arithmetic have to "exist" "in" places and times?
You can meaningfully say that 2+2=4 (considered as an abstract number-theoretic proposition rather than a string of symbols) is true, that it incorporates a two-place function, that it's quantifier-free, or that such-and-such is a proof of 2+2=4 in Peano Arithmetic.
But asking whether 2+2=4 is "only true in your mind" (or whether it was "true before people existed") is like asking whether an octopus is true or false.
Absolutely -- but what is missing is a discussion of context. It isn't enough to just say that 2+2=4 is true, or that a particular octopus is false; we need to know what the context of evaluation is.
We need context for the same reason that Bayes' theorem needs priors. Without context we don't have meaning. In many cases we assume context in a way that is transparent to us; but whether explicit or implicit, a context is still in use.
2+2=4 has a standard context, namely the natural numbers N. "2+2=4" (without qualification) asserts that N satisfies 2+2=4. So the fact that one can imagine a non-standard context where "2+2=4" means something false (like "Paris is the capital of the UK") doesn't really have any bearing.
In my use of the expression "2+2=4" I refer not merely to a function that maps contexts to propositions, but to one specific proposition, which has meaning in and of itself. (That's basically what a proposition is - a little chunk of semantics.)
And about that proposition it is meaningless to affirm or deny that it exists only in people's minds. To be fair, I think it's equally meaningless to say that the proposition "exists in" physical processes where someone puts two nuts next to two other nuts and then has four nuts.
Agreed. For efficiency in communication we often assume normative contexts. For the statement "2+2=4" it makes sense for us to rely on its implicit context. To make sense of a statement like "Is that octopus true or false?", we will need to make the context of evaluation explicit.
I'm not certain I understand this as you mean it, so I'll respond generally and see how you reply.
The idea that something can have "a meaning in and of itself" is false. This is equivalent to "objective truth". All meaning is relative to some context.
You can certainly have a conception of "a proposition that has meaning in and of itself", but that conception exists within the context of your mind, and the proposition with that nature is non-existent.
Perhaps you believe in dualism?