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JMiller comments on Mixed Reference: The Great Reductionist Project - Less Wrong
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I am somewhat confused about the nature of logical axioms. They are not reducible to physical laws, and physical laws are not reducible to logic. So then, it what sense are they (axioms) real? I don't think you are saying that they are "out there" in some Platonic sense, but it also seems like you are taking a realist or quasi-empirical approach to math/logic.
Logical Axioms are the rules that decide what can and can't happen. Then, our physical world is one application of these to some starting physical position (and that may be logical defined too, read this post, or Good and Real).
Logic is useful when we have uncertainty. If we are unsure about a certain variable, we can extrapolate to how the future will be given the different possibilites - the different variables that are logically consistent within a causal universe that fits with everything else we know. Of course, if we had no causal knowledge whatsoever, then we'd not have anything with which to apply logic (kinda like this post, with causal reference being emotions, and logic being logic).
So, I'm saying that logic can define how everything that could be would work, which we deduce from our universe's laws. If we have uncertainty, then logic defines the possibilites. If we pretend to have only the knowledge of one law, like '1 + 1 = 2', then we can find out more using logic. And this is the study of mathematics.
No, logical axioms are much too general from that. You need physical laws to projoect the future state of the world, and they are much more specific than logical axioms.
Could you provide an example please? I must apologise, I'm not competent with fundamental laws of physics, but why can't the most basic laws (the 'wave function' is apparently one of them) be specified logically? Wouldn't that just be a mathematical description of the first state of a universe? Then that whole universe, specified by the simplest law(s) would be one universe, and to those/us within that world would only be able to be affected by the things causally connected.
(I suppose I'm talking Tegmark's stuff, although I've only read Drescher's account)
You could, but that is not what is usually meant by "logical axiom". The rules that decide what can and can't happen are called physical laws.
Okay. I tried to respond here, but I'm not qualified to do so. I'll just state what I'm thinking, and then, if you could point out what I might be confused about, I'll leave it there and might go read some books.
I think this is a confusion of definitions. If every universe is described in logic, then the physical laws are a subset of those. So, logic describes everything that is consistently possible and then whichever universe we're in is a subset. Logic describes how our universe works. So the Great Reductionist Project is defining which branch of logical description space we are, and showing on the way that no part of the universe is not describable within logic.
yes, largely.
No, if you buy a book on logic, it doens't describe the universe.To get a description of our universe in mathematical/logical terms, you have to add in empirical information. There is a convenient shorthand for that: physics. Physics described how our universe works.
Huh? How can it show that? Whether there is part of our universe that is not describable by logic is an empirical claim. Science could encouner somethig irreducible are any point.
I think this may have been answered earlier. They are a set of ways you think a certain class of problem works. They're very much an element of your mental model of reality.
In other words, math (or logical axioms) are what adding two pebbles and three pebbles has in common with adding two apples and three apples.
Thank you. In that case, does math rely on at least one particular agent or computer having some [true] model that 2+3 = 5?
Uhm, not really. I'm not entirely sure what you mean by "math relies on things doing math". Math isn't about the thinking apparatus doing math. It's a way of systematically reducing the complexity of your mental models - it replaces adding pebbles and adding apples with just adding.
If you imagine a universe with 4 particles in it, then 2+3 is still 5.
I found Eliezer's post "Math is Subjectively Objective" which explains his position very clearly. Thanks for your help.
No it doesn't, since it ends "Damned if I know."
Right, which explains his position: math is real and 2+3 really is 5, but he does not know what that means, or where that is true.
You are right though, it isn't a fully fleshed out account. All I said is that it explains his position clearly, not that his position itself is perfectly clear.
I don't think it even makes it clear that math is real, just that mathematical truth is objective and timeless.
I think this question is somewhat ambiguous; you've gotten two correct answers that say "contradicting" (different) things and apparently answer different questions.
When you say math, are you talking about the way apples and stones interact and the states of the universe afterwards when the universe performs "operations" on them? If so, then math is agent-independent, as the world-state of 2+3 apples will be five apples regardless of the existence of some agent performing "2+3=5" in that universe.
If you're talking about the existence of the "rules of mathematics", our study of things and of counting, along with the knowledge and models that said abstract study implies, then it does rely on agents having 2+3=5 models, because otherwise there's just a worldstate with two blobs of particles somewhere, three blobs of particles elsewhere, and then a worldstate that brings the blobs together and there's a final worldstate that doesn't need "2+3=5" to exist, but requires an agent looking at the apples and performing "mathematics" on their model of those blobs of particles in order to establish the model that two and three apples will be five apples.
In other words, what-we-know-as "mathematics" would not have been invented if there were no agent using a model to represent reality, as mathematics are abstract methods of description. However, the universe would continue to behave in the same manner whether we invented mathematics or not, and as such the behaviors implied by mathematics when we say "2+3 apples = 5 apples" are independent of agents.
So when an agent or computing device performs an operation on real numbers, say division of 1200 by 7, that result is real, even though the instance of this division requires the agent to do it? The answer IS the only answer, but without an agent, there would not be a question in the first place?
That result is logically valid and consistent, but does not have any new physical real-ness that it didn't already have - that is, its correlation and systematic consistency with the rules of how the universe works.
Otherwise, yes, exactly.
Physical laws are no more real than logical axioms. Both are human constructs, started as models used to explain observations and grown to accommodate other interests. Just like the physical law F=ma is a model to explain why a heavier ball kicked with the same force does not speed up as much, the logical axiom of transitivity "explains" why if you can trade sheep X for sheep Y and sheep Y for sheep Z, it is OK to trade sheep X for sheep Z in many circumstances.
So is there any reason past regularities will continue into the future?