You seem to misinterpret what I mean, but that's my fault for explaining poorly. This post has been getting out of hand with all the clarifications, so I will retract it and post a hopefully clearer version later on. Maybe as I write it, I'll notice a problem with my view which I hadn't seen before, and I'll never actually post it.
Out there in reality, there are just atoms.
I know. But it's easier to talk about apples than atoms. And the apples are just another level of abstractions. From atoms emerge apples, and from apples emerge [natural] numbers.
Take a look at my response to tim. Replace god with Euclidean Geometry, and forget the fluff about god being inconsistent, and you can see that Euclidean Geometry is still coherent, because our minds can represent it with consistent rules, so these rules exist as an abstraction in the universe. So my view doesn't make Euclidean Geometry incoherent. I'm not sure what exactly you mean by validity, but the only thing that my view says is "invalid" about Euclidean Geometry is that it is not the same as the geometry of our universe.
Now it gets a bit difficult to write about clearly, I'm sorry if it's not clear enough to be understandable.... (read more)
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
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ETA: minds can contain gods, ...
No, I'm claiming that the idea of god exists physically.
In our universe, the map is part of the territory. So the concept of god which a human stores in his mind is something physical. God himself might not exist, but the idea of god, and the rules this idea follows, exist, despite being inconsistent. And these rules which the idea of god follows can be represented in many ways, all of them physical.
For example, in the human mind, in computers, in mathematical logic (despite inconsistencies), etc.... (read more)
Or, I could apply a constant force of 5 newtons to an object massing 25 kg for a duration of 8 seconds. I change it's velocity by 5N8S(1MKG/(NSEC^2))/25KG 1.6 M/S. By conserving units on all quantities, I convert force-time against a mass into acceleration.
Those units can be preserved through all mathematical operations, including exponentiation and definite integration.
Hmm... Another good argument. This one is harder. But that's just making this more fun, and getting me closer to giving in.
If I abstract a whole bunch of details about apples away, except the number and the fact that it's an apple, I get math with units. So if I have a bag of 3 apples,... (read more)
Bananas are constrained by the laws of physics, so when you reach the maximum number of bananas possible in our universe, the '+' operation becomes impossible to apply to it. So using physical bananas, it is impossible to talk about infinity.
But even if bananas aren't suited for talking about infinity, where does infinity come from?
Given that we reason about infinity, I infer that infinity can be represented using physical things (unless the mind is not physical). Also, given what I know about mathematics, I expect that infinity is thought about using rules/axioms/(your word of choice).
So to explain the properties of infinity, we simply defined it and some rules it follows, and from... (read more)
Very nice and convincing argument. There were some moments when thinking about it when I though your argument defeated my view. Sadly, we're not quite there yet.
Trying to add 2 miles to 2 apples does not make sense. There is no physical representation of such an operation. So you can't try to abstract that into numbers. Here's an example, to clarify:
Let's say I've got a bag of 2 apples, I add 2, and one falls through the hole in my bag. The number of apples in my bag is 2+2-1=3. The first 2 is an abstraction of the original number of apples in my bag, the second 2 is the number of... (read more)
If the physical facts of apples were to change such that 2 apples added to 2 more apples did not give you 4 apples, then removing the detail that it's an apple would not yield numbers. In such a case, you would not be able to abstract apples into numbers. They would abstract away into something else.
Likewise, if you changed the mental processes which makes Peano Arithmetic, you would not change numbers; you would merely have changed what Peano Arithmetic can be abstracted into.
The thing to get from my post is that numbers are an abstraction: they are apples, when you forget that it's apples that you're talking about. They are Peano... (read more)
Axiomatic Systems ... can all be reduced to physics. I think most LessWrongers, being reductionists, believe this.
I would be suprised if this were true. In fact, I'm not even sure what you mean by it.
Well, given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical.
However, I think you're confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
You can specify models much bigger than the physical universe. But that's just extrapolating the rules by... (read more)
In his latest post, Logical Pinpointing, Eliezer talks about the nature of math. I have my own views on the subject, but when writing about them, it became too long for a comment, so I'm posting it here in discussion.
I think it's important to clarify that I am not posting this under the guise that I am correct. I'm just posting my current view of things, which may or may not be correct, and which has not been assessed by others yet. So though I have pushed myself to write in a confident (and clear) tone, I am not actually confident in my views ... well, okay, I'm lying. I am confident... (read 1019 more words →)
That's true. Balls are very complex, so... (read more)