Taking your definition of an abstract model (so we don't squabble over mere definitions), I don't think that just by removing information you'll go from an actual baseball to the 'abstract concept' of a sphere. You'll also be adding information. For example, for your model you can provide the formula that will yield the exact volume of the sphere - you can't do that as precisely for your baseball. Will your abstract models typically be more compact / contain less information than your baseball, sure. However, the information may be partially different, not just a subset, which it would be if you were just ignoring information.

That's true. Balls are very complex, so there isn't actually much you can ignore about them without invalidating your results. But you can ignore a lot of things and get approximately correct results, which is usually good enough when talking about balls.

Numbers, however, tend to be a little more convenient. If there's a hole in the bag of apples which you don't take into account, you'll get bad results, because that's a detail which impacts the numeric aspect of the apples. But we don't really care that it's a hole when talking about the number of apples. All we need to keep in mind is that the number decreased. If 1 apple fell through the hole, you can abstract that to a simple -1.

Anyway, this post has gotten out of hand, mostly because I was unclear, so I'll retract it and use these comments to write a hopefully clearer version. Thanks for the feedback.

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. Things we figure out about numbers using Euclidean Geometry can still be valid, simply because when we abstract the details about Euclidean Geometry to be left with only numbers, we get the same thing as when we abstract apples to numbers, and the same thing is true about our mental representation of PA. So proofs from one can be "transferred" over to another. But "transfer" doesn't really describe it well. What's really happening is that from the abstract numbers, you can un-abstract them by filling them in with some details. So you can remember that the apples were in a bag, and that gravity was acting on them. If, when you add in the details, the abstract number behavior still holds, then the object follows the rules of numbers. So if the added details about apples don't affect the conclusions you make using PA, by abstracting PA into numbers, and then filling in the details about apples, you have shown that things that are true about PA are true about apples too. And all this is done using physical processes.

So my view doesn't entail anything about accepting or rejecting mathematical statements. What it says is that mathematical concepts are abstract concepts, which we obtain by ignoring details in things in this world, and thanks to our awesome simple and universal laws of physics, the same abstract concepts emerge again and again.

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. All these ways of representing god are done using completely different configurations of molecules. What is the common ground between them? Certainly not that the idea of god and it's rules are some special thing with special properties. So what do the hard drive and the human mind have in common when representing the idea of god?

By my theory, what they have in common is abstraction. Ignore all the specific details about how hard drives and human minds work, and just look at the specific abstract rules which we define as "god". These are complex, so we can't easily visualize this removal of details. It's much easier when talking about apples and numbers. You can see that when you have 2 apples, you can get the idea of 2 by ignoring the fact that it's apples, and that they're in a bag, and that gravity is affecting them. It's also easy to see when talking about balls. You get the idea of a ball by taking a sphere of matter, forgetting what it's composed of and forgetting it's radius. This abstract idea of a "ball" fits many things, because it's just ignoring details which vary from ball to ball.

So my claim is that the idea of axiomatic systems exists in the physical universe. In fact, all the ideas we ever have, and there rules, exist in the physical universe. But if we take PA as an example, the idea of PA exists in a mathematician's mind, and numbers emerge inside this idea of PA, because numbers do emerge inside PA. So by removing the details of how PA is stored in the mathematician's mind, we obtain numbers, which is just like getting numbers by removing the details about apples.

This still leaves the question of why numbers emerge in so many places. My best guess is that they do because the universe is built upon simple and universal laws of physics, so it's only natural that the same patterns would be appear everywhere.

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, I can abstract this to 3apples. I can add 1apple to this, and get 4apples. Why? Some ancient mathematicians have shown that when you retain this extra bit of information, the abstraction "math with units" is formed, which is pretty much exactly like normal math, except that numbers are multiplied by their units. "math with units" and plain old math are very similar because they are both numbers, but "math with units" happens to retain a few extra details about what the numbers are talking about.

1apple * 1apple=1apple^2 is true, but it doesn't make sense. You can't add in details to turn this into apples because of physical limitations, so 1apple * 1apple does not emerge from apples. But it is consistent with the rules of mathematics, which you obtain when you remove the complex details of what an apple is, so this is considered true.

1m * 1m=1m^2, however, does make sense, and we can add in details to transform this into an area. m/s can be transformed into the speed of an object by adding in details. m/s^2 can be transformed into an actual acceleration. kgm/s^2 can be transformed into an actual force by adding in details. I think this is true of all useful units, but if you can think of one for which this is not true, please share, because that will be a big blow to my theory. If this theory survives all the other arguments, I will still need to prove that this is true, and until I prove it, my theory is weak. Thank you for showing me this.

Anyway, a summary of this comment: math with units is an abstraction, one level below plain math, and it mostly follows the rules of math because it's just math+(some simple other thing). Units which don't make sense cannot be un-abstracted into things in the world, but because math is consistent, we can still manipulate them like anything else because we have abstracted the details which makes it not make sense away.

So I needed to extend my theory for this, but thankfully not by making it more complex at the base. This is all just emergent stuff in the universe. So if my theory holds up, I expect Occam's Razor will be in favor of it.

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 there proved other truths about infinity. Infinity may not exist in the real world, but it does exist as our definition, which is just physical stuff. If infinity exists in the real world and I don't know about it, we have probably observed it and created a model of it which follows the same rules as it. And then, by abstracting our model to ignore the fact that it's just a model, and abstracting the real infininty to ignore whatever it's composed of, we get the same thing, and that's what infinity is.

So while bananas are constrained by the amount of matter in our universe and can therefore not represent any number (some of the details about the bananas just can't be safely ignored), PA (or whichever set of axioms we use to think about infinity) can. And from this, we can find rules about infinity, using a purely physical process.

This is sort of guesswork on my part. I am not certain that this is how we have reasoned about infinity, though I would be surprised if it wasn't. So if it isn't, just say so, and I'll probably be convinced that I'm incorrect, and retract my post.

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 apples I added, and -1 is the number of apples that fell out through the hole, and 3 is the total. Everything in this equation can be mapped back to reality by adding in details. But in 2 apples + 2 miles, the + cannot be mapped back to reality. And so it is not representative of apples.

I can think of an easy attack on this, by saying that 2 apples + 2 oranges does make sense in our world. But this is a disguised incorrectness. The oranges do not fall into the criterion of "apple", so it does not actually make sense to add them using the '+apples' operator. Of course, we could generalize the '+' operation to mean '+fruit', but then it becomes 2 fruit + 2 fruit, which does make sense, and is true whether you reject my current views or not.

I'm looking forward to your next argument :).

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 Arithmetic, when you forget that it's mental processes you're talking about. They are bits, when you forget it's bits you're talking about. But this does not make them special, just like the concept "sphere" is not special. We're just lucky that numbers emerge all over the place in the universe, probably because the laws of physics are the same everywhere so everything is built upon the same base.

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 assuming they would keep on working. We do have good reason to believe that they would keep on working, though, because if they stop, then contradictions take place, and from contradictions anything is true, so we would be living in one very strange universe.

Edit:

It has also occurred to me that it doesn't even matter if a number is larger than the total amount of atoms in this universe. Because as I've said in my post, a number is what you get when you abstract away all the little details which aren't shared by all the places where numbers emerge, like the fact that it's atoms you're counting.

So a system for representing a number larger than the total number of atoms represents a number anyway, as long as it follows the rules of numbers. And a model of something much bigger than the universe works simply because the details of how large the universe actually is are ignored in the model.