Actual philosophy

[helios]

Numbers and operations are just the words, the syntax. Just like the word "dog", whether it is the sound you make, or the letters spelled out, isn't an actual dog, the number 5, isn't anything, it describe something.

If we agree that the string "dog" isn't a dog, then we must also agree that the string "the number 5" isn't the number 5. The number 5 is therefore not part of the syntax of mathematics in the same way that a dog isn't part of the vocabulary of English.

[htirwin]

I think I am starting to question whether this philosophical question can be answered. It may go right down to the most fundamental levels of existence, beyond what we know from quantum mechanics.

For example you could say dogs as we know them are just abstractions. Ultimately, at the fundamental level, they are just configurations of some very fundamental and mysterious things; the same things we are. Their dogness comes from the quantities, configurations, and states of what they are made of, and the best way we have to make sense of that is with mathematics. Without quantities and mathematical relationships, everything may more or less be the same thing. Similar to how in a computer, most information is just captured with high and low voltages, and we think of that as 1's and 0's, but in the programs encoded with them, you have all kinds of distinguishable objects.

In this sense, is it more true to say that dogs exist, or the math exists? At the most fundamental levels, there are no notions of dog's, but it's arguable that there is some notion of mathematics. At least you can say at the most fundamental level, things behave mathematically.

I think the real question is where/how does all of this mathematical structure and behavior arise from what we are made of at the most fundamental levels? But without knowing the true metaphysical nature of our reality, how can we really figure this out? We know from relativity that what is really going on is too strange for us to understand intuitively. And elements of quantum mechanics are equally strange. Behind even our best theories, is the truth, and while we can only model it, predict it, and sense it through a far, far, abstracted interface, it is there, and it is unintuitive.

Maybe, at some level, only one thing exists, and it's just all twisted up. The existence of thing that is twisted up is beyond our scope to measure in and of itself, but at the next layer of abstraction, that which exists and what we can measure, is the twistedness of it. Depending on how you like to define the word math, that word could be used as a reference for how it's twisted and arguably the only real thing we can say about distinguishable existing things.

I am past my comfort zone, starting to confuse myself, and deep into my gray area here, but I think we have to be careful about trying to make metaphysics into something intuitive. And as far as we have come along with physics, we still have a long way to go, and it may be actually impossible for us to measure the most fundamental essence of the universe, let alone describe in a way that makes sense to us.

[Mats]

Maybe, at some level, only one thing exists, and it's just all twisted up

This is exactly the conclusion all attempts of theories of everything are tending towards. They all boil down to everything in the universe is just a manifestation of gravity and at the most basic level, everything should be described with a single, extremely powerful mathematical formula, which would be the quantum theory of gravity. It may very well turn out in the end that this theory and prime numbers are related (at least from some lectures I've seen anyway).

We know from relativity that what is really going on is too strange for us to understand intuitively. And elements of quantum mechanics are equally strange.

I think it is more than equality. Quantum mechanics is just... Impossible: https://www.youtube.com/watch?v=w3ZRLllWgHI

[Mathhead200]

@htirwin
"Yes" to the idea that everything is a level of abstraction; that's exactly what I meant! When we talk about/do math or work with numbers, we are describing a quality of reality, but we are not trying to encapsulate everything in that description. An idealistic number five doesn't "exist" any more then an idealistic dog "exists". It's all an abstraction, math just goes a bit farther :)

As for you comment

htirwin wrote:

but it's arguable that there is some notion of mathematics. At least you can say at the most fundamental level, things behave mathematically.

I don't see it as things behave mathematically (although they do), I see it as things behave logically. I mean would you expect anything different? Logic is defined to be "natural," and math is (hopefully) a subset of deductive logic. In summary, things behave mathematically because we defined math to be like things are!

[helios]

When we talk about/do math or work with numbers, we are describing a quality of reality, but we are not trying to encapsulate everything in that description.

Sometimes (most times) the description doesn't encapsulate anything at all about reality. Is there some physical parallel for primality?

An idealistic number five doesn't "exist" any more then an idealistic dog "exists".

The reason why an archetypal dog couldn't exist is fairly clear. We don't know what set of characteristics define the dogness of something. Ultimately, it's a problem of language.
We do know the exact set of characteristics that define the fiveness of something and, given enough information, we can say unequivocally whether a thing is 5 or not. Why then, couldn't an archetypal 5 exist?
I will concede that, to my knowledge, there's no such thing as an archetypal number.

Logic is defined to be "natural," and math is (hopefully) a subset of deductive logic.

Actually, mathematics does things that logic can't. For example, propositional logic is both complete and consistent, while the Peano axioms are only consistent.

[Mathhead200]

helios wrote:

Is there some physical parallel for primality?

Of course. A prime number of things can't be arranged into a (fat) rectangle, only a line.

About the differences between a vague term like "dog" and a non-vague number like five
True. This is not what I was trying to point out however. Even if we could make "dog" a well defined concept like five is, we still couldn't map that back to reality without losing some of the power of the abstraction. The point of abstraction for me is to ignore the (contextually) irrelevant, so that we can see connections and patterns where they weren't obvious before. (E.g. modeling a Rubik's cube with group theory.)

helios wrote:

... propositional logic is both complete and consistent...

I'll have to look into that. I thought from Gödel's incompleteness theorems that you couldn't have complete and consistent axiomatic systems. [Off to read Wikipedia...]

[helios]

Gödel's theorems state that there are no complete and consistent axiomatic systems with the same or greater power than the Peano axioms.

Of course. A prime number of things can't be arranged into a (fat) rectangle, only a line.

Right. A prime number of things. Not prime things. Primality is a property of the number you get after you count how many things you have, not of the things you have.

True. This is not what I was trying to point out however. Even if we could make "dog" a well defined concept like five is, we still couldn't map that back to reality without losing some of the power of the abstraction. The point of abstraction for me is to ignore the (contextually) irrelevant, so that we can see connections and patterns where they weren't obvious before. (E.g. modeling a Rubik's cube with group theory.)

Okay, I think I see what you're getting at. Let me ask you this, then: are computer files abstract or concrete?

[Mathhead200]

helios wrote:

... are computer files abstract or concrete?

I guess both. They exist a physical "things" on some storage medium, but the idea of a file and the idea about its contents are abstracted. e.g. The file might be text in which case it could be using ASCII or UTF-8, or the file might be binary in which case it could be using 2's complement notation or maybe IEEE 754 floating point numbers. A computer file might be stored on a concrete (non-abstract) medium, but the magnetization of a hard drive or the divots in a CD are no more representative of the actual data/information in question then the squiggles you are reading right now. We use abstraction all the time, it's how we thing about the world. Math just takes it wholly to an extreme level.

helios wrote:

Right. A prime number of things. Not prime things. Primality is a property of the number you get after you count how many things you have, not of the things you have.

Isn't this akin to making the distinction between me as a human and my as a collection of atoms? The collection of things has the property of being prime, not the things in the collection.

[helios]

I guess both. They exist a physical "things" on some storage medium, but the idea of a file and the idea about its contents are abstracted.

Exactly.

A computer file might be stored on a concrete (non-abstract) medium, but the magnetization of a hard drive or the divots in a CD are no more representative of the actual data/information in question then the squiggles you are reading right now.

I disagree. The total internal state of any digital computer is on a 1-to-1 correspondence with some natural number, and its various processors are on a 1-to-1 correspondence with functions from N to N. Nothing has been lost from the concretization of these abstract objects.

Isn't this akin to making the distinction between me as a human and my as a collection of atoms?

No. Although that might be a valid distinction, depending on where your self resides.

The collection of things has the property of being prime, not the things in the collection.

The size of the collection is the one with the property.

[htirwin]

Right. A prime number of things. Not prime things. Primality is a property of the number you get after you count how many things you have, not of the things you have.

Maybe there is something that is analogous to prime, but I don't think that, whatever it would be, the word prime as we use it in mathematics would describe it.

I suppose you could say something is prime if it cannot be broken up into other things.

I don't see it as things behave mathematically (although they do), I see it as things behave logically. I mean would you expect anything different? Logic is defined to be "natural," and math is (hopefully) a subset of deductive logic. In summary, things behave mathematically because we defined math to be like things are!

But is the thing real, or is it the behavior that is real. The thing we describe with a noun, a single word has meaning to us as it allows us to classify it in our minds and memories. But what is the thing? It's just a collection of other things, and those collections of other things, and so forth. By the time you get down to what the thing actually is, you are left with more things, which are still just words we use for what we measure. And ultimately every source of information we have and everything we know is just a universe we have created to conceptualize the measurements we make. And what it is we are measuring, we don't really know, only that it is doing stuff.

If you are willing to consider the universe a gigantic calculator, then what is most real to us is how it calculates.

The size of the collection is the one with the property.

What makes a thing a thing? What is a thing; beyond it's name? To us it's some abstract collection of associated behaviors and properties. And those properties are really just behaviors. If the "size" of something isn't part of what it is, then the thing doesn't exist, at least in any form we can make ourselves aware of. Maybe things or thing exist/s that make/s it up, that we cannot perceive, but those things alone don't make it the thing we talk about, it's what we measure abstractly of it that makes it a thing. It's size, it's color, ...

I disagree. The total internal state of any digital computer is on a 1-to-1 correspondence with some natural number, and its various processors are on a 1-to-1 correspondence with functions from N to N. Nothing has been lost from the concretization of these abstract objects.

What are we calling the internal state? It depends on whether things like position and magnetism are discrete or continuous. Does the physical universe have frame-rate?

[Mats]

What are we calling the internal state? It depends on whether things like position and magnetism are discrete or continuous. Does the physical universe have frame-rate?

Magnetism is already considered discrete under quantum mechanics (http://en.wikipedia.org/wiki/Quantum_electrodynamics). Discrete energies for everything else, with a continuous model for gravity (which describes space-time) are mathematically incompatible. i.e. If the universe is mathematically consistent, then space does indeed have time in discreet amounts and there is a 'frame rate' of sorts. Will the universe be mathematically consistent though?

This thread has made for interesting reading so far!

[helios]

If the "size" of something isn't part of what it is, then the thing doesn't exist, at least in any form we can make ourselves aware of.

I think there's some logical gap in this sentence. One possible problem is that you may be confounding the word "size" as applied to collections with the word "size" as applied to shapes. The size of a shape is part of the identity of the shape; that is, you can have two shapes that would be equal if only their sizes were equal. But the size of a collection is only a consequence of it. You can't have two collections that are only differentiated by their size; necessarily, two collections of different sizes n and m will also have at least abs(n-m) elements unique to one collection.

[Mathhead200]

helios, about what I meant with respect to the hard drive question. Could you not have the exact same bit pattern representing two different types of data, two different piece of information* (depending on the decoding/encoding)? Could you not have two different bit patterns representing the same information? Part of what makes up the data is context and our interpretation of it.

* Information used in the vernacular, not as used in information theory.

[helios]

Could you not have the exact same bit pattern representing two different types of data, two different piece of information* (depending on the decoding/encoding)? Could you not have two different bit patterns representing the same information? Part of what makes up the data is context and our interpretation of it.

I think you grabbed my question from the opposite end I intended. I was asking about the abstraction of files relative to numbers, while you answered about the abstraction of files relative to the information they supposedly represent. While I agree with you that a file by itself is entirely disconnected from anything we would care to store in it, that's unrelated to the point I was trying to make.
I was trying to give an example of a mathematical abstraction that can be exactly mapped to reality without losing anything.

[Mathhead200]

(Ah, I thought you were talking about abstraction in general.) The fact that the exact numbers you are using to represent the file could be used to represent other concrete things leads me to conclude that you are losing something in the abstraction, even if I can't demonstrate exactly what.

[helios]

Your original argument was that something was lost by concretizing, though...
An abstraction must by necessity have less than the thing it abstracts, else it would just be the thing. A universal storage unit (bitstrings) can't also store what it is it's storing, because that metadata would itself be data. In OOP terms, you can't have a sequence of Objects where the Types of the Objects are themselves Objects and also stored in the sequence. You'd fall into an infinite regress.

So that's why I gave an example of an abstraction that can be made concrete without losing any of the properties of the abstraction. Numbers and files can be losslessly mapped back and forth to each other.
I suppose to advance your own argument you'd need to at least give reasons why, starting from an unambiguous description of "dogness", you couldn't create an object that's nothing but dog (with no undoggy properties). That might get us closer to why particular dogs exist despite being undefined, while numbers (supposedly) don't despite being defined.

[Mathhead200]

I meant hat you lose some of the power when you move away from the abstract. Things become more general and hence more widely applicable. I'm not so sure we could create a perfect quintessential "dog." But I also don't think the analogy is proper here. Dogs are not well defined like numbers are.