Each theorem is grounded in axioms (although, one is often working many, many levels above the most basic axioms). And each axiom is independent of physical reality, so it doesn’t have a definite truth value (as long as it is not inconsistent with itself).
And each axiom is independent of physical reality,
I’m not so sure about this. IMHO mathematics is more or less a formalization of practical intuitions, so it is at least somewhat grounded in physical reality. For example the concept of natural numbers and sum, subtraction, multiplication. Set theory, etc...
IMHO mathematics is more or less a formalization of practical intuitions
The mathematics used for describing the universe most certainly are, by construction; but any particular mathematical structure is not linked to or unique to this universe. That is, the properties of the universe force us to use one specific mathematical structure to describe it, but that doesn’t mean that this is the only possible mathematical structure. This paper explains something similar, in a better way.
For example the concept of natural numbers [...] [and] Set theory
The exact properties of the natural numbers are defined by a set of axioms, and there is no reason why mathematicians in a universe without practical intuition of the natural numbers (work with me...) couldn’t still propose the axioms and derive the consequences of them (prime numbers etc.). And similarly with set theory (This actually provides a good example: infinite sets don’t have a physical basis but we can still work with them abstractly).
but any particular mathematical structure is not linked to or unique to this universe.
How can you be sure? Every mathematical structure has to be represented in a physical brain. So the mathematical structures are constrained by the physicality of this universe.
I’m not sure I understand what you’re saying. I can imagine a lot of things that don’t exist in our universe, from magic flying ponies to Cthulhu. Some of those things are physically impossible; and yet, this imagination still takes place in my physical brain… Doesn’t it ?
That wasn’t my point. Whatever you are imagining has to be represented in your brain even if it is a phantasy. Can you imagine an infinite set in your brain? I’m not talking about the concept of an infinite set, but an actual infinite set?
I’m not entirely clear on the categorical difference between the concept of an infinite set, and an actual infinite set. Aren’t sets concepts to begin with, even finite ones ?
If it is invented, then any particular piece of mathematics doesn’t exist until someone thinks it up (i.e. requires a physical brain).
If it is discovered, then all mathematics exists (in some sense), but humanity can only see a small portion of the whole (and it being in a physical brain or not is irrelevant).
The “controversy” about infinite sets is about their existence/usefulness as physical objects, not their mathematical existence (you’ll note that I was careful to say that they were not physical objects in the grandparent). From that article:
“Infinite set atheism” is a tongue-in-cheek phrase used by Eliezer Yudkowsky to describe his doubt that infinite sets of things exist in the physical universe.
Thus, infinite sets are a perfect example of a mathematical object disconnected from physical reality/practical experience.
We can construct the natural numbers by starting with two symbols “0” and “1″ that are naturals, and saying that if n is a natural, then n+1 is too i.e. adding 1 over and over again. Part of the definition is each time we add 1, we get a number we haven’t seen before; and so we have an infinite set by construction. And we can make bigger ones by taking the power set (the power set always has a larger cardinality then the set it comes from).
So infinite sets are definitely mathematical objects because we can (and just have) construct them.
scroll to 4:40
I like his one argument: if we have finite neurons and thus cannot construct an infinite set in our “map” what makes you think that you can make it correspond to a (hypothetical) infinity in the territory?
scroll to 4:40 I like his one argument: if we have finite neurons and thus cannot construct an infinite set in our “map” what makes you think that you can make it correspond to a (hypothetical) infinity in the territory?
I don’t really see what this argument comes to. The map-territory metaphor is a metaphor; neural structures do not have to literally resemble the structures they have beliefs about. In fact, if they did, then the objection would work for any finite structure that had more members than there are synapses (or whatever) in the brain.
If he is saying that infinite sets are a mathematical impossibility then he is wrong.
But I’m fairly sure that he is saying they are a physical impossibility. Which is not at all unreasonable. (this is the “territory” I think he is talking about)
I have a feeling we are working with different definitions of the “mathematics”. I think your definition of “mathematics” might be “symbols that occur in physics and can be manipulated to give answers about the universe”.
My definition is something like “set of axioms ⇒ conclusions about the structure of the object generated by the axioms” (which includes things like the real numbers, which gives calculus, so the first version of “mathematics” is included the second).
Part of the definition is each time we add 1, we get a number we haven’t seen before; and so we have an infinite set by construction.
No. You have a rule that hypothetically would produce an infinite set if applied ad infinitum. This may seem like nitpicking but there is a difference between the concept of an infinite set and an actual infinite set, the latter can’t be represented in a finite brain(I suppose).
I can write down the rules of a turing machine, but this doesn’t produce a working computer to spring to life if you get my point.
No. You have a rule that hypothetically would produce an infinite set if applied ad infinitum.
Yep, exactly; no problem with that, that’s how mathematics works. There is only a problem if someone wants to write down every element of an infinite set.
there is a difference between the concept of an infinite set and an actual infinite set
This is mathematics. The concept of a mathematical object is the object, because the “concept” version satisfies all the same rules (axioms) as any “actual” version, and these rules completely describe its structure, and (broadly) mathematics is the study of structure/patterns.
One does not need a physical basis for these rules, and so one does not need a physical basis for structures generated by such rules.
Isn’t pure mathematics a counterexample?
Each theorem is grounded in axioms (although, one is often working many, many levels above the most basic axioms). And each axiom is independent of physical reality, so it doesn’t have a definite truth value (as long as it is not inconsistent with itself).
I’m not so sure about this. IMHO mathematics is more or less a formalization of practical intuitions, so it is at least somewhat grounded in physical reality. For example the concept of natural numbers and sum, subtraction, multiplication. Set theory, etc...
The mathematics used for describing the universe most certainly are, by construction; but any particular mathematical structure is not linked to or unique to this universe. That is, the properties of the universe force us to use one specific mathematical structure to describe it, but that doesn’t mean that this is the only possible mathematical structure. This paper explains something similar, in a better way.
The exact properties of the natural numbers are defined by a set of axioms, and there is no reason why mathematicians in a universe without practical intuition of the natural numbers (work with me...) couldn’t still propose the axioms and derive the consequences of them (prime numbers etc.). And similarly with set theory (This actually provides a good example: infinite sets don’t have a physical basis but we can still work with them abstractly).
How can you be sure? Every mathematical structure has to be represented in a physical brain. So the mathematical structures are constrained by the physicality of this universe.
I’m not sure I understand what you’re saying. I can imagine a lot of things that don’t exist in our universe, from magic flying ponies to Cthulhu. Some of those things are physically impossible; and yet, this imagination still takes place in my physical brain… Doesn’t it ?
That wasn’t my point. Whatever you are imagining has to be represented in your brain even if it is a phantasy. Can you imagine an infinite set in your brain? I’m not talking about the concept of an infinite set, but an actual infinite set?
Your question doesn’t make sense. Can you represent an actual elephant in your head?
I’m not entirely clear on the categorical difference between the concept of an infinite set, and an actual infinite set. Aren’t sets concepts to begin with, even finite ones ?
Would you say that mathematics is invented? Or that it is discovered?
Good question. I don’t know and honestly I don’t care. It is one of those deep philosophical question that can be debated ad nauseum.
But it is relevant to this discussion:
If it is invented, then any particular piece of mathematics doesn’t exist until someone thinks it up (i.e. requires a physical brain).
If it is discovered, then all mathematics exists (in some sense), but humanity can only see a small portion of the whole (and it being in a physical brain or not is irrelevant).
there is some controversy surrounding infinite sets for example: http://wiki.lesswrong.com/wiki/Talk:Infinite_set_atheism
The “controversy” about infinite sets is about their existence/usefulness as physical objects, not their mathematical existence (you’ll note that I was careful to say that they were not physical objects in the grandparent). From that article:
Thus, infinite sets are a perfect example of a mathematical object disconnected from physical reality/practical experience.
We can construct the natural numbers by starting with two symbols “0” and “1″ that are naturals, and saying that if n is a natural, then n+1 is too i.e. adding 1 over and over again. Part of the definition is each time we add 1, we get a number we haven’t seen before; and so we have an infinite set by construction. And we can make bigger ones by taking the power set (the power set always has a larger cardinality then the set it comes from).
So infinite sets are definitely mathematical objects because we can (and just have) construct them.
Watch Eliezers response to this question, http://www.youtube.com/watch?v=3dufqGC8X8c
scroll to 4:40 I like his one argument: if we have finite neurons and thus cannot construct an infinite set in our “map” what makes you think that you can make it correspond to a (hypothetical) infinity in the territory?
I don’t really see what this argument comes to. The map-territory metaphor is a metaphor; neural structures do not have to literally resemble the structures they have beliefs about. In fact, if they did, then the objection would work for any finite structure that had more members than there are synapses (or whatever) in the brain.
If he is saying that infinite sets are a mathematical impossibility then he is wrong.
But I’m fairly sure that he is saying they are a physical impossibility. Which is not at all unreasonable. (this is the “territory” I think he is talking about)
I have a feeling we are working with different definitions of the “mathematics”. I think your definition of “mathematics” might be “symbols that occur in physics and can be manipulated to give answers about the universe”.
My definition is something like “set of axioms ⇒ conclusions about the structure of the object generated by the axioms” (which includes things like the real numbers, which gives calculus, so the first version of “mathematics” is included the second).
No. You have a rule that hypothetically would produce an infinite set if applied ad infinitum. This may seem like nitpicking but there is a difference between the concept of an infinite set and an actual infinite set, the latter can’t be represented in a finite brain(I suppose).
I can write down the rules of a turing machine, but this doesn’t produce a working computer to spring to life if you get my point.
Yep, exactly; no problem with that, that’s how mathematics works. There is only a problem if someone wants to write down every element of an infinite set.
This is mathematics. The concept of a mathematical object is the object, because the “concept” version satisfies all the same rules (axioms) as any “actual” version, and these rules completely describe its structure, and (broadly) mathematics is the study of structure/patterns.
One does not need a physical basis for these rules, and so one does not need a physical basis for structures generated by such rules.