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It's surprising that this was written in 2007. This article just perpetuates the tired old myth that there is something special about set theory... At its heart

by fmap 9y ago

It's surprising that this was written in 2007. This article just perpetuates the tired old myth that there is something special about set theory... At its heart, set theory allows us to encode certain mathematical structures more or less naturally, but things are not "made out of sets". For example, the real numbers are not sets in the same sense that "the map is not the territory". Sets allow us to encode real numbers, but this encoding is mostly arbitrary. Is 3 an element of pi? If pi truly is a set, then this is a sensible question to ask, since sets are defined by their members. However, there is evidently an abstract concept of "real numbers" which exists without mentioning the elements of a real number. The technical reflection of this dilemma is the fact that in set theory there are multiple isomorphic copies of "the complete archimedean ordered field" which are all different as sets. This is one reason why type theory is superior to set theory. In type theory we can actually capture the abstract concept of real number and work with it, instead of always working with an encoding. This might not matter so much for something as simple as a real number, but it matters a lot when you talk about more complicated structures.

cpsempek 9y ago
So, while type theory may have the benefit of resolving the "isomorphism issue" you describe, how is type theory not also another encoding of, for example, the real numbers?
olfactory 9y ago
> This is one reason why type theory is superior to set theory. In type theory we can actually capture the abstract concept of real number and work with it, instead of always working with an encoding. Could you give a small example of how a simple bit of reasoning about a real (or even natural) number would work in both frameworks?
mathsandphysics 9y ago
Well, I wouldn't say type theory is superior. Real numbers can actually be logically constructed as sets. So yes, Pi is a set. For example, you can define the set of real numbers as the set of equivalent classes of rational cauchy sequences. In order words, Pi is a set of rational cauchy sequences that are equivalent (this set is infinite). In simple (non formal) words: Pi is the set of sequences of rational numbers that get closer and closer to the numerical value of Pi. >>>> Is 3 an element of pi? If pi truly is a set Pi being a set doesn't mean that it is a set of real numbers. In fact, that would be illogical (recursive). If Pi is a set, it must be a set of things that are not real numbers. So it is not logical to ask if 3 is an element of Pi. You can ask if 3 is equal to Pi, in which case the answer is no.
- cgmg 9y ago
  > Real numbers can actually be logically constructed as sets. So yes, Pi is a set. You missed the point. The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. That would entail that Cauchy sequences and Dedekind cuts, for example, are the exact same thing, which they are not. Rather, they are two different ways of encoding the same abstract structure (a Dedekind-complete ordered field). > Pi being a set doesn't mean that it is a set of real numbers. This is dodging the question. Is the set containing the empty set a member of pi? Such a question is meaningless without fixing a particular encoding, and it has nothing to do with the intrinsic properties of a real number.
  - mathsandphysics 9y ago
    Thanks for your reply. >The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is - unless you assume that mathematics should aim to be a formal encoding of some pre-existing 'intrinsic reality', in which case what you're doing is physics and not math. In physics, the debate is more open: Is the wave function literally the electron? Or is the wave function just an encoding of the electron, which is the intrinsic real thing? Or is the electron the way we humans perceive the wave function, which is the intrinsic real thing? Anyway this is probably way too theoretical of a debate :)
    - cgmg 9y ago
      > In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is Wrong. The real numbers are literally a Dedekind-complete ordered field. The real numbers are encoded as Cauchy sequences, or as Dedekind cuts. Cauchy sequences are not Dedekind cuts: The former are equivalence classes of sequences. The latter are downward-closed sets without a greatest element. They are literally different objects. Regarding your electron comment, the answer is straightforward: the electron is an excitation of the electron spinor field, nothing more. Fields are more fundamental than particles.
pron 9y ago
While I completely agree that set theory is not special nor the foundation of mathematics (there is no such thing; foundation of mathematics is a name given to any formal language that can express all or most of mathematics), you are also repeating a common misrepresentation of set theory. While it is true that people often present, say, the natural numbers as some arbitrary encoding in sets, that is not how the natural numbers must be represented in set theory, and set theory can nicely capture the idea of an abstract inductive definition of the natural numbers, as well as the isomorphism between different representation. Many set theories employ Hilbert's epsilon (choice) operator, that allows one to "choose" some set that satisfies a proposition[1]. The question of the members of the set is completely unanswerable, and so can be reasonably said to be nonexistent. You do not work with a particular encoding, and you can't even if you wanted to, because the choice operator does not "reveal" what it is. It is true that the fact that "there exists" (in a very nonconstructive sense) some unknowable, impenetrable encoding may offend the aesthetical sensibilities of some people, but type theory does not really "resolve" that in any way (and it is certainly not superior, just different, or rather, it is superior in some ways and inferior in others[2]); it simply constructs its axioms at a higher level of abstraction. The axiomatic existence of inductive data types in type theory is just a theorem about well-founded sets in set theory, but those sets are really defined in exactly the same abstract way. [1]: So the natural numbers can be defined as "epsilon set N, s.t. N is the minimal set s.t. an element zero exists in N and there exists a function succ in the set N -> N, such that for all n in N, succ(n) is in n, and there is no n in N s.t. succ(n) = zero". [2]: For example, because most type theories are based on the lambda calculus, they are on the precipice of inconsistency due to Curry's paradox, which makes some aspects of working with them extremely unpleasant compared to set theory.
- cgmg 9y ago
  You forgot to say that succ is injective.
  - pron 9y ago
    Yeah, whatever, maybe some other stuff too. I should have written `epsilon N s.t. N is a minimal set that satisfies Peano's axioms`. The point is that it is not true that set theory implies some encoding. When Hilbert's epsilon is used, there is no encoding in any meaningful sense. The main difference between type theories and set theories is that type theories have particular syntactic discipline. Everything else is or can be a feature of set theories, too.
roenxi 9y ago
I had to chuckle when I read "as simple as a real number". If we are being strict, anything beyond the natural numbers really introduces quite a puzzle over what on earth is going on. For example, we cannot represent the value of Pi in Hindu-Arabic notation; it stretches to some 'infinity' which is just bizarre and which we can't capture on paper. Saying it has a value opens up some interesting questions about what a number or a value actually is. I suspect people have very clear opinions which are all different from each other.

by fmap 9y ago

cpsempek 9y ago

So, while type theory may have the benefit of resolving the "isomorphism issue" you describe, how is type theory not also another encoding of, for example, the real numbers?

olfactory 9y ago

> This is one reason why type theory is superior to set theory. In type theory we can actually capture the abstract concept of real number and work with it, instead of always working with an encoding. Could you give a small example of how a simple bit of reasoning about a real (or even natural) number would work in both frameworks?

mathsandphysics 9y ago

Well, I wouldn't say type theory is superior. Real numbers can actually be logically constructed as sets. So yes, Pi is a set. For example, you can define the set of real numbers as the set of equivalent classes of rational cauchy sequences. In order words, Pi is a set of rational cauchy sequences that are equivalent (this set is infinite). In simple (non formal) words: Pi is the set of sequences of rational numbers that get closer and closer to the numerical value of Pi. >>>> Is 3 an element of pi? If pi truly is a set Pi being a set doesn't mean that it is a set of real numbers. In fact, that would be illogical (recursive). If Pi is a set, it must be a set of things that are not real numbers. So it is not logical to ask if 3 is an element of Pi. You can ask if 3 is equal to Pi, in which case the answer is no.

cgmg 9y ago
> Real numbers can actually be logically constructed as sets. So yes, Pi is a set. You missed the point. The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. That would entail that Cauchy sequences and Dedekind cuts, for example, are the exact same thing, which they are not. Rather, they are two different ways of encoding the same abstract structure (a Dedekind-complete ordered field). > Pi being a set doesn't mean that it is a set of real numbers. This is dodging the question. Is the set containing the empty set a member of pi? Such a question is meaningless without fixing a particular encoding, and it has nothing to do with the intrinsic properties of a real number.
- mathsandphysics 9y ago
  Thanks for your reply. >The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is - unless you assume that mathematics should aim to be a formal encoding of some pre-existing 'intrinsic reality', in which case what you're doing is physics and not math. In physics, the debate is more open: Is the wave function literally the electron? Or is the wave function just an encoding of the electron, which is the intrinsic real thing? Or is the electron the way we humans perceive the wave function, which is the intrinsic real thing? Anyway this is probably way too theoretical of a debate :)
  - cgmg 9y ago
    > In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is Wrong. The real numbers are literally a Dedekind-complete ordered field. The real numbers are encoded as Cauchy sequences, or as Dedekind cuts. Cauchy sequences are not Dedekind cuts: The former are equivalence classes of sequences. The latter are downward-closed sets without a greatest element. They are literally different objects. Regarding your electron comment, the answer is straightforward: the electron is an excitation of the electron spinor field, nothing more. Fields are more fundamental than particles.

pron 9y ago

While I completely agree that set theory is not special nor the foundation of mathematics (there is no such thing; foundation of mathematics is a name given to any formal language that can express all or most of mathematics), you are also repeating a common misrepresentation of set theory. While it is true that people often present, say, the natural numbers as some arbitrary encoding in sets, that is not how the natural numbers must be represented in set theory, and set theory can nicely capture the idea of an abstract inductive definition of the natural numbers, as well as the isomorphism between different representation. Many set theories employ Hilbert's epsilon (choice) operator, that allows one to "choose" some set that satisfies a proposition[1]. The question of the members of the set is completely unanswerable, and so can be reasonably said to be nonexistent. You do not work with a particular encoding, and you can't even if you wanted to, because the choice operator does not "reveal" what it is. It is true that the fact that "there exists" (in a very nonconstructive sense) some unknowable, impenetrable encoding may offend the aesthetical sensibilities of some people, but type theory does not really "resolve" that in any way (and it is certainly not superior, just different, or rather, it is superior in some ways and inferior in others[2]); it simply constructs its axioms at a higher level of abstraction. The axiomatic existence of inductive data types in type theory is just a theorem about well-founded sets in set theory, but those sets are really defined in exactly the same abstract way. [1]: So the natural numbers can be defined as "epsilon set N, s.t. N is the minimal set s.t. an element zero exists in N and there exists a function succ in the set N -> N, such that for all n in N, succ(n) is in n, and there is no n in N s.t. succ(n) = zero". [2]: For example, because most type theories are based on the lambda calculus, they are on the precipice of inconsistency due to Curry's paradox, which makes some aspects of working with them extremely unpleasant compared to set theory.

cgmg 9y ago
You forgot to say that succ is injective.
- pron 9y ago
  Yeah, whatever, maybe some other stuff too. I should have written `epsilon N s.t. N is a minimal set that satisfies Peano's axioms`. The point is that it is not true that set theory implies some encoding. When Hilbert's epsilon is used, there is no encoding in any meaningful sense. The main difference between type theories and set theories is that type theories have particular syntactic discipline. Everything else is or can be a feature of set theories, too.

roenxi 9y ago

I had to chuckle when I read "as simple as a real number". If we are being strict, anything beyond the natural numbers really introduces quite a puzzle over what on earth is going on. For example, we cannot represent the value of Pi in Hindu-Arabic notation; it stretches to some 'infinity' which is just bizarre and which we can't capture on paper. Saying it has a value opens up some interesting questions about what a number or a value actually is. I suspect people have very clear opinions which are all different from each other.