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Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual ans

by xamuel 8y ago

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing). In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too). The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to the reader: define appropriate arithmetical functions on the naturals. Define a relation ~ on pairs (a,b) of naturals by saying (a,b)~(c,d) if and only if b+c=a+d. Exercise to the reader: ~ is an equivalence relation. Its equivalence classes are called "integers". The equivalence class containing (a,b) represents the integer b-a. For example, the integer -1 is the equivalence class {(1,0),(2,1),(3,2),...}. Exercise: define appropriate arithmetical functions on the integers. Define a relation ~ on pairs (m,n) (n nonzero) of integers by saying (m,n)~(p,q) if and only if mq=pn. Exercise: Show ~ is an equivalence relation. Its equivalence classes are called "rationals". The equivalence class containing (m,n) represents the rational m/n. For example, 1/2 is the equivalence class {(1,2),(-1,-2),(2,4),(-2,-4),...}={(k,2k) for all nonzero integers k}. Exercise: define arithmetic operations on the rationals. To get from the rationals to the reals, see https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut

12elephant 8y ago
Your castle is built on sand. What, then, is {}?
- xamuel 8y ago
  One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset. Other axioms of set theory are used to formalize other steps in my post. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... For example, the Axiom of Infinity is used to formalize the handwavy part where I said "In general, the natural n+1 is {0,...,n}".
  - 12elephant 8y ago
    An axiom is simply something made up. At the end of the day, all of our Mathematics rests on foundations that are made up. It's difficult to say what the empty set is. Because it isn't really anything at all.
    - throwawaymath 8y ago
      There are legitimate criticisms you can levy against set theory, but I'm starting to lose you here. I'm not really following your point anymore - this seems like arguing about whether or not mathematics is invented or discovered. Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?
      - 12elephant 8y ago
        The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism". What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down. If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement: > In mathematics you don't understand things. You just get used to them.
        throwawaymath 8y ago
        In that case I'd agree with you. I'm not particularly keen on rehashing foundations of mathematics either.
    - xamuel 8y ago
      Do you complain similarly when Euclid states that for any two distinct points, there exists an infinitely long straight line which passes through those two points? We cannot point to that line, we can only point to a small portion of it within our field of vision, and have to extrapolate it to infinity. Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.
- dahfizz 8y ago
  > the natural 0 is the empty set.
  - zazen 8y ago
    That is a definition of "natural 0", not a definition of "empty set". GP was pointing out that GGP was using the concept of "empty set" to provide a definition for the natural numbers, without having first provided a definition for the empty set. Replies that consist solely of throwing a quote at someone are kind of rude even if you're in the right.
    - dahfizz 8y ago
      I don't see why GGP should have to reinvent set theory to use it in his construction of numbers. Set theory and the idea of {} is well defined, and hardly "built on sand".
      - zazen 8y ago
        He doesn't necessarily have to. But when he defines "natural zero" as "the empty set", and someone asks "what's the empty set?", and you say "natural zero", you're being circular.
- deleted 8y ago
  [deleted]
- throwawaymath 8y ago
  {} is the empty set, which as mentioned is natural 0.
  - zazen 8y ago
    GP understands the notation; he means to cast aspersions about the axiomatic project.
    - throwawaymath 8y ago
      Fair enough, though there weren't really any "aspersions" cast. I'm aware you can unravel set theory if you care enough about that particular exercise. But, in context, you could also interpret that comment as being written by someone who isn't aware that the empty set is defined as {}. Responding to someone's definition that natural 0 = {} that mathematics is a "castle built on sand" isn't exactly a cogent criticism of set theory.
- skh 8y ago
  All of mathematics is built on sand in the sense that there necessarily are going to be undefined terms. There’s no way around this. Think about the English language. Grab a dictionary. Look up any word in it. Look at the definition of that word. Pick a word in that definition. Repeat and eventually you’ll end up with a defintition that contains one of the words you previously looked up. So English is built on sand too and yet we are able to still communicate.
- deleted 8y ago
  [deleted]
Someone 8y ago
”an equivalent relation.” Auto-correct at work, I guess. For those not familiar with the term: if Google says did you mean equivalence relation, Google is right.
- xamuel 8y ago
  Thanks, fixed
deleted 8y ago
[deleted]
throwawaymath 8y ago
I'm getting flashbacks of Baby Rudin's first chapter from this comment :) For anyone who wants to learn more about this: what the parent comment is describing is the construction of natural numbers as cardinalities of sets. This is the (modified) Frege definition which avoids Russell's paradox - you can use that as a jumping off point for further dives into set theory and number systems. To define Peano arithmetic you start with a successor function, then an addition function, then multiplication, etc.
andrewla 8y ago
The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Not to mention that this is one possible construction of the natural numbers in set theory; in other formulations the theorem "is 2 a member of 4" would not be true, making things even more confusing. For example, we have the other standard approach of 0 = {}, 1 = {{}, 0}, n={{}, n-1}. The untyped lambda calculus and things like Church numerals have the same defect. If we can start assigning types, then we can reason about these things in a much more concrete manner without these nasty side effects of theorems that only make sense because of definitional shortcuts. We can even derive isometries between different derivations of the same objects to show that two different definitions (like the Peano numerals and binary numbers) of a "natural number" can be used exactly interchangeably, because theorems like "is 2 a member of 4" are simply not expressible given the derivation.
- throwawaymath 8y ago
  > The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent. > Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Out of curiosity, what is the category theoretic construction that avoids Russell's paradox? I don't know that there isn't one, but I can't think of it off the top of my head. I know there are category theoretic constructions in general (I responded to one someone else posted in this thread).
  - andrewla 8y ago
    The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory. Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category. Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.

by xamuel 8y ago

12elephant 8y ago

Your castle is built on sand. What, then, is {}?

xamuel 8y ago
One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset. Other axioms of set theory are used to formalize other steps in my post. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... For example, the Axiom of Infinity is used to formalize the handwavy part where I said "In general, the natural n+1 is {0,...,n}".
- 12elephant 8y ago
  An axiom is simply something made up. At the end of the day, all of our Mathematics rests on foundations that are made up. It's difficult to say what the empty set is. Because it isn't really anything at all.
  - throwawaymath 8y ago
    There are legitimate criticisms you can levy against set theory, but I'm starting to lose you here. I'm not really following your point anymore - this seems like arguing about whether or not mathematics is invented or discovered. Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?
    - 12elephant 8y ago
      The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism". What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down. If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement: > In mathematics you don't understand things. You just get used to them.
      - throwawaymath 8y ago
        In that case I'd agree with you. I'm not particularly keen on rehashing foundations of mathematics either.
  - xamuel 8y ago
    Do you complain similarly when Euclid states that for any two distinct points, there exists an infinitely long straight line which passes through those two points? We cannot point to that line, we can only point to a small portion of it within our field of vision, and have to extrapolate it to infinity. Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.
dahfizz 8y ago
> the natural 0 is the empty set.
- zazen 8y ago
  That is a definition of "natural 0", not a definition of "empty set". GP was pointing out that GGP was using the concept of "empty set" to provide a definition for the natural numbers, without having first provided a definition for the empty set. Replies that consist solely of throwing a quote at someone are kind of rude even if you're in the right.
  - dahfizz 8y ago
    I don't see why GGP should have to reinvent set theory to use it in his construction of numbers. Set theory and the idea of {} is well defined, and hardly "built on sand".
    - zazen 8y ago
      He doesn't necessarily have to. But when he defines "natural zero" as "the empty set", and someone asks "what's the empty set?", and you say "natural zero", you're being circular.
deleted 8y ago
[deleted]
throwawaymath 8y ago
{} is the empty set, which as mentioned is natural 0.
- zazen 8y ago
  GP understands the notation; he means to cast aspersions about the axiomatic project.
  - throwawaymath 8y ago
    Fair enough, though there weren't really any "aspersions" cast. I'm aware you can unravel set theory if you care enough about that particular exercise. But, in context, you could also interpret that comment as being written by someone who isn't aware that the empty set is defined as {}. Responding to someone's definition that natural 0 = {} that mathematics is a "castle built on sand" isn't exactly a cogent criticism of set theory.
skh 8y ago
All of mathematics is built on sand in the sense that there necessarily are going to be undefined terms. There’s no way around this. Think about the English language. Grab a dictionary. Look up any word in it. Look at the definition of that word. Pick a word in that definition. Repeat and eventually you’ll end up with a defintition that contains one of the words you previously looked up. So English is built on sand too and yet we are able to still communicate.
deleted 8y ago
[deleted]

Someone 8y ago

”an equivalent relation.” Auto-correct at work, I guess. For those not familiar with the term: if Google says did you mean equivalence relation, Google is right.

xamuel 8y ago
Thanks, fixed

deleted 8y ago

[deleted]

throwawaymath 8y ago

I'm getting flashbacks of Baby Rudin's first chapter from this comment :) For anyone who wants to learn more about this: what the parent comment is describing is the construction of natural numbers as cardinalities of sets. This is the (modified) Frege definition which avoids Russell's paradox - you can use that as a jumping off point for further dives into set theory and number systems. To define Peano arithmetic you start with a successor function, then an addition function, then multiplication, etc.

andrewla 8y ago

The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Not to mention that this is one possible construction of the natural numbers in set theory; in other formulations the theorem "is 2 a member of 4" would not be true, making things even more confusing. For example, we have the other standard approach of 0 = {}, 1 = {{}, 0}, n={{}, n-1}. The untyped lambda calculus and things like Church numerals have the same defect. If we can start assigning types, then we can reason about these things in a much more concrete manner without these nasty side effects of theorems that only make sense because of definitional shortcuts. We can even derive isometries between different derivations of the same objects to show that two different definitions (like the Peano numerals and binary numbers) of a "natural number" can be used exactly interchangeably, because theorems like "is 2 a member of 4" are simply not expressible given the derivation.

throwawaymath 8y ago
> The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model. Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent. > Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Out of curiosity, what is the category theoretic construction that avoids Russell's paradox? I don't know that there isn't one, but I can't think of it off the top of my head. I know there are category theoretic constructions in general (I responded to one someone else posted in this thread).
- andrewla 8y ago
  The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory. Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category. Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.