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So then of course the question arises, what is "two?" And is that a mathematical object? (Shorthand for S(S(0)) doesn't work either because S(S(0)) is clearly j

So then of course the question arises, what is "two?" And is that a mathematical object? (Shorthand for S(S(0)) doesn't work either because S(S(0)) is clearly just another numeral system, not a number itself, indeed arguably any formalism is just a numeral system, rather than the number itself) And how does that comport with the reality of natural numbers in our universe (i.e. the incompleteness argument)? > how do you know what a "sheep" is? Right the usual answer is to tackle the map-territory head-on and say some of it is essentially "out of scope" but I'm curious if you had another idea.

lisper 5y ago
> what is "two?" Two is the property shared by all of the things that behave the same as the collection of things which contain all collections of things which contain all collections of things which contain nothing, with respect to a set of rules that allow you to make collections of things that contain some things and not other things, and check to see if a given collection of things contains some things or no things (and with the proviso that collections count as things that can be parts of collections of things). :-) > I'm curious if you had another idea. See: http://blog.rongarret.info/2015/03/why-some-assumptions-are-better-than.html http://blog.rongarret.info/2015/03/why-some-assumptions-are-... This needs some revision, but it contains the essential idea.
- dwohnitmok 5y ago
  > Two is the property shared by all of the things that behave the same as the collection of things which contain all collections of things which contain all collections of things which contain nothing, with respect to a set of rules that allow you to make collections of things that contain some things and not other things, and check to see if a given collection of things contains some things or no things (and with the proviso that collections count as things that can be parts of collections of things). That is the standard Zermelo definition. But why not the von Neumann definition (or indeed a non-set-theory definition altogther?). By your definition, and presumably, your definition of "one" I could claim that the statement "two contains one" is true. And yet by the von Neumann definition (i.e. the property shared by the collection of things which contains the collection of things that contain nothing and which contains all collections of things that contain the collection of things that contain nothing), "two contains one" is false. And yet most mathematicians would tell you that this statement is a category error without a true or false value. How do you distinguish between all of them? What is the truth value of "two contains one?" All of these are hallmarks (turning category errors into sensible questions and having different answers to nonessential questions depending on the vagaries of the definition) that the definition you've provided is the definition of a numeral system, not a number itself. And again more to the central point, how can you reconcile all of this with Godel's incompleteness theorem and the reality of natural numbers as expressed by physical machines in our universe? > This needs some revision, but it contains the essential idea. Right I think you're making the same appeal in your article as the one made here: https://www.lesswrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple-truth https://www.lesswrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple... which is basically the map-territory definition is "out of scope" and more a question of prediction of reality. EDIT: I must make a modification to the "two contains one" statement to instead be "any collection that has the property called 'two' contains a collection that shares the property 'one'" since we are indeed talking about the properties rather than the collections themselves.
  - lisper 5y ago
    > the statement "two contains one" is true No. Look more carefully at my definition. On my definition, two is not a set, it is a property of a set.
    - dwohnitmok 5y ago
      Right sorry I just edited my comment (presumably just as you were making your reply). EDIT: As an aside, I think most mathematicians at this point would be shaking their heads vigorously at calling "two" a property and would be counting that as a category error.
      - lisper 5y ago
        > any collection that has the property called 'two' contains a collection that shares the property 'one' Yes. That is a true statement. It is (more or less) the observation that given a set S with two elements, you can make a subset of S that has one element. (Indeed you can make two such subsets :-) Note that the reason I put it in terms of a property was specifically go get around these kinds of objections. My definition of two is, essentially, "anything that behaves effectively the same as this definition of two". I picked the ZF definition of two as "this definition" but I didn't have to. I could have picked something else, and then the phraseology would have been different. But it all eventually bottoms out in the theory of computability. You can start with ZF or Turing Machines, or the lambda calculus, or Godel numbers, or Brainfuck. It doesn't matter. All starting points lead to the same destination.
        dwohnitmok 5y ago
        > It is (more or less) the observation that given a set S with two elements, you can make a subset of S that has one element. But your formal definition and your explanation do not match. Effectively you are conflating defining the "cardinality of a set is 2" with "2." To illustrate this, your formal definition can be satisfied by a set that only has one member ({{0}} with 0 as the empty set)! It has no strict subset with one element! Your formal definition is but a numeral system, your informal definition is the number itself. Indeed nothing changes in spirit if we start with "{0}" as 0 instead of the empty set itself, but your formal definition fails in another way. Another more dramatic illustration is just to define this entirely with natural number objects in category theory, which do away with properties of sets and collections entirely. It leads to statements which are also true, but cannot even be phrased in the language of set theory. > I picked the ZF definition of two as "this definition" but I didn't have to. I could have picked something else, and then the phraseology would have been different. But it all eventually bottoms out in the theory of computability. You can start with ZF or Turing Machines, or the lambda calculus, or Godel numbers, or Brainfuck. It doesn't matter. All starting points lead to the same destination. This is the entire point, Turing Machines, lambda calculus, Godel numbers are all different sets of rules and symbols. Yet we recognize that they are all the "same destination" somehow and that destination is the thing we want, not the rules and symbols themselves, which are, as you say, only the "starting point." In effect, none of these definitions of natural number can fully capture what a natural number is. They each come with their own infelicities and ultimately end up as numeral systems when fully formalized rather than the natural numbers themselves. I agree that computability is one of the key insights behind mathematics, but moving symbols around is only one encoding of computability. EDIT: Another thing you still haven't addressed is the issue of the reality of natural numbers and Godel's incompleteness theorems and how we are allowed to talk about "natural numbers" as mathematical objects in that scenario.

lisper 5y ago

> what is "two?" Two is the property shared by all of the things that behave the same as the collection of things which contain all collections of things which contain all collections of things which contain nothing, with respect to a set of rules that allow you to make collections of things that contain some things and not other things, and check to see if a given collection of things contains some things or no things (and with the proviso that collections count as things that can be parts of collections of things). :-) > I'm curious if you had another idea. See: http://blog.rongarret.info/2015/03/why-some-assumptions-are-better-than.html http://blog.rongarret.info/2015/03/why-some-assumptions-are-... This needs some revision, but it contains the essential idea.

dwohnitmok 5y ago
> Two is the property shared by all of the things that behave the same as the collection of things which contain all collections of things which contain all collections of things which contain nothing, with respect to a set of rules that allow you to make collections of things that contain some things and not other things, and check to see if a given collection of things contains some things or no things (and with the proviso that collections count as things that can be parts of collections of things). That is the standard Zermelo definition. But why not the von Neumann definition (or indeed a non-set-theory definition altogther?). By your definition, and presumably, your definition of "one" I could claim that the statement "two contains one" is true. And yet by the von Neumann definition (i.e. the property shared by the collection of things which contains the collection of things that contain nothing and which contains all collections of things that contain the collection of things that contain nothing), "two contains one" is false. And yet most mathematicians would tell you that this statement is a category error without a true or false value. How do you distinguish between all of them? What is the truth value of "two contains one?" All of these are hallmarks (turning category errors into sensible questions and having different answers to nonessential questions depending on the vagaries of the definition) that the definition you've provided is the definition of a numeral system, not a number itself. And again more to the central point, how can you reconcile all of this with Godel's incompleteness theorem and the reality of natural numbers as expressed by physical machines in our universe? > This needs some revision, but it contains the essential idea. Right I think you're making the same appeal in your article as the one made here: https://www.lesswrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple-truth https://www.lesswrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple... which is basically the map-territory definition is "out of scope" and more a question of prediction of reality. EDIT: I must make a modification to the "two contains one" statement to instead be "any collection that has the property called 'two' contains a collection that shares the property 'one'" since we are indeed talking about the properties rather than the collections themselves.
- lisper 5y ago
  > the statement "two contains one" is true No. Look more carefully at my definition. On my definition, two is not a set, it is a property of a set.
  - dwohnitmok 5y ago
    Right sorry I just edited my comment (presumably just as you were making your reply). EDIT: As an aside, I think most mathematicians at this point would be shaking their heads vigorously at calling "two" a property and would be counting that as a category error.
    - lisper 5y ago
      > any collection that has the property called 'two' contains a collection that shares the property 'one' Yes. That is a true statement. It is (more or less) the observation that given a set S with two elements, you can make a subset of S that has one element. (Indeed you can make two such subsets :-) Note that the reason I put it in terms of a property was specifically go get around these kinds of objections. My definition of two is, essentially, "anything that behaves effectively the same as this definition of two". I picked the ZF definition of two as "this definition" but I didn't have to. I could have picked something else, and then the phraseology would have been different. But it all eventually bottoms out in the theory of computability. You can start with ZF or Turing Machines, or the lambda calculus, or Godel numbers, or Brainfuck. It doesn't matter. All starting points lead to the same destination.
      - dwohnitmok 5y ago
        > It is (more or less) the observation that given a set S with two elements, you can make a subset of S that has one element. But your formal definition and your explanation do not match. Effectively you are conflating defining the "cardinality of a set is 2" with "2." To illustrate this, your formal definition can be satisfied by a set that only has one member ({{0}} with 0 as the empty set)! It has no strict subset with one element! Your formal definition is but a numeral system, your informal definition is the number itself. Indeed nothing changes in spirit if we start with "{0}" as 0 instead of the empty set itself, but your formal definition fails in another way. Another more dramatic illustration is just to define this entirely with natural number objects in category theory, which do away with properties of sets and collections entirely. It leads to statements which are also true, but cannot even be phrased in the language of set theory. > I picked the ZF definition of two as "this definition" but I didn't have to. I could have picked something else, and then the phraseology would have been different. But it all eventually bottoms out in the theory of computability. You can start with ZF or Turing Machines, or the lambda calculus, or Godel numbers, or Brainfuck. It doesn't matter. All starting points lead to the same destination. This is the entire point, Turing Machines, lambda calculus, Godel numbers are all different sets of rules and symbols. Yet we recognize that they are all the "same destination" somehow and that destination is the thing we want, not the rules and symbols themselves, which are, as you say, only the "starting point." In effect, none of these definitions of natural number can fully capture what a natural number is. They each come with their own infelicities and ultimately end up as numeral systems when fully formalized rather than the natural numbers themselves. I agree that computability is one of the key insights behind mathematics, but moving symbols around is only one encoding of computability. EDIT: Another thing you still haven't addressed is the issue of the reality of natural numbers and Godel's incompleteness theorems and how we are allowed to talk about "natural numbers" as mathematical objects in that scenario.