5 ms·

"the quantities that they correspond to are called 'numbers'." is rather circular here no? You haven't really defined quantities or numbers except in terms of e

"the quantities that they correspond to are called 'numbers'." is rather circular here no? You haven't really defined quantities or numbers except in terms of each other and yet at least one of these seems important enough to be called a mathematical concept that is distinct from a numeral. Put another way, we clearly can distinguish many different numeral systems as referring to the "same thing." We can encode that "same thing" symbolically via a set of logical symbols to form axioms, but even there the same issue arises. Encoding the natural numbers via an embedding of second-order Peano Axioms in first-order ZFC or via a straight FOL Peano Axioms definition also seems to fall flat, as we still have not been able to capture the idea that "these are all really the same thing" since these are now different sets of symbols and rules. And we haven't even touched the thorny issue of what "encoding" and more generally "mapping" really means if everything is just symbols and rules. But perhaps you are fine with infinite egress (which is perfectly defensible). The natural numbers have a more distressing problem when it comes to Godel's Incompleteness Theorems. There is presumably only one "true" set of natural numbers in our universe because the natural numbers have physical ramifications. That is for any symbolic, FOL statement of the natural numbers, we can create a corresponding physical machine whose observable behavior is dependent on whether that statement is true or false. And yet by Godel's incompleteness theorem we can never hope to fully capture the rules and symbols that determine these "natural numbers," and yet presumably most people would agree that these natural numbers exist and are a valid object of mathematical study. So how does that fit in? What do we call the subject that studies these objects? What do we even call these objects if not mathematical numbers? (As an aside I'm perhaps more of a formalist than I let on, but I find the realist side a fun playground to explore in.)

lisper 5y ago
> "the quantities that they correspond to are called 'numbers'." is rather circular here no? No. "Quantity" is different from "number". A quantity is a number plus a unit. "Two" is a number, not a quantity. "Two sheep" is a quantity. This is definitional progress because I can actually show you two sheep in order to explain to you what that means, whereas I cannot show you "two". > infinite egress It is not an infinite regress, though explaining why is much too long for an HN comment. It also makes an interesting little puzzle to figure out where and how the recursion bottoms out, but here's a hint: how do you know what a "sheep" is?
- dwohnitmok 5y ago
  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 quantities that they correspond to are called 'numbers'." is rather circular here no? No. "Quantity" is different from "number". A quantity is a number plus a unit. "Two" is a number, not a quantity. "Two sheep" is a quantity. This is definitional progress because I can actually show you two sheep in order to explain to you what that means, whereas I cannot show you "two". > infinite egress It is not an infinite regress, though explaining why is much too long for an HN comment. It also makes an interesting little puzzle to figure out where and how the recursion bottoms out, but here's a hint: how do you know what a "sheep" is?

dwohnitmok 5y ago
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.