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> "the quantities that they correspond to are called 'numbers'." is rather circular here no? No. "Quantity" is different from "number". A quantity is a numbe

by 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 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.
threatofrain 5y ago
> No, it isn't. You're just wrong about that. > Now, it is true that there are different kinds of numbers in math, but each different kind of number is formally defined. All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it. > though explaining why is much too long for an HN comment. I'm surprised you're still arguing here, basically from first principles and ad hoc arguments, as to whether there exists in current professional discourse a formal model for the concepts you want. You can just Name. The. Model. Imagine I said that a bacon McDonalds burger doesn't exist, and you decided to argue from internal and ad hoc reasonings as to whether McDonalds most assuredly has a bacon burger — when you could've just Named. The. Burger. You know, like the Bacon McDouble. Anyways, to be clear, "number" is an informal concept among mathematicians.
- lisper 5y ago
  If we were arguing about bacon burgers then I would not argue from first principles. I would just invite you to a restaurant and show you a bacon burger. You're right that there is no formal definition of "number" but there is a formal definition of "natural number" and "real number" and "imaginary number" and "complex number" and "rational number" and integers and quaternions and octonians, all of which are kinds of numbers even though their names don't contain the word "number"). All of those different kinds of formally defined numbers collectively are what a "number" is. If you want to call that informal, fine, but that seems to me like a pretty tiny garnish of informality on top of a mountain of formality.
  - threatofrain 5y ago
    > If you want to call that informal, fine, but that seems to me like a pretty tiny garnish of informality on top of a mountain of formality. Perhaps you forgot, but you decided that this sin of sprinkling a tiny garnish of informality was so circular and awful that you had to stop reading right there, and provide paragraphs of explanation as to how "number" is a formal concept. If you really thought this was just a tiny garnish of informality atop a mountain of formalism, then I don't know why this is the point you had to stop reading to warn people that this work sucks.

by lisper 5y ago

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 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.

threatofrain 5y ago

> No, it isn't. You're just wrong about that. > Now, it is true that there are different kinds of numbers in math, but each different kind of number is formally defined. All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it. > though explaining why is much too long for an HN comment. I'm surprised you're still arguing here, basically from first principles and ad hoc arguments, as to whether there exists in current professional discourse a formal model for the concepts you want. You can just Name. The. Model. Imagine I said that a bacon McDonalds burger doesn't exist, and you decided to argue from internal and ad hoc reasonings as to whether McDonalds most assuredly has a bacon burger — when you could've just Named. The. Burger. You know, like the Bacon McDouble. Anyways, to be clear, "number" is an informal concept among mathematicians.

lisper 5y ago
If we were arguing about bacon burgers then I would not argue from first principles. I would just invite you to a restaurant and show you a bacon burger. You're right that there is no formal definition of "number" but there is a formal definition of "natural number" and "real number" and "imaginary number" and "complex number" and "rational number" and integers and quaternions and octonians, all of which are kinds of numbers even though their names don't contain the word "number"). All of those different kinds of formally defined numbers collectively are what a "number" is. If you want to call that informal, fine, but that seems to me like a pretty tiny garnish of informality on top of a mountain of formality.
- threatofrain 5y ago
  > If you want to call that informal, fine, but that seems to me like a pretty tiny garnish of informality on top of a mountain of formality. Perhaps you forgot, but you decided that this sin of sprinkling a tiny garnish of informality was so circular and awful that you had to stop reading right there, and provide paragraphs of explanation as to how "number" is a formal concept. If you really thought this was just a tiny garnish of informality atop a mountain of formalism, then I don't know why this is the point you had to stop reading to warn people that this work sucks.