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Based on my reading you dont seem to fundamentally disagree with the author. I think hes just playing with the ambiguity of the word number for some effect. So

Based on my reading you dont seem to fundamentally disagree with the author. I think hes just playing with the ambiguity of the word number for some effect. So: (Some mathematical formalism for) numbers are objects you build to act like the (informal) concept of a number.

lisper 5y ago
No! There is nothing informal about the mathematical concept of a number. And numbers can be defined without resorting to circular definitions that include the word "number" in the definition. It just so happens that the symbols that denote numbers behave like certain things in the real world when manipulated according to the rules that are customarily associated with the symbols. Those things in the real world have properties that correspond to the informal idea of numbers. But that has absolutely nothing to do with the math (except insofar as it's one of the things that makes math worth doing).
- threatofrain 5y ago
  "Number" is an informal concept in mathematics. The real numbers are a specific concept with multiple formalizations. A model of a number system is an assembly of behaviors, rules, or objects which behave like the numbers you want. I don't see the circularity here, and given that they are tackling an informal concept of number, I don't see any loss of clarity either.
  - lisper 5y ago
    > A number is an informal concept in mathematics. 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.
    - threatofrain 5y ago
      > All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it. Then you could simply point to the formalism which unites various mathematical activity under some universal model of number. I will reiterate the claim again — "number" is an informal concept among mathematicians and they are fine with that.

lisper 5y ago

No! There is nothing informal about the mathematical concept of a number. And numbers can be defined without resorting to circular definitions that include the word "number" in the definition. It just so happens that the symbols that denote numbers behave like certain things in the real world when manipulated according to the rules that are customarily associated with the symbols. Those things in the real world have properties that correspond to the informal idea of numbers. But that has absolutely nothing to do with the math (except insofar as it's one of the things that makes math worth doing).

threatofrain 5y ago
"Number" is an informal concept in mathematics. The real numbers are a specific concept with multiple formalizations. A model of a number system is an assembly of behaviors, rules, or objects which behave like the numbers you want. I don't see the circularity here, and given that they are tackling an informal concept of number, I don't see any loss of clarity either.
- lisper 5y ago
  > A number is an informal concept in mathematics. 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.
  - threatofrain 5y ago
    > All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it. Then you could simply point to the formalism which unites various mathematical activity under some universal model of number. I will reiterate the claim again — "number" is an informal concept among mathematicians and they are fine with that.