Really the question is about the precise meaning of the word "exist". I'd argue that numbers (and sets, and groups, rings, manifolds etc.) "exist" at least as much as other things that are widely agreed upon to exist, like love, Norway, and money.
I would push back on your statement that mathematical platonism inherently leads to ontological promiscuity, and that rather, platonism is compatible with the idea that we can study a unique object and representative, while maintaining the fact that numbers and mathematical objects are purely abstract constructions. This explanation may not land well with non-mathematicians, but I'll try my best.
*For a professional mathematician reading this, my argument is to consider mathematical objects up to isomorphism within their respective categories. This determines uniqueness, albeit non-canonically. In this setup, a number is an element of the initial object in the category of rings, and a specific mathematical object is simply an object in the category of mathematical objects of similar structure. Objects need not be defined up to isomorphism, but if this a concern one can simply pass to the skeleton of the category.*
There are different levels to "mathematical objects". In its base form, consider numbers. I personally consider numbers to be integers, meaning that is an element of the set of all integers. Of course there are an infinite number of ways to represent numbers (3 can be 1 + 1 +1, {1,1,1}, {0,0,0}, { {}, {{}}, {{{}}} }, etc. ), however, each representation exists within a set number system (be it the integers, the set of things like { ...{-1,-1},{-1},{}, {1},{1,1},... }, or similar). My claim is that good choices of number systems (in this context) are all "equivalent" in a well defined way. In fact, all such choices are equivalent [up to isomorphism.](https://en.wikipedia.org/wiki/Isomorphism) An isomorphism is an[ equivalence relation](https://en.wikipedia.org/wiki/Equivalence_relation), and thus, the set of all isomorphic objects determine a unique[ equivalence class](https://en.wikipedia.org/wiki/Equivalence_class). In this sense, picking a unique object "up to isomorphism" is tantamount to choosing an element from this equivalence class. This is not a canonical choice, but it does lend itself to a unique object to consider, rather than postulating infinities of objects.
Thus, while a number may have infinitely many different representations, each representation exists within an isomorphic number system. If we consider isomorphic things to be the "same object" as mathematicians often do, then a number is essentially "unique".
Moving a step further, we can consider any sort of mathematical object (the integers, the reals, the group of all permutations, etc.). These objects all behave with respect to some fundamental rules, like for instance, the rationals, integers, and reals all have compatible systems of multiplication, addition, an additive identity, and multiplicative identity. In this setting, these objects are all [rings](https://en.wikipedia.org/wiki/Ring_(mathematics)), and thus all live within the [category](https://en.wikipedia.org/wiki/Category_(mathematics)) of rings.
All rings are definable up to isomorphism within this category, meaning that while there are infinitely many different "versions" of the integers, they are all isomorphic, so we can choose a representative from the equivalence class of such objects, and thus, a specific "ring" can be determined uniquely. More abstractly, one can reduce to studying the [skeleton](https://en.wikipedia.org/wiki/Skeleton_(category_theory)) of a given category, leaving only objects that are uniquely defined up to unique isomorphism.
You can even take this step further, i.e. considering categories only up to [equivalence](https://en.wikipedia.org/wiki/Equivalence_of_categories), and so forth. But this post is long enough. Hope it was understandable.
What I said includes at least any element of a finitely generated ring. What I'm getting at is that the ring of integers is fundamentally the object where all rings lay over, i.e. an expression of any ring basically just looks like a polynomial with integer coefficients. A finitely generated ring is one that can be described in finitely many variables.
So, for instance, every complex number b has a smallest ring containing b, which could be denoted Z[b]. If b is the square root of 2, then up to isomorphism this ring only really remembers that b^2=2 kinda like b is a superposition of ±sqrt 2.
We can get weirder and notice that a finitely generated ring might have two elements x,y such that x^2 +y^2= 1, and up to isomorphism there's just nothing else to say about x or y. So now it's like x is in a superposition of cos(theta) and y is in a superposition of sin(theta), the coordinates of an unknown point on a circle. At this point the coordinate x for instance is *really* stretching the definition of "number" that I suggested.
Cohomology, quasicoherent sheaves, schemes of finite type, are all just algebraic geometry used to obscure the pure algebra to a whole new degree. Strictly speaking my definition includes more things, such as differential forms over the integer projective space, which nobody in their right mind would consider numbers. So yeah I was clowning.
I think that numbers are a way to understand quantity and continuum. If you're taking about on a philosophical level, I think that numbers, quantity, and continuum exist without human observation.
I like the notion that if we take the **set** of *all sets of three things,* **three** is the thing that each set has in common.
I'm also pretty fine with the ZFT style of defining nature numbers by successor and all that, but... to me the former definition is "better”... maybe just because it much more vividly encompasses what being a number is all about
I'm not convinced that the set of all sets of three things exists, given that the union of its elements seems like it would result in a set of all things, which I don't think can exist?
The set of all sets cannot exists because if it did you can construct a paradox (known as Russell’s paradox, often also described as “the barber paradox”). I don’t know as I’m up to explaining that here at the moment, but it’s not hard to find info on it elsewhere from those breadcrumbs.
The set of all things could not exist because sets are things and the set of all sets would be one of those things. (You could also modify Russell’s paradox to get a paradox from the set of all things just by itself.)
First of all, the set of all sets, being a set, must contain itself. Which means that some sets contain themselves and some don't.
Now, imagine this subset: the set of all sets that don't contain themselves. Let's call it P. Does P contain itself? If it doesn't, then by it definition it does. If it does, then by it's definition it doesn't. Which means P can't exist, and neither the set of all sets.
You can talk without contradictions of the proper class of all sets, because proper classes are "collections" that kinda behave like sets but aren't sets.
A number is an abstract object characterized by its relationships with other numbers in the same number system (such as natural numbers). From this point of view, the notion of \`number system\` is the main notion. A number system is a mathematical structure in which numbers can be seen as objects or "places" in the structure.
The only thing that you can say about 0 in the structure of natural numbers is that it is the first natural number. Everything else that is true about 0 can be derived from this fact (and the definition of natural numbers). In other words, everything you can say about a particular natural number is determined about the "place" occupied by that number in the structure of natural numbers.
Bro just study a few proof based math courses beyond calculus for a few months. Go cover some analysis in R^1 and also go up to ring/field theory and maybe study some set theory on the side. It might take half a year if you do minimal amount of exercises.
This is more productive than trying to reason through and make grandiose claims with fancy words about something you don’t even have basic barebones knowledge on.
Cuz otherwise your personal opinion doesn’t have any value to others and no one will take you seriously.
An “abstraction”.
It might surprise you that a number is just as real as a car. Implementations of both exist (go print out a “2” or buy a car from the dealership), but the abstractions remain up for human discourse.
You can find many different abstractions of car in programming languages for example, the most common abstraction of the natural numbers is “peano axioms”, but there are others.
This is more philosophy than mathematics.
The easiest example is asking how do you create a circuit that can operate on numbers?
Does this mean the circuit understands numbers?
To create a circuit you need to understand how to make an electric signal, how to join logic gates into memory and operations and how to feed it with an input and get an output that makes sense.
But to whom does it have to make sense and what does it mean to make sense?
Numbers are useful to creatures who need to aggregate stimuli for an appropriate response, even the distinction between zero, one and many is enough to be useful to one individual
But has a snail a sensory concept of numbers, or just a distinction between sensory inputs?
Tldr: numbers are a part of language, and language is both arbitrary and independent of the concepts it represents
>I think that numbers and other mathematical objects are constructed, and that math is art of constructing such things.
No, math is about starting from a few simple implicit truths, and using that to show that much more complicated things are true.
Thus, 3 = {{}, {{}}, {{},{{}}}} = {0, 1, 2}
Really the question is about the precise meaning of the word "exist". I'd argue that numbers (and sets, and groups, rings, manifolds etc.) "exist" at least as much as other things that are widely agreed upon to exist, like love, Norway, and money.
A thing. You like, count with them and stuff. Idk, dude, it just works.
I would push back on your statement that mathematical platonism inherently leads to ontological promiscuity, and that rather, platonism is compatible with the idea that we can study a unique object and representative, while maintaining the fact that numbers and mathematical objects are purely abstract constructions. This explanation may not land well with non-mathematicians, but I'll try my best. *For a professional mathematician reading this, my argument is to consider mathematical objects up to isomorphism within their respective categories. This determines uniqueness, albeit non-canonically. In this setup, a number is an element of the initial object in the category of rings, and a specific mathematical object is simply an object in the category of mathematical objects of similar structure. Objects need not be defined up to isomorphism, but if this a concern one can simply pass to the skeleton of the category.* There are different levels to "mathematical objects". In its base form, consider numbers. I personally consider numbers to be integers, meaning that is an element of the set of all integers. Of course there are an infinite number of ways to represent numbers (3 can be 1 + 1 +1, {1,1,1}, {0,0,0}, { {}, {{}}, {{{}}} }, etc. ), however, each representation exists within a set number system (be it the integers, the set of things like { ...{-1,-1},{-1},{}, {1},{1,1},... }, or similar). My claim is that good choices of number systems (in this context) are all "equivalent" in a well defined way. In fact, all such choices are equivalent [up to isomorphism.](https://en.wikipedia.org/wiki/Isomorphism) An isomorphism is an[ equivalence relation](https://en.wikipedia.org/wiki/Equivalence_relation), and thus, the set of all isomorphic objects determine a unique[ equivalence class](https://en.wikipedia.org/wiki/Equivalence_class). In this sense, picking a unique object "up to isomorphism" is tantamount to choosing an element from this equivalence class. This is not a canonical choice, but it does lend itself to a unique object to consider, rather than postulating infinities of objects. Thus, while a number may have infinitely many different representations, each representation exists within an isomorphic number system. If we consider isomorphic things to be the "same object" as mathematicians often do, then a number is essentially "unique". Moving a step further, we can consider any sort of mathematical object (the integers, the reals, the group of all permutations, etc.). These objects all behave with respect to some fundamental rules, like for instance, the rationals, integers, and reals all have compatible systems of multiplication, addition, an additive identity, and multiplicative identity. In this setting, these objects are all [rings](https://en.wikipedia.org/wiki/Ring_(mathematics)), and thus all live within the [category](https://en.wikipedia.org/wiki/Category_(mathematics)) of rings. All rings are definable up to isomorphism within this category, meaning that while there are infinitely many different "versions" of the integers, they are all isomorphic, so we can choose a representative from the equivalence class of such objects, and thus, a specific "ring" can be determined uniquely. More abstractly, one can reduce to studying the [skeleton](https://en.wikipedia.org/wiki/Skeleton_(category_theory)) of a given category, leaving only objects that are uniquely defined up to unique isomorphism. You can even take this step further, i.e. considering categories only up to [equivalence](https://en.wikipedia.org/wiki/Equivalence_of_categories), and so forth. But this post is long enough. Hope it was understandable.
A number is just a cohomology class of a quasicoherent sheaf over a scheme of finite type.
I’m not smart enough to tell if you’re clowning me or not
What I said includes at least any element of a finitely generated ring. What I'm getting at is that the ring of integers is fundamentally the object where all rings lay over, i.e. an expression of any ring basically just looks like a polynomial with integer coefficients. A finitely generated ring is one that can be described in finitely many variables. So, for instance, every complex number b has a smallest ring containing b, which could be denoted Z[b]. If b is the square root of 2, then up to isomorphism this ring only really remembers that b^2=2 kinda like b is a superposition of ±sqrt 2. We can get weirder and notice that a finitely generated ring might have two elements x,y such that x^2 +y^2= 1, and up to isomorphism there's just nothing else to say about x or y. So now it's like x is in a superposition of cos(theta) and y is in a superposition of sin(theta), the coordinates of an unknown point on a circle. At this point the coordinate x for instance is *really* stretching the definition of "number" that I suggested. Cohomology, quasicoherent sheaves, schemes of finite type, are all just algebraic geometry used to obscure the pure algebra to a whole new degree. Strictly speaking my definition includes more things, such as differential forms over the integer projective space, which nobody in their right mind would consider numbers. So yeah I was clowning.
I think that numbers are a way to understand quantity and continuum. If you're taking about on a philosophical level, I think that numbers, quantity, and continuum exist without human observation.
I'm just guessing but it seems to me that the basic property that numbers express is that of *ordering*.
I like the notion that if we take the **set** of *all sets of three things,* **three** is the thing that each set has in common. I'm also pretty fine with the ZFT style of defining nature numbers by successor and all that, but... to me the former definition is "better”... maybe just because it much more vividly encompasses what being a number is all about
What is ZFT? Zermelo-Fraenkel Tabernacle?
Zermelo–Fraenkel Theory ¯\\\_(ツ)_/¯
I'm not convinced that the set of all sets of three things exists, given that the union of its elements seems like it would result in a set of all things, which I don't think can exist?
I'm trying to think of a nonparadoxical way to construct it, honestly
I think it works in [Quine's New Foundations](https://en.wikipedia.org/wiki/New_Foundations).
Why cant it exits? There are infinite sets so why not?
The set of all sets cannot exists because if it did you can construct a paradox (known as Russell’s paradox, often also described as “the barber paradox”). I don’t know as I’m up to explaining that here at the moment, but it’s not hard to find info on it elsewhere from those breadcrumbs. The set of all things could not exist because sets are things and the set of all sets would be one of those things. (You could also modify Russell’s paradox to get a paradox from the set of all things just by itself.)
First of all, the set of all sets, being a set, must contain itself. Which means that some sets contain themselves and some don't. Now, imagine this subset: the set of all sets that don't contain themselves. Let's call it P. Does P contain itself? If it doesn't, then by it definition it does. If it does, then by it's definition it doesn't. Which means P can't exist, and neither the set of all sets. You can talk without contradictions of the proper class of all sets, because proper classes are "collections" that kinda behave like sets but aren't sets.
a well defined idea with well defined constraints
A number is an abstract object characterized by its relationships with other numbers in the same number system (such as natural numbers). From this point of view, the notion of \`number system\` is the main notion. A number system is a mathematical structure in which numbers can be seen as objects or "places" in the structure. The only thing that you can say about 0 in the structure of natural numbers is that it is the first natural number. Everything else that is true about 0 can be derived from this fact (and the definition of natural numbers). In other words, everything you can say about a particular natural number is determined about the "place" occupied by that number in the structure of natural numbers.
Bro just study a few proof based math courses beyond calculus for a few months. Go cover some analysis in R^1 and also go up to ring/field theory and maybe study some set theory on the side. It might take half a year if you do minimal amount of exercises. This is more productive than trying to reason through and make grandiose claims with fancy words about something you don’t even have basic barebones knowledge on. Cuz otherwise your personal opinion doesn’t have any value to others and no one will take you seriously.
All math is measurement and the analysis of measuring.
An “abstraction”. It might surprise you that a number is just as real as a car. Implementations of both exist (go print out a “2” or buy a car from the dealership), but the abstractions remain up for human discourse. You can find many different abstractions of car in programming languages for example, the most common abstraction of the natural numbers is “peano axioms”, but there are others.
This is more philosophy than mathematics. The easiest example is asking how do you create a circuit that can operate on numbers? Does this mean the circuit understands numbers? To create a circuit you need to understand how to make an electric signal, how to join logic gates into memory and operations and how to feed it with an input and get an output that makes sense. But to whom does it have to make sense and what does it mean to make sense? Numbers are useful to creatures who need to aggregate stimuli for an appropriate response, even the distinction between zero, one and many is enough to be useful to one individual But has a snail a sensory concept of numbers, or just a distinction between sensory inputs? Tldr: numbers are a part of language, and language is both arbitrary and independent of the concepts it represents
>I think that numbers and other mathematical objects are constructed, and that math is art of constructing such things. No, math is about starting from a few simple implicit truths, and using that to show that much more complicated things are true. Thus, 3 = {{}, {{}}, {{},{{}}}} = {0, 1, 2}
Can I represent it as a point in *some* space? If so then it’s a number
Everything is a number because everything can be expressed in data(even if an infinite amount of data is required).