[andoando]

Is there any mathematics field that views mathematical objects as patterns of multiple numbers and considers order as a fundamental property? ex, 4 isnt just "4" but is either 1,1,1,1 or 3,1 or 1,3 or 2(a), 2(b) or 2(b), 2(a)? These all represent real life abstractions and I feel a lot of detail is lost when we only think about the structure of an object as its total count, and not its composition/order.

[ndriscoll]

In some sense, basically every field of math makes this kind of distinction. e.g. it's acknowledged that for a vector space V, the tensor products V⊗(V⊗V) and (V⊗V)⊗V are not equal, but they are equivalent in the sense of e.g. a linear isomorphism, or maybe in the sense of an algebra isomorphism or something, and mathematicians do have machinery to track in what sense they consider two non-equal things to be equivalent, and they take the time to prove that operations they want to define on equivalent but non-equal things give equivalent results. Category theory is partly concerned with this kind of machinery.

So they do care about this, but then invented ideas like equivalence relations and quotients so that they can not have to constantly care about it.

With programming languages or a proof assistant like lean, by default non-equal things are not the same, and you need to do some bookkeeping to carry around proofs that things are equivalent in the way you need them to be. The tricky thing is actually making it ergonomic to be able to think of (1+1)+1 and 1+(1+1) as "the same" without constantly needing to think about it. It's hard to do math in most programming languages partly because they lack the machinery necessary to have contextual notions of equivalence (but also because they lack dependent types).

In some fields, that extra bookkeeping is really important. E.g. in geometry you might be able to choose coordinate systems at each point of some space, but perhaps not in a way that is globally defined and consistent with the types of equivalence that you need, so you need to track everything through local coordinate changes as you move around.

Or a simpler example is a vector space and its dual: they're isomorphic, so in some sense equivalent, but not canonically so, so you need to track the equivalence you're using, and maps on one are "mirrored" when applied to the other. More generally, any finite dimensional vector space is characterized (in the sense of equivalence up to linear isomorphism) by a single number (its dimension), so in some sense it "is" that number, but we also think of them as each being different, depending on context. So f.d. vector spaces can in some sense be thought of as "numbers that remember which vector space they are".

[moritzwarhier]

Numbers have properties beyond their order, right, e.g. factorization.

Maybe the field you are looking for is Number theory? Or maybe algebra and group theory?

But I don't quite understand your sentence:

> [...] I feel a lot of detail is lost when we only think about the structure of an object as its total count, and not its composition/order.

Maths is concerned a lot with how numbers can be constructed or composed from other numbers, ir other mathematical structures.

Still, there is a difference between the abstract notion of 4 and objects where we assign some measurable quantity (e g. counting similar objects, let's say turtles, and saying in total it's "4 turtles").

Numbers are not concerned with counting alone, but that's the easiest way to construct numbers.

[andoando]

What I mean is that there are multiple structures that are equivalently "4". 1,3 or 2,2, etc. 1 AND 3 depicts two distinct ovjects (or one object split into 2 parts) and is fundamentally a different idea than the structure represented by 4. However in all the math Ive seen this are simply reduced and we just care about the end result.

I am saying this equivalence isnt a fundamental property, but one merely useful toward a purpose. I am interested in mathemathics where one can reason about and do operations on ordered sets of discrete numbers or booleans.

I suppose matrices, boolean algebra or category theory is the closest to what I am thinking of, but I need to learn more about that.

I am aware of vector spaces but keep in my mind by 1,1 or 1,3 I am not talking about points in a multidimensional coordinate space.

[moritzwarhier]

So you are thinking of the set of all integer pairs whose sum is equal to an integer n?

in other words

  a + b = n

?

The size and structure of this set (or these sets, or more precisely, the function A(n) that returns all such 2-element sets of integers for a given n) is trivial and well understood.

Even more precisely, all naturals, but I always get these terms mixed up in English. Including negative numbers just leads to infinitely many boring solutions of course.

I feel I may be missing something in your line of thought.

What about factorization? Do you find that interesting?

Or maybe the set of all sets of size 1..n whose members are integers and sum up to some other integer n?

These are all trivial examples of course.

But I feel you'll find plenty of material to study in maths if you get to be able to articulate the properties of numbers that interest you.

> What I mean is that there are multiple structures that are equivalently "4". 1,3 or 2,2, etc. 1 AND 3 depicts two distinct ovjects (or one object split into 2 parts) and is fundamentally a different idea than the structure represented by 4. However in all the math Ive seen this are simply reduced and we just care about the end result.

The first sentence of my answer describes the possible ways to construct integers by adding pairs of integers as you describe. I'm talking about sets here (instead of vectors) because addition is commutative and the ordering is irrelevant.

It might be you are interested more in ontology than in maths?

But I'm not very proficient in maths.

Very complex structures can emerge from simple structures.

Consider Fermat's theorem, which also deals with summarization of integers (although the multiplication/powers make this not as trivial as what I understood from your examples)

[andoando]

This is something I've been thinking about myself alone for a while so I don't have the language to properly communicate it with the common mathematical or philosophical language, but I appreciate your thoughts.

Effectively, I am interested in a mathematical representation of any conceivable structure in space-time, with operations that allow for transformations of those structures in space-time. Imagine for example a screen of a 1D grid of binary objects, where W is white/on and B is black/off.

I can have WWWB, which I can think of (by mentally breaking it down) as WWW,B or as WW,WB, or W,WWB, etc. I can also think of a structure like X,X where X is any other structure (effectively, encapsulation, and recursion). I can do operations like put something as X or remove something as X. I can take X,X,X and combine it with WWW to get WWW,WWW,WWW. Moreover, I am interested in applying these operations through time, so for example I can do X,Y,->X,X,Y and apply this abstract transformation to WWW,BBB to get WWW,WWW,BBB. So first and fore most I am interested in a data structure that can handle this information (having spatial relations, where any single node can be the sum of any other node. Like a matrix where each point can be a matrix, or better yet a 2D linked list, where each node can be a 2D linked list).

This is my interest in thinking about the composition of numbers. I am more interested in math that focuses on properties of the compositions, their inverses, operations defined on them, their relation to the whole number, etc. The number 4 in my line of thinking is simple XXXX, but this is far too limiting for what I want to do.

Its a little more interesting with 2D spaces, because my structure becomes something like X(right)X(right)X and I can apply a X(UP)Y to get a structure like.

YYY

XXX

I've been playing around with this idea and I can take some simple patterns and create much more complex 2D images. I can create two squares for example, one to the right of the other, and apply a pattern like XY->YX to switch them.

Whats interesting also, is that the square in my system of made of simpler constructions that effectively define the idea of a square (X1(up)X2(right)X3(down)X4(left)X1, which can be broken down to its simpler components of being X1(up)X2, X1(down)X2, etc which gets us very close to the human level notion of a square being "point going up x amount then right x amount then... or "two sets of (two points in direction A with a corresponding two points in the direction A(inverse) )). I want to define a single formal system by which with simple rules, I can manipulate structures in an analogous manner to human thought.

I find it really interesting but I am kind of stuck on where to go with this.

[naasking]

Aren't you basically talking about lisp? You're suggesting two core primitives, an atom and a pair constructor, and you're defining an equivalence across pairs, eg. WWWB modelled as (cons W (cons W (cons W B))) = (cons (cons W W) (cons WB)) = (cons (cons (cons W W) W) B).

I assume you can do this with set theory and a suitable definition of equality.

[moritzwarhier]

I would never ask this in a trolling way, but I'm a bit floored (ha), so I'm asking it seriously:

> Effectively, I am interested in a mathematical representation of any conceivable structure in space-time, with operations that allow for transformations of those structures in space-time.

Are you serious about that?

If so, this is not really about maths. It's the classical "theory of everything" question in physics.

But you can't say anything meaningful about space and time using only maths. It makes no sense.

[andoando]

Haha yeah I am very serious. I spent a ridiculous amount of time the last 3-4 years thinking about this. It might all be junk but I find it very interesting.

>But you can't say anything meaningful about space and time using only maths. It makes no sense.

Why do you think that? I am talking about entirely abstract mental conceptions. I am not saying anything about how real physical processes work. Theres several mathemathic formulations of space-time, so I am confused as to what your bewilderment is.

Also keep in mind this is not too different from the Turing machine, which is a row or a multidimensional row of squares, with operations that allow for manipulating any one starting configuration to another.

[hyperthesis]

Well, there's the fundamental theorem of arithmetic, that there is exactly one composition of prime factors for every natural number (excluding ordering). https://wikipedia.org/wiki/Fundamental_theorem_of_arithmetic

[whiterknight]

That’s how integers are defined in set theory.