Defining mathematical spaces (some ideas)

[sqishy]

So here I am again, with some obscure, haphazardly-timed, and mostly speculative mathematics (and whatever else related).

Basically, I've got some ideas on how to define spaces in a mathematical way, but in a way which tries to start 'from scratch', so to speak. In other words, I like not trying to use heavily other well-established things, and instead try something half-independent from my view. Don't expect anything really formal, or really informal either. Also, I am intentionally avoiding numbers (no 0s, 1s, 42s, etc).

Getting right to it, I don't have a clear definiton for a space in this idea-collection I have, but just imagine a space as a region where other objects can be in it.


A subspace is a region which is entirely contained within another space. The other space is a superspace to the subspace - in other words, a superspace is a region which entirely contains another region.
For the definitions here, all subspaces are FINITE and BOUNDED.


The magnitude of a space is the 'amount' it contains, how big it is. I won't try to formalise this further (at least for now).


In the context here, intersection means that a certain space is occupied by two or more other spaces 'at the same time' (inverted commas, as meant for metaphor - time is not involved here).


Every space has four properties which describe them (currently four, no reason to say there can only be four, I just have four defined at the moment).


FINITENESS:

A certain given space is infinite if: For any given subspace, there necessairily exists a subspace which has greater magnitude than the given subspace.

A certain given space is finite if: For any given subspace, there does not necessairily exist a subspace which has greater magnitude than the given subspace.

A certain given space is null if: For any given subspace, there necessairily does not exist a subspace which has greater magnitude than the given subspace.

CONTINUITY:

A certain given space is continuous if: For any given subspace, there necessairily exists a subspace which has lesser magnitude than the given subspace.

A certain given space is discrete (not continuous) if: For any given subspace, there does not necessairily exist a subspace which has lesser magnitude than the given subspace.

A certain given space is 'whole' if: For any given subspace, there necessairily does not exist a subspace which has lesser magnitude than the given subspace.

ORDEREDNESS:

A certain given space is unordered if: For any given subspace, every subspace of greater magnitude than it is not necessairily a superspace of it also.
Equivalently: For any given subspace, every subspace of lesser magnitude than it is not necessairily a subspace of it also.

A certain given space is ordered if: For any given subspace, every subspace of greater magnitude than it is necessairily a superspace of it also.
Equivalently: For any given subspace, every subspace of lesser magnitude than it is necessairily a subspace of it also.


BOUNDEDNESS:

A certain given space is unbounded if: For any given subspace, any other finite and bounded space of greater magnitude than the given subspace that intersects it, is necessairily also a subspace of the certain given space.

A certain given space is bounded if: For any given subspace, any other finite and bounded space of greater magnitude than the given subspace that intersects it, is not necessairily also a subspace of the certain given space.



I have defined finiteness and discreteness in 'negative' terms, as I want to hold the position that 'infinity','continuity' and 'unboundedness' come 'before' limitations; limitations are something added, not taken away, from them. I am trying to do the same with orderedness currently.

I'd love any reponses/input to this. All of these ideas are in flux, and could be changed with time. Thanks for reading!

[randomuser123]

How much do you know about set theory? It seems to me that you are trying to reinvent set theory a bit!

[sqishy]

How much do you know about set theory? It seems to me that you are trying to reinvent set theory a bit!

I know some set theory, like unions, intersections, cardinalities and such.

I don't want to think that I am trying to reinvent it, because I don't want to be egotistical or all-knowing about it. It's more that I want to personally try to explain or imagine some things that set theory happens to already be doing, or similarly so. I want to make sense of it myself, by doing it myself.

[darthearth]

Are you studying mathematics in college? I'm into university physics and would like to get to know some of this stuff better. Linear Algebra deals with spaces doesn't it? If you are in school do you have access to profs or just grad student TAs? What do they say about it? Like what's the difference between spaces and vector spaces? Just that one is supposed to have direction? And that would be a base vector right? Could your 4 properties be the 4 dimensions of space-time?

Orderedness and Boundedness were the hardest to follow, didn't you include in the definition of boundedness the term "bounded", is it appropriate to include the term you are defining as a critical part of the definition?

I mean I got finiteness and continuity better, those make some sense. Have to admit I just didn't follow the last two.

[sqishy]

Are you studying mathematics in college? I'm into university physics and would like to get to know some of this stuff better. Linear Algebra deals with spaces doesn't it? If you are in school do you have access to profs or just grad student TAs? What do they say about it? I found it a bit hard to follow. Like what's the difference between spaces and vector spaces? Just that one is supposed to have direction? And that would be a base vector right? Could your 4 properties be the 4 dimensions of space-time?

I mean I got most of it though. Sounds legit to me.

I actually study philosophy; the majority of all the maths I do are ideas I have, not from learning it. I've brought it up a few times when possibly relevant. I did do maths in school, but did not look much at what I am looking into now, which I'll just call set ideas for fun. It's not so much about the maths itself, it's the ideas behind it, for me. I do know lots of small random things here and there in the field, but it is more me just thinking things up from the knowledge 'scrap' of sorts. I know relatively extremely little mathematics, and won't act like I know stuff when I don't.

Vector spaces are spaces where each point within them has a magnitude and direction value 'attached' to them. Spaces in general can (guessing from the context with vector spaces) have either just magnitude at each point, or magnitude and direction at each point. In other words, vector spaces are particular kinds of spaces. I'm not sure if base vectors are relevant, I mean it literally. You probably know much more than me on this.

The four properties here are not to do with spacetime or any 4-dimensional structure. I am trying to not have dimensions get involved with the definition, along with forgetting about numbers too. So no dimensions, for now at least. One aim is that the definitions work for any number of dimensions.

I know my definitions are very abstract, but I am aiming to keep it as simple and understandable as possible at the same time. I don't see any point in archaic inaccessible ideas or views. It's going places, which is always good.

Thanks for reading too!
(I'll also be updating the definitions tomorrow, hopefully)


_____________________
EDIT 13/12: Updated definitions for finiteness and continuity.


Orderedness and Boundedness were the hardest to follow, didn't you include in the definition of boundedness the term "bounded", is it appropriate to include the term you are defining as a critical part of the definition?

I did yes, I'm working on that one - doubting it'll stay the way it is.


I mean I got finiteness and continuity better, those make some sense. Have to admit I just didn't follow the last two.

With orderedness, I thought of it in a intuitive way. If all subsets larger than a certain one are supersets to it, and all smaller ones are subsets to it, then it means there is some sort of concentric subset 'nesting'. They are ordered 'outwards' from a certain position - analogous to the number line, which is itself ordered. It is ordered in that sense, as opposed to there existing larger and smaller parts all over the place.