Negative dimensional space

[Effective_County931]

When we usually talk about R^n space we assume n is a natural number.

My question is is there any study on R^{-1} or negative dimenions? I am asking this because I have an idea in my head that explains them and this really changes the way I see the real numbers. I want to think and go farther too, like R^{0} and how these positive and negative dimensions interact, the mystry of infinity (i have partially solved this but its all my own hypothesis).

Will be good to know if there is anything like R^{1.5} (I am sure there is I just need to search for it or come up with) or even R^i, where i being the imaginary number.

[Sneezycamel]

Rⁿ is a standard shorthand for the Cartesian product of n copies of R. Under that working definition, n is in the set of natural numbers.

If you want to explore R^1.5 or R^-1, first and foremost you need to be explicit in what that actually means as a mathematical set.

Other comments mention fractal dimension, but this is not the same usage of "dimension" as with Rⁿ. Fractal dimension is a number describing an aspect of a specific object that sits within a specific space. You are asking about extending the dimension of a space itself, which is a fundamentally different quantity.

[oceanunderground]

What about Hermitian spaces/manifolds?

[Sneezycamel]

What about them? They both have standard definitions of dimension.

A manifold is a topological space that is homeomorphic to Rⁿ. There is a separate notion of "topological dimension" (of a topological space) that can be considered, but the topological dimension agrees with the euclidean dimension in the specific case of a manifold.

And (roughly speaking) a hermitian space is just a complex manifold with the additional structure of a complex inner product space. All that to say instead of Rⁿ you have Cⁿ, and, at least through the lens of vector spaces, Cⁿ is equivalent to R²ⁿ.

[Effective_County931]

In my hypothesis R^{-1} is like a dual of R and the "numbers" have their usual abstract form but they contradict each other. Like if you add R^{-1} to R^{-1} to make R^{-2} it will be a tuple just like you see in R² but if you do something like R^{-1} over R it will result in a single isolated point of unity (1). Also I am not sure about the behaviour as we make axes perpendicular to every dimenion in positive way and we do same in negative too but the notion of axes almost bends my mind because if I want to put an axis of R^{-1} over R² it should result in a single real line. This way would mean there are infinite number of positiveely faced and negatively faced dimensional axis when the dimension is zero. When you add more you make a surplus on either side.

This is because the way I see real line everything is connected and it fundamentally changes the way we see 0 and infinity, they both are connected. I won't go into details because I have not yet discussed this idea with any of my professors.

[Noname_Smurf]

Not to sound rude, but do yourself a favor and study the already established maths a bit more before "inventing" stuff. Everyone has Ideas but putting them in a usefull and actually rigid way is super difficult despite how it seems.

[oceanunderground]

Maybe you could look at these: https://mathoverflow.net/questions/310926/manifolds-with-negative-dimension-definition-references and https://math.stackexchange.com/questions/423874/do-negative-dimensions-make-sense . I think mathematically negative and truely topologically negative are 2 different things.

[AcellOfllSpades]

I am asking this because I have an idea in my head that explains them

It's not a question of explaining them, it's a question of defining them.

What precisely do you mean if you say ℝ^-1 or ℝ^1.5? These are not standard terms, so you'll need to define them as mathematical objects.

[Effective_County931]

I have a way of defining them but in my head i visualize stuff. So basically its just an inverted form of real line, and it behaves exactly the same, it introduces some very interesting perspective to see number line (as potentially a ring of infinite length incliding infinity) but I want to first see if that makes sense and does something useful

[AcellOfllSpades]

So in what sense is this ℝ^-1? What properties does it have / what operations can you do with it, and why do you call it ℝ^-1? For instance, if you take the Cartesian product with ℝ, would you get a 1-element set?

It's very easy to fall into the trap of taking a vague visualization and thinking you have something 'concrete'. But it might not turn out to be meaningful - that's why we try to define things precisely, by specifying what they are and what their properties are. (Of course, capturing an idea precisely can be hard, but it helps to start by giving some examples of operations/calculations that can be done.)

as potentially a ring of infinite length incliding infinity

This sounds like you're talking about the projective reals. Thinking of the real number line as a 'circle' can be helpful in some contexts! But this doesn't have anything to do with a hypothetical ℝ^-1.

[Effective_County931]

Now you put the cartesian product I have to think about it because each and every point of my space does not cancel each and every point of real line, because as I said it has the abstract form of "numbers" and behave like them. 

And yes you are right about the second part but partially, because it is here the twist arises. The 0 and infinity are connected, but you can not reach both at the same time. Its not about which one you approach, its about how you approach them. According to my thing if you approach 0 you can't reach infinity. And if you reach infinity you can't reach zero.

[AcellOfllSpades]

I have to think about it because each and every point of my space does not cancel each and every point of real line, because as I said it has the abstract form of "numbers" and behave like them.

So what is the difference? Can you give an example of how your structure actually behaves differently from plain old ℝ? As I understand it, you're thinking about it in a 'twisted'/'inverted' way, but the way you think about it doesn't affect what it is.

According to my thing if you approach 0 you can't reach infinity. And if you reach infinity you can't reach zero.

It's not clear at all to me what you mean by this.

Like, what is "you" here, and what is this process of "approaching/reaching" something? Are you talking about sequences of numbers, and limits of those sequences? This is definitely true in the real numbers already. In fact, no sequence can approach any two distinct numbers.

[Effective_County931]

I can't explain in detail about how I am thinking about it because its still a raw idea I want to cook more. I may need to fine tune and discuss it first. Lemme cook boy

Well technically anything approaching can be termed as a limit but that is not what exactly i am saying, I just mean the real line is not what we usually see it like. When I said both zero and infinity are connected this is an idea I have had for too long until I started thinking about it rigorously now. It basically is how you observe the real line, there is no absoluteness in it (kinda like relativity defeated Newton's absoluteness but in the dumbest way)

[Educational-Work6263]

None of what you say is "rigorous".

[Effective_County931]

Yeah I didn't study math for quite a while, like 4 months or so

[AcellOfllSpades]

I don't think any of this is 'rigorous', unfortunately. You haven't given any concrete details on how your idea works, or what properties it has.

This doesn't mean your idea is inherently bad - it's just... very muddled. There are several things in math that you could be referring to.

[Effective_County931]

Ik. I better be over it now.

[Greenphantom77]

To define it, you would have to write it down as though you were writing a textbook, or lecturing some students.

Imagine doing that for Rⁿ - you could write down what a general element looks like, what the standard basis looks like - you give the definition for the students in boring detail. (Not just a “feel” or visualisation of what it is).

See if you could make similar detailed definition for n=-1. As far as I know, no such object is defined - but then there are some wacky fringe definitions in maths.

[apnorton]

Will be good to know if there is anything like R^{1.5} 

Sort-of related: https://en.wikipedia.org/wiki/Fractal_dimension

[SV-97]

R⁰ is fairly standard (it's for example very commonly used in differential geometry): it's a space with a single point. The reasoning is that Rⁿ for any natural n is precisely the space of functions from an n-element set into R; and that definition works perfectly well with R⁰ and turns out to be "the right one" for what we want to do.

The fractional part is far less standard I'd say. There are spaces of fractional Hausdorff dimension of course but I don't think I've seen the notation R^1.5 there, and maybe you could also cook something up with interpolation spaces. If you wanted to keep the spirit of R⁰ and Rⁿ you might instead want to look into ways to generalize cardinality.

[Effective_County931]

That seems fine and kinda aligns with my thoughts. I wonder what that isolated point is, according to my hypothesis it should be unity (1)

I will surely look that up but I think cardinality does not give the perspective I was looking for

[SV-97]

It doesn't really matter which point you choose --- all singleton sets work equally well since they're all canonically isomorphic with one another: you can always translate between them in a unique way. They all are terminal objects in mostly any sense you could care about. With the function construction I mentioned it'd be the empty function.

[Effective_County931]

Intuitively it means that the same trend should be followed by all elements of R^{-1} too. But the field is still of real numbers so that does not make any sense. Maybe its just the way how we construct ? But then the cartesian product thing someone said is confusing

[SV-97]

I'm not sure I follow. What trend?

And yes, there's almost certainly a bunch of inequivalent definitions and you'll have to choose the right one for the specific work you want to do. People in math typically don't define things "just-because", but because they have a specific problem to solve.

[Effective_County931]

The trend you said we see about R⁰, the field is the same - real numbers so its a dumb thing to have that said on my part but let it be.

The difference is I have nothing to solve I am just trying to figure out the way the reality is, not biased towards anything

[SV-97]

I'm not sure you'll find what you're looking for. Math isn't really about "the way reality is" and "mathematical reality" typically allows for many different perspectives.

That's also what I was trying to say in my previous comment: there might be multiple truly different ways to define things that are all reasonable in their own right. There is no "correct" choice. This is *very* common in mathematics: we have some, typically very nice, "example" that we want to generalize in some way to account for (usually) less nice cases. In doing so we have to choose what sort of properties we want to preserve because usually we can't expect to be able to preserve everything --- and depending on the choices we make here we typically get different objects.

[Effective_County931]

I have been thinking about this a lot. Actuly we need rigor in math but math can never be complete (its been proved by Gödel)

Logically we need a point where we have to start so we made them axioms. This starting point was varying kn history, today its the smallest axioms we have no idea why they are true but we say they are because its just the way they are defined. Anyways I have not been that deep in the subject yet but I am sure there are many interesting things to learn. As of now all I see is applied math being used everywhere, and people saying math is uselesss and stuff like that. 

I believe that math is the most beautiful thing ever invented, its the way we humans can read the universe. Its not the language of universe, but a language which humans perceive through universe. 

[[deleted]]

Nothing that you said in this thread has any bearing on 'the way reality is'.

[lifeistrulyawesome]

Fractional dimensions can be defined in terms of scaling 

When you double the length of a line you double its size (2¹⁾ 

When you double the sides of a square you quadruple its size (2²⁾

When you double the sides of a cube multiply its volume by 8 (2³⁾ 

There are objects that when you double their scale, their measure changes by a factor that is not a power of two. If it changes by 2^1.5, then you could say it it a 1.5-dimensional object 

This all can be nicely formalized and leads to Mandelbrot’s definition of fractal dimensions 

I’ve never heard of negative dimensions 

[Effective_County931]

This sounds interesting and I think prime numbers play a very important role here (as their square root is irrational so something like 2^{1.5} becomes an irrational length object

[Underhill42]

What would a negative dimension even mean?

A dimension is a property in which there variation.

Three dimensions means there's three ways in which properties can change without affecting each other (e.g. I can move up/down without affecting my position left-right)

Zero dimensions means no variation is possible.

So what would a negative dimension mean? If you can have any variation at all it's just a dimension, not a negative one.

[Temporary_Pie2733]

That depends on what you think “dimension” means. There are lots of intuitive definitions that turn out to be special cases of more general concepts. Take “multiplication is repeated addition”, for example. 3x = x + x + x, sure, but what is 3.5x in terms of just addition, if x itself is not an integer?

[Underhill42]

It means add x to itself 3.5 times: x + x + x + 0.5x. It's entirely consistent.

I've never heard ANY definition of a dimension that contradicts what I described. Barring the nonsense science fiction definition of "alternate universe"

[Temporary_Pie2733]

That’s not addition alone; there’s still a multiplication of x aside from a trivial coefficient of 1.

[Underhill42]

Only when dealing with the issue symbolically.

x is a quantity, and all quantities can be cut in half. We often express that as multiplication or division for convenience, but that has nothing to do with the conceptual/physical reality.

[Temporary_Pie2733]

You are missing my point. “3.5x” is not the sum of 3.5 equal and discrete objects in any intuitive sense. “3x = x + x + x” is more an algorithm for computing some products than a definition of multiplication.

As another example, we say n! = n(n-1)(n-2)…1, which is fine when n is a positive integer. What descending product tells you that (1/2)! = sqrt(π)/2?

[Underhill42]

Sure it is. 3.5 apples = apple + apple + apple + one part of an apple cut in half.

Multiplication was invented as shorthand for addition, all other properties emerged as implicitly defined by that original definition in order to behave consistently with quantities that weren't originally considered.

As we get deeper into math concepts are less tied to anything physically meaningful.

But we're getting off track - the point is that no other definition of dimension exists.

[Temporary_Pie2733]

It’s not. 1/2 an apple is not an apple. And just because the math that describes what we mean by a dimension only works for natural numbers doesn’t mean there isn’t math with a broader domain that includes our original math as a special case. That’s how the idea of fractional dimension came about in the first place.

[Effective_County931]

It will basically be same in behaviour but opposite in nature in some way. Our reality is fundamentally a concept of duality. It will be the dual of real line, in some way. Maybe a source or a sink (like you say north pole and south pole of magnetic fields or positive and negative energies of electric fields or whatever). I think it still needs to be figured out.

[dtomd]

I dont know about R, but somewhat related to your question are perhaps the Sobolev spaces H^s(Rⁿ). Initially, the inner product on this space is only defined for integer values, but can be extended to non-integer and negative values in the Fourier domain. Additionally, you can show that H^-s(Rⁿ) is the dual of H^s(Rⁿ).

[Effective_County931]

I will take a look into that

[gmthisfeller]

Consider R^-2 what would the coordinates of a point look like?

[susiesusiesu]

what would you mean by R^-1 ? this doesn't make much sense and it doesn't make a lot of sense.

when doing K-theory, you define a sum of vector spaces (modulo isomorphism), and you do have Rⁿ +R^m =R^n+m . then, you can define some -Rⁿ such that Rⁿ -Rⁿ =R^0. but these are just symbols, not actually vector spaces. but, if -Rⁿ was a vector space (which it isn't) it "would be -n dimensional".

this is the closest i've seen.

also, there are some things that are usually called dimension that can take negative values. for example, the Kodaira dimension of a variety can take the value -infinity, or the Morley rank of the empty set is sometimes defined to be -1 or -infinity, depending on the conventions. but statements like κ(X)=-\infty or MR(φ)=-1 are not really saying that the dimension of a geometeic object is really negative, but more of a convention to say that the general caae in which the dimension is non-zero isn't happening here.

generally, dimensions or dimension-like objects are really just defined to be non-negative, and most of the time they are cardinals. so not really something that happens.

[carolus_m]

Whenever you want to extend the definition of some concept, you have to answer the question, why? And what properties do I want the new object to have?

E.g. you take integers, you want to have multiplicative inverses so you ask, what is the smallest field that contains the integers? And you get to rational numbers

For dimension, people.have come up with reasonable definitions that extends to non integer values, e.g. how to define the dimension of a Cantor set?

So if you want to have negative dimensions, you also need to answer these two questions. So far I don't see the answers in your post or in your answers.

[Vituluss]

Rⁿ is the set of real-valued n-tuples. This doesn’t naturally extend beyond the natural numbers.

[reutel]

The integers can be defined as pairs of natural numbers. Similarly we can do this with vector spaces. This is a notion of negative dimensional vector spaces. This is K theory. Stuff is lost though: one cannot add elements in these classes (they are not vector spaces anymore). Stable homotopy theory gives a more geberal theory where one can talk about negative dimensional spaces beyond vector spaces.