What is the Meaning of Dimension

[JOPearcy]

What is the meaning of dimension?

It is my understanding that dimension related to the measuring out of something. If there is some observable phenomena that we can measure by defining units of measure and counting the quantity of these units that there is an associated dimension which is not unit based but the units reside within or are composed of the dimension being measured.

Thus examples of dimensions include the SI base dimensions length, mass, time-duration, electric current, thermodynamic temperature, amount of substance and luminous intensity.

It is my understanding that while dimensions are typically referred to in the sense of 3 spatial dimensions and sometimes including the 4th dimension of time, that the number of dimensions are not limited to the dimensions of space and time but include all manner of observable phenomena which can be quantified and measured.

Am I correct in this or not?

 

[mgb_phys]

The word is used for two slightly different things ( both come from the latin word for measured )

1, The base units of some quantity, eg. you would say energy has dimensions of force*distance.

2, A direction you can measure something in independant of other changes, eg the normal X,Y,Z dimensions plus time.

 

[JOPearcy]

Is there a good detailed definition of what dimension means in science available on the net, something that explains it in depth, like a proof?


When I work through my understanding of the definition of dimension, I see a single definition for dimension from which a common subset of dimensions are typically referred to when the term dimension is used.

So, for example, length, mass, time, electric current, thermodynamic temperature, quantity of substance, luminous intensity and numerous derived dimensions from the first seven are all dimensions. But frequently when the term dimension is used, it is in specific reference to the dimension length, space defined in 3 dimensions of length or space-time in 3 dimensions of length and 1 dimension of time.

However, these dimensions of length and time are defined as dimensions through the same process as the other dimensions have been defined and the dimensions of length and time are a subset of a greater body of know and defined dimensions.



The following are statements which I believe are correct and I use understand what dimensions means.

Quantity is a property of a phenomenon, body, or substance, to which a magnitude can be assigned.

Quantities of the same kind are quantities that can be placed in order of magnitude relative to one another.

Quantities of the same kind within a given system of quantities have the same dimension.

The subdivision of quantities into quantities of the same kind is to some extent arbitrary. For example, moment of force and energy are, by convention, not regarded as being of the same kind, although they have the same dimension, nor are heat capacity and entropy.

System of quantities means a set of quantities together with a set of non-contradictory equations relating those quantities.

Base quantity means a quantity, chosen by convention, used in a system of quantities to define other quantities.

Derived quantity means a quantity, in a system of quantities, defined as a function of base quantities.

The unit of a physical quantity and its dimension are related, but not precisely identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but any length always has a dimension of L, independent of what units are arbitrarily chosen to measure it. Two different units of the same physical quantity have conversion factors between them. For example: 1 in = 2.54 cm; then (2.54 cm/in) is called a conversion factor (between two representations expressed in different units of a common length quantity) and is itself dimensionless and equal to one. There are no conversion factors between dimensional symbols.

“Quantity dimension” also phrased “dimension of a quantity” or simply phrased “dimension” is the dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

Quantities having the same dimension are not necessarily quantities of the same kind.

In deriving the dimension of a quantity, no account is taken of any numerical factor, nor of its scalar, vector or tensor character.

The dimension of a base quantity is generally referred to as ‘base dimension’, and similarly for a ‘derived dimension’.

 

[Renge Ishyo]

I prefer to attempt a more broad description.

1 Dimension - A straight line.

2 Dimensions - Take another line and arrange it across the first one at some angle (90 degrees in cartesian coordinates). The two together define a flat plane of some shape.

3 Dimensions - Take another line and place it across the previous two lines but not in the plane defined by the previous two. This will create depth.

The concept of the dimension can be explained without any physics in it. Once you define a certain unit as a base number for a measurement and decide that this unit of measurement will be plotted along one line and another measurement on another line, etc. then the units take on the form of a mathematical dimension when they are arranged in the way described above. But the units themselves are not necessary in order to explain why they are considered a dimension (in fact, numbers are not even necessary to describe the idea of a dimension if my description above can be trusted).

 

[cristo]

Is there a good detailed definition of what dimension means in science available on the net, something that explains it in depth, like a proof?

The definition of a dimension is exactly that-- a definition. You cannot prove a definition!

The mathematical definition of dimensions would be something like parameters used to describe position or other characteristics of an object within a space. The dimension of this space is the total number of parameters.

For example, in 3D space we need three parameters to define the position of an object in the space. Therefore, the space has three dimensions.

 

[rbj]

Wikipedia has a few pretty good explanations:

http://en.wikipedia.org/wiki/Physical_quantity
http://en.wikipedia.org/wiki/Dimensional_analysis
http://en.wikipedia.org/wiki/Fundamental_unit
http://en.wikipedia.org/wiki/Natural_units

 

[JOPearcy]

I've read through all those definitions before. If you can locate International vocabulary of basic and general terms in metrology (VIM) you might find it interesting.

By a definition that explains dimension in depth, like a proof, I did not mean a proof of the definition of dimension. If you are posting on this forum I am hoping that it means you have had experience with such things as writing a proof. To write a proof you have to be very detailed and cover all your bases.

To give such a rigorous definition of dimension and its use in science would answer my question because it would go into enough depth to show whether I am right or wrong in my view of dimension.

It would give a thorough enough of a definition to handle how it is used in science and then gives examples for both common uses and some uncommon uses. This might in fact include several actual proofs.

One such proof might be why and how length is found to be a dimension using the definition of dimension.

Another such proof might be why luminous intensity is found to be a dimension using the definition of dimension.

Yet another such proof might be why amount of substance is found to be a dimension using the definition of dimension.

Another such proof I’d like to see is how an effectively dimensionless dimension can be defined, the dimension of 1 such that it is derived as a ratio of dimensions of the same type, such as in deriving angle. When I have found such reference to the dimension of 1 I have found it particularly mind twisting and I’d really like to see an in-depth explanation for this.

 

[rbj]

...
Another such proof I’d like to see is how an effectively dimensionless dimension can be defined ...

at this point i just throw up my hands. i guess there there is a dimensionless dimension and such are just numbers.

 

[JOPearcy]

RBJ,

The place where I saw such a reference did not explain itself very well. I'm not sure the reference was correct or if it was worded poorly and I misunderstood it.

It goes like this.

The “quantity of dimension one” also worded as “dimensionless quantity” is a quantity for which all the exponents of the factors corresponding to the base quantities in the representation of its dimension are zero.

The values of quantities of dimension one are simply numbers.

The term ‘dimensionless quantity’ is for historical reasons commonly used. It stems from the fact that all exponents are zero in the symbolic representation of the dimension for such quantities. However, the term ‘quantity of dimension one’ reflects the convention in which the symbolic representation of the dimension for such quantities is the symbol 1 (see ISO 31-0 :1992, subclause 2.2.6).

EXAMPLES
Plane angle, solid angle, linear strain, friction factor, refractive index, mass fraction, amount-of-substance fraction, Mach number, Reynolds number, degeneracy in quantum mechanics, number of turns in a coil, number of molecules.

I’d like a better explanation about the ‘quantity of dimension one’ and how it is determined to be a dimension from the definition of dimension. I think one of the implications is that like other dimensions, just because something has the same dimension does not always mean they can be added.

For example, two lengths may not simply be added if they are perpendicular to each other, even though the both have the dimension of length.

Plane angle, solid angle, linear strain, friction factor and refractive index all have the dimension 1. Can you simply add them together like 5 scalar values?

I don’t clearly understand the how, why and associated implications of the dimension of 1.

 

[matt grime]

Dimension in physics refers to units. A dimensionless quantity is one with no units, such as the coefficient of restitution. This is nothing to do with mathematics.

Dimension in mathematics: if V is a vector space, then its dimension is the cardinality of a minimal spanning set or maximal linearly independent set of vectors. (What this is for infinite dimensional vector spaces depends on whether you want a Hamel basis, i.e. do you allow or disallow infinite direct sums).

In geometry, one can take the term dimension to mean this, roughly: if I have a geometric shape, with some associated quantity, and if I double all the lengths of the shape (scale by two), then if the associated quantity scales like 2^d, then d is the dimension. Example: in 2-d take an plane polygon. If you double its sidelengths you mulitply its area by 2^2, or for a polyhedron, doubling the sides gives a factor of 2^3 change in volume.

There are many other different definitions of dimension.

I hope that helps.

 

[matt grime]

Plane angle, solid angle, linear strain, friction factor and refractive index all have the dimension 1. Can you simply add them together like 5 scalar values?

if you felt like it - they're just numbers. The resulting answer may be physically meaningless. Don't forget you can multiply any physical quantity by 1 (unit x)/(unit y) to convert the associated 'units' into anything you choose.

Remembering what the units of a measurement are is just a good way of making sure you're doing something physically sound. It has no real mathematical content, as far as I'm concerned. (Although dimensional analysis is useful in applied mathematics, apparently).

 

[JOPearcy]

matt grime,

I agree that there are many definitions of dimensions, I believe there is one definition for dimension as it is used in science, math, physics, engineering and so on which is used for the process of defining any particular dimension or system of dimensions.

What does the dimension of length, mass, luminous intensity or an element of the coordinate space F^n?

In all cases you have some quantity you are measuring. In order to measure that quantity you need to define some base quantity you are measuring in relation to and thus you define a unit of the quantity you are measuring. The dimension of the quantity you are measuring is what the unit of quantity consists of or exists in.

So I can be measuring something in the units of inches, meters or yards. The units for making measurement are not the dimension but they lie in the dimension or consist of the dimension, the dimension in this instance being length.

The same holds true for thermal temperature. We begin with something we can measure. We then define some unit of quantity as a basis or comparison point for our measurement. The dimension of that measurement is then what our unit of quantity exists in and consist of. So you end up with some quantity of units of the dimension you are measuring.

 

[belliott4488]

JOPearcy - I think you are still confusing the issue by mixing the two uses of the word "dimension" in Physics. You could simply substitute the word "units" for one of those, the way people do when the state, for example, "energy has units of mass times length over time squared", and most people would understand exactly what you mean. (Strictly, "units" refers to a specific system, such as SI, but as I said, you will still be understood if you use that word in place of "dimension" when doing dimensional analysis.)

The other use of "dimension" is mathematical, and as far as I know, is unrelated to the first use. Mathematicians speak of, for example, a "5-dimensional Euclidean space", and they aren't thinking about units of anything. Such geometrical spaces have plenty of applications in Physics, for modeling space and time as well as other more abstract spaces, such as the Hilbert Space of Quantum Mechanics.

I'm not familiar with the use of "dimension one" to describe what I would have called a dimensionless number, but it clearly has nothing to do with a one-dimensional geometric space. It looks to me like a mere convention of terminology, stemming from the fact that if you express such a quantity in powers of basic units of length, mass, time, and electric charge, then all the exponents would be zero, thus giving you "one". Sort of. I'd just call it "dimensionless" and be done with it.

One more comment - you stated that you can't necessarily add quantities with the same dimensions, for example the lengths of two perpendicular lines. Of course you can. It just depends on the question you're asking and answering. If I were to ask the total length of the sides of a rectangle, you'd add up the lengths of the sides, even thought they're perpendicular. What you can't ever do is add a length to a time. Three meters plus two seconds makes no sense.

 

[JOPearcy]

Belliot4488,

From the tone of your reply, are you following me from another forum just to continue making statements of how wrong I am so I can continue to make statements of how wrong your are?

Your comment “I think you are still confusing the issue by mixing the two uses of the word "dimension" in Physics” implies to me you have knowledge of this argument from this other forum.

Are you only following me here to continue arguing without listening? If so, what is the point? To try and follow my discussion where ever I might go so you can shout me down as being wrong without listening or understanding what I am saying?

If you are not interested in understanding, why spend the effort to follow me?

 

[JOPearcy]

Belliot4488,

If you are not following me from this other forum them I apologize for being overtly defensive. I have had a discussion on another forum where a group of posters clearly did not understand what I was saying, incorrectly stated I was wrong over something I know I was correct about and then repeatedly insulted at which point I became insulting back.

I don’t want to get into a discussion with people where the substance of that discussion is no attempt to understand the other person backed by insults.

If you are really interested in understanding what I am saying, then I will try my best to explain it, piece by piece, so that hopefully there will not be a misunderstanding.

But I have no interest in having someone effectively shout me down saying “Wrong, Wrong, Wrong, How can you be educated” while they don’t even understand what I am saying.

I have spent 6 years of intensive study in electrical engineering, math and physics at Cal Poly Pomona. I excelled in the sciences. In other subjects, English, history and such I had to struggle. When I started my English Grammar skills were just good enough to get me by. But in areas of abstract and analytical skills of science I was top of my class.

The underlying principles of how dimensions are defined using the definition of dimension was taught to me at Cal Poly, first and foremost in the physics department but also in certain advanced math and engineering courses.

I do know what I am talking about. Whether or not I can explain it well enough for you, whether or not you are able to understand the underlying principle and/or whether or not I can find an authoritative online definition to point to explaining what I am talking about is all unknown.

 

[belliott4488]

JOPearcy,

Nope, I never saw your posts on any other forums. I honestly just thought you were not aware of the two separate uses of the word "dimension" in Physics.

You say that you are aware of these things, so could you please explain again why you are comparing them? How is the use of the word "dimensions" in the statement "velocity has dimensions of length over time" related to its use in the statement "Special Relativity can be described in terms of Minkowsky space-time with four dimensions"? I would have thought that it is an accident of language that we even use the same word for these two concepts.

 

[matt grime]

Belliott is entirely correct - you appear to be confusing the physical notion of 'unit' with the mathematical usages of dimension.

 

[JOPearcy]

Belliott4488,

I want to apologize again for being so snappy. You might figure out by now that I had some posts on another forum whose tone angered me.

I want to discuss this politely and rationally.

Velocity is a dimension. It is a derived dimension, but it is still a dimension. The dimensions that the dimension of velocity is derived from are a system of dimensions.

The dimensions of space-time are a system of dimensions.

What is space?

 

[cristo]

Velocity is not a dimension. You appear to be confused by the definitions. I suggest you review the mathematical definition of a dimension.

 

[belliott4488]

Um, okay ... but I still think your use of language is somewhat obscuring the point you're making. For example, I'm not sure how you define a "system of dimensions".

In any case, I've never heard any say that "velocity is a dimension," but it is not unusual to say, "the dimensions of velocity are length over time." Maybe they're the same thing to you?

What I don't see at all, however, is your connection between geometric dimensions, which are related to the coordinates needed to identify a point uniquely, and dimensions in the sense of units of measurement. For example, if we talk about a space of N dimensions, points are identified by N coordinates, in terms of which we can define distances (assuming the space has such a concept - I don't want to get side-tracked by spaces with no norm or anything like that), such that all the dimensions are measured by length units. The fact that space-time is commonly spoken about as having time as a dimension does not mean that the fourth dimension is measured in units of time. In fact, it is often introduced as being measured by ct (speed of light multiplied by t), so that it has units of length, just like the spatial dimensions. (You'll also see descriptions where the time dimension is measured by t alone, but then the metric, which defines how distances are measured, introduces c wherever t is used, so it comes out the same.)

The point is that for a geometric space to make sense, there must be an equivalence of all the dimensions, in the sense that you can rotate coordinate axes and mix the values measured along the axes. You cannot have a sensible geometric space where different geometric dimensions have different dimensions in the sense of units (e.g. one dimension of length and one of mass - that's not a geometrical space).

Does that make any sense to you? I hope so.

 

[JOPearcy]

Cristo,

Can you cite your definition?

What I’d really like to do is to work towards an understanding of what I am talking about, I want to simplify the topic towards space.

Can you define space?

 

[JOPearcy]

Belliott4488,

A "system of dimensions" is something which you view, work with, model, observe and so on which contains a particular defined set of dimensions.

I do understand that the dimensions from which the dimension of velocity is derived are not normally spoken about as a system of dimensions. What is the system of dimensions that the single dimension of velocity is derived from? If we are talking about a 3D velocity, then we are talking about the system of dimensions normally referred to as space-time.

The connection between the different types of dimensions is the meaning of dimension.

Time is a dimension. The function of ct is a two dimensional function with the resulting dimensions of length^1 and time^0 which is generally considered a one dimensional result. But it requires units in length/time multiplied by units in time. It would not work with let's say units in length/mass multiplied by units mass. Because of this, the function does require two dimensions, length and time.

To work on trying to explain what I am talking about, I want to simplify the topic.

Let us just talk about space for a bit.

What is space?

 

[rcgldr]

Dimension in physics and math is usually defined as it is in Wikipedia, associated with real or abstract geometric coordinates that define a location in a n-dimensional space. Wiki used a point as an example of a dimensionless variable. As posted, geometric dimensions are composed of the same units, so mass and location would be independent qualities of an object, but mass wouldn't be a "dimension".

In programming, dimension refers to the number of indexes a variable has. In this case the term "tuple" would be the equivalent mathematical term. A variable with no dimensions is called a "scalar", with one dimension, a "vector" or sometimes the more generic term "array", with two dimensions, a "matrix". APL programmers used the term "tensor" for variables with 3 or more dimensions, I'm not sure if there's a common term. Other languages use the term "n dimensional array".

I'm not sure if there's a consistent term for describing the number of qualities of a database. Normally the qualities to describe an instance within a collection of databases are independent, such as name, address, phone number, age, marital status, number of children. Sometimes there are redundancies, like including city and state with zip code (if located in the USA).

 

[JOPearcy]

Jeff Reid,

I understand what you are saying, but disagree with some of it. My problem has been my inability to simply explain what I am talking about in a way that is understood. My failure to do this leads me to try a different tack.

That is what I am now trying to do. First I want to simplify the discussion to one type of dimension and that which is related to that dimension. Then I want to work through a series of questions and hopefully understandings which in the end I hope we will both understand and agree with.

What is space? By space, I am referring to the simple physics view of the physical space that we exist in, the one we normally refer to as a flat 3 dimensional space.

 

[K.J.Healey]

I'm pretty sure that can be a tough question to answer.
We view spacetime as three "dimensions" of direction and one "dimension" of time.
When discussing this we basically mean "dimension" as a :
Measurement of a magnitude of a property of the "space".
By space I mean not outer-space, but "space" in a mathematical way.
We can view SOME of the properties of our SPACE as positional, which is where we get our 4-vectors from. Our "Dimensions" are our coordinates for this position.

A second USAGE of the word "Dimension" extends that to "Measurement of a magnitude of a parameter". A 2D graph can be X vs Y, OR it can be Happiness Vs. Season. Thats two "dimensions" or rather "magnitudes of measurement".

Also, the reason why I speak of Magnitudes rather than just coordinates, or locations, is that you have to have a reference point, and a reference metric. You have to say "starting at point P, and moving N Measurements away".
When measuring "Seasons", we don't have Winter,Spring,Summer,Fall. We have N Seasons from _Given Season_.


Remember too some "Dimensions" are linearly independent, some are not. Happiness and Seasons are dependent. X and Y are not. This leads you to the topic of different types of "spaces". Whether a Hilbert (all LI) or some other.


Are you running into trouble when you're using multiple methods of measurement as one measurement? (such as change in X over change in t is Velocity?) And calling something a dimension that has multiple metrics?


I would just say, for the sake of argument, that Theoretical Physicists generally us the term Dimension to mean a property of location. Then at that location you have other Properties, that while they can be "graphed" and "measured", they do not exhibit the property of location "spacially"

But I may be completely wrong as well.

 

[rcgldr]

I want to simplify the discussion to one type of dimension ... physical space

Then use the simple definition of dimension as used in the Wiki article. The position of object to other objects can be describe as a vector composed of 3 dimensional components, an X, Y, and Z component. All 3 dimensional components use the same units. Given this definition, time is not a dimension, because it's units are not the same, it's not used to describe the (relative to some origin) position of an object in standard 3d space, and time applies to everything in the universe, not individual objects.

This is the definition of dimension that I prefer. Objects can have other qualities, like age, velocity, color, but I don't consider these to be dimensions. Keep the definition of dimension simple and use other terms to describe "non-dimensional" qualities of an object.

Relative object positions can also be defined with spherical coordinates, but again I don't consider these to be dimensions, because the units are not the same (angle, angle, distance). Ditto for cylindrical coordinates (distance, angle, distance).

 

[JOPearcy]

K.J.Healey and Jeff Reid,

Both of you give good responses.

Can we agree the following is true?
“Three-dimensional space is the physical universe we live in. The three dimensions are commonly called length, width, and breadth; although any three mutually perpendicular directions can serve as the three dimensions. Some people consider space to be a fundamental quantity while others consider length to be a fundamental quantity, both being correct from their point of view.”

To further simplify the subject, this to narrow what we are talking about so as to make it more clear, let us talk about only one of the three mutually perpendicular directions that can serve as 1 dimension in space.

When discussing this we basically mean "dimension" as a:
Measurement of a magnitude of a property of the "space".QUOTE]

How do we know that one dimension of space exists? Is it because we can detect and measure it?

In order to measure out a length in 1 dimension, do you need to define units of quantity of that 1 dimensional length?

What does dimension mean in this 1 dimensional length of physical space?

Could you have the 1 dimension but not be able to define units of quantity of it and measure it with those units of quantity?

 

[K.J.Healey]

How do we know that one dimension of space exists? Is it because we can detect and measure it?

In order to measure out a length in 1 dimension, do you need to define units of quantity of that 1 dimensional length?

What does dimension mean in this 1 dimensional length of physical space?

Could you have the 1 dimension but not be able to define units of quantity of it and measure it with those units of quantity?

We know because we have 3 of them. That is a fundamental property of nature, 3 spatial dimensions, which CAN be measured as 3 orthogonal measurements for a location.

In order to measure out a length in 1 dimension, do you need to define units of quantity of that 1 dimensional length?

Yes, and we have. The meter is such a measurement. A meter is one dimensional, and we DEFINE it.

What does dimension mean in this 1 dimensional length of physical space?

Dimension means nothing to that length. That length is a measurement in one dimension, as it always is. All "basic" measurements are measurements of the space itself. They are all one-dimensional by definition.
That can lead us to state that "All multi-dimensional measurements are not measurements of a dimension by definition. They are measurements of multiple dimensions." So velocity cannot be a "dimension" of space since it is a measurement that relies on multiple dimensions.

I myself DO consider time a dimension since:
1. It exists.
2. No spatial dimension depends upon it. It is independent.
3. It is required to span all possible locations.




finally :

Could you have the 1 dimension but not be able to define units of quantity of it and measure it with those units of quantity?

YES! and NO! at the same time.
Imagine being out in the middle of space, and nothing exists. You have no knowledge of objects and you are a point particle. You have no measurement of thickness.
From our definition of a positional measurement requiring a definable origin location, as well as a metric (magnitude of measurement).
We can say that in order for there to be measurement of any kind of distance, you require at least 2 of something. That way you can "measure" the distance between those two, and use that as a measurement for distances to other areas near you where nothing exists.
Could you move somewhere else in space if it was just you, the point particle and nothign else? Once again yes and no. If you have no way of knowing if you've moved, there's no point in considering it. If you can't measure the change, the change didn't happen (as far as you know or care).

So I'd say yes, even in a 3D world, you could have dimensions that cannot be measured by not having any references.

 

[mathwonk]

one does not have to follow you from somewhere else to have the feeling you are mixing two different uses of the same word.

I do think yourr question is interesting however and that it may be possible to merge the two meanings. it is not easy however to merely postulate that all the uses that have been made of a word can somehow be brought into unified harmony.

i would suggest that dimension is a relative thing. for example if we consider time as one basic unit, then quantities of time are one dimensional in relation to that unit. i.e. given a unit of time, other quantities of time are given by that unit multiplied by one number.

then we can consider t as a variable, and think of a variable as one dimension, so a quantity expressed as t^2 would be 2 dimensionl wrt time.

if we have abasic unit of length we can consider other lengths as one dimensional wrt that unit, and hence given by one number. we can also consider area, either as a basic unit, or as derived from two independent lengths. thus areas are one dimensional in relation to a given unit of area but two dimensional in relation to a given unti of length.

similarly some physical units can be considered alone, or as derived from other more basic physical units. again, the number of the various units reuqired determiens the dimension in relation to those units.

just as in geometry, the numbers used to express something in realtion to given units may be subject to certain conditions, i.e. the chosen units may not be independent.

for example velocity or momentum may be chosen as units or derived from mass length and time units.

dimension usually refers to the number of independent units needed to determine something ofa given kind, again in realtion to certain fixed units.

it is a little easier to be clear if we stay within pure mathematics, but even there variations are possible. there is a notion of lineaR DIMESNION wrt a base field, but there is also algebraic dimension measured by transcendece degree. i.e,. there are two notions of independence, linear indeendence and algebraic independence. even here the notions are relative, as C has linear dimension one over C, and 2 over R, but algebraic dimension zero over both.

if we stick just to topological spaces, dimension ahs been thorougholy studied and is associated to connected ness, roughly speaking a space Y has dimension one more than X if removing X disconnects Y.

another geometric version of dimension is defined in terms of irreducible pieces. the dimension of a space is the length of a strictly ascending chain of irreducible pieces in the space, e.g. point, curve, surface,...

 

[mathwonk]

the simplest most naive notion of dimension is to just look at the letters used to express it, e.g. mv^2 is 3 dimensional wrt m and v, but if we express v in terms of t and distance we get a different dimension in those units, possibly fractional.

in math the naive notion was always the number of numbers needed to express somehting in terms of given units, minus the number of implict relations among those numbers.

poincare has written a famous essay on the meaning of dimension in his last essays on science. or perhaps science and hypothesis?

 

[JOPearcy]

K.J.Healey,

Good comments. I like your imaginative example about a point particle in space.

Mathwonk,

I do think you also make some good points, but, I want to focus on one narrow and simply example, to keep things focused and hopefully easier to understand for all of us. Kind of like trying to analyze a system be reducing it to as few of degrees of freedom as possible, preferably down to a single degree of freedom when starting.


Instead of picking a point particle in space, I want to pick a simple lab or experimental real physical setting.

We can observe real physical space around us. We know space exists. We can detect it, quantify it, measure it and analyze it. In the process we can, through experimentation and analysis, figure out that we can measure three dimensions in real physical space such that any three mutually perpendicular directions can function as the three dimensional lengths of real physical space.

To consider how this is done, we can look at beginning with a single dimension of length in physical space. We pick two points and draw a straight line between the two. We can then imagine the line extending infinitely from both ends of that line segment defined by the two points.

But, to define this length, we have to be able to detect it, quantify it and measure it. You unit of quantity can be arbitrary. But it is required. As soon as you place the second required point to define the one dimensional length, you have already created the basis for a defined quantity, the distance between the two defining points. Even if that it the only reference you have, it is still a required quantity. Without being able to define a quantity of length you can not define a dimension of length.

Quantity and dimension are inter-related. You must have one to define the other.

Think about the process of observing, quantifying, measuring and analyzing one dimension of real physical space.

Then consider the following formal definitions:

A “quantity” is the property of a phenomenon, body, or substance, to which a magnitude can be assigned.

“Quantities of the same kind” are quantities that can be placed in order of magnitude relative to one another.

A “system of quantities” is a set of quantities together with a set of non-contradictory equations relating those quantities.

A “base quantity” is a quantity, chosen by convention, used in a system of quantities to define other quantities.

A “derived quantity” is a quantity, in a system of quantities, defined as a function of base quantities.

A “quantity dimension”, equivalently phrased as a “dimension of a quantity”, equivalently simply phrased as a “dimension” is a dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.

Apply these definitions to the process of observing, quantifying, measuring and analyzing one dimension of real physical space.

Does this help you understand what dimension means with respect to one dimension of real physical space?

 

[mathwonk]

to a mathematician this is about as precise as a metaphysical article on god. we cannot easily communicate from such widely varying poles of language usage. this is like eighteenth or nineteenth century scientific writing, or even euclids version of geometry.

please forgive me but i am going to butt out.

 

[JOPearcy]

Mathwonk,

As a mathematician, have you not gone through a process of defining concepts in this manner?

Again, trying to keep things very simple, it is very common in math to graph a function, “y = f(x)”.

You graph this in 2 dimensions, one dimension being related to the x-axis and the other dimension being related to the y-axis. Is this a very simple understanding that we can both agree and visualize so that we both know we are talking about the same thing?

What does the x-axis dimension mean? The x-axis dimension does not exist without some quantity of x. A quantity of x can not exist without there being an x-axis dimension.

Apply the definitions I gave to this 2 dimensional xy graph. They apply and they hold together.

How much of math is based on making logical definitions and statements in words or equations which can be logically verified or tested?

 

[K.J.Healey]

The problem is you're trying to assign some specific definition to a pre-defined word (that happens to have multiple definitions already).

As physicists and mathematicians, when we hear this argument, we all say "dimension, measurement, whatever, you know what we mean when we write y=f(x) on a graph". Math is above language.

 

[JOPearcy]

Math is above language? Hmm, so you can define math without the use of language. Well, I don’t want to begin straying from the point.

K.J.Healey,

Given we are talking about one dimensional length in real physical space, one of three accepted dimensions of length for real physical space.

How do you define the meaning of dimension as it applies to this one dimension of length?