Insights Can Angles be Assigned a Dimension? - Comments

[Baluncore]

You are saying that "dimensions" are identified by the SI "units of measure" - i.e. that the "unit of measure" is more fundamental than the concept of "dimension".

No, you are saying that.
I am saying that dimension is fundamental to physics, but that in the everyday human world, dimension is implicit, and is hidden behind the units. I say that knowing the dimension of a numerical result should identify the appropriate SI unit for that result.
A force of 9.8 Newton has implicit dimension identified by both the term “force” = M⋅L⋅T^–2, and the unit “Newton” = kg⋅m⋅s^–2. That duplication can be used as a check on data inputs, and then on the integrity of the numerical computation system. To maximise the application of that integrity check requires that dimensions such as length, angle or temperature be somehow attached like a tag to the numerical data as it flows through the computational system.

Why make the assumption that adding different dimensions is an error

I refer to simple numerical addition. In a complex number, the operator i serves to keep two numbers apart and so precludes their immediate numerical addition, even though they have the same fundamental physical dimension. They remain independent members in a set, or a data structure.

Dimensional analysis is used as one check on the integrity of physical equations. It does not however detect all errors. My aim is NOT to reduce a dimension system to a divine physical fundamental minimum. It is to identify what dimensions are needed to maximise the possibility of integrity checks in computational systems.

Alexander Pope wrote in his Essay on Criticism, “To err is human, to forgive divine”. I argue here that; if the angle dimension did not need to exist in divine physics, humans would need to invent an angle dimension to detect human error.

 

[Stephen Tashi]

I refer to simple numerical addition.

So do I. Why is it necessarily an error?

Dimensional analysis is used as one check on the integrity of physical equations. It does not however detect all errors.

Dimensional analysis detects what dimensional analysis defines to be errors. However, as mentioned in previous posts, it is possible to report the results of an experiment precisely using equations that don't conform to the requirements of dimensional analysis.

 

[Baluncore]

I refer to simple numerical addition.

So do I. Why is it necessarily an error?

In The Physical Basis of DIMENSIONAL ANALYSIS, on page 10;

A base quantity is defined by specifying two physical operations:

a comparison operation for determining whether two samples A
and B of the property are equal (A=B) or unequal (A≠B), and

an addition operation that defines what is meant by the sum
C=A+B of two samples of the property.

Base quantities with the same comparison and addition operations are of
the same kind (that is, different examples of the same quantity). The
addition operation A+B defines a physical quantity C of the same kind as
the quantities being added. Quantities with different comparison and
addition operations cannot be compared or added; no procedures exist for executing such operations.

 

[Stephen Tashi]

In The Physical Basis of DIMENSIONAL ANALYSIS, on page 10;

OK, but that passage is a statement of assumptions. By the same conventional wisdom (i.e the usual assumptions of dimensional analysis) angles are dimensionless. The Insight under discussion challenges conventional assumptions. So I'm questioning the basis for the conventional assumptions.

Nobody as risen to the challenge of justifying the conventional assumptions, so I'll try answering my own question.

The assumption that the left and right hand sides of an equation describing a physical law must have the same dimension is essentially an empirical finding. If we look a given field of physics organized as mathematics, there are "fundamental laws" (equations that correspond to mathematical assumptions) and there are equations derived from them. The pattern in physics is that the fundamental laws (which are only "laws" because they are confirmed empirically) obey the assumptions of conventional dimensional analysis. In particular the dimensions on the left and right hand sides of the fundamental equations match. The mathematical consequence of this appears to be:

Any equation derived from the fundamental laws also obeys the assumptions of conventional dimensional analysis.

It would interesting to know if anyone has formulated a mathematical proof of that assertion. If we assume that assertion then an equation that violates the assumptions of conventional dimensional analysis is definitely not derivable from the fundamental laws. However, the fact that the equation isn't derivable from the fundamental laws doesn't imply that the equation is an inaccurate description of a physical situation.

So dimensional analysis can detect an "error" in an equation the sense of detecting that the equation isn't provable from the fundamental laws. But such an "error" doesn't imply that the equation describes an impossible experimental result.

 

[Baluncore]

The assumption that the left and right hand sides of an equation describing a physical law must have the same dimension is essentially an empirical finding.

The term equation implies mathematical equality. Equality of numbers, units and dimension.
1.
LHS = RHS. Divide both sides by LHS and you get 1 = RHS / LHS. The 1 on the left must now be a dimensionless ratio. Are you saying that the ratio RHS / LHS might have somehow suffered from a “little big bang” and grown some dimension ?
2.
LHS = RHS. Divide both sides by RHS and you get LHS / RHS = 1. The 1 on the right must now be a dimensionless ratio. Are you saying that the ratio LHS / RHS might have somehow suffered from a “little big bang” and grown some dimension ?
3.
Does RHS / LHS have the same dimension as LHS / RHS, or the reciprocal dimension of LHS / RHS.

So dimensional analysis can detect an "error" in an equation the sense of detecting that the equation isn't provable from the fundamental laws. But such an "error" doesn't imply that the equation describes an impossible experimental result.

You could not publish such a discordant result because it would not survive the dimensional analysis of peer review. The result would undermine the physics we describe with mathematics.

If that experiment could be done once, the result would instantly propagate throughout our universe, at the speed of mathematics, annihilating all dimensional analysis and physics as we thought we knew it.

With some minor mathematical manipulation, such an experiment could create free energy from a dimensionless angle.

 

[RockyMarciano]

I have a related question for everybody. Does the dimensional analysis belongs to mathematics? Or should it be considered as a part of physics?

Dimensional analysis is usually referred to physical magnitudes, and following this definition of dimension as a physical magnitude with units and measurable it belong to physics. Then again everything physical is usually analyzed mathematically.

Can the notion of dimension (like meter or second) make sense without referring to a physical measurement?

Actually meter or second are physical units, and there's a distinction between units like meter and second and there corresponding physical magnitudes like length and time referred to standards that are subject to physical conditions like a platinum bar or an atomic frequency.I think in this thread it is not so clear what the OP refers to as dimension, I think he means a measurable unit that adds more information to physical quantities with angular components when it is not simply treated as dimensionless real number since it seems odd to think that radians or degrees depend on physical conditions like for instance temperature in the length case.

In this last understanding certainly treating angle as a "dimension" adds information, it basically turns scalars into oriented pseudovectorsAlso the comments in the article and thread about the relation with i and complex notions when giving dimension to angles comes naturally as related when thinking that the idea of a conformal structure in the complex line(or complex manifolds in general) leads to thinking of angles as being more than dimensionless numbers, the complex structure(biholomorphic mappings) also introduces the orientation-preservation referred to above in the complex manifold.
Also as referenced in the first posts this has been thought of before to different degrees on different contexts, for instance in the WP page on dimensional analysis under "siano's extension orientational analysis", the idea is there also.

 

[DaTario]

Have you tried to consider the application of these notions to quaternion formalism? Historically, after the work of Hamilton, the dot and cross products are originated from this entity, which introduces four different unities : 1, i, j and k.

Another comment. it seems that an argument exists saying that we should avoid adding entities of different dimensions for the following reason. If one does so, the matematical shape of the formula would depend on the choice of units. I have never gone into the details of this analysis but it seems reasonable. Perhaps you should mention this.

Best wishes,
Congratulations for the initiative.

DaTario

 

[haruspex]

consider the application of these notions to quaternion formalism

No, but modelling it on 3 vectors, one could make the product of any two of i, j, k like a cross product, so the operator has dimension Θ, but the product of i with i etc. like a dot product.

it seems that an argument exists saying that we should avoid adding entities of different dimensions for the following reason. If one does so, the matematical shape of the formula would depend on the choice of units.

Not sure what you mean. It is already the case that the form of many equations would be different if we were to use a complete circle as the unit of angle instead of using radians.

 

[Baluncore]

It is already the case that the form of many equations would be different if we were to use a complete circle as the unit of angle instead of using radians.

I believe the form of the equations would be the same, but the π related coefficients would have a different value since angle dimension is then being measured in different units. Is that not why 2π often appears in physics formulas, because of the mathematically convenient radian unit we have chosen to use for the angle dimension.

 

[haruspex]

I believe the form of the equations would be the same, but the π related coefficients would have a different value since angle dimension is then being measured in different units. Is that not why 2π often appears in physics formulas, because of the mathematically convenient radian unit we have chosen to use for the angle dimension.

Yes, I was not sure what DaTario meant by a different "form". From the reference to units, I presumed DaTario would regard the appearance of a factor of 2π as being a different form, but, like you, I would consider that the same in form, just different in detail.

 

[DaTario]

Yes, I was not sure what DaTario meant by a different "form". From the reference to units, I presumed DaTario would regard the appearance of a factor of 2π as being a different form, but, like you, I would consider that the same in form, just different in detail.

Sorry, I was referring to the presumed fact that if we allow in physical equations the sum, for instance, of entities having different dimensions (as for example, summing meters to seconds) the equations would not be invariant through the change in units.

I have just found a site with a modest exposition of this idea:
http://www.johndcook.com/blog/2013/11/15/dimensional-analysis/

It seems to be in accordance to the " cloudy" reference I have claimed to have read a long time ago.

Best wishes,

DaTario

 

[haruspex]

I was referring to the presumed fact that if we allow in physical equations the sum, for instance, of entities having different dimensions (as for example, summing meters to seconds) the equations would not be invariant through the change in units.

That is one very good reason for not allowing such.
If the concept of attributing a dimension to angles has any validity, it must be possible to write any correct equation such that it is dimensionally consistent in that regard.

 

[DaTario]

That is one very good reason for not allowing such.
If the concept of attributing a dimension to angles has any validity, it must be possible to write any correct equation such that it is dimensionally consistent in that regard.

Two other questions:
1) Following your axioms, would it be correct to say that solid angles are truly adimensional?

2) the square root of an adimensional quantity has the angle dimension?

Best wishes,
DaTario

 

[haruspex]

Following your axioms, would it be correct to say that solid angles are truly adimensional?

Good question. I found a reason for saying they also have the angle dimension. E.g. if we consider two angle vectors $\vec {d\theta}$ and $\vec{d \phi}$, and a vector radius $\vec r$, the two arc elements are $\vec r\times\vec{d\theta}$ and $\vec r\times\vec{d\phi}$. The vector area element they form is $(\vec r\times\vec{d\theta})\times (\vec r\times\vec{d\phi})$. Counting the angle elements and cross products that has angular dimension.

the square root of an adimensional quantity has the angle dimension?

No, that would be ambiguous. Not unusual for square roots.

 

[DaTario]

Good question. I found a reason for saying they also have the angle dimension. E.g. if we consider two angle vectors $\vec {d\theta}$ and $\vec{d \phi}$, and a vector radius $\vec r$, the two arc elements are $\vec r\times\vec{d\theta}$ and $\vec r\times\vec{d\phi}$. The vector area element they form is $(\vec r\times\vec{d\theta})\times (\vec r\times\vec{d\phi})$. Counting the angle elements and cross products that has angular dimension.

But in the case of solid angles (stereoradians - sr) the operation is:
$$ \frac{s}{r} \frac{s}{r}. $$

So it is area divided by the square of the radius.

 

[haruspex]

But in the case of solid angles (stereoradians - sr) the operation is:
$$ \frac{s}{r} \frac{s}{r}. $$

So it is area divided by the square of the radius.

That's only after reducing it all to scalars. To see how angular dimension fits in, in my scheme, it seems to be necessary to work with vectors wherever appropriate.

 

[Stephen Tashi]

Sorry, I was referring to the presumed fact that if we allow in physical equations the sum, for instance, of entities having different dimensions (as for example, summing meters to seconds) the equations would not be invariant through the change in units.

Equations are not "invariant" under a change of units. For example F = MA is correct if F is in Newtons, A is in m/sec^2 and M is in kilograms. However if we measure M in grams, F = MA isn't correct.

In general, changing units changes equations by multiplying one or both sides of the equation by constant factors.

So the prohibition against adding unlike units can't be explained by the invariance of equations. We have to explain why a particular type of variation is the only permissible kind.

 

[Baluncore]

In general, changing units changes equations by multiplying one or both sides of the equation by constant factors.

Those constants actually have units, but not dimensions. For example, to convert metres to cm, multiply by 100 cm/metre. The metre units cancel, leaving the new unit cm in it's place. The units in the factor ratio must have the same dimension.

Notice also that conversion factors are always equal to one. For example 100 cm/metre = 1.
I would say that when all you are doing is multiplying something by one, it is invariant.

 

[Stephen Tashi]

I would say that when all you are doing is multiplying something by one, it is invariant.

But the numerical factor resulting from 1000 grams/kg is "1000", not "1". You are introducing a different concept of "1" than the purely arithmetic concept of "1". I agree that if we use a concept of "invariance" that says the equation F = MA in one system of units can be "the same equation" as F = 1000 MA in a different system of units then that definition allows equations that are not arithmetically invariant to be considered "invariant".

However, this extended concept of invariance doesn't explain why unlike dimensions can be multiplied and divided, but not added. Any explanation goes in the other direction - i.e. if we assume that conversion factors involving products and quotients of units of arbitrary dimension will be used, then particular conversion factors can be regarded as a type of "1" (e.g. "1000 g/ kg" is a such "1"). From that, we get a type of "invariance" of equations (different from arithmetical invariance) as a consequence.

 

[diogenesNY]

Submitted for your approval:

Benoit Mandelbrot seems to have had something to say on this issue, or at least in the same neck of the woods. In fact in part he seems to consider this effort a step towards getting Mathematics and Physics to play nice together without adult supervision. I am not sure that he actually accomplishes this, but that is another issue.

In this short and highly readable paper he addresses the Coastline Paradox and suggests a way of dealing with it that is borderline mind-expanding. Highly recommended.

Title: How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
This was originally published in Science in 1967.

Here is a wikipedia article _about_ the paper:

https://en.wikipedia.org/wiki/How_L...ical_Self-Similarity_and_Fractional_Dimension

Here is the article at the Science although you may need a subscription or academic site license to read it:

http://science.sciencemag.org/content/156/3775/636

Here is the article at JSTOR - this should be available at most (site licensed) academic institutions.:

http://www.jstor.org/stable/1721427

Okay, I just found a direct link to a (seemingly weirdly formatted) .pdf of the original article here:

http://users.math.yale.edu/~bbm3/web_pdfs/howLongIsTheCoastOfBritain.pdf

I hope y'all find this a worthy contribution.

diogenesNY

 

[Baluncore]

However, this extended concept of invariance doesn't explain why unlike dimensions can be multiplied and divided, but not added.

Please give an example of where you might need to add or subtract two differently dimensioned values?
What concept is involved and how is it meaningful to physics?

 

[Stephen Tashi]

Please give an example of where you might need to add or subtract two differently dimensioned values?
What concept is involved and how is it meaningful to physics?

My interest is in finding a correct justification for why conventional dimensional analysis allows multiplication of different dimensions, but not addition of different dimensions. I'm not advocating a revision of dimensional analysis to allow adding different dimensions. I'm advocating that we find a coherent statement of why multiplying different dimensions is allowed and adding different dimensions is not allowed.

The explanations we often hear are just dogma - e.g. "It would be nonsense to add apples and oranges". (Yet it does make sense to multiply apples by oranges ?)

Some methods for justifying a statement S are the following.

1) S is true because it is empirically true. We assert S is observed to be true without offering any proof or explanation of why S is true.

2) S is a theorem. S can be proven from other statements we accept as truths. This includes indirect proofs. e.g. Reasoning that begins "Suppose we did add apples and oranges then it would follow that ..."

3) S is an assumption or definition.

In my (current) opinion, the coherent justification for the principle "You can't add apples and oranges" is 1). It is an empirical fact.

People who are familiar using conversion factors become so familiar with the convenience they offer that they wish to use method 2) and offer words to prove that "You can't add apples and oranges" as a theorem. Dignified treatments of dimensional analysis used method 3), they simply assert "We may multiply apples by oranges but we do not add apples and oranges" as an assumption or make it a consequence of a definition.

Using method 1): Consider how often in physics it is sufficient to know only the product of two dimensional quantities in order to make a prediction. For example, to predict whether we can unscrew a bolt, it is often sufficient to know the available torque we have in ft-lbs. If we need a torque of 20 ft lbs, we can realized this torque by a 10 lbs force acting on a 2 ft lever, or a 20 lbs force acting on a 1 ft lever etc. The measurement of ft lbs is ambiguous as to how many ft and how many lbs are involved in the phenomena. ( In the particular case of "zero ft lbs", we at least know that that there were 0 ft or 0 lbs involved. ) The ambiguity in a measurement like 20 ft lbs often doesn't matter because there are important behaviors in Nature that are completely specified by the product of two dimensioned quantities and don't depend on how the factors in that product are implemented.

Dimensional analysis does not assert that it always makes sense to multiply quantities of different dimensions. It only asserts that one may multiply quantities of different dimensions. The justification is that we find empirically that there are many situations where we can make a useful prediction knowing the value of a product without knowing the value of its factors.

Empirically , we do not find any notable situations where knowing the value of a sum of differently dimensioned quantities (without knowing the values of the summands) allows us to make a useful prediction. For example a measurement of "20 apples+oranges" is ambiguous about whether there were 0 apples and 20 oranges , versus 10 apples and 10 oranges, versus -50 apples and 70 oranges, etc.

If someone can offer an explanation of why Nature operates so that the ambiguity of products is often useful, but the ambiguity of sums is not, then I'd like to hear it. As far as I can see, it is an empirical fact, not a theorem.

 

[Baluncore]

My interest is in finding a correct justification for why conventional dimensional analysis allows multiplication of different dimensions, but not addition of different dimensions.

We can simplify 3x*4y to 12xy.
Now explain why 3x+4y cannot be simplified.
That should satisfy your interest.

 

[haruspex]

Yet it does make sense to multiply apples by oranges ?)

It can. Tommy is allowed to take one apple and one orange from 3 apples and 4 oranges. What is the set of possible choices? 12 apple-orange pairs.
But fruit does not make a good analogy because you can also argue for adding apples and oranges. The discreteness creates a natural unit of measure.
Here is a possibility... Mutiplying makes sense because you can create new units to match. In some scenario, I take K kg and M metres to compute the product KM kgm. If you prefer to work in pound-inches, you know how to convert that without having to know K and M separately. But if we try to invent the concept of mass plus distance, and I tell you (K+M) "kg+m", you cannot convert the single number K+M to an equivalent number of pound+inches.

 

[Stephen Tashi]

We can simplify 3x*4y to 12xy.
Now explain why 3x+4y cannot be simplified.
That should satisfy your interest.

That makes no connection to physics.

 

[Stephen Tashi]

It can. Tommy is allowed to take one apple and one orange from 3 apples and 4 oranges. What is the set of possible choices? 12 apple-orange pairs.
But fruit does not make a good analogy because you can also argue for adding apples and oranges. The discreteness creates a natural unit of measure.

Dimensional analysis does not say that a physical unit can take a values corresponding to any given real number. So I see nothing wrong with your analogy.

Here is a possibility... Mutiplying makes sense because you can create new units to match. In some scenario, I take K kg and M metres to compute the product KM kgm. If you prefer to work in pound-inches, you know how to convert that without having to know K and M separately.

I'd put it this way: Changing the units in a product and "knowing how to convert" the result (using conversion factors in the standard manner" introduces no new ambiguity in the description of the physical situation.

We "know how to convert" because we know to use conversion factors. But the conventions of using conversion factors doesn't explain why those conventions are physically useful. In my opinion, it is just an empirical fact that the ambiguity in knowing the product of dimensions, but not knowing the individual factors often doesn't detract from the usefulness our knowledge in making physical predictions.

But if we try to invent the concept of mass plus distance, and I tell you (K+M) "kg+m", you cannot convert the single number K+M to an equivalent number of pound+inches.

That depends on what you mean by "equivalent". Perhaps you mean we have no obvious rules to convert it to a unique number of different units. For example, the conversion 4 apples+oranges to units of half_apples+oranges in a naive fashion is ambiguous because it might be done by converting 4 apples+oranges as (3 apples + 1 oranges) which is converted to (6 half_apples + 1 oranges) = 7 half_apples+oranges. Or it might be converted from (1 apples + 3 oranges) as ( 2 half_apples + 3 oranges) = 5 half_apples+oranges. So , if there is an empirical difference between the physical situation producing the measurement 7 half_apples+oranges and the situation producing the measurement 5 half_apples+oranges, this is a argument that converting units in such a manner introduces a harmful ambiguity.

 

[Baluncore]

That makes no connection to physics.

Yes it does. Physics requires that a mathematical equation be evaluated. If you cannot evaluate an equation to a single numerical value then you do not need to add the dimensions. The mathematical reason why differing dimensions are not added in physics is the same reason that 3x + 4y cannot be simplified. 3x + 4y = 3x + 4y.

 

[Stephen Tashi]

Yes it does. Physics requires that a mathematical equation be evaluated. If you cannot evaluate an equation to a single numerical value then you do not need to add the dimensions. The mathematical reason why differing dimensions are not added in physics is the same reason that 3x + 4y cannot be simplified. 3x + 4y = 3x + 4y.

"Equations" are mathematical statements that two functions are equal. Functions that can't be "simplified" can still be "evaluated".

 

[Baluncore]

Functions that can't be "simplified" can still be "evaluated".

So how do you evaluate 3x + 4y = ?

 

[Stephen Tashi]

So how do you evaluate 3x + 4y = ?

We don't "evaluate" 3x + 4y unless we are give numerical values for x and y. Likewise we don't evaluate 6xy unless we are given numerical values for x and y.