Insights Can Angles be Assigned a Dimension? - Comments

[robphy]

That depends on the units of the constants 1,2,6,24,120...

For example, the equation $y = 1 + 5x + 2x^2$ can describe a physical situation where $y$ is in units of Newtons, x is in units of meters, 1 is in units of Newtons, 5 is in units of Newtons per meter and 2 is units of Newtons per meter squared.

Can an object have a position given by $y = e^x$ ?

In the series expansion for exp(x/a), all of those numbers are pure [dimensionless] numbers... they are part of the definition of exp(z), where z is dimensionless.
Thus, the only thing that carries units is "x".
So, what are the units of the right-hand side?

No, $y = e^x$ cannot be a position equation...
You could have, say, $y=Ae^{(-t/\tau)}$, where $A$ has units of length, and $t$ and $\tau$ have units of time.

edit:
Your proposed equation: $y = 1 + 5x + 2x^2$ with units as you specified
possibly should be written as
$y = (1\ \rm{Newton})( 1 + 5 (\frac{x}{m}) + 2(\frac{x}{m})^2)$
[trying to conform to the exponential series expansion].
The point is... if there are units, they should be shown.

 

[Stephen Tashi]

In the series expansion for exp(x/a), all of those numbers are pure [dimensionless] numbers... they are part of the definition of exp(z), where z is dimensionless.

The mathematical definition of the function $f(x) = x^2$ likewise assigns no dimension to $x$. So the lack of dimension in the mathematical definition of a function don't prevent us from giving the argument of the function a dimension when we employ it in physics.

Thus, the only thing that carries units is "x"

I'm not making that assumption.

No, $y = e^x$ cannot be a position equation...
You could have, say, $y=Ae^{(-t/\tau)}$, where $A$ has units of length, and $t$ and $\tau$ have units of time.

I disagree. If an experimenter fits an equation of the form $y = e^t$ to his data where $y$ is in meters and $t$ is in seconds, he has described a physical relation unambiguously and a different experimenter who wishes to measure distance in centimeters and time in minutes can figure out how to create an equivalent equation using those units of measurement.

It may be true that it would more convenient for the second experimenter if the first experimenter had written his results in a different form.

 

[robphy]

If an experimenter fits an equation of the form $y = e^t$ to his data where $y$ is in meters and $t$ is in seconds, he has described a physical relation unambiguously and a different experimenter who wishes to measure distance in centimeters and time in minutes can figure out how to create an equivalent equation using those units of measurement.

It may be true that it would more convenient for the second experimenter if the first experimenter had written his results in a different form.

I agree it is unambiguous as long as all of those specifications of units are included with the equation in the sentence.
And, so, if someone uses t in units of hours, then they would expect to get the wrong answer.
Your equation would look different of course if someone wanted to use t in hours.
However, if you wrote
y=(1 meter)e^(t/(1 second))
then your equation would hold for a time t expressed in any units of time... and in fact would encode the requirements of your preferred choice of units, without forcing the person to use those units [as long as they performed the appropriate unit conversion].

This is an issue with curve-fitting software that I tell my students about. The software giving fit coefficients doesn't know how you are using them in an equation... that is, the physical interpretation of the data... it just sees a curve. So, the end user has to manually attach units to the fit coefficients.

By the way, we do this all the time with our calculators.
Presuming that our equations are in a consistent set of units, then we can just plug in the numbers and calculate. Then insert the appropriate unit at the end.

EDIT:
Often, we wish to focus on the law of physics [independent of coordinates, and independent of units].
It is another matter, if we wish to plug in special values for lookup in a table or a prepared chart.
In the latter case, there are units.. that one might want to implicitly assume... but they are there.

 

[Stephen Tashi]

Let's look at how much physics we must specify in order for the dimensions in a McLaurin series to work out.

Suppose I specify that $y$ has dimension length in units of meters and $t$ has dimension of time in units of seconds and $y = f(t)$ ( i.e. $f(t)$ has dimension length in units of meters.)

Then the Mclaurin series for $f(t)$ is $f(0) +f'(0) t + f''(0) t^2/2 + ...$

$f(0)$ has units of length in meters
$f'(0)$ means "take the derivative of $f$ and evaluate it at $t = 0$, so $f'(t)$ has units of meters per second
By a similar argument, $f''(0)$ has units of meters per second squared.

Each term in the McLaurin expansion that comes from a derivative of $f$ has the appropriate units in meter/ sec^k to produce the unit "meters" after it is multiplied by the power $t^k$ in seconds that is paired with it.

In the particular case of $y = e^t$, if we look at the mathematical derivation of the power series for $e^t$, the mathematics tells us that the constant terms have the appropriate units once we specify the units of $e^t$ and the units of $t$.

 

[Torbjorn_L]

It is a suggestion to look at. But my reaction is perhaps best summed up in short points:

1. The suggestion feels like an artificial attempt to fix something that isn't broken.
2. It looks to be analogous to the use of the mathematical dimension i and so doesn't add to the algorithms for solving physical problems
3. As seen from 2. it confuses mathematical dimensions (of angles, phases, non-euclidean spaces and functional spaces) with physical dimensions. It is perhaps best seen in the analysis of the Planck constant, which gives nonsensical results.

If the suggestion is spurred by students making unit mistakes that are invisible to the physics of dimensional analysis the solution should lie along Yggdrasil's observation. Treat angles (phases, ...) as fractions of a circle.

 

[haruspex]

It looks to be analogous to the use of the mathematical dimension i and so doesn't add to the algorithms for solving physical problems

In most of my examples there are no complex numbers, yet it adds a dimension. Checking that dimension would sometimes indicate algebraic errors, as with the other dimensions. E.g. I might wish to obtain an expression for the angular momentum of something. If the expression I get has dimension ML²T^-1 instead of ML²T^-1Θ then I know I have gone wrong.

It is perhaps best seen in the analysis of the Planck constant, which gives nonsensical results.

I've been working on that, and I believe I can make that work now.

Treat angles (phases, ...) as fractions of a circle.

As I pointed out, that solves nothing. You could equally make mass dimensionless by thinking of all masses as fractions of some standard mass. You may counter that the standard mass has dimension, so any fraction of it has dimension, but that is different. Saying A is some fraction of B means it is a fraction multiplied by B; it does not mean that A is that fraction as a mere number. Likewise, I would argue that a whole circle has dimension Θ, so any fraction of it has dimension Θ.
Also, I fail to see how that approach could be used in spotting algebraic errors. Seems more like it would hide them. Can you explain with an example?

 

[Baluncore]

I like the way this thread is going as it is covering some of those questions I have previously encountered.

A few years ago, I overloaded a computation language with dimensional analysis and unit conversions based on SI, just for the fun of it. I had problems with the dimension of angles, and with polynomial approximations which became too difficult at the time, so remained dimensionless. It is only when implementing a complete general dimension system that you encounter the deepest problems, like how do you represent a fractional dimension such as when you take a square root. Or how do you represent a dimensioned variable that is raised to a non-integer power.

I ended up needing dimensions of: length, mass, time, current, temperature, light, substance, angle, information and currency.
The advantage of including dimensions of temperature, angle and currency was that conversion between different inputs to output units could be more easily implemented. (e.g. celsius, kelvin, fahrenheit; degree, radian, grad; dollar, euro or yen.)

Unlike the the physical SI units, currency has a dynamic exchange rate, with inefficient conversions. There is no way that I can see to have a standard currency unit. Gold mining generates currency, but the demand for gold, and the cost of mining gold is variable. The closest physical unit to money is actually energy. My solar PV array could pay for itself. It is difficult to see immediately how inflation would be possible if our bank accounts held credit in joules. But then unregulated interest and taxation rates would be introduced by the bank and tax office.

 

[Stephen Tashi]

like how do you represent a fractional dimension such as when you take a square root. Or how do you represent a dimensioned variable that is raised to a non-integer power.

To answer how to represent something, we must say what we are trying to accomplish with the representation.

What we are trying to accomplish with a given theory of dimensions?

I see the the most basic requirement as:

If an experimenter states his results as an equation in one system of units, then a second experimenter who uses a different system of units must be able to interpret the results of the first experimenter in that different system of units.

This is a very relaxed requirement. For example, suppose there is a specific machine M. To operate it, an experimenter turns a crack through a given angle $\theta$ and holds it at that position for time $t$. The crank is released and the machine moves along the table for a distance $x$. The first experimenter states his results as $x = 3\sqrt{t} sin(\theta)$ where $x$ is in meters , $t$ is in seconds, and $\theta$ is in degrees.

I think a second experimenter who wishes to use a system of units consisting of centimeters, minutes, and radians can figure out how to state the results of the first experimenter in that system of units. So what is our theory of dimensions trying to accomplish in this situation? Are we seeking a theory where changing the units in an equation is always done by a particular procedure ? - conversion factors, for example. If expressing a result in different units cannot be done by using conversion factors, are we prepared to say the result is "not physically meaningful"?

 

[Baluncore]

A measurement without units is meaningless. Consider a measured value, complete with units as an input to a process. The units identify the dimension of the value. Convert that value to SI using known conversion factors. The dimension will not change.

Proceed with the computations while tracking any and all the combinatorial changes of dimensions. Adding or comparing apples and oranges will raise an immediate runtime error.

The final result will have dimensions that identifies the appropriate SI units of the result. If the resulting SI unit is silly, then dimensional analysis has identified an error is present. Either the wrong data has been input or the computational algorithm is wrong.

That is why for example, angle and temperature dimensions must exist in the system. Because they will pass through the dimension analysis system to verify integrity and identify the final SI unit, in this example as angle or as temperature.

For a calculator, the dimensional analysis module should follow all the data. If you press the wrong key it will detect your failure to use the correct algorithm.
To be most efficient in a computer, the dimensional analysis module might best be part of the compiler rather than a runtime module that tracks every repeated computation.

 

[Stephen Tashi]

A measurement without units is meaningless. Consider a measured value, complete with units as an input to a process. The units identify the dimension of the value. Convert that value to SI using known conversion factors. The dimension will not change.

That's backwards to the usual approach because the outlook of conventional dimensional analysis is that dimensions (e.g. time, mass) are the fundamental properties of nature and various units of measure (e.g. kilograms, seconds) are invented to quantify a dimension. 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".

Proceed with the computations while tracking any and all the combinatorial changes of dimensions. Adding or comparing apples and oranges will raise an immediate runtime error.

Why make the assumption that adding different dimensions is an error? As pointed out by others in the thread, there are two possible interpretations of "addition". One type of addition is "appending to a set" - for example, put 2 apples in a bag and then put 3 oranges in the bag. Another type of addition is "summation of numerical coefficients of units and creation of a new type of unit that does not distinguish the summands". An example of that would be: 2 apples + 3 oranges = 5 apples+oranges.

It's easy to say that "5 apples+oranges" makes no sense, but why do we say that? After all we don't object to products of units with different dimensions like 5 (ft)( lbs). What makes a unit representing a sum of dimensions taboo, but allows a unit representing a product of dimensions to be "the usual type of thing" ?

The answer might be that Nature prefers the ambiguity in products. For example, in many situations, the "final effect" on a process of a measurement 5 (ft)(lbs) is the same , no matter whether it came from a situation implemented as (1 ft) (5 lbs) or (2.5 ft) ( 2 lbs), etc. So the ambiguity introduced in recording data in the unit (ft)(lbs) is often harmless. However, it is not harmless is all physical situations. If a complicated experiment involves a measurement of 2 ft on something at one end of the laboratory and 2.5 lbs on something at the other end of the laboratory, summarizing the situation as 5 (ft)(lbs) may lose vital information.

Is it a "natural law" that products are the only permitted ambiguities? Allowing the ambiguity implied by a sum-of-units fails to distinguish situations that are (intuitively) vastly different. For example a measurement of 5 apples+oranges could have resulted from inputs of 3 apples and 2 oranges, or 0 apples and 5 oranges, or 15 applies and -10 oranges. However (taking the world view of a logician) it is possible to conceive of situations where this type of ambiguity has the same "net effect". We can resort to thinking of a machine with a slot for inputting apples and another slot for inputting oranges. The machine counts the total number of things entered and moves itself along the table for a distance of X feet where X is the total.

Is the argument in favor of products-of-units and against sums-of-units to be based only on statistics? - i.e that one type of ambiguity is often (but not always) adequate for predicting outcomes in nature, but the other type of ambiguity is rarely adequate ?

I suspect we can make a better argument in favor of products-of-units if we make some assumptions about the mathematical form of natural laws. For example, do natural laws stated as differential equations impose constraints on the type of ambiguity we permit in the measurements of the quantities that are involved ?

 

[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.

 

[Baluncore]

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.

If x was the unit metres and y was the unit seconds, what would you do then?

 

[Stephen Tashi]

If x was the unit metres and y was the unit seconds, what would you do then?

Are you trying to formulate an argument that says "We can't add unlike dimensions because we can't add unlike dimensions"?

As I said, I'm not advocating adding unlike dimensions. I'm investigating what justification we can give for declaring that adding 3 meters and 2 seconds to obtain a composite unit of 5 meters+seconds can never be done ( or should never be done) - no matter what interpretation we give to "meters+seconds".

Declaring "we can't add 3 meters to 2 seconds" or saying "adding 3 meters to 2 seconds doesn't make sense" isn't an explanation. If we want to demonstrate that there is no way to do something, then issuing the challenge "Somebody show me how to do it" isn't a demonstration. For example, "Show me some positive integers $a,b,c$ such such that $a^3 + b^3 = c^3$" isn't a proof of Fermat's theorem.

 

[haruspex]

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.

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.

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.

Maybe something along these lines...
We have two systems characterised by vectors of attributes, S=(x₁, x₂, ..) , S'=(x₁', x₂', ...). We can invent measures of attributes, i.e. maps to reals, m₁, m₂,... We believe (and this is the physics) that the attributes have an additive property such that in some sense we can partition an attribute and recover the whole by summing its parts. This being so, we require our measures to be linear in this sense.
Given two measures, m₁ and n₁ for X₁, if one is linear and there is a nonzero constant μ such that n₁(x)=μm₁(x) for all x, then clearly theother is linear. [Can we show that for two linear measures such a ratio exists?]

Anyway, if we want the product of measures of two attributes, we may choose m₁(x₁)m₂(x₂) or n₁(x₁)n₂(x₂). Armed with the knowledge of the measure ratios, μ, ν, we can compute the measure ratio μν for the measure products.

If we try to do the same summing two measures, there is no constant ratio that can convert one sum of linear measures to another.
This is much the same as I wrote before, but highlighting the dependence on an essential additivity in the underlying attributes, and the consequent requirement of linearity in the measures.

 

[Stephen Tashi]

that the attributes have an additive property such that in some sense we can partition an attribute and recover the whole by summing its parts. This being so, we require our measures to be linear in this sense.

I think that's a fundamental idea in dimensional analysis - but it's a concept that's hard to state only using terms from arithmetic. The notion that the whole attribute "can be recovered" from the "sum" of its parts isn't quite the same notion as the concept that the whole numerical value of something is equal to the arithmetic sum of its "parts".

One attempt to express the physics is that "the presence of an attribute x_1" has the same effect as the simultaneous presence of the mutually exclusive "parts" of the attribute. (That uses "has the same effect" instead of the arithmetic relation "equal" and "simultaneous presence" instead of the arithmetic sense of "sum".)

Alternatively, we could state the concept by embedding it in the definition of the "parts" of an attribute. For example "x_a and x_b are each "half" of attribute x_1" shall mean the following are satisfied: 1) The simultaneous presence of x_a and x_b has the same effect (in whatever phenomenon we are studying) as the presence of x_1 alone - and 2) The presence of x_a alone has the same effect as the presence of x_b alone.

Taking that approach, we could argue that is is useful to make an "isomorphism" between the physical language and mathematical language. i.e. "same effect" maps to "equal". "simultaneous presence" maps to "sum", "half of" in the physical sense maps to "half of" in the arithmetic sense.

As I understand what you are doing with a measure ratio, you are assuming the attribute x_1 is a numerical value at the outset.

If we try to do the same summing two measures, there is no constant ratio that can convert one sum of linear measures to another.

I agree that the use of ratios (i.e. conversion factors) for individual attributes doesn't define how to convert a sum of different dimensions measured in one set of units to a unique sum of the same dimensions measured in a different set of units. When people ask why we "can't" add different dimensions together, I think this inability is a good explanation of why we "don't" add different dimensions.

However, the statement that "it is impossible to add different dimensions" is claim that goes beyond what is convenient or inconvenient. Are we saying that no matter what scheme you make up for adding different dimensions, it won't work? What would it mean for a scheme (which may not be based on conversion factors) to work or not to work?

The general situation as I see it:

The product of xy of units x, y with unlike dimensions omits the information about the individual magnitudes of x and y. There are many physical situations where knowing the product xy is useful even though we don't know the individual values of x and y. (e.g. When a result of interest can be expressed as a function of one variable p = xy instead of function of two variables (x,y).) There are also physical situations where the knowledge of xy is not as useful as knowing the individual values of x and y. For example, if x is the diameter of your coffee cup in cm and y is the length of your cat's tail in cm then the product xy doesn't summarize the situation as well as knowing the individual values of x and y.

By contrast, empirically, there are few situations in physics where knowing the sum x + y of units of different dimensions is useful.

Can we go even further to say "Any physical situation where the result is expressed as a function of the sum of units of different dimensions can be rewritten as a different function of the same variables that does not involve the sum of units of different dimensions"?

 

[DaTario]

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.

Yes, I agree. Admissible invariances seems to be those which involves solely a scale factor.

 

[Baluncore]

By contrast, empirically, there are few situations in physics where knowing the sum x + y of units of different dimensions is useful.

Can you please give examples of those few situations in physics.

 

[haruspex]

@Stephen Tashi , @Baluncore :
I am pleased to have engendered such an interesting debate, but it does seem to have wandered a long way from my original article. How would you feel about having it moved to a different thread?
Meanwhile, you might be interested in http://nvlpubs.nist.gov/nistpubs/jres/65B/jresv65Bn4p227_A1b.pdf. Seems like much of my endeavour is a rediscovery of Page's work. See p 231 and Appendices 2 and 3. He did miss the trick of giving i angular dimension, and did not discuss Planck's constants. The rest of his article might be relevant to your own discussion.

 

[Stephen Tashi]

How would you feel about having it moved to a different thread?

If this thread is getting too long to be readable for new contributors then the discussion should be restarted in another thread. What would the topic of the new thread be?

It interests me to discuss the general foundations of dimensional analysis. I'm not particularly interested in debating the narrower issue of whether angles should be assigned a dimension until the big picture is clear. However, perhaps others would be interested in a new thread restricted to the topic of angles.

The discussion about whether angles should or shouldn't be assigned a dimension just meanders around because the reasons underlie dimensional analysis haven't been established. Some people just repeat the mantra "Angles are dimensionless" because it is a convention of standard dimensional analysis. More liberal participants say that angles can be assigned a dimension because such-and-such an aspect works out to be pleasing and others counter by saying that a different aspect of the situation isn't pleasing. Until the significance of the various aspects is clear, the importance of satisfying them is just a matter of individual tastes.

Discussions about whether angles may be assigned dimensions can become to similar to discussions about other "notorious" topics such as "Is 0.9999... = 1" or "Are dy and dx numbers?", "Can we divide by infinity?". Such discussions tempt us to ignore the arbitrary nature of definitions. We tend to assume that words and notations like "angles", "0.9999...", "dx", "infinity" are particular things that exist independently of anyone's definition of them. Then we use our intuitive concept of these pre-existing things in debating what their properties "are" instead of what we may define those properties to be.

 

[Baluncore]

How would you feel about having it moved to a different thread?

Sorry about being off topic, I plead a little bit guilty. Go ahead and split or fork the thread.
We need more discussion about those interesting bundles of {numbers, units and dimensions}.

 

[haruspex]

What would the topic of the new thread be?

It's your topic. Maybe "why does dimensional analysis work?"