Can Angles be Assigned a Dimension? - Comments

[Stephen Tashi]

In your Poisson example, yes. λ was specified as the average number of events in some unstated but fixed interval.

But if I am stating an equation that describes a physical situation, I can't get away with giving an equation that applies to an unstated interval.

Suppose the equation that fits my experimental data is $f(k) = \frac{ (2.3)^k e^{-2.3}}{k!}$ and an experimenter attempts to duplicate my results. He uses an interval of 10 seconds to define $\lambda$. In order to compare his results to mine, he needs to know what interval I used. He asks me and I tell him "My interval was 5 seconds long". The version of my equation that he can check against his data is $f(k) = \frac{(4.6)^k e^{-4.6}}{k!}$.

Are we to say that this conversion of equations takes place by some method other than by converting units using conversion factors ?

One may object: "You should have reported your equation in dimensionless form". That would side-step the need to convert units. However, reporting results in dimensionless form isn't a requirement in science.

 

[haruspex]

But if I am stating an equation that describes a physical situation, I can't get away with giving an equation that applies to an unstated interval.

Suppose the equation that fits my experimental data is $f(k) = \frac{ (2.3)^k e^{-2.3}}{k!}$ and an experimenter attempts to duplicate my results. He uses an interval of 10 seconds to define $\lambda$. In order to compare his results to mine, he needs to know what interval I used. He asks me and I tell him "My interval was 5 seconds long". The version of my equation that he can check against his data is $f(k) = \frac{(4.6)^k e^{-4.6}}{k!}$.

Are we to say that this conversion of equations takes place by some method other than by converting units using conversion factors ?

One may object: "You should have reported your equation in dimensionless form". That would side-step the need to convert units. However, reporting results in dimensionless form isn't a requirement in science.

I'm sorry, I am not grasping your point.
If the two experiments concern the same underlying process, presumably the rates should be the same. Therefore the "correct" version of the equation would make λ that rate and have λt everywhere that your equation has just λ. λt is dimensionless, as required.
The version of the equation in your post #25 can be likened to rating the top speed of a car as the number of kilometres it can go in a standard interval of one hour. That does not mean its speed has only a length dimension.

 

[Stephen Tashi]

I'm sorry, I am not grasping your point.

Lets try this: Suppose there is a random variable X , measured in meters, that has its density defined on interval $[0, \ln (2) ]$ by $f(x) = C( 2 - e^{x})$ where $C$ is the normalizing constant $\int_{ 0}^{ln (2)} {(2 - e^ {x})} dx$.

A experimenter who measures $X$ in centimeters can convert the above density function to the appropriate density for$X$ when $X$ is measured in centimeters. I agree that assigning units to the left and right hand sides of $f(x) = C( 2 - e^{x})$ is a confusing or impossible task. But I don't agree that the $e^x$ in the equation implies that the equation describes a physically impossible situation or that it makes it impossible for a experimenter measuring X in different units to convert the above density to his system of measurement.

 

[Demystifier]

I have a related question for everybody. Does the dimensional analysis belongs to mathematics? Or should it be considered as a part of physics? Can the notion of dimension (like meter or second) make sense without referring to a physical measurement?

 

[S.G. Janssens]

I have a related question for everybody. Does the dimensional analysis belongs to mathematics? Or should it be considered as a part of physics? Can the notion of dimension (like meter or second) make sense without referring to a physical measurement?

Preferably it belongs to physics. Otherwise, I am afraid that your next question will be: "To which category of mathematics does it belong?"

 

[Demystifier]

Otherwise, I am afraid that your next question will be: "To which category of mathematics does it belong?"

It would be definitely algebra.

 

[haruspex]

I'm sorry, I am not grasping your point.

Lets try this: Suppose there is a random variable X , measured in meters, that has its density defined on interval $[0, ln (2) ]$ by $f(x) = C( 2 - e^{x})$ where $C$ is the normalizing constant $int_{ 0}^{ln (2)} {(2 - e^ {x})} dx$.

A experimenter who measures $X$ in centimeters can convert the above density function to the appropriate density for$X$ when $X$ is measured in centimeters. I agree that assigning units to the left and right hand sides of $f(x) = C( 2 - e^{x})$ is a confusing or impossible task. But I don't agree that the $e^x$ in the equation implies that the equation describes a physically impossible situation or that it makes it impossible for a experimenter measuring X in different units to convert the above density to his system of measurement.

You don't need to assign units to either side of the f(x)= equation, they're dimensionless. But you can rewrite the $e^x$ as $e^{\lambda x}$ where $\lambda=1m^{-1}$.

 

[Stephen Tashi]

You don't need to assign units to either side of the f(x)= equation, they're dimensionless. But you can rewrite the $e^x$ as $e^{\lambda x}$ where $\lambda=1m^{-1}$.

In relation to issue of whether $e^x$ is an "error" in an equation describing a physical process when $x$ has a dimension:

In the first place, I don't see arguments of the form "You can rewrite ..." as having any bearing on question. Yes, an equation representing a physical process can be transformed to an equation in dimensionless form, but that doesn't show the original form of the equation is invalid.

Perhaps your complete thought is "Your original equation is wrong or meaningless and you should rewrite it as ...$.\$ In the example, I don't see that the original equation is wrong or meaningless in the sense of being uninterpretable or so ambiguous that a person doing in measurements in cm instead of meters couldn't figure out how to rewrite it as different equation $g(y)$ where $y$ has units of cm.

My equation may be wrong in the sense that the task of defining $g(y)$ can't be accomplished by the straightforward use of conversion factors. That's a topic we should investigate!

Let's pursue your suggestion of stating the equation as $p = f(x) = C( 2 - e^{\lambda x} )$ where $\lambda$ has units of $m^{-1}$ and $x$ has units of $m$. Can we convert that equation to a formula $p = g(y)$ where $y$ has units of cm by using conversion factors?

To convert to cm, we must convert both $\lambda$ and $x$ using the conversion factor (m/100 cm). We have $100 y \ (cm) = x\ (m)$ and $\lambda\ (m^{-1}) = \lambda\ (100\ cm)^{-1})$ So the equation converts to $p = g(y) = C(2 - e^{ \frac{\lambda}{100}100 y}) = C(2 - e^y)$ But the correct equation (for $y$ in cm) should be something like $p = g(y) = C(2 - e^{\frac{y}{100}})$.

I said "something like" that because we must change the value of $C$ from $\int_{0}^{ln(2)} {(2 - e^x)} dx$ to $C_2 = \int_0^{\ln(200)} { ( 2 - e^{\frac{y}{100}} ) } dy$ in order to normalize the probability distribution. We also must convert the interval on which the equation applies from $[0, ln(2)]$ to $[0, ln(200)]$.

Are we opposed to letting the function $ln(.)$ have an argument with a dimension? If so, how can we justify converting $ln(2)$ to $ln(200)$ ? A dimensionless constant like "C" or "2" can be converted to a different numerical value if it depends on several different dimensions. For example the "1" in F = (1)MA can convert to a different constant if we don't use MKS units. However, the only dimension that has been mentioned in this problem is length [L]. I don't see any way that a dimensionless constant that is define only in terms of lengths can be converted to a different numerical value by changing the unit of measure for length.

In contrast to the above difficulties if we take the viewpoint that the $x$ in $e^x$ and the "2" in $\ln(2)$ have dimension [L] length given in meters then the conversion from meters to cm gives results we need, namely $e^{\frac{y}{100}}$ and $\ln(200)$.From my point of view the probability density function $f(x)$ is not dimensionless. Like a linear density function for the density of physical mass, it represents "per unit length", so in my equation $f(x)$ has units of (1/meter). However, that consideration still leaves length as the only dimension represented in the equation.

 

[haruspex]

From my point of view the probability density function f(x) is not dimensionless. L

You are right.

 

[haruspex]

I don't see arguments of the form "You can rewrite ..." as having any bearing on question.

Then let me put it a different way. In the post in which you brought up this issue, λ was the average number of events in a specific, fixed time interval, and the algebraic expression featured e^λ. It seems to me that this way of defining λ makes it a pure number, so dimensionless, so no problem. It only becomes a problem if you then say, oh, but clearly it is really a rate, i.e. λ per that interval. But if it is to be thought of as a rate then that is how it should appear in the equation, e^λt.
Otherwise, you could apply the same thinking to e.g. KE: 1/2 ms², where s is the distance traveled per second. Dimension=ML².

 

[anorlunda]

I normally associate dimensions with degrees of freedom, as in 3D space or 4D spacetime.

A point-like particle can be described in 3D space with three coordinates. An asymmetric object needs 3 coordinates, plus 3 angular rotations to describe it's position-orientation. Aren't those rotations on an equal footing with translations as being dimensions?

p.s. I normally eschew semantic discussions, but this one caught my fancy. Nice thought provoking Insights article @haruspex

 

[haruspex]

I normally associate dimensions with degrees of freedom, as in 3D space or 4D spacetime.

A point-like particle can be described in 3D space with three coordinates. An asymmetric object needs 3 coordinates, plus 3 angular rotations to describe it's position-orientation. Aren't those rotations on an equal footing with translations as being dimensions?

p.s. I normally eschew semantic discussions, but this one caught my fancy. Nice thought provoking Insights article @haruspex

Thanks for the appreciation.
That's really a different usage of the term dimension. Dimensional analysis concerns what might be termed qualitative dimensions. All lengths are qualitatively the same, so just L. Area is different from length, but in a quantifiable way, as L², etc.
It is not just a semantic issue. The ability to represent angles as a dimension slightly increases the power of DA.

 

[robphy]

The ability to represent angles as a dimension slightly increases the power of DA.

So, it may be worthwhile exploring this, as you have done in your Insight.

However, it seems that to do so following your definitions
leads to not-so-slight modifications of how to do addition (in response to my question about 1+i in relation to your definitions).

This kind of addition can cope with adding items of different dimension. That is, to fit with the ϑ Dimension concept, I could define a complex number as an ordered pair, one of 0 dimension and one of dimension ϑ.

There may be other not-so-slight modifications.
So, maybe this isn't the way to do it [if it is at all possible to do it "slightly"].As a possible guide to a better approach,
this abstract discussion might be useful about what may be going on with regard to units (and dimensional analysis) in general:
https://golem.ph.utexas.edu/category/2006/09/dimensional_analysis_and_coord.html
In an abstract sense, it seems that our physically-dimensionful formulas
are mapping values from different spaces (somehow each associated with a "unit")
into another space of values (with a unit consistent with the algebraic operations).

 

[haruspex]

There may be other not-so-slight modifications.

I'm not suggesting any modification to the way we represent or perform complex addition. The consideration of alternative representations was to illustrate that, unlike regular addition, adding a real to an imaginary can cope with their having different dimensions.

 

[haruspex]

this abstract discussion might be useful about what may be going on with regard to units (and dimensional analysis) in general:
https://golem.ph.utexas.edu/category/2006/09/dimensional_analysis_and_coord.html

That's a fascinating thread. The comments most relevant to my article concern cycles.
I feel those parts get confused because we use the term both in a generic sense of repeating events and in the more physical sense of rotation. This is similar to the way distance was originally used in a Euclidean sense, but now is generalised to such as graphical distance, emotional distance, ... We are comfortable using the dimension L in the former but not the latter, so there is precedent for saying cycles as rotation can have dimension but not in the other uses.
Admittedly, this could lead to some tangled terminology. That could be avoided by agreeing that "cycle" always has the generic sense, and if we want to refer to a cycle in the rotational sense we should write "revolution". Thus, a rotating body rotates at one revolution per cycle, or 2π radians per cycle. Each of those would have dimension ϑ. This angular sense would also apply to phase angles in trig functions.

 

[Stephen Tashi]

We haven't managed to state precise mathematical properties for a "dimension". If we can't define what a "dimension" is, perhaps we can make definite statements about what it can't be.

For example, traditional dimensional analysis insists that the arguments to transcendental functions must be dimensionless. As a consequence, the transcendental functions themselves are dimensionless. Why is this assumed to be the case? If we let an argument to a transcendental function have a dimension, what is supposed to go wrong ?

 

[robphy]

Consider http://www.wolframalpha.com/input/?i=series(exp(x/a),x)
$\exp(\frac{x}{a})=1+\frac{x}{a}+(\frac{x}{a})^2/2+(\frac{x}{a})^3/6+(\frac{x}{a})^4/24+(\frac{x}{a})^5/120+...$
with $a$ as a dimensionless quantity but $x$ with units of length.

 

[Stephen Tashi]

Consider http://www.wolframalpha.com/input/?i=series(exp(x/a),x)
$\exp(\frac{x}{a})=1+\frac{x}{a}+(\frac{x}{a})^2/2+(\frac{x}{a})^3/6+(\frac{x}{a})^4/24+(\frac{x}{a})^5/120+...$
with $a$ as a dimensionless quantity but $x$ with units of length.

What do you want me to consider about it ?

 

[robphy]

What are the units on the right-hand side?

 

[Stephen Tashi]

What are the units on the right-hand side?

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 $y = e^x$ ?

 

[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 ?