What are units?

[sol invictus]

Why do we have separate units for time and space (seconds and meters), but not for length along the x axis and length along the y axis? Is there any maximum number of units we must define? Any minimum? Why are there seven base units in the SI system - is that merely a convention, or does it reflect something important about the universe? If we were much larger or smaller, or much faster or slower, might we define a different number?

It seems to me that we could, if we chose, define a separate unit for every distinct physical characteristic of the world. For example given a box of N distinguishable particles we could define p*N units, where p>6 would include the position and velocity components in each direction, as well as charge, mass, spin, and any other characteristics of that particle.

Each of those truly is a separate unit - in the sense that one does not add the position of one particle to that of another, or subtract the x position of a particle from its y position.

I think we do not define such units purely for reasons of convenience - it would be too hard to keep track of all those numbers. That's also why we do not define separate units of x-length and y-length: it's easy to rotate objects, so x-length and y-length are often rapidly changing. On the other hand rotating time into space by a detectable amount requires very high velocities that we cannot attain easily - so we think of time as measured in distinct units compared to space.

So as far as I can see, the choice of precisely which units we define and which we don't is essentially arbitrary, a matter of convenience and convention. Does anyone have a different view?

[DevilsAdvocate]

Originally Posted by sol invictus

It seems to me that we could, if we chose, define a separate unit for every distinct physical characteristic of the world. For example given a box of N distinguishable particles we could define p*N units, where p>6 would include the position and velocity components in each direction, as well as charge, mass, spin, and any other characteristics of that particle.

Sounds like Object Oriented Programming.

[Ririon]

If you want to measure time and space with the same units, just multiply the time by the speed of light. Fairly standard practice for people who wish to do that. That way you can make fun graphs out of special relativity problems.

[Vorticity]

Originally Posted by sol invictus

...For example given a box of N distinguishable particles we could define p*N units, where p>6 would include the position and velocity components in each direction, as well as charge, mass, spin, and any other characteristics of that particle.

Each of those truly is a separate unit - in the sense that one does not add the position of one particle to that of another, or subtract the x position of a particle from its y position.

I am certain that this bit, at least, wouldn't work. We know that particles interact via potentials. Translation invariance of the laws of physics implies that these potentials can depend only on the distances between pairs of particles. Therefore, the difference between particle 1's position vector and particle 2's position vector must be physically meaningful. Thus, there should be one unifying unit for all positions.

As for the seven base units of the SI system, I've never understood why the mole, the candela, or Kelvin are necessary.

[RdDrgn93]

Dear unconquerable sun god:
The meter is 1/10,000,000 the distance from the equator to the north pole. It was invented by the French after the revolution. They wanted a unit of measure that was not based on royalty (I.e. A yard is the length of the kings arm) and that could unify all the slightly different lengths that varied with each village across the countryside. Nothing cosmic.

[InfidelSavant]

Originally Posted by sol invictus

So as far as I can see, the choice of precisely which units we define and which we don't is essentially arbitrary, a matter of convenience and convention.

Well duh.

[Madalch]

Originally Posted by Vorticity

As for the seven base units of the SI system, I've never understood why the mole, the candela, or Kelvin are necessary.

The mole isn't- you could easily use the "unit" instead, and have the mole as 6.022 x 10^23 units. The kelvin is needed, since you do have to measure temperature somehow.

[DevilsAdvocate]

Originally Posted by RdDrgn93

Dear unconquerable sun god:
The meter is 1/10,000,000 the distance from the equator to the north pole. It was invented by the French after the revolution. They wanted a unit of measure that was not based on royalty (I.e. A yard is the length of the kings arm) and that could unify all the slightly different lengths that varied with each village across the countryside. Nothing cosmic.

I expect Sol knows the nature of the meter, both cosmic and non-cosmic, better than you or me.

I'm certainly out of my league here, but couldn't basic units be just space, time, and energy. Is there anything that does not ulimately break down to one or a combination of those units? E=Mc^2 is just energy, mass (space) and speed (time).

(I can't find the :shrug: smiley).

[shadron]

I have always thought about these 6 basic units as akin to an alphabet in a language; a convention to minimize the amount of flexibility vs usability; that leads naturally to the idea that their choice is neither necessary or sufficient to the task they are harnessed to. I cannot, however, determine what an equivalent but different set might be. It seems that your approach of defining a different unit for every variation might be possible - the ultimate in usability without any flexibility. Having more than just scalar quantities also works into this; you solution would subsume part of a vector, say, into its unit definition.

[shadron]

Originally Posted by Vorticity

I am certain that this bit, at least, wouldn't work. We know that particles interact via potentials. Translation invariance of the laws of physics implies that these potentials can depend only on the distances between pairs of particles. Therefore, the difference between particle 1's position vector and particle 2's position vector must be physically meaningful. Thus, there should be one unifying unit for all positions.

As for the seven base units of the SI system, I've never understood why the mole, the candela, or Kelvin are necessary.

Show me how you could define luminous flux density or heat transfer time without the candela or the Kelvin respectively. The mole is just a count; no units is really implied by it. Angle is self defining by a rotation. But could not electrical charge be likewise held to be a count, in this case of electrons collected together?

[I Ratant]

The metre (or meter), symbol m, is the base unit of length in the International System of Units (SI). Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level), its definition has been periodically refined to reflect growing knowledge of metrology. Since 1983, it is defined as the distance travelled by light in vacuum in 1⁄299,792,458 of a second.[1]
.
Now, THAT'S precise..

[Ririon]

Originally Posted by I Ratant

The metre (or meter), symbol m, is the base unit of length in the International System of Units (SI). Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole (at sea level), its definition has been periodically refined to reflect growing knowledge of metrology. Since 1983, it is defined as the distance travelled by light in vacuum in 1⁄299,792,458 of a second.[1]
.
Now, THAT'S precise..

Which means that c = 299 792 458 m/s. Exactly. If c was measured more accurately, the meter would change slightly, not this number.

[MattusMaximus]

We need units because pure numbers are, in and of themselves, merely abstractions with no practical connection to the universe around us.

[sol invictus]

Originally Posted by Vorticity

I am certain that this bit, at least, wouldn't work. We know that particles interact via potentials. Translation invariance of the laws of physics implies that these potentials can depend only on the distances between pairs of particles. Therefore, the difference between particle 1's position vector and particle 2's position vector must be physically meaningful. Thus, there should be one unifying unit for all positions.

By the same logic, Lorentz invariance implies that time and space must have the same units (because their squares must appear subtracted). And yet, in the SI system we regard length and time as having separate units (and c is not equal to 1), and we always get dimensionally consistent answers keeping them separate.

Based on that I think perhaps you're allowed to subtract positions of different particles only after you multiply each by an appropriate unit conversion factor (in the same way that x²-c²t² is allowed), and in SI units those conversion factors have all been set to 1. But I'm not very clear on this.

Originally Posted by Madalch

The mole isn't- you could easily use the "unit" instead, and have the mole as 6.022 x 10^23 units. The kelvin is needed, since you do have to measure temperature somehow.

Temperature is rms average kinetic energy per dof. Why does it need a unit that's distinct from that of energy?

Originally Posted by DevilsAdvocate

I'm certainly out of my league here, but couldn't basic units be just space, time, and energy. Is there anything that does not ulimately break down to one or a combination of those units? E=Mc^2 is just energy, mass (space) and speed (time).

Well, Einstein taught us that space and time are not really any more distinct than x-positon and y-position. That leaves you with space and energy. But we can convert from one to the other with appropriate powers of Planck's constant, the speed of light, etc. In fact, theoretical physicists very often work in a system of units where there are no units and a particular set of fundamental constants have been set to 1. That works just fine, you can restore them all at the end of the calculation if you like.

Originally Posted by shadron

The mole is just a count; no units is really implied by it. Angle is self defining by a rotation.

And yet, aren't radians and degrees separate "units"? You certainly don't want to add them together (absent a unit conversion factor of pi/180, that is). Same goes for dollars and euros.

[sol invictus]

Originally Posted by InfidelSavant

Well duh.

I'm glad someone finds this obvious...

Originally Posted by MattusMaximus

We need units because pure numbers are, in and of themselves, merely abstractions with no practical connection to the universe around us.

Yes, I think that's exactly it. Units are simply a convenient way to keep track of what the various numbers that appear in formulas are referring to.

Do you agree with me that we could (if we wanted) have a unit for every possible distinct quantifiable aspect of the universe, and we have reduced it to 7 for purely practical and conventional reasons?

[InfidelSavant]

Originally Posted by sol invictus

I'm glad someone finds this obvious...



Yes, I think that's exactly it. Units are simply a convenient way to keep track of what the various numbers that appear in formulas are referring to.

Do you agree with me that we could (if we wanted) have a unit for every possible distinct quantifiable aspect of the universe, and we have reduced it to 7 for purely practical and conventional reasons?

We could also have a name for every individual living organism on the planet if we wanted to. Such an exercise would be silly though.

[sol invictus]

Originally Posted by RdDrgn93

Dear unconquerable sun god:
The meter is 1/10,000,000 the distance from the equator to the north pole. It was invented by the French after the revolution. They wanted a unit of measure that was not based on royalty (I.e. A yard is the length of the kings arm) and that could unify all the slightly different lengths that varied with each village across the countryside. Nothing cosmic.

Dear mythical reptile with letters missing:
I'm not so much interested in the precise definition of the meter as I am in why we have units for length, but not for (say) color hue or width (as opposed to length), or even justice or pain.

It occurs to me that on earth, we do indeed have separate units for different spatial directions - degrees east or west versus degrees north or south, and then meters above sea level for the third direction.

[Ririon]

Originally Posted by sol invictus

Temperature is rms average kinetic energy per particle. Why does it need a unit that's distinct from that of energy?

-Nice day today!
-Yes, feels like about 4.14 zeptojoules per molecule on average.

I'll remember that the next time people talk to me about the weather.

Of course, it works fine in principle.

[InfidelSavant]

Originally Posted by sol invictus

Dear mythical reptile with letters missing:
I'm not so much interested in the precise definition of the meter as I am in why we have units for length, but not for (say) color hue or width (as opposed to length), or even justice or pain.

It occurs to me that on earth, we do indeed have separate units for different spatial directions - degrees east or west versus degrees north or south, and then meters above sea level for the third direction.

We do have units for width.

We don't have units for justice because it's an abstract concept, and that'd be like trying to measure, I don't know, how much love you're in with someone.

There are scales for pain but pain is a largely subjective phenomena and isn't entirely understood, so I don't think there are defined units for it.

There is, you may be interested to know, the Bristol Stool Scale^WP however.

[dasmiller]

Originally Posted by sol invictus

Dear mythical reptile with letters missing:
I'm not so much interested in the precise definition of the meter as I am in why we have units for length, but not for (say) color hue or width (as opposed to length), or even justice or pain.

It occurs to me that on earth, we do indeed have separate units for different spatial directions - degrees east or west versus degrees north or south, and then meters above sea level for the third direction.

I'm an engineer; for me, units are a pragmatic thing. If it makes it easier to describe something or analyze something by defining new units, I'm all for it, even if the new units reflect an unnatural combination of other more fundamental units (yes, I've had requests for moment in "snail-inches." I think a "snail" was defined as "1/12 of a slug," and, of course, a "slug" is the amount of mass that one lbf will accelerate by 1 ft/s^2). So my contribution to units philosopy is going to be pretty limited.

I'd argue that the various color scales (RGB, HSV, probably others) actually are unitizations of color.

I must say that I'm intrigued by this idea of units for justice. As far as pain, it's not a topic I wish to dwell upon, but there are a number of pain scales, but I'm not sure that the pain scales really describe pain in the form of a real unit one could, for example, meaningfully multiply or take a logarithm of.

[sol invictus]

Originally Posted by Ririon

-Nice day today!
-Yes, feels like about 4.14 zeptojoules per molecule on average.

I'll remember that the next time people talk to me about the weather.

Of course, it works fine in principle.

What do you have against zeptojoules per molecule?? It can't be any weirder than Fahrenheit. (And by the way 4 zeptojoules/molecule is not a nice day - don't forget the 5/2!)

Originally Posted by InfidelSavant

We do have units for width.

We do? That are different from length? What are they?

Quote:

There is, you may be interested to know, the Bristol Stool Scale^WP however.

Beautiful.

[Ririon]

Originally Posted by sol invictus

What do you have against zeptojoules per molecule?? It can't be any weirder than Fahrenheit. (And by the way 4 zeptojoules/molecule is not a nice day - don't forget the 5/2!)

Hey, whatever juices your piglet! Yes, I forgot the 5/2 (1/2 per degree of freedom according to the infallible Wikipedia?). But does this mean that two things at the same temperature would possibly have different energy-temperature-things-in-units-of-energy? Then it WOULD BE weirder than Fahrenheit.

But you could still have the Kelvin defined directly in terms of energy instead of being one of the base units. Then the Boltzmann constant would be a defined number just like the speed of light. I wouldn't mind that so much, I think.

[InfidelSavant]

Originally Posted by sol invictus

What do you have against zeptojoules per molecule?? It can't be any weirder than Fahrenheit. (And by the way 4 zeptojoules/molecule is not a nice day - don't forget the 5/2!)



We do? That are different from length? What are they?



Beautiful.

Why do the units for width have to be different for the units for length, especially since width and length are arbitary names for the dimensions of a shape determined primarily by the observers perspective anyway. One person's length is anothers width.

[Thabiguy]

Originally Posted by sol invictus

So as far as I can see, the choice of precisely which units we define and which we don't is essentially arbitrary, a matter of convenience and convention. Does anyone have a different view?

I don't disagree, but I think there may be slightly more to it than just that.

Originally Posted by sol invictus

It seems to me that we could, if we chose, define a separate unit for every distinct physical characteristic of the world. For example given a box of N distinguishable particles we could define p*N units, where p>6 would include the position and velocity components in each direction, as well as charge, mass, spin, and any other characteristics of that particle.

I guess we could do that, but if we actually did that, we would find that we can't really say much about a particle without specifying which one we're talking about. We could talk about distance between these two particles or between those two particles, but we would have trouble talking about distance between a general pair of particles. Perhaps we could put together some equation with an "unknown" conversion factor, but it would be worse than that, because expressions in that equation would be of unknown dimensions - and if we wanted to go around that and say that units of positions of two different particles are in fact alike and in a way interchangeable (with merely a numeric conversion factor), we would be undermining our initial intention of having separate units.

I think the problem is that such particle-specific units would be over-specified: giving measurement in a particular unit would not only convey information about the measurement itself, but also imply what particle we're talking about.

Similarly:

Originally Posted by sol invictus

It occurs to me that on earth, we do indeed have separate units for different spatial directions - degrees east or west versus degrees north or south, and then meters above sea level for the third direction.

If we wanted to specify distance between two objects on Earth's surface via longitude and latitude, we would necessarily be giving more information than just distance. In this case, we would also be giving some extra information about orientation of the two objects with respect to the Earth (and we would probably also have to specify at least one object's absolute latitude, because without that we couldn't really determine the distance).

So it seems to me that this is about more than just convenience - it's also about information. We use the kinds of units that we use because we recognize similarities in the world and we want to separate and abstract these common properties from other information, so that we can talk about stuff in general, without being more specific than necessary.

[Vorpal]

Originally Posted by sol invictus

Yes, I think that's exactly it. Units are simply a convenient way to keep track of what the various numbers that appear in formulas are referring to.

I think that statement is right but a bit ambiguous on just what is being referred to. A measurement of (say) 18in using a yard-stick gives a ratio of some aspect of thing being measured to that stick--a comparison that is actually unitless. Pondering questions like how a hypothetical change in the speed of light could be measured makes it fairly clear that the only way to do so would be to compare different physical (sub)systems in a way that makes a unitless ratio.

So let's go just a bit ahead on the above statement and treat that ratio as fundamental by saying that units "really are" signifiers representing some physical phenomenon that we happen to agree to compare other things to. Whether that phenomenon is some monarch's arms or the transitions of a certain kind of atom.

In that sense, we wouldn't even have to have waited for Einstein to show us that time and space are "the same" on a some very deep level in order to eliminate units of space or time. We could've just picked something (e.g., the orbit of the Earth around the Sun) as the kind of phenomenon to compare it to. Scientists could have used it to speak about distance in units of years with equal sensibility in the 19th century if they chose to do so (though perhaps not equal convenience).

[Uncayimmy]

I had to go look: This is the 9th thread you've started. Slow down a bit, eh?

Honestly, I don't know if this question is over my head or rather simple. It could be both. One of the reasons I don't really like the metric system (besides not being a scientist) is that I find the imperial measurements a bit more convenient in that they are well suited to the individual tasks for which I use them. It's only when you start relating things to each other that the true value of the metric system is seen.

While I'm no expert in physics, it seems like the same thing there. I would guess that each unit came into being as being suitable for the task at hand. It's actually a lot like programming. I've worked on code bases that were over 20 years old. Invariably you encounter things that made good sense at the time, but if you had it to do over again, you'd do it differently.

I suppose it's the same in physics. Knowing what we (well, you and others) know today, you'd probably build a completely different system from scratch that would kick the ass of what we have today. In the process, those who stay in limited areas might be a little annoyed just as I get annoyed that I can feel a temperature difference in my house of one degree Fahrenheit, so why would I want to use centigrade and set my thermostat to a half degree? I like that 0F is pretty damned cold and 100F is pretty damned hot whereas -18 and 38 are just, well, ugly numbers to me.

[DevilsAdvocate]

Originally Posted by sol invictus

Dear mythical reptile with letters missing:
I'm not so much interested in the precise definition of the meter as I am in why we have units for length, but not for (say) color hue or width (as opposed to length), or even justice or pain.

It occurs to me that on earth, we do indeed have separate units for different spatial directions - degrees east or west versus degrees north or south, and then meters above sea level for the third direction.

Well, there you go. Like I said, this is like Object Oriented Programming. Computer code stared with a whole bunch of defined things: commands, variables, types, values, events, etc. A “thing” like a “variable” would be like a certain type of “unit”. Programming languages became more refined and in OOP, every “thing” is an “object”. Every “object” has “properties” (variable values that define the object” and “methods” (things that the “object” can do, like commands and events).

Objects can be grouped and classed. A “class” defines an “object”. A class can be built on another class, or borrow parts of another class.

The fundamental units we know are simply the ones that have historically been the most convenient. If you want to restructure units according to a logical structure, look to OOP and build classes with properties (values) and methods (actions) that define objects. The basic “units” would be the properties of the most fundamental classes (and you should be more interested in the classes rather than the values (units)).

It works for computer programming. I think it will probably someday become the standard for other sciences. It is a logical way to structure information and can deal with any information in any capacity. In OOP, something like a “meter” would always relate back to a more primary class. So it would be something like All.SpaceTime.Space.Length.Meter.

The “units” used would not be (and probably currently are not) nearly as important as the structure. Someone could pull out certain units (properties) as the most commonly used, but within the structure they are not actually any more or less important than any other. They are just more common. In OOP, everything in an object, so everything is the same.

[shadron]

Originally Posted by sol invictus

Originally Posted by shadron

The mole is just a count; no units is really implied by it. Angle is self defining by a rotation.

And yet, aren't radians and degrees separate "units"? You certainly don't want to add them together (absent a unit conversion factor of pi/180, that is). Same goes for dollars and euros.

Sure 'nuff. I'm not saying that units are unimportant, but that they are basically just counts. But then, in retrospect, the same is true about all basic units; they can all be defined as counts of basic constants, such as Planck's lengths and intervals. So I suppose I was making a difference without a distiinction.

[Lukraak_Sisser]

If I recall my high school physiscs, the seven base units of the SI are the units that are the minimum necessary to define every other unit.
While its possible to give X and Y axis of length different definitions, they'd both still define length.
The mole is utterly necessary in chemistry. Without that type of definition every chemical reaction would have to be empircally determined as every molecule has a different weight. Its defined in such a way as to make caluclations easy, but without any form of definition we'd be back in alchemy times.

[Ziggurat]

Originally Posted by shadron

Show me how you could define luminous flux density or heat transfer time without the candela or the Kelvin respectively.

I'm not too familiar with luminosity, but temperature is easy. In fact, if you start from stat mech, then energy is most naturally expressed in units of energy. You can convert back and forth with Boltzmann's constant k_B, but as with time and c defining length, you could take k_B and energy as defining temperature.

Quote:

But could not electrical charge be likewise held to be a count, in this case of electrons collected together?

It appears so, yes.

[Ziggurat]

Originally Posted by Lukraak_Sisser

If I recall my high school physiscs, the seven base units of the SI are the units that are the minimum necessary to define every other unit.

That's if you don't want to make explicit reference to any physical constants within your definition. If you do, you can get away with fewer (ie, every reference to length can be substituted with a reference to time and c).

[Dilb]

Originally Posted by Ziggurat

I'm not too familiar with luminosity, but temperature is easy.

Luminosity is just radiant intensity multiplied by a weighting function. There's no real reason for it to have it's own unit. It's like how internal energy and enthalpy are both measured in Joules, despite them being different quantities. A light source could be described with a luminous intensity and an radiant intensity, both measured in W/steradian.

I imagine that getting interior designers to use W/steradian rather than candela is going to be an uphill battle, though.

[dlorde]

Originally Posted by sol invictus

I'm not so much interested in the precise definition of the meter as I am in why we have units for length, but not for (say) color hue or width (as opposed to length), or even justice or pain.

We have units for things we want to measure, quantify, convert, and compare. We may not have universal units for pain (although there are pain scales which aren't far off that), but we do, for example, have units for chilli heat (the Scoville Heat Unit), which are based on subjective criteria. This suggests it is a matter of convenience and utility.

[hodgy]

[quote=sol invictus;6577119]but not for length along the x axis and length along the y axis?QUOTE]

Length isn't defined along a specific axis but across both x and y simultaneously.

[sol invictus]

Originally Posted by InfidelSavant

Why do the units for width have to be different for the units for length, especially since width and length are arbitary names for the dimensions of a shape determined primarily by the observers perspective anyway. One person's length is anothers width.

Why are the units for time different from the units for space? (If you don't understand why I'm asking that in response to you, read up on special relativity.)

Originally Posted by Thabiguy

I guess we could do that, but if we actually did that, we would find that we can't really say much about a particle without specifying which one we're talking about. We could talk about distance between these two particles or between those two particles, but we would have trouble talking about distance between a general pair of particles. Perhaps we could put together some equation with an "unknown" conversion factor, but it would be worse than that, because expressions in that equation would be of unknown dimensions - and if we wanted to go around that and say that units of positions of two different particles are in fact alike and in a way interchangeable (with merely a numeric conversion factor), we would be undermining our initial intention of having separate units.

But we have the same situation with time and length - there are plenty of equations involving x^μ (where one component is time and the others space). The units get taken care of by the metric. I think I could write equations governing interactions between particles using a similar kind of "metric". I do agree that it would be very cumbersome.

Quote:

So it seems to me that this is about more than just convenience - it's also about information. We use the kinds of units that we use because we recognize similarities in the world and we want to separate and abstract these common properties from other information, so that we can talk about stuff in general, without being more specific than necessary.

You're saying the reason we've chosen the units we have (say in the SI system) is because the quantities they measure are in some sense more useful or more generic than most. Probably so - but what I'm trying to understand is precisely in what sense that's the case.

[sol invictus]

Originally Posted by Lukraak_Sisser

If I recall my high school physiscs, the seven base units of the SI are the units that are the minimum necessary to define every other unit.

That's what people often say, but it's plainly not true. For example, you'll have difficulty measuring dollars or euros in SI, yet prices are perfectly valid data about the world. Or for a more physical example, there's no SI unit for hue.

You might object that hue is the result of the wavelength of light, which can be measured with meters. But temperature is the result of kinetic energy, so why are there SI units for both temperature and energy?

So I think it's clear that there is nothing fundamental about those seven base units.

Originally Posted by Vorpal

I think that statement is right but a bit ambiguous on just what is being referred to. A measurement of (say) 18in using a yard-stick gives a ratio of some aspect of thing being measured to that stick--a comparison that is actually unitless. Pondering questions like how a hypothetical change in the speed of light could be measured makes it fairly clear that the only way to do so would be to compare different physical (sub)systems in a way that makes a unitless ratio.

Yes, those silly "changing speed of light" theories forced me to think about this a few years ago, and I came to that conclusion as well.

Quote:

So let's go just a bit ahead on the above statement and treat that ratio as fundamental by saying that units "really are" signifiers representing some physical phenomenon that we happen to agree to compare other things to. Whether that phenomenon is some monarch's arms or the transitions of a certain kind of atom.

In that sense, we wouldn't even have to have waited for Einstein to show us that time and space are "the same" on a some very deep level in order to eliminate units of space or time. We could've just picked something (e.g., the orbit of the Earth around the Sun) as the kind of phenomenon to compare it to. Scientists could have used it to speak about distance in units of years with equal sensibility in the 19th century if they chose to do so (though perhaps not equal convenience).

Somehow this must boil down to physics. It seems clear that the reason we do not ordinarily use different units for length and width is the rotation invariance of the laws of physics. But in situations where rotation invariance is broken, like the surface of the earth, we do use separate units (latitude and longitude). Similarly with time versus space - it's generally convenient to set c=1 only when describing relativistic phenomena.

So is the number of useful units related in some way to the number of approximate symmetries? If you had a bunch of distinct electric charges, each separately conserved, you'd want a unit for each. But we only have one unit for momentum even though it has three conserved components, presumably because rotations mix them. For locations on earth, where rotation invariance is broken, we're back to having three.

[Paul C. Anagnostopoulos]

Originally Posted by Madalch

The mole isn't- you could easily use the "unit" instead, and have the mole as 6.022 x 10^23 units.

But then Avogadro's constant would have the unit 1/unit, which is confusing. Abbott and Costello could do a bit about it.

~~ Paul

[rwguinn]

Originally Posted by sol invictus

Why do we have separate units for time and space (seconds and meters), but not for length along the x axis and length along the y axis? Is there any maximum number of units we must define? Any minimum? Why are there seven base units in the SI system - is that merely a convention, or does it reflect something important about the universe? If we were much larger or smaller, or much faster or slower, might we define a different number?

It seems to me that we could, if we chose, define a separate unit for every distinct physical characteristic of the world. For example given a box of N distinguishable particles we could define p*N units, where p>6 would include the position and velocity components in each direction, as well as charge, mass, spin, and any other characteristics of that particle.

Each of those truly is a separate unit - in the sense that one does not add the position of one particle to that of another, or subtract the x position of a particle from its y position.

I think we do not define such units purely for reasons of convenience - it would be too hard to keep track of all those numbers. That's also why we do not define separate units of x-length and y-length: it's easy to rotate objects, so x-length and y-length are often rapidly changing. On the other hand rotating time into space by a detectable amount requires very high velocities that we cannot attain easily - so we think of time as measured in distinct units compared to space.

So as far as I can see, the choice of precisely which units we define and which we don't is essentially arbitrary, a matter of convenience and convention. Does anyone have a different view?

That's why we invented Vectors!

[Thabiguy]

Originally Posted by sol invictus

You're saying the reason we've chosen the units we have (say in the SI system) is because the quantities they measure are in some sense more useful or more generic than most. Probably so - but what I'm trying to understand is precisely in what sense that's the case.

I think I understand the question here, but I don't think I have a fully satisfying answer. (It would be more surprising if I had it, since I haven't really thought about this stuff very deeply before you asked the question.) I can offer some quick thoughts, which might be helpful, or might not.

I think we can all agree that some conventions in physics are completely or almost completely arbitrary, like which charge is considered positive and negative, or the exact units for, say, length. (Although the scale of the exact units can be expected to be on a similar order as the phenomena we most commonly observe.)

The choice of which units are basic and which are derived is probably still mostly arbitrary, but perhaps somewhat less. If, as you suggest, we were very different in size, or for example sensitive to magnetic fields, we might have different opinions on which quantities are more basic than others. But we're a product of nature, and it would be difficult to imagine circumstances under which we wouldn't include distance and/or time among the basic quantities.

Putting aside the "basicness" of units and quantities, the set we use may still be somewhat arbitrary, but perhaps least so. I think the quantities (and corresponding units) we use follow more or less directly from what appears in the laws of nature as we understand them. (I mean laws of nature in the most general sense.) Now, there is probably still some effect of human perspective and interpretation in the formulation of those laws, but I think we'll agree that they're not entirely a product of human perspective and that they're also based on some fundamental properties of the universe.

But on the other hand, the exact number and character of our units probably doesn't entirely follow from some immutable characteristics of our universe. It's shaped by our knowledge which is incomplete; we can expect that there are laws of nature we don't know about yet, and when we learn about them, we may yet introduce new quantities and units or rearrange those we have. - So I suspect both fundamental and arbitrary factors are at play, in a mixture that may not be entirely clear.

[quarky]

You may need to factor in the arbitrariness.

I propose a new unit, the 'arbitrarium'.

I use to be a contractor. My partners and I came to the odd conclusion that we needed a new billing factor: The angst factor. Why would work be billed the same when one job was mostly angst-free, and another was 'ate-up' with angst?

The difficulty arises in formulating a mutually agreeable relative value (or anti-value) to the angst.
I thought it was quite daft that a new roof would cost the same, per surface unit, whether it was a first floor, or 100 feet up in the air. Justice, in a unit, would be easy to describe via wt/height. Some other qualities are more subtle:

Client is a jerk? Multiply the bid by the jerk factor.
Client is a beautiful single woman, and she brings lemonade to the crew? Factor in the anti-jerk unit.