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Does Temperature have a Metric Structure?∗
Bradford Skow
Abstract Is there anything more to temperature than the ordering of things from colder to hotter? Are there also facts, for example, about how much hotter (twice as hot, three times as hot...) one thing is than another? There certainly are—but the only strong justification for this claim comes from statistical mechanics. What we knew about temperature before the advent of statistical mechanics (what we knew about it from thermodynamics) provided only weak reasons to believe it.
- The tea in the mug on my desk is hotter than the air in my office. And the surface of the sun is hotter than the surface of the earth. Is there anything more to temperature than these kinds of facts—facts about how things are ordered with respect to temperature? Is it also true, for example, that the temperature difference between the surface of the sun and the surface of the earth is some definite number of times larger than the temperature difference between the tea and the air? Are there, in general, also facts about the ratios of lengths of temperature intervals? Or, instead, is it meaningless to ask what these ratios are? If it does make sense to ask then (I will say) temperature has a metric struc- ture. It is easy to see that temperature has a metric structure when one looks at what statistical mechanics says temperature is. But do we need to appeal to statistical mechanics to justify the claim that temperature has a metric structure? Statistical mechanics, I will argue, provides the only strong justification for this claim. What ∗Published in Philosophy of Science 78 (2011): 472-489.
we knew about temperature before the advent of statistical mechanics (especially what we knew about it from thermodynamics) provided at best only weak reasons to think that temperature has a metric structure.
- The thesis I am defending might seem outrageous at first. One might be tempted to argue against it like this: the temperature of the surface of the sun is close to 5700 Kelvin and the temperature of the surface of the earth averages something like 300 Kelvin. That is a separation of 5400 Kelvin. But the temperatures of the tea and the air are separated by less than 100 Kelvin. So the temperatures of the sun and the earth are separated by an interval at least 34 times larger than the one that separates the temperatures of the tea and the air. Someone who knows nothing about statistical mechanics can arrive at this conclusion. All he needs is a good way to measure temperature on the Kelvin scale (a scale that was established before statistical mechanics was developed). But the fact that the difference between the numbers the Kelvin scale assigns to the temperatures of the earth and the sun is 34 times larger than the difference between the numbers the Kelvin scale assigns to the tea and the air does not entail that the first temperature difference itself is 34 times greater than the second, or more generally that temperature differences stand in definite ratios. We have to be careful when drawing conclusions about temperatures themselves from features of the scale(s) we use to measure temperature. When we use real numbers to represent the values of a physical quantity like temperature we are using relations between the numbers to represent relations be- tween the values of the quantity themselves. (Quantities like temperature are deter- minables; by the value of a quantity I mean a property that is a determinate of that determinable. One and the same value for temperature may be assigned 0 on the Celsius scale, 32 on the Fahrenheit scale, and 273.15 on the Kelvin scale.) But not every relation among the numbers represents a relation among the values. Take, for example, the use of non-negative real numbers to represent masses on the kilogram scale. The number 1 is mathematically special: it is the only number y satisfying yx = x for all x. But there is nothing physically special about the mass value as- signed the number 1 on the kilogram scale. So it is possible (for all that has been
case there is a natural zero point for temperature. But the differences between these two structures will not play a role in what follows, so I lump them together as kinds of metric structure.
- One might maintain that there neither are nor could be any physical quanti- ties with metric structures. I shall call this thesis “conventionalism,” for the fol- lowing reason.^3 Whenever we set up a scale for measuring some quantity we must establish some conventions. Conventionalists think that setting up a scale requires more conventions than non-conventionalists do. Take, for example, length. Non-conventionalists think that there are scale-independent facts about the ratios of lengths. They take length to be a paradigm case of a quantity with a metric struc- ture. If this is right then we only need to make one conventional choice in order to set up a scale for measuring length. For a faithful scale for measuring length will have the property that the ratios of the numbers assigned to length values are equal to the scale-independent ratios of the corresponding values. So the only freedom we have when setting up a scale (if the non-conventionalist is right) is freedom to choose which length shall be assigned the number 1. Once we pick which length value S this is (say by legislating that S is the length value instantiated by a certain bar made of platinum-iridium alloy under certain conditions), anything else’s length will be l/S , the (scale-independent) ratio of that thing’s value l for length to S.^4 Conventionalists do not think that this procedure succeeds in setting up a scale for measuring length. For they deny that there are scale-independent facts about length ratios. So an utterance of “Let our scale assign to each length value (^3) I also call it “conventionalism” because it resembles claims that conventional- ists like Henri Poincaré made. I am not sure whether Poincaré actually accepted this thesis (see for example Poincare 1905, 17-28). Reichenbach (1957, 14-30) also claims to find more conventions than we might originally have thought existed in our measurement practices. (^4) I suspect that this non-conventionalist view of length is also the majority view. It is certainly the view of the International Bureau of Weights and Measures. Its brochure on the international system of units says “The value of a quantity is gener- ally expressed as the product of a number and a unit. The unit is simply a particular example of the quantity concerned which is used as a reference, and the number is the ratio of the value of the quantity to the unit” (2006, 103).
l the number that is the (scale-independent) ratio of l to S ” has a false presupposi- tion. Conventionalists think that we need to establish more conventions, besides a conventional choice of standard, in order to assign a unique number to each length value. Here is the sort of thing a conventionalist might do to set up a scale for mea- suring length. He starts, as the non-conventionalist does, but selecting a particular thing—a particular metal bar in this case—to be the scale’s standard for length. Then he stipulates that a length l shall be assigned a positive integer n iff n copies of the standard, laid end to end, reach exactly from one end to another of anything that instantiates l. (There is a natural way to complete this scale so that each length value is assigned some positive real number.) This last stipulation is an additional convention. Now, the conventionalist and the non-conventionalist who use the scales de- scribed above will agree on the lengths of things—as long as there is nothing funny going on. So it is worth emphasizing the differences between their views. The conventionalist will insist that even after a standard has been adopted he was free to adopt an alternative convention for assigning numbers to lengths and still have a scale that faithfully represents the structure of length. The non-conventionalist will deny this; those alternative conventions result in scales that are not faithful. The non-conventionalist will also say that the convention the conventionalist actu- ally adopted makes what is as a matter of contingent fact a reliable procedure for finding out the lengths of things into a procedure that is reliable by definition. (The conventionalist has made it true by definition that metal bars do not change their length when they are moved around.) Of course one might deny that length has a metric structure without being a conventionalist. Conventionalists believe that no possible quantity has a metric structure. If conventionalism is true then it follows, of course, that temperature in particular does not have a metric structure. So the debate over conventionalism is certainly relevant to the topic of this paper. But arguments for conventionalism are general, all-purpose arguments that apply to any possible quantity. And I am inter- ested in what we should think about the structure of temperature on the assumption that it is at least possible for a quantity to have a metric structure. So I am going
the mercury extends when the bulb is placed in thermal contact with melting ice and allowed to settle into thermal equilibrium. This is the 0 mark. The idea is that a given thermodynamic system’s temperature on this scale will be proportional to the ratio of two distances. Let p be the point on the cylinder coincident with the top of the mercury column when the bulb is placed in thermal contact with the system. We decree that that the number measuring the system’s temperature on this scale is equal to 100d d^12 , where d1 is the distance between the 0 mark and p and d2 is the distance between the 0 mark and the 100 mark. (That is its temperature if p is above the 0 mark; its temperature is − 100 d d^12 if p is below the 0 mark.)^5 We can see that the presuppositions about temperature made here are dif- ferent from the presuppositions made about length (by non-conventionalists). The numbers assigned to lengths on the meter scale are defined in part in terms of the (scale-independent) ratios of length values. But no ratios of temperature values, or ratios of lengths of intervals of temperature values, are used to define the numbers assigned to temperatures on this scale. We might, of course, believe that temperature has a metric structure, and wonder whether the mercury-centigrade scale faithfully represents that structure. (We might wonder, that is, whether the result of applying the function |x−y|/|z−w| to numbers this scale assigns to temperatures is always equal to the scale-independent ratio of the lengths of the corresponding temperature intervals). Many scientists in the late 18th and early 19th century did believe and wonder just these things. (Some of them wondered this because they accepted the caloric theory of heat, which (we will see below) says that temperature has a metric structure). For example, Johann Lambert wrote that “One hesitates as to whether the actual degrees of heat are really proportional to the degrees of the expansion” of the substance in the thermometer. Similarly, Joseph Gay-Lussac wrote that “The thermometer as it is today cannot (^5) The mercury-based centigrade scale is not the same as (what we currently call) the Celsius scale. Both assign 0 to water’s freezing point and 100 to its boiling point (under standard conditions), but they differ over how intermediate numbers are assigned. The first scale assigns intermediate numbers so that (by definition) mercury expands linearly with temperature. Mercury does not expand linearly with temperature on the Celsius scale. By definition, something’s temperature in Celsius is just its temperature in Kelvin, minus 273.15.
serve to indicate the exact proportion of heat, as we do not yet know what relation there is between the degrees of the thermometer and the quantity of heat which it is supposed to indicate. We generally believe, it is true, that equal divisions of the tube represent equal quantities of the caloric, but this opinion is not founded on anything positive.”^6 Still, even if scientists frequently wondered whether temperature scales like the mercury-based centigrade scale correctly represent the metric structure of temperature, it is nevertheless true that the presuppositions made when setting up these scales leave it open whether there is a question to be answered here at all. As a matter of historical fact, different centigrade scales were set up using different substances (mercury, alcohol, ...). And the result of applying the function |x − y|/|z − w| to the numbers the mercury scale assigned to some temperatures was not the same as the result of applying it to the numbers the alcohol scale assigned to the very same temperatures. So if temperature does have a metric structure at most one of these scales faithfully represents that structure. But without further theoretical input there is no way to decide which, if either, is faithful.
- So far I have discussed the presuppositions made when setting up “empiri- cal” temperature scales, scales that correlate temperature changes with properties (like volume changes) of particular substances (mercury, alcohol, air,...). Famously, William Thomson (later Lord Kelvin) proposed a scale for measuring temperature that did not make reference to the properties of any particular substance. (That is what is meant by the claim that his scale is an absolute temperature scale.) So what about Kelvin’s scale? Does that scale presuppose that temperature had a metric structure? Here is a brief description of the way the Kelvin scale is established. It is based on a theorem in thermodynamics. Consider heat engines that work in a cycle between two heat reservoirs at different temperatures: they absorb some amount (^6) Both of these quotations appear in Ernst Mach’s “Critique of the Concept of Temperature,” originally published 1896. An English translation appears as Ap- pendix I of (Ellis 1966). The quotations appear on page 190 of that version. Chang (2004, 60-62) has more on early scientists’ worries about the adequacy of their scales for measuring temperature.
the Kelvin scale’s special properties do provide reason to think that temperature has a metric structure. I will return to this thought below. Before proceeding with my argument I want to say something about why scientists who believed that temperature had a metric structure set up scales for measuring temperature that did not presuppose this. A temperature scale is just an assignment of numbers to temperatures, and a faithful scale is one that faithfully represents the structure of temperature. There is nothing in the definition of “scale for measuring a quantity” that entails that it is within our power to use that scale to measure the value anything has for that quantity. It is possible, therefore, to set up a scale for measuring temperature that is completely impractical: a scale which is such that it is impossible for anyone to learn what temperature anything has on that scale. It is not hard to see how to set up an impractical scale for measuring some quantity. We could set up an impractical length scale by choosing as our standard for length something to which we have no access—say, the height of the first human to set foot on Mars. Supposing humans do colonize Mars (and there is a single first human who gets there), each thing is assigned a length on this scale. But none of us can know how tall he is on this scale. No temperature scale presupposes that temperature has a metric structure sim- ply because (in the absence of theoretical advances) we could never determine what number any temperature value is assigned on such a scale. Even if we firmly be- lieved that temperature has a metric structure, it is completely impractical to set up a temperature scale that by saying that the number that measures something’s temperature on that scale is equal to 273. (^16) Wv , where v it that thing’s value for tem- perature and W is water’s value for temperature at its triple point. The problem is that temperature is not a quantity we can directly measure. The scales we do use do not suffer from this problem. On them we can measure temperature indirectly, by finding the value for some other quantity (the volume of a gas for example) that we can measure directly.
- It is time to turn to the main part of my argument. If there were reasons to think (before the advent of statistical mechanics) that temperature has a metric structure,
those reasons are to be found in the theories that existed prior to the development of statistical mechanics in which temperature plays a role. What do those theories tell us? After briefly discussing the caloric theory of heat I will turn to thermodynamics. Before we can know what a theory says about the structure of temperature we need to know where in a theory to look for this kind of information. And there are several ways in which a theory can indicate something about the structure of temperature. One way a theory can do this is by explicitly saying what temperature is. The caloric theory is an example of an early (and false) theory that does this. Actually, there were several different caloric theories; but for our purposes it is enough to work with a simplified version of one of them.^7 Now if one body is hotter than another then “heat will flow” from the first body to the second when they are put into thermal contact. The caloric theory explains this phenomenon by postulating the existence of a fluid—caloric—and identifying the temperature of a body with the density of the caloric fluid in that body. (Higher temperatures just are higher densities of caloric.) Caloric flows from regions of high caloric density to regions of low caloric density; so this theory has temperature behaving the way it should. If the caloric theory is true then temperature has a metric structure. For the density of caloric in a body is just the amount of caloric divided by the volume of the body, and both volume and (presumably) amounts of caloric are quantities with metric structures. But the caloric theory is false. So it is no guide to the structure of tempera- ture. The theory we should be looking at, of course, is thermodynamics. Like the caloric theory, thermodynamics is false. But statistical mechanics assigns the false statements of thermodynamics extremely high probabilities. So it is close enough to being true to take what the theory says about temperature as good evidence about the structure of temperature. But (unlike caloric theories) thermodynamics does not explicitly tell us what (^7) See (Chang 2004, 170-2) for brief descriptions of the dominant caloric theories. The simplified theory I present is closest to the Irvingist’s theory; the main differ- ence is that the Irvingists define a body’s temperature as the total caloric contained in the body divided by the body’s heat capacity, not by its volume.
have the same temperature. But two bodies have the same temperature if and only if they are in thermal equilibrium. These two claims are consistent only if thermal equilibrium is an equivalence relation. And that is what the zeroth law says. But the zeroth law does not say anything more about the structure of temperature. The third law of thermodynamics says that no thermodynamic system (or part of a thermodynamic system) may be cooled to absolute zero (in a finite number of operations). The third law, therefore, says that the (physically realizable) tempera- tures are bounded from below. This law places restrictions on the order type of the temperature values. But it does not presuppose that the temperature properties have a metric structure. So what about the second law of thermodynamics? It is more complicated to say what the second law of thermodynamics presupposes about the structure of temperature, because there are many statements of the second law. I will start by looking at two of the most famous statements of the second law: Kelvin’s statement and Clausius’s statement. Their statements may be paraphrased as follows:^8
(K) No cyclical process is possible whose sole results are the transfer of energy by heating out of some heat reservoir and the performance of work.
(C) No cyclical process is possible whose sole result is the transfer of energy by heating out of a colder body and the transfer of energy by heating into a hotter body.
It is clear that neither of these statements presuppose that temperature has a metric structure. Kelvin’s statement does not mention temperature differences at all. Clau- sius’s statement does, but it presupposes only that bodies are ordered from lower to higher temperatures. The magnitudes of temperature differences play no role in it.
- The most well-known statement of the second law of thermodynamics uses the concept of entropy:
(E) For any process undergone by an adiabatically isolated system that begins (^8) Uffink (2001) has a comprehensive discussion of the many versions of the sec- ond law.
and ends in an equilibrium state, the entropy of the final state is not less than the entropy of the initial state.
(A system is adiabatically isolated if energy cannot enter or leave it through heat- ing; energy is permitted to enter or leave only through the performance of work.) The word “temperature” does not even appear in (E) so it is not immediately ob- vious what, if anything, (E) presupposes about the structure of temperature. But (E) does employ the concept of entropy, which (in classical thermodynamics) is defined in terms of temperature, so (E) may indirectly presuppose something about the structure of temperature. However, “presuppose” is not really the right word here. If (E) makes it reasonable to believe that temperature has a metric structure it is not because (E) presupposes that temperature has that structure. It is for a different reason. And, in fact, figuring out what (E) indicates about the structure of temperature is not at all easy. Let me explain why. In standard presentations of classical thermodynamics (E) is not the funda- mental statement of the second law. Instead, some other statement of the second law that does not presuppose that temperature has a metric structure (like the Kelvin or Clausius statements, or some generalization of them, like the one Serrin [1979] uses) is used to prove that there is such a state function as entropy. Then it is proved further that (E) is equivalent to those statements.^9 Kelvin’s temperature scale plays an important role in this proof. I want to look at some of the details of this proof. But we are already in a position to say a few things about what (E) might indicate about the structure of temperature. There is some reason to think that (E) does not indicate that tempera- ture has a metric structure. For (E) is derived from other statements of the second law that do not presuppose that temperature has a metric structure. How could (E) say, or presuppose, more about the structure of temperature than the principles from which it is derived? If it did it would seem like something had come from nothing. On the other hand, there is also reason to think that (E) does indicate that (^9) Technically, (C) and (K) are not equivalent, so are not both equivalent to (E). But it took the recognition of the existence of negative absolute temperatures to see this (see Uffink 2001, 325-30).
where Qrc and Qrh are the energies absorbed and emitted by a reversible engine. That is, Qrc Qrh^ ≤^
Qc Qh^.^ (1) If we use Kelvin’s scale to measure temperature, then by definition
Qrc Qrh^ =^
Tc Th^.^ (2)
Equations (1) and (2) entail Qh Th^ ≤^
Qc Tc^ , or, replacing Qc with −Qc so that Qc denotes the amount of energy that the engine absorbs from the cold reservoir,
∑ (^) Q T ≤^0.
The next step (the details of which I omit) is to argue that, in general, for any cyclic process (not just for heat engines), ∮ (^) dQ T
Now for any reversible cyclic process, both
dQ/T ≤ 0 and
dQ/T ≥ 0. This (together with the claim that any two equilibrium states can be connected by a reversible process) implies that
dQ/T = 0, and this implies that
∫ (^) s 2 s 1 dQ/T^ is independent of path—entropy is a state function. What is important about this proof for our purposes is the following. Only if T stands for the numbers assigned to temperatures on the Kelvin scale or a scale similar to the Kelvin scale (a scale obtained from it by multiplying all numbers it assigns to temperature values by some constant) is
∫ (^) s 2 s 1 dQ/T^ independent of path. Now I earlier presented an argument that this proof provides a reason to be- lieve that temperature has a metric scale. That main premise at work in that argu- ment is the following conditional:
(*) If scales of temperature similar to the Kelvin scale are the only scales that
can be used to prove that entropy is a state function, then temperature has a metric structure that these scales faithfully represent.
This is a plausible claim. However, I do not think its antecedent is true. Consider the temperature scale that assigns to any temperature value the num- ber − 1 /T , where T is the number assigned to that temperature value on the Kelvin scale. This scale agrees with the Kelvin scale on how temperature values are or- dered, but not on their ratios, or on the ratios of temperature intervals.^10 I will use “τ” to stand for the numbers assigned to temperatures on this new scale. This scale can also be used to prove that entropy is a state function. For the integral −
∫ (^) s 2 s 1 τ^ dQ is independent of path, and is equal to the entropy difference between s 1 and s 2. As far as being of use to prove that entropy is a state function, then, the Kelvin scale and the τ scale work equally well. And that means we cannot appeal to (*) to justify belief that temperature has a metric structure. Of course if f (x) is any smoothly increasing (and so invertible) function from numbers to numbers we can define a new temperature scale that assigns the number f (T ) to the value to which the Kelvin scale assigns the number T. If we let “ ¯T ” denote numerical temperature values on this scale, then the integral
∫ (^) s 2 s 1 dQ/^ f^
− 1 ( ¯T )
will be independent of path. So for each such function f we can define a tem- perature scale that is a counterexample to the antecedent of (*). But if f is some extremely complicated function then this might look like a trick. We apply f to define the new scale, but then apply its inverse in the definition of entropy; it still looks like the Kelvin scale is doing the real work. But I do not think that the argument that appeals to the τ scale is a trick. The situation we are in with respect to τ is this: there is a temperature scale (the Kelvin scale) with the property that when you divide dQ by temperature on this scale you get a differential whose integral is independent of path (and we use this integral to define entropy); and there is another scale (the τ scale) with the property that when you multiply dQ by temperature on this scale you get a differential whose integral (^10) It also assigns all temperature values negative numbers. (Actually, it assigns negative numbers to values to which the Kelvin scale assigns positive numbers. But I am ignoring the existence of temperature values to which the Kelvin scale itself assigns negative numbers).
identical to the result of multiplying its value for temperature and Nk. You can only multiply numbers, and pressure values, volume values, and temperature values are not numbers. They are properties that are assigned numbers relative to scales of measurement. What PV = NkT means, then, is that the result of multiplying the number that is assigned to the gas’s pressure value (on a certain scale) and the number that is assigned to the gas’s volume value (on a certain scale) is identical to the result of multiplying the number assigned to its temperature value on the Kelvin scale and Nk.^11 So it does not follow from PV = NkT alone that doubling a gas’s volume doubles its temperature. This only follows if we also assume that temperature has a metric structure that is correctly represented by the Kelvin scale (since it is relative to that scale, and scales similar to it, that PV = NkT is correct). But this assumption is just what is in question. So this argument that temperature has a metric structure is circular. There is, however, more to be said. The questions I have been asking about the structure of temperature can (and have) been asked about other quantities. There are analogous debates about what laws involving those quantities indicate about the structure of those quantities. One famous example is duration: do durations have a metric structure?^12 Look at F = ma. There is an argument analogous to the one I just gave that starts from F = ma and concludes that accelerations have a metric structure (and if they do, so do durations, in terms of which they are defined). And there is an analogous reply: F = ma is true relative to one scale for measuring duration (or, better, one coordinate system for time). If we wanted to use a different coordinate system, we would have to use a different equation to express this law. Maybe duration does have a metric structure (that is faithfully represented by one of these coordinate systems), but that it is a law that F = ma does not give us reason (^11) Usually we do not want to write equations in physics so that they are only true relative to a particular scale for measuring a quantity. Instead write the equations so that they are true relative to a whole family of similar scales (scales that agree on the ratios of numbers assigned to quantity values). Equations like that are “dimen- sionally homogenous.” (^12) Newton (2004) famously argued that they do; Barbour (1999) is a recent pro- ponent of the view that they do not.
to think so. In the case of duration, however, this is not the end of the story. To settle the issue we write Newton’s theory (including his second law) in a way that is valid in any coordinate system whatever. This is the “generally covariant” formulation of the theory (Friedman [1983, 71-124] has the details.) When we do that the temporal metric appears explicitly in the equations expressing the theory. This is conclusive evidence that in Newtonian mechanics duration has a metric structure. One might hope that something similar can be done for the case of temper- ature. Write the ideal gas law (and other equations of thermodynamics) so that it is true no matter what scale we choose to measure P, V, and T. Then look to see if metric structure for T appears explicitly in that formulation of the theory. But I know of no formulation of thermodynamics like this. So we cannot use this strategy to settle the question. The best we can do, I think, is appeal to simplicity. The equations for thermo- dynamics take on a relatively simple form when written relative to the Kelvin scale. Perhaps this gives us some reason to think temperature has a metric structure that is faithfully represented by the Kelvin scale. But these reasons are weak. (For one thing, it is not obvious that these equations are less simple when written relative to the τ scale.)
- When we turn to statistical mechanics there is a clear and straightforward case to be made for the proposition that temperature has a metric structure. I will close by rehearsing it. Statistical mechanics, unlike thermodynamics but like the caloric theory, ex- plicitly says what temperature is. In statistical mechanics temperature is not treated as a fundamental quantity and is identified with a combination of more fundamental quantities. Statistical mechanics says that a body’s temperature is the derivative of its internal energy with respect to its entropy (with external parameters held fixed).^13 (^13) It does not say that a body’s temperature is the average kinetic energy of its constituent molecules. This claim is false for many reasons; for one thing, there are thermodynamic systems (ideal paramagnets, for example) that can have negative
