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Constructive Thermodynamics
CONSTRUCTIVE THERMODYNAMICS W. K. BURTON Department of Natural Philosophy, The University, Glasgow 1. The purpose of this note is to discuss the feasibility of formulating a fundamental part of physics in a constructive manner. As a starting point we take the formulation of thermodynamics given by Robin Giles [2]. In this book, Giles effects a complete separation between the physical and the mathematical aspects of the theory, and presents the latter as an informal axiomatic theory measuring up fully to the standards of rigour customary in contemporary mathematics. Its reformulation as a formal theory would present no particular difficulty, but there are reasons for believing it to be worth while to attempt this in a constructive sense, making slight modifications in the original theory if necessary. These reasons stem from the physical aspects of the theory. In addition to the various mechanisms for producing theorems (derived formulae) it is necessary, in a physical theory, to lay down certain rules of interpretation which connect at least some of the formulae with practical actions. In the past this kind of problem has not received much attention, and the further great merit of Giles’s approach is that for the first time questions of this sort are submitted to a precise analysis. The axioms of the theory contain just four primitive concepts which are called ‘state’, ‘union’ of states, the relation of a state ‘going to’ a state, and the relation of a state being ‘equal’ to a state. Giles’s theory being informal, there will of course be further primitive concepts, for example logical ones, which will have to be taken into account in a complete formalisation. As a matter of fact Giles himself appears not quite to count equality between states as one of his primitive concepts, perhaps feeling that it belongs to a different level from the others. Denoting states by small Roman letters with or without subscripts, we have primitive formulae of the form a = b, a + b = c , a + b (read as ‘state a equals state by,‘state a plus (union) state b equals state c’, and ‘state a goes to state b’, respectively). Formulae then result by combining primitive formulae by means of the logical particles. Giles’s idea is now the following: if rules 15
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are laid down which attribute unique meanings to the primitive formulae, all the formulae will acquire unique meanings. These rules, which he calls primitive rules of interpretation, permit other derived concepts to be introduced by means of explicit definitions, and these derived concepts are thereby ‘explained’ in terms of the primitive ones. No other concepts besides primitive and derived ones appear. The axioms of the theory, being formulae, also acquire an interpretation, and the question arises as to whether the axioms are true under this interpretation. If they are, then the theorems will also be true, providing the rules of inference lead from true formulae to true formulae. Giles selects the aspects of experience which are linked to the primitive concepts in the mathematical theory by the primitive rules of interpretation to be as ‘direct’ as possible. An experience is direct to the extent that it can be demonstrated rather than explained in terms of other (more direct) ones. The implied ordering of experience according to directness is admittedly rather crude: it corresponds roughly to an order of concept formation in a child as it matures. On the theoretical level the direct experiences are supposed to correspond in some way with primitive concepts in a theory, and the less direct ones to derived concepts. The theory then, as it were, ‘explains’ the indirect aspects of experience in terms of the direct ones. 2. Before presenting the axioms of Giles’s theory, we wish to summarise Giles’s own discussion of his rules of interpretation. We do this not only to give the theory some intuitive content, but also because we wish to consider later on some modifications in these rules. The main purpose of a physical theory is to make predictions. The basis on which these predictions are made consists of prior knowledge about the ‘system’ which is under investigation. This knowledge, in its turn, consists of information about what has happened to the system in the past: in other words of how the system has been prepared, Thus we consider that the basis on which predictions are made is the method of preparation (of a system), and it is this which we wish to call the state (of a system). We use capital Roman letters A, B, ..., to denote systems. Then a state a of a system A may be designated by adding a subscript to A: thus A,, A,, ..., are states of the system A. In the mathematical theory, systems are not alluded to at all, the method of preparation being taken as including a specification of how the system is selected or produced. Thus ‘system’will appear only, if at all, as an arbitrary collection of states. If we have two systems A and B then we can conceive of them jointly as
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forming a compound system, denoted by ‘A + B’, consisting of the conceptual union of systems A and B. In this union A and B are both considered as isolated. In fact a system can only be prepared in isolation, for if the method of preparation produced the system together with some ‘environment’, the position of the boundary between system and environment would have to be explained, and then the state would no longer be determined by the method of preparation alone. Accordingly, the term ‘state’ can only refer to conditions in which the system concerned is isolated. It is clear that + is associative and commutative. Given any system A it is possible in principle to construct a finite number of replicas of A. Thus ‘A + A’ has a meaning: it is the union of A with a replica of A. We denote it by ‘2A’. Similarly if m is a positive integer, ‘mA’ denotes the union of m replicas of A. Just as we can add systems, SO we can form in a natural way the union A, B, of any two states A, and B, of systems A and B. We define A, B, to be the state of the system A + B in which A and B are isolated and in the states A, and B, respectively. The addition of states is also associative and commutative, and as in the case of systems we can add replicas of the same state: we denote the union of m replicas of A, by ‘mA,’. Although A, +B, is always a state of the system A+B, not every state of A + B is of this form; only those in which the parts A and B are isolated. Thus the rule of interpretation for a + b is to be: $ a and b are states, then a b is that state whose method of preparation consists in the simultaneous and independent performance of the methods of preparation corresponding to the states a and b. The operation of addition of states may be regarded as defining a relation a + b = c between three states a, b and c. We now consider another relation between states connected with the natural evolution of a state with time. If, during some time interval, the state of a system A changes, a natural process is said to have occurred. In general, A will interact with other systems during such a process. Suppose A is part of a larger system I which remains isolated throughout the process. Thus I contains, together with A, every system with which A interacts during the process. Although these systems do not remain isolated during the process, it is possible that, for some of them, the initial and final states may coincide. If so, we say that they are not involved in the process. A system is involved in a process if and only if its initial and final states differ. If there exists a natural process involving only a system A which has initial and final states A, and A, respectively, then we write ‘A,-’A2’ (read “A1 goes to Az’’).
+
+
+
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Thus the rule of interpretation for a + b is to be: a-tb ifand only ifthere is a state k and a time interval z such that a + k evolves in isolation in the time z into the state b + k. With these explanations we have arrived at rules of interpretation for ‘state’, ‘+’ and ‘-+’.When are two states to be regarded as equal? Clearly if two states are prepared in the same way they should be regarded as equal. However, even if two states are not equal in this sense, but nevertheless any two experiments applied to these two states yield the same result (or rather the same statistical distribution of results) then these states need not be distinguished. This gives rise to a wider notion of equality, which in fact is the one which Giles uses in his book.
3. It is convenient [2]to characterise thermodynamics by making use of the concept of a primitive observer (for thermodynamics). Such an observer is a being whose direct experience embraces only the physical aspects of experience associated by the primitive rules of interpretation with the primitive concepts “state”, +, -+ and = . That is, he is directly aware of states and relations among them of the forms a = b, a + b = c and a-tb, but of nothing else. Thermodynamics can now be characterised as a physical theory which is meaningful to such an observer, and which could, indeed, have been developed by him. The specification of the concept of primitive observer for a theory amounts to the specification of a range of observational powers sufficient to guarantee that the theory can actually be applied in practice. As we shall see later, meagre though the powers of a primitive observer for thermodynamics may look, they transcend in important respects the powers of human observers. 4. We now present Giles’s axioms for thermodynamics as given in Appendix A of his book [ 2 ] . Consider a non empty set 6whose elements will be called states. We postulate in G an operation and a relation +. satisfying the following axioms. AXIOM1. In 6 (i) if a, be 6then a + b e 6 , a + b = b + a, and if a, b, c e 6 then a+(b+c) = (a b) c ; (ii) a-+a (iii) a+bA b-tc=>a+c a, b, CEG. (iv) a + c + b + c o a + b AXIOM2. If a, b, CEG a+b Aa+c=>b+cv c-tb.
+
+ +
1
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DEFINITION 1. A process is an ordered pair of states (a, b). Denote the set of all processes by 13; denote the elements of !@ by small Greek letters, a,P,y,....
Define an operation
a relation
--*
in ’p by
-
+ in ‘p by
+ (c, d) = (a + c, b + d) (a, b) (c, d ) o a + d b + c (a, b)
-+
+
in ‘p by setting (a, b)-(c, d) whenever there is a state and a relation x such that a + d + x = b + c + x . is an equivalence relation with respect to which It is easily shown that + and + are compatible. Henceforth equivalent elements in !@ are identified. In particular all processes of the form (a, a) are equal: denote any such process by 0. If CI is the process (a, b), denote the process (b, a) by - CI.Then 0 + a= CI and a+ (- a) = 0, and ‘p turns out to be an abelian group under + with zero element 0.
-
DEFINITION 2. CI isnaturalif a+O, antinaturalifO-+a,possibleifa-+O v O+a, reversible if a-0 A 0-m. It is irreversible if it is possible but not reversible, and impossible if it is not possible. The set of all natural (antinatural, possible, reversible) processes is denoted by ‘pN(’pA, pp,‘p,). It is easily shown that Ppand ‘p, are subgroups of ‘p.
DEFINITION 3. Given states a and b, if there exists a positive integer n and a state c such that (na + c, nb)E ‘ppwe write a c b (read “a is contained in b”). 4. A state e is an internal state if, given any state x, there DEFINITION exists a positive integer n such that x c n e .
AXIOM3. There exists an internal state. AXIOM4. Given a process a, if there exists a state c such that for any positive real number E there exist positive integers m, n and states x, y such that m/n<&,x c m c , y c m c and (x, y)+ncc+O, then a-0. Using methods from functional analysis Giles derives from these axioms : GILES’SMAINTHEOREM. There exists a positive additive function of state S, called the quasi-entropy, and a set of positive additive functions of state, called components of content, such that for any states a and b, a + b if and only if S(a)
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physical discipline of pure thermodynamics is accessible to a primitive observer in Giles’s sense. The significance of ‘positive’ in the Main Theorem is this: positive functions are bounded. DEFINITION 5. Define the norm IlalI of any state a relative to the internal state e by
The topology defined by the norm turns out t o be independent of the choice of e. DEFINITION 6. A real valued additive function Q(a) defined for every state a is bounded if there exists a constant k such that, for all a, /Q(a)l,
5. For reasons of space, we are obliged to refer the reader to Giles’s book [2] for an account of how the theorems are derived from the axioms. It will be found that what is done is in accord with the standards of rigour customary among contemporary mathematicians. It will, however, be clear that it is not just Giles’s explicitly stated axioms which are involved in establishing the theorems, but also inferential machinery of a logical and mathematical character not covered by the axioms themselves. That is also the situation in ordinary mathematics. But in a physical theory interpretational questions are involved in addition to purely mathematical ones. Giles deals with these interpretational questions by trying to establish some at least of the axioms as ‘true’ in some specific sense, which may be in need of further clarification, but which in any case necessarily calls into question the justifiability of the inferential machinery used. In other words the inferential machinery itself requires an interpretation. As his theory stands at present, there are at least two places where difficulties may be anticipated. One of these concerns the application of the axiom of choice in the proof of an extension theorem of the Hahn-Banach type which is used to extend an additive function on a subgroup to the whole group, and the other concerns an application of the law of excluded middle of a kind best illustrated by means of a quotation: “for each value of n (an integer) the process a1=(m,x, c)+m,n(a, b) is either natural or antinatural. Suppose first a1 is natural for an infinite number of values of n, then a-tb. If u1 is natural for only a finite number of values of n then it is certainly antinatural for an infinite number of values of n.
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Then a similar argument yields b+a”. This is reminiscent of the situation encountered in trying to prove that every rational sequence has a monotone subsequence. It seems that these difficulties can be satisfactorily resolved by making use of methods pointed out by Paul Lorenzen in his books [4,5] (see also [3]) where he sets up a form of constructive analysis which deviates only slightly from classical analysis as far as applications are concerned, and in which no axioms are assumed. If Lorenzen’s theory were axiomatised, then the axioms would be provable, because rules of interpretation could be introduced linking the primitive concepts with concretely specified actions in such a way that the axioms became true under these rules. Lorenzen does not, of course, express himself in these terms : he starts with inductive definitions which are used for defending prime propositions, out of which compound propositions can be defined by reference to obligations taken on in asserting them; then abstractions are introduced, and so on. As a matter of fact not all the formulae and theorems of Lorenzen actually receive an interpretation : instead the use of classical logic is justified by means of a consistency proof. This means, for example, that the use of the law of excluded middle is accepted as a fiction, but as a fiction which provably does no harm. If we grant that the primitive observer in thermodynamics has ‘established’ the Giles axioms as true, and that no more than a countable number of states can come into consideration, so that the use of the axiom of choice is avoided, then there would seem to be no difficulty in completing the Giles theory as a constructive theory in Lorenzen’s sense simply by adjoining Lorenzen’s considerations to those of Giles. However, it seems to the present writer that Giles’s primitive observer, although having only meagre-looking powers, has some which transcend those of any human observer. For example he can tell of any pair of states a and b whether a + b or not. Accordingly the proposition a - t b becomes truth definite [4]: there is a procedure which, when applied to the proposition, yields one and only one of two truth values, true or false. However, for any human observer all we are entitled to say is that the proposition a-tb is at most proof-definite [4] i.e., there exists a procedure which, when applied to another procedure which is applied to this proposition, yields a decision as to whether the second mentioned procedure constitutes a proof of a + b or not. What is required for establishing the truth of a + b is the following. We have to find a state k and a time interval z such that a + k evolves in time z into b + k. If we have found a suitable state k and time interval z so that this happens, then we have established that a+b; but if we have not, we have not shown that a + b is false. Perhaps further
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efforts to find suitable k and z will be successful, so that a + b after all. We can also describe the situation in terms of systems and apparatus like this: the system A can be in states A, and A,. Starting with system A in the state A, we place it in interaction with a piece of equipment K (the apparatus) which is in the initial state K,. This represents the start of an experiment on A. We then wait a time z,with A + K kept isolated, and at the end of that time we separate A and K so that both A and K, individually, are isolated. If A is now in the state A2 and K is again in the state K,, then we shall have shown by experiment that A,+A,. In any other case all we can say is that we have failed to decide whether A,-tA, or not. All that we have said so far about the problem of deciding about a-tb presumes that there are no problems connected with the notion of equality of states; but there are difficulties here as well, and these difficulties have repercussions on the rule of interpretation for a + b too. Expressed more fully, this rule would read: a-tb if there is a state k and a time interval z such that the state whose method of preparation is “apply simultaneously and independently the methods of preparation corresponding to a and k and then wait for a time 7’’is indistinguishable from the state whose method of preparation is “apply simultaneously and independently the methods of preparation corresponding to b and k” in the sense that any experiment applied to these states will yield the same result (or rather the same statistical distribution of results) in each case. Furthermore a consideration of microscopic systems suggests that, strictly speaking this can only apply to experiments which are concerned with only one of the parts of the state and which do not look for a correlation between these parts. Since the methods of preparation are independent there will be no such correlation for b + k ; but in the state evolved from a + k there will generally be some correlation - for instance a high energy for one part will tend to be accompanied by a low energy for the other. Thus in the above rule of interpretation, the phrase “into the state b + k” in the original formulation: ‘a-tb if there is a state k and a time interval z such that a + k evolves (in isolation) in the time z into the state b + k’ should strictly be replaced by “into a state which, ignoring the correlation between the parts, is indistinguishable from b + k”. In the model furnished by statistical mechanics it is just this procedure of ignoring correlation which accounts for the irreversible nature of thermodynamic processes. We see then that we are confronted with the problem of deciding whether or not two states are equal even when we know that they have arisen in quite different ways. If two states were counted equal only when their methods of
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preparation consisted of identical actions there would be no difficulty, but the foregoing discussion shows that we need a wider notion of equality than this: two states are still to be identified when no subsequent experiments which may involve deciding on the truth or falsity of propositions of the form a+b, once more - can reveal a difference. Thus some propositions a = b will only be refutation definite, i.e. there will only be available a decidable refutation concept. To put this another way, propositions of the form a # b, but not a = b, will be proof definite. This seems to threaten propositions of the form a + b with not even being proof-definite, and thus in turn to threaten propositions of the form a = b with not even being refutation definite. The same applies to propositions of the form a = b + c : if a is defined as b + c in a particular context, all is well, but if it is a question of deciding whether a state b + c is equal or not to an independently specified state, then tests of the form a+b may again be needed, and this proposition may be neither proof nor refutation definite. One way out of this dilemma would be to try to make equality truth definite by replacing the rule of interpretation which has been given for ‘state’ by another one. There is also a possibility of retaining equality as refutation definite and trying to make a + b dialogue definite [4] - and managing with that. We prefer however, at this stage, to examine the first of these possibilities, since an issue is involved which appears elsewhere in the natural sciences, and which seems very difficult to eliminate. This issue concerns the formation of policies for future behaviour. On the basis of past experience we decide to act in a certain way, but as a result of further experiences which have come about as a result of those actions, we may decide that we ‘should’ have performed other actions instead. That is, our present policies may be revised as a result of further experiences. This train of thought seems to suggest that we are concerned with ‘states of mind’ - a change of state of mind having occurred when a policy is changed. We now reformulate the revised rule of interpretation for ‘state’ as: a state is a state of mind induced by the available knowledge of a method of preparation. In formulating this rule of interpretation we have acknowledged the presence of a subjective element, and one of our tasks will be that of showing how conclusions can be reached which d o not depend on the personal peculiarities of those whose states of mind are referred to. Two ‘states’ are now to be regarded as equal just when the associated states of mind are the same. Let us imagine that two states a and b have arisen in connection with possibly different methods of preparation. Experiments are now carried out on these states and in every case the results
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BURTON
cannot be distinguished. Possibly, if further experiments were to be performed, differences would come to light. However, it may be presumed that the experiments will not be continued indefinitely; a point will come at which the experimenter says that all his efforts to distinguish a and b have failed, and that as far as he can see, he has reached a stage at which his further efforts are disclosing no fresh information. His state of mind is that a and b are equal. He is not at all saying that a and b are identical - he may well agree that further experiments may disclose differences which his experiments have not disclosed so far. By saying that a and b are equal he is merely saying that whatever predictions he is prepared to make about a he is prepared to make about b and conversely; if he were a betting man he would be prepared to lay exactly the same odds in each case. In saying that equality is truth definite we mean simply that the experimenter can decide, yes or no, whether his state of mind about a is the same as his state of mind about b, in so far as his predictions about a and b in future experiments are concerned, With this altered interpretation of state, a = b and a + b = c become truth definite, and a-b proof definite. 6. We return now to Giles’s axioms and ask to what extent can they be justified. Axiom l(i) runs: if a, b E 6 then a + b E G , a + b = b + a , and if a, b, CEGthen a+(b+c)=(a+b)+c. Thus the set 6 of states is required in particular to be closed under the operation . In a constructive theory the set G will not be ‘given’ in advance, but will have to be constructed from the states which will constitute its elements. We start with various states a, b, c, ... and declare them to be in 6. If a and b are in 6, then the rule of interpretation for shows that a + b is a state: accordingly it is put into 6 as well. Thus the set 6 is closed in a sense entirely analogous to the sense in which the set N of numerals is closed in Lorenzen’s arithmetic. In this case we have rules for producing numerals. -1 ii+nl.
+
+
Each figure, as it is produced, is ‘put’ in N. Just as the numerals 1, 11, 111, . .. are ‘generated‘ by these rules, so are the states generated from an initial stock of states by the operation of taking unions. The axioms a + b = b + a and a + (b +c) =(a + b) + c are again justified by the rule of interpretation for +. For example both a + b and b + a refer to the simultaneous and independent performance of the methods of prepa-
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ration associated with a and with b. The fact that a comes before b, or b before a, in a linear arrangement of symboIs is something which is forced on us only by the type of notation we have selected. Similarly in the case of the associative axiom. Thus we have ‘justified’ Axiom 1(i). The justification for Axiom 1 (ii) is found by reference to the rule of interpretation for +. If we take r=O in that rule then for a + a to be true we have to find a state k such that a k evolves in zero time into a + k. Any state k will do for this purpose. contains logical particles as Axiom 1 (iii), namely a + b A b+c*a+c, well as Giles’s primitive terms. If we interpret these operatively [3] or in terms of dialogues [4], then Axiom 1 (iii) would be justified by giving a winning strategy for it. Whoever asserts this axiom is obliged to assert and defend a 4 c if an opponent is prepared to assert a-+b and b+c and can successfully defend them. NOWa+ b and b+c are proof definite. Introducing temporarily the notation 7to mean “evolves in isolation in time z into”, proofs of these propositions involve establishing, with suitable k, and k, and r1 and r,:
+
+ k, 2 b + k,
(a)
a
(b)
b+k,;;tc+k,
and we can tell whether or not the opponent possesses proofs in this sense. If he does possess such proofs, the proponent now attempts to prove a+c by using the opponent’s (a) to produce the intermediate state b from a and then using the opponent’s (b) to produce c from b, so establishing a +k T c
(4
+k
for suitable k and r. The question now, however, is: what is the state k? Intuitively, the opponent may have employed two quite separate pieces of apparatus K(’) and K(’) with initial and final states k,, k, and k,, k, respectively. What should the proponent’s apparatus be? The most obvious suggestion is K(”+K(’); calling this K, it is a matter of arranging that the initial and final states of K be some state k. Since the opponent has produced states k, and k, for K(’) and K(’) respectively, this suggests that k should be k, + k,. The proponent would then try to arrange that a and
+ k , + k, 2 b + k, + k,
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so that he finally achieves (c')
a
+ k , + k2y-c
+
+ k, + k, ,
+
giving (c) with k = k, k, and t = t1 z2. But in order to do what is required of him, the proponent must arrange that in stage (a') only K(') interacts with the system under consideration, and in stage (bl) only K(') interacts with the system, because only that has been done by the opponent. Thus in order to succeed it looks as if the proponent must have the ability to maintain an arbitrary state constant - to freeze it, as we shall say. That is, in stage (a') the state k, has to be frozen while in stage (b') k, has to be frozen. So it appears that we must introduce a hypothesis - namely that an arbitrary state can be frozen - in order to be able to defend Axiom 1 (iii) hypothetically. This state of affairs certainly falls short of what we originally had in mind. It means that there will be an inherent restriction on the domain of applicability of the theory, the precise nature of which requires further elucidation. We are faced with the problem of distinguishing on the practical level whether or not we are confronted with a situation in which Axiom 1 (ii) is 'true'. There is, however, a possible way out of this difficulty: to modify the rule of interpretation for + once more. We define a new relation ias follows:
(D,)
a i b % a + b v V,V,
,,..., c,(a,+cl
A C ~ + C , A
... A C , - + ~ )
where the c,(i= 1, ..., n) are intermediate states. Under this definition a + b is still proof definite, and the transitivity law a i b
A
bic*aic
and Axiom 1 (i) is holds. Axiom 1 (ii) still holds with + replaced by i not affected by the replacement of + by 4. Thus with the new rule of interpretation for + (namely -+ means i with i defined by (D1) in which the '4' on the right is interpretated by the old rule) we have established that 6 is a partially ordered semigroup, i.e. we have proved Axioms 1 (i)1 (iii). The remaining clause of Axiom 1 amounts essentially to two axioms, namely a+c+b+c*a+b a+b +a+c+b+c.
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The first of these is established on the grounds that the state c can be considered as part of the state k in the rule of interpretation for -+ in either the new or the old sense. But the second axiom gives rise to the same difficulty as was encountered in connection with Axiom 1 (iii) : it asserts that the state c can be frozen, and there seems to be no obvious way out of the difficulty in this case. This axiom then, as well as Axiom 2, as it turns out, has to be treated differently from the others. So our axioms will be of two kinds: the ones which can be effectively defended in “all”cases, and those which require for their defence an additional hypothetical element which restricts the defensibility to those cases for which some method is available which shows the axioms in question to be ‘true’ in those cases, even if not in general. We refer to the first group of axioms as prototheoretically defensible and to the second group of axioms as hypothetically defensible. As far as the question of a state’s being freezable is concerned we can often, in practice, achieve what is required by the use of suitable apparatus, possibly in a rather far-fetched sense. For example if we had a sample of gas of non-uniform temperature and pressure as an instance of a state k, we could freeze k by ‘instantaneously’ inserting a large number of insulating partitions which would serve, approximately, to freeze k. If we were concerned with a solid rather than with a gas this would scarcely be feasible. One would have to call on the possibility of dividing up the solid into small pieces each of which is to be insulated from the others. At a later stage when the freezing is to cease the solid would be reassembled. What really matters here is that the freezing can be done ‘in principle’. The conviction in each case can only be arrived at by a consideration of the case in question. Usually the difficulty is not too pressing in practice. The situation for Axiom 2: a-+b~a+c*b+cvc+b is quite different. As the discussion given by Giles in his book shows, this axiom is not true for systems exhibiting hysteresis. So if Axiom 2 is to be justified at all, it must be justified by reference to tests carried out on the systems under consideration themselves. Each instance of Axiom 2 regarded as a proposition is proof-definite but not truth definite. After preliminary trials an experimenter may come to the conviction that Axiom 2 is true; but if he is mistaken in this connection, he will find that when he tries to get upper and lower bounds for the entropy, he will always find a finite gap between them, however many experiments and whatever experiments he performs. We
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might say that when Axiom 2 is ‘false’ only an upper and lower entropy (cf. outer and inner measures) is available, whose difference measures the degree of hysteresis exhibited in the system. So with Axiom 2 we are confronted not with a proper axiom at all; rather we might say that a system for which Axiom 2 is true is defined to be hysteresis-free, and that the problem reduces to the practical problem of recognising hysteresis-free systems. Axiom 2 then has ‘physical content’ in a much stronger sense than Axiom 1, and is decisive in settling the detailed structure of the theory.
7. The motivation underlying Giles’s Axioms 3 and 4 comes from the desire for boundedness in the entropy and components of content, and thus is mathematical rather than physical. Since we are now trying to replace Giles’s theory by a constructive one, these axioms may be expected to appear in a rather different light. In particular, even though the set of states 6 will in general be (potentially) infinite, in any given case 6 will be generated from a finite number of basic states by taking unions. In such a case the state which consists of the union of all the basic states will constitute an internal state in the sense of Definition 4. So Axiom 3 will be justified for that case. However, it is not possible to choose an internal state once and for all in this way, for it may be possible to enlarge 6 by enlarging the set of basic states which generate 6. If this is done our procedure would give rise to a new internal state. The use of this new internal state instead of the old one would, however, do no more than induce scale changes in the entropy and components of content. Thus in a constructive reformulation of the Giles theory we may perhaps regard the justification of Axiom 3 as unproblematical. The justification of Axiom 4 is then achieved by means of a further modification in the rule of interpretation for -+.If we define a>b+ there is a state c such that for any positive real (D,) number E , there are positive integers m, n and states x, y such that m/n
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As an ordinary physicist would no doubt express it, Axiom 2 stands out as ‘containing most of the physics’. It is somewhat remarkable that such a rich looking theory as thermodynamics should be supportable on such a meagre looking base. It is seen that while the Giles theory appears to suggest the feasibility of a fully constructive thermodynamics, and perhaps also of other physical theories, further work is required on the interpretational side (which may well lead to modifications in what appear above as axioms) before we shall be in a position to offer such a theory as a fitting companion to, say, Lorenzen’s constructive analysis. Acknowledgements It is a great pleasure t o thank Professor K. Schutte for his kind hospitality when I spent a year (1964-65) in his Department in Kiel, where the above work was started. I thank him also for the numerous illuminating discussions which clarified my own thinking considerably. It is a pleasure to acknowledge in the same sense the benefits accruing from discussions with Professor R. Giles. Finally I thank the Carnegie Trust for the Universities of Scotland and the then Directorate for Scientific and Industrial Research for Travel and Maintenance Grants, and The University of Glasgow for making my study in Germany possible. References 1. H. B. CURRY,Foundations of mathematical logic (McGraw-Hill, 1964) p. 48. 2. R. GILES,Mathematical foundations of thermodynamics (Pergamon, 1964). 3. P. LORENZEN, Einfiihrung in die operative Logik und Mathematik (Springer, 1955). 4. P. LORENZEN, Metamathematik (B. I. Hochschultaschenbiicher, 1962). 5. P. LORENZEN, Differential und Integral (Akademische Verlagsgesellschaft, 1965).
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are laid down which attribute unique meanings to the primitive formulae, all the formulae will acquire unique meanings. These rules, which he calls primitive rules of interpretation, permit other derived concepts to be introduced by means of explicit definitions, and these derived concepts are thereby ‘explained’ in terms of the primitive ones. No other concepts besides primitive and derived ones appear. The axioms of the theory, being formulae, also acquire an interpretation, and the question arises as to whether the axioms are true under this interpretation. If they are, then the theorems will also be true, providing the rules of inference lead from true formulae to true formulae. Giles selects the aspects of experience which are linked to the primitive concepts in the mathematical theory by the primitive rules of interpretation to be as ‘direct’ as possible. An experience is direct to the extent that it can be demonstrated rather than explained in terms of other (more direct) ones. The implied ordering of experience according to directness is admittedly rather crude: it corresponds roughly to an order of concept formation in a child as it matures. On the theoretical level the direct experiences are supposed to correspond in some way with primitive concepts in a theory, and the less direct ones to derived concepts. The theory then, as it were, ‘explains’ the indirect aspects of experience in terms of the direct ones. 2. Before presenting the axioms of Giles’s theory, we wish to summarise Giles’s own discussion of his rules of interpretation. We do this not only to give the theory some intuitive content, but also because we wish to consider later on some modifications in these rules. The main purpose of a physical theory is to make predictions. The basis on which these predictions are made consists of prior knowledge about the ‘system’ which is under investigation. This knowledge, in its turn, consists of information about what has happened to the system in the past: in other words of how the system has been prepared, Thus we consider that the basis on which predictions are made is the method of preparation (of a system), and it is this which we wish to call the state (of a system). We use capital Roman letters A, B, ..., to denote systems. Then a state a of a system A may be designated by adding a subscript to A: thus A,, A,, ..., are states of the system A. In the mathematical theory, systems are not alluded to at all, the method of preparation being taken as including a specification of how the system is selected or produced. Thus ‘system’will appear only, if at all, as an arbitrary collection of states. If we have two systems A and B then we can conceive of them jointly as
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forming a compound system, denoted by ‘A + B’, consisting of the conceptual union of systems A and B. In this union A and B are both considered as isolated. In fact a system can only be prepared in isolation, for if the method of preparation produced the system together with some ‘environment’, the position of the boundary between system and environment would have to be explained, and then the state would no longer be determined by the method of preparation alone. Accordingly, the term ‘state’ can only refer to conditions in which the system concerned is isolated. It is clear that + is associative and commutative. Given any system A it is possible in principle to construct a finite number of replicas of A. Thus ‘A + A’ has a meaning: it is the union of A with a replica of A. We denote it by ‘2A’. Similarly if m is a positive integer, ‘mA’ denotes the union of m replicas of A. Just as we can add systems, SO we can form in a natural way the union A, B, of any two states A, and B, of systems A and B. We define A, B, to be the state of the system A + B in which A and B are isolated and in the states A, and B, respectively. The addition of states is also associative and commutative, and as in the case of systems we can add replicas of the same state: we denote the union of m replicas of A, by ‘mA,’. Although A, +B, is always a state of the system A+B, not every state of A + B is of this form; only those in which the parts A and B are isolated. Thus the rule of interpretation for a + b is to be: $ a and b are states, then a b is that state whose method of preparation consists in the simultaneous and independent performance of the methods of preparation corresponding to the states a and b. The operation of addition of states may be regarded as defining a relation a + b = c between three states a, b and c. We now consider another relation between states connected with the natural evolution of a state with time. If, during some time interval, the state of a system A changes, a natural process is said to have occurred. In general, A will interact with other systems during such a process. Suppose A is part of a larger system I which remains isolated throughout the process. Thus I contains, together with A, every system with which A interacts during the process. Although these systems do not remain isolated during the process, it is possible that, for some of them, the initial and final states may coincide. If so, we say that they are not involved in the process. A system is involved in a process if and only if its initial and final states differ. If there exists a natural process involving only a system A which has initial and final states A, and A, respectively, then we write ‘A,-’A2’ (read “A1 goes to Az’’).
+
+
+
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Thus the rule of interpretation for a + b is to be: a-tb ifand only ifthere is a state k and a time interval z such that a + k evolves in isolation in the time z into the state b + k. With these explanations we have arrived at rules of interpretation for ‘state’, ‘+’ and ‘-+’.When are two states to be regarded as equal? Clearly if two states are prepared in the same way they should be regarded as equal. However, even if two states are not equal in this sense, but nevertheless any two experiments applied to these two states yield the same result (or rather the same statistical distribution of results) then these states need not be distinguished. This gives rise to a wider notion of equality, which in fact is the one which Giles uses in his book.
3. It is convenient [2]to characterise thermodynamics by making use of the concept of a primitive observer (for thermodynamics). Such an observer is a being whose direct experience embraces only the physical aspects of experience associated by the primitive rules of interpretation with the primitive concepts “state”, +, -+ and = . That is, he is directly aware of states and relations among them of the forms a = b, a + b = c and a-tb, but of nothing else. Thermodynamics can now be characterised as a physical theory which is meaningful to such an observer, and which could, indeed, have been developed by him. The specification of the concept of primitive observer for a theory amounts to the specification of a range of observational powers sufficient to guarantee that the theory can actually be applied in practice. As we shall see later, meagre though the powers of a primitive observer for thermodynamics may look, they transcend in important respects the powers of human observers. 4. We now present Giles’s axioms for thermodynamics as given in Appendix A of his book [ 2 ] . Consider a non empty set 6whose elements will be called states. We postulate in G an operation and a relation +. satisfying the following axioms. AXIOM1. In 6 (i) if a, be 6then a + b e 6 , a + b = b + a, and if a, b, c e 6 then a+(b+c) = (a b) c ; (ii) a-+a (iii) a+bA b-tc=>a+c a, b, CEG. (iv) a + c + b + c o a + b AXIOM2. If a, b, CEG a+b Aa+c=>b+cv c-tb.
+
+ +
1
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DEFINITION 1. A process is an ordered pair of states (a, b). Denote the set of all processes by 13; denote the elements of !@ by small Greek letters, a,P,y,....
Define an operation
a relation
--*
in ’p by
-
+ in ‘p by
+ (c, d) = (a + c, b + d) (a, b) (c, d ) o a + d b + c (a, b)
-+
+
in ‘p by setting (a, b)-(c, d) whenever there is a state and a relation x such that a + d + x = b + c + x . is an equivalence relation with respect to which It is easily shown that + and + are compatible. Henceforth equivalent elements in !@ are identified. In particular all processes of the form (a, a) are equal: denote any such process by 0. If CI is the process (a, b), denote the process (b, a) by - CI.Then 0 + a= CI and a+ (- a) = 0, and ‘p turns out to be an abelian group under + with zero element 0.
-
DEFINITION 2. CI isnaturalif a+O, antinaturalifO-+a,possibleifa-+O v O+a, reversible if a-0 A 0-m. It is irreversible if it is possible but not reversible, and impossible if it is not possible. The set of all natural (antinatural, possible, reversible) processes is denoted by ‘pN(’pA, pp,‘p,). It is easily shown that Ppand ‘p, are subgroups of ‘p.
DEFINITION 3. Given states a and b, if there exists a positive integer n and a state c such that (na + c, nb)E ‘ppwe write a c b (read “a is contained in b”). 4. A state e is an internal state if, given any state x, there DEFINITION exists a positive integer n such that x c n e .
AXIOM3. There exists an internal state. AXIOM4. Given a process a, if there exists a state c such that for any positive real number E there exist positive integers m, n and states x, y such that m/n<&,x c m c , y c m c and (x, y)+ncc+O, then a-0. Using methods from functional analysis Giles derives from these axioms : GILES’SMAINTHEOREM. There exists a positive additive function of state S, called the quasi-entropy, and a set of positive additive functions of state, called components of content, such that for any states a and b, a + b if and only if S(a)
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physical discipline of pure thermodynamics is accessible to a primitive observer in Giles’s sense. The significance of ‘positive’ in the Main Theorem is this: positive functions are bounded. DEFINITION 5. Define the norm IlalI of any state a relative to the internal state e by
The topology defined by the norm turns out t o be independent of the choice of e. DEFINITION 6. A real valued additive function Q(a) defined for every state a is bounded if there exists a constant k such that, for all a, /Q(a)l,
5. For reasons of space, we are obliged to refer the reader to Giles’s book [2] for an account of how the theorems are derived from the axioms. It will be found that what is done is in accord with the standards of rigour customary among contemporary mathematicians. It will, however, be clear that it is not just Giles’s explicitly stated axioms which are involved in establishing the theorems, but also inferential machinery of a logical and mathematical character not covered by the axioms themselves. That is also the situation in ordinary mathematics. But in a physical theory interpretational questions are involved in addition to purely mathematical ones. Giles deals with these interpretational questions by trying to establish some at least of the axioms as ‘true’ in some specific sense, which may be in need of further clarification, but which in any case necessarily calls into question the justifiability of the inferential machinery used. In other words the inferential machinery itself requires an interpretation. As his theory stands at present, there are at least two places where difficulties may be anticipated. One of these concerns the application of the axiom of choice in the proof of an extension theorem of the Hahn-Banach type which is used to extend an additive function on a subgroup to the whole group, and the other concerns an application of the law of excluded middle of a kind best illustrated by means of a quotation: “for each value of n (an integer) the process a1=(m,x, c)+m,n(a, b) is either natural or antinatural. Suppose first a1 is natural for an infinite number of values of n, then a-tb. If u1 is natural for only a finite number of values of n then it is certainly antinatural for an infinite number of values of n.
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Then a similar argument yields b+a”. This is reminiscent of the situation encountered in trying to prove that every rational sequence has a monotone subsequence. It seems that these difficulties can be satisfactorily resolved by making use of methods pointed out by Paul Lorenzen in his books [4,5] (see also [3]) where he sets up a form of constructive analysis which deviates only slightly from classical analysis as far as applications are concerned, and in which no axioms are assumed. If Lorenzen’s theory were axiomatised, then the axioms would be provable, because rules of interpretation could be introduced linking the primitive concepts with concretely specified actions in such a way that the axioms became true under these rules. Lorenzen does not, of course, express himself in these terms : he starts with inductive definitions which are used for defending prime propositions, out of which compound propositions can be defined by reference to obligations taken on in asserting them; then abstractions are introduced, and so on. As a matter of fact not all the formulae and theorems of Lorenzen actually receive an interpretation : instead the use of classical logic is justified by means of a consistency proof. This means, for example, that the use of the law of excluded middle is accepted as a fiction, but as a fiction which provably does no harm. If we grant that the primitive observer in thermodynamics has ‘established’ the Giles axioms as true, and that no more than a countable number of states can come into consideration, so that the use of the axiom of choice is avoided, then there would seem to be no difficulty in completing the Giles theory as a constructive theory in Lorenzen’s sense simply by adjoining Lorenzen’s considerations to those of Giles. However, it seems to the present writer that Giles’s primitive observer, although having only meagre-looking powers, has some which transcend those of any human observer. For example he can tell of any pair of states a and b whether a + b or not. Accordingly the proposition a - t b becomes truth definite [4]: there is a procedure which, when applied to the proposition, yields one and only one of two truth values, true or false. However, for any human observer all we are entitled to say is that the proposition a-tb is at most proof-definite [4] i.e., there exists a procedure which, when applied to another procedure which is applied to this proposition, yields a decision as to whether the second mentioned procedure constitutes a proof of a + b or not. What is required for establishing the truth of a + b is the following. We have to find a state k and a time interval z such that a + k evolves in time z into b + k. If we have found a suitable state k and time interval z so that this happens, then we have established that a+b; but if we have not, we have not shown that a + b is false. Perhaps further
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efforts to find suitable k and z will be successful, so that a + b after all. We can also describe the situation in terms of systems and apparatus like this: the system A can be in states A, and A,. Starting with system A in the state A, we place it in interaction with a piece of equipment K (the apparatus) which is in the initial state K,. This represents the start of an experiment on A. We then wait a time z,with A + K kept isolated, and at the end of that time we separate A and K so that both A and K, individually, are isolated. If A is now in the state A2 and K is again in the state K,, then we shall have shown by experiment that A,+A,. In any other case all we can say is that we have failed to decide whether A,-tA, or not. All that we have said so far about the problem of deciding about a-tb presumes that there are no problems connected with the notion of equality of states; but there are difficulties here as well, and these difficulties have repercussions on the rule of interpretation for a + b too. Expressed more fully, this rule would read: a-tb if there is a state k and a time interval z such that the state whose method of preparation is “apply simultaneously and independently the methods of preparation corresponding to a and k and then wait for a time 7’’is indistinguishable from the state whose method of preparation is “apply simultaneously and independently the methods of preparation corresponding to b and k” in the sense that any experiment applied to these states will yield the same result (or rather the same statistical distribution of results) in each case. Furthermore a consideration of microscopic systems suggests that, strictly speaking this can only apply to experiments which are concerned with only one of the parts of the state and which do not look for a correlation between these parts. Since the methods of preparation are independent there will be no such correlation for b + k ; but in the state evolved from a + k there will generally be some correlation - for instance a high energy for one part will tend to be accompanied by a low energy for the other. Thus in the above rule of interpretation, the phrase “into the state b + k” in the original formulation: ‘a-tb if there is a state k and a time interval z such that a + k evolves (in isolation) in the time z into the state b + k’ should strictly be replaced by “into a state which, ignoring the correlation between the parts, is indistinguishable from b + k”. In the model furnished by statistical mechanics it is just this procedure of ignoring correlation which accounts for the irreversible nature of thermodynamic processes. We see then that we are confronted with the problem of deciding whether or not two states are equal even when we know that they have arisen in quite different ways. If two states were counted equal only when their methods of
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preparation consisted of identical actions there would be no difficulty, but the foregoing discussion shows that we need a wider notion of equality than this: two states are still to be identified when no subsequent experiments which may involve deciding on the truth or falsity of propositions of the form a+b, once more - can reveal a difference. Thus some propositions a = b will only be refutation definite, i.e. there will only be available a decidable refutation concept. To put this another way, propositions of the form a # b, but not a = b, will be proof definite. This seems to threaten propositions of the form a + b with not even being proof-definite, and thus in turn to threaten propositions of the form a = b with not even being refutation definite. The same applies to propositions of the form a = b + c : if a is defined as b + c in a particular context, all is well, but if it is a question of deciding whether a state b + c is equal or not to an independently specified state, then tests of the form a+b may again be needed, and this proposition may be neither proof nor refutation definite. One way out of this dilemma would be to try to make equality truth definite by replacing the rule of interpretation which has been given for ‘state’ by another one. There is also a possibility of retaining equality as refutation definite and trying to make a + b dialogue definite [4] - and managing with that. We prefer however, at this stage, to examine the first of these possibilities, since an issue is involved which appears elsewhere in the natural sciences, and which seems very difficult to eliminate. This issue concerns the formation of policies for future behaviour. On the basis of past experience we decide to act in a certain way, but as a result of further experiences which have come about as a result of those actions, we may decide that we ‘should’ have performed other actions instead. That is, our present policies may be revised as a result of further experiences. This train of thought seems to suggest that we are concerned with ‘states of mind’ - a change of state of mind having occurred when a policy is changed. We now reformulate the revised rule of interpretation for ‘state’ as: a state is a state of mind induced by the available knowledge of a method of preparation. In formulating this rule of interpretation we have acknowledged the presence of a subjective element, and one of our tasks will be that of showing how conclusions can be reached which d o not depend on the personal peculiarities of those whose states of mind are referred to. Two ‘states’ are now to be regarded as equal just when the associated states of mind are the same. Let us imagine that two states a and b have arisen in connection with possibly different methods of preparation. Experiments are now carried out on these states and in every case the results
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BURTON
cannot be distinguished. Possibly, if further experiments were to be performed, differences would come to light. However, it may be presumed that the experiments will not be continued indefinitely; a point will come at which the experimenter says that all his efforts to distinguish a and b have failed, and that as far as he can see, he has reached a stage at which his further efforts are disclosing no fresh information. His state of mind is that a and b are equal. He is not at all saying that a and b are identical - he may well agree that further experiments may disclose differences which his experiments have not disclosed so far. By saying that a and b are equal he is merely saying that whatever predictions he is prepared to make about a he is prepared to make about b and conversely; if he were a betting man he would be prepared to lay exactly the same odds in each case. In saying that equality is truth definite we mean simply that the experimenter can decide, yes or no, whether his state of mind about a is the same as his state of mind about b, in so far as his predictions about a and b in future experiments are concerned, With this altered interpretation of state, a = b and a + b = c become truth definite, and a-b proof definite. 6. We return now to Giles’s axioms and ask to what extent can they be justified. Axiom l(i) runs: if a, b E 6 then a + b E G , a + b = b + a , and if a, b, CEGthen a+(b+c)=(a+b)+c. Thus the set 6 of states is required in particular to be closed under the operation . In a constructive theory the set G will not be ‘given’ in advance, but will have to be constructed from the states which will constitute its elements. We start with various states a, b, c, ... and declare them to be in 6. If a and b are in 6, then the rule of interpretation for shows that a + b is a state: accordingly it is put into 6 as well. Thus the set 6 is closed in a sense entirely analogous to the sense in which the set N of numerals is closed in Lorenzen’s arithmetic. In this case we have rules for producing numerals. -1 ii+nl.
+
+
Each figure, as it is produced, is ‘put’ in N. Just as the numerals 1, 11, 111, . .. are ‘generated‘ by these rules, so are the states generated from an initial stock of states by the operation of taking unions. The axioms a + b = b + a and a + (b +c) =(a + b) + c are again justified by the rule of interpretation for +. For example both a + b and b + a refer to the simultaneous and independent performance of the methods of prepa-
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ration associated with a and with b. The fact that a comes before b, or b before a, in a linear arrangement of symboIs is something which is forced on us only by the type of notation we have selected. Similarly in the case of the associative axiom. Thus we have ‘justified’ Axiom 1(i). The justification for Axiom 1 (ii) is found by reference to the rule of interpretation for +. If we take r=O in that rule then for a + a to be true we have to find a state k such that a k evolves in zero time into a + k. Any state k will do for this purpose. contains logical particles as Axiom 1 (iii), namely a + b A b+c*a+c, well as Giles’s primitive terms. If we interpret these operatively [3] or in terms of dialogues [4], then Axiom 1 (iii) would be justified by giving a winning strategy for it. Whoever asserts this axiom is obliged to assert and defend a 4 c if an opponent is prepared to assert a-+b and b+c and can successfully defend them. NOWa+ b and b+c are proof definite. Introducing temporarily the notation 7to mean “evolves in isolation in time z into”, proofs of these propositions involve establishing, with suitable k, and k, and r1 and r,:
+
+ k, 2 b + k,
(a)
a
(b)
b+k,;;tc+k,
and we can tell whether or not the opponent possesses proofs in this sense. If he does possess such proofs, the proponent now attempts to prove a+c by using the opponent’s (a) to produce the intermediate state b from a and then using the opponent’s (b) to produce c from b, so establishing a +k T c
(4
+k
for suitable k and r. The question now, however, is: what is the state k? Intuitively, the opponent may have employed two quite separate pieces of apparatus K(’) and K(’) with initial and final states k,, k, and k,, k, respectively. What should the proponent’s apparatus be? The most obvious suggestion is K(”+K(’); calling this K, it is a matter of arranging that the initial and final states of K be some state k. Since the opponent has produced states k, and k, for K(’) and K(’) respectively, this suggests that k should be k, + k,. The proponent would then try to arrange that a and
+ k , + k, 2 b + k, + k,
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W. K. BURTON
so that he finally achieves (c')
a
+ k , + k2y-c
+
+ k, + k, ,
+
giving (c) with k = k, k, and t = t1 z2. But in order to do what is required of him, the proponent must arrange that in stage (a') only K(') interacts with the system under consideration, and in stage (bl) only K(') interacts with the system, because only that has been done by the opponent. Thus in order to succeed it looks as if the proponent must have the ability to maintain an arbitrary state constant - to freeze it, as we shall say. That is, in stage (a') the state k, has to be frozen while in stage (b') k, has to be frozen. So it appears that we must introduce a hypothesis - namely that an arbitrary state can be frozen - in order to be able to defend Axiom 1 (iii) hypothetically. This state of affairs certainly falls short of what we originally had in mind. It means that there will be an inherent restriction on the domain of applicability of the theory, the precise nature of which requires further elucidation. We are faced with the problem of distinguishing on the practical level whether or not we are confronted with a situation in which Axiom 1 (ii) is 'true'. There is, however, a possible way out of this difficulty: to modify the rule of interpretation for + once more. We define a new relation ias follows:
(D,)
a i b % a + b v V,V,
,,..., c,(a,+cl
A C ~ + C , A
... A C , - + ~ )
where the c,(i= 1, ..., n) are intermediate states. Under this definition a + b is still proof definite, and the transitivity law a i b
A
bic*aic
and Axiom 1 (i) is holds. Axiom 1 (ii) still holds with + replaced by i not affected by the replacement of + by 4. Thus with the new rule of interpretation for + (namely -+ means i with i defined by (D1) in which the '4' on the right is interpretated by the old rule) we have established that 6 is a partially ordered semigroup, i.e. we have proved Axioms 1 (i)1 (iii). The remaining clause of Axiom 1 amounts essentially to two axioms, namely a+c+b+c*a+b a+b +a+c+b+c.
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The first of these is established on the grounds that the state c can be considered as part of the state k in the rule of interpretation for -+ in either the new or the old sense. But the second axiom gives rise to the same difficulty as was encountered in connection with Axiom 1 (iii) : it asserts that the state c can be frozen, and there seems to be no obvious way out of the difficulty in this case. This axiom then, as well as Axiom 2, as it turns out, has to be treated differently from the others. So our axioms will be of two kinds: the ones which can be effectively defended in “all”cases, and those which require for their defence an additional hypothetical element which restricts the defensibility to those cases for which some method is available which shows the axioms in question to be ‘true’ in those cases, even if not in general. We refer to the first group of axioms as prototheoretically defensible and to the second group of axioms as hypothetically defensible. As far as the question of a state’s being freezable is concerned we can often, in practice, achieve what is required by the use of suitable apparatus, possibly in a rather far-fetched sense. For example if we had a sample of gas of non-uniform temperature and pressure as an instance of a state k, we could freeze k by ‘instantaneously’ inserting a large number of insulating partitions which would serve, approximately, to freeze k. If we were concerned with a solid rather than with a gas this would scarcely be feasible. One would have to call on the possibility of dividing up the solid into small pieces each of which is to be insulated from the others. At a later stage when the freezing is to cease the solid would be reassembled. What really matters here is that the freezing can be done ‘in principle’. The conviction in each case can only be arrived at by a consideration of the case in question. Usually the difficulty is not too pressing in practice. The situation for Axiom 2: a-+b~a+c*b+cvc+b is quite different. As the discussion given by Giles in his book shows, this axiom is not true for systems exhibiting hysteresis. So if Axiom 2 is to be justified at all, it must be justified by reference to tests carried out on the systems under consideration themselves. Each instance of Axiom 2 regarded as a proposition is proof-definite but not truth definite. After preliminary trials an experimenter may come to the conviction that Axiom 2 is true; but if he is mistaken in this connection, he will find that when he tries to get upper and lower bounds for the entropy, he will always find a finite gap between them, however many experiments and whatever experiments he performs. We
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might say that when Axiom 2 is ‘false’ only an upper and lower entropy (cf. outer and inner measures) is available, whose difference measures the degree of hysteresis exhibited in the system. So with Axiom 2 we are confronted not with a proper axiom at all; rather we might say that a system for which Axiom 2 is true is defined to be hysteresis-free, and that the problem reduces to the practical problem of recognising hysteresis-free systems. Axiom 2 then has ‘physical content’ in a much stronger sense than Axiom 1, and is decisive in settling the detailed structure of the theory.
7. The motivation underlying Giles’s Axioms 3 and 4 comes from the desire for boundedness in the entropy and components of content, and thus is mathematical rather than physical. Since we are now trying to replace Giles’s theory by a constructive one, these axioms may be expected to appear in a rather different light. In particular, even though the set of states 6 will in general be (potentially) infinite, in any given case 6 will be generated from a finite number of basic states by taking unions. In such a case the state which consists of the union of all the basic states will constitute an internal state in the sense of Definition 4. So Axiom 3 will be justified for that case. However, it is not possible to choose an internal state once and for all in this way, for it may be possible to enlarge 6 by enlarging the set of basic states which generate 6. If this is done our procedure would give rise to a new internal state. The use of this new internal state instead of the old one would, however, do no more than induce scale changes in the entropy and components of content. Thus in a constructive reformulation of the Giles theory we may perhaps regard the justification of Axiom 3 as unproblematical. The justification of Axiom 4 is then achieved by means of a further modification in the rule of interpretation for -+.If we define a>b+ there is a state c such that for any positive real (D,) number E , there are positive integers m, n and states x, y such that m/n
CONSTRUCTIVE THERMODYNAMICS
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As an ordinary physicist would no doubt express it, Axiom 2 stands out as ‘containing most of the physics’. It is somewhat remarkable that such a rich looking theory as thermodynamics should be supportable on such a meagre looking base. It is seen that while the Giles theory appears to suggest the feasibility of a fully constructive thermodynamics, and perhaps also of other physical theories, further work is required on the interpretational side (which may well lead to modifications in what appear above as axioms) before we shall be in a position to offer such a theory as a fitting companion to, say, Lorenzen’s constructive analysis. Acknowledgements It is a great pleasure t o thank Professor K. Schutte for his kind hospitality when I spent a year (1964-65) in his Department in Kiel, where the above work was started. I thank him also for the numerous illuminating discussions which clarified my own thinking considerably. It is a pleasure to acknowledge in the same sense the benefits accruing from discussions with Professor R. Giles. Finally I thank the Carnegie Trust for the Universities of Scotland and the then Directorate for Scientific and Industrial Research for Travel and Maintenance Grants, and The University of Glasgow for making my study in Germany possible. References 1. H. B. CURRY,Foundations of mathematical logic (McGraw-Hill, 1964) p. 48. 2. R. GILES,Mathematical foundations of thermodynamics (Pergamon, 1964). 3. P. LORENZEN, Einfiihrung in die operative Logik und Mathematik (Springer, 1955). 4. P. LORENZEN, Metamathematik (B. I. Hochschultaschenbiicher, 1962). 5. P. LORENZEN, Differential und Integral (Akademische Verlagsgesellschaft, 1965).