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SCIENCE INFECTED WITH INCONSISTENCY
Pentcho Valev
Abstract
A theory can be regarded, in the abstract, as a set of propositions. If there are circumstances, either realistic or unrealistic, under which all the propositions in the set are true, the theory is consistent. If there are no such circumstances - e.g. the set contains the propositions "Henry is married" and "Henry is not married" - the theory can be called an inconsistency. An inconsistent theory is extremely dangerous since, somewhat paradoxically, it may possess a great deductive power and produce breathtaking conclusions. Moreover, if it occupies scientists’ minds for a long time, reasoning in terms of inconsistent propositions becomes "normal", the logical procedure reductio ad absurdum stops working and returning to a consistent conceptual framework proves impossible in the end. Special relativity and thermodynamics are perhaps the most successful inconsistencies ever created.
What would happen if a deductive theory proved inconsistent? Imagine that in special relativity, the postulate of constancy of the speed of light and the postulate of equivalency of frames contradict one another. Or that the 7 - 8 traditional statements of the second law of thermodynamics (the "less traditional" are perhaps more) are not equivalent. So what would happen? Nothing, if inconsistency had already become an essential feature of deductive science. Such a degeneration of the deductive approach is by no means improbable. Unlike induction which is firmly grounded in empirical reality, deduction is in a sense groundless (depends on a priori assumptions) and this makes it extremely vulnerable. Any foreign body - e.g. a "small" inconsistency that an author may have dropped and then forgotten inside the deductive construction - can eventually destroy the immune system of the theory so that the latter becomes susceptible to any inconsistency. The process is similar to an HIV infection.
In quest of symptoms of inconsistency, let us remember the famous twin paradox in special relativity. The twin coming back from a journey proves younger than his brother. "Einstein maintained the greater youthfulness of the travelling twin, and admitted that it contradicts the principle of relativity, saying that acceleration must be the cause" [1]. Here a non-infected mind sees two possibilities. Either, during the journey, acceleration is the cause of time dilation - then special relativity wrongly predicts time dilation for non-accelerating frames, or acceleration is not the cause of time dilation - then the principle of relativity (the postulate of equivalency of frames) is contradicted.
This (non-infected) argument naively presupposes that special relativity would not tolerate a conceptual framework in which acceleration both is and is not the cause of greater youthfulness. However, if the theory is essentially inconsistent (infected), it would tolerate inconsistencies because in fact it feeds on them. If acceleration is the cause of greater youthfulness special relativity would be unhappy. If acceleration is not the cause of greater youthfulness special relativity would be equally unhappy. Only the infected conceptual framework in which acceleration, like Hamlet, oscillates between "to be the cause" and "not to be the cause" makes special relativity perfectly happy.
One can also find symptoms of inconsistency in thermodynamics - another major deductive theory (at least it emerged as a deductive theory). Originally, entropy had been defined through dS=dQrev/T where dQrev is a small amount of heat absorbed by a system and the subscript "rev" ("reversible") implies that the system passes through a succession of equilibrium states. However, about a century ago, someone found it suitable to apply the self-same dS to states distant from equilibrium and this initiative gave birth to chemical thermodynamics. Imagine some non-infected mind desperately trying to fight the sprouting inconsistency: "Dear colleagues, since you define dS for a succession of equilibrium states please do not apply it to a succession of non-equilibrium states. The inconsistency destroys any rationality in your students’ minds." Any such wail must have been useless because what non-infected science sees as an inconsistency infected science sees as a norm.
Let us try to describe an inconsistency in general logical terms. A deductive theory can be presented, in the abstract, as a set of propositions. A proposition is either a premise (axiom, postulate, principle) or a conclusion (theorem) derived from the premises. Introducing the generic name "proposition" is appropriate since the distinction between premises and conclusions is not absolute – premises in one approach are conclusions in another. What logic strictly requires is that the set of propositions be consistent. An inconsistent set of propositions (briefly called an inconsistency) is one for which there are no circumstances, either realistic or unrealistic, that make all the propositions in the set true simultaneously. For instance, the following three propositions form an inconsistency:
John is rich.
All rich men are happy.
John is not happy.
It should be noted that, in characterizing the set as consistent or inconsistent, one ignores the actual truth or falsity of individual propositions. The second proposition above is most probably false but this is not the reason for declaring the set as inconsistent. In
John is rich.
All rich men are happy.
John is happy.
the second proposition remains false but now the triad is consistent.
An inconsistent theory may be breathtaking since, in a strict logical sense, anything at all can validly be derived from it. The following example is grotesque but clearly illustrates the point. Consider the two-proposition "theory":
Henry is married.
Henry is not married.
Since the propositions form an inconsistency, logic allows us to attach any conclusion to them:
Henry is married.
Henry is not married.
Therefore crocodiles like opera.
Any logician would agree that the argument is valid since under no circumstances can both premises be true and the conclusion false.
The effect the above example demonstrates is typical of any inconsistent theory. We must face the paradoxical fact that the interaction of inconsistent propositions increases the deductive power of the theory. A consistent theory usually offers a limited number of deductive paths which soon get depleted. In contrast, an inconsistency creates a large variety of paths which are artifacts but which nevertheless produce breathtaking conclusions. As a result, the inconsistent theory may become so fashionable that consistent but "dull" alternatives don’t even have a chance to be noticed.
Let us return to special relativity. Is it really an inconsistency? Its presentation often starts with a thought experiment involving two asymmetrical reference frames. Light emitted by a source on the floor of a train travels up to a mirror on the ceiling and then back down to the source, the roundtrip representing a "tick" of a light clock. According to an observer on the ground, the path of the light is relatively long since it is oblique with respect to the motion of the train (upper part of the picture):
In contrast, in light clocks on the ground, the path of the light is vertical all along and therefore shorter (lower part of the picture). If, according to the observer on the ground, the speed of light is the same along any path (this is a postulate in special relativity), the light on the train has made a smaller number of roundtrips than the light on the ground and therefore the clock on the train advances more slowly than clocks on the ground.
This conclusion is relevant for a situation in which the observer on the train has a single clock and measures no proper distance whereas the observer on the ground deals with at least two synchronized clocks at separate places and the distance between them is proper for him. Now remember that special relativity is based on the postulate of equivalency of frames according to which the observer on the ground sees things on the train in exactly the same way that the observer on the train sees things on the ground. That is obviously not the case as far as the setup is concerned. One would be right to suspect that the conclusions, contradictory or not, are artifacts resulting from the inconsistent application of the postulate of equivalency of frames. This suspicion may be followed by a hypothesis: If the principle of equivalency of frames were applied consistently, that is, if only symmetrical experiments were done, perhaps no artifacts would be possible and the theory would become falsifiable.
Let us see if this hypothesis is reasonable. Consider the way in which textbooks check whether there is a transverse length contraction (in addition to the longitudinal length contraction). Let two meter sticks, A and B, move past each other in the following way:
A has paint brushes at its ends pointing towards B so that if B is long enough (or A is short enough) then the brushes will leave marks on B.
Let us assume that A sees B short. Then B won’t reach out to the ends of A and there will be no marks on B. However, in accordance with the postulate of equivalency of frames, B must also see A short so there will be marks on B. The contradiction shows that the hypothetical transverse length contraction is inconsistent with the postulate of equivalency of frames and one of them should be rejected. It seems that symmetrical experiments do indeed make the theory falsifiable.
Let two sticks move towards one another with uniform relative velocity in the following way:
The proper length of B is L, that of A is L + X, where 0 < X < L(γ – 1).
As the end A1 reaches the end B2, it bumps into B2 and the two ends remain attached so that the sticks continue their existence as one body in a single frame. The question is: will A2 reach B1? It is easy to show that, if there is length contraction, the answer of B is yes – before the collision, B sees the length of A shorter than its own. In contrast, the answer of A is no – A always sees the length of B shorter than its own. We have an event – the meeting between A2 and B1 – which, if length contraction is a true prediction, both occurs and does not occur in the considered system.
In another example, the scenario is the same except that this time the proper lengths of A and B are equal. If there is length contraction, B will see A2 sweeping a segment of B (the left end of B) two times – before and after A1 bumps into B2. But B may have pawls on that segment preventing A2 from sliding on the surface of B to the left (the sticks almost touch one another so the pawls are efficient). This leads to a contradiction: on one hand, B sees the pawls keeping A2 in the most right position and thus preventing A from restoring its proper length. On the other, B must see A restore its proper length after the collision. Again, we have an event – A restores its proper length – that occurs and does not occur in the considered system.
Let us shift to thermodynamics where the inconsistent constructions are much more developed (after all, thermodynamics is older than special relativity). The discussion here will be restricted to the emergence and development of chemical thermodynamics. (Other problems concerning the inconsistent nature of thermodynamics are discussed in [2].)
Classical thermodynamics has started with a phenomenological analysis of the way in which a (Carnot) machine converts heat into work. But what does "phenomenological" mean? Since it may mean different things to different people, here is a definition relevant (in my view) to the original thermodynamic methodology. An approach will be called phenomenological if it is essentially independent of any assumptions about the internal constitution or dynamics of the considered (macroscopic or microscopic) system. One regards the system as a "black box" and deals with a specific set of quantities determinable at the boundary of the system. Any knowledge about the interior may be useful but is not indispensable by definition. The phenomenological approach characterizes the "behaviour" of the system, i.e. its functional relations with the surroundings, in contrast with approaches delving into the internal organization. (Similarly, in mathematics the category theory defines a set in terms of its external relations with other sets, in opposition to the set theory which studies the internal characteristics of the set [3]. This conceptual dualism is widespread. The two approaches, "external" and "internal", are complementary on one hand but self-sufficient on the other.)
Is today’s thermodynamics a phenomenological science? The following quotation is from a physical chemistry textbook ([4], p. 164):
Proving a result "thermodynamically" means basing it entirely on general thermodynamic relations and equations of state, without drawing on molecular arguments (such as the existence of intermolecular forces).
An approach to chemical systems that ignores molecular arguments does have some chance to be phenomenological or at least it is difficult to see how else it could be characterized. Yet the quotation is misleading since, in chemical thermodynamics, equations of state vitally depend on molecular arguments. The fundamental equation of chemical thermodynamics reads
dU = TdS - PdV + Σi mi dni (1)
where ni is the amount of the ith component contained in the system and m i is the respective chemical potential. (The other symbols have their usual meanings). Clearly, the terms m idni presuppose a dissection of the studied system into molecular constituents. Other terms (TdS and PdV) are phenomenological. So the right side of (1) can be likened to a grotesque centaur with a phenomenological body (TdS - PdV) and a non-phenomenological head (Simidni). This suggests that chemical thermodynamics must have deviated from the original phenomenological approach. When and why did that happen?
Initially, J. W. Gibbs introduced eq. (1) by assuming that the system is in internal equilibrium and that the amounts of components n1, n2 etc. are independent variables. If these assumptions are fulfilled, any change dni can only be the result of the ith component crossing the boundary of the system (i.e. dni is a small amount added to or removed from the system). This means that for an observer positioned at the boundary dni is a phenomenological quantity - like dS or dV, dni can be determined in the absence of any knowledge about the internal chemistry. The observer may even brake any link with the internal chemistry by replacing the original definition of ni, "amount of the ith component contained in the system", with "amount of the ith component added to the system". So a different, phenomenological variable is introduced. The point is that its change, dni, coincides with the change of the original variable as introduced by Gibbs. And since it is dni and not ni that matters, altering the definition of ni is justified. In fact, Gibbs’ assumptions implicitly make equation (1) phenomenological although the original definition of ni as "amount contained in the system" leaves a different impression.
Why have those assumptions been abandoned? Perhaps because systems obeying Gibbs’ assumptions are too simple from a chemical point of view. Since the variables ni are defined as independent, no chemical reaction between the respective components is allowed to take place. Understandably chemists would be much more interested in a situation where some of the components, e.g. the reactants A and B, are being converted into other components, e.g. the products C and D. However the respective system is not necessarily in equilibrium and the components are not independent, i.e. neither assumption of Gibbs is fulfilled. Yet the application of the fundamental equation has been extended so as to cover non-equilibrium chemical systems as well, in the hope that the terms in eq. (1) remain relevant.
Do they? The entropy change dS, before entering eq. (1), had been defined by dS=dQrev/T which means that it is only relevant for a system passing through a succession of equilibrium states. If the system passes through a succession of non-equilibrium states (as may be the case when the reaction A+B® C+D takes place), applying dS=dQrev/T is unreasonable. How can a quantity be explicitly defined for equilibrium changes and then again explicitly applied to non-equilibrium changes? If the inconsistency is so obvious, why has the fundamental equation been satisfactory to everybody for a century? The following quotation ([5], pp. 72-73) may provide part of the answer:
We regard equation (1) [eq. (1) referred to in this quotation and eq. (1) in the present paper are essentially identical] as an axiom and call it the fundamental equation for a change of the state of a phase a. It is one half of the second law of thermodynamics. We do not ask where it comes from. Indeed we do not admit the existence of any more fundamental relations from which it might have been derived. Nor shall we here enquire into the history of its formulation, though that is a subject of great interest to the historian of science. It is a starting point; it must be learnt by heart. It may be allowed to stand as an axiom until any single one of the host of equations that can be derived from it (with the help of other axioms of thermodynamics) has been shown experimentally to be false.
This ideology has been borrowed from logic. It allows one not to care about the truth or falsity of premises (axioms, postulates) until some of the conclusions prove false. However this logical liberalism has its dark and very dangerous side. If the premises form an inconsistency, the self-same liberalism becomes so devastating as to prevent any rational activity in the affected domain. I have already given some discussion to this.
Is the fundamental equation the result of an inconsistency? It is. Detailed discussion of the problem is beyond the scope of this paper but here is just one of the metastases, in the form of a couple of contradictory premises:
The system is not allowed to do non-expansion (non-PV) work.
The system is allowed to do non-expansion work.
On introducing the fundamental equation, some authors advance the former premise (see [6], p. 157 and [7]) whereas others advance the latter ([4], p. 172). Nevertheless, since the theory is essentially inconsistent, both premises are operative in any textbook so the confusion surrounding the concept of chemical (non-expansion) work is inextricable. Still I am going to describe an unrealistic scenario in which a shrewd student partially disentangles the confusion. (A realistic scenario would involve the student learning everything by rote).
Assume that, initially, the student bumps into the following text ([4], p. 172):
We saw in section 4.8 that, under the same conditions, dG = dWe,max. Therefore,
dWe,max= Simi dni (at constant p, T)
That is, non-expansion work can arise from the changing composition of a system that is not at internal equilibrium.
The student returns to section 4.8 ([4], p. 150) and reads:
… a system does maximum work when it is working reversibly.
Something is amiss. "A system that is not at internal equilibrium" can by no means work "reversibly" for the simple reason that "reversibly" implies that the system passes through a succession of equilibrium states. The student decides to resolve the problem by starting from the very beginning - from the definition of work. So he goes to p. 56 in [4] and reads:
A process that does work is one that could be used to bring about a change in the height of a weight somewhere in the surroundings……A chemical reaction that drives an electric current through a resistance is also an example of a process that does work because the same current could be driven through a motor and used to raise a weight.
It seems that the concept of chemical work can become clearer if electric motors are better understood, so the tenacious student goes to some physics textbook and looks for more explanation. There he sees a wire stretched between the poles of a magnet, perpendicularly to the magnetic field. As a current passes through the wire, a force pushes the wire in a direction perpendicular to both the wire and the magnetic field. The magnitude of this force is
F = BLI (3)
where B is the magnetic field strength, L is the length of the wire and I is the current. Essentially, this is the working force in electric motors - it can go to a pulley and cause the lifting of a weight.
The student realises that, if the wire connects the electrodes of an electrochemical cell (battery), the model of a chemical system doing work is complete. On one hand, there is a chemical reaction in the cell that drives the electric current through a wire. On the other, since the wire is suitably positioned in a magnetic field, the current generates a force which can be harnessed to lift a weight. The student even knows the magnitude of the work-producing force - it is given by eq. (3).
But something is amiss again. The system is not able to work reversibly and even to approach a state engaged in reversible work production. The force, F, is proportional to the current, I, which in turn is proportional to the reaction velocity Vr. (Both I and Vr reflect the amount of reactants converted into products per unit time.) So, in order for F to have a finite value and to be able to raise any weight, Vr must have a finite value too. This can only be the case if the chemical reaction is distant from equilibrium. The student has discovered a principle which contradicts the conventional wisdom according to which any work has a reversible maximum. A system distant from equilibrium does essentially irreversible work and this type of work production has no reversible maximum. States engaged in reversible work production cannot even be approached.
This example clearly demonstrates the difference between an inconsistent and a consistent theory. The latter would regard the statements "The system passes through a succession of non-equilibrium states", "The system is working reversibly" and "Working reversibly implies that the system passes through a succession of equilibrium states" as forming a silly combination that could only appear by virtue of mental aberration. In contrast, such combinations are seminal for an inconsistency. On the other hand, important and even obvious principles, such as the principle of essentially irreversible work production, may remain outside the conceptual framework of the inconsistent theory.
Since deductive science has been inconsistent for a very long time and reasoning in terms of inconsistent propositions has become "normal", efficient counteraction seems impossible for the moment. The logical procedure called reductio ad absurdum simply does not work. Let us return to the question advanced at the beginning of the paper. Assume one has derived both A > B and B > A from the same set of axioms. What would be the result? Rejection or glorification of the theory? I think the latter would be the case. Perhaps I am too pessimistic but I will remain so until that distant morning when I will see the following text written by some mainstream authority: "There are 7 - 8 second laws of thermodynamics and nobody cares whether they are equivalent or not. This situation cannot be tolerated anymore".
This will never happen because the inconsistent development of thermodynamics has already finished – all possible successful carriers have already been built and all the profits extracted – and now thermodynamics is just "obsolete" [8]:
«In the eyes of many modern physicists, the theory has acquired a somewhat dubious status. They regard classical thermodynamics as a relic from a bygone era… Indeed, the view that thermodynamics is obsolète is so common that many physicists use the phrase ‘Second Law of Thermodynamics’ to denote some counterpart of this law in the kinetic theory of gases or in statistical mechanics»
Similar symptoms can be detected in the relativity theory – one can often hear that special relativity cannot resolve the twin paradox (as if the calculation of the youthfulness of the traveler were not based on special relativity arguments) but general relativity can. One thing is sure - the inconsistency is an inexhaustible source of goods for science authorities.
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