If a way can be found, says de Selby, of discovering the ‘second direction’, i.e., along the ‘barrel’ of the sausage, a world of entirely new sensation and experience will be open to humanity. New and unimaginable dimensions will supersede the present order and the manifold ‘unnecessaries’ of ‘one-directional’ existence will disappear.

It is true that de Selby is rather vague as to how precisely this new direction is to be found.

— Flann O’Brien, The Third Policeman

String theory has received much attention as a consistent theory of quantum gravity, but criticism has been levelled at string theorists for an overemphasis on string theory as a unifying theory, and more especially at what some have viewed as the emphasis on the mathematical beauty of the theory over its predictive power [1]. In particular, since string theory is considered to fail to meet the Popperian falsifiability criterion for a scientific theory, some critics maintain that research efforts should be diverted away from it towards areas of physics with more immediate testability and application.

While the study of string theory has had positive outcomes for the development of science – see, for instance, much of the active debate on the black hole information paradox  [2,3] – and in particular for the development of mathematics – such as  [4] – its failure to provide predictions at currently workable energy scales raises questions for where it falls when one considers different answers to the demarcation problem.

All but the most zealous critics of the framework are unlikely, at a push, to brand string theory as a pseudoscience, and most critiques will usually still allow that some kind of consistent theory is required. While, for instance, Hossenfelder provocatively asks whether there is even a need for a ’theory of everything’, she maintains that there is no question whether the laws of physics should, in principle, be consistent [1]. It is interesting, however, to ask what advantages, if any, string theory has over simply admitting that general relativity and quantum field theory are exactly correct theories and yet irreconcilable.

Throughout most of the development of modern physics, logical inconsistency between mainstream theories has been a mainstay—one need look no further than the contradiction between Maxwell’s equations and Galilean relativity [5,6]. It would be an unuseful attitude at these points in history to ‘unanswer’¹ the problem of reconciling them by asserting that there is no need to – but would it necessarily be unreasonable? Much work has been done in the late 20th century to rigourously describe paraconsistent logics. In general a paraconsistent logic is a nontrivial system in which the statement $(P\land\lnot P)$ does not imply all other possible statements – that is, the principle of explosion cannot be invoked for these systems. Like in string theory, there is a sound mathematical framework for treating such nonexplosive logics; there are a large number of possible compactifications for the Calabi-Yau manifold [8], allowing for a myriad of string theories consistent with the facts, just as there exists a taxonomy of known classes of paraconsistent logics; and there are many problems in which a theory of quantum gravity is needed, just as there is a breadth of physical examples in which paraconsistency must (implicitly or explicitly) be invoked [5]. Inasmuch as one might attack an ‘unanswer’ for its ambiguity and non-falsifiability, as a ‘mathematical framework’ it would seem to possess an advantage over string theory in its historical basis and universality.

We would be suspicious of this ‘unanswer’ as being some combination of avoidant scepticism and intellectual laziness. “What is true in logic,” writes Popper, “is true in psychology … in scientific method, and in the history of science” [9]. Science and mathematics are usually considered to be predicated on logic, and an assumption that physical laws are logical is almost a tautology. Invoking paraconsistency to solve a problem would be, according to Quine, “changing the subject” [10]. Yet inconsistency nevertheless falls out of even simple, intuitive ideas like those in naïve set theory, and steadfast adherence to nonexplosive formal systems thus leads us to accept sets of axioms or axiom schemata which are not necessarily intuitively correct, citing reasons such as simplicity, effectiveness at trivially proving theorems, or that they formally reproduce facts which we do consider true from experience, such as the validity of Peano arithmetic [11,12]. Furthermore, a notion in one theory for some percept may not be compatible with the corresponding notion in another. Both the concept and mathematical formulation of space and time in Galilean relativity vs special relativity, for example, are not necessarily commensurate [13] with each other – and neither, for that matter, are the notions of time in GR and QFT. Do these not also ‘change the subject’ with respect to one another?

What motivation have we for choosing nonintuitive axioms in order to reproduce experience, but disavowing nonintuitive formal systems to reproduce the experimental observation? Outside of black holes, whose interiors we are prohibited from examining anyway, the standard model and general relativity are exquisitely experimentally accurate – and yet gravitation cannot be renormalised in quantum field theory [14]. Even simple calculations involving quantum field theory in curved spacetime (an approach which leaves out matter distribution as a cause of spacetime curvature and is thus notably not a full theory of quantum gravity) do not yield results which can be currently experimentally verified – the Hawking temperature of nearby black holes, for example, is several orders of magnitude below the temperature of the cosmic microwave background. Our two inconsistent theories appear to ‘just work’, as far as practical considerations are concerned.

Historically, of course, there is precedent for working with a theory in spite of contradictions (see  [13] and the myriad issues with different formulations of the early quantum theory), but this is different from an institutional ‘unanswering’ – rather it falls out of the practical concern not to put the epistemic cart before the horse. A tolerance of contradiction for the time being is a pre-emptive part of the time-worn physics tradition of approximation; an acceptance of contradiction is considered a pseudoscientific or at least unscientific enterprise. There is a reason, for instance, that little scientific attention is paid to Christopher Langan – infamous [15] for his personal ‘cognitive-theoretic model of the universe’ [16], and for ‘quantum meta-mechanics’ [17], both of which are contemporary ‘un-answering’ approaches: the former contains the very explicit contention that QFT and GR are just formal systems which are processed independently by the mind, the latter is similar but for various interpretations of quantum mechanics. The fact that string theory is ‘physics’ and quantum meta-mechanics is decidedly ‘pseudo-physics’ appeals to intuition.

But it is not clear that we can inductively proceed with an ‘answering’ approach rather than an ‘unanswering’ approach simply because one has worked in the past and the other does not currently work – and we should also explore exactly why one of the abovementioned is a physical theory while the other is not.

Popper’s approach to the demarcation problem uses falsifiability as the criterion for distinguishing between a scientific and pseudoscientific theory [18], but some authors make the distinction between Popper’s practical falsifiability and logical falsifiability. Practical falsifiability insists that there be an actual experiment, presently performable by a scientist, that can be used to observe an event which contradicts some statement of the theory; logical falsifiability, on the other hand, only asks that there be a logically possible event whose logically possible observation contradicts a sentence in the theory (paraphrased from  [19]). Interestingly, while string theory presently fails the practical falsifiability criterion, it interacts in an unusual way with the much weaker criterion of logical falsifiability. Consider the possibility that some theory contains a sentence describing the behaviour inside the event horizon of a black hole, and furthermore (and this is unlikely, but illustrative) that it contains no other statement which could logically contradict an empirical claim which would be logically possible to observe. Then the logical falsifiability is guaranteed – you are logically permitted to enter the black hole and make an observation – but the ability to share your results with other researchers is contingent on a unitary resolution of the black hole information paradox – that is, your research is useful if and only if it is encoded in outgoing Hawking radiation.

Suppose then that in fact information is lost in a black hole. It would appear, then, that while your theory (which may be string theory) is a scientific one according to the logical falsifiability criterion, it is not particularly useful. It is inaccessible to the rest of the scientific community for as long as the community stays outside of the black hole. On the other hand, your friend’s paraconsistent theory is eminently falsifiable – it contains contradictory sentences which, if treated logically, immediately contradict every empirical claim. And yet, since the theory itself allows for paraconsistent claims, it has, in a sense, been falsified without allowing itself to be false!

That falsificationism seems to treat some scientific theories much less fairly than it treats unscientific ones is not a new complaint [19,20]. But an ‘unanswer’ seems to resist demarcation from string theory even in light of newer extensions and responses to Popper’s criterion. An unanswer could be considered, for instance, a solution to a physical puzzle, as Kuhn [20,21] would like science to be – although strictly interpreting paraconsistency as a ‘solution’ to why our theories are not consistent is not a clear answer to any meta-scientific puzzle. “Whether [the existence of contradiction] is a consequence of the incorrect description of a contradictory world, or just a temporary state of our knowledge, or perhaps the result of a particular language that we have chosen to describe the world, conflicting observational criteria, or superpositions of world views, contradictions are apparently unavoidable in our theories” [5]. On this footing, the unanswer is as much a paradigm shift as any formalised program dealing with string theory or even other fringe TOEs like Geometry Unity or E8 Theory – but it would at least superficially appear to be more easily classified as a “metaphysical research program” [22] – but one that undermines the logical positivist premise of a number of classifications of science.

It would appear imperative, then, that paraconsistency be relegated to the role of an auxiliary science for formalising the ad hoc procedure of accepting inconsistencies until such time as they be ironed out, and that a realist interpretation of paraconsistent logic not be permitted in discussions of physics. To answer the question posed above, it would seem that the motivation for allowing freedom with respect to axiom choice but not with respect to rules of inference is to avoid a kind of Scotusian super-explosion – to avoid, we might say, lending genuine authority to Feyerabend’s tongue-in-cheek attitude that ‘anything goes’ [13]. Until such time as a sound realist explanation for inconsistency might be discovered, it should be assumed that the aim of physics, and the recipe for progress within physics, is concerned with the ironing out of inconsisencies; and if, furthermore, the time comes that we may say that logic can be supplanted (that the subject can be changed, that the world is in fact contradictory), it will be a result of the fruitful efforts of the parascientific study of paraconsistency.

To conclude, it may be noted that the differentiation between string theory and paraconsistency noted here is merely a conventional one, which arbitrarily places the study of paraconsistent logics in the category of non-science to simplify a description of the physics-metaphysics hierarchy. This arbitration for the description of science as a method of inquiry makes the claim that science should be predicated on logic, but does not necessarily lead to a logical-falsificationist view of science. The question whether string theory is a science is tentatively answered in the affirmative, but only since it appears to be lower in rank than this study of (non-)logic; the further suggestion is that simply being a science does not endow it with any special advantage – it may be falsifiable without being communicable, the actual academic fruits of the theory notwithstanding.

To the extent that the differentiation outlined permits paraconsistency as a program for illogical explanations for scientific phenomena, it also calls for a greater emphasis on logical explanation in order to make sense of theories, and calls into question some present interpretations of theories, such as for instance the orthodox interpretation of quantum mechanics. Assertions of ‘quantum logic’, such as unanswers to questions of the positions or momenta of particles, are not necessarily ‘illogical’, in the sense that the objects in any theoretical statement are states in a Hilbert space, and the notion of a particle is not then the same notion as is classically taken, but the extent to which this connects to an ontology should be scrutinized further. Inasmuch as string theory should be taken as physics because it pertains to physical phenomena rather than the language we use to describe them, so too should quantum mechanics and other theories be taken to describe objects rather than observations of them.

Bibliography #

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[12] P. Maddy, Believing the axioms. ii, The Journal of Symbolic Logic 53, 736 (1988).

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[17] C. Langan, Introduction to quantum metamechanics (QMM), Cosmos and History (2019).

[18] K. R. Popper, The Logic of Scientific Discovery (Hutchinson, London, 1959).

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