‘Beables' and Induction

When I was in graduate school in Scotland, I was told the following parable by my advisors. An economist, a mathematician, and a logician were on a train traveling north. Just after they passed the Scottish border they noticed a single cow standing in a field.

It is worth noting here that my advisors were both mathematicians. As such, I always had the impression that the parable, as they told it, was meant as a dig at both economists and logicians. Clearly the economist has over-extrapolated from the given

I. T. Durham (B)

Department of Physics, Saint Anselm College, Manchester, NH 03102, USA

e-mail: idurham@anselm.edu

© Springer Nature Switzerland AG 2019 105

A. Aguirre et al. (eds.), What is Fundamental?, The Frontiers Collection,

https://doi.org/10.1007/978- 3-030-11301-8_11 data. That point is hardly up for debate. But has the logician under-extrapolated? The fact is that our intrepid travelers do not know if the cow is the same color on both sides absent additional information. It is entirely possible that the mathematician, in casually suggesting that the cow was entirely brown, was wrong. Yet, in our own experience with cows, most of us would probably think it highly unlikely that a cow had such asymmetrical coloring as to be entirely different on either side.

In our effort to understand the world we inhabit, we wrestle with such questions of inference as a matter of course. At the heart of the problem of inference, more properly known as the problem of induction [24], is what John Bell referred to as the ‘subject-object distinction’ [6]. This distinction is best understood in the context of quantum mechanics but it is not limited to that realm. Quantum mechanics is ostensibly a theory about the results of measurements. Measurements are performed ‘on’ systems (object) and presuppose that something or someone (subject) must be doing the measuring. But as Bell pointed out, precisely where or when to draw a distinction between subject and object is not manifest in the theory itself. This inherent ambiguity continues to be the source of much debate.

As humans, we naturally tend to anthropomorphize. The very word ‘measurement’ suggests a human-centric outlook. So it is that, for many, the subject-object distinction is interpreted as concerning knowledge. By making a measurement on a system (object), the measurer (subject) has acquired knowledge about that system. Bell finds this unsatisfying. He suggests that any accurate, final theory of physics (should one ever be found) could not be about the acquisition of knowledge.

[It] could not be fundamentally about ‘measurements’, for that would again imply incom­pleteness of the system and unanalyzed interventions from outside. Rather it should again become possible to say of a system not that such and such may be observed to be so but that such and such be so [6], p. 41.

Rather than being about observables, such a theory would need to be about beables.

On its face this appears to be a bold prescription. Presumably any such final theory of physics would provide us with a means of obtaining complete knowledge of the world. But it’s not clear that objectively complete knowledge of the world is even attainable in theory let alone in practice. Bell is more practical. He recognizes that any final theory would need to somehow clarify or circumvent the ambiguities in the subject-object distinction that arise in any of our existing theories, quantum mechanics in particular.

It is worth noting exactly what Bell means by ‘beable’. He actually initially uses the word in two slightly different contexts. In the first context he suggests that beables within a given theory must be describable in classical terms since “they are there” [7], p. 51. Here he (oddly) seems to be motivated by Bohr, saying

[b]y ‘classical terms’ here Bohr is not of course invoking particular nineteenth century theories, but refers simply to the familiar language of everyday affairs, including laboratory procedures, in which objective properties—beables—are assigned to objects [6], p. 41. Such beables, he notes, must necessarily include things like the settings of switches and knobs, and the readings of instruments.

In the second context in which he initially employs the term, he suggests that beables in a given theory are expressly physical quantities in the sense “familiar already from classical theory” [7], p. 52. The example he cites in order to clarify this point is the contrast between the E and H fields in electromagnetism, which he suggests are physical, and the A and ô potentials, which are not. Make no mistake— Bell explicitly says that E and H are beables within the context of Maxwell’s elec­tromagnetic theory: “the fields E and H are ‘physical’ (beables we will say)...” [7], p. 52.

In both of these contexts, the beables form the ontology of the theory. It’s actually worth asking what we mean by this. All physical theories are ‘about’ something. One might say that the beables are what a theory is about. So Maxwell’s electromagnetic theory is about electric and magnetic fields and so those fields are beables within that theory. But that doesn’t quite capture the meaning implied in the first context where the beables are said to be objective properties, including instrument settings, that are applied to objects.

One response to this is to simply dismiss the neutrino as non-classical and thus not subject to the rules of classical electromagnetic theory. As a response to the subject­object distinction, this sort of thinking seems evasive at best. In addition, in his paper extending beables to the realm of quantum field theory (and thus what we might blithely call the ‘proper’ realm of the neutrino), Bell refers to the beables of a theory as “those elements which might correspond to elements of reality, to things which exist” [4], p. 174. One presumes that matters of reality and existence are independent of any particular theory. In other words, if beables are said to properly exist in that they are elements of reality, and they are understood to be objective properties of objects, then if a magnetic field is a beable in one theory, it ought to properly be a beable in any theory in which it appears. It doesn’t seem unreasonable to then ask that the nature of such beables be consistent across theories.

For the purposes of science, the existence of certain things is taken as self-evident. I may awake tomorrow to find that I am actually a Buddhist monk living in a monastery in the Himalaya and that my life as a physicist was nothing but a dream.

In fact the world of our direct experience is even more limited than that. We have no direct experience of electric fields in the sense that we have no way to directly measure one. We infer their existence from measurements of a scalar electric potential. This is curious. According to Bell, electric fields are beables in classical electromagnetic theory but scalar potentials are not. Our only experience of the beables which, to Bell represent what is ‘physical’, i.e. that which ‘exists’, is mediated by something Bell explicitly says is ‘nonphysical’ and thus, one would presume, does not actually exist (at least according to Bell).

Regardless of the physicality of scalar potentials we still have no known way of directly measuring an electric field. We must infer its existence from measurements of other properties. This is actually true of any field. We cannot measure a gravitational field directly either. We infer its existence from measurements of force, acceleration, mass, etc. Bell at least partially acknowledges this fact by noting that all physical theories are necessarily tentative in nature.

Such a theory is at best a candidate for the description of nature. Terms like ‘being’, ‘beer’,¹ ‘existent’, etc., would seem to me lacking in humility. In fact ‘beable’ is short for ‘maybe- able’ [4],p. 174.

Bell also recognizes that our fundamental window on the world is through observ­ables, but he says that our observables must be constructed from beables.^{[22] [23] Thus Bell acknowledges that at least some beables must be inferred. Certainly the settings of switches and knobs, and the readings of instruments, which Bell also considers beables [7], may be experienced directly. But at least some beables simply cannot be directly known.}

The problem of induction, then, is in knowing just how much we can reasonably and rationally infer from a set of sensory data that constitute our direct experience of the world. In a sense, the problem of induction is concerned with just how we identify what actually is fundamental. After all, one assumes that there is some minimum set of beables required for any final theory should such a theory even be attainable. To put it another way, one assumes that the universe, at its most fundamental level, consists only of those beables that are necessary to reproduce its manifest phenomena, i.e. there should be no extraneous beables. Are these fundamental beables knowable and, if so, how can we know them?

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