Carlo Rovelli's Relational Quantum Mechanics

i) Introduction

ii) Interactions

iii) Werner Heisenberg’s Electron

iv) Carlo Rovelli’s Electron

v) Systems, Systems, and More Systems

vi) Is RQM an Anti-Realist Position?

vii) Conclusion

Carlo Rovelli is an Italian theoretical physicist. He works mainly on quantum gravity. He’s also a founder of loop quantum gravity theory (along with Lee Smolin). Rovelli won the second prize in the 2013 FQXi contest “It From Bit or Bit From It?” for his essay on “relative information”. His book, Seven Brief Lessons on Physics, has also been translated into 41 languages and has sold over a million copies worldwide.

Rovelli introduced the relational interpretation of quantum mechanics in 1994.

Carlo Rovelli’s overall position is classed as relational quantum mechanics (RQM). This is an “interpretation” of quantum mechanics in which a quantum system is seen as being “observer dependent”. In terms specifically of the word “relational” in “relational quantum mechanics”, this means that there’s a relation between an observer and a quantum system.

In addition to the inclusion of observers into the quantum equation, we also have the many relations between physical systems and physical systems. Or as Rovelli puts it:

“Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world.”

The importance of relations is broader than one may initially think. Rovelli himself ties it to the theory of relativity.

Take the velocity of any object. The notion of the velocity of object O as it is “in itself” (as it were) is deemed to be meaningless. The velocity of object 0 is actually measured relative to — or in relation to — other objects.

Now take two events which are deemed to occur at the same point in time; even though they do so at vastly different places from one another. In the case of Special Relativity, both events must be measured relative to (or in relation to) something else. In addition, when it comes to General Relativity, objects in space and time (or in spacetime) need to be seen in relation to — or relative to — the gravitational fields in which they’re embedded (or to something which is “dynamical”).

However, Rovelli broadens out his relationism even more when he moves from quantum mechanics and Einsteinian relativity to — of all things! — what he calls “man”.

Here Rovelli gives a perfect description of what philosophers class as anti-realism. Oddly enough, he uses the words of Democritus to do so). Rovelli writes:

“Democritus gave a strange definition of ‘man’: ‘Man is what we all know.’”

This appears to be a poetic expression of philosophical anti-realism. That is, it stresses what we can know, not what is. That anti-realist position is applied to “man” in this way:

“The nature of a man is not his internal structure but the network of personal, familial and social interactions within which he exists.”

We can say, then, that Rovelli doesn’t believe that man has a substance (or an “internal structure”) — and he certainly doesn’t have what philosophers call an essence. Instead, man is “nothing more” than a network of relations. In this case, Rovelli’s relationism moves from the quantum scale to the “classical scale” — and there’s nothing more classical than man.

However, here we can argue that there must be something (or some things) which have these relations. So is it good enough to argue that whatever these things are, they are themselves made up entirely of relations?

Interactions

In the philosophical position called ontic structural realism (as usually advanced in the philosophy of physics), relations and (mathematical) structures are seen to be everything. (Therefore “every thing must go”.) Carlo Rovelli, on the other hand, stresses “interactions”. For example, he writes:

“The description of a system, in the end, is nothing other than a way of summerizing all the past interactions with it, and using them to predict the effect of future interactions.”

Having just mentioned ontic structuralism realism, there’s nothing in Rovelli’s account that immediately clashes with ontic structural realism. It seems to be a simple difference of stress. Indeed Rovelli himself also talks about relations. In any case, things (or objects) seem to be eliminated from the scene in Rovelli’s scheme.

We can now move to Nathaniel David Mermin (a solid-state physicist) to get a more explicit position on such ostensible thing-eliminitivism. In his “Ithica interpretation” we have what he calls “correlations without correlata” (i.e., instead of ontic structural realism’s “relations without relata”) .

Indeed David Mermin is even more explicit when he says that

“correlations have physical reality; that which they correlate does not”.

What’s more, he adds:

“[C]orrelations are the only fundamental and objective properties of the world.”

So whereas James Ladyman (an ontic structural realist) has sometimes said (or implied) that things aren’t actually eliminated from their ontic structural realism (simply because things are themselves structures), Mermin is far more explicit about his position.

Werner Heisenberg’s Electron

Much of this ties in with Werner Heisenberg’s well-known “uncertainty principle”.

In the updated language of Rovelli’s relationism, it can be said that Heisenberg argued that it’s only when the electron is “interacting” with another system that its position can be detected. When it’s not interacting, the electron is “spread out” over many different positions. That is, the electron is in a “quantum superposition” of various different positions.

What’s more, this is a relationism which not only includes interactions with other systems: it also includes interactions with human observers. That is, the position of the electron can be determined by an observer; by a “quantum reference system”; or by an experimental apparatus/experiment. So this is a set of relations “all the way down” — from particles to systems to human observers to experimental setups.

More technically, we have an “observer system” O (which can be seen as an epistemic system) which interacts with a quantum system S (which can be seen as an ontological system).

We therefore need to take into account the fact that there are (or can be) different accounts of the same quantum state or system. (This is, after all, a variation on the “underdetermination of theory by evidence” idea.)

In more technical detail, we can have a system which is in a superposition of two or more states. We can “collapse” this system to achieve an “eigenstate” which is (at least to some degree) determinately circumscribed. Thus if we have two or more “interpretations” of system (or state) S, then observers must have been brought into the story. And that means that there are additional relations (or “interactions”) to consider.

This also means (or can mean) that we need a second observer (O’) to observe the observer-system (O); who, in turn, observes the quantum system S. This multiplies relations indefinitely. Indeed don’t we have a possible infinite regress on our hands here?

Michio Kaku raises this issue in the case of human observers vs. cameras. Kaku writes:

“Some people, who dislike introducing consciousness into physics, claim that a camera can make an observation of an electron, hence wave functions can collapse without resorting to conscious beings. But then who is to say if the camera exists? Another camera is necessary to ‘observe’ the first camera and collapse its wave function. Then a second camera is necessary to observe the first camera, and a third camera to observe the second camera, ad infinitum.”

This is a concrete example of the problem of “Wigner’s friend”. That is, if I collapse the wave function for an electron, then my friend has to observe me collapsing the electron’s wave function. He also needs to collapse the wave function which is myself collapsing the electron’s wave function. Then a friend of my friend will need to observe my friend to collapse the wave function which is my friend; who, in turn, is collapsing the wave function which is myself collapsing the electron’s wave function... And so on.

Carlo Rovelli’s Electron

Rovelli also tackles the electron.

Rovelli puts an every-thing-must-go position for the electron. Or, at the very least, he argues that an electron is not a thing without its relations or interactions. In Rovelli’s own words:

“An electron is nowhere when it is not interacting… things only exist by jumping from one interaction to another.”

Of course in many respects this is a standard account of an electron and the wave function. Rovelli adds more to his account of an electron in the following:

“What if, effectively, electrons could vanish and reappear? What if these were the mysterious quantum leaps which appeared to underlie the structure of the atomic spectra? What if, between one interaction with something, and another with something else, the electron could literally be nowhere.”

Semantically, we can now ask this question:

If the electron does literally vanish, then how can it then reappear?

How how can something that ceases to be, then “reappear”? Surely if something vanishes (or ceases to be), then something entirely new (or something else) must appear. (Unless x vanishing isn’t the same as x ceasing to be.)

Of course we’re talking about quantum states/systems here so these questions may well be naïve or uninformed. Firstly, what exactly is meant by Rovelli’s word “vanish”? What we may have is various fields and forces which are “strong” or “excited” (as in “excitations of fields”) at one spacetime point; which become weak at other spacetime points; and then (due to Rovelli’s “interactions”) appear (or become stronger) at another spacetime point.

In terms of Rovelli’s own detail.

If this ostensible disappearance of an electron is accounted for by the “mysterious quantum leaps which appeared to underlie the structure of the atomic spectra”, then, between x and y (the “quantum leap”), there are still fields and forces. (Elsewhere, Rovelli says that “an electron is a combination of leaps from one interaction to another”.) However, the strength of the excitations of the fields and forces aren’t enough (i.e., between x and y) to constitute an electron. So when this something gets to y, the excitations of the fields and forces are indeed strong enough to constitute an electron. It can now be said that the electron at spacetime point x shouldn’t really be seen as “the same as” (or identical to) the electron at spacetime point y. Indeed why see it as the same electron at all?

Technically, this can be partly explained by reference to something Albert Einstein described way back in 1916.

An electron can be in (or actually be) an “excited” state of fields and forces. That partly means that the electron has extra energy to that which it had before. Indeed it may not be correct to see it as the same electron before it had that excited state of extra energy. One other consequence of this excited electron is that if a photon with a particular wavelength “passes” the electron, then that photon can make the electron leap/fall/jump into a lower energy state. That, in turn, will result in another photon being released with the same wavelength as the first photon. So then we would have two photons.

In terms of the theme of this piece, we can argue that the electron doesn’t exist at all between the the low-energy state and the high-energy state (i.e., there’s no electron — therefore no electron fall — between low-energy x and high-energy y). Indeed without any kind of excitation (therefore any kind of energy), the electron doesn’t exist at all. That is, there is no electron between x and y.

Despite all that, Rovelli himself doesn’t speak in terms of excitations of fields and forces: he talks, instead, in terms of “interactions”. That is, the electron reappears when it interacts with “something else”. However, this may be to say the same thing in a different way. That is, interactions in a quantum state (or system) are the same thing as excitations of fields and forces within a quantum state (or system). In other words, the interactions cause the excitations of fields and forces within a state/system.

Thus Rovelli believes that the electron is literally “nowhere” between the earlier x and later y. In that case, it makes no sense to say that the “electron is nowhere” because it’s (roughly) equivalent to saying, “My long-dead cat is nowhere.” Having said that, the conclusion that there is an electron at spacetime point x and another electron at spacetime point y (though no electron between spacetime points x and y), does seem to contradict certain positions in quantum mechanics.

So, again, when Rovelli says that

“the electron could be something that manifests itself only when it interacts, when it collides with something else; and that between one interaction and another it had no precise position?”

I would express it this way.

There is no electron between the interactions. If an electron exists at spacetime point x due to interactions, then when such interactions cease, the electron itself ceases to exist. And when we have another interaction at spacetime point y, then there’s an entirely new electron. In other words, in the spatiotemporal region between spacetime point x and y, there is literally no electron. Of course there must be something between x and y; though, still, there’s no electron.

Systems, Systems, and More Systems

To make this talk of systems simpler, it’s worth noting (again) that human observers are also classed as systems. What’s more, relationism actually appears to introduce an element of relativism when it comes to an electron: it only has a “meaning” relative to an observer or a system. Thus, if you are that system, then whatever it is you observe, then it only exists in that manner (or at all) for you. In Rovelli’s words, it only exists in that manner because you’ve “interacted” with it.

Although the stress on systems observing (or measuring) systems may seem to make any kind of realist ontology impossible (with regards to quantum mechanics). At least realism is impossible as it’s expressed in the following way:

Systems (including observer-systems) measure (or observe) systems because systems can’t measure (or observe) themselves.

Indeed even in the “classical world” it can be said that macro-objects don’t tell us about themselves. That is, the classical world as a whole doesn’t have its own favoured description. That’s why systems are required at both the quantum and classical levels. In anti-realist or epistemological terms, only systems can (metaphorically) know the world.

Simon B. Kochen (a Canadian mathematician) stressed another way in which systems themselves (rather than “the world”) are of importance in quantum mechanics. He wrote:

“The basic change in the classical framework which we advocate lies in dropping the assumption of the absoluteness of physical properties of interacting systems… Thus quantum mechanical properties acquire an interactive or relational character.”

That is, physical properties only become absolute when they interact. Or, instead, we can drop the notion of absoluteness altogether and say (from an anti-realist perspective) that we can only know physical things when they interact. And by “interact” (in this case) I mean that human observers (or observer-systems) are also doing the interacting. This, surely, is a new(ish) addition to the anti-realist’s armoury.

All this can be expressed technically in the following manner.

We have what’s called an “abstract vector space”. We make a measurement of it to get q. In terms of relationism, this is the probability that the system S being measured can affect the system S’ (or an observer-system) in a joint interaction in order to get q. Because of the importance of different interactions with S, the wavefunction Ψ must take into account different observer-systems and therefore different outcomes. All outcomes are probabilistic. And all observer-system interactions “collapse the wave function”.

Is RQM an Anti-Realist Position?

A philosophical bias can be displayed here if it’s argued that relational quantum mechanics is basically advancing the position that epistemology trumps ontology. RQM is about what we can know and what we can’t know. Thus, according to Rovelli, “relations [] ground the notion of ‘thing’”. Epistemically, we can know these “relations”: we can’t also know “things”. And, because of that, Rovelli also says that “[t]he world of quantum mechanics is not a world of objects: it’s a world of events”.

Now it can of course be said that it must be things (or some things) which interact. Yes; though, again, can we know anything about these things?

So if we’re dealing with what Rovelli calls “occurrences”, then, as a consequence, we’re also dealing with relations. That is, if anything, occurrences (or interactions) can only be occurrences (or interactions) between a system and another system. It can be argued that we know nothing of things. We only know about the relations between a system and another system. So things aren’t said not to exist: it’s simply a case of things actually being (at least in some sense) Kantian noumena.

(In this account we can use terms from Hegelian metaphysics. In this case, then, we’re only dealing with Becoming, not with Being — i.e., what a thing is. We don’t know about the nature of Being. We do know about Becoming.)

In fact Rovelli says something which appears to be mereological in nature. He believes that “[t]hings are built by the happening of elementary events”. That is, a thing is the sum of the “elementary events” which constitute it. Rovelli allows the (now dead) American philosopher Nelson Goodman to back him up on this. He quotes Goodman thus:

“‘An object is a monotonous process.’”

Here again we have the mereological reality of an “object” being constituted by “its” processes or events. If you take away the events, then you also take away the object. The object, then, is nothing more than the processes (or events) which constitute it — even if those processes (or events) are “monotonous”. So whereas Goodman expresses himself in philosophical terms by talking about an “object”, Rovelli himself elaborates by saying that an object (such as a stone or an electron) is actually “a vibration of quanta that maintains its structure for a while”.

Conclusion

It appears that there’s nothing much in Carlo Rovelli’s account that’s radically at odds with certain other interpretations of quantum mechanics. It also squares fairly well with the philosophical account offered by ontic structural realists; it squares very well with the physics-and-philosophy account of Lee Smolin; and it even squares with some other interpretations offered by pure (i.e., non-philosophical) physicists. It certainly squares with uninterpreted quantum mechanics. (Whether or not there’s a neat line in the sand which can be drawn between interpreted and uninterpreted quantum mechanics is hard to say.) As just hinted at, that’s because we’re talking about the interpretations of quantum mechanics here. And in terms of experiments, predictions and technology, this means that these interpretations — ultimately — don’t make that much of a difference to such things. This also means that it may not be possible to conclusively establish which interpretation is true. Indeed perhaps many rival interpretations are true at one at the same time. Or perhaps no interpretation is true. It may even be the case that truth shouldn’t really come into this because truth may require a metaphysically-realist stance. Yet, in the domain of quantum mechanics (perhaps elsewhere too), ontological realism may be impossible.

Ladyman and Ross: Every Individual Must Go? (5)

[This piece is an account of parts of James Ladyman and Don Ross's book, Every Thing Must Go: Metaphysics Naturalized.]

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Two simple things pose a problem for anyone wanting to embrace ontic structural realism: 1) The various two-slit experiments. 2) Problems with terminology.

Firstly, let's discuss the well-known two-slit experiments.

1

The Firing of Single Particles

How can we acknowledge the two-slit experiments without also acknowledging the nature and existence of single particles – or single things?

A photon (for example) is a single quantum of light. In fact it's referred to as a "light quantum" or as a “light particle”. It's also the case that single particles have been fired (in double-slit experiments) at relatively long intervals between each firing!

In terms of detail, let's forget here when particles-in-the-plural have been fired (e.g., by J.J. Thomson, Ernest Rutherford, Ernest Walton, etc.) and concentrate on the firing of single particles.

Take Pier Giorgio Merli, Giulio Pozzi and Gianfranco Missiroli who did so in the 1970s in order to demonstrate what's called “interference”. And, in 1989, Akira Tonomura and his colleagues also fired electrons one at a time at the Hitachi research laboratories. Then Herman Batelaan and his team did the same thing and the results were published in 2013.

(Alain Aspect's well-known experiment is actually said to have used many pairs of photons to demonstrate “spooky action at a distance” or “entanglement”.)

However, were single particles really fired in these experiments?

It's worth noting here that even atoms don't have rigid boundaries (or really any boundaries at all ). This is even more the case with particles. Thus these facts must surely raise particular questions as to how single particles can be fired in experiments. Having said that, this may well only be a technical (rather than scientific or philosophical) question.

Of course something must be fired.

Though is it the case that some individual thing is fired each time? Perhaps, instead, only spatiotemporal chunks (or spatiotemporal slices) of fields are fired. What can't be fired, however, are what philosophers call “individuals”, as we shall now see.

Terminology

All the above is partly a terminological issue.

Take the notions of a thing, an individual and a particle again.

The word “thing” is an everyday word. It can applied to anything (or to any thing). The word “individual” (as used by James Ladyman and other philosophers) is a philosophical technical term. And, finally, the word “particle” is mainly used within a scientific context. (It's true that Isaac Newton's use of of the word “particle” is at odds with many 20th-century uses; though all such uses are still fundamentally scientific.)

Particles (such as electrons) can't be individuals (as most philosophers see individuals) simply because all the particles of a specific kind share all their properties (i.e., spin, mass, charge, etc.). So, semantically, these properties can't be seen as intrinsic or essential because particles don't actually have contingent properties. However, that's unless we bring in the relations which particles have with other things/particles or with fields! And this is where ontic structural realism comes in.

Thus, on a philosophical reading, particles - by definition - can't be individuals. Every particle of a given kind has the same properties – that's if, again, we rule out relational or spatial properties. A single particle, then, isn't like a single human person, who can easily be distinguished from other human persons. Indeed a single human person may well have both essential and contingent properties. (That will depend on one's metaphysical position.)

In addition, what Ladyman, Ross and other philosophers call “identity dependence” and “existence dependence” doesn't seem to automatically rule out an entity being an individual. Or, if it does, we'd still need to know why that's the case. And then there's the added problem of using a philosophical technical term (i.e., “individual”) and then foisting it into discussions of the particles of physics.

This also applies to Ladyman and Ross's term “self-subsistent”. If that term is taken literally, then has any entity in the entire history of the universe ever been self-subsistent? Of course all that will depend on what each philosopher who uses that term actually takes it to mean.

So if a particle can't be an individual, can't it still be a thing?

Finally, the words “particle” and “electron” are scientific terms. That must surely mean that if physicists say that “electrons are particles”, then electrons are particles. Full stop. Thus since the word “particle” is seen by both laypersons and scientists as a technical scientific term, then naturalist philosophers shouldn't encroach on the territory of physicists by questioning their usage.

2

Individuals and Modern Logic

Ladyman and Ross stress the fact that modern logic (from Leibniz onward) has been fixated on “individual objects”. However, it may turn out that it's Ladyman and Ross who're fixated on individual objects.

In modern logic (at its most basic) we have individual objects which are symbolised by variables (such as x and y). Those variables (of objects or things) are the subjects of predication (or seen to be members of sets).

Ladyman and Ross, on the other hand, see “logical constants and variables as being mere placeholders” which are used for practical purposes. In other words, there are no “ontological commitments” to the things/objects the variables symbolise. Instead, the variables and constants are placeholders which plot relations and structures.

So it's fairly clear that quantum mechanics is on the minds of Ladyman and Ross when they cite the limitations of modern logic. However, modern logic wasn't designed to discuss quantum mechanics. Of course it can now be said that (in theory) modern logic must also be applicable to... well, everything. Therefore it must also be applicable to the phenomena of quantum mechanics.

Mathematical Structures and the Physical

What is fundamental?

That's a classic question of western metaphysics and it's been asked for over two thousand years. In both quantum mechanics and ontic structural realism, we're told that fields are fundamental, not particles. More precisely, particles in Quantum Field Theory (QFT) are seen as “excitations” of fields. Thus, to state the obvious, it's particles which are the excitations of fields, not fields which are the excitations of particles.

This makes fields fundamental. Or does it? Perhaps it's a difference which doesn't really make a difference – at least it doesn't to most hands-on physicists. (As the physicist John Polkinghorne once put it: “The average quantum mechanic is no more philosophical than the average motor mechanic.”)

Here again we can question this fixation on what is and what isn't fundamental. In certain respects, physics itself shows us that particles/things aren't fundamental, despite the long history of attempts to find the fundamental entities of the world.

So let's be specific about this. When Ladyman and Ross claim that physics shows us that physical objects aren't spatially located, aren't they only referring to particles? The same is the case when they say that things aren't “self-subsistent” (yet surely macro-objects aren't self-subsistent either); they lack “primitive indentity”; and that they aren't “ontologically fundamental”.

It's troublesome to say that things aren't “self-subsistent”. It's perhaps even more troublesome to claim that relational structures are “ontologically subsistent” and that relations are “primary to things”. At a prima face level, this appears to be a Platonic position not on numbers or mathematics, but on structures and relations. This isn't a surprise if the structures and relations in physics are themselves mathematical.

In addition, Ladyman and Ross say that “things are nonexistent” or that “things are dependent on relational properties for their existence”. Thus can we also argue that structure/relations are nonexistent or that structure/relations are dependent on things for their existence? In concrete terms, a pragmatist or instrumentalist may say that whether or not one stresses fields or particles depends on one's explanatory or experimental purposes.

So not only can we ask Ladyman and Ross how abstract mathematical structures relate to things/objects: we can also ask how they relate to anything physical (or concrete). However, Ladyman and Ross appear to reject these questions outright when they write:

“The ‘world-structure’ just is and exists independently of us and we represent it mathematico-physically via our theories.... the fact that we only know the entities of physics in mathematical terms need not mean that they are actually mathematical entities.”

Here we need to know what's meant (philosophically meant) by the word “represent”. That is, what is the ontological (i.e., not representational) relation between structures and the “entities of physics”?

So it's helpful (if only in a limited sense) that Ladyman and Ross explicitly state that they aren't eliminativists about physical entities when they say that

“the fact that we only know the entities of physics in mathematical terms need not mean that they are actually mathematical entities”.

So how does that admission (if it is an admission) help us? Nothing is said about physical entities. Indeed Ladyman and Ross more or less say (in a Kantian manner) that nothing can be said about physical entities (i.e., other than what's said via the medium of mathematical structures). Perhaps, then, we should bite the bullet and accept this limitation if there's no way around it.

Yet Ladyman and Ross are explicit about their Platonism (or Pythagoreanism). Or, at the very least, their position is Platonic/Pythagorean by default. They write:

“What makes the structure physical and not mathematical? That is a question that we refuse to answer. In our view, there is nothing more to be said about this that doesn't amount to empty words and venture beyond what the PNC allows. The 'world-structure' just is and exists independently of us and we represent it mathematico-physically via our theories.”

So whereas Platonists would be explicit and say it's all mathematics, Ladyman and Ross say that questions about their mathematical structuralism are “question[s] [they] refuse to answer”. Indeed they don't want to indulge in “empty words” in doing so. Ladyman and Ross are quite happy to express their Platonic and (structural) realist position by saying that the

“'world-structure' just is and [it] exists independently of us and we represent it mathematico-physically via our theories”.

Despite all that, the abstract and mathematical scheme of Ladyman and Ross does eventually give way to the physical (on concrete) when they say that the mathematical structures they endorse are “physically realized” and that the predicates they use are (as it were) attached to entities.

Thus this raises the question as to whether or not Ladyman and Ross are only realists about mathematical structures; or whether they're also realists about things - if via the route of mathematical structures. After all,

i) If Ladyman and Ross say that mathematical structures represent “real patterns”,
ii) then surely they can't also be saying saying that mathematical structures represent mathematical structures.

What's more:

i) If mathematical structure x represents a real pattern y,

ii) and this real pattern y represents a physical (or concrete) z,
iii) then mathematical structure x must also represent a physical (or concrete) z.

Structures: Syntax and Semantics

It's of course structures which are meant to save the day when it comes to both scientific realism and the well-known pessimistic meta-induction. That is, structures are real and they're passed on from (some) old scientific theories to (some) new scientific theories. But here too there's a problem.

We can say that it's the mathematical syntax of scientific theories which is passed on - not their semantics. That is, we have a possible (or actual) structural continuity; though that only takes the form of mathematical equations.

However, doesn't syntax (at least in this case) require a semantics? In other words, what is the subject matter of the syntax/mathematical equations? If the subject matter is a Lockean “something-I-know-not-what”, then how can things we can't know be the subject matter of equations (or of anything else for that matter)?

What's more, in physics the same equations can be mapped onto (or they can model) what are taken to be different physical phenomena. This is the inverse of the “underdetermination of theory by evidence”; in which the same evidence/observations (or the same physical phenomena) can give rise to different theories.

So this shows us some disjunctions between abstract mathematical structures and concrete physical phenomena. That is, the same physical phenomenon can be mapped by different mathematical structures; and the same mathematical structure can map different physical phenomena. Perhaps this must mean that there's always a remainder when it comes to any mapping of the concrete/physical by abstract mathematical structures.

In more relevant terms, in Quantum Field Theory, different structures are used to map the same spatiotemporal section of the physical world. Now it can also be added that different structures must surely have different ontologies. However, in practical term or in terms of prediction, it can be said that any different ontologies of a spatiotemporal x are differences which don't really make a difference.

(For those who buy string theory, we have the examples of different theories (such as type 1 and heterotic SO(₃₂)) which are mathematically equivalent. And even in the case of Maxwell's equations for electricity and magnetism, if one interchanges the electric fields for the magnetic fields and vice versa, then the resulting equations are almost identical.)

Conclusion

It would help if Ladyman and Ross explicitly stated that when they talk of “things”, “objects” and “individuals”, what they have in mind are the things, objects and individuals which exist within the domain of quantum mechanics. Indeed once that's acknowledged, the ontic structural realism of Ladyman and Ross is far less radical than it appears at first sight. Having said that, it's also true that Ladyman and Ross do sometimes talk about things, objects and individuals at the “classical” or macro-scale; though they do so far less often. Not only that: what they do say about the “classical world” will require supplementary arguments and data to that which is used to justify their philosophical positions on things, objects and individuals at the quantum-mechanical level.

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