A Critical Introduction to Mathematical Structuralism

Mathematical structuralism is a theory in the philosophy of mathematics which argues that mathematical objects are defined solely by their place in mathematical structures.

Mathematical structuralism adopts a position that's common to most other philosophical structuralisms in that it denies that there are any “intrinsic properties” of objects. It even denies the very existence of objects apart from structures.

This means that we don't have objects (or things): we only have structures and relations. In terms of mathematical structuralism only: we don't have numbers until we also have structures and relations. In other words, numbers are born of their structures and relations.

Mathematical structuralism has been dated back to David Hilbert’s The Foundations of Geometry of 1899. In that work Hilbert states:

“We think of . . . points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as 'are situated', 'between', 'parallel', 'congruent', 'continuous', etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.”

Here it can be seem that mathematical objects gain their identity relative to their “relations” to other things. Thus “points, straight lines, and planes” are defined by relational terms such as “between”, “parallel,” “congruent,” and “continuous”. This entire package then takes the form of a structure (or a system). Thus the points, straight lines and planes have relations of betweenness, congruence and being parallel to other things by virtue of being part of a whole structure (or system) in which these relations can occur.

Hilbert was even more explicit about his own (proto)structuralism in the following correspondence with Gottlob Frege (as quoted by Stewart Shapiro):

“Every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. If in speaking of my points, I think of some system of things, e.g., the system love, law, chimney-sweep . . . and then assume all my axioms as relations between these things, then my propositions, e.g., Pythagoras’ theorem, are also valid for these things . . . [A]ny theory can always be applied to infinitely many systems of basic elements.”

In terms of Paul Benacerraf's initial reasons for formulating mathematical structuralism.

Benacerraf firstly noted that algebraic theorists had no position on the ontology of mathematical objects. Such theorists were only concerned with their “structure”. Thus Benacerraf asked himself whether or not what is true of algebraic theories is also true of other mathematical theories.

As for mathematical structuralism itself, Hartry Field puts the mathematical structuralist position very clearly in the following:

“The core idea – which I'll call the structuralist insight – is that it makes no difference what the objects of a given mathematical theory are, as long as they stand in the right relations to one another.”

Clearly it's the case that in the passage above objects are played down and structures are played up. We can gain some purchase on what a structure is by talking about the “relations” (or the “right relations”) which “objects” need to have “to one another”. However, two obvious points need to be stated here:

i) It is objects which have these relations to one another.

ii) It is objects which are part of a structure.

These points will be tackled later. For now, let Benacerraf give a more detailed account of mathematical structuralism. He does so in the following:

“For arithmetical purposes the properties of numbers which do not stem from the relations they bear to one another in virtue of being arranged in a progression are of no consequence whatsoever. But it would be only these properties that would single out a number as this object or that.”

In simple terms, we can say that the number 1 is (partly) defined by being the successor of 0 in the structure determined by the/a theory of natural numbers. In turn, all other numbers are defined by their respective places in the number line. (See more detail on this later.)

It's worth noting here that “objective truth” (or at least truth) isn't rejected or denied by mathematical structuralists: it's just the account of how that truth comes about which is different to other accounts. Put simply, mathematical objects don't bring about objective truth: abstract structures do. Another way of putting this is to say that nothing is said about any mathematical object other than its place in a structure. Thus it seems to follow that there is no ontology of mathematical objects offered by mathematical structuralists.

Do Structures Give Birth to Numbers?

According to Benacerraf, the “structure” of a “particular sequence” provides the meat (as it were) of a number. Or, in an alternative phraseology, the structure (or the set of all similar structures) actually gives birth to the number. Firstly we have the structure, and only then do we have the number. We don't, in other words, firstly place a number within a structure because that would mean that the number already exists. Instead, the structure brings forth the number. What is important is the abstract structure, not the abstract object.

The idea that numbers are given birth to by structures is shown in the following passage:

“Only when we are considering a particular sequence as being, not the numbers, but of the structure of the numbers, does the question of which element is, or rather corresponds to, begin to make any sense.”

So firstly we have the structures. And then we have numbers as “elements” of these structures. But what can we make of these abstract structures before they have their elements or numbers? What do they “look” like in their naked form? To use a term and question from Quine (though he was referring to abstract propositions): what are the “identity conditions” of these abstract structures?

Again, how can we make sense of an abstract structure (or even of a “particular sequence”) without numbers (or at least without something) other than the abstract structure itself? In this case, Benacerraf's abstract structures appear to be like the bare substratums of certain ontologists or even like Kant's well-known noumena.

Of course others (including Benacerraf himself) have noted the problem with accepting abstract structures though not accepting abstract mathematical objects. The obvious problem is this:

If we can't gain epistemological access (whether causal or otherwise) to abstract numbers, then how can we gain access to abstract structures?

Thus, to repeat, Benacerraf's position is that he wants to get rid of abstract mathematical objects; though he's happy with abstract structures. Thus that means that abstractions (in and of themselves) aren't the problem. Take this passage:

“Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions.”

Thus if Benacerraf was attempting to get rid of numbers as abstract entities, then why didn't he also have a problem with “abstract structure”? Some or all of the questions asked about abstract (or Platonist) objects can now be asked about abstract structures. Indeed “progressions” themselves are also an abstraction.

In addition, if you take away the numbers from these abstract structures, what is left? Just a pure or naked abstract structure? But what is that? It's surely numbers that gives some kind of shape or reality to abstract structures or progressions. Indeed it's hard to even conceive what these structures could be without numbers or other mathematical objects (such as lines and planes in geometry, elements and operations in abstract algebra, etc.).

Types of Mathematical Structuralism

Prima facie, one may ask whether or not mathematical structuralists deny the existence of mathematical objects entirely or simply have a unique position on them.

One thing structuralists do share is that mathematical objects are “incomplete” in that only the structures they belong to make them complete. (Objects fill in the dots provided by structures.) However, an incomplete object is still an object of a sort.

Such objects are also said to lack “intrinsic properties”. So now we ask the following question:

Can an object with no intrinsic properties be an object at all?

(The denial of intrinsic properties in ontic structural realism, for example, does lead to the denial of objects/things themselves.)

Having said all that, certain brands of structuralism do endorse abstract mathematical objects of various kinds - not only structures. (Indeed it's hard to even imagine maths without abstract objects of some kind.) Some mathematical structuralist positions can even be deemed to be examples of Platonism.

Platonist Structuralism

Mathematical realists (or Platonists) believe that abstract mathematical objects exist independently of the mind. They also deem them to be eternal and incapable of change. (Of course this isn't to say that there's only an either/or situation when it comes to the philosophy of mathematics: i.e., either mathematical realism or mathematical structuralism.)

On the surface, mathematical structuralism appears to be radically at odds with Platonist (or realist) conceptions of numbers. That is because numbers in the Platonist scheme are (as it were) free-standing. That is, they exist apart from their relations to other numbers. Thus any relations numbers do have only occur after the fact. This means that any (necessary) relations between a given number and other numbers only come about because of the prior nature of the original number. That original number's nature makes those relations to other numbers possible.

Thus it's extremely hard to even conceive of what it would mean for numbers (or the number n) to have no relations to other numbers.

Nonetheless, it's entirely possible that a Platonist needn't necessarily believe that numbers are in fact free-standing (in either a basic or derivative sense of that word). Another way to put this is to say that numbers can still be seen as independent entities, yet they also have necessary relations to other numbers. In other words, even if number n is independent, it may still have necessary relations to other numbers.

In terms of specific Platonist positions, mathematical structures are deemed to both abstract and real. This position is classed as ante rem (“before the thing”) structuralism.

The Platonist position on structures can be characterised as the position that structures exist before they are instantiated in particular systems. The Aristotelian position on structures, on the other hand, has it that they don't exist until they are instantiated in systems.

The Platonist position can also be expressed by analysing the grammar of mathematical statements. Take the statement: 5 x 5 = 25. In this case, the numerals '5' and '25' refer to abstract objects. In other words, they are like (or even are) proper names.

To explain the Platonist position one can use Stewart Shapiro's own analogy. In his view, mathematical structures are akin to offices. Different people can work in a particular office. When one office worker is sacked or leaves, the office continues to exist. A new person will/can take his/her role in the office. Thus offices are like mathematical structures in that different objects can take a role within a given structure. What matters is the structure – not the objects within that structure.

Nonetheless, the people who work in offices are real. The idea of an office which is divorced from the people who work in it is, of course, an abstraction. Thus one may wonder why the office/structure is deemed to be more ontologically important than the persons/objects which exist in that office/structure. Surely it should be the other way around.

Aristotelian Structuralism

We also have Aristotelian structuralism. 

This is an in re ("in the thing") structuralism. Here structures are only “exemplified” in particular systems. That is, uninstantiated structures have no kind of existence.

A traditional problem (at least if it's seen as a problem) arises for this type of mathematical structuralism. Platonists claim that Aristotle's account of universals is problematic in that there may well be universals which have never been instantiated. Similarly with mathematical structuralism: there may well be bone fide structures which have never been exemplified or concretised (if we can use that latter term in this context) in a mathematical “system”. One reason cited for this possibility is that the world (or a part thereof) need not be tied to every mathematical structure. Therefore structures may exist and still not yet have been exemplified.

The earlier reference to universals isn't simply analogical or comparative. For example, just as the universal RED is to a particular red rose, so Stewart Shapiro (for example) believes that a universal STRUCTURE' is to a particular mathematical system. Thus traditional universals are instantiated by particulars; whereas a universal mathematical structure is exemplified by a mathematical system.

Paul Benacerraf's Structuralism

Finally we have Paul Benacerraf's position: post rem ("after the thing") structuralism.

In this case, abstract objects are completely rejected. This is why this position is sometimes classed as “eliminative structuralism”. That is, if “progressions” and relations are everything, then doesn't that mean that we can get rid of numbers altogether? (Or at least wouldn't it be possible to do so?) We may indeed have nodes (as it were) which exist within progressions and which plot relations and whatnot - though need they be numbers (as such)? That, of course, would depend on what we take numbers to be. And that's exactly the problem these philosophers are attempting to solve.

Benacerraf's position is also nominalist in nature. That is, even though it can be accepted that different structures have features in common, that commonality doesn't exist apart form its instances or exemplifications. Thus as with the nominalist position on red things: all red things don't share an identical property (or universal) that is RED (which, to Platonists, need not be instantiated). Instead the only things in common between different red things are their mutual similarities (which are taken to be “unanalysable” or “primitive”) and the fact that they're all classed as “red”. In other words, there is no universal RED or abstract mathematical STRUCTURE' biding its time in a Platonic domain waiting to be exemplified or concretised.

Nominalism?

Benacerraf's nominalist position is stressed by his reference to “notation” in the following:

“If what we are generating is a notation, the most natural way for generating it is by giving recursive rules for getting the next element from any element you may have...”

Thus all that matters (in this case) are “recursive rules”, not abstract or even concrete objects. In other words, if it's primarily about notation, then instead of using number symbols such as '1', '2' or '1001', we can use any symbol – such as 'cat', 'shlimp' or 'x*$*'. Indeed nominalists must go beyond the issue of the symbols we can use: the objects they refer to mustn't matter either. Or, rather, there are no objects at all – abstract or concrete. (In that sense, the symbols or numbers used are basically autonyms.)

Benacerraf puts this himself when he tells us that

“any recursive sequence whatever would do suggests that what is important is not the individuality of each element but the structure which they jointly exhibit”.

He continues:

“One would be led to expect from this fact alone that the question of whether a particular 'object' for example, [[[∅]]] – would do as a replacement for the number 3 would be pointless in the extreme, as indeed it is. 'Objects' do not do the job of numbers singly; the whole system performs the job or nothing does.”

We do have problems here. Firstly, surely we want to “generate a notation” for reasons which go beyond the notation itself. That is, we don't want to generate a notation solely for the sake of generating a notation. And, in this case, the notation is meant to capture what happens when we apply “recursive rules” to “elements” in order to get more elements. But then we're left with two questions:

i) What are recursive rules?

ii) What is an element?

For a start, recursive rules must be applied to (or operate upon) things which aren't themselves recursive rules. Benacerraf (in this instance at least) calls these things “elements”. So recursive rules have no meaning without these elements. (Just as Benacerraf elsewhere argues that numbers have no meaning outside of “progressions” or “structures”.)

In more detail, a recursive rule is displayed in the formal definition of the natural numbers according to Peano's axioms. Thus:

0 is a natural number, and each natural number has a successor, which is also a natural number.

This can be put more formally in the following manner:

0 is in N.

If n is in N,

then n + 1 is in N...

This means that we have the base case (i.e., “0 is a natural number”) and a recursive rule (“each natural number has a successor, which is also a natural number”). From that base case and recursive rule we can produce the set of all natural numbers.

However, we both start with numbers and end with numbers. That is, the base case is a number (i.e., 0) and the recursive rule both operates upon and produces numbers. Thus the recursive rule operates upon 0 and gets 1. Or can we say, instead, that the recursive rule actually creates the number 1? Perhaps we can. So does it create the number 0 too? Not if we're sticking to natural numbers because nothing comes before 0 - so no recursive rule could have created 0. And if the recursive rule somehow creates such numbers, it's still the case that it operates upon numbers too. So this is a two-way street we're discussing here.

Peano's axiom (at least grammatically) also seems to assume the prior existence of numbers. That is, the clause “0 is a natural number” and the clause “each natural number has a successor” seem to assume the prior existence of numbers. If that is correct, then recursive rules can be said to discover (not create) numbers; or, at the least, to codify and explain them.

Conclusion

Perhaps all this structure-object talk is an example of a “binary opposition”. Think here about the ontological debate about objects and events. Donald Davidson picks up on this in the following comment on P.F. Strawson's position:

“What does seem doubtful to me is Strawson's contention that while there is a conceptual dependence of the category of events on the category of objects, there is not a symmetrical dependence of the category of objects on the category of events.”

Put simply, this ontological hierarchy can easily be reversed (at least in most cases) without making any difference. In other words, there may well be no hierarchy at all: both objects and events are dependent on each other. Thus let's rewrite that passage from Davidson:

While there is a conceptual dependence of the category of numbers (as objects) on the category of structure, there is not a symmetrical dependence of the category of structure on the category of number.

In turn, does it really matter if we reverse that passage in this way? -

While there is a conceptual dependence of the category of structure on the category of numbers (as objects), there is not a symmetrical dependence of the category of numbers on the category of structures.

In addition to that, many mathematical structuralists seem to commit a logical sin that philosophers are always spotting in all sorts of others areas: that of assuming x in the very definition/description of x. In this case, they assume numbers in their definitions (or descriptions) of numbers. (Take as a comparison the notion of metaphysically-realist truth as seemingly presupposed in epistemological and coherentist accounts of truth.)

This can also be seen in something that Stewart Shapiro writes.

In defence of the structuralist position, Shapiro argues that “in the system of Arabic numerals, the symbol ‘2’ plays the two-role”. He also states that “anything at all can play the two-role in a natural number system”. Here the word “two” has been used to describe a structural “role” which is meant to provide the meat as to what a number is. It's true that Shapiro uses the word “two” rather than the symbol “2” - but does that make a difference? We can of course argue that the word “two” is a contingent natural-language expression; whereas the number 2 is an abstract object. Yet even when it comes to using the word “two”, the number 2 is still both presupposed and tacit.

**********************************

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.