# Is It Possible to Nominalize Quantum Mechanics?

### Abstract

Bueno argues that Balaguer’s approach to nominalizing Quantum Mechanics (QM) does not fit the nominalism framework. This paper shows that Balaguer provokes Bueno to arrive at such the conclusion through illustrating his claim that physical quantities used in QM are not abstract objects with examples from classical physics. In fact, they are real since in QM the notion of complete set of physical quantities relates to an actual quantum state. Besides, Bueno is not content with Balaguer’s treating an eigen value of the operator that corresponds to the physical quantity as the one that is not abstract. However, a finite number of such the eigen values completely determine the actual quantum state. Therefore, this paper asserts that physical quantities, which Balaguer calls propensities, suit the purpose of nominalizing QM. In other words, this piece of writing argues that Bueno is not right in stating that Balaguer’s strategy is not consistent with nominalism. As for Bueno’s claim that Balager’s strategy is not in agreement with a bunch of QM interpretations, its analysis is beyond the scope of this work.

## Is It Possible to Nominalize Quantum Mechanics?

According to Burgess and Rosen, normalization of an actual theory produces the inferior one. However, the latter is good enough to rely on. Since it can serve as an example of the piece of knowledge alien civilization acquired, it promotes philosophical comprehension of the distinguishing nature of science. Last decades of the past century were rich on the development of the philosophical thought regarding “reformulations of certain physical theories without any quantification over mathematical objects”. The trigger was Hartry Field’s offer of the strategy to reformulate the Newtonian gravitation theory, which assumes “quantification over space-time regions” as replacement for “quantifying over real numbers”. The plan prompted the attempts to reformulate Quantum Mechanics (QM) in a similar fashion. However, Malament argues that one cannot apply the strategy in pursuit to reformulate QM since this case lacks nominalist proxies appropriate for quantification. Nevertheless, “nominalistically acceptable version of” this theory has a value thanks to “the importance of QM to our overall picture of the world”. Mark Balaguer asserts that he “responded to Malament’s objection, by explaining how” one can normalize QM. However, Otávio Bueno opposes to this explanation criticizing Balaguer’s argumentation regarding the claim that “quantum probability statements are about physically real propensities of quantum systems”. In what follows, I show that Bueno does not take into account specifics of QM in his critique of the Balaguer’s work. Therefore, Balaguer’s strategy of normalizing QM can potentially give rise to a pattern of knowledge extraterrestrial life could acquire. However, Hilbert already came up with such the pattern for the first time.

### Showing That Bueno Does not Take into Account Specifics of QM

Geometry that does not deal with numbers. Specifically, Hilbert got rid of dealing with numbers in geometry. Instead of them, he introduced the concepts of betwenness, congruence, and point. Specifically, in his formalism Point 1 is between Point 2 and Point 3 if it lies on the piece of line which edges are Point 2 and Point 3. Moreover, two line-pieces are congruent if their lengths are identical. Remarkably, that Hilbert proves that this system is isomorphic to ordinary geometry if the following axioms are valid.

**Axiom 1**. A segment with edges at Points 1 and 2 and the one with edges at Points 3 and 4 are congruent if and only if dstnc1, 2 = dstnc3,4.

**Axiom 2**. Point 1 is between Point 2 and Point 3 if and only if dstn1, 2 + dstnc2, 3 = dstnc1, 3.

Here dstnc is the function that serves as length in the ordinary geometr. Hilbert’s success prompted Hartry Field to offer the strategy to reformulate the Newtonian gravitation theory, which assumes dealing with regions in space and time instead of expressing quantities in terms of real numbers as well as introduction of appropriate predicate involving comparison.

Propensities fit the framework of nominalism. However, Malament argued that such the idea is not going to work for QM since in this case there are no nominalist substitutes to deal with. Particularly, he points out that Jauch and Piron with collaborators introduced the grid of eventualities and tried to prove that it “is necessarily isomorphic to subspaces of some Hilbert space” and opposes the existence of such the entities. According to Bueno, Balaguer’s approach to nominalizing QM has the same flaw. Specifically, Bueno is not content with the following. Balaguer proposes to assume that propensity properties such as “the 0.5-strengthed propensity of a z+ electron to be measured spin-up in the x direction” are the “physically real” ones “ of quantum systems”. As for the rest of it, this approach mimics the Field’s one to nominalizing Newton’s theory of gravitation through assuming that these “propensity properties” correspond to “the closed subspaces of our Hilbert spaces”. In other words, this trick provides Balaguer with opportunity to change quantifying over vectors in Hilbert spaces to dealing with propensity properties. Nevertheless, he offers the reasoning in support that the notion of propensity fits the framework of nominalism.

Balaguer’s arguments. Firstly, propensities are qualities of specific real entities. Therefore, they exist in our universe and even can give rise to some outcome. Specifically, he indicates that the charge of the particle results in its specific trajectory in the external magnetic field and, therefore, it exists inside the particle. Although one cannot indicate its exact location, he or she cannot say that the charge is outside our universe either. Secondly, if one accepted the first argument, then the second one would be the statement that having physical properties at one’s disposal he or she needs nothing in addition to nominalize QM. To shed light on this argument, Balaguer explains why he agrees with Malamant regarding the fact that propositions Jauch, Peron, and their collaborators used do not fit the framework of nominalism. As Balaguer noted, “working with events” assumes dealing with “the appropriate orthomodular” grid “for a particular set of mutually incompatible observables”. However, coming up with such the grid entails referring to “the complete infinite set S(E) of events associated with these observables”. Such the treatment of events is analogous to treating abstract items since among the events associated with the observables there are the ones that have not occured yet. As for propensities, a specific quantum state produces “a set S(P) of propensities that” cause “the appropriate sort of structure”. Therefore, any quantum system “has an infinite collection of propensities that” result in “a (nominalistic) orthomodular lattice” thanks to the fact that every such the “system is always in a particular state”. Thirdly, since “propensities are just physical properties, like temperatures and lengths,” one can discard them in the Field’s style. Specifically, one may “eliminate references to r-strengthed propensities by introducing propensity-relations that hold between quantum systems”. In other words, instead of constructing complexes from stuff like the above mentioned electron propensity, one may “build structures from the quantum systems themselves”. For example, one may substitute “state-Ψ electrons are (A,Δ)-propensity-between state-Ψ1 electrons and state-Ψ2 electrons” for “state-Ψ electrons have r-strengthed propensities to yield values in Δ for measurements of A”. However, in view of the Borel set’s value for the theory of probability, it is challenging to explain how one can come up with describing “propensity between” relationship omitting operations on numbers due to figuring out specific values of probabilities.

Flaws in Bueno’s critique. As for Balaguer, to resolve this issue he offers to get rid of intervals in the statements. For instance, the statement “there is a probability of 0.75 that a state-Ψ electron will yield a value in the closed interval [m1, m2] for a measurement of momentum” he offers to replace with the following one “a state-Ψ electron has a 0.75-strengthed propensity to be momentum-greater-than-or-equal-to a state-Ψ1 electron and momentum-less-than-or-equal-to a state-Ψ2 electron”. Here “Ψ1 is the state of having a momentum value of m1 and Ψ2 is the state of having a momentum value of m2”. However, Bueno suggests that this excuse is not valid since m1 and m2 are specific numbers, and as such, they are abstract objects. Nevertheless, in QM there is the notion of complete set of physical quantities. One can simultaneously measure all quantities from a specific complete set of physical quantities. Moreover, results of these measurements are eigen values of the operators that correspond to physical quantities from the complete set and their values uniquely determine the quantum state. For instance, the wave function of free electron with momentum equal to m1 and energy equal to (m1)2t/(2me) where me is the mass of electron has the strictly specified form, which is the following: Ψ(t, x)=exp{i(m1x-(m1)2t/(2me))/ћ}/(2πћ)1/2. Hence, specifying the values of m1 and m2 is identical to specifying the respective quantum states that are not abstract. Besides, Bueno suggests that in the light of the above-mentioned Balaguer’s argument that events are abstract objects, the first Balaguer’s one regarding the claim that physical properties are not abstract objects is not valid too. Specifically, he offers reasoning that is the following. Propensity is just a particular tendency to conduct in a specified way. Obviously, exhibiting such the tendency relates to the possible world. Although most philosophers view a possible world as an abstract entity, there are the ones that regard such the worlds to be not abstract. However, these outstanding philosophers view all possible worlds except for the real one to be not actual either. Therefore, if Balaguer argued that events are abstract since they have not realized, he has to admit that physical properties are abstract too. For instance, if one mixed salt with water, the salt would pass into solution. Therefore, a portion of a salt on the table is prone to dissolving into added water due to chemical composition of this spice. Hence, this tendency of the salt is its physical property. However, it does not necessarily realize in the actual world since people need extra salt only on rare occasions. Unfortunately, this reasoning does not take into account the specific of QM such as the notion of a complete set of physical quantities. According to this notion, propensities or physical quantities relate to an existing quantum state. Therefore, in QM propensities are those physical properties that realize in the actual world. For instance, in the above-mentioned example of the free electron with specific values of its momentum and its energy these two quantities are propensities that are real in the actual world since the electron does have such the momentum and such the energy. Moreover, since these propensities cause the specific density of the energy flux, they are inside the electron and, therefore, belong to the actual world. As for coordinate of this electron, it is not a physical quantity in this case since it does not belong to a complete set of the quantum state in hand due to the uncertainty principle. At the same time, the electron’s momentum and energy are real since they determine the density of energy flux.

### Conclusion

Thus, nominalization of existing physical theories has a value for the development of the philosophical thought since it sheds light on the science nature allowing visualizing the possible piece of knowledge of an alien world. Hilbert’s achievement in coming up with the reformulation of geometry that fits nominalism framework inspired philosophers to check whether such the reformulation is feasible for various physical theories. Field delivered arguments in favor of the plausibility of the reformulation of the Newtonian gravitational theory that resembles the Hilbert’s one of geometry. One of the essences of Hilbert’s approach consists in introduction of points and segments that are real instead of point coordinates that are abstract. Another one is replacing values of distances that are abstract with relations between points and the ones between segments that are actual. As for Field, he offered to mirror the Hilbert’s ideas through introduction of regions in space-time instead of segments and comparative predicates instead of the Hilbert’s relations. Malament suggested that it is impossible to come up with the analog of a region in the case of QM that nominalists would accept. However, Balaguer pointed out that physical quantities comprising a complete set could serve as such the analog. He named these quantities as propensities. Balaguer’s illustration of this claim based on the example from classical physics provoked Bueno to argue that propensities do not fit the nominalism framework using the examples also from the classical physics. Specifically, one of his arguments indicates that propensities do not necessarily realize in the actual world. However, the notion of complete set of physical quantities in QM explicitly indicates that physical quantities relate to only actual quantum state. Another his argument suggests that the eigen value of the operator of the physical quantity is an abstract object since it is a number. However, such the number often fully determines the quantum state as in the case of free electron with specific values of its momentum and its energy. Moreover, in the worst case scenario a finite number of eigen values of operators that correspond to physical quantities comprising the complete set is enough to determine the actual quantum state. Therefore, the second Bueno’s argument is not valid too. Hence, Balaguer’s strategy could result in nominalist version of QM since Bueno’s critique is not accurate enough.