Strata of Real Being

To discuss ontology in general and to compare competing ontological proposals some structure is useful. This structure will help to place candidate beings in relation to others, understand their way of being and, for example, avoid the category mistake of identifying elements of physical reality with their mathematical representation.  Here we will explore the proposal that reality is indeed stratified and secondly what that stratification may be. In critical ontology it will be important to distinguish whether these levels are levels of reality per se or levels in a description of reality. 

The starting point cannot just be comprehensive scepticism about whether anything exists or not. That will get us nowhere. We will start by taking the phenomena around us as the they appear. There appear to be things that take up space and are stable. There appear to be things that take space and grow. There are appear to be things that have some purpose and make choices. There appear to be things that are made up of other things.   In addition, there are beings with emotional states and awareness of making choices. There is also awareness of resistance or pushback from other things. There is awareness of being the capacity of some beings to change the state of other things. We may be mistaken about particular things and their relationship to each other but the appearances are the appearances.  In talking about things and beings, we have implicitly used categories such a space, time, choice, emotion, awareness and substance. The categories will play an important role is characterising the levels in which beings exist. There will be separate post on categories.

Hartmann proposes a pluralistic view of reality, that covers all the things and beings listed above, that appear in four dynamically interrelated strata [1]:

1. Inorganic 
2. Organic 
3. Psychic 
4. Spiritual. 

The layers are presented in the order of decreasing ontological strength. That is, in each layer the beings depend on those in stronger layers to exist. However, the beings in the less strong layers cannot be reduced to beings in stronger layers. They have their specific categories of existence. There are also multi strata beings as will be illustrated at the end of this post. The question of whether these levels exist and the nature of their existence will not be examined here other than to recognise them as structural proposals and so will be seen to at least exist at the spiritual level. 

The name given to each level is not problematic except for the fourth.  Spiritual being is a translation from the German geistige Sein and the use of the term Geist follows the tradition influenced by Hegel and Dilthey—as it is used, for example, in Geisteswissenschaft—and its meaning covers both individual mind as well as superindividual culture, which is why it lacks an adequate English translation. No religious sense of the term is intended here. Beings at the spiritual level cannot exist without the other levels. 

Although the listing of the four strata suggests a hierarchy, to provide this thought with some substance something needs to be said about the relationship between the strata. The levels of reality are characterised (and therefore distinguished) by their categories. With regard to the levels of reality, it is obvious that the categories in question are ontological rather than epistemological. There are categories that pertain to reality as whole - such as time, whole, part. At the inorganic level the categories of physics are different from those of chemistry. Some examples follow.

1. Inorganic stratum, with categories such as space, time, process, substance, is organised through causality.
2. Organic stratum, with categories such as metabolism, assimilation, self-regulation, self-reproduction and adaptation, is organically organised - the details of the organising principles are unknown for the time being.
3. The psychic stratum, with categories such as consciousness, pleasure, act and content, has its own psychic network of relations. Equally unknown is the inner essence of the form of determination of mental acts.
4. Spirit includes categories of fear, hope, will, freedom, thought, personality, but also society, historicity, or intersubjectivity. It is organised as three modes of spirit;
• Personal - thought, freedom, conscious action and moral will
• Objective - superindividual but living as culture, collective or society
• Objectivated - superindividually existing but non-living. As in cultural artifacts and institutions

All categories at the inorganic level, such as space and time, recur at the organic level.  This recurrence of categories is called superinformation. There is also autonomy off levels and this is known as superposition - building above. For example, consciousness, a category of the psyche, is not spatial.

In addition there are the “fundamental” categories (common for all strata of real being) that come in pairs: 

principle–concretum; structure–modus; form–matter; inner–outer; determination–dependence; quality–quantity; unity–manifoldness; harmony–conflict; contrast–dimension; discretion–continuity; substratum–relation; element–structure.

The above only provides a sketch. For more detail on the categories of real being see Cicovacki [1] who provides a bridge to the detailed work by Hartmann.

The basic structure can be summarised in a diagram.

Hartmann's model of reality includes mind and spirit superposed on an ontologically stronger foundation of life and matter. There is, however,  a feedback loop that is not explicit in the diagram. The objectivated spirit gives rise to artifacts, such as books and architecture, that are also actualised as things at the material level. For example, a person, James Joyce say, has some thoughts (personal spirit) about Dublin society (objective spirit) and creates a novel, Ulysses (objectivated spirit), that is published as book (still objectivated spirit) and printed in several editions (inorganic objects).

The being James Joyce was made of inorganic matter organised as an inorganic life form with a mind and emotional life on which was superposed a spirit that could choose to work for years on intricate works of literature. He would also often choose to drink heavily after a day's work. He loved his immediate family.

A further example. This blog exists as objectivated spirit. It came into being because of my choice and was enabled by the infrastructure supplied by Google. Making and maintaining the blog are choices made by me at the level of my personal spirit. The blog infrastructure is dependent on organised matter (inorganic level) the organisation of it is objectivated spirit because it was designed. The blog and its posts are captured and encoded in the blog infrastructure. The posts are instantiated on the screen of a reader's device. The readers use their sense organs and brain (organic level) to access the information that is processed through their psychic level and, hopefully, understood at the level of their personal spirit.  The blog readers and I form a loose community that exists as objective spirit.

Simple beings such as electrons may only inhabit the inorganic level but the existence of complex beings may be across many or all strata. This complex existence is a process rather than a hierarchy.

[1] Cicovacki, Predrag, The Analysis of Wonder. Bloomsbury USA. 2014

The combination of quantum objects

 Quantum objects have been introduced in an earlier post. These objects, if not the most fundament entities, are quite basic to the constitution of the inorganic layer of reality. In contributing to the complexity of the real world these objects must combine. 

Many calculations in quantum mechanics can be carried out by considering a single isolated system and its the operator representing the total energy of the system - the Hamiltonian. That is how the spectrum of the Hydrogen atom is calculated and how tunnelling is analysed. However, it is especially important in examining contextual issues to consider mathematical representation of the combination and interaction of objects. Having introduced the concept of the σ-complex for an isolated system, the σ-complex of two systems \(S_1\) and \(S_2\) will now be constructed.

As is standard in the von Neumann formulation [1], if the Hilbert spaces \(\mathcal{H}_1\) associated with object \(S_1\)  and \(\mathcal{H}_2\) associated with object \(S_2\), the Hilbert space of  the combined object \(S_1 + S_2\) is the tensor product \(\mathcal{H}_1 \otimes  \mathcal{H}_2\).

 Given the combined system $S_1 + S_2$ with the $\sigma$-complex  $Q(\mathcal{H}_1)\oplus Q(\mathcal{H}_2)$, there is a unique Hilbert space $\mathcal{H}_1\otimes \mathcal{H}_2$ such that $Q(\mathcal{H}_1) \oplus Q(\mathcal{H}_2) \cong Q(\mathcal{H}_1\otimes\mathcal{H}_2)$, where \(\cong \) means equivalence unique up to isomorphism. For the proof, in finite dimensional Hilbert spaces, see Kochen [2]. 

The combination of two quantum objects gives a quantum object. This would build up a world consisting only of quantum objects. Is this the case or do the quantum charactertics weaken with many combinations? We will return to this question.

[1] John von Neumann. Mathematische Grundlagen der Quantenmechanik. German. Springer, Berlin, Heidelberg, 1932

[2] Simon Kochen. A Reconstruction of Quantum Mechanics. In: ArXiv e-prints (June 2015).

A Mathematical Formulation of Quantum Mechanics

The mathematical formulation to be discussed here is a modification of Kochen's Reconstruction of Quantum Mechanics [1]. It concentrates on a mathematical formulation using a union of σ-algebras that he calls a complex. In Kochen's reformulation it is through the interaction of a quantum system under consideration with another system that properties gain physical values. Such properties are said by Kochen to be relational or extrinsic, as opposed to the fixed intrinsic properties such as the charge of an electron.

While the mathematics depends heavily on the reconstruction by Kochen it makes use of von Neumann ∗-algebras (see Thirring [2]) rather than the Boolean algebras used by Kochen. This choice is made to avoid any hint of a logical interpretation. This blog also differs from Kochen's in the treatment and assignment of both intrinsic and extrinsic properties. The extrinsic properties will be treated here as an actualisation of potential, dispositional properties of the quantum entity that are shaped by its context.

In the mathematical formulation of quantum mechanics, a physical system, \(\mathbf{S}\), is characterized by a list of properties (called observables in the standard quantum mechanics), that can be represented by abstract self-adjoint operators, 

\[\mathcal{O}_\mathbf{S}=\{ \hat{X}_i = \hat{X}^*_i  | i \in \mathcal{I}_\mathbf{S}\} ,  \]

 with \(\mathcal{I}_\mathbf{S}\)  a set of indices depending on \(\mathbf{S}\), where every operator \(\hat{X} \in \mathcal{O}_\mathbf{S}\) represents a physical property characteristic of \(\mathbf{S}\), such as the total momentum, energy or spin of all particles localised in some bounded region of physical space and belonging to an ensemble of (possibly infinitely many) particles constituting the system \(\mathbf{S}\).

At every time \(t\), there is a representation of \(\mathcal{O}_\mathbf{S}\)  by self-adjoint operators acting on a separable Hilbert space \(\mathcal{H}_\mathbf{S}\): 

\[ \mathcal{O}_\mathbf{S} \ni  \hat{X}_i  \mapsto X_i =X^*_i \in \mathcal{A}(\mathcal{H}_\mathbf{S}) ,     \]

where \(\mathcal{A}(\mathcal{H}_\mathbf{S})\) is the \(*\)-algebra of all bounded operators acting on \(\mathcal{H}_\mathbf{S}\).

In general, the operators in the \(*\)-algebra do not all commute. This has deep physical consequences: 

• Properties that cannot appear together are represented in quantum mechanics by self-adjoint operators that do not all commute. 

• The sets of operators that do commute provide a set of properties for a measurable mathematical structure which is a \(\sigma\)-algebra.

The \(\sigma\)-algebra is essential to constructing mathematical probability models using the Kolmogorov axiomisation [3]. Therefore, each collection of commuting observables of the object has an associated \(\sigma\)-algebra.  So, a collection of \(\sigma\)-algebras that together capture all the property values associated with the object covers all possible events for that object in itself. This structure is the minimal one which contains all the $\sigma$-algebras arising from all the properties of the particle. 
Each dispositional property of an object, such as spin or position, has a \(\sigma\)-algebra of possible values that appear in interaction with other objects. All these \(\sigma\)-algebras form a complex property structure for the object under consideration. The formal definition of this notion is as follows:

 Let $F$ be a family of $\sigma$-algebras. The $\sigma$-complex $Q_F$ based on $F$ is the union, $\cup B$, of all $\sigma$-algebras $B$ lying in $F$. 

Generally, the family $F$ is left implicit, and reference is simply to a $\sigma$-complex $Q$ however the family \(F\) does tie the complex to an object that has properties that take values depending on the set of possible interactions that it can have with other objects. Usually $\sigma$-complexes that are closed under the formation of sub-$\sigma$-algebras are discussed. It is possible, in any case, to always close a $\sigma$-complex by adding all its sub-$\sigma$-algebras. This complex will include all the possible values that the dispositional properties of a particle can take in all possible interaction contexts.  \(B\) will denotes the \(\sigma\)-algebra relating to the property values of a particular (commuting) set of observables of the particle or system. 

In the mathematical formulation of Quantum Mechanics $\mathcal{H}$ is a Hilbert space. For the quantum object under consideration all properties are represented by self-adjoint operators and possible events are represented projection operators. For pair-wise commuting projection operators closed under the operation of orthogonal complement and countable product \(\prod_i \pi_i\) forms a \(\sigma\)-algebra. The family of all such \(\sigma\)-algebras form a \(\sigma\)-complex, \(Q(\mathcal{H})\), for the object in its possible interaction contexts. 

Proposition

In Quantum Mechanics, the operators representing the possible values of the properties of a system form the $\sigma$-complex $Q(\mathcal{H})$ of projections of the Hilbert space $\mathcal{H}$ of the system.       

It has been shown [1] that this proposition holds and the mathematical formalism using \(Q(\mathcal{H})\) is equivalent to the standard quantum formulation of the theory on \(\mathcal{H}\). However, the interpretation of the formalism here is that while a quantum object has a complete set of properties they appear only when the situation invokes the \(\sigma\)-algebra in the complex that allows that property to take a specific set of values.

[1] Simon Kochen. A Reconstruction of Quantum Mechanics. In: ArXiv e-prints (June 2015).
[2] WaIter Thirring. A Course in Mathematical Physics 3: Quantum Mechanics of Atoms and Molecules. Springer-Verlag, 1979.
[3] Andrey N. Kolmogorov. Foundations of Probability Theory. New York, 1950.

Quantum objects

Quantum mechanics is part of physics and experimentation is an activity in physics. Therefore, the entities of interest in physics must be observable to some extent. The must make an appearance. That is, entities of interest must be objects; they appear in at least one field of sense. 

The ontological framework for the discussion will be that quantum objects are physical objects that have some properties that can appear in one of a collection of fields of sense. To move from ontology to physical theory an explanatory structure is needed and, when possible, a mathematical formulation is adopted that describes the behaviour of the quantum objects in a precise way.  

There are puzzling and  paradoxal effects of quantum mechanics, be they the thought experiments of Einstein, Podolsky and Rosen (EPR) or Wigner's friend, Schrödinger's cat or the actual double slit experiment. These point to the need to re-examine what is meant by a physical property of an object.  In quantum mechanics it is known that all properties need a specific context in which to appear as quantified values. This is known to be the case in general, from the work of Kochen and Specker [1] but has been a common understanding among physicist based on the insights obtained from experiments such as that of Stern and Gerlach [2].  This experiment measures some consequences of the electron having spin. In quantum mechanics spin, while having the same dimensions as classical spin, is quite a different phenomenon and will be discussed below.

Before moving on to spin consider the Heisenberg Uncertainty Principle, which means an electron, for example, cannot have a precise value of its position and its momentum at the same time. Does this mean that it sometimes has position as a property and sometimes momentum but never both together? That understanding of the theory would mean that these properties are of the context rather than the object of the experiment. According to the ontology of objects and their properties being developed in this blog a particle, such as an electron, always has the properties of position and momentum but these properties take values by appearing in distinct Fields of Sense. This means that the context in which the particle position takes a value is different from that in which its momentum takes a value.

We now consider a particle with spin \(\frac{1}{2}\), an electron,  situated and free to move in three dimensions. Following Faddeev and Yakubovskii [3], the electron combines its spin Hilbert space (\(\mathbb{C}^2\)) with that associated with the electron's movement in three dimensions (\(\mathbb{R}^3\)). The Hilbert space \(\mathcal{H}\) for the electron's position and momentum is the space \(L^2(\mathbb{R}^3)\) of square-integrable complex-valued functions. This Hilbert space is the state space that can represent the electron in position coordinates (neglecting spin). In mathematical notation, the state space with spin becomes the enlarged Hilbert space

\[\mathcal{H}_S = L^2(\mathbb{R}^3)\otimes\mathbb{C}^2.\] This captures the intuition that the state function becomes a pair of complex valued \(L^2(\mathbb{R}^3)\) functions. For a pure state \({\Psi(x)}\) in \(\mathcal{H}\) there are two orthogonal states in \(\mathcal{H}_S\):

\[\Psi_{\uparrow} = \binom{\Psi(x)}{0} , \, \, \, \Psi_{\downarrow} =  \binom{0}{\Psi(x)}.\]

Focussing on the spin property, \(\hat{S}\), a self-adjoint operator \(S\) on \(\mathbb{C}^2\), representing \(\hat{S}\) can be represented as the linear combination of four linearly independent matrices. A convenient set of such matrices are the identity matrix and the well-known Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\)

\[I = \begin{pmatrix} 1 & 0\\0 & 1\end{pmatrix}, \sigma_1 = \begin{pmatrix}0 & 1\\1 & 0\end{pmatrix},\sigma_2 = \begin{pmatrix}0 & -i\\i & 0\end{pmatrix} \sigma_3 = \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix}\]

Quantum physics is brought into this mathematics by introducing the Plank constant, \(\hbar\), and defining the self-adjoint spin-\(\frac{1}{2}\) operators as  

\[ S_j = \frac{\hbar}{2} \sigma_j, j =  1, 2, 3 \label{eq:Sj} ~~~~~~~~~~~~~(1)\]

and these have the angular momentum commutation relations

\[[S_1,S_2]= iS_3, \,  \, [S_2,S_3]= i S_1, \,  \, [S_3,S_1]= iS_2.\]

The operators \(S_j\) have eigenvalues \(\pm \hbar/2\), which are the admissible values of the projections of the spin on some direction. The vectors \( \Psi_{\uparrow} = \binom{\Psi(x)}{0}\) and \(\Psi_{\downarrow} = \binom{0}{\Psi(x)}\) are eigenvectors of the operator \(S_3\) with eigenvalues \(+\hbar/2\) and \(-\hbar/2\). The eigenvectors of \(S_1\) and \(S_2\) are linear combinations of \(\Psi_{\uparrow}\) and \(\Psi_{\downarrow} \).  Therefore, these vectors describe states with a definite value of the third projection of the spin. Similar constructions can be made for the other spin components. But as the components do not commute, they cannot simultaneously take precise values in all components. The spin will appear to interact with some other object, involving a magnetic interaction, to take a value of \(+\hbar/2\) or \(-\hbar/2\). Independent of the direction defined by the interaction, the possible values the spin property can take are \(\pm \hbar/2 \), with probabilities determined by the state of the electron. The probabilities of \(\pm \hbar/2 \) appearing sum to one.  

The property values appear when the appropriate Field of Sense is available. However, in relation to this point it should be noted, that the isolated system that has properties represented by Hermitian operators, such as the spin-\(\frac{1}{2}\) \( S_j\) , equation (1), that have eigenvalues \(\pm \frac{1}{2}\hbar\) in the absence of any interaction term coupling it to another object. The spin values are quantitative characteristics of the spin property, but the values will only actually appear once the appropriate Field of Sense is available to provide a context for an interaction with another object.

A Field of Sense affords a context for the appearance of properties of the quantum object, whether, position, momentum or spin components.

[1] Simon Kochen and Ernst P. Specker. The Problem of Hidden Variables in Quantum Mechanics. In: J. Math. & Mech. 17 (1967), p. 59.

[2] W. Gerlach and O. Stern. Der experimentelle Nachweis der Richtungsquantelung. In: Zeitschrift für Physik, (1922), pp. 349-352.

[3] L D Faddeev and O A Yakubovskii. Lectures on Quantum Mechanics for Mathematics Students. Student Mathematical Library. 47 vols. American Mathematical Society, 2000.

The ontology of dispositions

 The motivation for analysing the ontology of dispositions is to understand partial causes and objective chance. It is objective chance that is the stronger interest here, motivated by wanting to understand the behaviour of basic entities at the inorganic ontic level. That is, entities that fall within the remit of quantum physics.

 Dispositional properties, as described by Anjum and Mumford [1], will be used to clarify how there can be properties of quantum entities that cause quantifiable property values to appear and influence other objects. This will develop into a dispositional understanding of probabilities in quantum mechanics.

While it is easy to appreciate dispositional properties intuitively it can be harder to make the concept precise enough to be useful in constructing a scientific theory. To help the following definitions will be used: 

Definition 1

A disposition is a property (such as solubility, fragility, elasticity) whose actualisation entails that the thing which has the property would change, or bring about some change, under certain conditions. 

Definition 2

A dispositional power is a physically existing dispositional property of an entity. 

In everyday use to say that some object is soluble is to say that it would dissolve if put in some liquid. To say that something is fragile is to say that it would break if, for instance, dropped in suitable circumstances; to say that something is elastic is to say that it would stretch when pulled. The fragility (solubility, elasticity) is a disposition; the breaking (dissolving, stretching) is the actualisation of the disposition. These examples are intuitively uncontroversial, but position and momentum are not classical dispositional properties of a particle. Results such as that of Kochen and Specker [2] show that the quantum situation is more subtle than dispositional properties such as solubility and elasticity because they demonstrate a contradiction in assuming that it is possible to assign simultaneously values to all observable properties in all states. Their result, although called a paradox in their paper, indicates the need for an ontology that deals with this more subtle reality. 

There is still ongoing debate about the nature of dispositions and key elements of this debate are outlined in Mumford [3]. However, the application in physics makes the ontology of causal powers the more attractive interpretation of the nature of dispositions [4].  These powers exist and can be independent of an observer. 

With the motivation to provide an objective interpretation of chance in quantum mechanics Popper [5], [6] took a significant step towards a theory of dispositional properties. He took the position that quantum mechanics is a statistical description of the behaviour of physical entitles and introduced dispositional properties through his propensity interpretation of probability:

 ...waves (even those of the second quantization) are mathematical representations of propensities, or of dispositional properties, of the physical situation (such as the experimental set-up), interpretable as propensities of the particles to take up certain states. 

Popper proposed that the wave aspect of quantum mechanics determines the tendency of the particles to assume specific states under certain conditions but with the strength of this disposition given by a probably. However, Popper developed his thinking prior to the demonstration that it is impossible, in general, to assign intrinsic values to all observable properties in all states and he did not completely develop the ontological consequences. Popper remained committed to a physics in which particles have properties that take values as in classical physics but that the quantum theory provides a statistical description of that reality.  

It is proposed here that properties of a physical entity need not all take values simultaneously for them to be physically existing properties. Dispositional properties are properties that govern how the entity tends to appear in various contexts. For example, a separate physical system may be of the form that couples to an electron's spin in a specific direction and instantiate a value for it. That would be how the electron spin appears to that system. 

In the dispositional theory developed in this blog there are properties, objective causal powers, and in addition the probabilistic appearance of the properties as property values.  The concepts required to clarify what is meant by `to exist' and `to have a property' are set out in "Why critical ontology?" and Fields of Sense. 

Dispositional properties cannot themselves appear in a Field of Sense, but their effects can.  This is the point where the modification of the Field of Sense ontology is used that recognises that there are properties of objects that affect appearances but do not themselves appear. This modification means that Fields of Sense do not provide a complete ontology, contrary to what seems to be proposed by Markus Gabriel.

A dispositional property such as fragility appears if the fragile object breaks, on falling from a table, for example. This is how it appears in a domestic Field of Sense. The fragile object, for example a porcelain cup, exits and appears in the domestic scene even if it does not break. It also has properties that are not dispositional for example it has a mass. So, an object can have non-dispositional properties that pin it down in a context. The electron has the properties of charge and (rest) mass that pins it down in the physical universe in which it has, as will be explained elsewhere, dispositional spin, position, and momentum. 

So, the position taken on the ontology of dispositions is:

• Physical objects have at least one property that appears or causes a value to appear in some Field of Sense. 
• Dispositional powers cause property values to appear in Fields of Sense.
• Physical dispositions are properties of some physical objects.

[1] Rani Lill Anjum and Stephen Mumford. What Tends to Be. The Philosophy of Dispositional Modality. Routledge, 2018.

[2] Simon Kochen and Ernst P. Specker. The Problem of Hidden Variables in Quantum Mechanics. In: J. Math. & Mech. 17 (1967), p. 59.

[3] Stephen Mumford. Routledge Encyclopedia of Philosophy. Dispositions. 2011.

[4] Rani Lill Anjum and Stephen Mumford. Causation in Science and the Methods of Scientific Discovery. Oxford University Press, 2018.

[5] Karl R. Popper. Quantum Mechanics without 'The Observer'. In: Quantum Theory a

nd Reality. Ed. by Mario Bunge. New York: Springer-Verlag, 1967, p. 32.

[6] Karl R. Popper. Quantum Theory and the Schism in Physics. Unwin Hyrnan Ltd, 1982.

[7] Markus Gabriel. Fields of Sense. A New Realist Ontology. Edinburgh University, 2015.