Reductionism of Numerical Complexity: A Wittgensteinian Excursion

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Wittgenstein’s criticism of Russell’s logicist foundation of mathematics contained in (Remarks on the Foundation of Mathematics) consists in saying that it is not the formalized version of mathematical deduction which vouches for the validity of the intuitive version but conversely.

If someone tries to shew that mathematics is not logic, what is he trying to shew? He is surely trying to say something like: If tables, chairs, cupboards, etc. are swathed in enough paper, certainly they will look spherical in the end.

He is not trying to shew that it is impossible that, for every mathematical proof, a Russellian proof can be constructed which (somehow) ‘corresponds’ to it, but rather that the acceptance of such a correspondence does not lean on logic.

Taking up Wittgenstein’s criticism, Hao Wang (Computation, Logic, Philosophy) discusses the view that mathematics “is” axiomatic set theory as one of several possible answers to the question “What is mathematics?”. Wang points out that this view is epistemologically worthless, at least as far as the task of understanding the feature of cognition guiding is concerned:

Mathematics is axiomatic set theory. In a definite sense, all mathematics can be derived from axiomatic set theory. [ . . . ] There are several objections to this identification. [ . . . ] This view leaves unexplained why, of all the possible consequences of set theory, we select only those which happen to be our mathematics today, and why certain mathematical concepts are more interesting than others. It does not help to give us an intuitive grasp of mathematics such as that possessed by a powerful mathematician. By burying, e.g., the individuality of natural numbers, it seeks to explain the more basic and the clearer by the more obscure. It is a little analogous to asserting that all physical objects, such as tables, chairs, etc., are spherical if we swathe them with enough stuff.

Reductionism is an age-old project; a close forerunner of its incarnation in set theory was the arithmetization program of the 19th century. It is interesting that one of its prominent representatives, Richard Dedekind (Essays on the Theory of Numbers), exhibited a quite distanced attitude towards a consequent carrying out of the program:

It appears as something self-evident and not new that every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers [ . . . ] But I see nothing meritorious [ . . . ] in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers.

Perec wrote a detective novel without using the letter ‘e’ (La disparition, English A void), thus proving not only that such an enormous enterprise is indeed possible but also that formal constraints sometimes have great aesthetic appeal. The translation of mathematical propositions into a poorer linguistic framework can easily be compared with such painful lipogrammatical exercises. In principle all logical connectives can be simulated in a framework exclusively using Sheffer’s stroke, and all cuts (in Gentzen’s sense) can be eliminated; one can do without common language at all in mathematics and formalize everything and so on: in principle, one could leave out a whole lot of things. However, in doing so one would depart from the true way of thinking employed by the mathematician (who really uses “and” and “not” and cuts and who does not reduce many things to formal systems). Obviously, it is the proof theorist as a working mathematician who is interested in things like the reduction to Sheffer’s stroke since they allow for more concise proofs by induction in the analysis of a logical calculus. Hence this proof theorist has much the same motives as a mathematician working on other problems who avoids a completely formalized treatment of these problems since he is not interested in the proof-theoretical aspect.

There might be quite similar reasons for the interest of some set theorists in expressing usual mathematical constructions exclusively with the expressive means of ZF (i.e., in terms of ∈). But beyond this, is there any philosophical interpretation of such a reduction? In the last analysis, mathematicians always transform (and that means: change) their objects of study in order to make them accessible to certain mathematical treatments. If one considers a mathematical concept as a tool, one does not only use it in a way different from the one in which it would be used if it were considered as an object; moreover, in semiotical representation of it, it is given a form which is different in both cases. In this sense, the proof theorist has to “change” the mathematical proof (which is his or her object of study to be treated with mathematical tools). When stating that something is used as object or as tool, we have always to ask: in which situation, or: by whom.

A second observation is that the translation of propositional formulæ in terms of Sheffer’s stroke in general yields quite complicated new formulæ. What is “simple” here is the particularly small number of symbols needed; but neither the semantics becomes clearer (p|q means “not both p and q”; cognitively, this looks more complex than “p and q” and so on), nor are the formulæ you get “short”. What is looked for in this case, hence, is a reduction of numerical complexity, while the primitive basis attained by the reduction cognitively looks less “natural” than the original situation (or, as Peirce expressed it, “the consciousness in the determined cognition is more lively than in the cognition which determines it”); similarly in the case of cut elimination. In contrast to this, many philosophers are convinced that the primitive basis of operating with sets constitutes really a “natural” basis of mathematical thinking, i.e., such operations are seen as the “standard bricks” of which this thinking is actually made – while no one will reasonably claim that expressions of the type p|q play a similar role for propositional logic. And yet: reduction to set theory does not really have the task of “explanation”. It is true, one thus reduces propositions about “complex” objects to propositions about “simple” objects; the propositions themselves, however, thus become in general more complex. Couched in Fregean terms, one can perhaps more easily grasp their denotation (since the denotation of a proposition is its truth value) but not their meaning. A more involved conceptual framework, however, might lead to simpler propositions (and in most cases has actually just been introduced in order to do so). A parallel argument concerns deductions: in its totality, a deduction becomes more complex (and less intelligible) by a decomposition into elementary steps.

Now, it will be subject to discussion whether in the case of some set operations it is admissible at all to claim that they are basic for thinking (which is certainly true in the case of the connectives of propositional logic). It is perfectly possible that the common sense which organizes the acceptance of certain operations as a natural basis relies on something different, not having the character of some eternal laws of thought: it relies on training.

Is it possible to observe that a surface is coloured red and blue; and not to observe that it is red? Imagine a kind of colour adjective were used for things that are half red and half blue: they are said to be ‘bu’. Now might not someone to be trained to observe whether something is bu; and not to observe whether it is also red? Such a man would then only know how to report: “bu” or “not bu”. And from the first report we could draw the conclusion that the thing was partly red.

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Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

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Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation SiRSj between the two ‘individual terms’ (subjects) Sj and Si. In a short hand notation we represent this relation by Rij. Relations may be composed: whenever we have relations of the form Rij, Rjl, a third transitive relation Ril emerges following the rule

RijRkl = δjkRil —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

Si = ∑j Rij —– (2)

The individual Si thus defined is an eigenstate of the Rii relation

RiiSi = Si —– (3)

The relations Rii are idempotent

R2ii = Rii —– (4)

and they span the identity

i Rii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, Rij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ Si is Rii.

Rij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of Rij

Rij = |ri⟩⟨rj| —– (6)

where state ⟨ri | is the dual of the state |ri⟩ and they obey the orthonormal condition

⟨ri |rj⟩ = δij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

n|ri⟩⟨ri|=1 —– (8)

All relations remain satisfied if we replace the state |ri⟩ by |ξi⟩ where

i⟩ = 1/√N ∑n |ri⟩⟨rn| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |ri⟩ is derived as the sum of all its interactions with the other states. Rij acts as a projection, transferring from one r state to another r state

Rij |rk⟩ = δjk |ri⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Qij = |qi⟩⟨qj | —– (11)

with

Qij |qk⟩ = δjk |qi⟩ —– (12)

and

n |qi⟩⟨qi| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

RijQkl = |ri⟩⟨rj |qk⟩⟨ql | = ⟨rj |qk⟩ Sil —– (14)

with

Sil = |ri⟩⟨ql | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(Sil) = ∑n ⟨rn |Sil |rn⟩ = ⟨ql |ri⟩ —– (16)

From the above relation we deduce

Tr(Rij) = δij —– (17)

Any operator can be expressed as a linear superposition of the Rij

A = ∑i,j AijRij —– (18)

with

Aij =Tr(ARji) —– (19)

The individual states could be redefined

|ri⟩ → ei |ri⟩ —– (20)

|qi⟩ → ei |qi⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨ri |qj⟩ changes and therefore we need an invariant formulation for the transition |ri⟩ → |qj⟩. This is provided by the trace of the closed operation RiiQjjRii

Tr(RiiQjjRii) ≡ p(qj, ri) = |⟨ri |qj⟩|2 —– (22)

The completeness relation, equation (13), guarantees that p(qj, ri) may assume the role of a probability since

j p(qj, ri) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.

Quantum Groupoid

Hopf algebra

A (finite) quantum groupoid over k is a finite-dimensional k-vector space H with the structures of an associative algebra (H, m, 1) with multiplication m : H ⊗k H → H and unit 1 ∈ H and a coassociative coalgebra (H, ∆, ε) with comultiplication ∆ : H → H ⊗k H and counit ε : H → k such that:

  1. The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that

(∆ ⊗ id)∆(1) = (∆(1) ⊗ 1) (1 ⊗ ∆(1)) = (1 ⊗ ∆(1)) (∆(1) ⊗ 1) —– (1)

2.  The counit is a k-linear map satisfying the identity:

ε(fgh) = ε(fg(1))ε(g(2)h) = ε(fg(2))ε(g(1)h), (2) ∀ f, g, h ∈ H —– (2)

3.   There is an algebra and coalgebra anti-homomorphism S : H → H, called an antipode, such that, ∀ h ∈ H ,

m(id ⊗ S) ∆(h) = (ε ⊗ id) ∆(1)(h ⊗ 1) —– (3)

m(S ⊗ id) ∆(h) = (id ⊗ ε)(1 ⊗ h) ∆(1) —– (4)

A quantum groupoid is a Hopf algebra iff one of the following equivalent conditions holds: (i) the comultiplication is unit preserving or (ii) the counit is a homomorphism of algebras.

A morphism of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.

The set of axioms is self-dual. This allows to define a natural quantum groupoid  structure on the dual vector space H’ = Homk (H, k) by “reversing the arrows”:

⟨h,φ ψ⟩ = ∆(h), φ ⊗ ψ —– (5)

⟨g ⊗ h, ∆'(φ)⟩ = ⟨gh, φ⟩ —– (6)

⟨h, S'(φ)⟩ = ⟨S(h), φ⟩ —– (7)

∀ φ, ψ ∈ H’, g, h ∈ H. The unit 1ˆ ∈ H’ is ε and counit ε’ is φ → ⟨φ,1⟩. The linear endomorphisms of H defined by

h → m(id ⊗ S) ∆(h), h → m(S ⊗ id) ∆(h) —– (8)

are called the target and source counital maps and denoted εt and εs, respectively.

From axioms (3) and (4),

εt(h) = (ε ⊗ id) ∆(1)(h ⊗ 1), εs(h) = (id ⊗ ε) (1 ⊗ h)∆(1) . (9)

In the Hopf algebra case εt(h) = εs(h) = ε(h)1.

We have S ◦ εs = εt ◦ S and εs ◦ S = S ◦ εt. The images of these maps εt and εs

Ht = εt (H) = {h ∈ H | ∆(h) =∆(1)(h ⊗ 1)} —– (10)

Hs = εs (H) = {h ∈ H | ∆(h) = (1⊗h) ∆(1)} —– (11)

are subalgebras of H, called the target (respectively source) counital subalgebras. They play the role of ground algebras for H. They commute with each other and

Ht = {(φ ⊗ id) ∆(1)|φ ∈ H’,

Hs = (id ⊗ φ) ∆(1)| φ ∈ H’,

i.e., Ht (respectively Hs) is generated by the right (respectively left) tensorands of ∆(1). The restriction of S defines an algebra anti-isomorphism between Ht and Hs. Any morphism H → K of quantum groupoids preserves counital subalgebras, i.e., Ht ≅ Kt and Hs ≅ Ks.

In what follows we will use the Sweedler arrows, writing ∀ h ∈ H , φ ∈ H’:

h ⇀ φ = φ(1)⟨h, φ(2)⟩,

φ ↼ h = ⟨h, φ(1)⟩φ(2) —– (12)

∀ h ∈ H, φ ∈ H’. Then the map z → (z ⇀ ε) is an algebra isomorphism between Ht and H. Similarly, the map y → (ε ↼ y) is an algebra isomorphism between H and H’t. Thus, the counital subalgebras of H’ are canonically anti-isomorphic to those of H. A quantum groupoid H is called connected if Hs ∩ Z(H) = k, or, equivalently, Ht ∩ Z(H ) = k, where Z(H) denotes the center of H. A k-algebra A is separable if the multiplication epimorphism m : A ⊗k A → A has a right inverse as an A − A bimodule homomorphism. When the characteristic of k is 0, this is equivalent to the existence of a separability element e ∈ A ⊗k A such that m(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a), (1 ⊗ a)e = e(a ⊗ 1) ∀ a ∈ A. The counital subalgebras Ht and Hs are separable, with separability elements et = (S ⊗ id)∆(1) and es = (id ⊗S)∆(1), respectively. Observe that the adjoint actions of 1 ∈ H give rise to non-trivial maps

H → H : h → 1(1)hS(1(2)) = Adl1(h), h → S(1(1))h1(2) = Adr1(h), h ∈ H —– (13) …….

 

Conjuncted: Unitary Representation of the Poincaré Group is a Fock Representation

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The Fock space story is not completely abandoned within the algebraic approach to Quantum Field Theory. In fact, when conditions are good, Fock space emerges as the GNS Hilbert space for some privileged vacuum state of the algebra of observables. We briefly describe how this emergence occurs before proceeding to raise some problems for the naive Fock space story.

The algebraic reconstruction of Fock space arises from the algebraic version of canonical quantization. Suppose that S is a real vector space (equipped with some suitable topology), and that σ is a symplectic form on S. So, S represents a classical phase space . The Weyl algebra U[S,σ] is a specific C∗-algebra generated by elements of the form W(f), with f ∈ S and satisfying the canonical commutation relations in the Weyl-Segal form:

W(f)W(g) = e−iσ(f,g)/2W(f + g)

Suppose that there is also some notion of spacetime localization for elements of S, i.e. a mapping O → S(O) from double cones in Minkowski spacetime to subspaces of S. Then, if certain constraints are satisfied, the pair of mappings

O → S(O) → U(O) ≡ C{W(f) : f ∈ S(O)},

can be composed to give a net of C∗-algebras over Minkowski spacetime. (Here C∗X is the C∗-algebra generated by the set X.)

Now if we are given some dynamics on S, then we can — again, if certain criteria are satisfied — define a corresponding dynamical automorphism group αt on U[S,σ]. There is then a unique dynamically stable pure state ω0 of U[S,σ], and we consider the GNS representation (H,π) of U[S,σ] induced by ω0. To our delight, we find that the infinitesimal generators Φ(f) of the one-parameter groups {π(W(f))}t∈R behave just like the field operators in the old-fashioned Fock space approach. Furthermore, if we define operators

a(f) = 2−1/2(Φ(f) + iΦ(Jf)),
a∗(f) = 2−1/2(Φ(f)−iΦ(Jf)),

we find that they behave like creation and annihilation operators of particles. (Here J is the unique “complex structure” on S that is compatible with the dynamics.) In particular, by applying them to the vacuum state Ω, we get the entire GNS Hilbert space H. Finally, if we take an orthonormal basis {fi} of S, then the sum

i=1 a∗(fi)a(fi),

is the number operator N. Thus, the traditional Fock space formalism emerges as one special case of the GNS representation of a state of the Weyl algebra.

The Minkowski vacuum representation (H00) of A is Poincaré covariant, i.e. the action α(a,Λ) of the Poincaré group by automorphisms on A is implemented by unitary operators U(a,Λ) on H. When we say that H is isomorphic to Fock space F(H), we do not mean the trivial fact that H and F(H) have the same dimension. Rather, we mean that the unitary representation (H,U) of the Poincaré group is a Fock representation.

 

Tortile Category and Philosophy of Non-Commutative Geometry

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In terms of Feynman diagrams, a quantum field theory is nothing but a finitely generated subcategory QFT of the ribbon category Rep(G). A ribbon category (also called a tortile category) is a braided pivotal category, or equivalently a balanced autonomous category, which satisfies  θ*=θ, where  θ is the twist. This is a kind of category with duals. QFT is generated by the fundamental particles (irreducible representations) and all possible combinations of the fundamental interactions coming from the Feynman rules (intertwiners). Thus one is led to consider generalized quantum field theories QFT′ living inside arbitrary Hermitian ribbon categories R, where the braiding and twist need not be trivial.

Now Tannaka-Krein duality tells us that one can recover the group G from the category Rep(G). In a certain sense, every ribbon category is a category of representations – in the general case not of a group, but of a quantum group. When we do quantum field theory in ribbon categories, we are replacing the symmetry group by a quantum group.

We encounter this phenomenon in Chern-Simons theory, where the Lie group G is replaced by its quantum deformation, Uq(g). The words “Chern–Simons theory” can mean various things to various people, but it generally refers to the three-dimensional topological quantum field theory whose configuration space is the space of principal bundles with connection on a bundle and whose Lagrangian is given by the Chern-Simons form of such a connection (for simply connected  G, or rather, more generally, whose action functional is given by the higher holonomy of the Chern-Simons circle 3-bundle. The reason for this deformation of the underlying symmetry group, as one passes from the classical to the quantum theory, has not been altogether elucidated, and remains an interesting problem. In Witten’s approach, three dimensional Chern-Simons theory defines a two dimensional conformal field theory on the boundary, the Wess-Zumino-Witten (WZW) model. The corresponding affine lie algebra g of the WZW model defines, for each k ∈ Z+, a category Ck(g) of integrable modules of level k, and these categories are modular.

On the other hand, in Turaev’s approach, one deforms the lie algebra g into a quantum group Uq(g), where q = eπi/k, which for k ∈ Z+ is a root of unity. The representation categories Rep(Uq(g)) of these quantum groups are also modular, and are the starting point in Turaev’s approach.

Despite this theorem, the relationship between the Witten and Turaev approaches is still not completely understood. Ordinary Lie groups are the symmetry groups of manifolds. Quantum groups are the symmetry groups of noncommutative spaces – deformed, noncommutative versions of the commutative algebra of functions on a manifold. Thus the process of passing from QFT to QFT′ is associated with the philosophy of noncommutative geometry, a relatively recent trend in physics.

Algebraic Representation of Space-Time as Esoteric?

If the philosophical analysis of the singular feature of space-time is able to shed some new light on the possible nature of space-time, one should not lose sight of the fact that, although connected to fundamental issues in cosmology, like the ‘initial’ state of our universe, space-time singularities involve unphysical behaviour (like, for instance, the very geodesic incompleteness implied by the singularity theorems or some possible infinite value for physical quantities) and constitute therefore a physical problem that should be overcome. We now consider some recent theoretical developments that directly address this problem by drawing some possible physical (and mathematical) consequences of the above considerations.

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Indeed, according to the algebraic approaches to space-time, the singular feature of space-time is an indicator for the fundamental non-local character of space-time: it is conceived actually as a very important part of General Relativity that reveals the fundamental pointless structure of space-time. This latter cannot be described by the usual mathematical tools like standard differential geometry, since, as we have seen above, it presupposes some “amount of locality” and is inherently point-like. The mathematical roots of such considerations are to be found in the full equivalence of, on the one hand, the usual (geometric) definition of a differentiable manifold M in terms of a set of points with a topology and a differential structure (compatible atlases) with, on the other hand, the definition using only the algebraic structure of the (commutative) ring C(M) of the smooth real functions on M (under pointwise addition and multiplication; indeed C(M) is a (concrete) algebra). For instance, the existence of points of M is equivalent to the existence of maximal ideals of C(M). Indeed, all the differential geometric properties of the space-time Lorentz manifold (M,g) are encoded in the (concrete) algebra C(M). Moreover, the Einstein field equations and their solutions (which represent the various space-times) can be constructed only in terms of the algebra C(M). Now, the algebraic structure of C(M) can be considered as primary (in exactly the same way in which space-time points or regions, represented by manifold points or sets of manifold points, may be considered as primary) and the manifold M as derived from this algebraic structure. Indeed, one can define the Einstein field equations from the very beginning in abstract algebraic terms without any reference to the manifold M as well as the abstract algebras, called the ‘Einstein algebras’, satisfying these equations. The standard geometric description of space-time in terms of a Lorentz manifold (M,g) can then be considered as inducing a mathematical representation of an Einstein algebra. Without entering into too many technical details, the important point for our discussion is that Einstein algebras and sheaf-theoretic generalizations thereof reveal the above discussed non-local feature of (essential) space-time singularities from a different point of view. In the framework of the b-boundary construction M = M ∪ ∂M, the (generalized) algebraic structure C corresponding to M can be prolonged to the (generalized) algebraic structure C corresponding to the b-completed M such that CM = C, where CM is the restriction of C to M; then in the singular cases, only constant functions (and therefore only zero vector fields) can be prolonged. This underlines the non-local feature of the singular behaviour of space-time, since constant functions are non-local in the sense that they do not distinguish points. This fundamental non-local feature suggests non-commutative generalizations of the Einstein algebras formulation of General Relativity, since non-commutative spaces are highly non-local. In general, non-commutative algebras have no maximal ideals, so that the very concept of a point has no counterpart within this non-commutative framework. Therefore, according to this line of thought, space-time, at the fundamental level, is completely non-local. Then it seems that the very distinction between singular and non-singular is not meaningful anymore at the fundamental level; within this framework, space-time singularities are ‘produced’ at a less fundamental level together with standard physics and its standard differential (commutative) geometric representation of space-time.

Although these theoretical developments are rather speculative, it must be emphasized that the algebraic representation of space-time itself is “by no means esoteric”. Starting from an algebraic formulation of the theory, which is completely equivalent to the standard geometric one, it provides another point of view on space-time and its singular behaviour that should not be dismissed too quickly. At least it underlines the fact that our interpretative framework for space-time should not be dependent on the traditional atomistic and local (point-like) conception of space-time (induced by the standard differential geometric formulation). Indeed, this misleading dependence on the standard differential geometric formulation seems to be at work in some reference arguments in contemporary philosophy of space-time, like in the field argument. According to the field argument, field properties occur at space-time points or regions, which must therefore be presupposed. Such an argument seems to fall prey to the standard differential geometric representation of space-time and fields, since within the algebraic formalism of General Relativity, (scalar) fields – elements of the algebra C – can be interpreted as primary and the manifold (points) as a secondary derived notion.

Frege’s Ontological Correlates of Propositions

For Frege there were only two ontological correlates of propositions: the True and the False. All true propositions denote the True, and all false prepositions denote the False. From an ontological point of view, if all true propositions denote exactly one and the same entity, then the underlying philosophical position is the absolute monism of facts.

Lets disprove what Suszko called ‘Frege’s axiom’: namely the assumption that there exist only two referents for propositions.

Frege’s position on propositions was part of a more general view. Indeed, Frege adopted a principle of homogeneity (Perzanowski) according to which there are two fundamental categories of signs (Bedeutungen and truth-values) and two fundamental categories of senses (Sinn and Gedanken).

Both categories of signs (names and propositions) have sense and reference. The sense of a name is its Sinn, that way in which its referent is given, while the referent itself, the Bedeutung, is the object named by the name. As for propositions, their sense is the Gedanke, while their reference is their logical value.

Since the two semiotic triangles are entirely similar in structure, we need analyze only one of them: that relative to propositions.

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Here p is a proposition, s(p) is the sense of p, and r(p) is the referent of p. The functional composition states that s(p) is the way in which p yields r(p). The triangle has been drawn with the functions linking its vertexes explicitly shown. When the functions are composable, the triangle is said to commute, yielding

f(s(p)) = r(p), or f ° s(p) = r(p)

An interesting question now arises: is it possible to generalize the semiotic triangle? And if it is possible to do so, what is required? A first reorganization and generalization of the semiotic triangle therefore involves an explicit differentiation between the truth-value assigning function and the referent assigning function. We thus have the following double semiotic triangle:

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where r stands for the referent assigning function and t for the truth-value assigning function. Extending the original semiotic triangle by also considering utterances:

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Suszko uses the terms logical valuations for the procedures that assign truth-values, and algebraic valuations for those that assign referents. By arguing for the existence of only two referents, Frege ends up by collapsing logical and algebraic valuations together, thereby rendering them indistinguishable.

Having generalized the semiotic triangle into the double semiotic triangle, we must now address the following questions:

  1. when do two propositions have the same truth value?
  2. when do two propositions have the same referent?
  3. when do two propositions have the same sense?

Sameness of logical value will be denoted by (logical equivalence), while sameness of referent will be indicated with (not to be confused with the equiform to express indiscernibility) and sameness of sense (synonymy) by . Two propositions are synonymous when they have the same sense:

(p ≈ q) = 1 iff (s(p) = s(q)) = 1

Two propositions are identical when they have the same referent:

(p ≡ q) = 1 iff (r(p) = r(q)) = 1

Two propositions are equivalent when they have the same truth value:

(p ↔ q) = 1 iff (t(p) = t(q)) = 1

These various concepts are functionally connected as follows:

s(p) = s(q) implies r(p) = r(q), r(p) = r(q) implies t(p) = t(q)

In general, the constraints that we impose on referents correspond to the ontological assumptions that characterize the theory. The most general logic of all is the one that imposes no restriction at all on r valuations. Just as Fregean logic recognizes only two referents so the most general logic recognizes more than numerable set of them. Between these two extremes, of course, there are numerous intermediate cases. Pure non-Fregean logic is extremely weak, a chaos. If it is to yield something, it has to be strengthened.