Category theory mereologically considered

Suppose consider arrows $f :B \rightarrow A$ in a category as expressing generalized parthood and read them as $A$ is an $f$-part of $B$.   But what if we also consider a $g :  C \rightarrow B$ ? Then by composition $C$ is a $ f \circ g$-part of $A$.  What is the concept of sieve mereologically speaking ?

A sieve on $x$ is a set (class) of parts $y$ of $x$, $Pyz$, such that if $z$ is a part of $y$ then $z$ also belongs to this set.

\[Siev(A,x) \equiv \forall y. ((y \in A \rightarrow Pyx) \& \forall z. Pzy \rightarrow Pzx) \]

A sieve on $x$ is a set of parts of $x$ plus all the parts of those parts. It is a $P$-filter on $z$.

A covering sieve is a generalization of fusion: $x$ is to be seen as a generalized fusion of any one of its covering sieves.  The axioms for a Grothendieck topology make sense for fusions. If $z$ is a fusion of the $\phi(x)$ and $Pyz$ then we can consider $\phi_y(x)$ (the $y$-restriction of $\phi(x)$) expressing elements which are overlaps of elements which satisfy $\phi(x)$ with $y$. Then we should have that $y$ is the fusion of the $\phi_y(x)$.

Suppose $z$ is a fusion of the $\phi(x)$. And consider a $P$-filter on $z$ given by $\psi(y)$. If for every element $x$ which satisfies $\phi(x)$ we have that restriction $x$ is a fusion of the $\psi_x(y)$ then $z$ is also a fusion of the $\psi(y)$.

Finally each $z$ is the fusion of all its parts.

The Aristotelic concept of the topos of $x$ involves considering a $z$ with $Pxz$ such that for all $y$ such that $Pxy$ we have $Pzy$ (a minimal cover). A modal version would be more appropriate.  Then the topos itself should be a kind of completement $t$ such that $t$ and $x$ do not overlap and $t + x = z$.

C. Bo, A Descriptivist Refutation of Kripke's Modal Argument and of Soames's Defence

 https://onlinelibrary.wiley.com/doi/10.1111/j.1755-2567.2012.01136.x

 https://plato.stanford.edu/entries/two-dimensional-semantics/

Kipper and Soysal, A Kripkean argument for descriptivism

 https://onlinelibrary.wiley.com/doi/epdf/10.1111/nous.12378

Raclavský on Salmon's new Frege's Puzzle

The following discussion is from pp. 99-100 of J. Raclavský's book Belief Attitudes, Fine-Grained Hyperintensionality and Type-Theoretic Logic. We refer the reader to N. Salmon's chapter Reference and Information Content: Names and Descriptions in vol. 10 of the Handbook of Philosophical Logic. There we learn that John Stuart Mill proposed as very similar theory of proper names to that which we presented in our previous post on the subject.

Let us consider  again Tullius is Cicero.   First, our own perspective. To us, for proper names N1 and N2, sentences of the form 'N1 is N2' are ambiguous.  In ordinary discourse their pragmatic aspect is clearly informative. The sentence is expressing metalinguistic information about the reference of a given proper name by employing a proper name whose reference is known in the context. That is, 'N1 is N2' is read: the reference of expression 'N1' is N2 (the reference of 'N2' being known).  That is, in this case N1 is treated qua expression but N2 is treated according to its reference. However ambiguity occurs because the roles of N1 and N2 can easily be switched without any syntactic distinction. The intention of the speaker and the interpretation of the the interlocutor are not determined. In Tichý system we could, perhaps, read the sentence something like ' exec N1 $\cong$ exec ${}^0 C$' where $C$ represents the individual Cicero and $N1/\star_1$. Note that a sentence's meaning is way a presenting a state of affairs while its denotation or reference can be considered as the state of affairs itself or an equivalence class between modes of presentation of states of affairs. Husserl's example is that  $a < b$ represents the same state of affairs as $b > a$. Bealer's T2 equality is equality between modes of presentation while his T1 equality is modal and corresponds to states of affairs themselves: $[a < b] = [b > a]$ precisely because it is true in all possible worlds.

Now consider this type of sentence as subject of a propositional attitude 'aK (Tullius is Cicero)'. Obviously this cannot be obtained by substitution from the validity 'aK(Cicero is Cicero)' and the Kripkean reading of Tullius is Cicero as true - this is Salmon's so-called 'new Frege's puzzle'.  This puzzle can be approached according to our theory of proper names presented in a previous post. But let us consider Raclavsky´s view.  Raclavský takes a Kripkean view of proper names: they are univocal tags. His view is that THL (transparent intensional logic) is a semantic doctrine and thus abstracts from the linguistic incompetence of speakers. Thus he accepts 'aK(Cicero is Cicero)' and 'Tullius = Cicero' and sees the falsity of the conclusion as a mere expression of a's linguistic incompetence,  'insufficient competence to understand English'.  What would states of affairs or the modes of presentation be in TIL ? These might be terms whose executions yield functions from times to worlds to truth values, of type $(\omicron\omega\rho)$.

Natural language is syntactically highly ambiguous. There are of course many distinct types of sentences which present the form 'N1 = N2'.  We are dealing here with the metalinguistic type. There are of course nonmetalinguistic types such as 'Phosphorous = Hesperus' which are not about language at all but about the world.

The Church-Turing Thesis, Kripke and Kant

If we consider the abundance of hypothetical and counterfactual elements embedded irrevocably in our linguistic discourse, then a possible worlds semantician might be inclined to view the existence of Kripke's rigid designators as the transcendental conditions for the consistency and intelligibility of our discourse about the world.  But here we wish to discuss a Kantian turn in a different domain. What is it exactly that it means to follow a rule, a set of rules, to play a game,  learn how to use a language, carry out a logical debate, or in general to engage in the world ? For a subject  or mind to do this, it must be computationally competent, in other words, (at least) Turing-complete.  Secondly, it must be able to do this cross-platform, in an indefinite number of domains.  Thus the Church-Turing thesis, like Kripke rigid designators, appears as a transcendental condition for the possibility of our engaging in the world. It also suggests the a priori nature of a basic but fair portion of arithmetic, combinatorics and recursion theory.

Equality and sameness from Frege to Martin-Löf to Voevodsky

C. Ortiz Hill's book Word and Object in Frege, Husserl and Russell contains an interesting discussion on equality and identity in Frege and Husserl.

This question is exactly what Martin-Löf type theory and the contemporary developments in homotopy type theory and Voevodsky's univalent foundations of mathematics are all about.

Gödel once referred to Husserl as 'the true Kant'. I sometimes think of Per Martin-Löf as 'the true Frege' due to his very similar role as founder of a radical new system of mathematical logic and foundations of mathematics. In Martin-Löf type theory we have two notions of equality. One is extrinsic and corresponds to sameness and satisfies Leibniz's law. The other is internal, and corresponds to equality, sameness according to some aspect. To use a geometric analogy, one notion corresponds to two segments completely coinciding, being in fact, the 'same' segment, whilst the other corresponds to two segments having the same length (and hence having the possibility of one being moved in the plane so as to coincide with the other). To be able to have Leibniz's law for internal equality is not something that is just given, it needs to be proven for special conditions. ML type theory also allows us to work with reification in a controlled way, specially regarding how internal equality between higher types is treated.

Voevodsky's univalent foundations was conceived specifically for mathematics, thus it postulates extra axioms on top of ML type theory for the purposes of doing mathematics. These extra axioms evidently can be questioned in different contexts or as candidates for a universal logic.
Voevodsky views a certain property involving extensional equality of functions - a third kind of equality, weaker than internal equality, called equivalence - in a geometric light. He does not postulate internal (let alone extrinsic) equality for equivalent functions. Thus he is not a Fregean. Thus while internally equal functions are equivalent, equivalent functions are not necessarily internally equal. How then is mathematics possible ? Voevodsky's univalence axiom reads:

Internal equality itself is equivalent to equivalence.

So Husserl criticized Frege for conflating sameness and equality (and thus questioned the extensionality principle, Law V). Sameness is a much stronger notion than equality.

Martin-Löf and Voevodsky say: let us keep sameness outside and distinct and instead work with a strong and weak kind of equality. Let us take versions of Law V, and an analogue of that useful but fatally flawed identification of sameness and equality, and transpose them instead to strong and weak equality. In this way we can do mathematics.

h-types and holology

 

In Martin-Löf type theory,  or more specifically standard dependent type theory (DTT),  there is an extrinsic and an intrinsic concept of equality.  The intrinsic concept (written $\Gamma \vdash  t : A = A$) is what we shall mainly consider here. It can be given a topological (and higher-categorical) interpretation which is of some interest to mereology.  Here we only give a rough description of some of the intuitions involved.  Think of a collection of points in the plane.   Internal 'equality' between two (externally unequal) points $a$ and $b$, written $a=b$,  is seen as being witnessed by there being a continuous path from $a$ to $b$.  In this way reflexivity, symmetry and transitivity have natural interpretations (in fact we obtain a 'groupoid' or more correctly a 1-groupoid). Note that there can be more than one path between two points.  For a space $C$ consisting of two disjoint 'chunks' $A$ and $B$ there is no path from a point of $A$ to a point of $B$.  There are two extreme situations. One is the discrete case in which no two externally distinct points can be connected by a path - this is the classical concept of a set. The other extreme is a connected (or 1-connected) space in which we can find a single point $p$ such that $p$ can be connected to any other point in the space.  Thus the discrete space has the minimum of internal unity and the contractible space the maximum of internal unity. In general we can have a space with a certain number of connected components ('chunks'). If we view internal equality as an equivalence relation (and thus the space as a setoid)  then the number of components is the number of equivalence classes.

But being connected is only the maximum of unity at the lowest level. Given two points $a$ and $b$ there is a space of all paths from $a$ to $b$, $Path(a,b)$. Only now that 'points' are paths and the 'paths' are deformations between paths (homotopies). So given two paths $p1,p2$ in $Path(a,b)$ we have that $p_1=p_2$ is witnessed by there being a 'path' (i.e. deformation/homotopy) between $p_1$ and $p_2$. Now the same story is repeated at a higher octave. $Path(a,b)$ itself can possess the extremes of being discrete (set-like) or connected or anything between.  Indeed from elementary algebraic topology we get examples of spaces which are connected but which for two points $a$ and $b$ there are paths $p_1$ and $p_2$ in $Path(a,b)$ which cannot be deformed into each other. In the case where $Path(a,b)$ is connected for every $a$ and $b$ the space is called contractible but we now see why we could also call it, for example, 2-connected. We note that there are also closed paths or 'loops', elements in $P(a,a)$, and there is no a priori reason why these could not be discrete, not mutually deformable into another. Thus we could add to our characterization of classical 'set'.  If we think of a set as a collection of points or atoms, then these will not have interesting internal structure or symmetries and thus we could add to the definition of classical set the condition that $Path(a,a)$ be connected for every $a$: that is, all loops are internally the same.  We have seen the recurring motifs: connected and discrete. To this we can add external notions: there being no elements whatsoever in the space(type), there being exactly one according to external equality (singleton) and there being exactly two (for instance the type of classical Boolean values).

We can now continue this process considering deformations of deformations and so forth. If a deformation is a path between paths in $Path(a,b)$, i.e. an element in $Path(p_1,p_2)$ where $p_1,p_2: Path(a,b)$, then now we are given for $d_1,d_2 : Path(p_1,p_2)$ elements of $Path(d_1,d_2)$, deformations of deformations.  A structure like what we have here, in which we can keep iterating the path construction, is called an $\infty$-groupoid (this is only a rough description).  We get thus a notion of $n$-connectivity for each level $n$. What kind of $\infty$-groupoid will have the maximum of connectivity or unity, i.e. how can we define a contractible $\infty$-groupoid ? Clearly by requiring at each level that $Path(x,y)$ for any $x$ and $y$ be connected, that is, that we have $n$-connectivity at each level $n$.

A main idea of   the Univalent Foundations of Mathematics (also called Homotopy Type Theory) is that mathematics is better done over $\infty$-groupoids rather than over sets and that this not only gives place to a profound unity between logic, category theory and geometry but is also the most natural way of developing computer software for formalizing and checking proofs  -  something which can be very useful when the proofs are very long and complex. The concept of 'h-type'  in the title of this post is directly related to what we called $n$-connectivity in this post.

This might have applications to ontology and metaphysics, in particular to the philosophy of biology and mind.  There is a lowest level of inorganic matter which corresponds to classical sets, thus to  0-discreteness. Then living beings have an organic unity which corresponds to 1-connectedness. Each part of a living body is connected to every other one on the lowest level, but can have quite distinct mode of connection.  We can also think of this as there being the same life present in every part but functionally  specialized. Next we come to mind which is a higher-order organism in that the same mind is present in every part in an essentially similar way (perhaps related to metacognition or self-luminosity), thus we would postulate  both 1- and 2-connectedness for instance. The failure of 3-connectivity could reflect the presence of time or sensation for instance. We could also follow ancient metaphysics such as Plotinus and characterize the nous (seen as a unified space of all forms) as being $n$-connected for all $n$, thus a contractible $\infty$-groupoid, the ontological maximum of unity.

Categorical holology

We are missing is the proper abstract treatment of the category of Essence (Wesen) whereby Being negates itself to enter more deeply into itself and prepare the way for becoming in and for itself in Concept. The category of Essence corresponds to the categories of classical metaphysics, but it also corresponds to objective spirit, to nature. In modern terms the proper abstract treatment of the category of Essence (if it is not to lapse back into Being) must involve the abstract formalization of a missing theoretical biology and theoretical biochemistry. For it is the living organism which gives us light on the categories of existence (as organism and system) and active substance. Living beings manifest, they have appearance and phenomena. They preserve themselves in their own negation. Biology cannot be reduced to physics and is essentially more than physics in the same way that mediated Essence cannot be reduced to immediate Being. Life is process and mediation.  The theory of computation on one side is connected to formal logic. But on the other side it is a crude approximation of biocomputation, the computation of nature (which probably will be linked to quantum computation via topology and anyons, etc.). Our hardware and operating system software are based on many historical, social and economic contingent factors and thus have limited philosophical import as well as ad hoc technical restriction.. However many concepts in theoretical computer science, specially those dealing with concurrency, are important for the category of Essence. What is a suitable mathematical model for biological systems that cannot be reduced to the general scheme of mathematical physics (sheaves, gauge potentials, jet bundles, PDEs, etc.) ? Here are some ideas.

1) Let physics correspond to a temporal process $\phi: T \rightarrow \Gamma(S)$ where $\Gamma(S)$ are sections over some bundle-like structure over a 'space' $X$. This structure forms a certain kind of (commutative) algebraic structure. Then biology corresponds to a temporal process $\beta : T \rightarrow B$ in which $B$ is a (non-commutative) algebraic structure which is not isomorphic to sections of a bundle-like object over any 'space' $X$. However $B$ may have phenomenological 'representations' or 'projections' $p: B \rightarrow \Gamma(S_p)$ for $S_p$ a bundle-like structure over some space $X_p$. Thus living organisms have physical measurable manifestations but cannot be reduced to such.

2) Biology cannot be reduced to physics because it cannot be described by sections of a bundle which evolve according to a closed manifold in a jet bundle, that is, exhibiting the local causality of PDEs or integral equations based on these assumptions. The 'space' involved may be such that the stalk at any 'point' (and the subsequent evolution thereof) depends on the information on the entire space. For instance $X$ can be seen as a groupoid and we can think of the groupoid algebra product which for two elements is defined in terms of the entire groupoid.

3) Consider process algebras describing concurrency and communication or object specification logic wherein we distinguish between local constraints and global constraints which cannot be reduced to the sum-total of local constraints and which does not simply 'emerge' for them. In computer engineering such global constraints come from human agency. But in biological systems it is intrinsic and essential.

Consider a subcategory $i: \mathcal{B} \rightarrow \mathcal{A}$. Let this subcategory be seen as the mind and the category be seen as the world. Representation means that we perceive within our mind an image which expresses the relationship between ourselves and the world. A morphism  $f: A \rightarrow i(B)$ in $\mathcal{A}$ expresses how aspect $A$ of the world is affecting $\mathcal{B}$ with respect to aspect $B$. $A$ and $f$ are then represented by an object $r(A)$ and morphism $r(f): r(A) \rightarrow B$ in $\mathcal{B}$. Thus the subcategory mirrors external relationships within itself. The classical concept of reflective subcategory, where the situation above is expressed as an adjunction $r \vdash i$ is related to the Hegelian concept of reflection in second part of the Science of Logic, Essence. Shining within itself, $r: \mathcal{B} \rightarrow \mathcal{A}$ negates $\mathcal{A}$ and preserves itself within itself in a way which subsumes the relation between the two. The category of existence as well as other categories in the objective part of the section on Notion suggests that we are in the domain of general systems theory (concurrent interacting agents).

We note that Hegel in discussing objective notion mentions a system in which the whole is implicit in each particular determination: this is exactly the case of holomorphic functions: the germ at each point completely determines the whole by analytic continutation and the whole completely determines the germ at each point. In general we can consider a sheaf in which the restriction morphisms are isomorphisms. Holomorphic functions really are holistic because they are uniquely determined by their germs. Change a holomorphic function locally and you change it globally. The global information on the function is contained implicitly in any local restriction or germ (via analytic continuation). Thus in a way the same germ is spread out across the whole domain.
The case of holomorphic function corresponds to the global being contained in the particular (or local). The global is totally local. But there is also the opposite situation in which the local is emptied of any global information. The global is totally non-local. This is the case for the space of Penrose tilings. It is impossible to distinguish between two tilings by looking at local patterns.