Conjuncted: Internal Logic. Thought of the Day 46.1

So, what exactly is an internal logic? The concept of topos is a generalization of the concept of set. In the categorial language of topoi, the universe of sets is just a topos. The consequence of this generalization is that the universe, or better the conglomerate, of topoi is of overwhelming amplitude. In set theory, the logic employed in the derivation of its theorems is classical. For this reason, the propositions about the different properties of sets are two-valued. There can only be true or false propositions. The traditional fundamental principles: identity, contradiction and excluded third, are absolutely valid.

But if the concept of a topos is a generalization of the concept of set, it is obvious that the logic needed to study, by means of deduction, the properties of all non-set-theoretical topoi, cannot be classic. If it were so, all topoi would coincide with the universe of sets. This fact suggests that to deductively study the properties of a topos, a non-classical logic must be used. And this logic cannot be other than the internal logic of the topos. We know, presently, that the internal logic of all topoi is intuitionistic logic as formalized by Heyting (a disciple of Brouwer). It is very interesting to compare the formal system of classical logic with the intuitionistic one. If both systems are axiomatized, the axioms of classical logic encompass the axioms of intuitionistic logic. The latter has all the axioms of the former, except one: the axiom that formally corresponds to the principle of the excluded middle. This difference can be shown in all kinds of equivalent versions of both logics. But, as Mac Lane says, “in the long run, mathematics is essentially axiomatic.” (Mac Lane). And it is remarkable that, just by suppressing an axiom of classical logic, the soundness of the theory (i.e., intuitionistic logic) can be demonstrated only through the existence of a potentially infinite set of truth-values.

We see, then, that the appellation “internal” is due to the fact that the logic by means of which we study the properties of a topos is a logic that functions within the topos, just as classical logic functions within set theory. As a matter of fact, classical logic is the internal logic of the universe of sets.

Another consequence of the fact that the general internal logic of every topos is the intuitionistic one, is that many different axioms can be added to the axioms of intuitionistic logic. This possibility enriches the internal logic of topoi. Through its application it reveals many new and quite unexpected properties of topoi. This enrichment of logic cannot be made in classical logic because, if we add one or more axioms to it, the new system becomes redundant or inconsistent. This does not happen with intuitionistic logic. So, topos theory shows that classical logic, although very powerful concerning the amount of the resulting theorems, is limited in its mathematical applications. It cannot be applied to study the properties of a mathematical system that cannot be reduced to the system of sets. Of course, if we want, we can utilize classical logic to study the properties of a topos. But, then, there are important properties of the topos that cannot be known, they are occult in the interior of the topos. Classical logic remains external to the topos.

Hegel and Topos Theory. Thought of the Day 46.0

The intellectual feat of Lawvere is as important as Gödel’s formal undecidability theorem, perhaps even more. But there is a difference between both results: whereas Gödel led to a blind alley, Lawvere has displayed a new and fascinating panorama to be explored by mathematicians and philosophers. Referring to the positive results of topos theory, Lawvere says:

A science student naively enrolling in a course styled “Foundations of Mathematics” is more likely to receive sermons about unknowability… than to receive the needed philosophical guide to a systematic understanding of the concrete richness of pure and applied mathematics as it has been and will be developed. (Categories of space and quantity)

One of the major philosophical results of elementary topos theory, is that the way Hegel looked at logic was, after all, in the good track. According to Hegel, formal mathematical logic was but a superficial tautologous script. True logic was dialectical, and this logic ruled the gigantic process of the development of the Idea. Inasmuch as the Idea was autorealizing itself through the opposition of theses and antitheses, logic was changing but not in an arbitrary change of inferential rules. Briefly, in the dialectical system of Hegel logic was content-dependent.

Now, the fact that every topos has a corresponding internal logic shows that logic is, in quite a precise way, content-dependent; it depends on the structure of the topos. Every topos has its own internal logic, and this logic is materially dependent on the characterization of the topos. This correspondence throws new light on the relation of logic to ontology. Classically, logic was considered as ontologically aseptic. There could be a multitude of different ontologies, but there was only one logic: the classical. Of course, there were some mathematicians that proposed a different logic: the intuitionists. But this proposal was due to not very clear speculative epistemic reasons: they said they could not understand the meaning of the attributive expression “actual infinite”. These mathematicians integrated a minority within the professional mathematical community. They were seen as outsiders that had queer ideas about the exact sciences. However, as soon as intuitionistic logic was recognized as the universal internal logic of topoi, its importance became astronomical. Because it provided, for the first time, a new vision of the interplay of logic with mathematics. Something had definitively changed in the philosophical panorama.

Badiou, Heyting Algebras cross the Grothendieck Topoi. Note Quote.

Let us commence by introducing the local formalism that constitutes the basis of Badiou’s own, ‘calculated phenomenology’. Badiou is unwilling to give up his thesis that the history of thinking of being (ontology) is the history of mathematics and, as he reads it, that of set theory. It is then no accident that set theory is the regulatory framework under which topos theory is being expressed. He does not refer to topoi explicitly but rather to the so called complete Heyting algebras which are their procedural equivalents. However, he fails to mention that there are both ‘internal’ and ‘external’ Heyting algebras, the latter group of which refers to local topos theory, while it appears that he only discusses the latter—a reduction that guarantees that indeed that the categorical insight may give nothing new.

Indeed, the external complete Heyting algebras T then form a category of the so called T-sets, which are the basic objects in the ‘world’ of the Logics of Worlds. They local topoi or the so called ‘locales’ that are also ‘sets’ in the traditional sense of set theory. This ‘constitution’ of his worlds thus relies only upon Badiou’s own decision to work on this particular regime of objects, even if that regime then becomes pivotal to his argument which seeks to denounce the relevance of category theory.

This problematic is particularly visible in the designation of the world m (mathematically a topos) as a ‘complete’ (presentative) situation of being of ‘universe [which is] the (empty) concept of a being of the Whole’ He recognises the ’impostrous’ nature of such a ‘whole’ in terms of Russell’s paradox, but in actual mathematical practice the ’whole’ m becomes to signify the category of Sets – or any similar topos that localizable in terms of set theory. The vocabulary is somewhat confusing, however, because sometimes T is called the ‘transcendental of the world’, as if m were defined only as a particular locale, while elsewhere m refers to the category of all locales (Loc).

An external Heyting algebra is a set T with a partial order relation <, a minimal element μ ∈ T , a maximal element M ∈ T . It further has a ‘conjunction’ operator ∧ : T × T → T so that p ∧ q ≤ p and p ∧ q = q ∧ p. Furthermore, there is a proposition entailing the equivalence p ≤ q iff p ∧ q = p. Furthermore p ∧ M = p and μ ∧ p = μ for any p ∈ T .

In the ‘diagrammatic’ language that pertains to categorical topoi, by contrast, the minimal and maximal elements of the lattice Ω can only be presented as diagrams, not as sets. The internal order relation ≤ Ω can then be defined as the so called equaliser of the conjunction ∧ and projection-map

≤Ω →^e Ω x Ω →_π1^∧ L

The symmetry can be expressed diagrammatically by saying that

is a pull-back and commutes. The minimal and maximal elements, in categorical language, refer to the elements evoked by the so-called initial and terminal objects 0 and 1.

In the case of local Grothendieck-topoi – Grothendieck-topoi that support generators – the external Heyting algebra T emerges as a push-forward of the internal algebra Ω, the logic of the external algebra T := γ ∗ (Ω) is an analogous push-forward of the internal logic of Ω but this is not the case in general.

What Badiou further requires of this ‘transcendental algebra’ T is that it is complete as a Heyting algebra.

A complete external Heyting algebra T is an external Heyting algebra together with a function Σ : PT → T (the least upper boundary) which is distributive with respect to ∧. Formally this means that ΣA ∧ b = Σ{a ∧ b | a ∈ A}.

In terms of the subobject classifier Ω, the envelope can be defined as the map Ω^t : Ω^Ω → Ω¹ ≅ Ω, which is internally left adjoint to the map ↓ seg : Ω → Ω^Ω that takes p ∈ Ω to the characteristic map of ↓ (p) = {q ∈ Ω | q ≤ p}27.

The importance the external complete Heyting algebra plays in the intuitionist logic relates to the fact that one may now define precisely such an intuitionist logic on the basis of the operations defined above.

The dependence relation ⇒ is an operator satisfying

p ⇒ q = Σ{t | p ∩ t ≤ q}.

(Negation). A negation ¬ : T → T is a function so that

¬p =∑ {q | p ∩ q = μ},

and it then satisfies p ∧ ¬p = μ.

Unlike in what Badiou calls a ‘classical world’ (usually called a Boolean topos, where ¬¬ = 1_Ω), the negation ¬ does not have to be reversible in general. In the domain of local topoi, this is only the case when the so called internal axiom of choice is valid, that is, when epimorphisms split – for example in the case of set theory. However, one always has p ≤ ¬¬p. On the other hand, all Grothendieck-topoi – topoi still materially presentable over Sets – are possible to represent as parts of a Boolean topos.