nLab Hegel's "Logic" as Modal Type Theory

under construction

Abstract While analytic philosophy famously rejected the speculative metaphysics of Hegel in favor of the analysis of concepts by means of mathematical logic, in particular predicate logic, recent developments in the foundations of mathematics via homotopy type theory offer a way to re-read Hegel as having useful formal meaning not in predicate logic, but in ‘modal type theory’. The essence of this suggestion has been made by Lawvere in 1991, which however remains largely unnoticed. Here we aim to give a transparent account of this perspective both philosophically as well as category-theoretically. We then further expand on Lawvere’s formalization of Hegel’s “Science of Logic” in terms of the categorical semantics given by cohesive higher toposes. We discuss how there is a useful formalization of a fair bit of modern fundamental physics, in fact of local gauge quantum field theory, to be found here.

Introduction

With the rise of analytic philosophy in Britain, and its adoption and development of the new logic forged by Frege and Peano, there came a radical rejection of the existing philosophical paradigm, the idealism of T. H. Green? and F. H. Bradley?, and those considered to have provided its primary source, notably Hegel. Bertrand Russell was adamant that the new logic would thoroughly transform philosophy, as he made clear at the end of his chapter Logic As The Essence Of Philosophy:

The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds are beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion. (Logic As The Essence Of Philosophy, 1914)

In the same chapter, Hegel was taken to task for his attempt to widen logic to a form of metaphysics. Much of it appears to Russell to result from muddled thinking, for instance, a failure to distinguish the ‘is’ of predication from the ‘is’ of identity.

This set the tone for analytic philosophy’s opinion of Hegel for several decades. In 1951, the logical empiricist Hans Reichenbach wrote

Hegel has been called the successor of Kant; that is a serious misunderstanding of Kant and an unjustified elevation of Hegel. Kant’s system, though proved untenable by later developments, was the attempt of a great mind to establish rationalism on a scientific basis. Hegel’s system is the poor construction of a fanatic who has seen one empirical truth and attempts to make it a logical law within the most unscientific of all logics. Whereas Kant’s system marks the peak of the historical line of rationalism, Hegel’s system belongs in the period of decay of speculative philosophy which characterizes the nineteenth century. (Reichenbach 1951, p. 72)

However, despite the initial lengthy rejection of Hegelian ideas by analytic philosophy, there have recently been increasing signs of a willingness to re-engage with Hegel’s writings, in particular on the part of John MacDowell? and Robert Brandom (Redding, 2007). Interestingly, one feature of analytic philosophy which needs reconsidering according to Redding is the primacy of first-order predicate logic, not, however, that he is promoting an alternative formalization based on Hegel’s ideas.

It took a mathematician William Lawvere to find something formalizable of value in Hegel’s Logic, in particular the concept of a ‘Unity of Opposites’. In this paper, taking our lead from Lawvere in seeing common elements between categorical logic and Hegel’s logic, we will pursue this connection right up to modal forms of homotopy type theory.

Now, Hegel’s understanding of the term ‘Logik’ was much broader than is normally the case. Indeed, the ‘Objective logic’ of Science of Logic is more akin to a kind of metaphysics. While the Vienna Circle, admirers of Russell, had adopted the positivist attitude towards metaphysics, dismissing it as meaningless (Carnap 32), later in the twentieth century there was a resurgence of analytic metaphysics. One important source of this resurgence was the rise of modal logic as a tool for philosophy, especially due to Saul Kripke in Naming and Necessity. If the necessity of a proposition consists in its holding in all neighboring possible worlds, sense had to be made of these possible worlds. To what would have been Carnap’s amazement, philosophers such as David Lewis proposed the reality of all possible worlds, the actual world being in no way special, except for its being our world.

Today, standard topics for analytic metaphysics today include causality, necessity, space and time, identity, and mental states. Little attention is paid to the question of explaining the kind of physical principles which operate in our universe. It is noteworthy, however, that the need for some such questioning seems to be as strongly felt among modern physicists as it must have been to Hegel and his predecessors, as witnessed by … debates such as critically reviewed in (Albert 12). (…) Attempts to re-install this kind of metaphysics are these days undertaken by particle physicists and cosmologists themselves, whose success at formally describing remote aspects of the observable universe has emboldened many to feel as superior to modern philosophy as modern philosophy typically feels towards Hegel’ idealism.

(…)

Modern foundations – Types, Judgements and Deduction

(…) something like:

analytic philosophy was certainly right to ask for a formal basis of philosophical reasoning, in order to make genuine intellectual progress possible. Back then predicate logic and set theory was the state of the art concerning the foundations of mathematics and so this is what analytic philosophy adopted and developed. But since then there has been much progress in the foundations of maths. … intuitionistic type theory … homotopy type theory…

Accordingly, the original starting point of analytic philosophy needs to be re-examined. While predicate logic has no way of making formal sense of Hegel’s “Logic”, for type theory the situation is rather different…

Categorical semantics and Categorical logic

… type theory proper is a formal system for symbol manipulation … useful for computer encoding, and much more readable than full formalization in, say, ZFC, but still not really what a discussion such as our here would directly work with…

… instead, the syntax of type theory has semantics in categories – categorical semantics … relation between category theory and type theory – and this semantic level is what we will be concerned with here

… originates with Lawvere’s thesis, see also “Adjointness in Foundations”

Modalities, moments and opposites

(… basics of modalities, modal operators, monads, leading up to adjoint modality/unity of opposites…)

Formal determinations of Being

… using this and building on Lawvere’s lead, we are led to objectively investigate the following:

starting with bare homotopy type theory and adding adjoint modalities to it, what is the nature of the new propositions that can be proven with these modalities, hence what is the new “quality” of the types in the presence of these new axioms…

…

$$\array{ reines\;Sein && Dasein && Realitaet \\ being && existence && reality \\ pure being && determinate being \\ \\ && && \Re \\ && && \bot \\ && \int & \subset & \Im \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \& \\ \bot & & \bot && \\ \ast & \subset& \sharp }$$

symbol	$\;\;$ name
$\int$	shape modality
$\flat$	flat modality
$\sharp$	sharp modality
$\Re$	reduction modality
$\Im$	infinitesimal shape modality
$\&$	infinitesimal flat modality

(…)

References

• Rudolf Carnap, Überwindung der Metaphysik durch logische Analyse der Sprache, Erkenntnis Vol II (1932). Translated by A. Pap as The Elimination of Metaphysics Through Logical Analysis of Language (pdf)

• Georg Hegel, Science of Logic (1812)

• William Lawvere, Some Thoughts on the Future of Category Theory (1991)

• William LawvereCohesive Toposes and Cantor's "lauter Einsen" (1994)

• William LawvereUnity and Identity of Opposites in Calculus and Physics (1996)

• Higher toposes of laws of motion

• Homotopy-type semantics for quantization

• Paul Redding, Analytic Philosophy and the Return of Hegelian Thought, Cambridge University Press, 2007 (review by Willem A. deVries)

• Bertrand Russell, ‘Logic As The Essence Of Philosophy’, chap. 2 of Our Knowledge of the External World, (1914)

• Hans Reichenbach, ‘The Rise of Scientific Philosophy’, 1951.

• David Albert, On the Origin of Everything, book review of

  “A Universe From Nothing”, by Lawrence M. Krauss, in the New York Times from March 25, 2012, on page BR20 of the Sunday Book Review (web)