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.

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Irrationality. Note Quote.

To mathematics it is unique, that two absolutely contrary opinions do not logically exclude each other but exist simultaneously while there seems to be no chance to pick out a false one and to establish a remaining truth. This case is realised by the philosophy and mathematics of the infinite. While transfinite set theory is impossible without different degrees of infinity, constructivists and intuitionists deny this notion without running into inconsistencies as is admitted by some of the foremost set theorists:

… the attitude of the (neo-)intuitionists that there do not exist altogether non-equivalent infinite sets is consistent, though almost suicidal for mathematics. [p. 62]

It would not be astonishing if in different axiomatic systems different results were obtained with respect to peculiarities of those systems. But set theorists on one side and constructivists and intuitionists on the other are certainly believing to address the same entities when speaking of “rational numbers” or of “irrational numbers”. In spite of that, the former are convinced that there are infinitely many more irrational numbers than rational numbers while the latter deny that:

Hence the continua of Weyl, Lebesgue, Lusin, etc. are denumerable … [p. 255]

This situation yields bewildering results:

Feferman and Levy showed that one cannot prove that there is any non-denumerable set of real numbers which can be well ordered. … Moreover, they also showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. [p. 62]

Nevertheless, the great majority of mathematicians refuse to accept the thesis that Cantor’s ideas were but a pathological fancy. Though the foundations of set theory are still somewhat shaky. Most surprising and by no means to be expected of a pupil of Fraenkel’s is that Robinson states:

Infinite totalities do not exist in any sense of the word (i.e. either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. Nevertheless, we should act as if infinite totalities really existed. [3]

Does there exist a correct and an incorrect position? And, if so, who is right, who is wrong?

Following the advice of Fraenkel, namely to judge about the value and necessity of the basic axioms, in particular of the axiom of choice, by considering its consequences, in order to settle this question. These consequences will turn out to entail what, in an euphemistic way, by set theorists usually is called a “paradoxical result”, in order to avoid the term self-contradiction.

Apart from the well-ordering theorem some statements of quite different character – in particular geometrical statements – have been proved by means of the axiom of choice, which because of their paradoxical character induced some mathematicians to reject the axiom. Presumably the earliest statement of this kind is Hausdorff’s discovery that half of the sphere’s surface is congruent to a third of it. … It may surprise scholars working in the field … that even after more than half a century of utilising the axiom of choice and well-ordering theorem, a number of first-rate mathematicians (especially French) have not essentially changed their distrustful attitude.

Transfinite set theory arises from Cantor’s observation that the set of all irrational numbers has infinitely many more members than the set of all rational numbers. While the latter has the same cardinality χ₀ as the set N of all natural numbers n, the cardinality χ of the set of all irrational numbers is larger, χ = 2^χ₀. It is proven to be uncountable, i.e., any bijection with N can be excluded.