basic constructions:
strong axioms
further
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
This page collects material related to the article
Cohesive Toposes and Cantor’s “lauter Einsen”,
Philosophia Mathematica (3) Vol. 2 (1994) pp. 5-15.
on adjoint cylinders and cohesive toposes.
The article claims that Georg Cantor‘s original use of terminology (Cantor 1895) was distorted by his editor to become what we now take to be the standard meaning of sets and their cardinality in set theory. But that instead Cantor really meant “cohesive types” when saying “Menge” and used “Kardinale” really for what we call the underlying set $\flat X$ of a cohesive type $X$.
Beyond this, the article is in part motivated from attempts to formalize thoughts appearing in Georg Hegel‘s Science of Logic on unity of opposites.
Concretely it is about the unity of opposites that in a set all elements are “distinct, but indistinguishable”. Lawvere argues that this and Cantor’s original notion of cardinality is captured by the adjoint modality given by flat modality $\dashv$ sharp modality.
This seems to match well with Hegel‘s Pure quantity both informally as well as formally (as discussed there).
Notice that where the English translations of Hegel‘s Science of Logic say “the One”, the original has “Das Eins”, which might just as well be translated with “The Unit”. In view of this and looking through Hegel’s piece on discreteness and repulsion, it seems clear that Hegel’s “Einsen” is precisely Cantor’s “Einsen” as recalled by Lawvere. Namely: copies of the unit type.
Notice that the German “lauter” is pretty much the English “lots of”. When you say “A lot o'” as in the slang “A lot o' stuff” or else say “lauter” with a Bavarian accent, it comes out pretty much alike.
(? The German expression ‘lauter’ predominantly denotes universal quantification, so that the Cantor quote translates ‘all (of) units’. There might in fact be an etymological connection to English ‘lot’ preserved in the meaning ‘(specific) quantity’ - German ‘Posten’ like in e.g. ‘yesterday arrived two lots of cd-players’)
On page 7, Lawvere makes a distinction between objective and subjective cohesion. The subjective version involve the “coming to know” aspects of points in a space. “A good example of a type of set possessing subjective cohesion is that of recursive set, which is ‘traced’ by various threads generated by particular recursive functions.”
In view of the relationship between cohesion and modality, Lawvere’s distinction should correspond to that between metaphysical and epistemic modalities (“As far as I know, it may be the case that…”).
On page 11, Lawvere discusses categories intermediate between the original cohesive category and the base. This relates to the discussion, pp. 10-11 of Some Thoughts on the Future of Category Theory, which remarks on dimensional levels of increasing determinateness, including infinitesimal increases.
It is noteworthy that Cantor’s conception of cardinal number is a mildly disguised form of the definition of number as ‘a multitude made up of units’ in Euclid, book VII, def.2. This was until the end of the 19th century a standard definition of the finite cardinals and in Cantor’s time was e.g. used by the highly influential Karl Weierstrass in his lectures^{1} and defended by Weierstrass’ student Edmund Husserl in his ‘Philosophie der Arithmetik’ (1891).
Cantor comments on Euclid in ‘Mitteilungen zur Lehre vom Transfiniten’ (1887-88, Cantor (1932) pp.378-438)). His comment (p.380f.) makes it clear that he views cardinals actually as the cohesive ‘united’ objects whereas he stresses the discreteness of the concrete elements in the ‘Mengen’. He criticizes Euclid for not emphasizing that numbers are actually more unified objects than the material sets they are abstracted from. Another key passage (with unmistakingly Aristotelian overtones) in this direction reads:
‘Die Elemente der uns gegenüberstehenden Menge $M$ sind getrennt vorzustellen; in dem intellektualem Gebilde $\overline{M}$, welches ich ihren Ordnungstypus nenne, sind dagegen die Einsen zu einem Organismus vereinigt. In gewissem Sinne lässt sich jeder Ordnungstypus als ein Kompositum aus Materie und Form ansehen; die darin enthaltenen begrifflich unterschiedenen Einsen liefern die Materie, während die unter diese bestehende Ordnung das der Form entsprechende ist.’ (p.380)
In a letter to Peano he writes:
‘I conceive of numbers as ‘forms’ or ‘species’ (general concepts) of sets. In essentials this is the conception of the ancient geometry of Plato, Aristotle, Euclid etc.’ (Cantor, Briefe, 1991, p.365; quoted after Hill (2000))
What underlies the contrast between Mengen and Kardinalen, is a platonist conception of mathematics. By abstracting from the contingencies of the material sets one reaches a realm of ideas of higher reality as they achieve a better unification of the many into unity; in particular, the introduction of the units acts as a uniformisation and homogenisation of the diversity-matter of the concrete sets which are then able to participate in the idea of the cardinal number. The pertinent contrast in Cantor is between concrete-particular vs. abstract-general. Moreover, it seems, that though cardinals are indeed (related to) ‘abstract sets’, a general concept of ‘abstract set’ is absent here, as on Cantor’s view there can only be one abstract set of a given cardinality i.e. according to Cantor there is something like THE number 2. It seems also somewhat paradoxically to come up with rudimentary structural set theory in order to do transfinite arithmetic !
The views advanced in R. Dedekind’s ‘Was sind und was sollen die Zahlen?’ (1887) appear to be a more decisive step towards the modern structural approach to set theory and mathematics. Though he similarly defines the natural numbers in §6.73 by abstraction as a certain ‘simple infinite set’, he clearly has a concept of such a system up to isomorphism: he has also a more general view of ‘abstract set’ (see also the quote in the footnote at ETCS) and of the importance of mathematics ‘up to isomorphism’. Furthermore, he stresses the fundamental importance of maps between sets as basic concept in general.
The abstractionist view was severely criticized by Gottlob Frege in ‘Die Grundlagen der Arithmetik’ (1884) and his reviews of Cantor and Husserl (1892,1894). Frege rejected the process of ‘abstraction’ involved in such accounts and held views on identity that made the ‘distinct yet indistinguishable units’ an inconsistent concept and spoke therefore sarcastically of ‘unglückliche Einsen’ - ‘unfortunate units’: the adjective has stuck in this context as we find it repeated in Zermelo’s and Tait’s discussion of Cantor’s conception.
Lawvere takes this interpretation of Cantor up again in
He gives prominence to the essentiality of the double negation topos and the “Cantor comonad” that comes with it in the more recent
perspective: A simple example_ , Categories and General Algebraic Structures with Applications 4 no.1 (2015) pp.1–7. (pdf)
Judging from the preface to (Lawvere-Rosebrugh 2003) Myhill made his observation presumably in Lawvere’s 1985 course. C. McLarty interprets Cantor’s cardinals as abstract sets in a well-pointed topos in
Colin McLarty, Defining Sets as Sets of Points of Spaces , JPL 17 (1988) pp.75-90.
Colin McLarty, Elementary Categories, Elementary Toposes , Oxford UP 1992.
The Cantor text Lawvere refers to is
The text is reprinted on pp.282-351 of Cantor’s collected works and the revelant comment by Zermelo is in the second annotation on p.351:
An English version of Cantor’s 1885 review of Frege appears as
For an extensive discussion and relevant quotes of Cantor’s platonism and abstractionism see
For a recent controversial discussion of Cantorian abstraction, the second article using MLTT:
K. Fine, Cantorian abstraction: A reconstruction and defense , J. Phil. 95 no.12 (1998) pp.599-634.
W. Hinzen, Constructive versus Ontological Construals of Cantorian Ordinals , Hist. Phil. Logic 24 no.1 (2003) pp.45-63.
The key concepts of ‘one’ and ‘many’ taken from Plato’s ‘Philebus’ dialogue reappear in Hegel’s “set theory”:
For more on the Frege-Cantor debate on the concept of cardinal number, which touches upon the status of structural mathematics more generally, see
For a proposal that contrasts Cantorian set theory with Zermelo-Fraenkel set theory (though leaving out the question of abstract sets) see
It seems somewhat ironical that Weierstrass whose quest for arithmetical foundations of analysis was one of the main driving forces behind 19th mathematics’ move towards set-theoretic foundations actually held views on his fundamental concept ‘number’ that are closely related to a structural view! ↩
Last revised on July 7, 2020 at 06:03:01. See the history of this page for a list of all contributions to it.