The Kreisel Lawvere Debate on Categories and the Foundations of Mathematics

[@bernardofitzpatrick5403]

Just what I was looking for. Thank you!

[@bernardofitzpatrick5403]

Thank you very much. Will look up the sources you mention.

[@marshacd]

Exceptionally interesting to me, too, so also thank you for posting! Are there other videos from this conference? I would like very much to see them if they exist. (I have just subscribed to your postings, so perhaps other videos will be suggested by KZbin.)

[@xuan-gottfriedyang5094]

Thank you for the interesting. This lecture was of the conference "Journée Aspects historiques et philosophiques de la théorie des Catégories". You may find the other lectures of it here www.diffusion.ens.fr/index.php?res=cycles&idcycle=229"

[@xuan-gottfriedyang5094]

and a new conference on similary subject(also with many videos) sites.google.com/site/logiquecategorique/autres-seminaires/ihes/ihestopos

[@bernardofitzpatrick5403]

Thank you! was looking for something on this topic

[@xuan-gottfriedyang5094]

The talker is Jean-Pierre Marquis who is also an author of "The History of Categorical Logic: 1963-1977". One could find this paper on his homepage I think

[@bernardofitzpatrick5403]

Xuan YANG thank you very much for the link. Looking foward to reading the paper later today.

[@xuan-gottfriedyang5094]

Bernard Fitzpatrick and there is also “the development of categorical logic” of John L. Bell

[@bernardofitzpatrick5403]

Xuan YANG that you very much for your help. Much appreciated.

[@deRoland87]

This is excellent stuff! To my mind what could be emphasized more is that category theory qua foundation leaves behind one very important desideratum of all logical foundations that had been pursued so far. The desideratum I am talking about is that the primitive notions have a naive counterpart in ordinary language. ZFC as a foundation of mathematics is always justified by referring back to the primitive understanding of collection (this is implicit in Feferman's 1977 paper). "X is a member of the set Y" is the primitive notion of set theory which can be explained to an infant and which everyone can be made to see the importance of. One need not dwell on this notion for a long time to see that it is used in all rational thought! To replace ZFC with Lawvere's ETCS means to ditch this desideratum. All the primitives of the theory are multi-interpretable! By this I mean that composition of arrows has an entirely different interpretation depending on what category is our intended interpretation! And this is a good thing. But as such category theory is another big step away from the idea that there exist some primitive concepts that are privileged in a way that they should be the basis for all of mathematics. Rather, we privilege those concepts which are multi-interpretable, because this way we'll know that they will make sense in many categories. In other words, category theory separates mathematics by another step from the analysis of ordinary language. The hegemony of the desideratum I mentioned just now is probably explained by the strong influence of analytic philosophy on the philosophy and foundations of mathematics in the 20th century (my guess).

[@MathProofsable]

For me, the problem with material set theory (eg. ZFC) is how many axioms need to be introduced to avoid contradictions. Structural set theory (eg. ETCS) boils down to working with a well-pointed topos with a natural numbers object and which satisfies the axiom of choice. This is the concept of "set for mathematicians" where the relevant structures of set are structures which can exist in any category. I don't care so much about logical foundations when I am working on a problem, but I do care about constructing limits, colimits, taking subobjects, and forming exponentials. I have no problem with ZFC if foundations is your thing. But for many mathematicians, ETCS gives you what you need. Since a model of ZFC is a well-pointed topos with a natural numbers object satisfying the AC, why should we be concerned about the axiom of regularity for example. It just muddies the understanding in my opinion.

[@deRoland87]

that is another good reason in my opinion. In a way, moving from ZFC to ETCS is to leave one thing behind. Epsilon in ZFC is one of the meanings of the copula ("is"). It is the meaning of "is" that you find in sentences such as "Henry is a horse". We can take this to mean "Henry is a member of the set of horses". This is a relation between objects (Henry and the set of horses) and this way of talking is ubiquitous in ordinary language. Hence, ZFC contributes something to the clarification of the logic present in ordinary language. In ETCS, this aspect is lost, because we no longer put objects in relation to each other. But we gain many things. First of all, sets qua abstract structure are (arguably) much better understood in ETCS than in ZFC. The features of the category of sets are features that many other categories (namely topoi) have too. etc. etc.

[@MathProofsable]

Roland Bolz What is lost is not elementhood, but distinctions between nonequal isomorphic sets. For ETCS, the only invariant that matters is cardinality. Elementhood in ETCS is a set map from a terminal set to another set. So "Henry is a horse" is the name of a map from 1 to the set of horses. In this sense, elementhood and subset are placed on equal footing. The statement "Men are mortal" becomes a monomorphism from the set of men to the set of mortals. Lawvere discusses how this is similar to what Cantor had in mind.

[@deRoland87]

Yes I appreciate that elementhood can be reintroduced using arrows from the terminals. But this has little do with how set membership is ordinarily expressed as a relation between two objects (as in the example). So I am merely pointing out that ETCS also says goodbye to the idea that the primitives of a logical system need to reflect our usage of them in ordinary language. This is what interests me as a philosopher of language and logician.

[@justin9571]

Bill is goated

[@shoopinc]

White boy goated with the sauce bustin it down sexual style

[@xuan-gottfriedyang5094]

books.google.de/books?id=bvy0aANuhPYC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=true

[@lopezb]

Thank you!

[@xieyuheng]

看完了这个 talk 第一个观众的提问竟然还是强烈排斥 category 的 ... 第二个人提问用的法语 ... 哇啦哇啦破夸破夸 回答也是法语 ... (那我用中文好了)