Philosophy of mathematics is still philosophy and has nothing to do with mathematics, since philosophy does not adhere to any mathematical principles. Philosophy of logic is the same, since philosophy does not adhere to any logical principles. However, the logic of mathematics and the mathematics of logic is more interesting in particular as a major part of informatics can learn from logic. It is somehow interesting that the philosophy of set theory is never on any agenda, even if set theory, logic and mathematics is very much intertwined. Early 20th century work and development in G??ttingen, Vienna and Warsaw, and other places, of course, is often said to be very well known, but surprisingly few actually still read work from that era. Why, for instance, is it so clear that G??del's Incompleteness Theorem is a "theorem" and not a "paradox"? After all, it is nothing but a bit more subtle version of the Liar paradox. I paradox means Fix it!, whereas a theorem means Don't touch!. In logic, why do we make a giant leap from Aristotle (who was a philosopher, not a logician) to Boole/Peano/Frege, ignoring whatever happened logically in between? In math we don't do that. Category theory can play a role in all this, in particular in more strict definitions of the notion of logic. Type theory is good example, where type constructors are still allowed to dangle around any formalism adopted, and then something magic comes in from the outside and provides a "solution". HoTT and its predecessors are doing that all the time. The phrase "Philosophy of Practice of Mathematics & Informatics", I guess, is as good as any variation of it. WE could also debate about the "Mathematical Practice of the Informatics of Philosophy", or the "Informatics if Mathematics of Philosophy & Practice", or even the "Mathematical Practice of Informatics without any interference whatsoever of Philosophy". Best, Patrik www.glioc.com On 2015-07-25 16:57, Graham White wrote:

And (continuing "why on the categories mailing list?") it seems to some people (such as me) that what category theory actually is, is a formal description of the practice of mathematics, rather than a foundation for mathematics. It may do the latter as well (though I don't really believe so), but an account of the practice of mathematics would be far more philosophically interesting than a foundation. It would, for example, allow a dialogue between the philosophy of mathematics and the rest of philosophy, which has, for 30 or 40 years now, been much less foundational than it used to be. And it may even make category theory an important tool in philosophy generally.

Graham

On 24/07/15 05:12, Ralph Matthes wrote:

...

[why on the categories mailing list? some of the courses are strongly based on category theory, and it seems that quite some subscribers to this list are interested in connections between mathematics and philosophy]

Dear colleagues,

The thematic trimester CIPPMI "Current Issues in the Philosophy of Practice of Mathematics & Informatics" will be held from 4th April to 1st July 2016 at the Centre International de Math??matiques et d'Informatique de Toulouse (CIMI).

This thematic trimester is organised by an interdisciplinary team of researchers in Mathematics, Philosophy, and Computer Science from the Institut de Math??matiques de Toulouse (IMT) & the Institut de Recherche en Informatique de Toulouse (IRIT).

It will feature course sessions, workshops, and a thematic school on themes at the interface of Philosophy, Mathematics and Computer Science.

You will find all relevant information on the website of the thematic trimester that will be regularly updated: http://www.cimi.univ-toulouse.fr/cippmi/en

A mailing list allows you to receive the different announcements from CIPPMI: https://sympa.math.ups-tlse.fr/wws/info/cippmi

You can register at http://www.cimi.univ-toulouse.fr/cippmi/fr/inscriptionregistration

A funding for accommodation is available in priority for junior researchers and for some senior researchers without funding from their laboratory. For further information, please consult the page: http://www.cimi.univ-toulouse.fr/cippmi/fr/boursesgrants

With apologies for cross-posting, best regards, the CIPPMI scientific organisation committee.

---

Ralph Matthes

IRIT (CNRS & Univ. Toulouse) http://www.irit.fr/~Ralph.Matthes/

-- Prof. Patrik Eklund Ume?? University Department of Computing Science SE-90187 Ume?? Sweden ------------------------- mobile +46 70 586 4414 website www8.cs.umu.se/~peklund [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On 7/26/2015 12:33 PM, Patrik Eklund wrote:

Philosophy of mathematics is still philosophy and has nothing to do with mathematics, since philosophy does not adhere to any mathematical principles.

Philosophy of logic is the same, since philosophy does not adhere to any logical principles.

By an argument such as this, it would appear that you could say that bacteriology has nothing to do with bacteria, because you cannot grow bacteriology on a Petri dish or sequence its DNA -and obviously this would be absurd. I think the source of the confusion here is that mathematics is reflexive in a way that bacteriology isn't: mathematics/logic _does_ feed back into itself and become a tool for doing more mathematics/logic. It's thus tempting to think that anything outside this loop is not part of mathematics/logic. However, the loop is not closed, and cannot be. There are questions which are legitimate parts of mathematics/logic that cannot be answered internally. I'm not talking about Goedel incompleteness here (though one might), but about why we do what we do. If we want to say what constitutes mathematics worth doing - to say why Fermat's Last Theorem or the Riemann Hypothesis are more important that the (3n+1) problem or finding palindromic sequences in the decimal expansion of pi - we cannot do this by calculation and proof. This is an example of a place where philosophy of mathematics can have a genuine connection. -Robert Dawson [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

On Hewitt's Inconsistency Robustness, and given that my original posting was related to category theory as a metalanguage for logic in the sense of explaining extensions of the Goguen-Burstall and Meseguer approaches, phrases like "mathematics is inconsistent" and "proof does not intuitively increase our confidence in the consistency of mathematics" (appear in Hewitt's paper) really make no sense at all. Martin Escardo in his first reply to my posting says "I am not sure why these questions are being asked in this list". My simple reply is because my posting was suggesting to debate the use of category theory as a the underlying language for logic. Basically none of the replies to my posting has so far anything to do with category theory. Hewitt says "A mathematical theory is an extension of mathematics whose proofs are computationally enumerable." which basically means he says logic is an extension of mathematics. Logic is part of mathematics as a discipline, so logic cannot be the external canon for mathematics as little as philosophy can be the external canon for science. Nothing is global or canonic as far as mathematics is concerned. Many have tried to do so, but their is no consensus in that direction. Hewitt's note confirms that quite clearly. "Philosophy and mathematics can have a genuine connection", but that doesn't lead to anything useful. It just adds to fragmentation of the understanding of foundations. With category theory underlying type theory we believe we can manage type constructors more formally without restarting foundations a la HoTT. In Hewitt's note I am surprised not to see anything written about the 'iota' and 'o' types when speaking about Church. These are key types in the categorical description of the "lativity" of logic I mentioned earlier, and universal algebra comes short to deal with them. That lativity is important also otherwise. Proof mechanisms come after (not before or during) the bags of terms and sentences have been closed and sealed. So statements and enumerations involving proofs are not sentences in that sealed bag. This is the principle of lativity not respected by traditional logic where the metalanguage is absent. With categories in the meta for logic, we have term monads because we need substitutions to compose, but we only have sentence functors. A sentences functor being a monad simply means those sentences are terms. We have also said that an unquantified proposition P is not a sentence. It's a term. Quantifying it makes it a sentence, but then we have problems with the negation. The "not" before a P (as a term) is really different from a "not" before a quantifier, isn't it? Traditionally not, but when we start to construct terms and sentences otherwise than using natural language, we see it more clearly. All this is overlooked in traditional logic, and having category theory as a metalanguage for constructing terms and sentences, respectively as monads and functors, gives us something new to think about. This is our credo. Languages are more or less formal. Languages are more or less mathematically defined. Whatever the situation, if an object language allows less formal outsiders to be invoked side by side with its underlying metalanguage, ugly gets even worse. Best, Patrik [For admin and other information see: http://www.mta.ca/~cat-dist/ ]