Current Issues in the Philosophy of Practice of Mathematics & Informatics
[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/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/
-- Graham White Electronic Engineering and Computer Science Queen Mary, University of London http://www.eecs.qmul.ac.uk/~graham/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
I am not sure why these questions are being asked in this list: On 26/07/15 16:33, Patrik Eklund wrote:
Why, for instance, is it so clear that G??del's Incompleteness Theorem is a "theorem" and not a "paradox"?
I am not sure what you mean by a paradox. But let me take this as a possible interpretation: A paradox is a statement P such that both P and not P are theorems (or, equivalently, such that P holds iff not P holds). As far as current mathematical knowledge goes, Goedel's Incompleteness Theorem is just a theorem, with significant further work needed to elevate it to the status of 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!.
For comparison, in naive set theory, the set of all sets that don't belong to themselves does lead to a paradox, corresponding to the Liar Paradox: this set belongs to itself if and only it doesn't. Naive set theory is inconsistent (and hence deserves its name). In ZFC, however, the same argument *proves* that there is no set of all sets, and no set of sets that don't belong to themselves. It is important here that ZFC can actually formulate the question of whether there is a set of all sets. And the answer is no. Of course, in principle, the possibility is open that ZFC, too, has a some paradox. This involves exhibiting a statement P and two proofs, following strict rules of mathematical rigour, one of the statement P and another of the statement not P. Nobody has so far managed to exhibit such three things. This is so hard that probably deserves a Fields Medal (followed by immediate eviction from the mathematical community). M. [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/ ]
While agreeing wholeheartedly with Robert, I would like to point a finger at what I think of as the "monolithic mathematics mob", MMM. These are the people who treat mathematics as a single theory. Carl Hewitt, with whom I shared an admin at MIT for a decade, has a proof of the consistency of mathematics based on that premise at https://docs.google.com/file/d/0B79uetkQ_hCKbkFpbFJQVFhvdU0/edit?usp=sharing along with a little more that so far I've been unable to pin down. But I rather suspect that pretty much everyone who finds G??del's second incompleteness theorem paradoxical shares Hewitt's view of mathematics as a single theory. I find the following difficulties with the MMM view. 1. You can't have a theory without a language. What is the language of mathematics? Judging by appearances it would seem to be a living thing that grows in different directions following the many varied and evolving interests of mathematicians, pure, applied, Arcturan, or whatever. This leads to: Principle 1. There will never come a time when mathematicians have settled on the language of mathematics. 2. In the unlikely event that Principle 1 is violated, namely by collecting every mathematical symbol that will ever be needed in mathematics into a single language possessed of a single consistent theory T, there is no reason to expect any such thing to be recursively enumerable. With the requisite assumptions this is G??del's first rather than his second incompleteness theorem. But this raises the imponderable question of whether mathematics is what mathematicians know, or what they could ever know (by enumeration of theorems), or the aforementioned theory T, which they can never know at any future time t even as all the consequences-in-principle of whatever finite axiomatization of T they have agreed on by time t. Which leads to: Principle 2. There will never come a time when mathematicians have settled on what constitutes mathematics. G??del's first incompleteness theorem suffices for Principle 2. Principle 1 is even more elementary. Much as I appreciate G??del's second incompleteness theorem, it seems to me that his first is all that's needed to answer those who find the second paradoxical. Vaughan Pratt On 7/29/2015 6:56 AM, Robert Dawson wrote: ...
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/ ]
A week or more ago Patrik Eklund <peklund@cs.umu.se> wrote of
... a "lativity" in logic ... .
Not wishing to broadcast my illiteracy in the matter, I searched high and low for the meaning of the quoted term, to no avail: neither Google, nor Wikipedia, nor the other contemporary search mechanisms I tried, offered any insight whatsoever into that term. So I ask you now, in public, where my shame can be greatest: what do you mean by "lativity"? Anagram for "vitality"? (Yes, I have seen -- but been mystified by -- Eklund's use of that term in an older Categories posting, of Feb 05 2014, 09:55, with Subject: : categories: Re: Martin-Lof type theory gentle introduction please ... .) Many thanks. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Fred E.J. Linton -
Graham White -
Martin Escardo -
Patrik Eklund -
Ralph Matthes -
Robert Dawson -
Vaughan Pratt