Re: pragmatic foundation

Re: pragmatic foundation

[Vaughan Pratt]

Eduardo J. Dubuc wrote:
> I wish you (V.P.) were more clear. I can not see what is your point.
> Witty msages for the Illuminati don't serve any pourpose, except
> amusement to some.

Yes, sorry about that.  A couple of others wrote privately with the same
request.

I have no problem with the notion of mathematical truth per se, which I
imagine to be what all mathematicians seek, along with mathematical
tools and a consensus thereon by their colleagues.

What I had in mind by the "haunting" remark is that the implications of
Goedel's incompleteness results don't immediately leap out at one, and
there is a certain optimistic tendency to minimize those implications
and continue to argue the issues as though Goedel's theorems weren't
relevant.

We can't *define* mathematical truth (Tarski may have been the first to
enunciate that implication most clearly), yet we can often recognize it
when we see it.  Learning to do mathematics amounts to learning how to
find and communicate those mathematical truths that are easily
recognized as such by other mathematicians according to community standards.

We imagine that mathematics on Arcturus must be like ours, but
mathematics is an intrinsically cultural subject and I don't see why
Arcturan mathematics should be like ours.  Do Arcturans have logic?  Do
they have algebra?  Do they draw a distinction between the two?  Do they
believe in either?  Add category theory as a third framework and ask the
same questions of it.  Do they know about initial algebras and final
coalgebras, and if so which came first for them?  Do they know about
monads and adjunctions, and if so which came first?

That's surely too brief to be clear.  I'd be happy to engage further in
this sort of speculation on the practice of mathematics as a cultural
issue.  I have less to contribute on the intrinsic nature of mathematics
itself for lack of insight into its scope.

Vaughan Pratt


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Re: pragmatic foundation

[Eduardo J. Dubuc]

Vaughan Pratt wrote:
>
> Eduardo J. Dubuc wrote: [regarding Manin]
>> He does not place himself
>> within
>> any philosophical or political frame. He feels free to say what it
>> crosses his
>> mind just as it comes. beautiful !
>
> Right, but clearly we cannot extend the same freedom to the hoi polloi,
> who cannot be trusted not to abuse it.
>
> Vaughan Pratt
>

of course, only a few can make of such freedom a meaningful discourse  e.d.


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Re: pragmatic foundation

[Vaughan Pratt]

Eduardo J. Dubuc wrote: [regarding Manin]
> He does not place himself
> within
> any philosophical or political frame. He feels free to say what it
> crosses his
> mind just as it comes. beautiful !

Right, but clearly we cannot extend the same freedom to the hoi polloi,
who cannot be trusted not to abuse it.

Vaughan Pratt


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Re: pragmatic foundation

[Zinovy Diskin]

>>
>>> I invite everyone to read the interesting interview of Yuri Manin
>>> published in the November issue of the Notices of the AMS:
>>
>> Manin is always entertaining but not very careful about what he says.
>>

Hm, Manin is never just entertaining: he wrote several papers
concerning physics, linguistics, psychology, and his  thinking is an
example of how a true mathematical mind works in complex areas like
the humanities, generates unexpected views, reveals deep connections
etc. If the results are readable and enjoyable, it just shows the
literary talent of the author... :)

I also wouldn't say that Manin is not very careful about what he says.
The parts of the interview about foundations and physics say,
basically, this. After Bourbaki, a correct mathematical text should
consist of two parts:
(a) definition of the structure  in question (structure in the sense
of Bourbaki),
(b) deductions about this structure in some logic (perhaps, non-classical).
Manin says that texts generated by physicists do have (b) but not (a).
These are deductions about something that has not been defined and
hence, for a mathematician, that does not exist at all  (the Eiffel
Tower is in the air). This situation is not unique, of course: Manin
mentions Cantor's set theory at the time of invention, and it was and
is so for engineering theories. Software engineering should be of
special interest for this list because modern software executes
deductions about categorical structures.

It is not in the interview explicitly, but the following model of a
mathematical text would be probably close in spirit to what Manin
says. Mathematical texts form a span:
PM <--- MM --->FM
with
PM -- the universe of "physical" mathematical texts (physics, computer
science, engineering etc),
MM -- the mathematician's universe of mathematical texts; they are
written in a special subset of the natural language (nowadays, in
accordance with Bourbaki or category theory),
FM -- the universe of formal (machine-readable) mathematical texts.

A physicist is interested in the left foot, a logicist  -- in the
right one, but mathematics is about the entire span (well, for a true
mathematician, P stands for Platonic rather than Physics). If you
want: the logicist view is more normative because it insists on the
right right leg, but Bourbaki concerned about the entire span and did
not want to fix neither right nor the left legs (unless P is for
Platonic). So, they proposed a reasonable structure for MM for which
the left and right sides of the whole could be added (if needed). It's
indeed more about practical foundations...

After all, Eduardo said it best:

> Well, the fact that he is not very careful is precisely what makes his
> saying
> meaningful, interesting, fresh and enjoyable. He does not place himself
> within
> any philosophical or political frame. He feels free to say what it crosses
> his
> mind just as it comes. beautiful !
>


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Re: pragmatic foundation

[Eduardo J. Dubuc]

Colin McLarty wrote:
> 2009/11/6 Andre Joyal <joyal.andre@uqam.ca>:
>
> writes
>
>> I invite everyone to read the interesting interview of Yuri Manin
>> published in the November issue of the Notices of the AMS:
>
> Manin is always entertaining but not very careful about what he says.
>

Well, the fact that he is not very careful is precisely what makes his saying
meaningful, interesting, fresh and enjoyable. He does not place himself within
any philosophical or political frame. He feels free to say what it crosses his
mind just as it comes. beautiful !


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Re: pragmatic foundation

[Andre.Rodin]

Selon Colin McLarty <colin.mclarty@case.edu>:


> I myself am also confident that people will calm down and notice that
> axiomatic categorical foundations such as ETCS and CCAF work perfectly
> well, in formal terms, and relate much more directly to practice than
> any earlier foundations.  One hundred and fifty years of explicitly
> foundational thought has made this progress possible.  By now, that
> can hardly qualify as "extraordinary"!
>
I do NOT believe that ETCS and CCAF "work perfectly well". Each of these involve
two foundational "layers", namely, the classical "bottom" and a categorical
"superstructure". By the classical bottom I mean NOT an underlying Set theory
but the "Elementary theory of categories" (ETC), i.e. a theory of categories
using the usual First-Order Logic (FOL) and relying on the standard
Hilbert-Tarski-style axiomatic method. I agree with John Mayberry and some
other people who argue that this aximatic method alone assumes a basic notion
of set or collection. Unlike Mayberry I don't think that this fact implies that
the project of categorical foundations, as a alternative to and replacement for
set-theoretic foundations, is futile. Recall that the axiomatic method we are
talking about (which is, of cause, quite different from Euclid's method and
other earlier versions of axiomatic method) emerged together with Set theory.
In order to make categorical foundations into a viable alternative of
set-theoretic foundations we still need to provide Category theory with a new
axiomatic method rather than use the older axiomatic method as do ETCS and
CCAF. Elements of this prospective axiomatic method are found in what I just
called the "categorical superstructure" of ETCS and CCAF but as far as these
theories are concerned the classical background (FOL+ETC) is indispensable.
This is why I say that ETCS and CCAF do NOT work perfectly weel as categorical
foundations.
Building of "purely categorical" foundations remains an open problem. It is not
a matter of a ideological purity but a matter of complete "rebuilding" (Manin's
word) of foundations: in my view, such a rebuilding is healthy and refreshing
in any circumstances (unless it clashes severely with practice).




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Re: pragmatic foundation

[Andre.Rodin]

I agree with WHAT Yuri Ivanovitch Manin says about foundations of mathematics
but DISagree when he calls these foundations "pragmatic". I might be mistaken
(in this case, I hope, Yuri Ivanovitch will correct me) but I don't think that
in the given context the word "pragmatic" is supposed to be understood with a
philosophical seriousness. In the given context "pragmatic" is rather
synonymous to "practical" as opposed to "theoretical" - and perhaps also to
"purely mathematical" as opposed to "philosophical". I fully share with Yuri
Ivanovitch his disappointment about what he calls the "normative foundations
that logicists or constructivists tried to impose". But I see a solution in a
new dialectical philosophy of foundations (tightly connected to mathematical
practice), NOT in developing foundations purely "pragmatically" without
theoretical and philosophical grounds. The history teaches us that
philosophical thinking is crucial for what Yuri Ivanovitch calls the
"rebuilding" of foundations, and I don't see any reason why this might cease to
be true today.
On the contrary, I think that the acceleration of mathematical progress
necessitates the acceleration of rebuilding of foundations - and this makes
philosophy more relevant to mathematical research than ever. "Logicists and
constructivists" don't have centuries to come to eternalise their findings by
establishing a new Scholastic tradition in philosophy - even if some of them
would wish it.

Andrei

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Re: pragmatic foundation

[Vaughan Pratt]

Colin McLarty wrote:
> [responding to Manin thoughts] I myself am also confident that people will calm down and notice that
> axiomatic categorical foundations such as ETCS and CCAF work perfectly
> well, in formal terms, and relate much more directly to practice than
> any earlier foundations.

Thanks, Colin.  There I was nicely calmed down and then you got me all
worked up again.  :)

I prefer the Euclidean plane over sets as a suitable starting point for
understanding mathematics.  What advantage is there to making geometry
rest on set theory as opposed to vice versa?

What is wrong with starting from a geodesic space as a place where it is
always determined, given two points, what is the next one, subject to
some simple equational principles?  This is a common basis for the
second postulate of Book I of Euclid's *Elements*, Newton's first law
of motion, Einstein's theory of general relativity that a falling body
is merely following a geodesic in a space curved by a nearby mass, and
the notion of Hamiltonian flow of a vector field for an energy function
defined on the cotangent space of a manifold as an expression of the
principle of least action.

In this framework a *set* is simply a geodesic space where the next
point after x and y is x.  (So if I ask what is the next element in the
sequence 3,4,... the answer is 3, not 5.)

More on this at http://boole.stanford.edu/pub/consgeom.pdf .  A geodesic
space or geode, aka kei, is related to a quandle (see
http://en.wikipedia.org/wiki/Quandle ), the difference being that for
abelian groups, quandles are merely sets whereas flat geodes (those
satisfying Euclid's 5th postulate) form a symmetric monoidal closed
category fully and reflectively extending Set (properly of course).
Moreover its subdirect irreducibles are those of Ab except for those of
even order as per the last slide.  Quandles are for knot theory, not
geometry.

The difference between sets and geodesic spaces in foundations is like
the difference between scales and Fur Elise for piano students.  Both
are good ways to get started but the second is more interesting.
(Apologies again to Eduardo for my impenetrable writing, in this case I
can only counsel patience since these ideas seem to come with a certain
viscosity that inhibits any royal road of the kind Eduardo would like.)

Best,
Vaughan


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Re: pragmatic foundation

[Colin McLarty]

2009/11/6 Andre Joyal <joyal.andre@uqam.ca>:

writes

> I invite everyone to read the interesting interview of Yuri Manin
> published in the November issue of the Notices of the AMS:

Manin is always entertaining but not very careful about what he says.

André says:

> The foundational framework of Bourbaki is very much in the tradition
> of Zermelo-Fraenkel, Godel-Bernays and Russell.
> I am aware that Bourbaki was more interested in the development of
> mathematics than in its foundation.

I agree.  Naturally Bourbaki was in a better situation to make up a
system that would work, since they had the others behind them.  And
still their system did not work in fact.

Russell was more concerned with philosophic issues of logic, but his
touchstone for logic was that it should work!  (He was very clear
about this by 1919, in his Principles Of Mathematical Philosophy.)  He
knew a lot less than Zermelo about what would work for two reasons:
Russell got into it much earlier, and Russell studied math as a
philosopher at Cambridge while Zermelo studied it as a mathematician
with Hilbert in Göttingen and in debates with Poincaré.

All these people sought a foundation that would make sense in itself
and would work.  Naturally they had different emphases, partly shaped
by the different resources they could draw on.  Russell, Zermelo, and
Gödel all read each other (recalling that Russell was 59 years old,
and two decades past his work on logic, when Gödel published the
incompleteness theorem, and everyone took years absorbing it).

> In the interview, Manin also said that:
>
>>And so I don’t foresee anything extraordinary
>>in the next twenty years.

Of course we do not expect to *foresee* extraordinary things.

>> Probably, a rebuilding of what I call the “pragmatic
>> foundations of mathematics” will continue.

That is a pretty safe bet.

>>By this I mean simply a
>>codification of efficient new intuitive tools, such
>>as Feynman path integrals, higher categories, the
>>“brave new algebra” of homotopy theorists, as
>>well as emerging new value systems and accepted
>>forms of presenting results that exist in the minds
>>and research papers of working mathematicians
>>here and now, at each particular time.

Yes, there will be progress on all of these things.

I myself am also confident that people will calm down and notice that
axiomatic categorical foundations such as ETCS and CCAF work perfectly
well, in formal terms, and relate much more directly to practice than
any earlier foundations.  One hundred and fifty years of explicitly
foundational thought has made this progress possible.  By now, that
can hardly qualify as "extraordinary"!

best, Colin


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Re: pragmatic foundation

[Eduardo J. Dubuc]

Andre touches an interesting problem.

I wish you (V.P.) were more clear. I can not see what is your point. Witty
messages for the Illuminati don't serve any purpose, except amusement to
some.

I would like Bill (Lawvere) send us his opinion about Maning's views.

e.d.

Vaughan Pratt wrote:
>> Any comments?
>> AJ
>
> Hi Andre.  I read the Manin interview in the AMS Notices with much
> interest myself last week.
>
> With regard to your request for comments, I can only repeat von
> Neumann's remark after Goedel's lecture as cited recently in Logicomix:
> "It's all over."  I couldn't agree more.  Like a rat trap slamming shut
> on a rat that we no longer need fret about.
>
> That the rat's ghost continues to haunt so many is an interesting
> commentary on human nature.
>
> Cheers,
> Vaughan
>
>


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Re: pragmatic foundation

[Vaughan Pratt]

> Any comments?
> AJ

Hi Andre.  I read the Manin interview in the AMS Notices with much
interest myself last week.

With regard to your request for comments, I can only repeat von
Neumann's remark after Goedel's lecture as cited recently in Logicomix:
"It's all over."  I couldn't agree more.  Like a rat trap slamming shut
on a rat that we no longer need fret about.

That the rat's ghost continues to haunt so many is an interesting
commentary on human nature.

Cheers,
Vaughan


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pragmatic foundation

[Andre Joyal]

Dear category theorists,

I invite everyone to read the interesting interview of Yuri Manin 
published in the November issue of the Notices of the AMS:

http://www.ams.org/notices/200910

http://www.ams.org/notices/200910/rtx091001268p.pdf

One the ideas discussed by Manin is that of a "pragmatic foundation" of
mathematics as opposed to a "normative foundation" by logicists or constructivists. 
He attributes the former to Bourbaki.

I disagree.

The foundational framework of Bourbaki is very much in the tradition 
of Zermelo-Fraenkel, Godel-Bernays and Russell.
I am aware that Bourbaki was more interested in the development of 
mathematics than in its foundation. 
My guess is that the foundation was too problematic to be given a proeminent place 
in the treaty, not for logical reasons but for conceptual reasons.
I claim that nobody truly understand set theory, even today!
The emperor has no clothes!
I mean that the hierarchy of infinite cardinals is so profoundly mysterious 
that it looks pathological.
What is the value of a theory if it leads to meaningless problems and structures? 
Having no good answer to offer, Bourbaki decided to diminish the importance of 
foundation rather than leaving it open. 
It may explain why category theory was not incorporated in the foundation later.

In the interview, Manin also said that:

>And so I don’t foresee anything extraordinary 
>in the next twenty years. Probably, a rebuilding of 
>what I call the “pragmatic foundations of math- 
>ematics” will continue. By this I mean simply a 
>codification of efficient new intuitive tools, such 
>as Feynman path integrals, higher categories, the 
>“brave new algebra” of homotopy theorists, as 
>well as emerging new value systems and accepted 
>forms of presenting results that exist in the minds 
>and research papers of working mathematicians 
>here and now, at each particular time. 

Any comments?


AJ


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