Re: Explanations

Re: Explanations

[peasthope]

Fred,

The process of resolving misconceptions and comprehending an
intended meaning is part of learning.  So I view this matter
of "other O" as helping me to understand the equaliser.

Yes, Wikipedia has careless editing and that is one of the
motivations for building the Citizendium.  [http://en.citizendium.org/]
It has careless editing also; hopefully not so prevalent as
in the Wikipedia.  Even with the limitations, the Wikipedia
and the Citizendium have helped me.

For publishing with immunity to outside interference, nothing is better than
a personal Web page.

Thanks for the responses,          ... Peter E.

-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



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Re: Explanations

[Clemson Steve]

I think this is a perfect example of when something is or is not an
explanation. It's both in this case: to the cogniscenti, it is perfectly
clear; the novice is going to head off on the wrong track.

On 4/30/11 15:58, Charles Wells wrote:
> In the expression "any x:T->X" the T depends on x.  If you use the
> arrow notation you seem bound to name the domain of the morphism.  You
> could say "for any x with codomain X there is an e:dom x ->  X ..." but
> in the rest of the sentence you will have to mention the domain again.
>
> My impression is that notation "any x:T->X" where T depends on x
> without that fact being mentioned is common in category theory
> writing.  There is nothing wrong with this if a reader understands the
> intent.
<snip>

-- 
Dr. D. E. Stevenson
Associate Professor
Director, Insitute for Modeling and Simulation Applications
School of Computing, Clemson University


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Re: Explanations

[peasthope]

From:	Charles Wells <charles@abstractmath.org>
Date:	Sat, 30 Apr 2011 14:58:14 -0500
> In the expression "any x:T->X" the T depends on x.

Why not x depends on T?  Having objects prior to maps seems
more natural than maps prior to objects.

> I would call it "suppression of dependence".

Will try to store that concept away for future reference.

Thanks,            ... Peter E.

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Re: Explanations

[peasthope]

Fred,

From:	"Fred E.J. Linton" <fejlinton@usa.net>
Date:	Sat, 30 Apr 2011 17:09:22 -0400
> ... every reason *not* to wish to restrict
> attention only to objects O *other* than E or T or X ...

Nice catch.  According to the history, readers have been complacent
about "other" for at least seven years!  Gone now,  ... Peter E.

"http://en.wikipedia.org/wiki/Equaliser_(mathematics)"



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Re: Explanations

[F. William Lawvere]

Dear Peter
We thought of "other" of course. But that word has no agreed on
mathematical definition. Students who think it means "distinct"
will be confused when told that it is a special application of the UMP
that yields the graph of a map as a section of a projection.
(Perhaps best if "self" is a special case of "other" ?)

Sammy always scolded Jon, Fred, Myles, and me that such "helpful"
explanations make difficult the digestion and mathematical use
of simple clear definitions. (I don't think this excludes explanation
in a separate paragraph or footnote).
Bill

> From: peasthope@shaw.ca
> Date: Fri, 29 Apr 2011 11:56:48 -0800
> To: categories@mta.ca
> CC: peasthope@shaw.ca
> Subject: categories: Re: Explanations
> 
> Charles & everyone,
> 
> Earlier peasthope wrote,
> "...changing a few words of a sentence can make a concept obvious rather
> than nebulous".  Revise that to "obvious rather than difficult".
> 
> From:	Charles Wells <charles@abstractmath.org>
> Date:	Fri, 22 Apr 2011 09:37:44 -0500
>> Can you give specific examples?  I suspect that in most cases the change
>> introduces a useful metaphor that was hidden before.
> 
> Here is a small example from the _Conceptual Mathematics_ of
> Lawvere and Schanuel.  No offense to the authors or the book.
> It's an indispensible and invaluable resource.
> 
> L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there is
> exactly one e:T-->E ... ."    "For all T" is implicit.
> 
> http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory
> ... defined by a universal property, ... object E and morphism eq ... such that,
> given any other object O and morphism m ... ."
> 
> For me, the reference to "any other object O" helps.  The definition in  the
> Wikipedia seems to reveal the "universality" of the equalizer better.  The
> diagram also helps.
> 
> A trivial issue for most readers but a small detail can make a difference  for
> a student.
> 
> Regards,                     ... Peter E.
> 
> -- 
> Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
> Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
> Personal pages http://members.shaw.ca/peasthope/ .
> 
> 

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Re: Explanations

[Fred E.J. Linton]

Hi, Peter,

Actually, the word "other" below introduces a red herring:
there is in fact every reason *not* to wish to restrict 
attention only to objects O *other* than E or T or X -- indeed, 
I can imagine that there might be settings in which there are 
*no* objects "other than" E or T or X, in which case the 
Wikipedia verbiage quoted paints you into a corner you really 
*don't* want to be in :-) .

Cheers, -- Fred

------ Original Message ------
Received: Sat, 30 Apr 2011 03:30:49 PM EDT
From: peasthope@shaw.ca
To: categories@mta.ca
Cc: peasthope@shaw.ca
Subject: categories: Re: Explanations

> Charles & everyone,
> 
> Earlier peasthope wrote,
> "...changing a few words of a sentence can make a concept obvious rather
> than nebulous".  Revise that to "obvious rather than difficult".
> 
> From:	Charles Wells <charles@abstractmath.org>
> Date:	Fri, 22 Apr 2011 09:37:44 -0500
>> Can you give specific examples?  I suspect that in most cases the change
>> introduces a useful metaphor that was hidden before.
> 
> Here is a small example from the _Conceptual Mathematics_ of
> Lawvere and Schanuel.  No offense to the authors or the book.
> It's an indispensible and invaluable resource.
> 
> L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there
is
> exactly one e:T-->E ... ."    "For all T" is implicit.
> 
> http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory
> ... defined by a universal property, ... object E and morphism eq ... such
that,
> given any other object O and morphism m ... ."
> 
> For me, the reference to "any other object O" helps.  The definition in  the
> Wikipedia seems to reveal the "universality" of the equalizer better.  The
> diagram also helps.
> 
> A trivial issue for most readers but a small detail can make a difference
for
> a student.
> 
> Regards,                     ... Peter E.
> 
> -- 
> Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
> Shop pages http://carnot.yi.org/ accessible as long as the old drives
survive.
> Personal pages http://members.shaw.ca/peasthope/ .
> 
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]




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Re: Explanations

[Charles Wells]

In the expression "any x:T->X" the T depends on x.  If you use the
arrow notation you seem bound to name the domain of the morphism.  You
could say "for any x with codomain X there is an e:dom x -> X ..." but
in the rest of the sentence you will have to mention the domain again.

My impression is that notation "any x:T->X" where T depends on x
without that fact being mentioned is common in category theory
writing.  There is nothing wrong with this if a reader understands the
intent.

I would call it "suppression of dependence".  In the Handbook I talked
about suppression of parameters, but this is not suppression of
parameters.  It is something I had not noticed before.   Are there
other situations in math where this happens?

On Fri, Apr 29, 2011 at 2:56 PM, <peasthope@shaw.ca> wrote:
>
> Charles & everyone,
>
> Earlier peasthope wrote,
> "...changing a few words of a sentence can make a concept obvious rather
> than nebulous".  Revise that to "obvious rather than difficult".
>
> From:   Charles Wells <charles@abstractmath.org>
> Date:   Fri, 22 Apr 2011 09:37:44 -0500
>> Can you give specific examples?  I suspect that in most cases the change
>> introduces a useful metaphor that was hidden before.
>
> Here is a small example from the _Conceptual Mathematics_ of
> Lawvere and Schanuel.  No offense to the authors or the book.
> It's an indispensible and invaluable resource.
>
> L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there is
> exactly one e:T-->E ... ."    "For all T" is implicit.
>
> http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory
> ... defined by a universal property, ... object E and morphism eq ... such that,
> given any other object O and morphism m ... ."
>
> For me, the reference to "any other object O" helps.  The definition in  the
> Wikipedia seems to reveal the "universality" of the equalizer better.  The
> diagram also helps.
>
> A trivial issue for most readers but a small detail can make a difference  for
> a student.
>
> Regards,                     ... Peter E.
>
> --
> Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
> Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
> Personal pages http://members.shaw.ca/peasthope/ .
>

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Re: Explanations

[peasthope]

Charles & everyone,

Earlier peasthope wrote,
"...changing a few words of a sentence can make a concept obvious rather
than nebulous".  Revise that to "obvious rather than difficult".

From:	Charles Wells <charles@abstractmath.org>
Date:	Fri, 22 Apr 2011 09:37:44 -0500
> Can you give specific examples?  I suspect that in most cases the change
> introduces a useful metaphor that was hidden before.

Here is a small example from the _Conceptual Mathematics_ of
Lawvere and Schanuel.  No offense to the authors or the book.
It's an indispensible and invaluable resource.

L&S page 292, "Definition ... equalizer ... and for each x:T-->X ... there is
exactly one e:T-->E ... ."    "For all T" is implicit.

http://en.wikipedia.org/wiki/Equalizer_(Mathematics) , "In category theory
... defined by a universal property, ... object E and morphism eq ... such that,
given any other object O and morphism m ... ."

For me, the reference to "any other object O" helps.  The definition in the
Wikipedia seems to reveal the "universality" of the equalizer better.  The
diagram also helps.

A trivial issue for most readers but a small detail can make a difference for
a student.

Regards,                     ... Peter E.

-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



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Explanations

[Ellis D. Cooper]

In direct response to "computation alone is not mathematics, and
neither is intuition alone. The former is typical of machines,
whereas the latter is typical of artists" by Marta Bunge I would say
rigor cleans the window through which intuition shines. Category
theory is a house with many windows.

Ellis D. Cooper



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Re: Explanations

[Marta Bunge]

Dear Andre,
In connection with the current discussion on explanatory proofs, and more  particularly about the difference between computer generated proofs and human ones, I have something to add to what you wrote --drawn from my own experience. Humans do not have either the speed or the ability to hold the  enormous amounts of information that machines do. On the other hand, the  only way to see the big picture is speed of reasoning coupled with intuition - naturally without checking the details at each step. That way, two things may happen to humans of which computers are free of: (1) errors are made, and (2) new ideas originate. Errors are not a good thing,  of course, but they are a possible outcome from taking risks, without which no new ideas would ever surface. Working out the details of that first glimpse of the truth may be painful, but necessary.  It may lead to truth (hardly ever), or to further glimpses. I agree with you that the sequence intuition --> computation---> intuition---> computation-->.... is the only available course of action for a good mathematician. With luck, the sequence terminates eventually, and it does in truth. But it must begin with intuition. Some of my collaborators have expressed surprise at my starting any investigation with a title and an abstract, when they would leave both  for the end. Naturally, that title and abstract may very well change at the end of the investigation, but if I were incapable to see the big picture at first, I would not begin any work at all. A final trivial thought - computation alone is not mathematics, and neither is intuition alone. The former is typical of machines, whereas the latter is typical of artists. 

Best regards,Marta





----------------------------------------
> Date: Mon, 25 Apr 2011 22:01:05 -0400
> Subject: categories: Re: Explanations
> From: joyal.andre@uqam.ca
> To: jds@math.upenn.edu; ronnie.profbrown@btinternet.com; graham@eecs.qmul.ac.uk
>
> Dear Jim,
>
> You are perfectly right!
> I am always amazed by the fact that a computation
> can yield a surprising result.
> It is as if the formal system knew more than me!
> Actually, I find a computation boring when the result is not surprising.
> Computing is probably the main vehicule by which we can move
> beyond a given body of intuitive knowledges.
> But after the initial surprise, we try hard to
> integrate the new result in a larger body,
> where it may become less surprising.
> It may even become obvious!
>
> The chain
>
> intuition--->computation---->intuition--->computation.....
>
> is probably more important than the chain
>
> proof--->method---->proof--->method.....
>
> André
>

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Re: Explanations

[Mattias Wikström]

> On 26/04/2011 06:55, Timothy Porter wrote:
> In category theory, many proofs are transparent and of the form: what do
> we know about the situation, just one fact, so we have to use that....
> it works. (I am thinking of classical Yoneda lemma type situations,
> since the only elements in hom-sets that we can be sure exist are the
> identities.) [...]

The question is: "What facts and what objects do we have?", "What is
given?". In general we need to work with what is directly given in
order to arrive at some indirectly given thing that we are seeking
(what we are seeking has to be given in some sense, or else the
problem cannot be solved).

In category-theoretic terms we can think of what is given as a
subobject A of some larger object B in an allegory (where B represents
things that "exist" but which we may or may not be able to refer to).
A subobject of B is then given just in case it factors through A. (For
this to work well the allegory we are working with should contain lots
of different objects so that "subobject of X" and "part of X" become
practically synonymous.)

We may also view things in terms of symmetry and invariants. We are
directly given certain subobjects A1, A2, ..., An of some object B in
an allegory and we are indirectly given any subobject of B which stays
invariant as we apply isomorphisms to B that fix A1, A2, ..., An. (The
earlier object A would be the smallest subobject of B containing A1,
A2, ..., An as parts.)

Finally, we can view what we are given as a theory (axiom system), and
the question is what can be defined/specified/referred to in that
theory. Of course, mathematics inevitably involves axiom systems, but
any theorem which starts "for all ..." can be thought of as involving
an axiom system of its own.

Mattias

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Re: Explanations

[Uli Fahrenberg]

Let me mention here the essay "Mathematics, morally" by Eugenia which deserves to be more widely known:
http://cheng.staff.shef.ac.uk/morality/

If I may summarize, one of the most interesting points in this essay is that Eugenia claims that category theory is
"morally complete": everything which is morally true, is also provable.  Or in the terms of this conversation's
subject: Every categorical truth has an explanatory, "moral" proof.

Cheers,
Uli


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Re: Explanations

[Ronnie Brown]

I put it in an email to Jean-Pierre Marquis that a proof is conducted in
a conceptual landscape: an analogy is that to describe the way to the
station we may say: go right, then left and turn right again at the
first traffic light. We do not describe the cracks in the pavement; but
we might say: beware of the road works.

Constructing (finding?) landscapes was Grothendieck's amazing
contribution, as suggested by Bill's comments.

I am told that computer assisted proofs have had some success, for
example in Boolean algebra (every Robbins algebra is Boolean.): we can
see that a computer can find its way through a maze, and analogous
things should be useful in mathematics generally. The problem seems to
be that the computer needs a way of prescribing the appropriate
conceptual landscape, analogously to the way we do mathematics, and
there is here a need for programming languages  which can manage the
construction of the variety of hedges in the appropriate high level
maze!  (see Bill's comments).  Even at a given level, it is easy to give
problems which are too hard to do by hand: I gave a course at Bangor in
which one of the exercises was to find a polynomial in x,y over the
reals which had at least 5 critical points, to classify them, and to
produce an illustration of the surface (using Grobner basis methods in
Maple).

There is also an aesthetic element: what do we mean by a good proof? My
own route to groupoids (and then higher groupoids), came about because I
was trying to write a book on topology in the 1960s including the
fundamental group and got fed up with having to get the fundamental
group of the circle by an entirely different method, which really needed
a development of its own.

Raoul Bott said that Grothendieck was amazing in that he was prepared to
work very hard to make things tautological: this was also by playing
with concepts and producing remarkable things. (I overheard this at the
1958 ICM in Edinburgh.)

But Grothendieck wrote to me saying that
"Throughout my whole life as a mathematician, the possibility of making
explicit, elegant computations has always come out by itself, as a
byproduct of a thorough conceptual understanding of what was going on. "

Ronnie

On 26/04/2011 14:45, William Messing wrote:
> I agree with what Andre wrote concerning proofs.
> Ronnie, you will certainly recall Grothendieck's letter to you in
> which he recalled that at the first Seminaire Cartan he was initially
> quite perplexed as to how the singular chain complex of a topological
> space, gigantic in size, could possibly lead to concrete computations
> and applications.  As he said, he soon realized that it is not the
> size that matters, but understanding things properly, that is, in the
> correct order or manner.  In the same letter he recalled that
> initially he was mystified as to how one would ever be able to make
> concrete calculations in etale cohomology, until, after, as he1 put
> it, several days of intense thought, he saw that understanding the
> cohomology of curves, with perhaps arbitrary constuctible torsion
> sheaves (torsion prime to the characteristic of the field over which
> one is working) was the key.
>
> Concerning proofs constructed by people as opposed to computer
> assisted proofs, many years ago Deligne remarked that while he did not
> believe in computer assisted proofs, he was not going to look for a
> counterexample to the proof of the four color theorem.
>
> Best regards,
>
> Bill Messing
>

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Re: Explanations

[William Messing]

I agree with what Andre wrote concerning proofs.

Ronnie, you will certainly recall Grothendieck's letter to you in which
he recalled that at the first Seminaire Cartan he was initially quite
perplexed as to how the singular chain complex of a topological space,
gigantic in size, could possibly lead to concrete computations and
applications.  As he said, he soon realized that it is not the size that
matters, but understanding things properly, that is, in the correct
order or manner.  In the same letter he recalled that initially he was
mystified as to how one would ever be able to make concrete calculations
in etale cohomology, until, after, as he1 put it, several days of
intense thought, he saw that understanding the cohomology of curves,
with perhaps arbitrary constuctible torsion sheaves (torsion prime to
the characteristic of the field over which one is working) was the key.

Concerning proofs constructed by people as opposed to computer assisted
proofs, many years ago Deligne remarked that while he did not believe in
computer assisted proofs, he was not going to look for a counterexample
to the proof of the four color theorem.

Best regards,

Bill Messing

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Re: Explanations

[Timothy Porter]

Steve says: `Since mathematics is a formal system ...' is it?  I gave a
talk some years ago at the national college of the UK women's institute
and the title I used was:  Mathematics, a human activity.  My point was
that mathematics is done by mathematicians (amongst others).  Until we
find to the contrary, mathematicians are more often than not human (in
the widest sense of the word!!!!!). The form and direction of
mathematical investigation is determined by curiosity, and similar human
emotions., (sometimes also by rivalry, hatred, envy ,  and other ones of
less beauty).

   A (subjective view) good proof convinces the `reader' that the
statement is true. The 'explanation' behind a proof by contradiction
explains  somewhere along the lines: the result is trapped, it cannot
get away, therefore we have it. That is a human judgement and is
sometimes accompanied by the sentiment of `but that argument leaves me
dissatisfied as I do not see why'.  (The level of belief in the use of
contradiction is sometimes an issue but not always.) `Explanation' can
be modelled by a worldview approach, but then you have the problem of
the  teaching situation where the teacher gives an explanation of some
mathematical result, but has to say that the proof has to take a
different route.

In category theory, many proofs are transparent and of the form: what do
we know about the situation, just one fact, so we have to use that....
it works. (I am thinking of classical Yoneda lemma type situations,
since the only elements in hom-sets that we can be sure exist are the
identities.)  A thorough understanding of the proof does give an
explanation of why the result holds. (The problem I have with the
original request for examples is that explanation requires understanding
of the situation so is dependent on the knowledge of the `codomain'/
reader!)


Tim


On 25/04/2011 14:17, ClemsonSteve wrote:
> Quoted from Jean-Pierre Marquis email: "yes, of course, Salmon is
> certainly one of the important contributors to the field [scientific
> explanation]. In mathematics, Paolo Mancosu has been pushing the issue
> for the past 10 years or so, following the paths of Steiner, Resnik and
> a few others."
>
> In science, the issue of explanation has been discussed for at least a
> century. Since mathematics is a formal system and not a physical system,
> we have to be more careful about what *explanation* means. This makes it
> a "worldview" problems? As a constructionist/computationalist I would
> say no constructive, computational proof then there is no explanation.
> Platonist have their own. Is their explanation useful to me? Don't know
> because if I can figure a constructive technique out from the plationic
> technique, I'm good.
>
> steve stevenson
> clemson
>
>

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Re: Explanations

[jim stasheff]

On 4/25/11 9:51 AM, Joyal, André wrote:
> The goal of a human proof is to convince other peoples of the truthfulness
> of a proposition.
> It is by nature explanatory and it can lead to new insights.
I disagree mildly.

convincing other people of the truthfulness
of a proposition can involve staightforward computation
which I don't find explanatory

jim





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Re: Explanations

[Joyal, André]

I cannot resist adding my grain of salt.
 
Maybe we should distinguish between a human proof and a mechanical proof.
The goal of a human proof is to convince other peoples of the truthfulness
of a proposition.
It is by nature explanatory and it can lead to new insights.
A mechanical proof can be checked by computer but may not produce new insights.
It is establishing a fact. Of course, it is better than none.
Mathematics is above all a human activity.
The value of a proof depends very much on its method.
A new proof may suggest a new method.
A method is a kind of toolbox for proving a large class of propositions.
Commutative algebra is a method in geometry
Category theory is a method in mathematics.

proofs --->methods ----> proofs ----> methods .......


André


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Re: Explanations

[ClemsonSteve]

Quoted from Jean-Pierre Marquis email: "yes, of course, Salmon is
certainly one of the important contributors to the field [scientific
explanation]. In mathematics, Paolo Mancosu has been pushing the issue
for the past 10 years or so, following the paths of Steiner, Resnik and
a few others."

In science, the issue of explanation has been discussed for at least a
century. Since mathematics is a formal system and not a physical system,
we have to be more careful about what *explanation* means. This makes it
a "worldview" problems? As a constructionist/computationalist I would
say no constructive, computational proof then there is no explanation.
Platonist have their own. Is their explanation useful to me? Don't know
because if I can figure a constructive technique out from the plationic
technique, I'm good.

steve stevenson
clemson



On 4/23/11 17:52, Dusko Pavlovic wrote:
> a friend told me that there was a conference where music critics and professors discussed the visual content of music. on the other side, there are learned essays about the deep links between music and architecture. (hegel in particular wrote about that.)
>
> the question whether a mathematical proof explains the theorem might be of a similar kind. while most proofs of the pythagoras theorem do explain why it is true, wyles' proof of the great fermat theorem (just a slightly different statement) does not seem to be explaining it to too many people.

-- 
Steve Stevenson Clemson University


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Re: Explanations

[Dusko Pavlovic]

a friend told me that there was a conference where music critics and professors discussed the visual content of music. on the other side, there are learned essays about the deep links between music and architecture. (hegel in particular wrote about that.)

the question whether a mathematical proof explains the theorem might be of a similar kind. while most proofs of the pythagoras theorem do explain why it is true, wyles' proof of the great fermat theorem (just a slightly different statement) does not seem to be explaining it to too many people.

some proofs yield an explanation, some explanations lead to a proof - but there is a sense in which the *attitudes* leading to one and to the other are *opposite*: while explanations tend to increase the number of words in the world, many people prove things so that we can stop talking about them. 

maybe the situation resembles the discussion between zeno and parmenides: "as parmenides argued that the movement does not exist in the universe, zeno stood up and walked around".

-- dusko

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Re: Explanations

[Ronnie Brown]

As a good example of a proof giving an explanation using categorical
methods  like the proof of the Seifert-van Kampen Theorem for the
fundamental group in
Crowell, R.~H.
\newblock \enquote{On the van {K}ampen theorem}.
\newblock \emph{Pacific J. Math.} \textbf{9} (1959) 43--50.

It does not quite use modern categorical language but in essence it
proves a colimit theorem by verifying the required universal property.
This then leads to specific calculations.  Previous proofs were
difficult to understand (e.g. van Kampen's account) or restricted to the
simplicial case. The value of the proof was also that it could be
generalised to the groupoid (many base point) case, and to higher
dimensions, using higher homotopy groupoids.


Ronnie Brown



On 22/04/2011 14:55, Graham White wrote:
> And the folklore is (I haven't checked this in a proper history book)
> that Gauss proved quadratic reciprocity numerous times because he didn't
> consider the proofs sufficiently explanatory. It's certainly true that
> modern proofs (i.e. those using the methods of algebraic number theory)
> generalise it, and thereby explain, for example, what it is about the
> rationals, and the number two, that makes primes in the rationals obey
> quadratic reciprocity. I think one conclusion here is that, if you say
> "explanatory", I am entitled to answer "so what do you want explained?"
>
> Another point is this: there are lots of combinatorial
> identities of the form
>
> big ugly formula_1 = big ugly formula_2
>
> which can be proved directly (for example, by induction
> and a lot of algebra), but where the proof is utterly unilluminating.
> And in many cases there are more conceptual proofs which people
> generally find more illuminating (depending on taste, of course).
>
> Graham
>

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Re: Explanations

[David Yetter]

My private reply to the original query from Jean-Pierre Marquis pointed
to the style of combinatorial proof you refer to:  they are called "bijective" or
"combinatorial"  proofs depending on the author, and rely on giving
interpretations of "big ugly formula_1" and "big ugly formula_2" as enumerating
the same thing by different means.

For instance on can prove that

n*2^{n-1} = \sum_{k=1}^n k*C(n,k)

(writing C(n,k) for the binomial coefficient "n chose k") by differentiating the
binomial theorem and evaluating at 1, but this hardly seems to explain it.

Better is to observe that both sides count the number of ways to select a 
subset with a distinguished element from an n element set, the LHS by
selecting the distinguished element, then the rest of the subset, the RHS
by choosing a cardinality k for the subset, selecting the subset then selecting
the distinguished element from the subset.

David Y.


On 22 Apr 2011, at 08:55, Graham White wrote:

> And the folklore is (I haven't checked this in a proper history book)
> that Gauss proved quadratic reciprocity numerous times because he didn't
> consider the proofs sufficiently explanatory. It's certainly true that
> modern proofs (i.e. those using the methods of algebraic number theory)
> generalise it, and thereby explain, for example, what it is about the
> rationals, and the number two, that makes primes in the rationals obey
> quadratic reciprocity. I think one conclusion here is that, if you say
> "explanatory", I am entitled to answer "so what do you want explained?"
> 
> Another point is this: there are lots of combinatorial
> identities of the form
> 
> big ugly formula_1 = big ugly formula_2
> 
> which can be proved directly (for example, by induction
> and a lot of algebra), but where the proof is utterly unilluminating.
> And in many cases there are more conceptual proofs which people
> generally find more illuminating (depending on taste, of course).
> 
> Graham
> 

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Re: Explanations

[Graham White]

And the folklore is (I haven't checked this in a proper history book)
that Gauss proved quadratic reciprocity numerous times because he didn't
consider the proofs sufficiently explanatory. It's certainly true that
modern proofs (i.e. those using the methods of algebraic number theory)
generalise it, and thereby explain, for example, what it is about the
rationals, and the number two, that makes primes in the rationals obey
quadratic reciprocity. I think one conclusion here is that, if you say
"explanatory", I am entitled to answer "so what do you want explained?"

Another point is this: there are lots of combinatorial
identities of the form

big ugly formula_1 = big ugly formula_2

which can be proved directly (for example, by induction
and a lot of algebra), but where the proof is utterly unilluminating.
And in many cases there are more conceptual proofs which people
generally find more illuminating (depending on taste, of course).

Graham

-------- Forwarded Message --------
> From: peasthope@shaw.ca
> Reply-to: peasthope@shaw.ca
> To: categories@mta.ca
> Cc: peasthope@shaw.ca
> Subject: categories: Re: Explanations
> Date: Thu, 21 Apr 2011 11:09:36 -0800
>
> Fred & all,
>
>> My goodness! I'd turn that question around: is there any proof (apart
>> from an "indirect" proof, or "proof by contradiction") that one would
>> *not* "consider as being explanatory in this sense?"
>
> Speaking as a novice: yes, certainly.  Isn't it a question of degree?  Some
> proofs explain beautifully while others are clear as mud; most are
> between.  Ideally a proof shouldn't depend upon natural language but
> most do.  Striking sometimes how changing a few words of a sentence
> can make a concept obvious rather than nebulous.
>
> For example, I've proven some of the power laws for map objects.  There
> should be a way to reduce the definition of a map object and the power
> laws to analogues in arithmetic.  Still eludes me.  My proofs have yet to
> help.  So my understanding is incomplete and my power law proofs are
> poor.
>
> Best regards,                      ... Peter E.
>
> --
> Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
> Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
> Personal pages http://members.shaw.ca/peasthope/ .



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Re: Explanations

[peasthope]

Fred & all,

> My goodness! I'd turn that question around: is there any proof (apart 
> from an "indirect" proof, or "proof by contradiction") that one would 
> *not* "consider as being explanatory in this sense?"

Speaking as a novice: yes, certainly.  Isn't it a question of degree?  Some 
proofs explain beautifully while others are clear as mud; most are  
between.  Ideally a proof shouldn't depend upon natural language but 
most do.  Striking sometimes how changing a few words of a sentence 
can make a concept obvious rather than nebulous.

For example, I've proven some of the power laws for map objects.  There 
should be a way to reduce the definition of a map object and the power 
laws to analogues in arithmetic.  Still eludes me.  My proofs have yet to 
help.  So my understanding is incomplete and my power law proofs are 
poor.

Best regards,                      ... Peter E.

-- 
Telephone 1 360 450 2132.  bcc: peasthope at shaw.ca
Shop pages http://carnot.yi.org/ accessible as long as the old drives survive.
Personal pages http://members.shaw.ca/peasthope/ .



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Re: Explanations

[Fred E.J. Linton]

On Wed, 20 Apr 2011 08:04:09 AM EDT, Jean-Pierre Marquis
<jean-pierre.marquis@umontreal.ca> asked:

> ... some people claim that there are mathematical proofs that are 
> explanatory, that is, not only do they establish the claim they prove,
> but they also show why the given result holds.
> 
> ... is there any proof, involving categories or not (but preferably so),
> that you would consider as being explanatory in this sense? ...

My goodness! I'd turn that question around: is there any proof (apart 
from an "indirect" proof, or "proof by contradiction") that one would 
*not* "consider as being explanatory in this sense?"

Cheers, -- Fred



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Explanations

[Jean-Pierre Marquis]

Hi,

I have a general question which is not strictly speaking about categories, but I thought I would ask the members of the list anyhow. Here is the context: some people claim that there are mathematical proofs that are explanatory, that is, not only do they establish the claim they prove, but they also show why the given result holds.

Here is my question: is there any proof, involving categories or not (but preferably so), that you would consider as being explanatory in this sense? Please answer off-list.

Thanks,

Jean-Pierre


Jean-Pierre Marquis
Professeur titulaire
Responsable du premier cycle
Département de philosophie
Université de Montréal
jean-pierre.marquis@umontreal.ca

Tel: 514-343-6111 (33445)
Télécopieur: 514-343-7899

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