Re: Another question on Grothendieck
All I am saying is that one need not read Galois in order to learn Galois theory. When a new idea is introduced, even if it is not explained so clearly that everyone understands it right away, as long as someone understands it and can rephrase it in a helpful way, the impact of the idea has been not only felt but disseminated. Dissemination is not always a single step. Vaughan On 9/3/2010 6:03 PM, Eduardo J. Dubuc wrote:
I confess that I am a little bit confused about what Vaughan is saying.
This promps me to repeat my posting in other words:
If a mathematical statement is understood by a reader (the hypotesis, the conclusion and the proof)
then the mathematical meaning of any particular notation used should come up by itself to this reader (that is, it should be clear for him that only one possible meaning for this particular notation would make the things work).
Eduardo
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Yes Vaughan, I agree with your point "up to a point", and this is an interesting topic to discuss (see Point 2) below). But I was raising another point, much more simple. Point 1) It was about the notation "U_{i_0..i_p}". If you understand the mathematics, then whether this stands for the intersection, the union, or any other known construction with the U_i_j, it should be clear which one is. So it seemed to me ridiculous that people in the list, all mathematicians, start discussing and speculating about possible meanings of "U_{i_0..i_p}". Then, I just wrote: "I am wondering, nobody can read the mathematics and come up with what Grothendieck meant !!!" (I should have added "by U_{i_0..i_p}". Point 2) Concerning Vaughan point, I tend to think that making the effort to read and understand the originals gives you an ADVANTAGE. Thus, even if I agree with "that one need not read Galois in order to learn Galois theory", I also think that if you read Galois, you will know Galois theory better. And of course, this goes without saying concerning Grothendieck. I suspect that all the great mathematicians had learned and/or Known the work of their great predecessor (or at least the ones not to distant in time) from the original sources. e.d.
Vaughan
Vaughan Pratt wrote:
All I am saying is that one need not read Galois in order to learn Galois theory. When a new idea is introduced, even if it is not explained so clearly that everyone understands it right away, as long as someone understands it and can rephrase it in a helpful way, the impact of the idea has been not only felt but disseminated. Dissemination is not always a single step.
Vaughan
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Dear Eduardo, I have written papers that deliberately have two possible meanings: one classical point-set and one constructive point-free. That is to say, the development in terms of points is done under logical (geometric) constraints that enable it to be interpreted in topos-valid point-free topology (locales), but it can be interpreted directly in point-set topology if one accepts classical logic. I did this for expositional reasons, to help classical topologists understand the topological content of what I was doing. See: "Localic completion of generalized metric spaces I" "The connected Vietoris powelocale" Is this compatible with what you were saying about "only one possible meaning"? Regards, Steve Vickers. On Fri, 03 Sep 2010 22:03:19 -0300, "Eduardo J. Dubuc" <edubuc@dm.uba.ar> wrote:
I confess that I am a little bit confused about what Vaughan is saying.
This promps me to repeat my posting in other words:
If a mathematical statement is understood by a reader (the hypotesis, the conclusion and the proof)
then the mathematical meaning of any particular notation used should come up by itself to this reader (that is, it should be clear for him that only one possible meaning for this particular notation would make the things work).
Eduardo
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Dear Steve, I was already aware that my statement "only one possible meaning"? was much too general and I myself speculated (at the time of posting the msage) about many possible exceptions when literally interpreting my statement. But I decided to leave it like that. Luckily it was understood as I meant (private msages). I clarify to you and to those that may rise similar exceptions: The Tohoku paper is just plain old classical mathematics (*), and nothing of the sort of your example is to be found there. I imagine on the other hand that in your papers you do not let the reader stay in the doubt about the meaning of these two possible meanings. (*) where you can of course point out if some reasoning is constructively valid (an exceptional example of this is the chapter on field extensions in the second edition of the classical Van der Waerden book). e.d. Steven Vickers wrote:
Dear Eduardo,
I have written papers that deliberately have two possible meanings: one classical point-set and one constructive point-free.
That is to say, the development in terms of points is done under logical (geometric) constraints that enable it to be interpreted in topos-valid point-free topology (locales), but it can be interpreted directly in point-set topology if one accepts classical logic.
I did this for expositional reasons, to help classical topologists understand the topological content of what I was doing.
See:
"Localic completion of generalized metric spaces I" "The connected Vietoris powelocale"
Is this compatible with what you were saying about "only one possible meaning"?
Regards,
Steve Vickers.
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participants (3)
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Eduardo J. Dubuc -
Steven Vickers -
Vaughan Pratt