Timeline for Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)
Current License: CC BY-SA 2.5
13 events
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| Jun 14, 2019 at 16:46 | comment | added | LSpice |
s/I have rarely seen infinite amounts of data/I have yet to see an infinite amount of data/ :-)
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| Sep 14, 2016 at 5:29 | comment | added | Tim Porter | Today someone just voted up my answer so I looked again at it. I forgot one obvious word that is almost begging to be mentioned but was not: `duality' The dual of a Boolean algebra is a Stone space. The dual of a C*-algebra is a compact Hausdorff space and so on. Perhaps a quick answer to the question is Algebra is dual to Topology. That answers nothing but is somehow true and neat! | |
| Aug 12, 2011 at 13:22 | comment | added | Tim Porter | ANALOGY! That reminds me of.... These are terms that need a lot more exposure in popularising maths. Ronnie Brown and I wrote a paper on the Methodology of mathematics and analogy was a central issue there. | |
| Aug 11, 2011 at 1:03 | comment | added | Jon Bannon | All of what I wrote essentially appeared in the comments following the question, in spirit. I just like the Poincaré image! | |
| Aug 11, 2011 at 1:00 | comment | added | Jon Bannon | @Tim: Perhaps this chicken-egg thing doesn't matter. This all reminds me of Poincaré painting the image of our perception of regularity in the noise that passes before us. We see something that repeats, and that repetition suggests stability (actual or perceived is irrelevant). If a `structure' shows up frequently enough, it is good sense to try to understand new structure in terms of the familiar structure. (I can't help but think of Grothendieck group here...) This may be at the heart of the fecundity of mathematical analogy, in Polya's sense. | |
| Feb 10, 2010 at 7:58 | comment | added | Tim Porter | I think that my answer to your query is thus: bits of both worlds. (I am not sure about what `external' means.) This is probably getting off thread, although I would love to continue... but where. I do not want to clog this, but can give some references that explore the analogies etc. a bit. | |
| Feb 10, 2010 at 7:55 | comment | added | Tim Porter | My thoughts are, sort of, pragmatic, i.e. what works! When looking at something, we tend to use `models'. I think there is something 'out there' to model, but the models are not the same as the 'reality' we seek to understand. A lot of mathematics involves 'relations' in the non-technical as well as the technical sense. You can model relations in various ways including 'spatial' ones. Look at Chu spaces (Vaughan Pratt), or the old theorem of Dowker on simplicial complexes associated to relations. Those ideas help us organise things in a useful way, and are inherent in the idea of relation. | |
| Feb 10, 2010 at 5:46 | comment | added | François G. Dorais | Tim, you seem to take an anthropocentric view at first, but in the end you suggest that there might be external reasons too. This is sort of a chicken & egg question, but I would appreciate a clarification of your point of view. Do you think that topology is a nice way that we came up with to organize things in our environment, or that natural topological organization is something we grew to like enough that we feel a need to emphasize it whenever possible? | |
| Feb 9, 2010 at 13:27 | comment | added | Neel Krishnaswami | That's just the one I read! It's a charming little book. | |
| Feb 9, 2010 at 11:39 | comment | added | Tim Porter | People may know of the book 'Topology via logic' by Steve Vickers. That is very relevant to this question and this comment. | |
| Feb 9, 2010 at 9:04 | comment | added | Neel Krishnaswami | Just as a data point: I was scared of topology until I learned about locales. At that point the light went on, and I said, "Oh, topologies are representations of Heyting algebras! Intuitionistic logic isn't scary, so I shouldn't be scared of topology, either!" | |
| Feb 8, 2010 at 19:49 | history | edited | Tim Porter | CC BY-SA 2.5 |
added an improved wording mentioning 'organising'
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| Feb 8, 2010 at 15:21 | history | answered | Tim Porter | CC BY-SA 2.5 |