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I like Charles Wells' distinction between what is physical, real and wobbly on the one hand (section 1) and what is abstract and unchanging on the other (section 2). The teacher's task lies in using 1) to promote 2). That is why it is difficult or perhaps impossible! Bob Burn On Tue, 12 December, 2006 9:22 pm, Charles Wells wrote: > Several people commented on my remarks about math objects that were in > response to an email from a reader who mentioned examples of sets > varying over time such as the set of US Presidents. Some of them > suggested using topos theory (in different ways) to model sets varying > over time, which can be a valid approach. I am writing this note to > make clear the context of my remarks. > > One of the chapters on my abstractmath.org site discusses the way we > think about math objects using mental images and metaphors. One > particular point I make (perhaps the most important point in the whole > section on understanding math) is that > > 1) We are free to use all sorts of images and metaphors when we are > trying to understand or communicate math (which I call the “rich view”) > provided that we are aware that such things can also be misleading, BUT > > 2) When we are constructing a proof, the ONLY safe image / metaphor is > what I call the “rigorous view”, which views all math objects as > unchanging and inert: they don’t change with time and they don’t affect > anything else. That viewpoint makes any thought of causation impossible > and is the only view that allows material implication to work according > to its truth table. > > But that of course makes it difficult to talk about sets of physical > objects (among other things). So I have now rewritten the section on > math objects and the section on images and metaphors to require that on > the abstractmath site all constructors (set of…, relation between …, etc) > apply only to math objects. The site simply does not talk about sets of > physical objects, relations between people, etc. There is nothing wrong > with doing that, but for the website I want to keep things simple. > Abstractmath.org is forever teetering toward discussions of complex > philosophical questions that have no business being mentioned on the > site, and this is my way to avoid one of them! > > Charles Wells > > > abstract math website: http://www.abstractmath.org/MM//MMIntro.htm > professional website: http://www.cwru.edu/artsci/math/wells/home.html > personal website: http://www.abstractmath.org/Personal/index.html > genealogical website: > http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/ > NE Ohio Sacred Harp website: > http://www.abstractmath.org/fasola/index.html > > > -- Bob Burn Research Fellow, Exeter University Sunnyside Barrack Road Exeter EX2 6AB 01392-430028