Unfortunately I am still having sound quality issues, but people may still be interested in looking at the screencast I have made of this year’s edition of “Why do we do proofs?”

This is available at:

http://wirksworthii.nottingham.ac.uk/webcast/maths/G12MAN-09-10/Why-Proofs-09-10-b/Why-Proofs-09-10-b.html

Joel Feinstein

7/10/09

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*Related*

See http://mathforum.org/kb/forum.jspa?forumID=185 for a lot of discussion arising from the innocent-seeming question I posed: “Are equilateral triangles isosceles?”

In particular, see:

http://mathforum.org/kb/thread.jspa?threadID=1993195&tstart=0

http://mathforum.org/kb/message.jspa?messageID=6868596&tstart=0

http://mathforum.org/kb/thread.jspa?threadID=1994530&tstart=0

http://mathforum.org/kb/message.jspa?messageID=6869053&tstart=0

http://mathforum.org/kb/thread.jspa?threadID=1994577&tstart=0

Let me just say that I think everyone is wrong (including me, of course). However, at least amongst the list members, I appear to have achieved the stated aim of my session of convincing people of the importance of having clear definitions! It’s just that people don’t agree on what the “correct” clear definition of triangle is.

My current view is that there are at least two issues here:

– Which is the appropriate definition to work with for students at a particular stage of study?

– Which definition do you actually believe captures the true nature of the named concepts you are talking about?

I do not think that these two always have the same answer.

When discussing subsets of two-dimensional space in my 2nd-year mathematical analysis module, I have to distinguish carefully between discs and their boundary circles, and between open discs and closed discs. When I discuss open rectangles, it is clear that I am not referring to the boundary, but to the interior. But it may be that most people would say that a rectangle consisted only of the boundary, not the interior.

So, what is a triangle? Is it a labelled triple of vertices? Joined by edges? Excluding or including the interior? Is a triangle a set, or is it a labelled set? Would the term “triangular region” be clearer when describing a subset of the plane? What about other polygons?

What is the area of a polygon? If you think the polygon refers to the boundary only, then the Lebesgue area measure of the set is 0!

This is silly, of course, because the “area of a polygon” is really shorthand for “area of the region enclosed by a polygon”. But the more one thinks about these issues, the more one sees how imprecise is the language that we use so casually. Trying to make everything one hundred per cent precise all of the time is not going to be practical, though!

So I suggest that the appropriate definitions to use are generally context-dependent, and that the in-use definitions should be re-clarified in the context of any taught course/module that they appears in.

This is somewhat related to the question of whether 0 is a natural number or not. I have to tell the students at the start of the module which version of the natural numbers I am using. I don’t claim that everyone else is wrong! I just want it to be clear to the students which definition WE are using, and that they need to ensure that they know which definition others use in their modules.

Joel Feinstein

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“Why do we do proofs?” is now also available on the Nottingham You-Tube channel at http://www.youtube.com/watch?v=JqZ3yhBJmWA

It seems to be quite popular!

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