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Topic: WM's argument disassembled - If you are his student, please PLEASE
comment here

Replies: 19   Last Post: Jul 23, 2014 1:01 PM

 Messages: [ Previous | Next ]
 Ben Bacarisse Posts: 1,972 Registered: 7/4/07
Re: WM's argument disassembled - If you are his student, please PLEASE comment here
Posted: Jul 22, 2014 9:49 AM

Jim Burns <burns.87@osu.edu> writes:
<snip>
> Another image that I have seen cause problems is successor-as-step.
> WM seems to make a lot of use of that one, with his "complete"
> and "incomplete" and "complete/incomplete" sets. It is easy to
> imagine proof by induction as showing that we can walk (imagine
> walking) to any natural number, one application of the successor
> operation at a time.

Yes, his time metaphors -- "after all have been used" for example --
will get in the way of his students getting a good metaphor for
unbounded things.

> However, that misrepresents the justification for induction.
> We (sometimes) refer to the _minimal_ inductive set, and it is
> by it being _minimal_ that we justify using induction to prove
> things about that set. I like to say that the one-ended chain
> _has no extra links_ , no extra side-chains said to be part
> of the main chain, but which aren't really. So, couldn't
> one form of the Basic Metaphor of Infinity be a one-ended
> chain with no extra links?

One of mine is unbounded computation. It's rather more abstract than a
chain, but it might chime more readily with today's students.

In this metaphor, the rationals can be enumerated because there is a
program like this:

let string r = "1/1";
loop {
manipulate r in some finite way to get r'
set r = r'
}

in which every rational will be represented by some r. WM's claim that
there is a "potential" bijection between N and R maps to the existence
of such a program for R. (I'm not saying that he'd be unwise enough to
make that claim -- I'm sure he wouldn't -- but that's how this metaphor
represents it.)

It's not perfect by any means, but it has the merit of (a) obviously not
ending, so all talk of what happens "after" is ruled out; and (b) making
concrete sense of "every rational" without anything being "completed".

<snip>
--
Ben.

Date Subject Author
7/22/14 Ki Song
7/22/14 JT
7/22/14 Ki Song
7/22/14 Jim Burns
7/22/14 Ben Bacarisse
7/22/14 mueckenh@rz.fh-augsburg.de
7/22/14 Virgil
7/22/14 Jeff S.
7/22/14 Virgil
7/23/14 mueckenh@rz.fh-augsburg.de
7/23/14 Virgil
7/22/14 Ki Song
7/22/14 mueckenh@rz.fh-augsburg.de
7/22/14 Ki Song
7/22/14 mueckenh@rz.fh-augsburg.de
7/22/14 Virgil
7/22/14 Virgil
7/22/14 Ben Bacarisse
7/22/14 Ki Song