On Tue, Sep 11, 2012 at 6:00 PM, Joe Niederberger <email@example.com> wrote: > Why not just address the challenge: give us a detailed mathematical account of the "process" (Devlin) that multiplication is that is not flat-out circular. > > Note that I haven't used the word scaling, or property, or model, or any of these complications. Just give a straightforward account of the process (Devlin's word -- and he used it explicitly) of multiplication of any two real numbers that's not circular and shows how it produces an answer. >
> "Repeated addition" refers to something better thought of as a computational procedure.
> Math that is computable is all of the math that the vast majority of students will ever need. >
On your challenge further above:
Define "answer". That is, in your challenge I detect the hidden "challenge" to compute that which cannot be computed. In other words, your claim of circularity assumes that a produced answer counts as an answer only if it is the result if a computable process, that a model counts only if it is computable.
On your claim that we can ban almost all of the real numbers and everything will just fine:
I reiterate everything I said in my post in this thread
"Are you saying that the "proper" response to being confronted with mathematics that is or is based on that which is continuous or non-computable is to deny it?
Are you saying that we should accept only mathematics that is or is based on that which is discrete or computable?
If so, then you are denying essentially all of precalculus and calculus and then just about almost everything that comes afterwards in analysis and topology and even a good chunk of abstract algebra - in other words, most of abstract mathematics in general. That is, you are saying that the only "real" mathematics is discrete mathematics or computable mathematics.
This is one very, very extreme position.
Perhaps this denial of all of mathematics that is or is based on that which is continuous or non-computable is an example of what Devlin had in mind when suggested that irreparable harm is being caused to kids by teaching kids that repeated addition is what multiplication *is* - it creates later on such a cognitive dissonance at the subconscious or unconscious level when the person finds out otherwise that the person has no choice but to come up with such extreme answers to the problem."
Regardless, to meet your challenge above that scaling cannot be a model for the multiplication of arbitrary real numbers - given that I utterly reject your assumption above that a model must be computable - and I note that your assumption above denies all of geometry based on real numbers because you deny almost all real numbers:
Here is a real number based geometric construction using similar triangles that gives point (product) ab on the real number line given point 1 and arbitrary points (arbitrary real numbers) a and b on the real number line:
Take a standard x-axis and y-axis in the Cartesian plane, and for sake of simplicity, name the points on these lines according to the fact they are each a real number line. Then:
Connect 1 on the x-axis to b on the y-axis, and, parallel to that drawn line segment, connect a on the x-axis to the y-axis, and this point on the y-axis is ab. That is, in terms of distance from 0 or magnitude or absolute value: ab is to b as a is to 1 - that is, written in terms of ratios or proportions: ab:b as a:1.
And via commutativity we have the other way as an alternative:
Connect point 1 on the x-axis to point a on the y-axis, and, parallel to that drawn line segment, connect b on the x-axis to the y-axis, and this point on the y-axis is ab. That is, in terms of distance from 0 or magnitude or absolute value: ab is to a as b is to 1 - that is, written in terms of ratios or proportions: ab:a as b:1.