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        general >> truth >> Discovering multiplication
        
(Message started by: Mickey1 on Nov 26^th, 2012, 6:46am)

Title: Discovering multiplication
Post by Mickey1 on Nov 26^th, 2012, 6:46am

Let us assume we are equipped with the natural numbers, the Peano axioms and one composition rule: addition. This will allow us to count objects, numbers for example. We first discover the one-to-one correspondence between the natural numbers such as n, and the geometrical concept of rectangles with length n and width 1, by associating the number n with a rectangle of length n consisting of n unit area element.

We now consider the number of unit area elements of a rectangle of length n and width m, we can count the unit areas line by line and we can denote the result n*m (assuming * to be an unused symbol). We can now see, by applying a physical or geometrical principle of relativity, that (imagining the rectangle placed on the floor) if we move about 90 degrees in a circle around the rectangle it will look like another rectangle of m*n units of area. Assuming our movement has not changed the rectangle, we have n*m=m*n.

We can also make a cut in a n*m rectangle perpendicular to the length dimension, after k units of length ( 0 < k < n) which will give us n*m = k*m + (n-k)*m so that a*(b+c) = a*b + b*c. I haven’t described all the components of multiplication here but they don’t seem too difficult to “discover”.

Title: Re: Discovering multiplication
Post by Mickey1 on Dec 2^nd, 2012, 3:33am

Perhaps I should add that this is a speculation about the Presburger arithmetics, the Peano axioms with only addition added, vis-à-vis the same axioms with both addition and multiplication added.

In the first case any proposition is decidable but not in the second. My problem is: how is it possible that I can sneak the multiplication in - without any other actions than appealing to the reader’s feel for geometrical figures – and seemingly go from one system to the other?

Observe that prime numbers – numbers that fail to appear as the number of rectangles - can be found easily. If we wonder about 7 for example we can start by observing that any rectangle with length 7 or higher, and width >1 will correspond to a number >7. We therefore have a finite number of rectangles to consider, with length < 7 and width < 7 since n*m=m*n (interpreted as numbers of unit areas for a rectangle), i.e. not too different from solving the same problem in Peano arithmetic.

Title: Re: Discovering multiplication
Post by peoplepower on Dec 2^nd, 2012, 7:31pm

It seems imperative that one defines (unordered) pairs formally within the structure of a Presburger arithmetic before we can talk about the geometric forms corresponding such pairs.

Universally it is impossible. However, having only dealt with Presburger arithmetic for 30 minutes today, I would not rule out the possiblity to pair the elements of some infinite subset (given a model of course).

P.S. I like the geometric interpretation of multiplication and might have a lot more to say on that.