Dear Rob

Thank you for writing such a thought-provoking essay.

I had the pleasure of having dinner with David Hestenes last week and we discussed your essay (along with ours) at some length.

You have convinced me that multiplication as repeated addition is not the same as multiplication of scale or ratios. While different arguments will have different effects on people, your statement that in the problem

2 2 2 = 3 x 2

the 3 on the RHS acts as an adjective and the 2 on the RHS acts as a noun, really struck me. I get it! Very subtle! Bravo!

Now, you have read my essay, so this may make some sense to you:

I *know* that you can show that the x operator above is associative and commutative which leads to it being an invertible transform of additivity. However, you can also show that x distributes over in repeated addition, which constrains the quantification to being a log so that x must be multiplication.

Now what I bet you can show, is that the symmetries of ratios also lead to multiplication. So I am willing to bet that both problems "repeated addition" and "scaling" are quantified by the same function but for different reasons.

I am going to look into this as that would be really cool!

I also wanted to say that your circle-square problem is mesmerizing and I think that it effectively highlights the fact that there are some unrecognized subtleties still lingering in the metaphors that lead to the mathematics that we use (see Hestenes' essay and mine).

Thank you for a very enjoyable and thought-provoking essay!

Kevin Knuth

I am glad you had a wonderful time talking to my good buddy David: such an outstanding individual.

The hardest thing to grasp is that there really are serious fundamental foundational issues with mathematics. The circle square illustration is just one of thousands. Recognizing that there are issues, is only part the problem, finding ways to illustrate them and solutions to them are others.

What I tried to point out in this paper is that dropping units may seem to be a logical approach to a generalized mathematics. However, so much is tossed out, requiring weird machinations to make it work. There are three different types of numbers: ordinal, cardinal and relationship. Currently mathematics switches back and forth between these without any indication that it has done so.

To be able to clearly see the issues requires what I call a "mathematics without numbers". What becomes obvious is that numbers do not encode quantitative information but rather structural information. This then opens a doorway into a world of thinking, reasoning and logical connections. I believe that if Russell, Whitehead, Frege, etc. had discovered this way of thinking about number, math would be vastly different today.

Rob

BTW: The difficulty between the three's and two's was I believe (although I can' t find it) pointed out by Hermann Weyl, as a difficulty with set theory.

2 years later

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3 months later

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