## Philosophers love mathematicians, confusions aside

Over at Starts with a Bang, Ethan made a very interesting post about diverging/converging series. Aside from the mathematical interest, philosophers understand the need for logic and thus, find a different sort of interest.

From Bernard Lonergan’s *Insight* Chapter 1, Elements:

A fourth step will be the discovery of the need of a higher viewpoint. This arises when the inverse operations are allowed full generality, when they are not restricted to bringing one back to one’s starting point. Then, subtraction reveals the possibility of negative numbers, division reveals the possibility of fractions, roots reveal the possibility of surds. What is multiplication when one multiplies negative numbers or fractions or surds? What is subtraction when one subtracts a negative number? Etc., etc., etc. Indeed, even the meaning of ‘one’ and of ‘equals’ becomes confused, for there are recurring decimals, and it can be shown that point nine recurring is equal to one.

A fifth step will be to formulate a higher viewpoint.

Distinguish (1) rules, (2) operations, and (3) numbers.

Let numbers be defined implicitly by operations, so that the result of any operation will be a number, and any number can be the result of an operation.

`Let x = 0.9...`

then 10x = 9.9...

hence 9x = 9

and so x = 1

The real key that Lonergan roots out in the early chapters of Insight is that mathematics are not real, nor are they imagined. They are concepts that exist without reference imagery. A mathematician does not need to picture the operations beyond the formulations, but the human mind needs to imagine it first before moving on to the *higher viewpoint*.

Also, because it converges on 1, it is equivalent to 1. Because… *drum roll* it’s infinite, right? It’s infinite-ness necessitates that it will at some point (after infinity) reach 1 by adding just that one little piece that is missing. For all practicality, it is already equivalent. At the same time, in the realm of the conceptual, it goes on for infinity and will never reach 1.

Ethan’s post is about math, although he does briefly touch on the difference between empirical and statistical residues at the end:

I can see that my series sums up to one whole pie! So who’s right: is it zero pies, one pie, or does the solution simply not exist? To get it right, you have to ask yourself what happens on average.

Well, for half of the terms you have a pie, and for the other half of the terms you have no pies. On average? You have half a pie. Even though at no point do you actually have half-a-pie, this series sums to one-half. (For more rigor, you might want to read this.)

1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + … = 1/2.

This is a hard one to wrap your head around, and many mathematicians and physicists that I knew in graduate school were unable to do it.

Very interesting stuff.

Tags: Bernard Lonergan, mathematicians, negative number, negative numbers, operations