Find the Fallacy

Two mathematicians begin a discussion by assuming that A=B. Then they begin making changes. They add an equal term to both sides of the equation. A+A=A+B or 2A=A+B. Then they subtract an equal number from both sides. 2A -2B = A+B -2B. They look at this and agree that this all makes sense. Then they agree that this equation can be simplified by a simple operation. 2(A-B) = A+B-2B. This again simplifies to 2(A-B) = A-B. The result astounds them. Two times (A-B) equals (A-B)? Two equals one????

The answer is that the fallacy occurs only at the final step.

We get to 2(A-B) = A-B. But at this point we should recall that if you subtract something from itself you always get zero. So A-B = 0 and 2(A-B) also equals 0.

It is surely the case that 0 = 0, but thus stated the air of paradox has disappeared.

You obviously can’t get from there to the conclusion 2=1, only to the conclusion that

2(0) = 1(0).

The final step would be division by zero, which the basic rules prohibit. Precisely to avoid such paradoxes as this!

My point? Just that zero, and the rules governing its use are more exotic human conceptual inventions than one might think. The casualness with which we usually treat zero comes from familiarity, not from simplicity. This confirms the point I sought to make yesterday, that our most successful inventions are also discoveries, and vice versa.

This turns out to be, not especially Kantian, but certainly Jamesian, or well within the area of their overlap.

Invention and Discovery

Continuing my thought from last week, a thought instigated by Ciceronianus: 

Do humans actively shape the world?  Do we invent reality? Or do we merely discover it?

Surely we build skyscrapers and bridges, in much the same way that birds make nests. We seek to shape our environment for the sake of our own survival.

Is there some sense that doesn’t immediately involve motor activity in which humans invent the world? Something more constructivist?

A “yes” answer seems to make more sense to me than it does to Ciceronianus. In part this is because of reflection on the history of mathematics A very short statement is this: mathematics is a series of outrageous re-definitions of what it means to be a number. We learn to count when very young, and I suppose it has always been thus. The first conception of number derives from the act of counting.

But through our lives, if we receive any sort of education, we learn about ever more outlandish sorts of number. The strangeness of zero, for example.  Or irrational numbers, those wild things like pi that never repeat and never end.  How uncanny!

We may also wrestle with negative numbers. Then the idea of an "infinitesimal." I remember an old Sesame Street episode with the question whether a circle is “all one side” or whether a circle has “a whole lot of very little sides.” Ernie was raising the question of infinitesimals.  Circles (or other curves) can be thought of as an infinite number of tangent straight lines, each line always receding in size, with the Euclidean point as a limit.

Beyond even that, there is the notion of imaginary numbers. In the real number system, the basic rules of multiplication and division make it impossible that there should be such a thing as the square root of a negative number. But forget about that and invent the square root of -1 anyway! Call it i.

These increasingly absurd seeming steps of human reason are also steps of human imagination.  They seem as sheerly inventive as anything else we as a species can do. The paradox, then, is that the inventions of these outlandish notions by clever humans working at a very high level of abstraction, and often unconcerned with practical consequences, always turns out to have enormous practical consequences.

Interacting with the World, and Ciceronianus

This weekend, Ciceronianus, author of a wonderful neo-Stoic blog, posted on epistemology. He wrote: "I'm bemused from time to time by the view that the world (as in "reality" or the universe) is, in part at least, our creation, or perhaps is created by each of us for himself/herself."

He named first Kant and later Wilfred Sellars as examples of the sort of epistemologist he has in mind. Let us use the broad term "constructivism" for the broad PoV that Kant and Sellars share, and that bemuses our Stoic.

Ciceronianus, if I understand him, then proceeds to the assertion that such constructivism is either pointlessly obvious or wildly wrong. The obvious and uninteresting point is that "we are human beings, and as such interact with the world as human beings do." Yet those who are most serious about urging that "they shape the world" seem to want to go much further than this, and that furtherness is what bothers Ciceronianus.

I contributed a thought of my own to his comment section.
 

There is a tee shirt that bears upon some of the issues you raise. It shows somewhat anthropomorphized versions of the Greek letter pi (Π) and of the expression √-1.

Pi is saying to √-1, “Get real.” And√-1is replying, “Be rational!”

I’ll pause now while you slap your knees.

The joke, of course, is that pi is an example of an “irrational” but real number, while √-1 is the definition of i, the foundation of the imaginary numbers.

Fermat's Last Tango

I recently watched a DVD of the musical comedy Fermat's Last Tango,  produced by The Clay Mathematics Institute.

I may discuss the production in greater depth at some point next week.  Right now I'd just like to record the lyrics of one of the songs: Your Proof Contains a Hole.

Just a bit of background information first. This scene takes place in a quasi-heaven called "The AfterMath," where famous mathematicians live out their immortality. The bulk of the song is sung by Fermat, who believes his own immorality requires the insolubility of his theorem. He is singing to Daniel Keanes, a fictionalization of Andrew Wiles, who though still very much alive is visiting the AfterMath.

The chorus consists of AfterMath regulars: Euclid; Pythagoras; Newton; Gauss.

Without further ado, then....

Fermat: Your proof contains a flaw, Professor Keane,
               It destroys the whole foundation of your finely tuned machine:
               I hate to be a spoilsport, I know it was your goal
               But your proof contains a big fat hole.

Keane: A hole?
Chorus: A hole?
Keane: My proof contains a hole?

Fermat: I didn't want to be the one to saaaay
               I know this is upsetting, please show some self-control
               But your proof contains a big fat hole.

Random Quote: History of Philosophy

Henri Bergson, CREATIVE EVOLUTION (1913).

"There is, then, immanent in the philosophy of Ideas" [Platonism and its kin], "a particular conception of causality, which it is important to bring into full light, because it is that which each of us will reach when, in order to ascend to the origin of things, he follows to the end the natural movement of the intellect....The affirmation of a reality implies the affirmation of all the degrees of reality intermediate between it and nothing. The principle is evident in the case of number: we cannot affirm the existence of the number 10 without affirming the existence of the numbers 9, 8, 7, ... , etc., -- in short of the whole interval between 10 and zero."

Emphasis in the original.