Thursday, January 8, 2015

The no-math math book

This is a story about an inability to accept change that ended as you might predict. -- or perhaps it's a case study in organizational change management --

And, we find this story in a history book, or perhaps it's a math book. In any event, it's not too often that I feature a math book here on Musings. But, making an exception, let me describe a math book with no math -- or, actually, it's a history of math book with no math. I'm talking about "Infintesimal, How a dangerous mathematical theory shaped the modern world" by Alexander, an easy-to-read history of the early life of calculus (I hear you yawning!)

And we should care?

Actually, about the consequences of not accepting change, the book is very instructive, with obvious lessons for the big organization, project or business.

About "infintesimals", the concept is used almost every day: without it, project managers couldn't make simple statements -- though certainly important statements -- like 'the confidence of finishing the project on time is 60 to 80%'. Confidence intervals are the consequence of a bit of calculus on the underlying probability distribution.

So, what do we find out in this book:
  • The concept we now call calculus began emerging in the early 15th century in Italy. Emerging insights... now there's a concept for you!
  • The Catholic church -- actually, the Jesuits -- fought it tooth and nail for nearly 200 years. The Jesuits had a full-blown monopoly on universities and university curriculum, so they didn't need or want changes
  • One of the most accomplished mathematician of early calculus was an Italian who fled the Jesuit grip, settled in France in the 18th century to develop his theories -- and French-fried his name to Joseph-Louis LaGrange. Thus ended Italian dominance of mathematics.
For me, the most revealing thing was the iron grip that the Church -- and the Jesuits -- had on all of mathematics for so long a time, not waning until the 18th century. What was the Church's interest, and why so protective? Control of their university monopoly for one thing; but the real agenda was control of the public thinking of virtually all the mathematicians and scientists, some of whom challenged the  orthodoxy of the Church.

Ooops! Challenge a top-down rigidly hierarchical organization with a lot to lose if toppled? That's trouble, with a capital T for sure. So, what's going on here?

First, in the 15th and 16th century, math and science was part of the philosophy curriculum -- that is, beliefs and secular truths. Though we might find religious control of math an oddity today, the intersection of science and religion continues to vex in the 21st century, some 500 years on.

Second, Paradoxes ... they, the Jesuits in charge -- couldn't accept paradoxes because everything was supposed to be neatly and hierarchically ordered by the Devine. Paradoxes were viewed as a form of disorder that no Divinity would sanction. Thus, no earthly curriculum could sanction them either.

And the paradox that caused all the angst? Infinitesimals! Or, the infinitely small. Why should the Church care what mathematicians were thinking about really small stuff? Because at time, the principal math was geometry, based on very orderly and hierarchical bottoms-up proofs. But, infinitesimals are not geometric with bottom-up proofs.

And, a bottoms-up orthodoxy supported the top of the heap. Thus, any challenge to thinking or constructing things from lower order proofs was a challenge to the hierarchical constructed organization. To challenge one is to challenge the other. 

So, what is this infinitesimal idea? Using the 16th century notion -- still valid -- it is the idea that any geometrical figure, like a line, or a surface, or a solid, can be subdivided endlessly into ever smaller increments. A meter long line can be divided into decimeters, and then into centimeters, and then into millimeter-long increments. If we want the line back, we merely multiply 1000 increments of 1 millimeter each.

So, from millimeters, we can go on subdividing  until.... Until in the limit, there are a nearly-infinitely large number of such increments, each of nearly-infinitely small size (read: all but 0 size). Thus arises vexing paradoxes, in the limit:
  • Infinity of increments x 0 size of each increment = something finite (the dimension of the original geometric figure)
  • Infinity / Infinity = something finite (the ratio of two similar geometric figures)
The Jesuits were having none of it; but, as we know now, infinitesimals are the foundation calculus. Something like Einstein who famously rejected quantum physics because he declared that the Devine would never sanction a probabilistic location of small particles. But, now we have physical proof of that very thing. Just as calculus is accepted as mainstream math.

And, what happened to the Jesuit's monopoly and the Church's grip on the thinking of mathematicians? Well, the monopoly is certainly all gone; and the mathematicians of the day fled to Protestant states, like France, England, and Germany. And, pressed on.

And, the thus the change management lesson is written: Beware the ankle biters with emerging and new concepts to offer. Change or perish!

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