Box 1: Pandora's Box

Introduction

Mathematics has an amazing ability to escape Houdini-like from any box it gets put into, and so doing develops whole new techniques, whole new fields of inquiry.  This blog explores mathematics as the art and science of thinking outside the box.  In this exploration, we shall look at several of the "boxes" that attempted to enclose areas of mathematics, and we shall see how mathematics escaped each box and grew more and more powerful in the process!

As the great mathematician James Joseph Sylvester noted:

  
Box 1:  Pythagoras and the Natural Numbers

 Pythagoras of Samos

Pythagoras believed that "all is number" and is credited with discovering the mathematical basis of music harmony:  that notes that sound well together were found at intervals that could be expressed in ratios of natural or counting numbers.

Pythagorean Hammers

According to Iamblichus, Pythagoras made an important discovery after observing that the hammers being used by blacksmiths in town made a ringing sound when striking iron. He realized that the weight of two hammers bore a simple numerical relationship to each other.

If the weight of two hammers were in ratio to each other, they were harmonious. If the hammers didn’t have a simple weight ratio in common, when they struck the iron, the sounds weren’t harmonious. This story is considered influential because it demonstrates an important relationship between music and math.

The finding is that tones with low-integer relationships are harmonious and thus pleasing to the human ear.

-  from the Moment of Science: The Harmony of Hammers web page.

Being an avid lyre player, Pythagoras could see how this discovery also applied to the lengths of plucked strings and many other musical instruments:

Woodcut showing Pythagoras with hammers, bells, a kind of glass harmonica, a monochord and (organ?) pipes in Pythagorean tuning. From Theorica musicae by Franchino Gaffurio, 1492 (1480?)  This image comes from Gallica Digital Library and is available under the digital ID bpt6k58171q.f36

The History of Music Theory website has an excellent page devoted to Pythagoras and his contributions to music theory:

Pythagoras: Music, Geometry and Mathematics 

It seemed common sense that the natural numbers were all the numbers that were required, since you could get as big a natural number as you wished (eg. 1,345,678,900) or invert it to get as small a number as you would ever want (eg. 1/1,345,678,900).

But when the Pythagoreans used their famous theorem to find the length of the diagonal of a square

they found it to be the square root of 2, but then discovered that this number could NOT be expressed as the ratio of two natural numbers.  Some credit the Pythagorean named Hippasus with proving the square root of 2 was IRRATIONAL.  He cleverly used an indirect proof by contradiction, or reductio ad absurdum.  For this method of proof you begin by assuming that there is a ratio of natural numbers p/q that is equal to the square root of 2 and then show that this assumption leads to a contradiction.   This proof rocked the foundation of the Pythagorean faith in the natural numbers, and legend has it that other Pythagoreans had Hippasus drowned so that irrational numbers would remain unknown:

What are irrational numbers anyway and do we really need them?

 An entertaining look at proving the square root of 2 is irrational:

  
Are there other ways to prove that the square root of 2 is irrational?

A fascinating visualization of irrational numbers:

With numbers behaving in such an irrational manner, Greek mathematicians changed their focus to geometry.  Geometers could easily construct line segments of irrational length, even if they were incommeasurable with other lines of rational length.  

 As Wikiwand explains:



Irrational numbers remained troublesome and controversial.  Even as late as the 19th century, the great German mathematician Leopold Kronecker was eager to banish them and put mathematics back in the Pythagorean Box:

"Interestingly, the topics Kronecker studied were restricted by the fact that he believed in the reduction of all mathematics to arguments involving only the integers and a finite number of steps. Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs. It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature. Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist."

- from web page on Kronecker:  God made the integers, all the rest is the work of man – Leopold Kronecker 


But Pandora's Box was now opened and could not be closed even to please the great Kronecker. 

From the cover of Brian Bolt's great collection of math puzzles:

A Mathematical Pandora's Box

 Even more exotic and provocative numbers would be found that would further enrage the followers of Finitism, but keep mathematicians entranced and excited for eons to come!  We shall deal with more of them in future posts.