A Short Look at Fifths and Fives: Joltin’ Joe, Legal Sausage and the Structure of Mathematics

JF Ptak Science Books LLC  Post 925 

I’m going to take a drive through five fifths—all of which don’t come close to making a whole—to get to my ultimate destination, a ribbon of numbers circling the earth.

In American law, “taking the fifth (Amendment)” allows you the prohibition of incriminating yourself, giving the supposed ability to look innocent whole not saying anything about anything for fear of legal repercussions, though this sausagey implosion from guilt seldom looks pretty. 

There’s the fifth column (as in Hemingway’s play by this name), the not-fourth-but-fifth force, the musical perfect fifth of highest consonance, the Keplerian nestling of the five Platonic solids,  Beethoven’s Fifth, a fifth of (good) scotch, the fifth cardinal direction (the center), the five books of the Torah, the five pillars of Islam, the worst categorization of hurricanes and tornadoes, living in the fifth Mayan world, the fifth Fibonacci number, and on and on.

And aside from being the number that Joe DiMaggio wore on his pinstripes, it is also the number of the  Euclidean postulate that called to all those who were interested in the possibilities of the postulate not being so—that there exists as perfect a non-Euclidean system as there is for the Euclidean.  The fact of the matter is that this calls into question the foundation, the very basis, of mathematics.  Or used to.

All of this was brought into mind by this 1957 ad for the RECOMP computer produced by Autonetics.  The image of a sheet of numbers circling the earth¹, the product of digital computation, suggested structure, mathematical imperative and solidification of numbers.  But the point of this was the bottom line: what was the point, and for that matter, what was the point?  They were hardly physical things floating in space, but ideas (except of course in the worlds of Flatland). 

Euclidean structure was the infallible, rigid, predictive, explanatory structure and the most accurate descriptor of physical space.  This had certainly been the case for fifteen hundred years or so, or at least until the struggling fifth postulate of Euclid’s Elements became really, well, unwieldy.  More to the point, the axiom, the parallel postulate axiom, wasn’t as self-evident as the first four, and wasn’t provable from the other nine of the first ten axioms.  And so came the unbelievable question, what if the fifth postulate wasn’t true?  What if there were other geometries—non-Euclidean geometries—that were as provable as that of the Master?  The possibility of relegating the understanding of nature and the physical surroundings to something other than the existing geometry was simply, positively, extraordinary. 

The origins for this movement away from Euclid started with Girolamo Saccheri (his book published in 1733 but not “discovered” for its non-Euclidean importance for another 150 years) Georg Kluegel (1763), Johann Lambert (Theorie der Parallellinien, 1766) and Adrien-Marie Legendre (1752-1833, who was sort of in a similar boat with Saccheri in that his work wasn’t actually published until after that of the next two mathematicians); and coming to a point of real invention in the work of Nikolai Lobachevsky (1792-1856) and Janos Bolyai³ (1802-1860). (It would be incorrect to credit these last two men [plus Gauss] with the creation of non-Euclidean geometry given the longish and complex history of its development.)

This work brought into question the way in which the world was seen, and the very foundation of recording visualized space.  This is all much messier than had been planned (so to speak) by almost all previous mathematicians, questioning the very foundations of mathematics.  The orderliness of the earth-circling numbers doesn’t seem to be quite as they were, but as Henri Poincare stated, the “new” system isn’t anything better or worse, just different,  a new and more convenient way of looking at things⁴.

Notes

1. To be honest about it, the foundations of math bit didn’t suggest itself right away.  The first thing I thought when looking at what this paper trail was supposed to represent—a string of numbers 41 billion zeroes long—didn’t look right.  Given the size of the numbers coming out of the “printer” there would need to be five times as many rings around the earth.  But when you read the text the numbers are supposed to be “hand written”.  Therefore if you assume the numbers to be less than an inch high and with no spaces in between, then this 2.5 times around the earth works out to be accurate to the number of inches in the circumference. 
2. Still earlier work is found in the work of  Omar Khayyam's Discussion of Difficulties in Euclid, John Wallis and Nasir al-Din al-Tusi... among others.
3. Bolyai’s work—published as an appendix to his father’s work on the foundations of geometry—was privately reviewed by the great and impossibly smart Carl Gauss, who gave it a backhanded series of compliments while at the same time saying “I thought so…”  Which was true, evidently, as it looks as though he was working on the problem beginning in 1799.
4. "If geometry were an experimental science, it would not be an exact science. it would be subject to continual revision ... the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate. In other words the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What then are we to think of the question: Is Euclidean geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates are false. One geometry cannot be more true than another; it can only be more convenient.” --from:  M J Greenberg, Euclidean and non-Euclidean geometries: Development and history. (1980).

Comments

[Jeff Donlan]

Five is also the number of fingers you see when someone says, "Talk to the hand." And there's the Five-Fingered Discount. But then there's Fifth Avenue and Fifth Business and the portal to the Fifth Dimension that is supposed to exist here. The existence of Five has given us the marvelous word Quincunx. A fine word all around.