Evaluating our ignorance about ancient Egyptian mathematics  

As the above list of the eight only surviving mathematical documents from ancient Egypt demonstrates, the unfilled gaps in our knowledge of pharaonic mathematics are large enough to drive through them a herd of Giza- size sphinxes with giraffe necks grafted on.

To put these meager remnants in their proper perspective, imagine for a moment some future archaeologists trying to reconstruct the achievement level of our mostly paper- based[18] mathematics from a comparable random handful of pre- comet- impact finds:

·       a week’s worth of daily receipts and disbursements from a Chaucerian tavern;

·       three dozen mildew- spotted pages with the planning of labor hours, stone supplies, and beer rations for the building of a medieval village church;

·       a few scraps from an early logarithm table riddled with errors;

·       two thirds of a slim arithmetic primer for Victorian trade school students;

·       a cardboard box from an upper stratum with the date 6/75 marked on its cover, filled with punched cards from the payroll department of a bank; and finally

·       a brownish brittle, here and there illegible stock quotation page from the Wall Street Journal found in the same vicinity.  The headings are torn off, but experts will tentatively identify its columns of numbers as a record of Hudson river levels, despite the several entries outside the possible range which they will ascribe to our notoriously bad observations and sloppy transcriptions.  

Although we have found not one of the myriad calculations required for any of the colossal construction projects of the Old Kingdom, the surviving structures themselves show clearly that the people who organized the resources of a large and prosperous country to put up the pyramids knew quite well how to work with numbers. 

Their building supervisors and administrators dealt with precise fits and orientations, and they solved the staggering logistics problems of piling up enormous volumes of hewn stone, often transported from far away, on tight schedules, with crowds of workers who all had to be fed and housed and provided with chisels and ropes and wood for sleds and other materials, and coordinated and either credited for their corvée services or paid in bread and beer. 

The currently promoted image of ancient Egyptian mathematics as primitive and strictly utilitarian implies that the people involved in this massive massaging of numbers never felt the desire to ponder and explore these mysterious entities which none could see or touch but which decided questions everywhere and determined everyone’s activities. 

If those ancient number crunchers were human, then it seems more probable that some of them would have wanted to know more about these body-less sources of power.  The ability to manipulate numbers was a prized skill for many scribes, and nothing prevented them from delving deeper into the magic of mathematics than what the job at hand required. 

We know from later ancient and even modern times that mathematical theory was usually far ahead of its applications, so we should expect to find a similar pattern in ancient Egypt since its inhabitants were obviously skilled in many forms of computing and did a lot of it.  We can deduce this from looking at the highly sophisticated geometry of the buildings they put up[19], and from the many other pioneering achievements of their civilization. 

So what if those postulated future archaeologists behaved like many of today's Euro- centric mainstream scholars who cannot accept that any ancient Egyptians developed an interest in mathematics beyond their needs for distributing bread and beer among workmen and similar practical tasks?   

Would these experts give our civilization credit for Euler’s discovery of e^{p i} + 1 = 0, the stunning formula that some call the most beautiful in all of mathematics [20]?  Would they suspect the steady stream of mathematical brain teasers that Martin Gardner and his many colleagues past and present kept flowing, and could they guess that we studied Cantor’s transfinite numbers and Gödel’s proof? 

Or would these future folks rather mock us for our “very, very primitive” ways of measuring in pounds and gallons and inches, and of dividing these into the cumbersome fractions of that quaint ½, ¼, 1/8, 1/16 ...  geometrical progression which we inherited from the even more antique pyramid people and which we did not replace with a better system in more than five thousand years? 

Would they shake their head at the archaic binary code in those computer instruction cards and point out its structural similarities with the “two- times table” that said pyramid people had used to multiply by successive doubling and adding, with multipliers that, like binary numbers, could always be written as a power of two[21]? 

Moreover, would they conclude with the same fundamentalist certitude from this patent lack of progress that none of us had enjoyed mathematical thinking or innovation?  Would they assert authoritatively that any resemblance between the stainless steel remnants of Eero Sarinen’s Gateway Arch in Saint Louis and a catenary curve must be entirely coincidental? After all,

·       the primitive state of our mathematics precluded us from knowing such equations, and

·       it would have made no sense whatsoever for anyone to build a structure shaped like a hanging chain, but upside down.

Considering this hypothetical scenario, it seems somewhat incautious to claim from our small sampling that the Egyptians had no interest in or curiosity about numbers[22] because they left us no written proof of it, or that the one sadly incomplete[23] school text for apprentice scribes among those few finds contained, as the mathematician Underwood Dudley wants us to believe, 

“much, if not all, of Egyptian mathematical knowledge of the time”[24]. 

Dudley cites Richard K. Guy’s Law of Small Numbers to warn us that “capricious coincidences cause careless conjectures” and “early exceptions eclipse eventual essentials”[25]. 

I must add to these warnings that skimpy samplings systematically suggest skewed stories, so shrewd scholars should scrupulously shun such shaky shortcuts: if we adopt as axioms all adamant, although arbitrary, academic assertions about an alleged absence of apparently advanced and astoundingly astute, albeit admittedly arcane, arithmetical and algebraic achievements among an ancestral and artistically accomplished ancient arch- civilization, then, alas, we ought to also accept any amazing amalgam of far- fetched fringe follies which affirm that, in fact, the phantom of Elvis the ageless Atlantean artfully fashioned the Sphinx from an alien astronaut’s abandoned UFO to form a feline friend for the future Bigfoot. 

It is important to remember the old truism that absence of direct evidence does not equal evidence of absence.  For instance, until very late times[26] none of the many detailed foundation scenes on Egyptian temple walls ever mentions or depicts foundation deposits, nor any of the elaborate ceremonies that must have accompanied their burial.  Yet, archaeologists keep finding such deposits[27]. 

Similarly, no surviving papyrus or other inscription ever alludes in words to the Egyptian artists’ and builders’ ritual canon of proportions that was so characteristic of their arts; yet, their buildings[28] and sculptures and paintings and furniture[29] and leftover sketching grids consistently display the use of those proportions for all to see. 

Besides the poor survival of the ancient Egyptians' mathematical writings, an additional reason for our finding no mention of these proportions could well be the great secrecy with which the priests protected their knowledge.