Crawl out of your personal pit and read economic system weights and measure units from 1650 BCE that priced a loaf of bread, a glass of beer, and pen feed dove, duck, good and quail based on the amount of grain contained therein. The inverse proportion named pesu was the arithmetic operator as the NY Times clearly reported.
Egyptian meta math began with theoretical building blocks, i.e. rational numbers scaled to unit fraction series by optimized least common multiples, and applied practical formulas that allowed double checking of every calculation insuring accuracy.
Wait a few weeks and a formal paper will be completed that better links the fragmented aspects of Egyptian fraction math into a unified whole.
Egyptian meta math scaled rational numbers n/p by a LCM m to mn/mp. Medieval scribes used a modified subtraction context to convert n/p by LCM m considering
(n/p -1/m) to (mn -p)/mp, with (mn -p) = 1, whenever possible.
Certain issues of Platonic and Classical Greek arithmetic have not been confirmed for several reasons. The biggest reason is the unavailability of Greek arithmetic texts. What we do know relates to Egyptian, Greek and medieval scribes scaling rational numbers in almost identical ways. It is highly likely that the medieval scaling method was created by Greeks and/or Arabs by modifying the ancient Egyptian scaling method.
One summary point is that Egyptian, Greek, Arab and medieval texts like the Liber Abaci are made readable by meta arithmetic.
Concerning the word hekat, it meant volume, defined by a modern looking formula
RMP 41, 42, 43, 44, 45, and 46 discussed the hekat's division into a two-part number. In RMP 47 100 hekat was multiplied by 1/10, 1/20, 1/30, 1/40,1/50, 1/60, 1/70, 1/80, 1/90 and 1/100, a method that divided (6400/64) by 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 to
(6400/64)/n = Q/64 + (5R/n)ro
with Q =quotient and R = remainder.
an interesting formula.
Yes, there can be several classes of pits to climb out of. Modern folks, like yourself, fall into pits of your own mental creation. Read actual Egyptian, Greek, Arab and medieval arithmetic texts by crossing over meta bridges and never fall into a related pit again.
Conversation two (2)
I did read the NY article and fell into a pit of 'no compute' when I read this:
"Seven houses have seven cats that each eat seven mice that each eat seven grains of barley. Each barley grain would have produced seven hekat of grain. (A hekat was a unit of volume, roughly 1.3 gallons.)
The goal: to determine how many things are described. The answer: 19,607. (The method: 7 + 7² + 7³ + 74 + 75.)
???? I'll probably have to think in a different way to understand that but you and I are somewhat on the same wave length about 'meta' data.
Perhaps the term 'hekat' means 'an object' , A 'thing' not a quantity, a weight, etc? To me it's a quantity that's not been counted yet, so the answer is not verifiable and doesn't make sense. There's probably a benefit to the idea: . "Mathematics makes us think of things that cannot be seen." The attachment is William Blake's Ancient of Days, the added feature is my imagination at work.
I appreciate your explanation of 'meta', its exactly an idea I have but in a different context, ie, that there's a mindset that does not have any understanding other than 'forests' i.e. concepts. Details are not part of their understanding, its as though every 'detail' in a forest is assumed somehow, not spoken, written, or relevant, as though it's so obvious it needn't be addressed in any way. That seems to me to have been a 'stage' when a new mindset (or a new system of things, as the bible puts it) replaced the ancient symbolic mindset. As I understand it now, it's all part of a grand scheme to make us aware we are informed about the past, as well as our 'now', and the future sometimes, through dreams but probably another function of mind that in in itself, has to establish a foundation to understand it, through mathematics where 'functions' have become a 'fact'.
To explain exactly what I mean is how many books I read about quantum physics that didn't mention the specific 'detail', in a concept (forest); that all human activity is derived from that hidden world behind matter physics before reading Atom, Matter, Physics by a Seattle physician/science fiction author, Alan Nourse!!! He made very distinct, detailed descriptions that the floor I stand on, the orange I eat, the tree I plant and see maintained in place, growing, as well as the impulses and motives, and activities of 'me', weather, all movement or stability are the result of transactions and interactions of particles.
Since nanotechnology has sprung into our material world, I wonder if we really know anything except that 'number', counting, establishing equivalents began long before we have adequate records, so trying to decode what we do have is our work. I lived 5+ decades without suspecting that, then.... that's another topic, not connected to mathematics.
It may be that this makes no sense to you, in connection to what I wrote, but quite a variety of incidents and what I read one day when I was reading the book I mentioned made me think about your interest in codes. The alchemist way of thought, and seeking caused them to recognize their mind was included in what they observed, is probably installed in some mindsets, or it can be 'downloaded', that reads as weird, but I know it myself. Nothing really mysterious or mystical, but very difficult to prove.
"What happened in the past is determined by what is to happen in the future" Norman O. Brown
Conversation three (3)
By the way ... the NY Times published a article on 12/7/10 needs to be read: