Milo Gardner wrote something I just want to quote as a holistic take on a modern hot topic: a dual definition of multiplication. This comes from a thread in “Math, Math Education, Math Culture” on LinkedIn.

Modern mathematics including paper folding offers distractions from the central dual multiplication definition conflict. Multiplication defined as been repeated addition and scaling of rational numbers co-existed as main stream Western Tradition ideas 4,000 years ago, and maintained the tradition for 3,500 years.

Math historians report Egyptian fraction cultures formally used the paired multiplication definitions by 2050 BCE. Specifically, the Egyptian Middle Kingdom. Ahmes, a 1650 BCE scribe, recorded a 2/n table that scaled 2/3, 2/5, 2/7, …, to 2/101 to concise unit fraction series that followed a dual multiplication method.

Modern scholars scratched their collective heads during the 20th century when only reporting the additive side of the paired dual set of multiplication definitions. Ahmes 2/n table introduced 87 arithmetic, algebraic, geometric and weights and measures problems that required a dual understanding of the multiplication definitions.

Both sides of the multiplication definitions were needed by Ahmes, and Egyptian scribes, as scribes as late as Fibonacci in 1202 AD used to record the Liber Abaci, Latin speaking/writing Europe’s arithmetic, algebra, geometry and weights and measures instruction book for 250 years.

Of course, with the death of Egyptian fractions, and the birth of modern base 10 decimal arithmetic in 1600 AD, the ancient dual definition of multiplication conflict seemed to disappear. But has it?

I think not. Modern mathematical physics reports the same dual conflict in ways that would have made ancient Egyptian fraction scribes shake their heads.