I read Karl Menninger’s work after reading Where Mathematics Comes From by George Lakoff and Rafael Nunez, of which my interest stemmed from reading about the Piraha, the Amazonian tribe with no numbers.
I kinda wonder whenever I hear/read someone say that math is the universal language and can explain the universe.
If you think about it from an evolutionary standpoint, how did we as a species even come up with numbers in the first place? There’s no natural pattern, that is something I’d figure out without socialization, I know of that says the word “one” should correspond to a symbol “1” or the word “two” should correspond to the symbol “2” and represent two things in a unit. There is no pattern in nature that makes me think of assigning numbers across a line to indicate position. I could use other symbols for that.
The defining quotes in a book of defining quotes:
Cultural history shows over and over again that regardless of how close a subject matter may be to its ultimate and perfect expression, only the intellectual readiness of a culture to take this step will ensure its development into complete maturity.
Our language is Germanic, our writing is Roman, our numerals are Indian!
Reads like an encyclopedia chock full of etymologies of number words like “one”, “two”, and a “thousand”. For example, he says about the word number “two”:
The word two can either join “(twin, “twain”) or divide (German Zwist, ‘dispute, discord”) Man experience both quite early in his development. Thus there is a vast family of related words in which the number “two” is latent in various forms: German zwi, Norse tvi, Latin bis, dis-, vi, and ambo, and Greek dis, dia, and ampho.
Meninger’s work also contains swaths of information on different methods of counting and computational methods in different cultures. He spent quite a bit of paper talking about tally sticks, tally notches, knots on ropes, reckoning boards, the abacus, the Japanese soroban, the Chinese suan pan as the different methods used to count and account.
In this context of counting and accounting for, he brought up the origins of the check, as in the paycheck, the check you sign and endorse before giving it to a bank.
The check as a certificate of demand for payent goes back to the notched tally sticks. The English Royal Treasury, as we shall see in the Exchequer tallies, kept its records of income, such as taxes, and expenditures on notched tally sticks. But the Excheqer also issued the stocks of double notched sticks or tallies (perhaps, for instance, with a L20 notch) to its citizens as certificates of payment…In English to check still means to compare an origial document or piece of wriing against a copy to see if the copy is correct.
The most essential point of this work is that numbers and number words as we know them in English, as we know them in China, as we know them in India, went through an evolutionary process of its own that spanned centuries of collaborative knowledge across regions and across peoples.
If we address ourselves directly to our number words, “one, two, three…nine, ten, hundred, thousand,” they remain strangely silent about their inner meanings; in this respect they behave like but few other words of our language. It is remarkable that in our daily lives we encounter the constantly and use them as the most reliable bearers of concepts in our possession. Yet we allow ourselves to be content with their usefulness while knowing them only by name. They pass us by mute, like alien slaves valued only for their servies, and we do not dignify them by inquiring into their “person” or their homeland. And yet they do have “personalities”…We are aware of course that the number sequence did not spring from a single brain, as a perfected system, but that it developed slowly, like a tree, keeping pace with man’s gradual development. Thus we cannot view the first few number words as being artifically contrived formations invented, like Esperanto, by one person alone and then adopted by others.”
In the course of our search for the meanings of our own number words, we have found both encouragment: that images enter into concepts of number and magnitude — and disappointment: that words may gradually slough off their meanings until they become mere gibberish.
…The form of the number is closely bound to the concept. Ancient measures are likewise strongly tied to the things they measured: cloth was measured in ells, height in feet, depths are stated in German folk tales in Klaftern and by seamen in fathoms.
One example of this multi-cultural, multi-regional, multi-central development of numbers: the concept of zero.
In roman numerals, the concept was hard to fathom. Roman numerals are not conducive to making computations nor representing large numbers.
A further obstacle to the development of the primitive Roman numerals into a “mature” form of numerals is revealed in the way the number 123456789 was written in the year 1792.
i = 1
xxiij = 23
iiij = 4
lvj = 56
Vij = 7
lxxxix = 89
If you were to write 123456789 into roman numeral form it looks like this: ixxiijiiijlvjvijlxxxix. Every unit has to have a symbol, and it gets quite messy once you start mixing numbers together. They did not make computations with those things. There was no system for them to multiply V x II and get X. Instead, Romans used the abacus, a type of counting board if they actually wanted to do calculations, accounting or what not. Writing number symbols was another activity altogether. They didn’t figure shit on scratch paper with bulky Roman numerals, they figure shit out on tallyboards and/or the abacus.
It took the Arabs via the Indians to come up with the number system that we know of today, and that took a while too, starting from Brahmi, to Gavalior to Sanskrit-Devanagari to West and East Arabic. The number system is what we call a place-value system with the concept of zero in it. That lets you do a few things: you can travel up and down the number line from -1 to infinity! You can now write down those number symbols, the 123456789 as 123456789, and do computations while you’re at it. 123456789 + 123456789 = 246913578. Yes, I used calculator for that, but I could figure that out on a sheet of paper if I wanted to.
The book is exhausted with tidbits about the development of numbers and scientific revolution-esque food for thought. Difficult to absorb in its encyclopedia-esque entirety and breadth, but still packed with importance.
After finishing this work, one question I now have is how Indians had a language that was conducive to place-value in the first place. What social, political, and linguistic conditions made it possible for them to ignite the concept of zero as well as the concept of place-value?