"To say the least, the above picture is one example of the mathematics I guess you could say our cavemean ancestors wrote on bones. They've found more than one pair of these bones dating from thirty thousand years to eight thousand years ago. A major point I suppose could be that agricultural civilization was not the impetus for the advanced mathematics shown on these bones; yes, the matheamtics on these bones is not even as advanced as classical algebra, but arithmetic is more advanced than just knowing number words for the first few numbers(usually never beyond five) that many anthropologists have found throughout hunter-gatheror tribes of just a hundred years ago.
Mathematics is not denotative; a mathematical discovery is not of the kind like looking up and noticing the moon is there or going out and noticing a stone age spear flint. A piece of mathematics is much more like the discovery of the solar system; we figured it out by rearranging our perspectives. We derive the concept from looking at underlying structural relations of our assumed current perspectives/concepts. A basic example proves the point.
When we want to calculate the tiling of a plane, we could certainly place one tile after another onto a given plane till we've tiled it; but this is not mathematics. Mathematics is when we notice some underlying structural relations that we can then find some analogies between either the elemetns or the relations of the given structural relations, at which point, we have a conceptual machine(what Jacob Bronowski calls constructive analogies) like an equation which spits out the answer. In our geometric example, we note that there are rows and columns that we can multiply; two numbers as oppossed to the mere empirecal exercise of tiling a given plen one tile after antoher.
Getting back to the point about agriculture not being the stimulus for mathematics in Homo Sapeiens, could it be that mathematics was the stimulus for Agriculturalism? At least played a part? When Pythagoras discovered his musical scale by applying numbers, he was struck by the thought of doing the same to the rest of nature. These Caveman number bones seem to correspond to phases of the moon. These people had fire for hundres of thousands of years, and now comparably advanced arithmetics on their bones for keeping track fo the days and the cycle of the moon. They may have been inspired to apply mathematics to . . . much more; the universe . . . and life itself. They may have played with planting plants and then seeds in a geometric analogy to the number multiples they were playing with on their bones; surelly, they would have noticed the growing seasons as they played these geometic games with their advanced arithmetic?!
This could just be one possibility; i'd have to think somemore to find others, and right now, I want to continue to reading some books I recently bought for myself . . . I guess for this christmas! I should go ahead and tell you what I bought because I'm going to talk about what I read in these books. I bought Morris Kline's "Mathematics in Western Culture" and Edna Kramer's "The Nature and Growth of Modern Mathematics.' I also bought a movie; i guess I can go ahead and tell you that one also! I bought 'Operation Crossbow'; a movie about Peenemunde back in the early 1960's; i've already seen it; i suppose there are a few more sci-fi movies I want to buy, but not many! About those books . . . .
I read Morris KLine's "Mathematics in Western Culture' and his "Mathematics: The Loss of Certainty' in high school. So, I'm just getting that last book to reread and for reasons of, I like it, i want it, and so on. I also saw Edna Kramer's "The Nature and Growth of Modern Mathematics' at my high school; but, it was a far larger book and I never got around to finishing it off; to say the least, I got all this stuff for thity dollars on Amaozon! Why? Because nobody is intersted in these things! One more great tidbit about the Edna Kramer book I bought; on the opening title page is a stamp of the Cambridge engineering professor tha towned it and died around 1973 - Enrico G. Volterra. When I saw that I wondered if it was the functional analyses mathematician Volterra of the early 1900's, even though it says 'Engineering professor.' Well, I was able to shoot down that idea, but., maybe I should go back and see the family history . . . !"
If you want the illustrated version, you'll have to go to my blog which I give at my profile;