Jim D. > What is the reality of "2" or "4" if it is not 2 or 4 OF something?
Yes. I'm told that the original numbering systems were developed by people with herds of sheep and the like. It made sense to count them, whereas it seldom made sense to count the number of grains of wheat. ^^^^^ CB; I've been trying to think through that it was production of commodities that initiated arithmetic. In other words, the analysis in the first chapter of _Capital_ is also the schema of the origin of arithmetic. The main need to count originates with the need to establish equivalences between different things. Ten sheep are "the same thing" as forty pounds of wheat. Before that all that is needed is "many" , "few" , more, less,i.e. ordinal numbers. There is modern anthropology that corroborates this wherein certain groups don't "count" past 3 or 4. Cardinal numbers originate with commodity exchange, is the hypothesis I'm trying to articulate. Note writing originates at the "same" time in the big picture of history. So, the Tigris-Euphrates cunieform wedges originate in taxes (Michael P.)/commodities. Algebra would predate arithmetic because , as Levi-Strauss suggests, kinship relations organized are none other than _group_ theory algebra. (See _The Elementary Structures of Kinship); and kinship predates commodity exchange. ^^^^^ The basic idea is that (as Martin Gardner once said), mathematics represents the abstract aspect of empirical reality. To me, this means abstracting from the inherent heterogeneity of that reality: if you count your sheep, you ignore the differences among the sheep. (You also ignore the idea that they represent merely parts of the greater whole of sheepdom, and of the Animal Kingdom, and of Nature.) ^^^^^^ CB: Yes, treating two sheep as exactly "the same thing", as fungible, is an abstraction. My hypothesis is that what causes people to do this is the need to establish equivalences between different things, to imagine that 10 sheep are the same thing as 40 pounds of wheat _in order to exchange them for each other. The exchange establishes an equation : 10 sheep equal 40 pounds of wheat. They are "the same thing" by the act of trading them for each other. The first equation is establishe by the exchange. Marx and Engels give the idea that labor time in producing is the basis for this. Engels' has the longer essays claiming that commodity exchange goes back to approximately the same time as the origin of the family, private property and the state. Arithmetic originates with "trade". And it is among the first dometicators of animals. This is contra Piaget and those who see the first counting as being by an individual of grains of sand on a beach. That's a Robinsonade, a positivist approach to the origin of counting. Ask me a critical question this :>)
