Well said Eckard,
I like the idea of starting from nothing, when talking about number. There is an increase in comprehension when children are taught to count 0,1,2.. instead of 1,2,3... But when I was that age, the latter formalism was the most common or conventional. If we take a constructivist view of this matter, it forces us to find a minimal set of assumptions which allow counting and/or measuring. I talk about this in Quantum Biosystems Vol 1 no. 1, but briefly - measurement proceeds by procedural steps of observation, change, and comparison. If we repeat 'observe, move, compare' enough times a sense of size, distance, and proportion becomes apparent. So no one observation can give the total picture, but an understanding arises as the number of observations increases. But when talking about a number of objects, things are a little different.
You see; when we see a single image containing a small number of similar objects (beans or oranges perhaps), we can immediately distinguish between a field of view with three - and a field with four or five objects. On the other hand, a planet-sized ball cannot be easily distinguished from a marble, unless our field of view contains both objects at once - enabling us to make a comparison. So size is relative, in the sense where the scale is determined by a size comparison with other objects. But the number of objects in a given space is in a sense more absolute. However, a scale change in the field of view may admit quite a few more oranges or beans to enter the picture. So there is a distinct relation between the two dissimilar concepts.
So this invokes or involves a set-theoretic view, where the boundary is arguably a topological distinction between objects in the set (within the field of view under consideration) and those outside it. Taking this view allows us to represent oranges on the screen with points lying within a circle. But of course; a real orange has size and might be half in our field of view, with the rest outside it. So at some stage of any real determination of number and measure, the sense of proportion arising from the concept of size comes into play.
Anyhow; I greatly enjoyed your recounting the various definitions of number past thinkers have brought us. To a degree, the concept of number is natural - or arises naturally from observation and comparison. However; there is a lot of definition that we accept on faith, or reject at our own peril. But I think that's what Isham is talking about when he says we need to revamp Math in a major way, because it is exactly this kind of definition by hidden assumptions that must be re-examined. And I think the constructivists had the right idea, but until now have lacked some of the tools that category theory offers. Using various topoi allows us to approach the concept of number differently, or variously, it would seem.
But I really like what Rudy Rucker has written about the subject of numbers and numeration. I would heartily recommend 'Mind Tools' or 'Infinity and the Mind' if you can find them, but I know he has covered the same topic elsewhere too.
All the Best,
Jonathan
