It's showing that this pattern applies to fingers and to toy cars and lollies and so on - divorcing the pattern from the things being counted. Only after this pattern is understood does the child begin to ask about bigger numbers, and eventually to realise there is no biggest number.

So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games.

Math as we know it piggy-backed the development of money. Money, first invented in Lydia, was the first abstract object, typifying value, but not specifying the value of what.

So abstraction isn't a philosophical folly. It's the result of an astounding innovation.

the first abstract object — frank

I might have said property - this counts as being mine. Basic idea is right.

I might have said property - this counts as being mine. Basic idea is right. — Banno

That existed before money. They bartered. The problem was that corruption in bartering was rampant. They would put the good dates on the top of the caravan, and it was just mud-balls below that. It was so bad that it inhibited trade.

Money set trade free from corruption because it was these little pieces of gold which were stamped to assure a specific weight and purity.

Next came banking, which was mainly invented by the Italians. Now we have virtual money, which allows economies to grow past their present means. The human world as we know it today is a result of money and banking.

I think the tricky bit is that philosophers hear "1 finger and 2 fingers make 3 fingers because 1 + 2 = 3," or even "1 finger and 2 fingers must make 3 fingers, because ..." and this sounds to them like the natural world obeying the "laws of mathematics" or some such. As if the fingers might try to add up some other way, but they would always fail, because there's a law.

But it's actually more like this: if I'm already committed to saying 1 and 2 make 3, then I'm also committed to saying 1 finger and 2 fingers make 3 fingers; if I didn't, I'd be inconsistent. Similarly, I can't say it works with fingers but not with train cars.

Children do have to learn, through trial and error, how much they're supposed to generalize. (Calling cows "doggies" and all that. And learning the difference between count nouns and mass nouns.) And of course what counts as success or failure is determined not by nature alone but also by the adults that mediate a child's understanding of nature.

What's difficult for us, in talking about mathematics, or about language, or about concepts, is that we want to pass over the generation upon generation of practice and refinement, to recreate the primordial scene in which someone, however far back, came up with a way of doing this sort of thing that worked, and we want to identify the features of the environment that enabled it to work, very much as if we expect there would only be one way. Some aspects of our thinking we find relatively easy to change, but some are so deeply embedded that we cannot quite imagine an alternative, so we think this way uniquely fits how the world is.

But it's not just a question of whether other ways of thinking were adequate to "our" needs, but recognizing that there was already adaptive behavior and already learning before there was any conception at all, and even our first conceptual steps were built on that.

Math as we know it piggy-backed the development of money. — frank

Are you saying there could have been a period when people had money, but didn't have amounts of money?

I agree with the spirit of your history lesson, that abstraction was a practical, observable, behavioral thing, but I don't understand the idea that money is the basis of math.

Do you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically? — Ludwig V

No, I've repeated this numerous times now, "space" is purely conceptual. it doesn't make sense to talk about dividing space physically. Physically there is substance, and that's what is divided. And representing that substance as "space" which is infinitely divisible is what I called the false premise which produces Zeno's paradoxes.

But a physical limit to the process of division doesn't undermine the conceptual description. — Ludwig V

It means that the conceptual description is false. And, this falsity, because it is a falsity, produces the absurd conclusions which Zeno demonstrates.

We've already left Meta behind, since he has claimed numbers are not ordered... — Banno

As usual, a completely false and utterly ridiculous representation. I said it doesn't make sense to use "next" in a way which is not either spatial or temporal. If we switch the term to "order" rather than "next", this allows all types of hierarchy such as good/bad, big/small, etc.. But the principle of the hierarchy, and the order of things within the category still needs to be defined. There is no such thing as simply "order" in the general sense. And to have a next implies a direction, which implies either a temporal or spatial ordering.

Therefore we cannot avoid expressing the order itself in spatial or temporal terms. If the scale is big and small for example, then for there to be an order one of the two extremes must be prior to the other, and this turns out to be a temporal order. If there was a supposed order which was infinite in all ways it could not be an order, because infinite possibility is disorder.

I was thinking some days ago that, though I'm not sure what the favored way to do this is, if pressed to define the natural numbers I would just construct them: 1 is a natural number, and if n is a natural number then so is n+1. I would define them in exactly the same way we set up mathematical induction. (Which is why I commented to Metaphysician Undercover that the natural numbers "being infinite" is not part of their definition, as I see it, but a dead easy theorem.) — Srap Tasmaner

You just show that it is limitless which is how "infinite" is defined, so there is no difference and you are not getting away from it being so, by definition.

But we need another step - "1 counts as a number" - to get the procedure moving. — Banno

The prerequisite platonist premise.

It's not platonic. — Banno

The usual denial. That "1 counts for a number" rather than signifying a quantitative value, is platonic. That's what platonism does, it makes values which are inherently subjective mental features, into countable independent objects. This is a faulty attempt to portray what is fundamentally subjective (of the subject) as something objective (of the object)

So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games. — Banno

Your statement "every number has a subsequent number" is a stipulation. Therefore it is something produced by design, definition, it is not something that we "discover". So you continue in your misguided attempt to justify mathematical platonism.

No, I've repeated this numerous times now, "space" is purely conceptual — Metaphysician Undercover

Are you a cartoon character? Do you know SpongeBob?

Are you a cartoon character? Do you know SpongeBob — frank

People trapped in a perpetual vortex are those likened to Plato's famously quoted analogy of 'The Cave:.