Polynesian people used a kind of binary number system 600 years ago (2013)

How do you know I don't understand math? And if a high schooler could figure this out, then Leibnitz own discussion or contribution to this topic is then meaningless?

The main point is that the finding in the article has cultural significance, rather than purely in terms of mathematics. I think you either don't understand that or are not willing to see that. The article is not trying to elevate a Polynesian people's contribution to mathematics. There is little mathematical significance there as these people from 600 years did not celebrate or promote their number system. Rather, the article explores how mathematical knowledge arises and how it was used. Could there an underlying commonality in the way humans learn and organize knowledge? That's an interesting question to ask and is not diminished - but is in fact supported - by the fact that the same knowledge gets rediscovered in multiple isolated instances.

> ...which lack the simplicity of 2 states...

If you're not working with computers, does this matter?

Also, in terms of computers, binary is only "simple" when you have the ability to create, assemble, and "warehouse" the "mechanics" necessary to represent them.

Babbage arguably didn't use base-10 for his Analytical Engine because it was more complex, but rather that it lended itself to gearing mechanisms (both in being more compact, and somewhat efficient (at least in how Babbage implemented it).

On the other hand, Zuse came up with an ingenious method of representing binary states using simple and efficient (and somewhat compact) sliding rod-like mechanisms in his first computer, the Z1.

Prior to the application of electrical and electronic circuits to computation, coming up with a mechanical system to represent binary (and be able to calculate with useful numbers, given friction and such) wasn't a trivial task. That said, it took a while once potential circuitry came into existence; I've always found it strange (to a certain point) that Babbage didn't use relays and binary (as he was contemporary with Boole, and had to know about his work, as well as relays as used in telegraphy - the only explanation I've been able to surmise is that they were still too early in their development for them to be reliable for calculation, but that's just my reasoned opinion).

Hollerith used electricity for his calculating machines, but they weren't binary-based (being essentially tallying machines they didn't have to be).

Strangely enough, the ENIAC was also a base-10 machine (it's ring counters consisted of 10 flip-flops more or less); again, this might be a reliability issue with the vacuum tubes of the period...

They aren't required, it's certainly possible to have a number system and not a writing system. But for anything where you need to do non-trivial arithmetic or accounting you can't just do all that in your head and memorize the results. You need some way of recording the results in a persistent way, even if it's just tallies carved into a piece of wood. And I imagine any culture that figures out a recording system for numbers would pretty quickly (in historical terms, so like years or decades) develop a writing system. I could be wrong of course, I haven't really formally studied ancient history.

Aside: Is Quipu not a writing system? I thought there was some evidence indicating it was.

> Our raw mental capabilities may have always been the same, but how smart we are also depends on our learning.

FWIW, I tend to equate "smart" with "cleverness," as a separate measure distinct from "experience," for the same reason you describe--so what I usually go by is something along the lines of:

Cleverness is a measure of "what can you do with what you have," whereas experience is a measure of "what do you have?"

> Over the centuries we've made great gains in mathematical tools, scientific knowledge, in democratizing education and in disseminating knowledge. And this means over the centuries we've (as a species) obtained more leverage that we can apply to our raw mental capabilities, giving us (overall) greater intellectual capabilities.

I don't think we've gained greater intellectual capabilities--our intellectual capabilities are the same, we just operate in a completely different mental environment than our priors did.

Moreover, having a different view of the world allows for different connections to be made, and different potentials to be expressed (irrespective of one's individual level of cleverness).

So, for example, by placing a priority on stories & views that encourage greater investigation of the physical world, we get to where we are today. And we can teach the next generation slightly different stories that optimize for different kinds of usefulness.

To bring it to the HN contingent--if I learn a new programming language, I've gained experience in different ideas and operate in a different mental landscape. But I'm not smarter afterwards, and I wasn't dumber before.