I'm not trying to claim the "internal representation" is somehow more correct or even prior-to the real #s, etc.; I do think (though admit I haven't proven) tha

I'm not trying to claim the "internal representation" is somehow more correct or even prior-to the real #s, etc.; I do think (though admit I haven't proven) that a great many common examples of innumeracy can be explained by assuming the innumerate are using a "reals + infinitesimal" representation...which doesn't make the innumerate claims right, but possibly helps get at why they make those particular mistakes and not others. In the previous post I was hoping to make (only) the following claims(s): - in my experience, the theory "people who claim .9... != 1 are (unknowingly) equating .9... to 1 - delta" has explanatory power (in that the results of doing calculations with deltas are aligned with what these people claim) - I suspect that the internal human numerical representation is often a cruder, less rigorous form of nonstandard analysis, and that the reason you encounter otherwise-smart people who won't budge on .9... != 1 is that for them the question goes straight to that internal representation The just-so story for why I think the internal numerical representation is approximately real #s + some infinitesimal would go like this: - small distances aren't reliably perceivable under natural conditions: if I have two identical 8L pots and one has 4L and the other has 4.001L in it I probably can't tell which is which visually; if I'm not in very controlled conditions artifacts of ambient lighting or perspective or variations in the pot material will overrule any perceivable visual difference between the two - actions with nonzero actual effect often have no perceivable effect: if i have an 8L pot with 4L in it and I insert 1 ml from a pipette I probably can't see a difference even after the ripples settle down, but of course I will know that there's more water in the pot after I put some in than before ...and thus it'd make sense (at least for items with a continuous scale of size) to keep track of both: - what size do I perceive this to be? (the real component) - if I know extra information (eg: that I added a .001L) about the size does that information say that the actual size is going to be > or < the size I perceive it to be (the "infinitesimal" component) ...and it's not unreasonable that for ad-hoc, intuitive reasoning the mind would (roughly speaking) make the "correct" adjustments when combining size measurements (eg: (x + delta) + y > x + y, (x + delta) + (y + delta) > x + y + delta, (x + delta) + (y - delta) 'is a wash between' x + y, etc.)...after all even babies and animals apparently can do simple sums intuitively/automatically.