3 ms·

And if you did 'solve' X = (.9... + 1)/2 using the standard 'algorithms' for doing it by hand you will wind up with X = .9... + .0...1 (more on this in a sec);

by frig 17y ago

And if you did 'solve' X = (.9... + 1)/2 using the standard 'algorithms' for doing it by hand you will wind up with X = .9... + .0...1 (more on this in a sec); you get there by noticing 1.9... / 2 looks like it ought to be .9..., but "of course" 2 x .9... == .9...8, ergo you can deduce you have .9... + a remainder of .0...1. If you don't believe .0...1 exists (infinite 0s followed by a 1) then this ought to convince you that .9... == 1 (as (x+y)/2 == x => x == y) but since we're (hypothetically) arguing with someone who believes .9... != 1 it's quite plausible that the person is also going to believe in .0...1 existing, also. What I suspect is actually going on (I've also met otherwise-savvy people who will debate this issue also) is that the human numerical-cognition system uses something like a crude form of nonstandard analysis internally; and, under this theory, you see this internal system (and its intuitions) leak out in some people smart enough to get that .9... really does have 'infinitely many' 9s but who also for whatever reason are both undereducated on this point of mathematics and generally self-assured. Briefly: you can extend the real #s with a new entity (call it delta) such that: - delta > 0 - if r is a nonzero # then delta < |r| - (for now) delta^2 = 0 (in effect: delta is an infinitesimal, it's nonzero but smaller than any other positive #, and delta^2 is 0 is mainly an algebraic convenience). Once you've extended things by including delta you can write extended #s in the form r + s x delta (eg: 1 - delta). If we re-define ".9..." to be 1 - delta, then the following all drop out of that: - .9... != 1 (1 - .9... == delta != 0) - .9... < 1, (1 - .9... == delta > 0) - and we even get: (1 + .9...)/2 = 1 - (delta/2) ==> (1 + .9...)/2 > .9... And more generally in my experience most of the spinning-in-circles around people with beliefs about .9... != 1 is that the people who vociferously believe .9... != 1 don't even know about non-standard analysis (or infinitesimals, etc.) and thus can't articulate what their intuition is telling them, but what their intuition is using is basically (a nonrigorous form of) analysis with infinitesimals.

slackenerny 17y ago
(for now) delta^2 = 0 Oh, that's another wonderful story — of Clifford algebras: http://en.wikipedia.org/wiki/Dual_number http://en.wikipedia.org/wiki/Dual_number , useful in automatic differentiation. As to your general point of what constitutes "internal representation", someone below tries to convince people that definitions involved with Reals are arbitrary and can be made differently thus forming Hyperreals. But no. One first have to construct Reals anyways, then one arrives at infinity, which is not a number, hyperreal or otherwise, but a consequence of numbering per se. Nothing ad hoc here. Hyperreals are that or another notation, but the main thinking is still the same, just paradoxes pop in different places, the only gain is increase in confusion as hyperreals, tuns out, do not conform to that "internal representation" in many more places than the poor 0.(9) .
- frig 17y ago
  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.
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by frig 17y ago

slackenerny 17y ago

(for now) delta^2 = 0 Oh, that's another wonderful story — of Clifford algebras: http://en.wikipedia.org/wiki/Dual_number http://en.wikipedia.org/wiki/Dual_number , useful in automatic differentiation. As to your general point of what constitutes "internal representation", someone below tries to convince people that definitions involved with Reals are arbitrary and can be made differently thus forming Hyperreals. But no. One first have to construct Reals anyways, then one arrives at infinity, which is not a number, hyperreal or otherwise, but a consequence of numbering per se. Nothing ad hoc here. Hyperreals are that or another notation, but the main thinking is still the same, just paradoxes pop in different places, the only gain is increase in confusion as hyperreals, tuns out, do not conform to that "internal representation" in many more places than the poor 0.(9) .

frig 17y ago
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.
- deleted 17y ago
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