I’m not sure about the “three” or the “applies more to questions of logic than questions of knowledge”, but yeah, pretty much. Smarts gets you to better answers faster.
I’m not sure about the throwaway ‘three’ either but the ‘crystal vs fluid’ is something that is true if I am considering “demonstrate to me...” I find that this varies a lot based on personality. What people know doesn’t impress me nearly as much as seeing how they respond to new information, including how they update their understanding in response.
That makes sense. Those two bit are probably fairly good approximations to correct, but I can smell a possibility of better accuracy. (For example: “logic” is probably overspecific, and experience sounds like it should land on the “knowledge” side of the equation but drawing the correct conclusions from experience is an unambiguous sign of intelligence.)
I generally agree, I’m merely less-than-confident in the wording.
, and experience sounds like it should land on the “knowledge” side of the equation but drawing the correct conclusions from experience is an unambiguous sign of intelligence.)
Absolutely.
I generally agree, I’m merely less-than-confident in the wording.
I would quickly start believing someone was smart if they repeatedly drew conclusions that looked wrong, but which I would later discover are correct. I would believe they were smarter than me if, as a rule, whenever they and I are presented with a problem, they reach important milestones in the solution or dissolution of the problem quicker than I can, even without prior knowledge of the problem.
Concrete example: xkcd #356 includes a simple but difficult physics problem. After a long time (tens of minutes) beating my head against it, letting it stew (for months, at least), and beating my head against it again (tens of minutes), I’d gotten as far as getting a wrong answer and the first part of a method. Using nothing but a verbal description of the problem statement from me, my dad pulled out the same method, noting the problem with that method which I had missed finding my wrong answer, within five minutes or so. While driving.
(I’ve made no progress past that insight—rot13: juvpu vf gung lbh pna (gel gb) fbyir sbe gur pheerag svryq sebz n “fbhepr” be “fvax” bs pheerag, naq gura chg n fbhepr-fvax cnve vagb gur argjbex naq nqq Buz’f-ynj ibygntrf gb trg gur erfvfgnapr—since the last time I beat my head against that problem, by the way.)
The canonical method is to nggnpu n pheerag qevire gb rirel abqr.
Jevgr qbja gur Xvepubss’f ynj ynj rirel abqr va grezf bs gur vawrpgrq
pheerag, gur ibygntr ng gung ybpngvba, naq gur ibygntr ng rnpu nqwnprag
cbvag. Erjevgr gur nqwnprag ibygntrf va grezf bs genafyngvba bcrengbef,
gura qb n (frzv-qvfpergr) Sbhevre genafsbez (gur qbznva vf vagrtref,
pbqbznva obhaqrq serdhrapvrf, fb vg’f gur bccbfvgr bs n Sbhevre frevrf),
chg va gur pbaqvgvbaf sbe n havg zntavghqr fbhepr naq fvax, naq vaireg
vg, juvpu jvyy tvir lbh gur ibygntrf rireljurer. Gur qvssreraprf va
ibygntrf orgjrra gur fbhepr naq fvax vf gur erfvfgnapr, orpnhfr gurer vf
havg pheerag sybjvat npebff gurz.
I’m not wnoise, but yeah, you probably wouldn’t want to read the (now great-)great-grandparent. Put it this way: the first 5 words are ‘The canonical method is to.’ (I read it anyway ’cuz I’m spoiler resistant. I don’t think my math/EE aptitude is enough to carry out the method wnoise gives.)
I know little about it, but if I knew how to compute equivalent resistances beyond the basics of resistors in parallel and in series, I’d fbyir n ohapu bs rire-ynetre svavgr tevqf, fbeg bhg gur trareny rkcerffvba sbe na A-ol-Z tevq jvgu gur gnetrg abqrf nyjnlf ng gur pragre, naq gura gnxr gur yvzvg nf A naq Z tb gb vasvavgl.
You can try Xvepubss’f pvephvg ynjf, but at the moment I’m thinking of nffhzvat gung nyy pheeragf sybj njnl sebz zl fbhepr naq qrgrezvavat ybjre naq hccre yvzvgf ba gur pheeragf V arrq onfrq ba gubfr vardhnyvgvrf.
The problem doesn’t say anything about sources, so I’m not sure what I’m supposed to assume for voltage or current. Can you recommend a good instructional primer? I need something more that Wikipedia’s infodump presentation.
I’m using the term as a metaphor from fluid dynamics—n fbhepr vf n abqr va juvpu pheerag vf vawrpgrq jvgubhg rkcyvpvgyl gnxvat vg bhg naljurer ryfr—orpnhfr gur flfgrz vf vasvavgr, pheerag nqqrq ng n abqr pna sybj bhg gb vasvavgl gb rssrpgviryl pbzcyrgr gur pvephvg, naq orpnhfr gur rdhngvbaf ner flzzrgevp, gur svany fbyhgvba sbe gur pheerag (sebz juvpu Buz’f ynj sbe ibygntr naq ntnva sbe rssrpgvir erfvfgnapr) pna or gubhtug bs nf gur fhz bs n cbfvgvir fbhepr ng bar raqcbvag naq n artngvir fbhepr (n fvax) ng gur bgure.
I don’t know how this compares to wnoise’s canonical method—it might be that this is a less effective path to the solution.
Mathematical ability seems to be a high sensitivity test for this. I cannot recall ever meeting someone who I concluded was smarter than me who was not also able to solve and understand math problems I cannot. But it seems to have a surprisingly low specificity—people who are significantly better at math than me (and this includes probably everyone with a degree in a math heavy discipline) are still strangely very stupid.
Hypotheses:
The people who are better at math than me are actually smarter than me, I’m too dumb to realize it.
Intelligence has pretty significant domain variability and I happen to be especially low in mathematical intelligence relative to everything else.
My ADHD makes learning math especially hard, perhaps I’m quite good at grasping mathematical concepts but lack the discipline to pick up the procedural knowledge others have.
Lots of people of smart people compartmentalize their intelligence, they can’t or won’t apply it to areas other than math. (Don’t know if this differs from #2 except that it makes the math people sound bad instead of me)
The easiest of your hypotheses to examine is 1: can you describe (suitably anonymized, of course) three* of these stupid math wizzes and the evidence you used to infer their stupidity?
* I picked “three” because more would be (a) a pain and (b) too many for a comment.
can you describe (suitably anonymized, of course) three* of these stupid math wizzes and the evidence you used to infer their stupidity
Of course the problem is the most memorable examples are also the easiest cases.
1: Dogmatic catholic, knew for a long time without ever witnessing her doing anything clever.
2: As a nuclear physicist I assume this guy is considerably better at math than I am. But this is probably bad evidence as I only know about him because he is so stupid. But there appear to be quite a few scientists and engineers that hold superstitious and irrational beliefs: witness all the credentialed creationists.
3: Instead of just picking someone, take the Less Wrong commentariat. I suspect all but a handful of the regular commenters know more math than I do. I’m not especially smarter than anybody here. Less Wrong definitely isn’t dumb. But I don’t feel like I’m at the bottom of the barrel either. My sense is that my intellect is roughly comparably to the average Less Wrong commenter even though my math skills aren’t. I would say the same about Alicorn, for example. She seems to compare just fine though she’s said she doesn’t know a lot of math. Obviously this isn’t a case of people being good at math and being dumb, but it is a case of people being good at math while not being definitively smarter than I am.
I’m going with number 2 on this one (possibly a result of doing 4 either ‘actively’ or ‘passively’).
I have a very high error rate when doing basic math and am also quite slow (maybe even before accounting for fixing errors). People who’s ability to understand math tops out at basic calculus can still beat me on algebra tests. This effect is increased by the fact that due to mathematica and such, I have no reason to store things like the algorithm for doing polynomial long division. It takes more time and errors to rederive it on the spot.
At the higher levels of math there were people in my classes who were significantly better at it than I, and at the time it seemed like they were just better than me at math in every way. Another classmate and I (who seem to be relative peers at ‘math’) would consistently be better at “big picture” stuff, forming analogies to other problems, and just “seeing” (often actually using the visual cortex) the answer where they would just crank through math and come out with the same answer 3 pages of neat handwriting later.
As of writing this, the alternative (self serving) hypothesis has come up that maybe those that I saw as really good as math weren’t innately better than me (except for having lower error rate and possibly faster) at math, but had just put more effort into it and committed more tricks to memory. This is consistent with the fact that these were the kids that were very studious, though I don’t know how much of the variance that explains.
If you can’t ever recall meeting someone who you concluded was smarter than you who wasn’t good at X, and you didn’t use any kind of objective criteria or evaluation system to reach that conclusion, then you’re probably (consciously or otherwise) incorporating X into your definition of “smarter.”
There’s a self-promotion trap here—you have an incentive to act like the things you’re good at are the things that really matter, both because (1) that way you can credibly claim that you’re at least as smart as most people, and (2) that way you can justify your decision to continue to focus on activities that you’re good at, and which you probably enjoy.
I think the odds that you have fallen into this self-promotion trap are way higher than the odds for any of your other hypotheses.
It’s not really all that simple, and it’s domain specific, but having someone take the keyboard while pair programming helped to show me that one person in particular was far smarter than me. I was in a situation where I was just trying to keep up enough to catch the (very) occasional error.
Really? You’re easily impressed. I can’t think of one teacher from my first 12 years of education that I am confident is smarter than me. I’d also be surprised if not a single one of the people I have taught was ever smarter than me (and hence mistaken if they apply the criteria you propose). But then, I’ve already expressed my preference for associating ‘smart’ with fluid intelligence rather than crystal intelligence. Do you actually mean ‘knows more stuff’ when you say ‘smarter’? (A valid thing to mean FWIW, just different to me.)
They were smarter than you then, in the topic area in which you learned something from them.
When you’ve caught up with them, and you start being able to teach them instead of them teaching you, that’s a good hint that you’re smarter in that topic area.
When you’re able to teach many people about many things, you’re smart in the sense of being abie to easily apply your insights across multiple domains.
The smartest person I can conceive of is the person able to learn by themselves more effectively than anyone else can teach them. To achieve that they must have learned many insights about how to learn, on top of insights about other domains.
It sounds like you do mean (approximately) ‘knows more stuff’ when you say ‘smarter’, with the aforementioned difference in nomenclature and quite probably values to me.
I don’t think that’s a fair restatement of my expanded observations. It depends on what you mean by “stuff”—I definitely disagree if you substitute “declarative knowledge” for it, and this is what “more stuff” tends to imply.
If “stuff” includes all forms of insight as well as declarative knowledge, then I’d more or less agree, with the provision that you must also know the right kind of stuff, that is, have meta-knowledge about when to apply various kinds of insights.
I quite like the frame of Eliezer’s that “intelligence is efficient cross-domain optimization”, but I can’t think of a simple test for measuring optimization power.
The demand for “the simplest credible way” sounds suspiciously like it’s asking for a shortcut to assessing optimization power. I doubt that there is such a shortcut. Lacking such a shortcut, a good proxy, or so it seems to me, is to assess what a person’s optimization power has gained them: if they possess knowledge or insights that I don’t, that’s good evidence that they are good at learning. If they consistently teach me things (if I fail to catch up to them), they’re definitely smarter. So each thing they teach me is (probabilistic) evidence that they are smarter.
Hence my use of “teach me something” as a unit of evidence for someone being smarter.
I don’t think that’s a fair restatement of my expanded observations. It depends on what you mean by “stuff”—I definitely disagree if you substitute “declarative knowledge” for it, and this is what “more stuff” tends to imply.
That’s reasonable. I don’t mean to reframe your position as something silly, rather I say that I do not have a definition of ‘smarter’ for which the below is true:
They were smarter than you then, in the topic area in which you learned something from them.
When you’ve caught up with them, and you start being able to teach them instead of them teaching you, that’s a good hint that you’re smarter in that topic area.
I agree with what you say here:
The demand for “the simplest credible way” sounds suspiciously like it’s asking for a shortcut to assessing optimization power. I doubt that there is such a shortcut. Lacking such a shortcut, a good proxy, or so it seems to me, is to assess what a person’s optimization power has gained them: if they possess knowledge or insights that I don’t, that’s good evidence that they are good at learning. If they consistently teach me things (if I fail to catch up to them), they’re definitely smarter. So each thing they teach me is (probabilistic) evidence that they are smarter.
..but with a distinct caveat of all else being equal. ie. If I deduce that someone has x amount of more knowledge than me then that can be evidence that they are not smarter than me if their age or position is such that they could be expected to have 2x more knowledge than me. So in the ‘my teachers when I was 8’ category it would be a mistake (using my definition of ‘smarter’) to make the conclusion: “They were smarter than you then, in the topic area in which you learned something from them”.
What would be the simplest credible way for someone to demonstrate that they were smarter than you?
If they disagree with me and I (eventually?) agree with them, three times in a row. Applies more to questions of logic than questions of knowledge.
I’m not sure about the “three” or the “applies more to questions of logic than questions of knowledge”, but yeah, pretty much. Smarts gets you to better answers faster.
I’m not sure about the throwaway ‘three’ either but the ‘crystal vs fluid’ is something that is true if I am considering “demonstrate to me...” I find that this varies a lot based on personality. What people know doesn’t impress me nearly as much as seeing how they respond to new information, including how they update their understanding in response.
That makes sense. Those two bit are probably fairly good approximations to correct, but I can smell a possibility of better accuracy. (For example: “logic” is probably overspecific, and experience sounds like it should land on the “knowledge” side of the equation but drawing the correct conclusions from experience is an unambiguous sign of intelligence.)
I generally agree, I’m merely less-than-confident in the wording.
Definitely.
Ditto.
Absolutely.
So am I. Improve it for me?
I would quickly start believing someone was smart if they repeatedly drew conclusions that looked wrong, but which I would later discover are correct. I would believe they were smarter than me if, as a rule, whenever they and I are presented with a problem, they reach important milestones in the solution or dissolution of the problem quicker than I can, even without prior knowledge of the problem.
Concrete example: xkcd #356 includes a simple but difficult physics problem. After a long time (tens of minutes) beating my head against it, letting it stew (for months, at least), and beating my head against it again (tens of minutes), I’d gotten as far as getting a wrong answer and the first part of a method. Using nothing but a verbal description of the problem statement from me, my dad pulled out the same method, noting the problem with that method which I had missed finding my wrong answer, within five minutes or so. While driving.
(I’ve made no progress past that insight—rot13: juvpu vf gung lbh pna (gel gb) fbyir sbe gur pheerag svryq sebz n “fbhepr” be “fvax” bs pheerag, naq gura chg n fbhepr-fvax cnve vagb gur argjbex naq nqq Buz’f-ynj ibygntrf gb trg gur erfvfgnapr—since the last time I beat my head against that problem, by the way.)
Bah. I was hoping your dad gave the actual answer. That’s as far as I got too. :)
He suggested fbyivat n frevrf grez-ol-grez zvtug or arprffnel but I didn’t know precisely what he meant or how to do it.
The canonical method is to nggnpu n pheerag qevire gb rirel abqr. Jevgr qbja gur Xvepubss’f ynj ynj rirel abqr va grezf bs gur vawrpgrq pheerag, gur ibygntr ng gung ybpngvba, naq gur ibygntr ng rnpu nqwnprag cbvag. Erjevgr gur nqwnprag ibygntrf va grezf bs genafyngvba bcrengbef, gura qb n (frzv-qvfpergr) Sbhevre genafsbez (gur qbznva vf vagrtref, pbqbznva obhaqrq serdhrapvrf, fb vg’f gur bccbfvgr bs n Sbhevre frevrf), chg va gur pbaqvgvbaf sbe n havg zntavghqr fbhepr naq fvax, naq vaireg vg, juvpu jvyy tvir lbh gur ibygntrf rireljurer. Gur qvssreraprf va ibygntrf orgjrra gur fbhepr naq fvax vf gur erfvfgnapr, orpnhfr gurer vf havg pheerag sybjvat npebff gurz.
Buggrit. Build a grid of resistors a few meters square and pull out the multimeter.
That works fairly well, as things converge quickly.
Wait, so if I want to solve it myself, I shouldn’t read the text in the great-grandparent of this comment?
Well, yes, that’s why I rot13d it. I’ll unrot13 the beginning which will provide a clear warning.
I’m not wnoise, but yeah, you probably wouldn’t want to read the (now great-)great-grandparent. Put it this way: the first 5 words are ‘The canonical method is to.’ (I read it anyway ’cuz I’m spoiler resistant. I don’t think my math/EE aptitude is enough to carry out the method wnoise gives.)
Thanks!
I know little about it, but if I knew how to compute equivalent resistances beyond the basics of resistors in parallel and in series, I’d fbyir n ohapu bs rire-ynetre svavgr tevqf, fbeg bhg gur trareny rkcerffvba sbe na A-ol-Z tevq jvgu gur gnetrg abqrf nyjnlf ng gur pragre, naq gura gnxr gur yvzvg nf A naq Z tb gb vasvavgl.
You can try Xvepubss’f pvephvg ynjf, but at the moment I’m thinking of nffhzvat gung nyy pheeragf sybj njnl sebz zl fbhepr naq qrgrezvavat ybjre naq hccre yvzvgf ba gur pheeragf V arrq onfrq ba gubfr vardhnyvgvrf.
At this rate I’m going to be proficient at reading rot13 within a week!
I’m intentionally not reading anything in rot13 and always using the electronic translator, with hopes that I will not become proficient.
The problem doesn’t say anything about sources, so I’m not sure what I’m supposed to assume for voltage or current. Can you recommend a good instructional primer? I need something more that Wikipedia’s infodump presentation.
I’m using the term as a metaphor from fluid dynamics—n fbhepr vf n abqr va juvpu pheerag vf vawrpgrq jvgubhg rkcyvpvgyl gnxvat vg bhg naljurer ryfr—orpnhfr gur flfgrz vf vasvavgr, pheerag nqqrq ng n abqr pna sybj bhg gb vasvavgl gb rssrpgviryl pbzcyrgr gur pvephvg, naq orpnhfr gur rdhngvbaf ner flzzrgevp, gur svany fbyhgvba sbe gur pheerag (sebz juvpu Buz’f ynj sbe ibygntr naq ntnva sbe rssrpgvir erfvfgnapr) pna or gubhtug bs nf gur fhz bs n cbfvgvir fbhepr ng bar raqcbvag naq n artngvir fbhepr (n fvax) ng gur bgure.
I don’t know how this compares to wnoise’s canonical method—it might be that this is a less effective path to the solution.
That is definitely part of the method.
I can see why my idea is incompatible with your approach.
Mathematical ability seems to be a high sensitivity test for this. I cannot recall ever meeting someone who I concluded was smarter than me who was not also able to solve and understand math problems I cannot. But it seems to have a surprisingly low specificity—people who are significantly better at math than me (and this includes probably everyone with a degree in a math heavy discipline) are still strangely very stupid.
Hypotheses:
The people who are better at math than me are actually smarter than me, I’m too dumb to realize it.
Intelligence has pretty significant domain variability and I happen to be especially low in mathematical intelligence relative to everything else.
My ADHD makes learning math especially hard, perhaps I’m quite good at grasping mathematical concepts but lack the discipline to pick up the procedural knowledge others have.
Lots of people of smart people compartmentalize their intelligence, they can’t or won’t apply it to areas other than math. (Don’t know if this differs from #2 except that it makes the math people sound bad instead of me)
Ideas?
The easiest of your hypotheses to examine is 1: can you describe (suitably anonymized, of course) three* of these stupid math wizzes and the evidence you used to infer their stupidity?
* I picked “three” because more would be (a) a pain and (b) too many for a comment.
Of course the problem is the most memorable examples are also the easiest cases.
1: Dogmatic catholic, knew for a long time without ever witnessing her doing anything clever.
2: As a nuclear physicist I assume this guy is considerably better at math than I am. But this is probably bad evidence as I only know about him because he is so stupid. But there appear to be quite a few scientists and engineers that hold superstitious and irrational beliefs: witness all the credentialed creationists.
3: Instead of just picking someone, take the Less Wrong commentariat. I suspect all but a handful of the regular commenters know more math than I do. I’m not especially smarter than anybody here. Less Wrong definitely isn’t dumb. But I don’t feel like I’m at the bottom of the barrel either. My sense is that my intellect is roughly comparably to the average Less Wrong commenter even though my math skills aren’t. I would say the same about Alicorn, for example. She seems to compare just fine though she’s said she doesn’t know a lot of math. Obviously this isn’t a case of people being good at math and being dumb, but it is a case of people being good at math while not being definitively smarter than I am.
I suspect that “smarter” has not been defined with sufficient rigor here to make analysis possible.
I’m going with number 2 on this one (possibly a result of doing 4 either ‘actively’ or ‘passively’).
I have a very high error rate when doing basic math and am also quite slow (maybe even before accounting for fixing errors). People who’s ability to understand math tops out at basic calculus can still beat me on algebra tests. This effect is increased by the fact that due to mathematica and such, I have no reason to store things like the algorithm for doing polynomial long division. It takes more time and errors to rederive it on the spot.
At the higher levels of math there were people in my classes who were significantly better at it than I, and at the time it seemed like they were just better than me at math in every way. Another classmate and I (who seem to be relative peers at ‘math’) would consistently be better at “big picture” stuff, forming analogies to other problems, and just “seeing” (often actually using the visual cortex) the answer where they would just crank through math and come out with the same answer 3 pages of neat handwriting later.
As of writing this, the alternative (self serving) hypothesis has come up that maybe those that I saw as really good as math weren’t innately better than me (except for having lower error rate and possibly faster) at math, but had just put more effort into it and committed more tricks to memory. This is consistent with the fact that these were the kids that were very studious, though I don’t know how much of the variance that explains.
If you can’t ever recall meeting someone who you concluded was smarter than you who wasn’t good at X, and you didn’t use any kind of objective criteria or evaluation system to reach that conclusion, then you’re probably (consciously or otherwise) incorporating X into your definition of “smarter.”
There’s a self-promotion trap here—you have an incentive to act like the things you’re good at are the things that really matter, both because (1) that way you can credibly claim that you’re at least as smart as most people, and (2) that way you can justify your decision to continue to focus on activities that you’re good at, and which you probably enjoy.
I think the odds that you have fallen into this self-promotion trap are way higher than the odds for any of your other hypotheses.
If you haven’t already, you may want to check out the theory of multiple intelligences and the theory of intelligence as information processing
It’s not really all that simple, and it’s domain specific, but having someone take the keyboard while pair programming helped to show me that one person in particular was far smarter than me. I was in a situation where I was just trying to keep up enough to catch the (very) occasional error.
Teach me something I didn’t know.
Really? You’re easily impressed. I can’t think of one teacher from my first 12 years of education that I am confident is smarter than me. I’d also be surprised if not a single one of the people I have taught was ever smarter than me (and hence mistaken if they apply the criteria you propose). But then, I’ve already expressed my preference for associating ‘smart’ with fluid intelligence rather than crystal intelligence. Do you actually mean ‘knows more stuff’ when you say ‘smarter’? (A valid thing to mean FWIW, just different to me.)
They were smarter than you then, in the topic area in which you learned something from them.
When you’ve caught up with them, and you start being able to teach them instead of them teaching you, that’s a good hint that you’re smarter in that topic area.
When you’re able to teach many people about many things, you’re smart in the sense of being abie to easily apply your insights across multiple domains.
The smartest person I can conceive of is the person able to learn by themselves more effectively than anyone else can teach them. To achieve that they must have learned many insights about how to learn, on top of insights about other domains.
It sounds like you do mean (approximately) ‘knows more stuff’ when you say ‘smarter’, with the aforementioned difference in nomenclature and quite probably values to me.
I don’t think that’s a fair restatement of my expanded observations. It depends on what you mean by “stuff”—I definitely disagree if you substitute “declarative knowledge” for it, and this is what “more stuff” tends to imply.
If “stuff” includes all forms of insight as well as declarative knowledge, then I’d more or less agree, with the provision that you must also know the right kind of stuff, that is, have meta-knowledge about when to apply various kinds of insights.
I quite like the frame of Eliezer’s that “intelligence is efficient cross-domain optimization”, but I can’t think of a simple test for measuring optimization power.
The demand for “the simplest credible way” sounds suspiciously like it’s asking for a shortcut to assessing optimization power. I doubt that there is such a shortcut. Lacking such a shortcut, a good proxy, or so it seems to me, is to assess what a person’s optimization power has gained them: if they possess knowledge or insights that I don’t, that’s good evidence that they are good at learning. If they consistently teach me things (if I fail to catch up to them), they’re definitely smarter. So each thing they teach me is (probabilistic) evidence that they are smarter.
Hence my use of “teach me something” as a unit of evidence for someone being smarter.
That’s reasonable. I don’t mean to reframe your position as something silly, rather I say that I do not have a definition of ‘smarter’ for which the below is true:
I agree with what you say here:
..but with a distinct caveat of all else being equal. ie. If I deduce that someone has x amount of more knowledge than me then that can be evidence that they are not smarter than me if their age or position is such that they could be expected to have 2x more knowledge than me. So in the ‘my teachers when I was 8’ category it would be a mistake (using my definition of ‘smarter’) to make the conclusion: “They were smarter than you then, in the topic area in which you learned something from them”.