The Limits of Intelligence and Me: Domain Expertise

by ChrisHallquist5 min read7th Dec 201379 comments

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General IntelligenceUnderconfidence
Personal Blog

Related to: Trusting Expert Consensus

In the sequences, Eliezer tells the story of how in childhood he fell into an affective death spiral around intelligence. In his story, his mistakes were failing to understand until he was much older that intelligence does not guarantee morality, and that very intelligent people can still end up believing crazy things because of human irrationality.

I have my own story about learning the limits of intelligence, but I ended up learning a very different lesson than the one Eliezer learned. It also started somewhat differently. It involved no dramatic death spiral, just being extremely smart and knowing it from the time I was in kindergaarden. To the point that I grew up with the expectation that, when it came to doing anything mental, sheer smarts would be enough to make me crushingly superior to all the other students around me and many of the adults.

In Harry Potter and the Methods of Rationality, Harry complains of having once had a math teacher who didn't know what a logarithm is. I wonder if this is autobiographical on Eliezer's part. I have an even better story, though: in second grade, I had a teacher who insisted there was no such thing as negative numbers. The experience of knowing I was right about this, when the adult authority figure was so very wrong, was probably not good for my humility.

But such brushes with stupid teachers probably weren't the main thing that drove my early self-image. It was enough to be smarter than the other kids around me, and know it. Looking back, there's little that seems worth bragging about. I learned calculus at age 15, not age 8. But that was still younger than any of the other kids I knew took calculus (if they took it at all). And knowing I didn't know any other kids as smart as me did funny things to my view of the world.

I'm honestly not sure I realized there were any kids in the whole world smarter than me until sophomore year, when I qualified to go to a national-level math competition. That was something that no one else at my high school managed to do, not even the seniors... but at the competition itself, I didn't do particularly well. It was one of the things that made me realize that I wasn't, in fact, going to be the next Einstein. But all I took from the math competition was that there were people smarter than me in the world. It didn't, say, occur to me that maybe some of the other competitors had spent more time practicing really hard math problems.

Eliezer once said, "I think I should be able to handle damn near anything on the fly." That's a pretty good description of how I felt at this point in my life. At least as long as we were talking about mental challenges and not sports, and assuming I wasn't going up against someone smarter than myself.

I think my first memory of getting some inkling that maybe sufficient intelligence wouldn't lead to automatically being the best at everything comes from... *drum roll* ...playing Starcraft. I think it was probably junior or senior that I got into the game, and at first I just did the standard campaign playing against the computer, but then I got into online play, and promptly got crushed. And not just by one genius player I encountered on a fluke, but in virtually every match.

This was a shock. I mean, I had friends who could beat me at Super Smash Bros, but Starcraft was a strategy game, which meant it should be like chess, and I'd never had any trouble beating my friends at chess. Sure, when I'd gone to local chess tournaments back in grade school, I'd gotten soundly beat by many of the older players then, but it's not like I'd ever expected all older people to be as stupid as my second grade teacher. But by the time I'd gotten into Starcraft, I was almost an adult, so what was going on?

The answer of course was that most of the other people playing online had played a hell of a lot more Starcraft than me. Also, I'd thought I'd figured the game designer's game-design philosophy (I hadn't), which had let me to make all kinds of incorrect assumptions about the game, assumptions which I could have found out were false if I'd tested them, or (probably) if I'd just looked for an online guide that reported the results of other people's tests.

It all sounds very silly in retrospect, and it didn't change my worldview overnight. But it was (among?) the first of a series of events that made me realize that trying to master something just by thinking about it tends to go badly wrong. That when untrained brilliance goes up against domain expertise, domain expertise will generally win.

A whole bunch of caveats here. I'm not denying that being smart is pretty awesome. As a smart person, I highly recommend it. And acquiring domain expertise requires a certain minimum level of intelligence, which varies from field to field. It's only once you get beyond that minimum that more intelligence doesn't help as much as expertise. Finally, I'm talking about human scale intelligence here, the gap between the village idiot and Einstein is tiny compared to the gap between Einstein and possible superintelligences, so maybe a superintelligence could school any human expert in anything without acquiring any particular domain expertise.

Still, when I hear Eliezer say he thinks he should be able to handle anything on the fly, it strikes me as incredibly foolish. And I worry when I see fellow smart people who seem to think that being very smart and rational gives them grounds to dismiss other people's domain expertise. As Robin Hanson has said:

I was a physics student and then a physics grad student. In that process, I think I assimilated what was the standard worldview of physicists, at least as projected on the students. That worldview was that physicists were great, of course, and physicists could, if they chose to, go out to all those other fields, that all those other people keep mucking up and not making progress on, and they could make a lot faster progress, if progress was possible, but they don’t really want to, because that stuff isn’t nearly as interesting as physics is, so they are staying in physics and making progress there...

Surely you can look at some little patterns but because you can’t experiment on people, or because it’ll be complicated, or whatever it is, it’s just not possible. Partly, that’s because they probably tried for an hour, to see what they could do, and couldn’t get very far. It’s just way too easy to have learned a set of methods, see some hard problem, try it for an hour, or even a day or a week, not get very far, and decide it’s impossible, especially if you can make it clear that your methods definitely won’t work there. You don’t, often, know that there are any other methods to do anything with because you’ve learned only certain methods...

As one of the rare people who have spent a lot of time learning a lot of different methods, I can tell you there are a lot out there. Furthermore, I’ll stick my neck out and say most fields know a lot. Almost all academic fields where there’s lots of articles and stuff published, they know a lot.

(For those who don't know: Robin spent time doing physics, philosophy, and AI before landing in his current field of economics. When he says he's spent a lot of time learning a lot of different methods, it isn't an idle boast.)

Finally, what about the original story that Eliezer says set off his original childhood death spiral around intelligence?:

My parents always used to downplay the value of intelligence. And play up the value of—effort, as recommended by the latest research? No, not effort. Experience. A nicely unattainable hammer with which to smack down a bright young child, to be sure. That was what my parents told me when I questioned the Jewish religion, for example. I tried laying out an argument, and I was told something along the lines of: "Logic has limits, you'll understand when you're older that experience is the important thing, and then you'll see the truth of Judaism." I didn't try again. I made one attempt to question Judaism in school, got slapped down, didn't try again. I've never been a slow learner.

I think concluding experience isn't all that great is the wrong response here. Experience is important. The right response is to ask whether all older, more experienced people see the truth of Judaism. The answer of course is that they don't; a depressing number stick with whatever religion they grew up with (which usually isn't Judaism), a significant number end up non-believers, and a few convert to a new religion. But when almost everyone with a high level relevant experience agrees on something, beware thinking you know better than them based on your superior intelligence and supposed rationality.

Edit: One thing I meant to include when I posted this but forgot: one effect of my experiences is that I tend to see domain expertise where other people see intelligence. See e.g. this old comment by Robin Hanson: are hedge fundies really that smart, or have they simply spent a lot of time learning to seem smart in conversation?

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A rather out-of-context takeout of:

I recently read of a certain theist that he had defeated Christopher Hitchens in a debate (severely so; this was said by atheists). And so I wrote at once to the Bloggingheads folks and asked if they could arrange a debate. This seemed like someone I wanted to test myself against. Also, it was said by them that Christopher Hitchens should have watched the theist's earlier debates and been prepared, so I decided not to do that, because I think I should be able to handle damn near anything on the fly, and I desire to learn whether this thought is correct; and I am willing to risk public humiliation to find out. Note that this is not self-handicapping in the classic sense—if the debate is indeed arranged (I haven't yet heard back), and I do not prepare, and I fail, then I do lose those stakes of myself that I have put up; I gain information about my limits; I have not given myself anything I consider an excuse for losing.

I don't expect I can handle 'anything' domain-unspecifically on the fly. I thought I should be able to handle arguments William Lane Craig made, or tactics he used, on the fly. The entire article is about "Don't guess ... (read more)

I assumed you didn't mean literally anything. But I'm also assuming you know very little about debate tactics, is that correct? If so, the sentence I quoted seems to imply you seem to think you should be able to handle quite a wide range of things on the fly.

Craig is selective about who he debates, but mostly he seems to be optimizing for how big of names his opponents are. As much as I respect your work, Eliezer, you simply aren't as big of a name as Sam Harris or Christopher Hitchens (who've debate Craig) or Jerry Coyne (who recently declined a debate invitation from Craig).

I can suspect him of cowardice when he refuses to debate Jeffrey Jay Lowder (who like Craig and unlike most of Craig's opponents, has a background in college debate), but unless you've got some amazing debate performances under your belt that I don't know about, I don't think fear of losing is the reason he refused to debate you.

5Eliezer Yudkowsky7yNnnoo, it implies I thought I should be able to handle a wide range of events inside a public conversation (about religion) on the fly. This is a tiny slice of human endeavor. I probably wouldn't say something similar nowadays, and I worry whether that might be due to decrease of energy rather than calibration of confidence.
0Laoch7yThinking is metabolically expensive I guess(?)
0RichardKennaway7yNot necessarily [http://www.scientificamerican.com/article.cfm?id=thinking-hard-calories].

Filling in some details of the quote, since I was involved...

I was one of the atheists who said that (theist) William Lane Craig beat (atheist) Christopher Hitchens soundly, here and here. Also, I doubt the "Bloggingheads folks" said that Hitchens should have prepared more — that's what I and perhaps other public atheists said.

Also, if someone doesn't already believe that debates are mostly about debating skill (e.g. using the clock) rather than argument quality, a quick review of how Craig routinely dominates atheists in debates, while arguing not just for theism but also for Christian particularism, might change one's mind.

0passive_fist7yThanks for the link, I had never seen that debate before. I agree with the assessment that Hitchens was rambling and incoherent. Most of Craig's points could have had very simple, one-sentence refutations.

Good post.

When I notice someone outperforming me in figuring out some part of the world, I like to ask why they're able to do that.

Sometimes the most obvious explanation is sheer IQ. There's just no way I'm going to master model theory as quickly as Paul Christiano or Eliezer Yudkowsky.

In other cases, the most obvious explanation is superior rationalist practice, though this happens less often because applied rationality isn't yet a well-developed field with a high ceiling of human performance, and I'm fortunate enough to be pretty involved with one tentpole of the field.

And with other people, the standout explanation is that they just know way more shit than I do. As it turns out, many of these people are Carl Shulman.

My current strategy for catching up with Carl, or at least for falling behind him less quickly, is to listen to nonfiction audiobooks at 2x speed whenever I'm traveling, and whenever my eyes have given up for the day but my brain still wants more. Then I flip open the Audible app on my phone, put on my eye mask, and start learning about Cold War nuclear security.

Of course, I can't remember the exact details of most of the content, but I remember enough to know where... (read more)

5hyporational7yI recently discovered I can use text-to-speech on my android smartphone to listen to pdf files. The result is intelligible, and speed can be adjusted. The robotic sound doesn't bother me too much, in fact in some sense it's easier to follow because of the lack of emotional noise. I wouldn't recommend listening to fiction with it though. Audiobooks are obviously superior, but unfortunately limited in variety. My eyes tire faster than my brain too. If I could use Anki with my eyes closed, that would be awesome.
4Dr_Manhattan7ySweet! I've been unknowingly following your not-falling-behind-Carl strategy for a while and occasionally extolled virtues of 2-, then 3x listening in places here (I still only do 1.5x for fiction books, as I get more enjoyment that way). I've recently realized I've listened to more than 500 books over the years. Thanks for sharing your list. I should be doing something similar but need to improve my organizational skills first :)
2ChrisHallquist7yThanks, Luke. I tend to burry my eyes in my Kindle when I'm in transit, and I've gotten a lot of books read that way. But that has drawbacks, so I may try to make a point to listen to audiobooks more.

I think I can somewhat identify with this.

Growing up I was always told I was really smart; I had teachers telling me I was the smartest in class, right in front of the other students. I got into the gifted children program and I got into the national-level math competition.

What did that all do? Did it make me grow into the next Einstein or Witten? Nope. All that praise just got to my head. At 15 my head was larger than the Hindenburg, and just as doomed to catastrophic failure.

By high school I didn't even study. I didn't put effort into anything. And soon... (read more)

I had somewhat of your problem, but fortunately my comparison group consisted of famous scientists and mathematicians, and also competitive chess players -- so instead of thinking how smart I am compared to other people, I was more amazed at the stupidity I saw around me. Specific things that help put me in my place were being clobbered at chess by a kid half my height (much younger than me), getting clobbered at chess by a guy playing against 20 other people simultaneously (this happened on a regular basis with different such people), and realizing that even with the advantage of hindsight, I couldn't understand quantum and general relativity. So I've long known that I'm not memorably smart.

[-][anonymous]7y 4

On the subject of embarrassing teachers: I had a college professor who thought endorphins were cells and another who thought only humans ever made tools. A claim he repeated even after I showed him videos of crows making tools.

I did not feel especially good about knowing better regarding these things (and I have many other stories on this topic). I never felt like I had all that much knowledge or special talents and, as I know actual geniuses, I was pretty sure I was not one.

I guess surrounding yourself with people smarter than you are could serve as a sort of humility preserver but apparently that doesn't always work.

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4Jiro7yIt's generally stupid to immediately change your mind about facts when a single person presents you with a single counterexample. If you don't think animals make tools, and are shown a video of animals making tools, the first thing you should think of is "if that evidence is so definitive, there would be more than just a single video; I should be able to find it in an encyclopedia, a news article, or some other source which indicates that experts have accepted it". Then you look it up, discover that that's true, and then you accept it. After all, there could always be something wrong with the single example that isn't immediately obvious. (For instance, do you remember offhand the definition of a tool in sufficient detail that you could determine that the video actually shows tools, just by looking at it?)
6[anonymous]7yThat is a lot more reasonable than what he actually did though. He just didn't seem to acknowledge it at all and did not appear to be skeptical of it either. He was just like "Oh, okay. So as I was saying humans are special because we are the only ones who make tools." If he had wanted articles or something I could have found some for him very fast but he didn't even ask or anything.

As someone who thinks a lot about AI, the fact that domain expertise almost always trumps "raw" IQ seems to me to have very significant implications about the relationship between human learning, intelligence, and the structure of real world problems.

You left out discussion of domain specific intelligence, and I'd like to know why that is. Maybe it was intentional, maybe not. Because of this reason, I suggest your early misconceptions could have just as easily stemmed from misunderstanding what intelligence is (it's not like even the experts understand it well).

For example, it's easy to think you're the best at everything if you really are prodigiously talented just at math and merely very talented in other things like language. On top of this, consider the fact you're more likely to seriously compete ... (read more)

3ChrisHallquist7yThis is a good point. A big example of this for me (which I left out because it didn't seem to fit naturally into the flow of the piece) was that I've always hated memorization. And maybe some other people have a knack for memorization that I lack - in college, I had a friend who insisted that memorizing genetic pathways was easy, which baffled me. But my hating memorization may just be a product of the fact that, outside a few savants, rote memorization is always labor-intensive no matter how smart you are, and finding other things so easy made me resent having to put in the effort to memorize things, and maybe never learn certain study skills very well. It is true that, up through the end of high school and beginning of college, I saw myself as a math person above all else. In retrospect, I think I was mainly drawn to math because there were clear right and wrong answers which made it easy to demonstrate my intelligence. (There's a great quote about this that I can't seem to find at the moment...) In middle school English, for example, I got an assignment to write a an essay with a very specific structure which, from reading newspaper op-eds, I was pretty sure nobody actually used in real life. I wrote my essay about why the assignment was stupid. But rather than making me think, "Hey, I know a hell of a lot about writing for someone my age," the experience mostly made me hate English. It wasn't until college that I started to really see myself as having a talent for writing.
8Emile7yHave you tried Spaced Repetition Software, like Supermemo or Anki? I used to hate memorieation (who enjoys being forced to repeatedly dona boring task?), but eventually I realized that I could memorize some stuff just fine, and that if I had just memorized some stuff already (like german grammar rules) instead of whining I would have saved time in the long run. Eventually I started relying on Anki and now memorization is more like my secret weapon...
6TheOtherDave7yThere are more and less labor-intensive ways to handle rote memorization. When I started doing a lot of amateur theatre, I developed the practice of recording my lines on my phone and listening to them over and over as a background task, which allowed me to learn them with relatively little effort. It's a very inefficient way to learn lines if I'm measuring time-spent, but I would much rather spend three times as long memorizing lines if I can play video games or go on long walks or do push-ups while I'm doing it. Given how many people easily memorize the lyrics of popular music and commercial jingles without even meaning to, I suspect I could make it even less effortful by setting them to music, though of course that would create some difficulties when it came time to deliver the lines... that would work better for something I wanted to memorize but not necessarily declaim.
0hyporational7yI never quite made this connection, not sure why. Thanks. Might stop working [http://en.wikipedia.org/wiki/Interference_theory] if you set enough memories to music. Perhaps music works for memorization because it's rare?
-2Eugine_Nier7yWell, in the days before writing, setting things to music was the standard way of memorizing things.
0hyporational7yTrue, but perhaps it wasn't the only one, and there were fewer things to memorize.
0VAuroch7yIt and the Method of Loci were the main memorization methods, from what I know of i.e. Ancient Greece. And measured by total duration of recital of content memorized, I doubt we have that much larger stores of information considered important worth memorizing.
0hyporational7yHow would you estimate something like that? Who's we? What kinds of people are you comparing?
0VAuroch7yI'm comparing a modern generalist to Ancient Greek professional thing-knowers, which is to say their court bards, who memorized weeks length of poetry and song. Now that I think more carefully, I agree that any estimation is probably far off the mark.
4Ishaan7yI don't think this is true. I think people vary in the number of repetitions they need to be able to recall something. (conclusion formed via anecdotal evidence only) Hyporational's "domain specific intelligence" refers to when two different people put in the same amount on effort and knowledge on two different tasks, and person 1 performs better on task A and person 2 performs better on task B. Meaning, someone could be better than you not because they worked harder and not because they are more knowledgeable and not because they're just smarter in general for every dimension, but because they are smarter in the dimensions relevant to the task. If you attribute someone beating you because they worked harder (memorization) or were more knowledgeable (starcraft) then you effectively don't attribute their better performance to intelligence. The idea behind bringing up "domain specific intelligence" is to make it harder to exclude the hypothesis that they are better at activity X because they are smarter at activity X based on the fact that they were clearly dumber when it came to activity Y. I think it makes sense not to include this in the post, though...it seems like a rather different topic.
2hyporational7yThis sounds like rationalization. Ease of memorization is a spectrum, and none of my interactions with humans suggest otherwise. It varies between domains too, different people find different things difficult to memorize. I find memorization of language quite effortless for example. Other things vary from easy to difficult, but Anki helps with that, which is a bit ironic since it was initially developed for memorization of language.
0ChrisHallquist7yFascinating. How effortless are we talking? Do you even need flashcards for vocab lists or complicated conjugations?
8hyporational7yI wouldn't call myself a language prodigy, merely natively very good. My childhood development didn't make the news or anything. It's almost 10 years since high school when my proficiency at languages was tested, and I never used flashcards back then. Memorizing vocab lists mostly happened minutes before the exam in the hallway. I didn't just read through the words but also minimally tested myself for the more difficult words. I learned Swedish, English, and German this way, but have forgotten most of the Swedish and German vocabulary because I never use them. Grammar and pronunciation I just somehow absorb by listening and reading, in almost all the cases I couldn't name the rules I'm using and never tried to explicitly learn them. Verb conjugation in Finnish [https://en.wikipedia.org/wiki/Finnish_verb_conjugation], my native language, is pretty complicated so me being good at that in other languages wouldn't tell you much. I'm pretty sure I don't forget grammar with time like I forget vocabulary. These days I only need Finnish and English, and if I encounter an unfamiliar English expression, I check it in the English wiktionary, and rarely forget it. I suppose medical language for my work as a doctor counts too, but that's mostly just simple vocabulary, and I can't say I've put much effort into learning that either, although it's thousands of words. I started learning to speak Russian for fun a couple of years ago using an audio program, but found the program too slow and repetitive and got bored. I know I make mistakes in my English, and wish people would correct me more often. It's difficult to judge your own proficiency and I find I'm the laziest at improving in stuff I find easy, so pointing out a few stupid mistakes might motivate me to improve a lot. A couple of mistakes I know I make consistently: I forget which prepositions I should use with certain verbs, and probably use commas more often than I should. By the way, I think my episodic memory is als
0ChrisHallquist7ySo now your case seems much less impressive: Holding a list of vocab words in my head for an hour based on a few minutes of study is something I might be able to do too, depending on the length of the list. Terrible approach for long-term retention, though. I can definitely pick up basic grammar (for example the most commonly used very forms) just by getting a lot of experience with the language. But in high school Spanish class (I'm American), we had to know grammar rules explicitly, as well as knowing more obscure conjugations. In more advanced classes, this literally knowing first/second/third person singular and plural for a half-dozen tense-mood combinations for multple irregular verbs. And that kind of thing is very difficult to just pick up naturally, if for no other reason than that you're unlikely to come across the most obscure verb forms in conversation or media. Don't get me wrong, I got As in Spanish. It just required a lot of time with homemade flash cards and time spent staring at a blank sheet of paper, testing myself on how much of the table of conjugations I could write out, then studying the ones I missed extra-hard. I didn't know anything about fancy spaced repetition software back then. And I didn't even hate Spanish all that much. Especially not compared to (English) spelling. English spelling is the worst.
0hyporational7yJust sharing data points here, the impressiveness was in your head to begin with :) You said memorization is labour intensive and I don't find that to be true. Well, I didn't say I forgot them in an hour and those exams did include conjugations. We had bigger exams including grammar on top of those vocab exams, and I didn't really study for those excluding the classes. Earlier vocabulary was naturally needed for the later classes. The lack of serious initial repetition could be the reason I don't remember them 10 years afterwards, but I doubt anything can be forever remembered without repetition and long term retention really can't be called the labour intensive part of memorization. I have other experiences that suggest initial repetition for a couple of months doesn't help much in the long term. In Finnish you spell it almost exactly the way you say it and I don't think many languages do that. I've always enjoyed English and one of the reasons is the spelling and the pronunciation. Vocal acrobatics was one of the reasons I wanted to learn Russian for fun :) An example from medicine: I make flashcards in bursts, and sometimes make hundreds of them in a day. At the end of the day I usually remember something like 90% of them on the first try. Should I call memorization difficult?
0ygert7yHmm, I too have the feeling I use commas more often than I should. I wonder how others would judge themselves in that regard...
1hyporational7yI just love commas, can't help it.

This is a good picture of the value of experience.

If the system you're looking at is simple enough for you to entirely analyse and understand it when encountered, experience won't add anything. However even for smart humans this restricts you to pretty simple systems!

Experience allows you to observe and make models of the emergent patterns of high-level behaviour of a system, without understanding the details of how they follow from the lower-level details. In many complex systems this high-level behaviour can be very simple. For example the sun rises once every day. You could understand this in terms of the physics of the situation, but you can simply learn the pattern from experience.

Just as an aside, and not to criticize your frustration at your grade school math teacher, it may be worth spending some time thinking about whether negative numbers in fact exist and what exactly do you mean when you confidently assert that they do.

1scav7yI expect the math teacher wasn't making any kind of philosophical argument such as "do any numbers exist, and if so in what sense?" There is a different connotation, for my idiolect anyway, between "no such thing as X" and "X does not exist". It's possible that the only numbers that exist are the complex numbers, and that more familiar subsets such as the hilariously named "real" and "natural" numbers are invented by humans. I appreciate that this story is usually told the other way round.
4shminux7yAll numbers are abstractions and are therefore in the map. Positive integers have no more claim for existence than quaternions or what have you.
-1Eugine_Nier7yDisagree. Mathematical objects exist in the same way physical objects do at the very least, i.e., the standard anti-solipsist arguments for the existence of the physical world apply equally well to mathematical objects.
2shminux7yI sort of agree, but probably not in the way you mean. In the above I followed the map/territory meme, where you can find sheep in the territory, but the numbers in the map. However, I am interested if you outline or link to the arguments you mention.
-7Eugine_Nier7y
3Anatoly_Vorobey7yYeah, I'm sure the teacher wasn't making a philosophical argument. I can easily devil's-advocate for the teacher who may have thought, with some justification, that you first need to explain to children why "3 - 4" doesn't make sense and is "illegal", before you introduce negative numbers. A lot depends on the social context and the behavior of little Chris Hallquist, but it's not unusual that precocious little know-it-alls insist on displaying their advanced knowledge to the entire class, breaking up the teacher's explanations and confusing the rest of the kids. What Chris saw as a stupid authority figure may have been a teacher who knew what negaive numbers were and didn't want them in their classroom at that time. Re: the existence of negative numbers - I was thinking more of the status of negative numbers compared to natural numbers. Negative numbers are an invention that isn't very old. A lot of very smart people throughout history had no notion of them and would have insisted they didn't exist if you tried to convince them. While natural numbers seem to arise from everyday experience, negative numbers are a clever invention of how to extend them without breaking intuitively important algebraic laws. Put it like this: if aliens come visit tomorrow and share their math, I'm certain it'll have natural numbers, and I think it likely it'll also have negative numbers, but with much less certainty.
0scav7yAs to the teacher, yeah that sounds plausible. If Chris wants to satisfy our curiosity he can expand a little on how that conversation went. In my experience, teachers can really be dicks about that kind of thing. AFAIK, integers (including negative integers) occur in nature (e.g. electrical charge) as do complex numbers. Our everyday experience isn't an objective measure of how natural things are, because we know less than John Snow about nearly everything. I'd bet any aliens who get here know more than us about the phenomena we currently describe using general relativity and quantum mechanics. If they do all that without negative or complex numbers I'll be hugely surprised. But then I'd be super surprised they got here at all :)
0Anatoly_Vorobey7yElectric charge is precisely the sort of example that makes me think aliens could conceivably be doing OK without negative numbers. There are two kinds of charges, call them white charge and red charge. White charges create white fields, while red charges create red fields, and the white and red fields coexist in space. These fields exert forces on white and red charges according to well-defined equations. We find it very convenient to identify white with + and red with -, and speak of a single electromagnetic field, but I don't think (though I might be missing something) that this description is physically essential. That is, not only is the choice of electron as - and proton as + arbitrary, but the decision to view these two kinds of charges as positive and negative halves of a single notion of charge is arbitrary as well. It does seem very convenient mathematically, but without that convenience the equations of motion would not be significantly more difficult.
-1Lumifer7yCharge conservation makes a lot more sense in the + and - context than in the red and white context.
-1Anatoly_Vorobey7yIt's more convenient, but "a lot more sense"? I don't know. I have bread and cheese in my kitchen, which I only use to make cheese sandwiches. I don't have a bread conservation law, and I don't have a cheese conservation law, but I have a "bread and cheese conservation law", which says that the amount of bread that will go missing is the same as the amount of cheese that will go missing, up to a constant factor. Do I really need to introduce a notion of "beese", viewing bread as positive beese and cheese as negative beese? I could do that, and I will then have a beese conservation law, but it's not evident to me that my "bread and cheese conservation law" is less suitable for solving practical problems than the "beese conservation law". If I didn't need negative numbers for other things and didn't already know about them, I suspect I could get by with my "bread and cheese conservation law".
1[anonymous]7yIndeed, people talk about the conservation of bee minus ell [https://en.wikipedia.org/wiki/B_%E2%88%92_L] without labelling it anything else. So what?
-2Lumifer7yYou can, but if you get a guest who's gluten-intolerant and who will eat your cheese ignoring the bread, the "beese conservation law" will be broken. If you can show that the charge conservation law could be broken, the argument for positive/negative would become much weaker. That's a pretty large "if", however, more or less Nobel-sized :-)
-1Anatoly_Vorobey7yThat's just the poverty of my analogy, not of the underlying argument. In the white/red formulation of electromagnetism, the law of white and red charge conservation says that whenever any amount of red charge goes missing, the same amount of white charge must disappear with it. There's no inherent need to use negative magnitudes and sum up anything to 0.
3Lumifer7yIn a similar way you can call numbers less than zero red numbers and numbers greater than zero white numbers. So you've changed the labels, but did anything more important happen?
0Anatoly_Vorobey7yhttp://arxiv.org/pdf/1303.6576 [http://arxiv.org/pdf/1303.6576] I came across this in a Hacker News discussion. It's a rigorous derivation of (positive) real numbers without using 0 or negative numbers at all. In other words, pretend that you don't know what 0 and negative numbers are, come up with a slightly different axiom set for what is essentially a positive part of an ordered field, etc. Interestingly, this isn't stated as an explicit goal in the article, you need to read it between the lines. The paper is weak evidence of what I was talking about in this thread; weak because actual aliens probably wouldn't discover real numbers this way. But it does show it's possible to quite easily talk and reason about them w/o ever employing negative numbers, or even 0.
0Anatoly_Vorobey7yThe point is - is it possible to get to a working theory without inventing negative numbers. So with charges and my white-red charge conservation law, I never need to subtract 5 reds from 3 reds. Unlike e.g. loaning money, this sort of problem doesn't seem to arise with charges. When we use positive and negative charge, a large part of the algebraic machinery made available to us by negative numbers sits unused (we don't multiply charges either; that is we do in terms of Coulomb's law, but that's a notational convenience). That's why I said that electric charge is a good example of why aliens could conceivably get by w/o negative numbers. If they didn't have them for other reasons by the time they got around to investigate electricity, they might get by with the white-red formalism just fine. If you already know negative numbers, then sure, it's easy to imagine just relabeling them and nothing much changes. But to people in the first millennium AD, they were a very real and tangible invention. When ancient Greeks said that something like "x+4=2" is an obviously absurd equation w/o a solution, they meant it. They didn't go "oh, I have these I-OWE-U numbers that I use to count my debts but don't call them "negative", anyway, the solution is I-OWE-U-2".
-2Gurkenglas7yGetting physics to space travel tier without doing subtractions is implausible, and the integers are the subtraction-closure of the natural numbers. Look at any oscillation function, say one of the solutions of the simplest (most universally considered) differential equations, sin(x). It has negative values, and if you were to work only with positive numbers [edit: I meant taking its absolute value], it wouldn't be differentiable everywhere, which would be a pain. Complex numbers come from closing the real numbers algebraically, and if you don't have complex numbers, you won't go to space today.
0Anatoly_Vorobey7yYou seem very confident about all these assertions, but I don't understand where the confidence is coming from. Subtraction is clearly needed, but having imaginary entities that you use for the results of subtraction operations that are clearly invalid seems fanciful. It took a lot of time and effort for people to realize that it only looks fanciful, but in fact is very convenient and useful. You don't need negative numbers to define a limit of a sequence; in fact the notion of a distance between a and b is more natural than the formula |a-b|. You can define the distance between a and b as the subtraction of the smaller between them from the larger. Therefore you don't need negative numbers to define the derivative of a smooth function at x. You just say that at x the function is growing with derivative such-and-such, or shrinking with derivative such-and-such, or holding steady. Working with such a definition might be more cumbersome and might break down to more special cases, but I don't think it's hugely more cumbersome. You can do quantum mechanics without complex amplitudes [http://lesswrong.com/lw/j8g/the_limits_of_intelligence_and_me_domain_expertise/a6gy] .
-2Lumifer7yWhich assertions would that be? Fanciful? So, which criteria are we using to decide whether something like negative numbers are a reasonable concept? By the way, I think people used negative numbers for a very long time, it's just that they called them "debt" or "shortfall".
0Lumifer7yIntegers, sure, but can you give some examples for complex numbers occurring in nature?
6ialdabaoth7yWave functions [http://en.wikipedia.org/wiki/Wave_function] are complex, as are impedance [http://en.wikipedia.org/wiki/Electrical_impedance] values. (The former might be closer to "ontologically basic" than the latter)
0Anatoly_Vorobey7yHowever, I believe there [http://www.mrao.cam.ac.uk/~clifford/introduction/intro/intro.html] are [http://geocalc.clas.asu.edu/html/GAinQM.html] alternatives [http://www.amazon.com/Geometric-Algebra-Physicists-Chris-Doran/dp/0521715954].
-2Lumifer7yYou are talking about maps -- about human minds finding it convenient to describe certain natural phenomena through complex numbers. I read the original claim as saying that complex numbers are part of the territory. Are there square roots of -1 in nature?
8pragmatist7yWhy do you think wave functions are part of the map but electric charge is part of the territory? (I'm assuming you agree with scav's claim that electric charge is an example of negative integers occurring in nature.)
4Lumifer7yHmm... I don't have a good answer. My intuition is that integers are "simple enough" and, in particular, sufficiently unambiguous, to be part of the territory, but complex numbers are not. However even a tiny bit of reflection shows that my idea of "simple enough" is arbitrary. I guess we've fallen into the "is mathematics real?" tar pit. Probably shouldn't thrash around too much :-)
2ialdabaoth7yUmm... I'm gonna punt [http://lesswrong.com/lw/si/math_is_subjunctively_objective/] this one.
2shminux7yNumbers don't occur in nature.
0[anonymous]7ys/negative//

This post reminded me of this interview with Jeremy Howard, multiple-times winner of Kaggle data prediction contest. The article is titled "Specialist Knowledge Is Useless and Unhelpful" and includes this:

Q. How have experts reacted?

A. The messages are uncomfortable for a lot of people. It's controversial because we're telling them: "Your decades of specialist knowledge are not only useless, they're actually unhelpful; your sophisticated techniques are worse than generic methods."

It's only anecdotal evidence, but still; I think tha... (read more)

Experience was mentioned in the comments, but one thing that is missing is emphasizing the value of the subconscious brain(System 1 in Kahneman terminology). A lot of things experts "know" they cant verbalize. Consider a chess master, through thousands of games he has internalized a lot of patterns and "intuitively" knows what moves to consider. He doesnt have to be particularly rational in his thinking or practicing of chess, just sheer repetition is what gave him the skills.

The real power is in the brain(System 1) automatically picking up the patterns without we being aware of it.