Consider the following
Those are weak examples. The counterpart to Brin is Zuckerberg who's a programmer but haven't shown any flashes of brilliance in math or CS; the counterpart to Gates is Ellison (of Oracle) who certainly understands databases but again, not a genius in math or CS; the counterparts to Simons are a large number of hedge fund founders (e.g. Ray Dalio); and the obvious counterpart to Munger is Warren Buffett himself.
What "major disadvantages" do you have in mind? Brilliant mathematicians are rarely wealthy or enjoy high social status.
Those are weak examples.
You want to take base-rates into account. Of intellectually gifted people outside of academia, what fraction do you think have high technical proficiency in a quantitative subject? How does that compare with the fraction of super-rich people who have such proficiency?
What "major disadvantages" do you have in mind? Brilliant mathematicians are rarely wealthy or enjoy high social status.
No, but to a large degree, that's not what they're motivated by. If you look at revealed preferences, the most intellectually capable people do math disproportionately. Gauss is the one who developed the theory of linear regression and p-values, and he described math as being the queen of science and number theory as the queen of math. Alexander Grothendieck wrote:
I well recall the power of my emotional response ( very subjective naturally); it was as if I'd fled the harsh arid steppes to find myself suddenly transported to a kind of "promised land" of superabundant richness, multiplying out to infinity wherever I placed my hand in it, either to search or to gather... This impression, of overwhelming riches has continued to be confirmed and grow in substance and depth down to the present day.The phrase "superabundant richness" has this nuance: it refers to the situation in which the impressions and sensations raised in us through encounter with something whose splendor, grandeur or beauty are out of the ordinary, are so great as to totally submerge us, to the point that the urge to express whatever we are feeling is obliterated.
All else being equal, wouldn't you want the option to feel that way? :D
Those who do decide that they want to do something outside of math often do very well (again, Simons is virtually the only elite mathematician to have left academia), there just aren't very many of them.
Of intellectually gifted people outside of academia, what fraction do you think have high technical proficiency in a quantitative subject?
I have no idea and I suspect that this strongly depends on the definition of "intellectually gifted".
How does that compare with the fraction of super-rich people who have such proficiency?
Looking here it doesn't strike me that "math, physics, theoretical computer science and statistics" are a path to the riches. I'd bet on business ability instead.
the most intellectually capable people do math disproportionately
Do you have data (as opposed to anecdotes) to show that?
wouldn't you want the option to feel that way?
It depends on the price. Options are expensive :-P
However pick any description of satori, or nirvana, or even a mystical union with Jesus. It will sound very similar, perhaps even better. Wouldn't you want the option to feel that way?
I am not quite sure of your point. It seems to be "if you are intellectually gifted -- study math". However your examples do not support your assertion. Brin, Gates, etc. achieved their station in life not through studying mathematical proofs.
I have no idea and I suspect that this strongly depends on the definition of "intellectually gifted".
But this is highly relevant :D. As I say elsewhere:
"There are so few people who do publishable TCS research as college sophomores that it's very unlikely that the wealthiest person in the world is one of them by chance. I acknowledge that the correlation may be entirely spurious, but it still warrants a Bayesian update."
I am not quite sure of your point. It seems to be "if you are intellectually gifted -- study math". However your examples do not support your assertion. Brin, Gates, etc. achieved their station in life not through studying mathematical proofs.
See my response to D_Malik and to shminux. I'm not making an argument, I'm just presenting a perspective, and some evidence, intended as food for thought.
There are so few people who do publishable TCS research as college sophomores that it's very unlikely that the wealthiest person in the world is one of them by chance.
Of course not by chance: there is a common cause -- high IQ. This tells you that high IQ is very useful and can be used both for CS and for business success. However this tells you nothing about the relationship between publishing CS papers and becoming a billionaire.
Bill Gates also famously dropped out of college. Does that warrant a Bayesian update, too? (Peter Thiel probably thinks so :-D)
I'm not making an argument,
You mean you are not offering a proof. But you are still asserting things. For example, this
...as a practical matter, intellectually gifted people who haven't developed very strong technical ability in a quantitative subject tend to be at a major disadvantage relative to those who have.
looks very much like a point you're trying to make.
Of course not by chance: there is a common cause -- high IQ.
Supposing his IQ to be at the 1 in 10k level (at the level of standard deviations, it's very unlikely that he's much higher than this, on priors). There are 400 such Americans of each age. What fraction do you think do publishable math or TCS research as college sophomores?
Bill Gates also famously dropped out of college. Does that warrant a Bayesian update, too? (Peter Thiel probably thinks so :-D)
Yeah, it's an update in the direction of it being a good idea for people of his demonstrated ability by that age and with his ambitions to drop out.
Yeah, it's an update in the direction of it being a good idea for people of his demonstrated ability by that age
Well, let's look at Mark Zuckerberg... and we're updating right back to where we started. And there is Sergey Brin...oh! we're now updating in a different direction?
Cherry-picking examples is not a good way to go.
Every example (Brin, Gates, Zuckerberg) should inform the implicit statistical model that we create: every time we learn about about one of them, we should update our model. If you don't do that, you're not fully utilizing the evidence available to you! ;-). Also, the model isn't just of "is this a good idea or isn't it?", what we're doing implicitly is determining probability distributions... And factors specific to individuals matter, the update is just of the type "all else being equal, this now looks to be a better idea."
Every example ... should inform the implicit statistical model that we create: every time we learn about about one of them, we should update our model. If you don't do that, you're not fully utilizing the evidence available to you!
This is a popular banner to fly at LW. I don't agree with it.
The problem is that "evidence available to us" is vast. We are incapable of using all of it to update our models of the world. We necessarily select evidence to be used for updating -- and herein lies the problem.
Unless your process of selecting evidence for updating is explicit, transparent, and understood, you run a very high risk of falling prey to some variety of selection bias. And if the evidence you picked is biased, so would be your model.
There is a well-known result of an experiment which asks people to name some random numbers. To the surprise of no one at LW, the numbers people name are not very random. In exactly the same way you may think that you're updating on a randomly selected pieces of evidence and that the randomness should protect your from bias. I am afraid that doesn't work.
I would update on evidence which I have reason to believe is representative. Updating on cherry-picked (even unconsciously) examples is worse than useless.
Ok, so I have to put more work in to externalizing my intuitions, which will probably take dozens of blog posts. It's not as though I haven't considered your points: again, I've thought about these things for 10,000+ hours :-). Thanks for helping me to understand where you're coming from.
I'm a bit confused about the point that you are trying to make here. As far as I can see there is nothing in this about social class in the traditional meaning of the phrase. It's about your view that people who study quantitative subjects (rather than poor benighted arts students like me) do better in "business", make more money, and become more successful.
You've cited some examples of people who, it is undeniable, are successful, but who also happen to fit your argument. But equally there are many successful businesspeople who did not study maths/CS/physics (John Paulson, hedge fund manager, started NYU doing film studies) and there are many examples of people with qualifications that you would probably argue show them to be intellectually gifted, who have completely failed in business (the example par excellence here is Long Term Capital Management, stuffed full of PhDs from top schools).
A key in this whole discussion is to define success. If you are just using money to keep score, then consider Tom Cruise, who didn't attend any university and is worth around half a bill.
You've cited some examples of people who, it is undeniable, are successful, but who also happen to fit your argument. But equally there are many successful businesspeople who did not study maths/CS/physics
If only there were some way of quantifying this.
I'm a bit confused about the point that you are trying to make here.
I'm not making a point. I do have responses to some of what you say, which I'll be writing about later.
You mean you are not offering a proof. But you are still asserting things.
What I wrote is vague. (What do I mean by "tend" and major" and "disadvantage"?) It's very hard to convey quantitative effect sizes in words. I'm pointing at a phenomenon – the particulars of how I'm pointing to it are not of the essence. Bruce Lee said:
It is like a finger pointing a way to the moon. Don't concentrate on the finger or you will miss all that heavenly glory.
If you doubt that there's anything to what I'm saying then we can talk about that.
It's very hard to convey quantitative effect sizes in words.
Huh? No, it's not hard at all. Besides it's not like something prohibits you from using, y'know, numbers on LW.
What I wrote is vague ... there's anything to what I'm saying
Hold on. You're a smart guy and there are a bunch of smart people on LW. You wrote a top-level post which implies that you want to communicate something and, moreover, you think it's important. What's all this backsliding into vagueness and the very very low bar of "anything"?
As far as I can read you, you are saying that studying math, in particular among the peer group of mathematicians, is desirable for high-IQ people. You implied two reasons for that: it might lead to riches (and so you mentioned Brin, Gates, Munger, etc.), and it might lead to awesome internal experience and, to use a Maslowian term, self-actualization (and so in the follow-up comments you quoted e.g. Grothendieck).
Whether math is a good way to achieve wealth or internal rapture is debatable, but your position seemed to be fairly defined. Why are you backing away from it now?
Huh? No, it's not hard at all. Besides it's not like something prohibits you from using, y'know, numbers on LW.
It's hard to convey the quantitative effect sizes as encoded in our intuitions: they're not stored in our brains as numbers.
What's all this backsliding into vagueness and the very very low bar of "anything"?
:D. The threshold that I'm trying to clear is "get readers to think seriously about whether or not developing strong proficiency with a quantitative subject would be good for those intellectually gifted people who they know (possibly including themselves), based on the considerations that I raise.
I have no idea whether you should try to develop strong proficiency in a quantitative subject: there's so much that I don't know about you and your situation, so I'm not going to try to make an argument on that front. What I have to say is actionable only with a lot of individual-specific context that I don't have. I'm trying to present information that individuals can use to help them make decisions.
quantitative effect sizes as encoded in our intuitions
I don't understand what that means.
get readers to think seriously about whether or not developing strong proficiency with a quantitative subject would be good for those intellectually gifted people who they know (possibly including themselves), based on the considerations that I raise.
Cool. That's an entirely reasonable position which can be discussed. Now these "considerations" that you raise, can you make them more coherent and explicit as well? Then the discussion can proceed about whether they are valid, whether there are more considerations which support them or, maybe, counterbalance them, etc.
I don't understand what that means.
I mean, e.g. that your brain doesn't have a numerical answer to the question "What's the probability that I'll get into a car crash if I drive to work tomorrow morning?" - it has information that can be used to derive a numerical answer, but the number itself isn't there.
Yes, I need to make the considerations more coherent and explicit. Thanks for the feedback.
Is there any particular reason to focus on anecdotes and not focus on the percentage of millionaires who have specific degrees?
http://time100.time.com/2013/11/11/these-are-the-most-popular-college-degrees-earned-by-millionaires/ gives for example a list.
Agreed that statistics > anecdotes. That said, the list here leaves me wondering about the direction of causation. I'm less interested in current-net-worth than average annual-change-in-net-worth (both % and $) in the years since graduation.
I don't think this list tells you much.
The most obvious reason why the list might look this way is that there are simply more people getting MBAs or engineering degrees. So of course there would be more of those becoming millionaires than there are math students etc.
I'm also not sure if the study actually measured how many people with a given degree actually got this level of wealth by themselves or if they also asked those who had wealthy parents. I don't have statistics on this but it does sound reasonable (not to say that it necessarily is) that wealthy parents try to direct their children towards those degrees which are 'useful' - in their opinion - in a business environment. Since most of these people have a very poor concept of what math actually is, math probably won't be among that class.
People who study math, theoretical physics or theoretical computer science are also much more likely to stay in academia than do those who studied engineering etc.
For all I know those people who did become millionaires by themselves who got the degrees they got might actually have done much better if they had studied math or theoretical physics.
So a statistic which would really be telling here would be one which measured how many of those getting a certain degree and going into the market are becoming rich.
First, I'd predict that much of the observed correlation between technical proficiency and wealth is just because both of them require some innate smarts. In general, I'm suspicious of claims that some field develops "transferable reasoning abilities", partly because people keep using that to rationalize their fiction-reading or game-playing or useless college degrees. I'm worried that math and physics and theoretical CS are just nerd-snipery / intellectual porn, and we're trying to justify spending time on them by pretending they're in line with our "higher" values (like improving the world), not only with our "lower" values (like intellectual enjoyment).
Second, if technical proficiency does build transferable reasoning ability, I'd expect the overall benefit to be small, much smaller than from, say, spending that time working on whatever contributes most to your goals (which will usually not be building technical proficiency, because the space of all actions is big). You should always be trying to take the optimal action, not a random "beneficial" action, or else you'll spend your time mowing lawns for $10/hour.
Edit: I think this comment is too hostile. Sorry. I do agree that learning technical skills is often worthwhile.
I'm worried that math and physics and theoretical CS are just nerd-snipery
No way, especially not physics. We as a civ need to do more of this stuff, not less, compared to what we are doing now.
I can't think of any category of human activity that did more to improving the world than the hard sciences. Maaaaybe some religions in the "convince people to stop killing each other and cooperate long enough to get science off the ground" sense.
Hehe, it's rare that intellectual satisfaction is called a 'low' value.
Yes, there's also the theory that success in hard math/science fields are strong signals of intellect, and thus smart people flock to them to signal their intelligence.
As someone about to start a CS degree and figuring out what to do with his life, this is a sobering line of thought.
My view is that some degree of technical facility helps a lot. As I recently wrote, I think that learning to read very carefully and not make unwarranted assumptions is a very important skill, and one way to get it is by studying proof-based math. I don't have strong views on how much studying pure math and TCS help after the first 1-2 years. I think that the case for learning advanced statistics & machine learning is much stronger.
Separately, I benefited a huge amount from reading and interacting with elite mathematicians. Even though they weren't thinking about the things that I'm doing now directly, I was able to transfer what I had learned from them to the things that I'm currently focused on. That's the peer group effect.
Graduates of elite law schools are mostly "intellectually gifted people who haven't developed very strong technical ability." How do you think most of them would achieve better life outcomes by developing such an ability? Keep in mind that most of them are relatively much better at verbal and social stuff than math.
Yes, that might be a good nontechnical counterpart. I grew up in San Francisco and live in the Bay Area where the wealthy / privileged people tend to be in tech, and my perceptions are correspondingly skewed. I added a footnote linking your comment.
You could include most branches of engineering as well, not just electrical. Or at the very least certain subsets of these branches, e.g., fluid dynamicists in mechanical and aerospace engineering tend to be no different intellectually from physicists, in my experience. (Probably largely because they are basically physicists. Physics as a field has neglected classical mechanics lately, so if fluids are your interest, you won't get a degree in physics.)
Yeah, the reason why I didn't include engineering is that it seems that most engineers are in administrative-like roles in practice, but I'd definitely include the ones who do research and development in the reference class that I had in mind.
Well law includes a lot of logic, in the form of verbal reasoning. Bertrand Russell formalised how math can be reduced to logic...kinda. I pray for the haters of a lawyer who can substantiate his reasoning with hard data. Imagine a lawyer who can substantiate an argument for legalising assisted suicide by referring to quality adjusted life years and grounding his/her reasoning in some kind of felsific calculus.
On second thoughts, the poor reception that hedonic calculus (according to that one Charles Dickins book at least) got from the world is evidence against this line of thinking.
:/
edit 1: evidence that lawyers aren't great at quantitative evidence and by extension, some applications of logic.
I can't give a brief justification for this, but I have good reason to believe that the ~10000x+ differential in net worth comes in large part from the people having had unusually good opportunities to conducive to becoming very technically proficient, that resulted in them developing transferable reasoning abilities and having had an intellectually elite peer group to learn from.
Either it's a hunch, or you should try to provide the justification.
My counterargument to this claim is summed up by the Dilbert cartoon. For every Gates/Simon/Brin, there are hundreds of technically adept but socially naive Dilberts who are working for dumb pointy-haired bosses.
Bill Gates took Math 55 as a freshman at Harvard, which is the class designed for the most mathematically talented students at Harvard. During his sophomore year he did research which he later published a paper on with well known theoretical computer science professor Christos Papadimitriou.
Bill Gates's father was an antitrust lawyer (focusing on the corporate, rather than government, side), and Gates is proficient at poker--the calculational, strategic, and human elements. Those seem stronger influences on his being able to create a monopolistic software juggernaut and get the better of his counterparties in several critical deals.
As Leo Atreides puts it, "A Mentat Duke would be formidable indeed." But merely being a mentat is not enough! One must also train to be a duke, which is likely the harder part to convince this crowd of.
Sure, that's plausible. Gates had other advantages as well. But even if those do dominate, his involvement in math is still (weak) Bayesian evidence for my claim.
There are so few people who do publishable TCS research as college sophomores that it's very unlikely that the wealthiest person in the world is one of them by chance :D. I acknowledge that the correlation may be entirely spurious, but it still warrants a Bayesian update.
But even if those do dominate, his involvement in math is still (weak) Bayesian evidence for my claim.
Can you quantify what you mean with "weak"?
Weak bayesian evidence E is something which you can reasonable expect to find given either hypothesis (e.g. "math is useful" vs "math is useless"), but nevertheless results in P(H|E)/P(~H|E) > P(H)/P(~H).
Strong bayesian evidence would pretty much kill the alternative hypothesis, i.e. P(~H|E) ~ 0.
The world has more than just two categories. It's useful to know whether we talk about updating by 1% or 0.001%.
Certainly this is true for some "theoretically gifted" people. But others do indeed have little to no "technical ability". The "Grothendieck prime" is an extreme example of it, but there are many accounts of the genius theorists told to stay out of the labs, or else.
It is worthwhile for any "intellectually gifted people who haven't developed very strong technical ability in a quantitative subject" to give it an honest try, but not to be too surprised if they find themselves lacking.
there are many accounts of the genius theorists told to stay out of the labs, or else.
What do you mean by "technical ability" ? I intended to include everything in math / physics / TCS / statistics under that umbrella, not just computation / engineering.
The most significant thing to my mind is actually the peer group – even if one doesn't become quantitatively sophisticated oneself, just being around people who are can make a big difference.
What do you mean by "technical ability" ?
Hmm, I guess I don't understand what you meant by that. What is not a technical ability in your model?
What is not a technical ability in your model?
I think that I erred in using the word "technical" at all. What I had in mind was "the academic subjects that are referred to as 'technical'", but I already used "quantitative," which covers that.
Sorry the confusion :P.
I think you mean income, not social class. An academic mathematician might have higher social status but lower income than they would as a startup founder, for example.
Social status is relative to a particular subculture. In certain circles startup founders have higher status than math professors.
I can't give a brief justification for this, but I have good reason to believe that the ~10000x+ differential in net worth comes in large part from the people having had unusually good opportunities to conducive to becoming very technically proficient
I would think this would be obvious. Social class and life situations are a big freaking deal.
I just wanted to mention that your last few posts have been very nice. You have clearly worked on exposition, and it has paid off.
I became interested in this sequence after your first post on mathematical ability, and I'm glad to see you factoring your ideas into digestible pieces and writing them up.
My impression significantly differs, though I'm far from confident. I'd be interested in seeing an expanded version of this point because it seems potentially very valuable to me.
If your claim about the difference between having training in a mathematical field and not having it or are you talking about the difference between people with strong achievements in those domains vs. those that don't have them?
Sorry, this was an useless post so now it's gone
My view is that some degree of technical facility helps a lot. As I recently wrote, I think that learning to read very carefully and not make unwarranted assumptions is a very important skill, and one way to get it is by studying proof-based math. I don't have strong views on how much studying pure math and TCS help after the first 1-2 years. I think that the case for learning advanced statistics & machine learning is much stronger.
Separately, I benefited a huge amount from reading and interacting with elite mathematicians. Even though they weren't thinking about the things that I'm doing now directly, I was able to transfer what I had learned from them to the things that I'm currently focused on. That's the peer group effect.
Something that I've come to realize is that as a practical matter, intellectually gifted people who haven't developed very strong ability in a quantitative subject tend to be at a major disadvantage relative to those who have. The quantitative subjects that I have in mind as "quantitative subjects" are primarily math, physics, theoretical computer science and statistics, though others such as electrical engineering may qualify. [1]
This point is usually masked over by the fact that people who don't have very strong technical ability are often reasonable functional by the standards of mainstream society, and don't realize how far they're falling short of their genetic potential. They tend to have jobs that don't fully use their strengths, and experience cognitive dissonance around being aware on some level of far they are from utilizing their core competencies.
Consider the following:
I can't give a brief justification for this, but I have good reason to believe that the ~10000x+ differential in net worth comes in large part from the people having had unusually good opportunities to conducive to becoming very technically proficient, that resulted in them developing transferable reasoning abilities and having had an intellectually elite peer group to learn from.
I know a fair number of brilliant people who didn't have such advantages. The situation actually seems to me like one in which amongst intellectually gifted people, there's an "upperclass" of people who had opportunities to develop very strong technical ability and an "underclass" of people who who could have developed them under more favorable environmental circumstances, but haven't. Many intellectually gifted people who didn't have the chance to develop the abilities mistakenly believe that they lack the innate ability to do so. And people who did have the opportunities to develop them often look down on those who didn't, unaware of how much of their own relative success is due to having had environmental advantages earlier in their lives.
[1] James Miller points out that graduates of elite law schools may have analogous advantages – that's a population that I haven't had exposure to.