I went a similar path (doing physics but not really excelling at it) and ended up a programmer. I'm pretty happy with programming overall. Note that in real-world applications, most of the effort goes into the engineering-like side of making sure your code is clean and maintainable, rather than the comp-sci-like side of having clever data structures and algorithms. It certainly doesn't feel "too easy" most days, though it can sometimes be frustrating when you end up spending time struggling with tools rather than what you're really trying to do.
Perhaps I should've said, hard in the wrong ways. The long term goal for a good professional programmer seems to be understanding what the client wants. Some math is needed to understand the tools, so you can give some context for options. But I spend most of my creative energy making sure my programs do what I want them to do, and that is really hard when each language has it's own prejudice motivating its design.
I seriously considered looking into real time high risk software applications. But I just decided that instead of learning new languages until I ran out of youth, it'd be more fun learning general relativity, or even measure theory. The ideas in those subjects will probably hold out a lot longer then python.
It's not clear to me from your post if you have any specific terminal-like goals in mind. Do you want to just "do research"? Or to teach? Or to "do good"? Or what?
My between-the-lines impression of the post was that this is the (subconscious?) lingering question.
I'm not sure. I'm trying to work towards a career path which uses as much of my ability as I can. The most important job for a professional programmer, was understanding what your client wanted. This is a fine job, but being good at algorithms isn't necessarily a requirement.
When talking to an engineer at Google, I asked what he thought a good career choice was for working on hard problems. His immediate first thought was graduate school, then he sort of mentioned robotics.
My ideal dream isn't being a professor, it's working on something that needs inference, that uses my mathematical abilities. So I'm leaning towards research, but that's the implication not necessarily the goal.
Teaching isn't the goal, hands on altruism isn't the goal. Fitting into a place where I'm using as much of my skill set as possible, is the goal.
And that is a terminal goal, I can do boring stuff in the mean time. My point for jumping out of programming, was exactly that the math wasn't the important part, it was the picture. The math is important to someone else. I'd like to be that someone else.
I try to explain this to people though, and almost all of them think I'm being way to vague (or they don't understand). You go to school because that's the only way you're going to study the distribution of zeroes for the Wronskian of orthogonal polynomials. I've had maybe one professor discourage me from being too picky...
Fitting into a place where I'm using as much of my skill set as possible, is the goal.
This is one of the harder problems out there, in my experience. Many extremely intelligent people spin their wheels on this one for years. Some indefinitely.
Especially for a person who is talented at inferential or analytical problem-solving, looking for acceptable institutions first may be a case of putting the cart before the horse. Those places- the universities, research groups, etc.- tend to be looking for researchers that think of the institution as a tool, not as a goal. This is at least partly because it's very hard to quantify successful research, and they're looking for assurances that work will continue. If you value 'being a member of the Center for Advanced Studies', then you can succeed at that goal without actually doing any work.
Imagine a dissertation for yourself, the multi-year project that you would be working on during your time in the Platonic University where everyone is accepted and everything is perfect and you study exactly what you want to study. What is that project? What can you do, right now, to pursue that project? What factors will get in your way, and what steps do you need to take to minimize or eliminate those factors? If you can't picture one just yet, that's fine. Talk with your professors if you want; not about grades but about what their own research is, and why they care about it. Ask a lot of professors about that; they almost always think that their work is important, and are happy to describe it. Get a sense of the conversation as it currently exists, and then find a niche that interests you.
In other words, don't look for a pleasant societal position as its own end- just do the thing, become more skilled in thing-doing, and grab help when you need it. The more you do the thing, the more you will quite naturally build a positive reputation among a widening network, and this will grant you access to a surprising number of institutions as you need them. But at the end of the day, it is the privilege of the researcher to encounter a reality that is not built by social consensus. It doesn't make a whole lot of sense to break your back accumulating social status markers like grades and test scores just to convince somebody really respectable to tell you what questions to ask.
intelligent enough to make a meaningful contribution
If you're doing it to make a "meaningful contribution", not for fun, it can be hard to stay motivated without outside assurances that you're doing "well".
throw in the towel? grow up?
It's hard not to identify as a child prodigy anymore. If you want to do something, there are things to do to increase your odds of success. For example, if you want to do really well on the Math GRE, do practice problems until you know all the concepts and get the score you want on practice tests. Unfortunately, this takes a lot of focus over a long time. If you want to make a meaningful contribution, look at Mark Andreessen's advice: get pretty good at two separate areas/fields, then do something that uses both of them. The other option, being the best at one specific area, requires competing with people who enjoy studying so much it damages their health. Good luck.
This type of question is difficult to answer here because my answer necessarily relies mostly on relatively non-academic sources, largely personal experience, and it's hard to discuss this topic for the same reasons it's hard to discuss politics.
Have you used any career counseling? If you are an effective altruist or are just interested in optimizing your career along specific criteria there is 80,000 hours. That link has a list of common choices for people in your demographic (intelligent, good grasp of math). They also have more personalized advising I believe.
Pure IQ isn't as important for success as you might think. See e.g. Wired: Which Traits Predict Success? (The Importance of Grit)
If you read Duckworth's study closely, IQ matters somewhat more than conscientiousness for income. And grit and conscientiousness are closely correlated.
How old are you? 78% percentile of what demographic?
But more importantly, why are you in mathematics in the first place? What's your motivation for being there? From an effective altruism standpoint I don't see academic math or physics a very valuable field.
I think it's much better to take the skills you learned in math and then go into some field like biology, engineering or programming and actually apply the knowledge to important real world problems.
That said when it comes again to math skills I'm not sure whether tests measure the right thing. Contributions to math as a field are not done in a few hours. They are done by working focused on a problem for days, weeks and months. That needs certain skills.
There also the model that advancing the field is basically about knowing a bunch of techniques and a bunch of problems. If you are the only person who knows a certain mathematical technique and a certain mathematical problem that's solvable with that technique you advance the field.
That means it's very much about your ability to spend a lot of time learning complex mathematical techniques. It's more about endurance than about solving a bunch of problems in the time span of a test.
I've already "given up" once before and tried programming, but the average actual problem was too easy relative to the intellectual work (memorizing technical fluuf).
Don't make the mistake of assuming that the work you do when you learn a field is the same as the job in the field.
While university classes on programming might involve memorizing fluuf that's not about what working as a programmer is about. Trying to understand how baldy documented legacy code or API work is something different than solving mathematical problems but it's also a highly intellectual challenge. Designing software architecture is also highly intellectual.
I think it's much better to take the skills you learned in math and then go into some field like biology, engineering or programming and actually apply the knowledge to important real world problems.
+1
For example, psychology is too important to be left to the sort of people who typically become psychologists in my opinion.
If you have a few hundred dollars (or it's covered by insurance) you could get an IQ test from a qualified professional. Your college might even offer this service for students attempting to see if they have learning disabilities.
And the compromise is to study economics. The academic market for economists with Phds is great.
It seems unlikely that an IQ test gives very much extra information about polymer's prospects as a mathematician or physicist, on top of his/her experiences and scores studying mathematics and physics at university.
I agree, that I have a wealth of information to work with right now. Just trying to honestly balance it (felt like LW fit the theme somewhat).
On the one hand, both of those scores are my first time, and they were taken cold. And, I could argue I thought a lot of homework in school was unimportant and unnecessary (because of a poor philosophical attitude).
But of the 26 questions wrong or incomplete on the practice Math subject test, roughly 16 of them I had sufficient knowledge, but I simply wasn't fast enough. And the Algebra class, was really hard, and I did do homework eventually.
It's not like I haven't been very successful in some courses. Graduate Complex Analysis, and stochastic processes come to mind. And the admissions director at my undergrad (University of Oregon) has told me directly I am ready for graduate school, but he would prefer I went to a better school.
I'm just lost. It seems in this context, failure speaks louder then success. Even if I was smart enough, perhaps I simply haven't worked hard enough (or on the right things). The practical consequence would be the same. I wish I knew what the admissions officer saw, it's hard to suppress the feeling he's only saying that because I did well in his courses.
A few disorganized remarks that may or may not be any help:
(My own background: got the PhD, did a couple of years of postdoc, was quite staggeringly unproductive, got out of academia and into industry, have been reasonably happy there. Probably happier than I'd have been as a struggling academic. Most academics are struggling academics, especially for, say, the first 5-10 years after getting their PhDs.)
Some questions you might want to answer for yourself:
if you're good at analysis and not so good at algebra
Then he might also want to consider applied mathematics programs, especially if he also excels at programming and engineering but feels they're too easy.
So, my point regarding the speed.
In the middle of working out a problem, I had to find the limit of
S = 1/e + 2/e^2 + ... + n/e^n + ...
I had never seen this sum before, so now cleverness is required. If I assumed guess C was true, that would imply
e/(e - 1) = (e - 1)S
This claim is much easier to check,
(e - 1)S = 1 + 1/e + 1/e^2 + ... = 1/(1 - 1/e) = e/(e - 1)
We know what S is, and the solution to the problem follows. In retrospect, I understand one method for how I could find the answer. But during the test, I can't see through the noise fast enough (although I can smell the clue). I could go through each guess one by one, but I'm just too slow. Maybe there's something else I'm missing that would've made the guess simpler, but that's what I'm basing the slow opinion off of.
I don't know if being slow at inference in this sense is a barrier, or indicative, of deeper creativity issues (or if I'm just suffering from the availability heuristic.)
Anyways your questions all very good, I don't care for academia perse, I care about the questions. If I don't keep doing academic stuff, I would hope I would've formed enough connections to find some route towards practical problems that still require some creativity.
Your last question is very interesting. I'm not sure how to answer it. My unhealthy worry, I think, is I really don't like wasting peoples time. I suppose I don't care about either being "just OK", if "just OK" isn't wasting peoples time, but I still get to be creative.
I guess I don't want to be a pundit? I mean I'll teach, but I'd be much happier if I was doing something theoretically. If this is impossible for me, I'd like to know the reasons why, and fail out as soon as possible.
Your questions are very interesting though, I still need to think about them more. Thank you for your thoughts, they give very good context to think about this, and its clear you've worried about analogous issues.
Different people are good at different things. In particular, the algebra/analysis dichotomy is a pretty standard one and if you're good at analysis and not so good at algebra, it probably matters how good you are at what you're best at.
(I've heard people talk of branches of maths the way gender essentialists such as EY or Ozy Frantz would talk of gender identity.)
One of my pet theories is that math and (applied) statistics require very different brains. People whose brains are wired for math make poor (applied) statisticians and people who are really good at stats tend to be poor at math.
This is partly an empirical observation and partly, I think, is a consequence of the fact that math deals with "hard" objects (e.g. numbers) that might not be known at the time, but they are not going to mutate and change on you, while statistics deals with uncertainty and "soft"/fuzzy/nebulous objects (e.g. estimates). Moreover, for applied statistics the underlying processes are rarely stable and do mutate...
You took the GREs cold. I'm surprised you did half as well as you did. Why? Because anyone who is not mentally handicaped can pay tutor a large sum of money, do exactly what the tutor says, and get a perfect score. I'm not exagerating -- I have friends who tutor in this business, and every year they sit for the GRE as a requirement for their job, and get a perfect score. It's a teachable skill, and one which has very little to do with the subject matter.
Now consider that most of the other people who took the GRE knew about this weakness. Especially internationally in places like China and India where (1) there are a lot of test takers, (2) a much larger test prep industry, and (3) massive incentives to do well (so as to get into an American or European university). Now keeping all that in mind, you still scored better than 72 / 68 percent of the competition despite having absolutely no preparation whatsoever.
Why are you not congratulating yourself?
I'm not convinced this is a good argument. You're certainly over-stating how teachable the GRE is, and I have a least anecdotal evidence of lots of people who scored above 90% on the general GRE quantitative section without tutors. This includes at least one person who "took it cold." Maybe those are super exceptional folks, but I think the statement that most of the people scoring in the top 30% had tutors is a really strong statement. I have worked for a test prep agency before and there aren't a lot of top tier students in those classes, and indeed the courses and techniques really geared towards the bottom/middle-tier students. Also, you can't do well on the GRE, especially the subject tests, without knowing the subject matter.
Your argument is plausible, but it's all conjecture. I'm curious as to whether you think the GRE percentages mean anything at all, and if so, what the 'adjustment' for taking it cold should be,
on the practice Math subject test, ... I simply wasn't fast enough
Here's some heterodox advice: Take stimulants. Before I wrote the SAT, I stayed off caffeine for a couple of weeks. Then I drank lots of coffee right before the test, and in the break between sections. Caffeine affects me very strongly, so you can use some other stimulant. It might shorten your life by a week, but it's probably worth it.
If you're not used to using stimulants, including ones as common as caffeine, don't experiment for the first time on test-day. My cousin had to get a dean's excuse to postpone a test when he (who never/rarely drank coffee) drank way to much while studying and wound up trembling too hard to hold a pen.
And the compromise is to study economics.
A similar strategy worked successfully for my friend. As a student, he was very enthusiastic about math and programming, but a big part of that was influence of his friends, me included. Later he saw evidence that he was in these topics, let's say, above average, but not great enough. (He would be able to write simple programs, and he would get the job, but the more complex parts would be too abstract for him.) He tried studying informatics, and dropped out.
So he switched to economics, choosing some study that also included maths. Hanging out with different kinds of people he discovered he had good social skills (he didn't notice that while hanging out with nerds). These days he is a consultant, and his specialization could be described as applying database tools to examine or improve economical stats of companies. (Imagine a huge company which has a lot of data in dozen different systems, including Excel sheets; those systems are not connected, they don't even use a similar structure, and the company actually doesn't even know which divisions or products are profitable. So my friend comes, and uses different tools to connect all those data sources together, and then creates easy-to-read reports. Which is not as easy as it seems, because those data sources describe the data differently, so he must examine the underlying territory to understand what can be connected with what. Also he must reduce all the available information into cca seven very simple graphs, so that even the most stupid managers could understand that easily.) So, he has some IT things there, enough to make him feel happy for living his dream of working in IT, but no lambda calculus or anything like that. On the other hand, travelling and debating with clients is okay for his extraverted nature.
If you have a few hundred dollars (or it's covered by insurance) you could get an IQ test from a qualified professional. Your college might even offer this service for students attempting to see if they have learning disabilities.
Btw, is an IQ test by Mensa usually as trustworthy and worthwhile than a similar test by a psychologist?
Mensa tests are administered by psychologists. At least in Germany that is, but I assume Mensa has standards on this.
Is tutoring a low-income student, or someone in prison, or someone with disabilities, or someone new to your country (each having few high-caliber resources) "a meaningful contribution to math or physics" to you?
Have you done mathematics research? It's pretty different from taking a class.
but the average actual problem was too easy relative to the intellectual work (memorizing technical fluff)
I'm curious whether this is true of all non-academic professions. It's even true of some academic professions. Probably not math though.
I wouldn't rule out professions that have a lot of technical fluff, it is pretty ubiquitous and you will rule out a lot of good options.
If you face the challenge of finding purpose in life, then maybe this site (which I found on the map of the rationalist community might help you.
Who is doing the type of work that you want to be doing right now, and what would it take to work with them? If you don't have an answer to that question, I suggest that you look into it.
If you go to grad school, focus on finding the right person, (your supervisor.) It's not as important which department, or program or school you go to.
I wouldn't base any major life decisions on the results of a standardized test. Even if it was measuring intelligence and aptitude well, (which it is not,) there are other things that are more important.
Giving up seems like a really bizarre thing to do, given what you've posted. What appeals to you about the idea of giving up? What would be the benefits of doing so?
High mate, I've just finished reading "David and Goliath" and my impression is that you have low self-esteem approximately because your classmates were doing slightly better than you. If you had chosen university/college with lower requirements you would be more motivated to do science because you would be on top of the leaderboard, shame that you are on crossroads now, but it is your way to go
I think you may be over-estimating the intelligence required to be a physicist. I don't know what constitutes a meaningful contribution to physics, but there are certainly productive tenured professors who are not in the top 25% of quantitative ability.
Also, if you're just motivated by interesting math problems, there are lots and lots of those in the world. Research is a lot less about interesting math problems than taking classes is. Research a lot of times is more about finding the right problem than just being able to solve it. It's not clear to me that raw quantitative ability or even IQ is particularly well-correlated with the ability to ask the right questions.
Also, if you are primarily interested in solving problems, your grades may not be quite as good as people who are primarily interested in getting good grades. It's usually possible to turn in a superficially good, but deeply flawed solution/derivation that gets significant partial credit. This is usually much easier that a full solution, but worth almost the same amount of points.
I think you may be over-estimating the intelligence required to be a physicist. I don't know what constitutes a meaningful contribution to physics, but there are certainly productive tenured professors who are not in the top 25% of quantitative ability.
This is not true. According to Kaufman, Alan S. (2009). IQ Testing 101. New York: Springer Publishing MDs, JDs and PhDs have an average IQ of 125+. PhDs in Physics are going to have higher quantitative scores than that. I regret that being behind the Great Firewall I can't source this but Steve Hsu wrote on his blog, infoproc.blogspot.com that he thinks the average professor of Physics is 1 in 100,000 in intelligence. I may be misremembering badly and the 1 in 100,000 could be anything from has a doctorate in Physics to is an outstanding contributor to Physics.
The idea that there are tenured professors of Physics who are just barely in the top quartile of quantitaive IQ is mindboggling. There will be few enough professors of English with quantitative IQs that low.
Is this the post you're looking to cite? From what I've been able to tell looking at that post and some related sources physicists seem to be about 2 standard deviations above the norm on average, with an uncertainty in that average measurement of a couple IQ points. This makes a physicist something like 1 in 50, rather than 1 in 100,000. Unfortunately it's not really clear what the standard deviation is for those numbers. An average, by itself tells you very little about a distribution. If we assume that the standard deviation of IQ scores of physics Ph.D.s is close to that of the rest of the population (this may be a bad assumption) then we expect 10% of physics doctorate holders to have an IQ of 110 or less, which would put them outside of the top quartile of the IQ distribution. You could play around with the parameters for the distribution here and come up with a bunch of different results, and I'm not totally sure which ones are meaningful.
I may be wrong. I may be framing the problem wrong. I may not have a good picture of just how bad merely average quantitative skills are. I do work with a lot of people who have Ph.Ds in physics and while some of them certainly have remarkable quantitative skills, others really don't. I know people who have gotten doctorates in physics who didn't score particularly well on the GRE. This is especially true in experimental physics (in physics the standard joke is that as a theorist you don't have to study anything that exists in the real world, and as an experimentalist you don't have to get the math right). Physicists are pretty smart people, but they're not all 1 in 100k and that's certainly not a requirement for being a physicist. Also, I think the level of quantitative ability necessary for physics research is often below the level needed to muddle through the coursework.
Be sure to make some choice.
I've thought for a while about what I could learn from my own experiences after graduating and so far the main broadly applicable thing is that whenever you make a big choice, you should discuss it in detail with other people and ask their opinions. That helps avoid bias.
So what are your other options?
I'm sorry if this is the wrong place for this, but I'm kind of trying to find a turning point in my life.
I've been told repeatedly that I have a talent for math, or science (by qualified people). And I seem to be intelligent enough to understand large parts of math and physics. But I don't know if I'm intelligent enough to make a meaningful contribution to math or physics.
Lately I've been particularly sad, since my score on the quantitative general GRE, and potentially, the Math subject test aren't "outstanding". They are certainly okay (official 78 percentile, unofficial 68 percentile respectively). But that is "barely qualified" for a top 50 math program.
Given that I think these scores are likely correlated with my IQ (they seem to roughly predict my GPA so far 3.5, math and physics major), I worry that I'm getting clues that maybe I should "give up".
This would be painful for me to accept if true, I care very deeply about inference and nature. It would be nice if I could have a job in this, but the standard career path seems to be telling me "maybe?"
When do you throw in the towel? How do you measure your own intelligence? I've already "given up" once before and tried programming, but the average actual problem was too easy relative to the intellectual work (memorizing technical fluuf). And other engineering disciplines seem similar. Is there a compromise somewhere, or do I just need to grow up?
classes:
For what it's worth, the classes I've taken include Real and Complex Analysis, Algebra, Differential geometry, Quantum Mechanics, Mechanics, and others. And most of my GPA is burned by Algebra and 3rd term Quantum specifically. But part of my worry, is that somebody who is going to do well, would never get burned by courses like this. But I'm not really sure. It seems like one should fail sometimes, but rarely standard assessments.
Edit:
Thank you all for your thoughts, you are a very warm community. I'll give more specific thoughts tomorrow. For what it's worth, I'll be 24 next month.
Double Edit:
Thank you all for your thoughts and suggestions. I think I will tentatively work towards an applied Mathematics PHD. It isn't so important that the school you get into is in the top ten, and there will be lots of opportunities to work on a variety of interesting important problems (throughout my life). Plus, after the PHD, transitioning into industry can be reasonably easy. It seems to make a fair bit of sense given my interests, background, and ability.