Most people don’t need to understand evolution. Maybe we should distinguish between “harmful to self”, “harmful to society”, and “harmful to a democratic society”.
If you can’t do math at a fairly advanced level—at least having competence with information theory, probability, statistics, and calculus—you can’t understand the world beyond what’s visible on its (metaphorical) surface.
I’ve got a tangential question: what math, if learned by more people, would give the biggest improvement in understanding for the effort put into learning it?
Take calculus, for example. It’s great stuff if you want to talk about rates of change, or understand anything involving physics. There’s the benefit; how about the cost? Most people who learn it have a very hard time doing so, and they’re already well above average in mathematical ability. So, the benefit mostly relates to understanding physics, and the cost is fairly high for most people.
Compare this with learning basic probability and statistical thinking. I’m not necessarily talking about learning anything in depth, but people should have at least some exposure to ideas like probability distributions, variance, normal distributions and how they arise, and basic design of experiments—blinding, controlling for variables, and so on. This should be a lot easier to learn than calculus, and it would give insight into things that apply to more people.
I’ll give a concrete example: racism. Typical racist statements, like “black people are lazy and untrustworthy,” couldn’t possibly be true in more than a statistical sense, and obviously a statistical statement about a large group doesn’t apply to every member of that group—there’s plenty of variance to take into account. Basic statistical thinking makes racist bigotry sound preposterously silly, like someone claiming that the earth is flat. This also applies to every other form of irrational bigotry that I can think of off the top of my head.
Remember when Larry Summers suggested that maybe part of the reason for the underrepresentation of women in Harvard’s science faculty was that women may have lower variance in intelligence than men, and so are underrepresented in the highest part of the intelligence bell curve? What almost everybody heard was “Women can’t be scientists because they’re stupid.” People heard a statistical statement and had no idea how to understand it.
There are important, relevant subjects that people just can not understand without basic statistical thinking. I would like to see most people exposed to basic statistical thinking.
Are there any other kinds of math that offer high bang-for-the-buck, as far as learning difficulty goes? (I’ve always thought that the math behind computer programming was damn useful stuff, but the engineering students I’ve talked with usually find it harder than calculus, so maybe that’s not the best idea.)
(I’ve always thought that the math behind computer programming was damn useful stuff, but the engineering students I’ve talked with usually find it harder than calculus, so maybe that’s not the best idea.)
Tangential question to your tangential question: I’m puzzled, which math are you talking about here? The only math relevant to programming that I can think of that engineering students would also learn would be discrete math, but the extent needed for good programming competency is pretty small and easy to pick up.
Are we talking numerical computing instead, with optimization problems and approximating solutions to DE’s? That’s the only thing I can think of relevant to engineering for which the math background might be more difficult than calculus.
I was thinking more basic: induction, recursion, reasoning about trees. Understanding those things on an intuitive level is one of the main barriers that people face when they learn to program. It’s one thing to be able to solve problems out of a textbook involving induction or recursion, but another thing to learn them so well that they become obvious—and it’s that higher level of understanding that’s important if you want to actually use these concepts.
I’m not sure about all the details, but I believe that there was a small kerfuffle a few decades ago over a suggestion to change the apex of U.S. ``school mathematics″ from calculus to a sort of discrete math for programming course. I cannot remember what sort of topics were suggested though. I do remember having the impression that the debate was won by the pro-calculus camp fairly decisively—of course, we all see that school mathematics hasn’t changed much.
Calculus might not be the best example of a skill with relatively low payoff, because you need some calculus to understand what a continuous probability distribution is.
I do? I thought I understood both calculus and continuous probability but I didn’t know one relied on the other. You are probably right, sometimes things that are ‘obvious’ just don’t get remembered.
For example, suppose you have a biased coin which lands heads up with probability p. A probability distribution that represents your belief about p is usually a non-negative real function f on the unit interval whose integral is 1. Your credence in the proposition that p lies between 1⁄3 and 1⁄2 is the integral of f from 1⁄3 to 1⁄2.
Most people who learn it have a very hard time doing so, and they’re already well above average in mathematical ability.
Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud. You multiply by the number up to the top right of the letter then reduce that number by 1. Or you do the reverse in the reverse order. You know, like you put on your socks then your shoes but have to take off your shoes then take off your socks.
Sometimes drawing a picture helps prime an intuitive understanding of the physics. You start with a graph of velocity vs time. That is the ‘acceleration’. See… it is getting faster each second. Now, use a pencil and progressively color in under the line. that’s the distance that is getting covered. See how later on more when it is going faster more distance is being traveled at one time and we have to shade in more area? Now, remember how we can find the area of a triangle? Well, will you look at that… the maths came out the same!
has to do with the derivative as a rate of change or as a slope of a tangent line. And from the perspective of a calculus student who has gone through the standard run of American school math, I can understand. It does require a level up in mathematical sophistication.
from the perspective of a calculus student who has gone through the standard run of American school math,
That’s the problem. See that bunch of symbols? That isn’t the best way to teach stuff. It is like trying to teach them math while speaking a foreign language (even if technically we are saving the greek till next month). To teach that concept you start with the kind of picture I was previously describing, have them practice that till they get it then progress to diagrams that change once in the middle, etc.
Perhaps the students here were prepared differently but the average student started getting problems with calculus when it reached a point slightly beyond what you require for the basic physics we were talking about here. ie. they would be able to do 1. and but have no chance at all with 2:
I’m not claiming that working from the definition of derivative is the best way to present the topic. But it is certainly necessary to present the definition if the calculus is being taught in math course. Part of doing math is being rigorous. Doing derivatives without the definition is just calling on a black box.
On the other hand, once one has the intuition for the concept in hand through more tangible things like pictures, graphs, velociraptors, etc., the definition falls out so naturally that it ceases to be something which is memorized and is something that can be produced ``on the fly″.
Doing derivatives without the definition is just calling on a black box.
A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan “brought to you by apostrophe, the letters LIM and an arrow” but you can go on to tell them “this is how impressive mathematcians say you should write this stuff that you already understand” and they’ll get it.
I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn’t easy.
you can go on to tell them “this is how impressive mathematcians say you should write this stuff that you already understand” and they’ll get it
That is close to saying “this stuff is hard”. How about first showing the students the diagram that that definition is a direct transcription of, and then getting the formula from it?
(Actually, many would struggle with 1. due to difficulty with comprehension and abstract problem solving. They could handle the calculus but need someone to hold their hand with the actual thinking part. That’s what we really fail to teach effectively.)
Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud.
People get the simple concepts mixed together with a bunch of mathy-looking symbols and equations, and it all congeals into an undifferentiated mass of confusing math. Yes, I know calculus is actually pretty straightforward, but we’re probably not a representative sample. Talk with random bewildered college freshmen to combat sample bias. I did this, and what I learned is that most people have serious trouble learning calculus.
Now, if you want to be able to partially understand a bunch of physics stuff but you don’t necessarily need to be able to do the math, you could probably get away with a small subset of what people learn in calculus classes. If you learned about integration and differentiation (but not how to do them symbolically), as well as vectors, vector fields, and divergence and curl, then you could probably get more benefit-per-hour-of-study than if you went and learned calculus properly. It leaves a bad taste in my mouth, though.
what I learned is that most people have serious trouble learning calculus.
When taught well the calculus required for the sort of applications you mentioned is not something that causes significant trouble, certainly not compared to vector fields, divergence or curl. By taught well, if you will excuse my lack of seemly modesty, is how I taught it in my (extremely brief—don’t let me get started on what I think of western school systems) stint teaching high school physics. The biggest problem for people learning basic calculus is that people teaching it try to convey that it is hard.
I’m only talking here about the level of stuff required for everyday physics. Definitely not for the vast majority of calculus that we try to teach them.
It has been said that democracy is the worst form of government except all the others that have been tried.
--Sir Winston Churchill
I’m not quite going to make that analogy but I will hasten to assert that there are far worse systems of education than ours. Including some that are ’like ours but magnified”.
In terms of healthy psychological development and practical skill acquisition the apprenticeship systems of various cultures have been better. Right now I can refer to the school system on one of the Solomon Islands. The culture is that of a primitive coastal village but with western influences. Western teaching materials and a teacher are provided but occurs in the morning for 4 hours a day. No breaks are needed and nor is any pointless time wasting. The children then spend their time surfing. But they surf carrying spears and catch fish while they are doing it.
What appeals to me about that system is:
The shorter time period.
Most of the time kids spend at school is a blatant waste. in particular, in the youngest years a lot of what the kids are doing is ‘growing older’. That is what is required for their brains to handle the next critical learning skills.
Much more than 4 hours a day of learning is squandered on diminishing returns. The Cambridge Handbook of Expertise and Expert Performance suggests that 4 hours per day of deliberate practice (7 days a week for 10 years) is a good approximate guide for how to gain world-class expert level performance in a field. It is remarkably stable across many domains.
The children’s social lives are not dominated by playground politics and are not essentially limited to same age peers.
Not only are the extracurricular activities physically healthier than more time wasted in classes they are better for brain development too. What is the formula for increased release of Neurotrophic Growth Factors, consolidation into stable Neurogenesis and optimized attention control and cognitive performance? Aerobic exercise + activities requiring extensive coordination + a healthy diet including adequate Omega-3 intake. That’s right. Spending hours swimming, surfing and catching fish with spears is just about perfect.
I agree, and this is a tragedy in that it makes it so students don’t have marketable skill by 14 as they would in an apprentice system, and so are dependent on mommy and daddy. This “age of genuine independence from parents” is increasing all the time, and there’s no excuse for it. It disenfrachises children more than any legal age restrictions on this or that.
If you can’t do math at a fairly advanced level—at least having competence with information theory, probability, statistics, and calculus—you can’t understand the world beyond what’s visible on its (metaphorical) surface.
While as a mathematician I find that claim touching, I can’t really agree with it. To use the example that was one of the starting points of this conversation, how much math do you need to understand evolution? Sure, if you want to really understand the modern synthesis in detail you need math. And if you want to make specific predictions about what will happen to allele frequencies you’ll need math. But in those cases it is very basic probability and maybe a tiny bit of calculus (and even then, more often than not you can use the formulas without actually knowing why they work beyond a very rough idea).
Similar remarks apply to other areas. I don’t need a deep understanding of any of those subjects to have a basic idea about atoms, although again I will need some of them if I want to actually make useful predictions (say for Brownian motion).
Similarly, I don’t need any of those subjects to understand the Keplerian model of orbits, and I’ll only need one of those four (calculus) if I want to make more precise estimations for orbits (using Newtonian laws).
The amount of actual math needed to understand the physical world is pretty minimal unless one is doing hard core physics or chemistry.
The amount of actual math needed to understand the physical world is pretty minimal unless one is doing hard core physics or chemistry.
For example… trying to work out what happens when I shoot a neverending stream of electrons at a black hole. The related theories were more or less incomprehensible to me at first glance. Not being able to do off the wall theorizing on everything at the drop of the hat has to at least make 49!
The human-scale physical world is relatively easy to understand, and we may have evolved or learned to perform the trickier computations using specialized modules, such as perhaps recognizing parabolas to predict where a thrown object will land. You get far with linear models, for instance, assuming that the distance something will move is proportional to the force that you hit it with, or that the damage done is proportional to the size of the rock you hit something with. You rarely come across any trajectory where the second derivative changes sign.
The social world, the economic world, ecology, game theory, predicting the future, and politics are harder to understand. There are a lot of nonlinear and even non-differentiable interactions. To understanding a phenomenon qualitatively, it’s helpful to perform a stability analysis, and recognize likely stable areas, and also unstable regions where you have phase transitions, period doublings, and other catastrophes, You usually can’t do the math and solve one of these systems; but if you’ve worked with a lot of toy systems mathematically, you’ll understand the kind of behaviors you might see, and know something about how the number of variables and the correlations between them affect the limits of linear extrapolation. So you won’t assume that a global warming rate of .017C/year will lead to a temperature increase of 1.7C in 100 years.
I’m making this up as I go; I don’t have any good evidence at hand. I have the impression that I use math a lot to understand the world (but not the “physical” world of kinematics). I haven’t observed myself and counted how often it happens.
I’d like this to be true, as I want the time I spend learning math in the future to be as useful as you say, but I seem to have come rather far by knowing the superficial version of a lot of things. Knowing the actual math from something like PT:LOS would be great, and I plan on reaching at least that level in the Bayesian conspiracy, but I can currently talk about things like quantum physics and UDT and speed priors and turn this into changes in expected anticipation. I don’t know what Kolmogorov complexity is, really, in a strictly formal from-the-axioms sense, nor Solomonoff induction, but I reference it or things related to it about 10 times a day in conversations at SIAI house, and people who know a lot more than I do mostly don’t laugh at my postulations. Perhaps you mean a deeper level of understanding? I’d like to achieve that, but my current level seems to be doing me well. Perhaps I’m an outlier. (I flunked out of high school calculus and ‘Algebra 2’ and haven’t learned any math since. I know the Wikipedia/Scholarpedia versions of a whole bunch of things, including information theory, computer science, algorithmic probability, set theory, etc., but I gloss over the fancy Greek letters and weird symbols and pretend I know the terms anyway.)
A public reminder to myself so as to make use of consistency pressure: I shouldn’t write comments like the one I wrote above. It lingers too long on a specific argument that is not particularly strong and was probably subconsciously fueled by a desire to talk about myself and perhaps countersignal to someone whose writing I respect (Phil Goetz).
I flunked out of high school calculus and ‘Algebra 2’ and haven’t learned any math since.
I have a belief that I can fix things like this, having spent time working with other students in high school. If I ever meet you in person, will you assist me in testing that belief? ;)
I’m pretty sure that most people around lesswrong have about the same level of familiarity with most subjects (outside whatever field they actually specialize). Although I do think that you are relatively weak in mathematics, but advanced math just really isn’t that important, vis-a-vis being generally well educated and rational.
Most people don’t need to understand evolution. Maybe we should distinguish between “harmful to self”, “harmful to society”, and “harmful to a democratic society”.
If you can’t do math at a fairly advanced level—at least having competence with information theory, probability, statistics, and calculus—you can’t understand the world beyond what’s visible on its (metaphorical) surface.
I’ve got a tangential question: what math, if learned by more people, would give the biggest improvement in understanding for the effort put into learning it?
Take calculus, for example. It’s great stuff if you want to talk about rates of change, or understand anything involving physics. There’s the benefit; how about the cost? Most people who learn it have a very hard time doing so, and they’re already well above average in mathematical ability. So, the benefit mostly relates to understanding physics, and the cost is fairly high for most people.
Compare this with learning basic probability and statistical thinking. I’m not necessarily talking about learning anything in depth, but people should have at least some exposure to ideas like probability distributions, variance, normal distributions and how they arise, and basic design of experiments—blinding, controlling for variables, and so on. This should be a lot easier to learn than calculus, and it would give insight into things that apply to more people.
I’ll give a concrete example: racism. Typical racist statements, like “black people are lazy and untrustworthy,” couldn’t possibly be true in more than a statistical sense, and obviously a statistical statement about a large group doesn’t apply to every member of that group—there’s plenty of variance to take into account. Basic statistical thinking makes racist bigotry sound preposterously silly, like someone claiming that the earth is flat. This also applies to every other form of irrational bigotry that I can think of off the top of my head.
Remember when Larry Summers suggested that maybe part of the reason for the underrepresentation of women in Harvard’s science faculty was that women may have lower variance in intelligence than men, and so are underrepresented in the highest part of the intelligence bell curve? What almost everybody heard was “Women can’t be scientists because they’re stupid.” People heard a statistical statement and had no idea how to understand it.
There are important, relevant subjects that people just can not understand without basic statistical thinking. I would like to see most people exposed to basic statistical thinking.
Are there any other kinds of math that offer high bang-for-the-buck, as far as learning difficulty goes? (I’ve always thought that the math behind computer programming was damn useful stuff, but the engineering students I’ve talked with usually find it harder than calculus, so maybe that’s not the best idea.)
Probability theory as extended logic.
I think it can be presented in a manner accessible to many (Jaynes PT:LOS is not accessible to many).
Tangential question to your tangential question: I’m puzzled, which math are you talking about here? The only math relevant to programming that I can think of that engineering students would also learn would be discrete math, but the extent needed for good programming competency is pretty small and easy to pick up.
Are we talking numerical computing instead, with optimization problems and approximating solutions to DE’s? That’s the only thing I can think of relevant to engineering for which the math background might be more difficult than calculus.
I was thinking more basic: induction, recursion, reasoning about trees. Understanding those things on an intuitive level is one of the main barriers that people face when they learn to program. It’s one thing to be able to solve problems out of a textbook involving induction or recursion, but another thing to learn them so well that they become obvious—and it’s that higher level of understanding that’s important if you want to actually use these concepts.
I’m not sure about all the details, but I believe that there was a small kerfuffle a few decades ago over a suggestion to change the apex of U.S. ``school mathematics″ from calculus to a sort of discrete math for programming course. I cannot remember what sort of topics were suggested though. I do remember having the impression that the debate was won by the pro-calculus camp fairly decisively—of course, we all see that school mathematics hasn’t changed much.
Calculus might not be the best example of a skill with relatively low payoff, because you need some calculus to understand what a continuous probability distribution is.
I do? I thought I understood both calculus and continuous probability but I didn’t know one relied on the other. You are probably right, sometimes things that are ‘obvious’ just don’t get remembered.
For example, suppose you have a biased coin which lands heads up with probability p. A probability distribution that represents your belief about p is usually a non-negative real function f on the unit interval whose integral is 1. Your credence in the proposition that p lies between 1⁄3 and 1⁄2 is the integral of f from 1⁄3 to 1⁄2.
Yes, extremely obvious now that you mention it. :)
Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud. You multiply by the number up to the top right of the letter then reduce that number by 1. Or you do the reverse in the reverse order. You know, like you put on your socks then your shoes but have to take off your shoes then take off your socks.
Sometimes drawing a picture helps prime an intuitive understanding of the physics. You start with a graph of velocity vs time. That is the ‘acceleration’. See… it is getting faster each second. Now, use a pencil and progressively color in under the line. that’s the distance that is getting covered. See how later on more when it is going faster more distance is being traveled at one time and we have to shade in more area? Now, remember how we can find the area of a triangle? Well, will you look at that… the maths came out the same!
I teach calculus often. Students don’t get hung up on mechanical things like (x^3)′ = 3x^2. They instead get hung up on what
has to do with the derivative as a rate of change or as a slope of a tangent line. And from the perspective of a calculus student who has gone through the standard run of American school math, I can understand. It does require a level up in mathematical sophistication.
That’s the problem. See that bunch of symbols? That isn’t the best way to teach stuff. It is like trying to teach them math while speaking a foreign language (even if technically we are saving the greek till next month). To teach that concept you start with the kind of picture I was previously describing, have them practice that till they get it then progress to diagrams that change once in the middle, etc.
Perhaps the students here were prepared differently but the average student started getting problems with calculus when it reached a point slightly beyond what you require for the basic physics we were talking about here. ie. they would be able to do 1. and but have no chance at all with 2:
I’m not claiming that working from the definition of derivative is the best way to present the topic. But it is certainly necessary to present the definition if the calculus is being taught in math course. Part of doing math is being rigorous. Doing derivatives without the definition is just calling on a black box.
On the other hand, once one has the intuition for the concept in hand through more tangible things like pictures, graphs, velociraptors, etc., the definition falls out so naturally that it ceases to be something which is memorized and is something that can be produced ``on the fly″.
A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan “brought to you by apostrophe, the letters LIM and an arrow” but you can go on to tell them “this is how impressive mathematcians say you should write this stuff that you already understand” and they’ll get it.
I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn’t easy.
That is close to saying “this stuff is hard”. How about first showing the students the diagram that that definition is a direct transcription of, and then getting the formula from it?
(Actually, many would struggle with 1. due to difficulty with comprehension and abstract problem solving. They could handle the calculus but need someone to hold their hand with the actual thinking part. That’s what we really fail to teach effectively.)
People get the simple concepts mixed together with a bunch of mathy-looking symbols and equations, and it all congeals into an undifferentiated mass of confusing math. Yes, I know calculus is actually pretty straightforward, but we’re probably not a representative sample. Talk with random bewildered college freshmen to combat sample bias. I did this, and what I learned is that most people have serious trouble learning calculus.
Now, if you want to be able to partially understand a bunch of physics stuff but you don’t necessarily need to be able to do the math, you could probably get away with a small subset of what people learn in calculus classes. If you learned about integration and differentiation (but not how to do them symbolically), as well as vectors, vector fields, and divergence and curl, then you could probably get more benefit-per-hour-of-study than if you went and learned calculus properly. It leaves a bad taste in my mouth, though.
When taught well the calculus required for the sort of applications you mentioned is not something that causes significant trouble, certainly not compared to vector fields, divergence or curl. By taught well, if you will excuse my lack of seemly modesty, is how I taught it in my (extremely brief—don’t let me get started on what I think of western school systems) stint teaching high school physics. The biggest problem for people learning basic calculus is that people teaching it try to convey that it is hard.
I’m only talking here about the level of stuff required for everyday physics. Definitely not for the vast majority of calculus that we try to teach them.
Aw, please ? I’d be interested in hearing about the differences with other systems :)
It has been said that democracy is the worst form of government except all the others that have been tried. --Sir Winston Churchill
I’m not quite going to make that analogy but I will hasten to assert that there are far worse systems of education than ours. Including some that are ’like ours but magnified”.
In terms of healthy psychological development and practical skill acquisition the apprenticeship systems of various cultures have been better. Right now I can refer to the school system on one of the Solomon Islands. The culture is that of a primitive coastal village but with western influences. Western teaching materials and a teacher are provided but occurs in the morning for 4 hours a day. No breaks are needed and nor is any pointless time wasting. The children then spend their time surfing. But they surf carrying spears and catch fish while they are doing it.
What appeals to me about that system is:
The shorter time period.
Most of the time kids spend at school is a blatant waste. in particular, in the youngest years a lot of what the kids are doing is ‘growing older’. That is what is required for their brains to handle the next critical learning skills.
Much more than 4 hours a day of learning is squandered on diminishing returns. The Cambridge Handbook of Expertise and Expert Performance suggests that 4 hours per day of deliberate practice (7 days a week for 10 years) is a good approximate guide for how to gain world-class expert level performance in a field. It is remarkably stable across many domains.
The children’s social lives are not dominated by playground politics and are not essentially limited to same age peers.
Not only are the extracurricular activities physically healthier than more time wasted in classes they are better for brain development too. What is the formula for increased release of Neurotrophic Growth Factors, consolidation into stable Neurogenesis and optimized attention control and cognitive performance? Aerobic exercise + activities requiring extensive coordination + a healthy diet including adequate Omega-3 intake. That’s right. Spending hours swimming, surfing and catching fish with spears is just about perfect.
I agree, and this is a tragedy in that it makes it so students don’t have marketable skill by 14 as they would in an apprentice system, and so are dependent on mommy and daddy. This “age of genuine independence from parents” is increasing all the time, and there’s no excuse for it. It disenfrachises children more than any legal age restrictions on this or that.
While as a mathematician I find that claim touching, I can’t really agree with it. To use the example that was one of the starting points of this conversation, how much math do you need to understand evolution? Sure, if you want to really understand the modern synthesis in detail you need math. And if you want to make specific predictions about what will happen to allele frequencies you’ll need math. But in those cases it is very basic probability and maybe a tiny bit of calculus (and even then, more often than not you can use the formulas without actually knowing why they work beyond a very rough idea).
Similar remarks apply to other areas. I don’t need a deep understanding of any of those subjects to have a basic idea about atoms, although again I will need some of them if I want to actually make useful predictions (say for Brownian motion).
Similarly, I don’t need any of those subjects to understand the Keplerian model of orbits, and I’ll only need one of those four (calculus) if I want to make more precise estimations for orbits (using Newtonian laws).
The amount of actual math needed to understand the physical world is pretty minimal unless one is doing hard core physics or chemistry.
For example… trying to work out what happens when I shoot a neverending stream of electrons at a black hole. The related theories were more or less incomprehensible to me at first glance. Not being able to do off the wall theorizing on everything at the drop of the hat has to at least make 49!
The human-scale physical world is relatively easy to understand, and we may have evolved or learned to perform the trickier computations using specialized modules, such as perhaps recognizing parabolas to predict where a thrown object will land. You get far with linear models, for instance, assuming that the distance something will move is proportional to the force that you hit it with, or that the damage done is proportional to the size of the rock you hit something with. You rarely come across any trajectory where the second derivative changes sign.
The social world, the economic world, ecology, game theory, predicting the future, and politics are harder to understand. There are a lot of nonlinear and even non-differentiable interactions. To understanding a phenomenon qualitatively, it’s helpful to perform a stability analysis, and recognize likely stable areas, and also unstable regions where you have phase transitions, period doublings, and other catastrophes, You usually can’t do the math and solve one of these systems; but if you’ve worked with a lot of toy systems mathematically, you’ll understand the kind of behaviors you might see, and know something about how the number of variables and the correlations between them affect the limits of linear extrapolation. So you won’t assume that a global warming rate of .017C/year will lead to a temperature increase of 1.7C in 100 years.
I’m making this up as I go; I don’t have any good evidence at hand. I have the impression that I use math a lot to understand the world (but not the “physical” world of kinematics). I haven’t observed myself and counted how often it happens.
I’d like this to be true, as I want the time I spend learning math in the future to be as useful as you say, but I seem to have come rather far by knowing the superficial version of a lot of things. Knowing the actual math from something like PT:LOS would be great, and I plan on reaching at least that level in the Bayesian conspiracy, but I can currently talk about things like quantum physics and UDT and speed priors and turn this into changes in expected anticipation. I don’t know what Kolmogorov complexity is, really, in a strictly formal from-the-axioms sense, nor Solomonoff induction, but I reference it or things related to it about 10 times a day in conversations at SIAI house, and people who know a lot more than I do mostly don’t laugh at my postulations. Perhaps you mean a deeper level of understanding? I’d like to achieve that, but my current level seems to be doing me well. Perhaps I’m an outlier. (I flunked out of high school calculus and ‘Algebra 2’ and haven’t learned any math since. I know the Wikipedia/Scholarpedia versions of a whole bunch of things, including information theory, computer science, algorithmic probability, set theory, etc., but I gloss over the fancy Greek letters and weird symbols and pretend I know the terms anyway.)
A public reminder to myself so as to make use of consistency pressure: I shouldn’t write comments like the one I wrote above. It lingers too long on a specific argument that is not particularly strong and was probably subconsciously fueled by a desire to talk about myself and perhaps countersignal to someone whose writing I respect (Phil Goetz).
I have a belief that I can fix things like this, having spent time working with other students in high school. If I ever meet you in person, will you assist me in testing that belief? ;)
I’m pretty sure that most people around lesswrong have about the same level of familiarity with most subjects (outside whatever field they actually specialize). Although I do think that you are relatively weak in mathematics, but advanced math just really isn’t that important, vis-a-vis being generally well educated and rational.