What do grad students in math do all day?(gist.github.com)
gist.github.com
What do grad students in math do all day?
https://gist.github.com/stoutbeard/4158578
23 comments
I used to say what I did every day consisted of counting, coloring, drawing pictures, and talking to people. And mostly, it did.
As a math professor, I was enthusiastic when I read this short piece in Math Horizons, and promptly put it on my office door. The metaphor is a great hook, is well carried out, and leaves the reader with a pretty much spot-on idea of what math research is all about. I am delighted to see it reach a wider audience on the web.
math research would be much more relevant if they talked to engineers (and hopefully are listened to).
In college a I was told "quantum mechanics = linear algebra" then I hear "electrical engineering = linear algebra".
They were right.
The PhD is a little different. It becomes so specialized... your papers are geared towards only a handful of experts. Literally you and a few other people are the only in the world who understand it. I thought mathematicians had a lot more common ground, but they don't.
In college a I was told "quantum mechanics = linear algebra" then I hear "electrical engineering = linear algebra".
They were right.
The PhD is a little different. It becomes so specialized... your papers are geared towards only a handful of experts. Literally you and a few other people are the only in the world who understand it. I thought mathematicians had a lot more common ground, but they don't.
Yeah, if you're lucky enough to never have to teach, pass quals, take classes of your own, etc. In reality, research takes up less than 30% of your average day.
I thought they worked it out with a pencil as they saying goes.
"Can use use a vacuum cleaner to clean a cat?" ... Yes, but I've found that the cat might not like it. I'd also recommend using the upholstery tool and whatever you do, don't allow the cats tail (or any other part) to come into contact with the rotating brushes. For a moderately dirty cat, the vacuum's suction should be enough. If you have a really dirty cat, or a lot of cats to clean at once, try an automatic car wash instead (just make sure it's the kind where the doors close before the wash starts, otherwise too many escape).
My cats hate the vacuum but one of them absolutely loves to be swept on his back with the broom. He's old school like that.
You sound like you've done some serious research about bulk cat washing. You should write a paper.
Fun read, well-written. I can relate to this somewhat. I recently proofread a paper discussing new applications for hermite polynomials (for english mistakes), and I was blown away by the years of study it would take me to even begin to understand what was going on with my amateur math skills. I guess it can be similar to what a non-programmer feels like when looking at code.
An incredibly accurate depiction of research in any theoretical field, I'd say. Compound that with the fact that during your education you're mostly presented with texts that summarize decades or more of research into a scant few pages as if the people involved had just flowed naturally from one idea to the next, from a problem statement to the incredibly complex idea that unlocks the proof.
When it's finally your turn to try your hand at actual research, it turns out that your contributions are barely a couple of side notes on a restricted subset of a problem in the hope that someone will use that information to find out something that is actually relevant in practice.
Rather than turning me off from academia, it makes me marvel at the tower of minuscule pebbles upon which our modern civilization rests. One day, I might get to place a few more of them on top.
When it's finally your turn to try your hand at actual research, it turns out that your contributions are barely a couple of side notes on a restricted subset of a problem in the hope that someone will use that information to find out something that is actually relevant in practice.
Rather than turning me off from academia, it makes me marvel at the tower of minuscule pebbles upon which our modern civilization rests. One day, I might get to place a few more of them on top.
Agreed. After doing 2 years I left with a masters and never looked back. What is even scarier than the minuscule contributions you might make is the few that have done anything but minuscule. Those people are amazingly smart.
By the way, cats do like vacuums... http://www.youtube.com/watch?v=bCzkm2z4-6g
By the way, cats do like vacuums... http://www.youtube.com/watch?v=bCzkm2z4-6g
That's the part of the article that stuck out to me. I mean, I used to have a lovely ginger cat who liked nothing more than to be vacuumed. I think that the author should ensure that his cat vacuuming experiments are more rigorous in future.
Well, my cat gets scared to death by a vacuum cleaner. She darts out of the room as soon as it's turned on.
Its the motor noise. Once the cat is 20 years old and totally deaf (depends on breed...), I can verify the cat will love the vacuum cleaner. It feels like being brushed and massaged at the same time. I'm surprised day spas don't do this to humans. Needless to say this is using the hose and brush attachment, not the carpet beater.
If the nozzle is too large, or the cat too skinny, it risks getting sucked into the machine. Fair enough the cat is scared. Cats don't do math either. Well, perhaps Schrödinger taught his cat, but I hear it had a sticky end. Nobody is really sure.
If they're so smart, why aren't they any better at bringing newcomers up to speed?
Some thing are hard no matter how you try to explain them. There's lots of stuff that seems to require working through it yourself to understand.
Oh, I don't disagree; some topics are inherently intractable. It's just that, in all my experiences, the numbers are more like 1 truly intractable topic for every thousand that some expert will mistakenly claim is intractable simply because they haven't put the effort into tracing out the connections to other topics.
The woman top row of this xkcd pretty much sums up my life:
http://xkcd.com/566/
The woman top row of this xkcd pretty much sums up my life:
http://xkcd.com/566/
It's the newcomers' faults, clearly.
Probably because it's hard to distill years of work connecting the dots (which at first don't appear to be connected in an obvious way) in a way that is easily digestable by somebody new to the field.
In any case, I imagine there's more glory in digging further into intellectual stuff than spending time making it palatable for newbies. :)
In any case, I imagine there's more glory in digging further into intellectual stuff than spending time making it palatable for newbies. :)
Your description of research as "pebbles" reminds me of this representation of PhD research (posted on HN a couple years ago). I like how it captures the idea of pushing out the boundaries of what we know just a tiny bit at a time.
http://matt.might.net/articles/phd-school-in-pictures/
http://matt.might.net/articles/phd-school-in-pictures/
Thanks for posting this. His comment immediately reminded me of this and I was about to try to find this exact page.
At my university, the administration is really pushing to have business undergrads active in research - I don't think most of them have the patience to contribute. Makes me really wish they would read articles like this
I admit that I have little knowledge of how many fields work and probably know the least about academia but I still have to ask (or maybe that's why I have to ask) what research does a business undergrad participate in?
My dad is an accounting professor, and while I don't think he normally involves undergrads in his research, it seems like he could. This is just one small segment of business research, but he tests how rules and incentives impact decision making.
For example, make a game where two people are teamed up. You give person A $10 and tell them they can give as much as they want to person B, and that's it. How much does person A share? Ok, now what if you give instructions explaining to person A that person B leaves empty handed if they don't share? What if you say that they're supposed to share 50%? What if 10% of whatever is given to person B is lost as a "tax"? The idea is to figure out ways to get people to act honestly and fairly in the hopes that it can be applied to business.
Undergrads could help find test subjects. They could help design the tests (they generally involve simple software that administers the games). Some could probably analyze the results in the hopes of finding interesting and unexpected takeaways.
For example, make a game where two people are teamed up. You give person A $10 and tell them they can give as much as they want to person B, and that's it. How much does person A share? Ok, now what if you give instructions explaining to person A that person B leaves empty handed if they don't share? What if you say that they're supposed to share 50%? What if 10% of whatever is given to person B is lost as a "tax"? The idea is to figure out ways to get people to act honestly and fairly in the hopes that it can be applied to business.
Undergrads could help find test subjects. They could help design the tests (they generally involve simple software that administers the games). Some could probably analyze the results in the hopes of finding interesting and unexpected takeaways.
That sounds more like Sociology.
As an aside, I believe my first post and this one could sound like I'm trying to be a dick, I'm honestly not. I am wondering and interested. Business majors and related (like your accounting professor father) always seemed to me to be more practical degrees and not really research fields.
As an aside, I believe my first post and this one could sound like I'm trying to be a dick, I'm honestly not. I am wondering and interested. Business majors and related (like your accounting professor father) always seemed to me to be more practical degrees and not really research fields.
Former Sociology Ph.D. candidate here.
Yes, a lot of b-school research crosses fields. I've worked with b-school profs doing work across sociology, economics, law (to a lesser degree), computer science, and mathematics. The difference between research in those fields and b-school research seems to be that the b-school profs had more of an applied bent. The b-school research was more motivated by practical business problems.
Of course, much like programs within a discipline can be radically different from university to university, I imagine there's also a range of research styles across b-schools.
Yes, a lot of b-school research crosses fields. I've worked with b-school profs doing work across sociology, economics, law (to a lesser degree), computer science, and mathematics. The difference between research in those fields and b-school research seems to be that the b-school profs had more of an applied bent. The b-school research was more motivated by practical business problems.
Of course, much like programs within a discipline can be radically different from university to university, I imagine there's also a range of research styles across b-schools.
Your impression is pretty accurate. Professional schools like business, social work, education or law do a weird mix of vocational courses and cross-disciplinary/applied work. The post you're responding to sounds more like behavioural eceonomics than sociology to me (it's the ultimatum game or a similar one) but you could do very similar research in psychology or in anthropology doing cross cultural comparisions of how people play this game. It has been and is being done.
I'm not familiar with accounting research but my impression is that there's quite a lot of overlap with some areas of microeconomics. After all, accounting is either about modelling and prediction or about accurate reporting and analysis if it's not to be book keeping.
Sometimes the cross-disciplinary/applied thing just means that garbage is published by people who aren't familiar with research in another field that covered the same ground decades before. The worst though is when something is researched, disproved and they keep on teaching it in these professional schools for reasons of ideology or politics. Stuff like Howard Gardner's multiple intelligences is still taught in Ed schools and researchers like Linda Gottfredson have to deal with witch hunts because they do work worthy of a real psychology department while in an Ed school. The perils of doing work on on intelligence in a political faculty.
I'm not familiar with accounting research but my impression is that there's quite a lot of overlap with some areas of microeconomics. After all, accounting is either about modelling and prediction or about accurate reporting and analysis if it's not to be book keeping.
Sometimes the cross-disciplinary/applied thing just means that garbage is published by people who aren't familiar with research in another field that covered the same ground decades before. The worst though is when something is researched, disproved and they keep on teaching it in these professional schools for reasons of ideology or politics. Stuff like Howard Gardner's multiple intelligences is still taught in Ed schools and researchers like Linda Gottfredson have to deal with witch hunts because they do work worthy of a real psychology department while in an Ed school. The perils of doing work on on intelligence in a political faculty.
"what research does a business undergrad participate in"
Listening to a coworker of mine:
Marketing, obviously. This is how industry wide 99% of SUVs will never leave suburban blacktop but most SUV commercials have the vehicle speeding up and down dirt trails in a national park. Sometimes similar things in business come from random convergent individual evolution, but more often from the outside. Somebody released a paper on the topic of Americans being germophobes about 5-10 years ago, predictably we had to sit thru a product cycle where anything that can theoretically include triclosan, had to include triclosan (and other anti-bacterials, etc). You may remember a similar fad a decade earlier for citrus oils.
Generalized, generic version of what they'll do on an individual case basis with numerical metrics in private sector. Talk to buddies in the field to get raw randomized data and turn the data into a report from a business perspective if you pay $x bounty for finding a security bug that results in a supply demand curve shaped exactly as such, with this effect on product development times, and this effect on total cost of production / ownership, sales figures, etc, all theoretically "industry wide" or at least "subindustry wide"
We don't have a free market system, more of a centrally controlled system. So there's a basically infinite collection of govt laws, rules, regs, where you can gather the raw data for game theoretic analysis of what the central controllers have decided for us. Is it better, purely economically, to hire illegals to staff fast food restaurants? Identifying all the economic costs is actually pretty hard. Someone out there is probably researching every law/rule/reg out there... costs of obeying vs disobeying, cost to purchase alternative legislation via election funds / lobbying, and repeat each along the short term / long term axis.
For accounting the rules are often "what rule can you purchase" or "what you can get away with" but there's also real world accounting. You need to keep two sets of books, one with how fast does the IRS allow you to depreciate "XYZ" for tax purposes, and another set of books for how fast "XYZ" actually depreciates in the real world. Then you have to live off the real world books or else you can get in a horrible cash crunch if your cashflow depends on selling a used "XYZ" to make payroll this month or as a downpayment on a new "XYZ".
Finally criminals spend a lot of time inventing new crimes. New forms of control fraud, etc. Basically, classic social engineering without computer involvement.
Listening to a coworker of mine:
Marketing, obviously. This is how industry wide 99% of SUVs will never leave suburban blacktop but most SUV commercials have the vehicle speeding up and down dirt trails in a national park. Sometimes similar things in business come from random convergent individual evolution, but more often from the outside. Somebody released a paper on the topic of Americans being germophobes about 5-10 years ago, predictably we had to sit thru a product cycle where anything that can theoretically include triclosan, had to include triclosan (and other anti-bacterials, etc). You may remember a similar fad a decade earlier for citrus oils.
Generalized, generic version of what they'll do on an individual case basis with numerical metrics in private sector. Talk to buddies in the field to get raw randomized data and turn the data into a report from a business perspective if you pay $x bounty for finding a security bug that results in a supply demand curve shaped exactly as such, with this effect on product development times, and this effect on total cost of production / ownership, sales figures, etc, all theoretically "industry wide" or at least "subindustry wide"
We don't have a free market system, more of a centrally controlled system. So there's a basically infinite collection of govt laws, rules, regs, where you can gather the raw data for game theoretic analysis of what the central controllers have decided for us. Is it better, purely economically, to hire illegals to staff fast food restaurants? Identifying all the economic costs is actually pretty hard. Someone out there is probably researching every law/rule/reg out there... costs of obeying vs disobeying, cost to purchase alternative legislation via election funds / lobbying, and repeat each along the short term / long term axis.
For accounting the rules are often "what rule can you purchase" or "what you can get away with" but there's also real world accounting. You need to keep two sets of books, one with how fast does the IRS allow you to depreciate "XYZ" for tax purposes, and another set of books for how fast "XYZ" actually depreciates in the real world. Then you have to live off the real world books or else you can get in a horrible cash crunch if your cashflow depends on selling a used "XYZ" to make payroll this month or as a downpayment on a new "XYZ".
Finally criminals spend a lot of time inventing new crimes. New forms of control fraud, etc. Basically, classic social engineering without computer involvement.
[deleted]
> "It is a tool that does suck up dust to make what you walk on in a home tidy."
This is essentially the idea behind Natural Semantic Metalanguage[1], Andrzej Bogusławski's theory of cross-cultural semantics from the 70s.
[1] http://en.wikipedia.org/wiki/Natural_semantic_metalanguage
This is essentially the idea behind Natural Semantic Metalanguage[1], Andrzej Bogusławski's theory of cross-cultural semantics from the 70s.
[1] http://en.wikipedia.org/wiki/Natural_semantic_metalanguage
I did my PhD in the database area, and I really relate to some of the stuff in this article - most particularly the overwhelming difficulty (and pressure involved) in doing something genuinely new. I came in with all these wonderful ideas, and within a few months realised that not only had they all been done, most of them had been done decades ago. Quite the comedown!
It does give you a bit of a cynical outlook on new tech in general.
It does give you a bit of a cynical outlook on new tech in general.
One of the things I've found is that, to really consider myself a "master" of a particular field, I have to know where the research frontier is. "What are the current unanswered problems? How do you extend method X to cover case Y?"
I'm just finishing a math PhD. Talking to CS PhD's my impression is that we do pretty much the same thing, we just work on different domains.
I'm a CS PhD, and I agree. I always tell people I'm studying math because it gives them much more of a flavor for what I'm actually doing everyday. When I say computer science, people think I'm just programming something.
Also, when people ask me, "So What do you actually do studying math?" I reply, "Sit and stare at the wall for several hours a day. Occasionally I write something down."
Also, when people ask me, "So What do you actually do studying math?" I reply, "Sit and stare at the wall for several hours a day. Occasionally I write something down."
It does depend a lot on the sub-field of CS.
For systems work you spend a lot of time implementing ideas and running experiments. In a lot of cases it isn't necessarily that difficult to find an interesting idea that's probably viable. The real work is spending a lot of time writing code and work out all the annoying details to get to the point where you have a decent proof of concept.
For systems work you spend a lot of time implementing ideas and running experiments. In a lot of cases it isn't necessarily that difficult to find an interesting idea that's probably viable. The real work is spending a lot of time writing code and work out all the annoying details to get to the point where you have a decent proof of concept.
Presumably, when you tell people you're doing research in math they imagine you're multiplying long numbers all day.
Yes, when as a math grad student I saw some of that, I developed some opinions and ideas to look for more productive approaches.
Somewhere I read: "There is a famous recipe for rabbit stew that starts out, 'First catch a rabbit'.", and I changed that to, "There is a recipe for how to do applied mathematics, first get an application.".
For more, commonly the main criteria for 'research' is that it be "new, correct, and significant". And quite broadly in some powerful places, e.g., a famous David report, there were complaints that a result in math that met the first two but had no visible applications, inside or outside math, was likely short on "significant". So, eventually it dawned on me that if start with an application (something significant in the real world, although inside math would do also but tends to be more difficult and less highly valued outside math) and get a good solution for that application, then have "significant" handled. Yes, these thoughts did occur to me, but they were only secondary: My real interest was 'significant' outside of math and, in particular, in my bank account.
So, on to "new, correct": In math, "correct" is comparatively easy -- just work in the style of definitions, theorems, and proofs where it is fairly easy to check math correctness.
That leaves the part "new": Surprise! If start with a significant problem from the real world, then likely there is no solid solution for that problem on the shelves of the research libraries. Why? Because it's a really complicated real world out there! So, find in your real problem where current math doesn't really provide a solution and then do some more math to get some math for a better solution for the real problem. Now maybe the math just did that was "new" is not as earth shaking for pure math as, say, resolving the Riemann hypothesis, or, now, P versus NP, but still have covered "new, correct, and significant" and, besides, may have something powerful and valuable for the real problem outside math.
And that new math result got for that one real problem has a nice property: Given a new result in 'pure' math, the probability of an application in the next 12 months is small. Given a new result in math that has an application, the probability of another application in the next 12 months is nicely higher. Moreover, that probability appears to be monotone increasing with the number of known applications. Indeed, one skeptical way to evaluate such a result is to look for two significant applications instead of just one!
There is more going for this approach: Are taking math directions and 'values' based heavily on what solves some problems outside math. Well, where'd we get calculus? Sure, trying to make sense out of elliptical orbits of planets. And calculus is the main well spring of the part of math called 'analysis' that is so far by a wide margin the most applicable part of math (I know, number theory can do good things for computer security; maybe some people studying string theory in physics will want some topology; and people in logic may value work in foundations). But, tough not to notice that calculus led to the study of heat flow and Fourier theory which did great things for signal processing.
But in part the OP is correct: When I went through measure theory, it seemed fantastic stuff, especially since finally I had a better theory of integration for applications. But in Rudin's 'Real and Complex Analysis' he discussed regular Borel measures, and I never saw just why he cared about the 'regular'. Maybe if I'd go back and think about those few pages for a few days I'd see it. Yet, if I did see it, then I'd write it down so I wouldn't have to work to see it again, and I wish that Rudin had done that in his book. The precise definitions, theorems, and proofs are crucial, but too often pure math is written with too little explanation of the view from 50,000 feet.
My view is that the key to much more value from computing over the next few decades will be some novel uses of math and its techniques of definitions, theorems, and proofs. Why? Because for what to do in building our hardware and system software, applications, and larger systems, we need more powerful tools than intuitive heuristics or just programming what in principle we see how to do manually.
So, right, in the short term, my approach to math is to do 'applied' math where essentially we start with an application. Then in the longer term my hope is that such math, as calculus did, will lead to new, grand, powerful structures in pure math.
Yes, if the pure mathematicians can make good progress as isolated from applications as in the OP, then good for them, but I concluded that, in effect, good math needs some good applications from outside math.
Much of this 'philosophy' has come to pass whether deliberately or not: Quite broadly it is accepted that the best research 'mathematizes' its field. So, yes, the leading example is mathematical physics, but math is now just crucial in mechanical, electrical, electronic, and civil engineering, statistics, and operations research. Other fields that try to be more mathematical include finance, economics, psychology, sociology, and, now, genetics. And of course computer science is becoming increasingly mathematical.
As powerful as math has been for these other fields, pure math has essentially been left suffering as the applications, grants, and students based on applications of math go to fields outside math. So, if I were a chair or a dean over a math department, then I would welcome serious attention on important problems from outside math. I would keep fully high standards of definitions, theorems, and proofs. But, for more, first cut it would seem that the criteria "new, correct, and significant" would be easier than those criteria with also "applicable, powerful, and valuable" outside of math, but my view is that this is false, that being applicable is easier just to publishable papers but more importantly to real significance both inside and outside math.
Somewhere I read: "There is a famous recipe for rabbit stew that starts out, 'First catch a rabbit'.", and I changed that to, "There is a recipe for how to do applied mathematics, first get an application.".
For more, commonly the main criteria for 'research' is that it be "new, correct, and significant". And quite broadly in some powerful places, e.g., a famous David report, there were complaints that a result in math that met the first two but had no visible applications, inside or outside math, was likely short on "significant". So, eventually it dawned on me that if start with an application (something significant in the real world, although inside math would do also but tends to be more difficult and less highly valued outside math) and get a good solution for that application, then have "significant" handled. Yes, these thoughts did occur to me, but they were only secondary: My real interest was 'significant' outside of math and, in particular, in my bank account.
So, on to "new, correct": In math, "correct" is comparatively easy -- just work in the style of definitions, theorems, and proofs where it is fairly easy to check math correctness.
That leaves the part "new": Surprise! If start with a significant problem from the real world, then likely there is no solid solution for that problem on the shelves of the research libraries. Why? Because it's a really complicated real world out there! So, find in your real problem where current math doesn't really provide a solution and then do some more math to get some math for a better solution for the real problem. Now maybe the math just did that was "new" is not as earth shaking for pure math as, say, resolving the Riemann hypothesis, or, now, P versus NP, but still have covered "new, correct, and significant" and, besides, may have something powerful and valuable for the real problem outside math.
And that new math result got for that one real problem has a nice property: Given a new result in 'pure' math, the probability of an application in the next 12 months is small. Given a new result in math that has an application, the probability of another application in the next 12 months is nicely higher. Moreover, that probability appears to be monotone increasing with the number of known applications. Indeed, one skeptical way to evaluate such a result is to look for two significant applications instead of just one!
There is more going for this approach: Are taking math directions and 'values' based heavily on what solves some problems outside math. Well, where'd we get calculus? Sure, trying to make sense out of elliptical orbits of planets. And calculus is the main well spring of the part of math called 'analysis' that is so far by a wide margin the most applicable part of math (I know, number theory can do good things for computer security; maybe some people studying string theory in physics will want some topology; and people in logic may value work in foundations). But, tough not to notice that calculus led to the study of heat flow and Fourier theory which did great things for signal processing.
But in part the OP is correct: When I went through measure theory, it seemed fantastic stuff, especially since finally I had a better theory of integration for applications. But in Rudin's 'Real and Complex Analysis' he discussed regular Borel measures, and I never saw just why he cared about the 'regular'. Maybe if I'd go back and think about those few pages for a few days I'd see it. Yet, if I did see it, then I'd write it down so I wouldn't have to work to see it again, and I wish that Rudin had done that in his book. The precise definitions, theorems, and proofs are crucial, but too often pure math is written with too little explanation of the view from 50,000 feet.
My view is that the key to much more value from computing over the next few decades will be some novel uses of math and its techniques of definitions, theorems, and proofs. Why? Because for what to do in building our hardware and system software, applications, and larger systems, we need more powerful tools than intuitive heuristics or just programming what in principle we see how to do manually.
So, right, in the short term, my approach to math is to do 'applied' math where essentially we start with an application. Then in the longer term my hope is that such math, as calculus did, will lead to new, grand, powerful structures in pure math.
Yes, if the pure mathematicians can make good progress as isolated from applications as in the OP, then good for them, but I concluded that, in effect, good math needs some good applications from outside math.
Much of this 'philosophy' has come to pass whether deliberately or not: Quite broadly it is accepted that the best research 'mathematizes' its field. So, yes, the leading example is mathematical physics, but math is now just crucial in mechanical, electrical, electronic, and civil engineering, statistics, and operations research. Other fields that try to be more mathematical include finance, economics, psychology, sociology, and, now, genetics. And of course computer science is becoming increasingly mathematical.
As powerful as math has been for these other fields, pure math has essentially been left suffering as the applications, grants, and students based on applications of math go to fields outside math. So, if I were a chair or a dean over a math department, then I would welcome serious attention on important problems from outside math. I would keep fully high standards of definitions, theorems, and proofs. But, for more, first cut it would seem that the criteria "new, correct, and significant" would be easier than those criteria with also "applicable, powerful, and valuable" outside of math, but my view is that this is false, that being applicable is easier just to publishable papers but more importantly to real significance both inside and outside math.
Mathematizing a fields is a bit of a double edged sword though and I think needs a great deal of skepticism and caution. Yes, if you do it right you can get insights and prove things more rigorously. If you do it wrong, the mathematics can easily become divorced from reality: it really does require ongoing skepticism and awareness of the connection between whatever mathematical abstractions you've built and reality.
In computer science, assuming constant factors in algorithms don't matter is a useful approximation that makes it much easier to do mathematical analysis. But constant factors are very important in practice.
Also, plenty of academic economics research has gone off the deep end with more and more elaborate mathematical models based on assumptions that don't jive with reality.
In computer science, assuming constant factors in algorithms don't matter is a useful approximation that makes it much easier to do mathematical analysis. But constant factors are very important in practice.
Also, plenty of academic economics research has gone off the deep end with more and more elaborate mathematical models based on assumptions that don't jive with reality.
Excellent observation. In many ways, mathematization has killed many branches of not just academic economics, but practical economics as well. Just look at how many failed policies in the past were driven by the false assumption that if we change X by so many units, such and such economic indicator will move by Y.
I have only one answer for you: Yup!
Computer science isn't something that is becoming increasingly more mathematical, but quite the opposite way: computer science was seen as just another branch of mathematics. Alan Turing didn't quite have physical machines at his disposal to develop a theory of computation on, so he invented an abstract machine. Von Neumann thought that the foundational trouble in mathematics with Gödel meant that pure mathematics would soon be a sterile field, so he went on to design the Von Neumman architecture of computing machines we still basically use now. Read some Knuth or Dijkstra. They're mathematicians first, specifically, computer scientists.
It was only in the 80's or so that computer science started to be seen as its own thing instead of a branch of mathematics. It's around this time that university departments separate from the mathematics department started to be created.
When modern "CS" graduates come out of university thinking that an "algorithm" is some irrelevant theoretical tool that is only useful for implementing standard libraries of programming languages and forever forgotten, I weep a little.
It was only in the 80's or so that computer science started to be seen as its own thing instead of a branch of mathematics. It's around this time that university departments separate from the mathematics department started to be created.
When modern "CS" graduates come out of university thinking that an "algorithm" is some irrelevant theoretical tool that is only useful for implementing standard libraries of programming languages and forever forgotten, I weep a little.
Didn't know the von Neumann story about his influence from poor Kurt.
For 'more mathematical', really should say from when! Right, not from von Neumann!
If pick when carefully, if only from yesterday, then since then the field has become more mathematical! Or pick from the start of Stanford Professor Ng's class on 'machine learning' which seems to borrow a lot from maximum likelihood estimation in statistics and steepest ascent optimization from mathematical programming or operations research. Or, CLRS -- they went to mathematical programming and borrowed linear programming. Once at IBM's Watson lab I saw some guys attacking load leveling via some carefully done work in stochastic optimal control. Design of distributed computer systems has long been one of the better applications of queuing theory. Once I needed to find nearest neighbors in R^n so started by doing essentially binary search on each axis (yes, after that need to do some tree backtracking with some cutting planes). I heard that there was such a thing, k-D trees, and, yup, found it in Sedgewick and later discovered that such work is called 'computational geometry'. And, for my last example (drum roll), finally there is some interest in the computer and network monitoring community in regarding monitoring as some continually applied statistical hypothesis tests, maybe! We will see if KPCB sends a copy of my paper to Endgame!
In the short run, to attack a field with math can be one heck of a publishing opportunity. So in the long run, if only from 'academic competitiveness', a field doesn't have much hope as 'science' but to 'mathematize'.
For 'more mathematical', really should say from when! Right, not from von Neumann!
If pick when carefully, if only from yesterday, then since then the field has become more mathematical! Or pick from the start of Stanford Professor Ng's class on 'machine learning' which seems to borrow a lot from maximum likelihood estimation in statistics and steepest ascent optimization from mathematical programming or operations research. Or, CLRS -- they went to mathematical programming and borrowed linear programming. Once at IBM's Watson lab I saw some guys attacking load leveling via some carefully done work in stochastic optimal control. Design of distributed computer systems has long been one of the better applications of queuing theory. Once I needed to find nearest neighbors in R^n so started by doing essentially binary search on each axis (yes, after that need to do some tree backtracking with some cutting planes). I heard that there was such a thing, k-D trees, and, yup, found it in Sedgewick and later discovered that such work is called 'computational geometry'. And, for my last example (drum roll), finally there is some interest in the computer and network monitoring community in regarding monitoring as some continually applied statistical hypothesis tests, maybe! We will see if KPCB sends a copy of my paper to Endgame!
In the short run, to attack a field with math can be one heck of a publishing opportunity. So in the long run, if only from 'academic competitiveness', a field doesn't have much hope as 'science' but to 'mathematize'.
Your argument was raised and given a historical context in an interesting article that was posted here a week ago.
http://news.ycombinator.com/item?id=5353315
http://news.ycombinator.com/item?id=5353315
Yes, I saw that and considered commenting but didn't because it seemed that the thread was mostly about bringing the 'revolution' to teaching in K-12. A one word description of the revolution is Bourbaki, and I'm not thrilled about seeing that influence K-12. For the 'revolution' good courses in plane geometry or trigonometry are about as close to the revolution as I would want to see. And for first calculus, I believe that the traditional ways are fine -- lots of water flowing from tanks, lots of solids of rotation, lots of pictures and intuition. An alternative would appear to be to drag compact sets and uniform continuity into first calculus, and I would discourage that. Another alternative would be to dispense with Riemann integration and just teach measure theory -- I'd discourage that also.
For Bourbaki and 'telegraph style' definitions, theorems, and proofs, I see them as crucial ingredients in the soup but not the whole soup.
For Bourbaki and 'telegraph style' definitions, theorems, and proofs, I see them as crucial ingredients in the soup but not the whole soup.
Sounds like what math grad students did 50 years ago, before we solved all the pure math even tenuously related to the 4 dimensional reality we live in.
You are just wrong. For example, algebraic topology is one of the most abstract subjects I hear conversations about week. It has applications in physics, data analysis, and numerical computing.
Also, saying that reality is 4-dimensional is hooding yourself. The machine learning techniques we use everywhere depend on the math of higher-dimensional spaces, and they wouldn't work if reality couldn't be meaningfully viewed as many-dimensional.
Also, saying that reality is 4-dimensional is hooding yourself. The machine learning techniques we use everywhere depend on the math of higher-dimensional spaces, and they wouldn't work if reality couldn't be meaningfully viewed as many-dimensional.
Machine learning casts a problem into higher dimensions in order to linearize it. Linear problems are easier solve, and can be projected back into the original space. An extra dimension is literally just another variable, which doesn't necessarily have a physical meaning.
Please share a link to an example paper published by a pure math department.
http://arxiv.org/pdf/1303.2807.pdf
Went to the Arxiv into the Algebraic Topology category. Any comments?
Went to the Arxiv into the Algebraic Topology category. Any comments?
Here's one I've been struggling with lately: Parallel coordinates representations of smooth hypersurfaces. I'm not aware of extensive reseach of the parallel coordinates geometry prior to 50 years ago.
http://www.cs.usc.edu/assets/004/83248.pdf
http://www.cs.usc.edu/assets/004/83248.pdf
I prefer the explanation that I developed back when I was a math student:
Imagine that all of mathematics is represented as a solid sphere (ball). At the core (origin) of this sphere are the most basic concepts in math, that we all learn in elementary and high school. On top of the core are many layers of knowledge that are all interconnected, but lead in different directions from the origin. These layers have names like algebra, calculus, and geometry. On top of those are other layers with names like topology, set theory, number theory, analysis, etc. These layers continue, like an onion, all the way to the surface of the sphere.
All mathematics students begin at the core and climb outward, towards the surface of the sphere. They choose different directions and set out to learn and practice everything that they encounter on their paths through the sphere. Eventually, after many years of study, the students who survive finally reach some point on the outside surface of the sphere. Which exact point on the surface each student reaches depends on the direction that he chose at the start. Once the students reach the surface of the sphere, they have understood everything that is known to mankind about some specific series of subjects within their speciality. Standing on the surface of the sphere, they must then must work to add a new layer to the sphere, to add something new and original to human knowledge.
As time goes by, new theories are developed and added as new layers onto the sphere. In ancient times, a great mathematician could learn everything in the whole of the sphere within a single lifetime. The sphere has grown exponentially, however, and in modern times no one person could ever visit every place in the sphere within a single lifetime. The sphere has become so vast as to defy comprehension, as generation after generation has expanded it with new layers.
When the mathematics students were starting out in the core of the sphere, they were all in the same place. They could easily see and speak to one another. However, when they reach the surface of the sphere, it is as if they are scattered across different points of the surface of a huge planet. A student might be lucky to find himself at a popular spot on the surface, where there are perhaps a handful or even a few dozen other students who he can talk to. Another student, less fortunate, may find himself stranded in a deserted place where there is no one that can hear him and there is no one for him to speak to. The surface is a lonely place, where few souls are encountered and if you do encounter some wandering traveller then you are unlikely to speak to the same language and must communicate by crude gestures like waving of hands.
As more and more matter is added to sphere, the dwellers on its surface drift further and further apart, as the surface area expands. Furthermore, the surface of the sphere grows farther and farther way from the core of the sphere. It takes longer and longer for the students to reach the surface of the sphere, as there is more and more volume to traverse.
Imagine that all of mathematics is represented as a solid sphere (ball). At the core (origin) of this sphere are the most basic concepts in math, that we all learn in elementary and high school. On top of the core are many layers of knowledge that are all interconnected, but lead in different directions from the origin. These layers have names like algebra, calculus, and geometry. On top of those are other layers with names like topology, set theory, number theory, analysis, etc. These layers continue, like an onion, all the way to the surface of the sphere.
All mathematics students begin at the core and climb outward, towards the surface of the sphere. They choose different directions and set out to learn and practice everything that they encounter on their paths through the sphere. Eventually, after many years of study, the students who survive finally reach some point on the outside surface of the sphere. Which exact point on the surface each student reaches depends on the direction that he chose at the start. Once the students reach the surface of the sphere, they have understood everything that is known to mankind about some specific series of subjects within their speciality. Standing on the surface of the sphere, they must then must work to add a new layer to the sphere, to add something new and original to human knowledge.
As time goes by, new theories are developed and added as new layers onto the sphere. In ancient times, a great mathematician could learn everything in the whole of the sphere within a single lifetime. The sphere has grown exponentially, however, and in modern times no one person could ever visit every place in the sphere within a single lifetime. The sphere has become so vast as to defy comprehension, as generation after generation has expanded it with new layers.
When the mathematics students were starting out in the core of the sphere, they were all in the same place. They could easily see and speak to one another. However, when they reach the surface of the sphere, it is as if they are scattered across different points of the surface of a huge planet. A student might be lucky to find himself at a popular spot on the surface, where there are perhaps a handful or even a few dozen other students who he can talk to. Another student, less fortunate, may find himself stranded in a deserted place where there is no one that can hear him and there is no one for him to speak to. The surface is a lonely place, where few souls are encountered and if you do encounter some wandering traveller then you are unlikely to speak to the same language and must communicate by crude gestures like waving of hands.
As more and more matter is added to sphere, the dwellers on its surface drift further and further apart, as the surface area expands. Furthermore, the surface of the sphere grows farther and farther way from the core of the sphere. It takes longer and longer for the students to reach the surface of the sphere, as there is more and more volume to traverse.
Agreed, but as a slight counterpoint, some research is about stripping away unnecessary middle layers, to find abstractions, a quicker way to the surface in other words.
If we didn't do this, research would inevitably stop, unless eternal life lies inside a 100 year radius of the sphere (and we can keep learning and getting smarter forever, which I doubt)
If we didn't do this, research would inevitably stop, unless eternal life lies inside a 100 year radius of the sphere (and we can keep learning and getting smarter forever, which I doubt)
And, some stuff has been forgotten. Even if it used to seem pretty important.
Speaking about spheres, there's lots of stuff about spherical trigonometry that you can find in old books (say, 1880s to 1920s [1]). They used to think it was math, but it has been determined to not be math any more. It's now stuff that "everyone knows".
[1] http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?...
Speaking about spheres, there's lots of stuff about spherical trigonometry that you can find in old books (say, 1880s to 1920s [1]). They used to think it was math, but it has been determined to not be math any more. It's now stuff that "everyone knows".
[1] http://ebooks.library.cornell.edu/cgi/t/text/pageviewer-idx?...
It kind of reminds me of Matt Might's "The illustrated guide to a Ph.D." [1]
(Which I now see have been posted in an earlier comment)
[1]: http://matt.might.net/articles/phd-school-in-pictures/
(Which I now see have been posted in an earlier comment)
[1]: http://matt.might.net/articles/phd-school-in-pictures/
> Fortunately, math has an incredibly powerful tool that helps bridge the gap. Namely, when we come up with concepts, we also come up with very explicit symbols and notation, along with logical rules for manipulating them. It's a bit like being handed the technical specifications and diagrams for building a vacuum cleaner out of parts.
Just to play the devil's advocate ... math has wonderful symbols and methods, but they don't really assist in comprehension unless everyone agrees on their meaning, and then only if everyone already understands the underlying concepts, the axioms. For example, starting in 1910, Bertrand Russell and Alfred North Whitehead published "Principia Mathematica" (a borrowed title):
http://en.wikipedia.org/wiki/Principia_Mathematica
But, notwithstanding their high intellectual level and the ambitions behind the project, and notwithstanding the system of symbols used, Russell and Whitehead missed a crucial, central point -- their plan to systematize mathematics, place it on a solid logical foundation, make it immune from uncertainty and doubt, was doomed from the start. Kurt Gödel demonstrated this a few years later:
http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo...
So much for the power of symbols. The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds.
Just to play the devil's advocate ... math has wonderful symbols and methods, but they don't really assist in comprehension unless everyone agrees on their meaning, and then only if everyone already understands the underlying concepts, the axioms. For example, starting in 1910, Bertrand Russell and Alfred North Whitehead published "Principia Mathematica" (a borrowed title):
http://en.wikipedia.org/wiki/Principia_Mathematica
But, notwithstanding their high intellectual level and the ambitions behind the project, and notwithstanding the system of symbols used, Russell and Whitehead missed a crucial, central point -- their plan to systematize mathematics, place it on a solid logical foundation, make it immune from uncertainty and doubt, was doomed from the start. Kurt Gödel demonstrated this a few years later:
http://en.wikipedia.org/wiki/G%C3%B6dels_incompleteness_theo...
So much for the power of symbols. The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds.
> The consequences of the Incompleteness theorems are often overstated, but they do falsify the idea that mathematics is logically consistent, or that a set of symbols, however clear and unambiguous, will prevent basic misunderstandings in even the most fertile minds.
The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency.
They also don't remove the possibility that there is some metamathematical justification for an unambiguous interpretation of the concept of the natural numbers or even of a set (see all of the work in large cardinals, culminating in Hugh Woodin's recent work on extender models for supercompact cardinals).
The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency.
They also don't remove the possibility that there is some metamathematical justification for an unambiguous interpretation of the concept of the natural numbers or even of a set (see all of the work in large cardinals, culminating in Hugh Woodin's recent work on extender models for supercompact cardinals).
> The incompleteness theorems don't falsify the idea that mathematics is logically consistent; they do falsify the idea that any reasonable mathematical theory could prove its own logical consistency.
Yes, true, but I think that amounts to the same thing. Our inability to prove the thesis casts in doubt our right to assert it at all. That's certainly true for any other mathematical idea. No one was willing to say that Fermat's Last Theorem, or the Four-Color Map Theorem, were proven, until they were.
Nevertheless, I shouldn't have said that the Theorems "falsify" the logical consistency of mathematics. They prevent the notion from being demonstrated, but doesn't invalidate its existence as a hypothesis.
> They also don't remove the possibility that there is some metamathematical justification for ...
Yes, but that's not a positive claim, it's the assertion that it can't be ruled out. And such an effort might fall afoul of the "sufficiently complex" criterion of Godel's Theorems, which brings us full circle.
Yes, true, but I think that amounts to the same thing. Our inability to prove the thesis casts in doubt our right to assert it at all. That's certainly true for any other mathematical idea. No one was willing to say that Fermat's Last Theorem, or the Four-Color Map Theorem, were proven, until they were.
Nevertheless, I shouldn't have said that the Theorems "falsify" the logical consistency of mathematics. They prevent the notion from being demonstrated, but doesn't invalidate its existence as a hypothesis.
> They also don't remove the possibility that there is some metamathematical justification for ...
Yes, but that's not a positive claim, it's the assertion that it can't be ruled out. And such an effort might fall afoul of the "sufficiently complex" criterion of Godel's Theorems, which brings us full circle.
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I know a lot of guys who started Math Phds...I know no one who actually finished one.
I read the title wrong and became curious what a grad student in math does all day after they graduate? Is there a lot of work in that field?
In the US, the NSA hires a lot of math folks: https://www.nsa.gov/careers/career_fields/mathematics.shtml
The blog post certainly illustrates the importance of a good copywriter...
"Tool to get rid of dust."
"Tool to get rid of dust."
[deleted]
Tool not get rid of all dust, only dust tool can get to.
You're focusing on the specifics too much. This is actually very akin to a startup. How can you convey what you do in a clear and concise sentence without muddying it with details. My point in the original reply is that conveying the idea of a vacuum cleaner even with the harsh restrictions set, can still be done without sounding like a robot or caveman.
There's a very big difference between a tool that gets rid of all dust, and a tool that only gets rid of a small subset of dust.
It may be a specific, but so is the fact that it gets rid of dust...
Today on HN: Math analogy spawns discussion on marketing copy and the properties of vacuum cleaners.
It may be a specific, but so is the fact that it gets rid of dust...
Today on HN: Math analogy spawns discussion on marketing copy and the properties of vacuum cleaners.
Funny stuff, and no doubt often accurate, but it wasn't my experience. I wanted to work in game theory, and there weren't an game theorists in the Harvard mathematics department; there were, however, in the business school. So I went over there, and Elon Kohlberg told me of an outstanding problem, and gave me a couple of papers to explain. I worked until I had a proof of the conjecture, and that was that, even though it had in fact been proved elsewhere (the Mertens-Neyman Theorem) a couple of months before I finished.
The whole thing took about a year, after which I stared my post-doc in public policy. After that, I became a stock analyst. After that, I tried some startups (not successfully). Then I became a self-employed industry analyst.
What could be more straightforward than that? :)
The whole thing took about a year, after which I stared my post-doc in public policy. After that, I became a stock analyst. After that, I tried some startups (not successfully). Then I became a self-employed industry analyst.
What could be more straightforward than that? :)
The Man Who Almost Invented The Vacuum Cleaner
The man officially credited with inventing the vacuum cleaner is Hubert Cecil Booth. However, he got the idea from a man who almost invented it.
In 1901 Booth visited a London music-hall. On the bill was an American inventor with his wonder machine for removing dust from carpets. The machine comprised a box about one foot square with a bag on top.
After watching the act -- which made everyone in the front six rows sneeze -- Booth went round to the inventor's dressing room.
"It should suck not blow," said Booth, coming straight to the point. "Suck?", exclaimed the enraged inventor. "Your machine just moves the dust around the room," Booth informed him. "Suck? Suck? Sucking is not possible," was the inventor's reply and he stormed out. Booth proved that it was by the simple expedient of kneeling down, pursing his lips and sucking the back of an armchair. "I almost choked," he said afterwards.
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There's a story here somewhere that aught to relate to mathematics, if only I could find it...