When we do that, there is some maths happening in our brains, it just isn't conscious.
No, there isn't. Recognizing spatial structures and symmetries is a strong feature of the brain, but it's not mathematical reasoning, i.e. it does not necessarily lead to an unassailable logical truth. As an example, there are many "geometric truths" that one can convince one self of by drawing suitable geometric diagrams, but which turn out to be false if we attempt to prove them for example in Euclid's plane geometry.
You're right, that is exactly what is happening in the spider's case. However to "just point" to an intercept seems like an incredibly simple thing to us, but to do it with the amount of brain cells a spider has is quite a trick. Bear in mind this all has to come from sensory data - it has to find branches, blades of grass or whatever and make a decision whether it is feasible to spin a web there, using very rough input from it's eyes.
The eyes are not all the senses that a spider uses to spin its web.
The article doesn't mean 'linear' in the sense of 'linear dependence on a set of variables', but rather 'linear' as in 'sequence of events that follow one another as a direct consequence of the previous one'.
During the last month, one of our customers accused us of providing another customer with their specification.
Forget about fancy industrial espionage scenarios with evil Chinese crackers. If this really happened and isn't just paranoia on the part of your customer, chances are it was someone in your company who had authorized access to the specs and, probably out of stupidity or by accident, forwarded the confidential information to someone they shouldn't have.
Sadly your most effective approach is to comb through e-mail logs of people with access to this document, and see what attachments they've been forwarding recently.
As others have already explained, there's no way to prevent this kind of thing from happening again, either. Just educate your people to keep confidential documents secure and get rid of people who disregard this rule.
That's because they recalled that a previous Democracy in Athens had killed one of mankind's greatest thinkers, Socrates, simply because they didn't like him. They did not want the right to life to be taken-away by a simple 50% +1 vote
Did you realize that to kill a man in many states it takes not 50% of people + 1 vote, but just 12 people? Well, maybe also the judge and DA.
Is there any guarantee that Berlusconi will still be Prime Minister in January?
Hilarious. There's more chance of Vladimir Putin being deposed than Berlusconi. He's prime minister for the fourth time already, and he's not going anywhere.
The fact is that if you have the brains and discipline to get a PhD in science and become a prof at MIT or Harvard, you could probably make a *ton* more money, and quicker, and have much more job security, going into medicine, law, or business. Or you could be slave-labour grad student for 7 years, then serfdom post-doc for 6 years, slave as an untenured professor for 6 years and then be fired... er sorry, "denied tenure" and be looking for a brand new job or an entirely new career at the age of 35.
That's a harsh way to treat someone who finished their bachelor's at 16!
Seriously, academia tends to attract people who aren't into medicine, law, or business for the sake of making a lot of money (not to mention being unsuitable for those fields because of differing personalities and social skills). I could make more money not playing a grad student, but it's a decision I made because I got bored of the pressure and tedium of working with people who didn't really know what they were doing (usually called business). Don't assume everyone is driven by lust for money or a luxurious career.
I recommend "Mathematical Methods of Classical Mechanics" by V.I. Arnold
Seconded, but make sure you have another textbook in mechanics handy for the inevitability that you get confused by Arnold's presentation. Goldstein is probably a good choice.
When someone has a 40-page purported proof of a famous difficult theorem, any mathematician will stop reading it after the first blatant error. It is up to the author to fix his own mistakes, and before they do so the whole thing is worthless.
For those who get excited about Millennium problem "solutions" on arXiv.org, there's also what I interpret as an attempt to solve Navier-Stokes posted recently:
http://arxiv.org/abs/0806.4902
Have fun finding the first error!
This thread is so full of misinformation I don't know where to start.
Newtonian mechanics (no matter if you dress it up as Lagrangian or Hamiltonian mechanics) is basically just solving a second order ODE with constraints. Depending on how you set up the constraints and discretize the system, you end up solving a linear system of equations on each time step. Oh, and forget analytical solutions. There are like a handful of mechanical systems that you can solve analytically (called integrable), the rest can be shown to be impossible.
This is the approach used in real mechanics simulations. Guess what, it's too expensive for real-time computer games. That's when you get creative and start bending the rules in such a way that the physics is no longer strictly correct, but the methods work incrementally in such a way that from the state of the system at the previous time step you compute the state at the next time step, update the forces, and then maybe do some correction steps. No linear systems of equations to solve, much faster algorithms, but not strictly physical.
Then you have a whole world of elastic bodies and fluid simulations that I haven't even touched on. Again there the operating principle is: "Cut corners to make it fast but not too unrealistic".
In my experience even most engineers don't do math beyond simple algebra and maybe trig. The most useful math for any engineer is statistics, which gives you a way to quantify the gap between theory and reality, because reality is just too damn complicated.
Forget cranking numbers or solving equations. Engineers (and most people who work in technical or scientific areas) need logic and abstractions, both of which you can learn while doing abstract mathematics.
Calculus is useful to read research papers, but it's real world application is limited. As my vector calc teacher put it, "half of your grade will be based on setting up the equation right, because once you get out of this classroom you'll just put the equation into a computer and it will come up with a faster and more accurate answer."
People who fail their Calculus I course will not be able to set up the equation nor will they have any idea which equation they actually have to set up.
Um, So? Why should a teacher master differential equations to teach algebra? I'd rather have a good teacher that knows enough math, than a great mathmetician that can't teach. When you require degrees, you restrict more than you enable.
Because math teachers who do not understand for example what a derivative is cannot teach limits, teachers who do not understand what a polynomial is cannot teach algebra, etc. Failure to understand fundamental topics translates to an inability to clearly explain even the less advanced things.
What really annoys me is the anachronistic faculty who force their grad students to use Fortran 77...
I doubt anyone requires F77 given the newer versions. And Fortran isn't an anachronism any more than C is. Together they are two of the de facto languages in scientific computing.
Not quite sure why people assume all of the Africa is starving or lacks critical infrastructure. Take a look at the pictures on the wikipedia entry for Johannesburg, for comparison sake.
Obviously! I mean, look : one apple, two apples, three apples. There. Numbers. See that funny relation between the diameter and circumference/area of a circle? There's pi. And so on...
That reasoning fails when you realize mathematics is not the same as arithmetic. To claim that purely abstract concepts that have no representation in the physical world are discovered pretty much necessitates belief in a Platonic 'world of ideas'.
I would say a few things are invented (like axioms of plane geometry), then the rest are deduced based on those things.
If you are certain that you want a career in CS (possibly as a researcher or at a place like Google), then pick the school which has the best CS program. Nothing beats working with motivated people who will challenge you.
If however you are not totally certain yet that CS is your future then pick the school that also offers alternatives you are interested in, preferably as wide a range as possible. If you get burned out over your potential major it is important to be able to switch to something else so that you're not stuck in a field you have no interest in.
Wikipedia will never be a reliable source for sociological, economical, historical, or political content. Take the article about Continuation War for example. It is alternatively biased towards the Finns (miraculous victory of 1944, superiority of Finnish troops), the Russians (Finnish Nazi affiliation, concentration camps, and genocide), or the British (an RAF squadron in Murmansk that purportedly shot down dozens of Finnish planes despite no records of such engagements or losses existing).
It is however a pretty good starting point for physical sciences and mathematics. There isn't going to be bias in the article about commutative algebras. I still wish there were more references to textbooks in most of the scientific articles.
Yeah, thanks. Come over here and leech off of the system we have built on our (very high) taxes.
I would rather have foreign students who are intelligent and hard working (and who hopefully choose to live here even after getting their degree) than stupid locals.
Is it reasonable to assume that every student will carry out their homework assignment in isolation? I don't think it is.
Speaking as a TA in mathematics at a technical university (plus my personal experiences as a student), most people probably do their homework as teams. "Study groups" where a handful of friends get together and share solution fragments are quite ubiquitous. Instead of returning your homework for credit you are (sometimes) called upon to explain "your solution" on the blackboard in front of the TA, which discourages blatant copying. The homework problems are geared towards this system, as they can be rather challenging and slogging through them on your own will take a lot of time. We also organize official study groups where students can ask a faculty member for hints and tips on their homework. Presumably for the benefit of people who do not know many people in their class to work with.
...and the people interested in them generally have journal subscriptions and such to access details. I think a decent criteria to start with is if the proof takes less than a page, and uses high school level mathematics. University students or faculty have access to the university's subscriptions so a cite on Wikipedia suffices.
Not all mathematical knowledge is currently encoded in research journals or textbooks. There are proofs and tools that are known to work but are either too elementary to publish in a journal or too obscure to print out in a textbook in detail. It's very common to go through a load of textbooks and have each and every one basically explain the same thing exactly the same way, which is annoying if you want a deeper understanding on some particular detail. Frequently the details are left out as an exercise for the reader, so unless you can figure it out yourself you're stuck.
Even if such a known mathematical fact would be interesting to include in a WP article this becomes impossible because there is no source to use and presumably unsourced statements are bad. The only recource is then to add a short subarticle explaining the proof or method, but this then gets deleted by the deletionists.
Wikibooks is a project under the umbrella of wikipedia which aims to create "a free library of educational textbooks that anyone can edit". Were it for me, the mathematical demonstrations will go there... (disclaimer: I have a degree in maths)
A one-page collection of elementary proofs is not a textbook. Presumably if someone was to create a Wikibook that contained these proofs it would get deleted (by the same deletionhounds?).
At the same time, they could link to Mathworld - they do have a good assortment, and they annotate nicely. Wikipedia probably can't do as good a job. The Mathworld entry on the Pythagorean theorem is surprisingly engaging - even including pop culture references like the Scarecrow reciting it incorrectly in the Wizard of Oz.
I hope not. Mathworld is, for the most part, awful as any kind of reference. Half the topics are empty or nonexistent. There is a heavy slant towards certain specialized topics and especially "pure" mathematics at the expense of anything that smacks of applications. Half the articles are simply long and winding lists of minutiae and funky identities culled from research literature, but usually do not properly motivate or even define the actual concept being discussed.
Also, we're in good company; Isaac Asimov's autobiography mentions that he cruised through algebra, slogged when he got to calculus, and came to a screeching halt on differential equations. That's another way of phrasing what others have said here about getting your basics nailed down -- a knack for sloppy visualizations may let you bluff your way through geometry and linear algebra, but will let you down badly once you reach higher math.
Sidestepping the question whether DEs are higher math than linear algebra or geometry, the reason many students find DEs hard is because they're usually so badly taught:
"Here is an equation. Here is the trick to solve it. Here is another equation. Here is another trick to solve it. Here is yet another equation and yet another trick."
"What if you have an equation like this or that?"
"You can't solve that. Anyway, here's another equation..."
How many students will understand the basic properties of initial value problems or have any idea what to do when confronted with a DE that has no analytical solution? These things are fundamental to all kinds of science and modelling, and people are basically being made to memorize tricks on objects they have no understanding of whatsoever.
Slashdot Mirror
When we do that, there is some maths happening in our brains, it just isn't conscious.
No, there isn't. Recognizing spatial structures and symmetries is a strong feature of the brain, but it's not mathematical reasoning, i.e. it does not necessarily lead to an unassailable logical truth. As an example, there are many "geometric truths" that one can convince one self of by drawing suitable geometric diagrams, but which turn out to be false if we attempt to prove them for example in Euclid's plane geometry.
You're right, that is exactly what is happening in the spider's case. However to "just point" to an intercept seems like an incredibly simple thing to us, but to do it with the amount of brain cells a spider has is quite a trick. Bear in mind this all has to come from sensory data - it has to find branches, blades of grass or whatever and make a decision whether it is feasible to spin a web there, using very rough input from it's eyes.
The eyes are not all the senses that a spider uses to spin its web.
The article doesn't mean 'linear' in the sense of 'linear dependence on a set of variables', but rather 'linear' as in 'sequence of events that follow one another as a direct consequence of the previous one'.
During the last month, one of our customers accused us of providing another customer with their specification.
Forget about fancy industrial espionage scenarios with evil Chinese crackers. If this really happened and isn't just paranoia on the part of your customer, chances are it was someone in your company who had authorized access to the specs and, probably out of stupidity or by accident, forwarded the confidential information to someone they shouldn't have.
Sadly your most effective approach is to comb through e-mail logs of people with access to this document, and see what attachments they've been forwarding recently.
As others have already explained, there's no way to prevent this kind of thing from happening again, either. Just educate your people to keep confidential documents secure and get rid of people who disregard this rule.
That's because they recalled that a previous Democracy in Athens had killed one of mankind's greatest thinkers, Socrates, simply because they didn't like him. They did not want the right to life to be taken-away by a simple 50% +1 vote
Did you realize that to kill a man in many states it takes not 50% of people + 1 vote, but just 12 people? Well, maybe also the judge and DA.
Is there any guarantee that Berlusconi will still be Prime Minister in January?
Hilarious. There's more chance of Vladimir Putin being deposed than Berlusconi. He's prime minister for the fourth time already, and he's not going anywhere.
The fact is that if you have the brains and discipline to get a PhD in science and become a prof at MIT or Harvard, you could probably make a *ton* more money, and quicker, and have much more job security, going into medicine, law, or business. Or you could be slave-labour grad student for 7 years, then serfdom post-doc for 6 years, slave as an untenured professor for 6 years and then be fired... er sorry, "denied tenure" and be looking for a brand new job or an entirely new career at the age of 35.
That's a harsh way to treat someone who finished their bachelor's at 16!
Seriously, academia tends to attract people who aren't into medicine, law, or business for the sake of making a lot of money (not to mention being unsuitable for those fields because of differing personalities and social skills). I could make more money not playing a grad student, but it's a decision I made because I got bored of the pressure and tedium of working with people who didn't really know what they were doing (usually called business). Don't assume everyone is driven by lust for money or a luxurious career.
You do know they don't arrest you and convict you based solely on DNA evidence
Oh they will if the crime is heinous/high-profile enough and there are no other leads/suspects. It also helps if the DA is up for re-election.
I recommend "Mathematical Methods of Classical Mechanics" by V.I. Arnold
Seconded, but make sure you have another textbook in mechanics handy for the inevitability that you get confused by Arnold's presentation. Goldstein is probably a good choice.
When someone has a 40-page purported proof of a famous difficult theorem, any mathematician will stop reading it after the first blatant error. It is up to the author to fix his own mistakes, and before they do so the whole thing is worthless.
For those who get excited about Millennium problem "solutions" on arXiv.org, there's also what I interpret as an attempt to solve Navier-Stokes posted recently: http://arxiv.org/abs/0806.4902 Have fun finding the first error!
This thread is so full of misinformation I don't know where to start.
Newtonian mechanics (no matter if you dress it up as Lagrangian or Hamiltonian mechanics) is basically just solving a second order ODE with constraints. Depending on how you set up the constraints and discretize the system, you end up solving a linear system of equations on each time step. Oh, and forget analytical solutions. There are like a handful of mechanical systems that you can solve analytically (called integrable), the rest can be shown to be impossible.
This is the approach used in real mechanics simulations. Guess what, it's too expensive for real-time computer games. That's when you get creative and start bending the rules in such a way that the physics is no longer strictly correct, but the methods work incrementally in such a way that from the state of the system at the previous time step you compute the state at the next time step, update the forces, and then maybe do some correction steps. No linear systems of equations to solve, much faster algorithms, but not strictly physical.
Then you have a whole world of elastic bodies and fluid simulations that I haven't even touched on. Again there the operating principle is: "Cut corners to make it fast but not too unrealistic".
In my experience even most engineers don't do math beyond simple algebra and maybe trig. The most useful math for any engineer is statistics, which gives you a way to quantify the gap between theory and reality, because reality is just too damn complicated.
Forget cranking numbers or solving equations. Engineers (and most people who work in technical or scientific areas) need logic and abstractions, both of which you can learn while doing abstract mathematics.
Calculus is useful to read research papers, but it's real world application is limited. As my vector calc teacher put it, "half of your grade will be based on setting up the equation right, because once you get out of this classroom you'll just put the equation into a computer and it will come up with a faster and more accurate answer."
People who fail their Calculus I course will not be able to set up the equation nor will they have any idea which equation they actually have to set up.
Um, So? Why should a teacher master differential equations to teach algebra? I'd rather have a good teacher that knows enough math, than a great mathmetician that can't teach. When you require degrees, you restrict more than you enable.
Because math teachers who do not understand for example what a derivative is cannot teach limits, teachers who do not understand what a polynomial is cannot teach algebra, etc. Failure to understand fundamental topics translates to an inability to clearly explain even the less advanced things.
What really annoys me is the anachronistic faculty who force their grad students to use Fortran 77...
I doubt anyone requires F77 given the newer versions. And Fortran isn't an anachronism any more than C is. Together they are two of the de facto languages in scientific computing.
Not quite sure why people assume all of the Africa is starving or lacks critical infrastructure. Take a look at the pictures on the wikipedia entry for Johannesburg, for comparison sake.
Would that be the "most dangerous city in the world with respect to street crime" Johannesburg? Hardly a place where international researchers would flock to live and work in.
I've also heard Internet connections are plentiful and inexpensive in Mogadishu, but I wouldn't start a research center there.
it will most likely just be used for more nuclear weapons simulations.
s/nuclear weapons simulations/homeland security boondoggles
Obviously! I mean, look : one apple, two apples, three apples. There. Numbers. See that funny relation between the diameter and circumference/area of a circle? There's pi. And so on...
That reasoning fails when you realize mathematics is not the same as arithmetic. To claim that purely abstract concepts that have no representation in the physical world are discovered pretty much necessitates belief in a Platonic 'world of ideas'.
I would say a few things are invented (like axioms of plane geometry), then the rest are deduced based on those things.
If you are certain that you want a career in CS (possibly as a researcher or at a place like Google), then pick the school which has the best CS program. Nothing beats working with motivated people who will challenge you.
If however you are not totally certain yet that CS is your future then pick the school that also offers alternatives you are interested in, preferably as wide a range as possible. If you get burned out over your potential major it is important to be able to switch to something else so that you're not stuck in a field you have no interest in.
Wikipedia will never be a reliable source for sociological, economical, historical, or political content. Take the article about Continuation War for example. It is alternatively biased towards the Finns (miraculous victory of 1944, superiority of Finnish troops), the Russians (Finnish Nazi affiliation, concentration camps, and genocide), or the British (an RAF squadron in Murmansk that purportedly shot down dozens of Finnish planes despite no records of such engagements or losses existing).
It is however a pretty good starting point for physical sciences and mathematics. There isn't going to be bias in the article about commutative algebras. I still wish there were more references to textbooks in most of the scientific articles.
Yeah, thanks. Come over here and leech off of the system we have built on our (very high) taxes.
I would rather have foreign students who are intelligent and hard working (and who hopefully choose to live here even after getting their degree) than stupid locals.
Is it reasonable to assume that every student will carry out their homework assignment in isolation? I don't think it is.
Speaking as a TA in mathematics at a technical university (plus my personal experiences as a student), most people probably do their homework as teams. "Study groups" where a handful of friends get together and share solution fragments are quite ubiquitous. Instead of returning your homework for credit you are (sometimes) called upon to explain "your solution" on the blackboard in front of the TA, which discourages blatant copying. The homework problems are geared towards this system, as they can be rather challenging and slogging through them on your own will take a lot of time. We also organize official study groups where students can ask a faculty member for hints and tips on their homework. Presumably for the benefit of people who do not know many people in their class to work with.
Not all mathematical knowledge is currently encoded in research journals or textbooks. There are proofs and tools that are known to work but are either too elementary to publish in a journal or too obscure to print out in a textbook in detail. It's very common to go through a load of textbooks and have each and every one basically explain the same thing exactly the same way, which is annoying if you want a deeper understanding on some particular detail. Frequently the details are left out as an exercise for the reader, so unless you can figure it out yourself you're stuck.
Even if such a known mathematical fact would be interesting to include in a WP article this becomes impossible because there is no source to use and presumably unsourced statements are bad. The only recource is then to add a short subarticle explaining the proof or method, but this then gets deleted by the deletionists.
Wikibooks is a project under the umbrella of wikipedia which aims to create "a free library of educational textbooks that anyone can edit". Were it for me, the mathematical demonstrations will go there... (disclaimer: I have a degree in maths)
A one-page collection of elementary proofs is not a textbook. Presumably if someone was to create a Wikibook that contained these proofs it would get deleted (by the same deletionhounds?).
At the same time, they could link to Mathworld - they do have a good assortment, and they annotate nicely. Wikipedia probably can't do as good a job. The Mathworld entry on the Pythagorean theorem is surprisingly engaging - even including pop culture references like the Scarecrow reciting it incorrectly in the Wizard of Oz.
I hope not. Mathworld is, for the most part, awful as any kind of reference. Half the topics are empty or nonexistent. There is a heavy slant towards certain specialized topics and especially "pure" mathematics at the expense of anything that smacks of applications. Half the articles are simply long and winding lists of minutiae and funky identities culled from research literature, but usually do not properly motivate or even define the actual concept being discussed.
Also, we're in good company; Isaac Asimov's autobiography mentions that he cruised through algebra, slogged when he got to calculus, and came to a screeching halt on differential equations. That's another way of phrasing what others have said here about getting your basics nailed down -- a knack for sloppy visualizations may let you bluff your way through geometry and linear algebra, but will let you down badly once you reach higher math.
Sidestepping the question whether DEs are higher math than linear algebra or geometry, the reason many students find DEs hard is because they're usually so badly taught:
"Here is an equation. Here is the trick to solve it. Here is another equation. Here is another trick to solve it. Here is yet another equation and yet another trick."
"What if you have an equation like this or that?"
"You can't solve that. Anyway, here's another equation..."
How many students will understand the basic properties of initial value problems or have any idea what to do when confronted with a DE that has no analytical solution? These things are fundamental to all kinds of science and modelling, and people are basically being made to memorize tricks on objects they have no understanding of whatsoever.