Thank you for your input; it has proved invaluable to the topic at hand, as my initial offering was tragically unclear. I have taken it upon myself to look up each word of this post to ensure that it is semantically correct.
I fail to see the grounds for being sarcastic about the worthy endeavor to use words correctly. Indeed, I for one make sure I understand the meaning of each word I use, independently of the topic being discussed, and I do believe the fact that words be used accoding to their senses to be invaluable while discussing any topic.
Hmph. I mentioned the representation theory of semisimple Lie groups and the fact that it basically "reduces" to "elementary" manipulations of diagrams and tableaux as an example of another theory which "is in the end reduced to simple operations" yet as a theory "is not trivialized"; in the same vein, I mentioned number theory, which in some respects strives (with various levels of success...) to reduce its answers to elementary manipulations with integers.
I was trying to show both that it is not only signal processing which achieves this "reduction" and that this "reduction" by no means renders the theory uninteresting or useless. In fact, these examples suggest the very opposite!
The fact that you did not understand this is only a reflection of the patently poor intelligibility of my post.
Hmm. Did I ever say that the theorem was relevant to the design of audio codecs? In the post I replied to originally, you said
The result he proved is nice mathematics, but you don't need it for iPods or audio coding.
and I did not even mention that point. On the other hand you continued to say:
First of all, for many engineering purposes, it only matters that it works, not that you can prove that it works theoretically. Secondly, audio coding is done over discretely sampled signals, and most of those theorems become simple linear algebra in that case.
Of these points, I chose to ignore the first one (which is a truism) and contested the second one.
As for the last line in this last post of yours, well, since I fail to see what you are referring to, I'll ignore it: honestly it's a tad boring, figuring out where you see name-dropping and such.
The "in the discrete domain" is just a phrase, which means "after discretizing". The use of the word "domain" in that phrase is completely unrelated to the technical use of the word "domain" as used, say, in "the domain of a function".
BTW, if you are "talking about the kinds of objects engineers deal with: real-valued functions over the integers, representing regularly sampled values", then you are not only talking abour functions with discrete range, but also with a discrete domain (and now, yes, I am using the term "domain" in the technical sense)
Even when you are ultimately only interested in dealing with Fourier analysis of discretized signals, the kind of mathematics Carleson worked with is useful. Working with discretizations in a meaningful way imposes the need of being sure that discretizing will actually aproximate the real, non-discrete, phenomenon, and how that aproximation will work. The kind of information you get from the theory is how finely you need to sample to preserve the amount of detail in the original signal you are interested in preserving, how you can manipulate sampling parameters when the nature of the signal you are working changes, and what not.
Those questions, which are fundamental in signal processing of any kind, by no means reduce to finite linear algebra!
In any case, and more deeply, pretending, as in the blurb on top of this thread, to trace the influence of one particular theorem on any aspect of reality (iPods, say) is pretty naive, except in very special cases. There are many ways in which Carleson can have influenced the design of the iPod other than someone explicitely saying "...and by Carleson's theorem..." during the design process of the thingy.
I do not find any big words in my post. I do, though, find words that one might no be familiar with if you are not a mathematician (even an undergrad student). I'll gladly explain any term you might not understand.
Saying that Fourier analysis becomes, when you go to the discrete domain, "simple linear algebra" shows very clearly that you do not understand the complexities of discrete fourier analysis. That something is in the end reduced to simple operations does not mean that the subject has been trivialized. Most of number theory "reduces" to finite, discrete computation with integer numbers; the representation theory of semisimple Lie groups (and that of the symetic groups) "reduces" to "mere" handling of Ferrers diagrams and Young tableaux. No one who knows anything about these subjects would ever use the word "simple"...
In any case, if you think that one could do with just the discrete version of Fourier theory, and that somehow the original, 'continuous' theory becomes irrelevant when dealing with iPods and what not, well, you need to go back and read up on those subjects...
Proposing prizes as the motivation for scientists shows that you do not understand how scientific work gets done.
Of course, there are prizes around, offered to people that solve specific unsolved problems (like, say, the Clay Milennium Prize Problems), and, more usually, prizes given out to people for their achievements.
But relevant scientific work is not only the solution of those "prize-worthy" problems like the ones you list, but the every day toil in the quest for information, without which those big problems would never ever be solved. If for some reason we were left only with the geniuses, science would essentially stop.
Rewarding geniality only is blindness, as science is not built upon the work of geniuses. Geniuses do show the way, but the roads are not usually built by them. And, do note, it is quite rare that society can benefit from the mere knowing of where the way is: society needs to get to where the way leads to.
Indeed, every single study made on this subject shows that the correlation is strong. Not only in the US, but everywhere. This Newsweek journalist seems not to have done his homework.
What is the "officially provided" and commonly agreed upon way to access a sftp:// URL? One that will interact with the ssh-agent, and look into my keychain for keys, of course. What about fonts:///? What is the standard way of opening a resource located by an http URL? cp(1) seems not to grok
If they don't use the proper APIs, it simply doesn't work.
The GnomeVFS API may well be difficult to work with (*), but this statement of yours applies to absolutely every API out there.
(*) Actually, most programs rarely need much more than rather small subset of the API, which basically mimics the POSIX file API; hence, your use of the word `impossible' in this context is, at the very least, dubious.
Among the many things a compositing manager can do is precisely that, scaling windows. A window manager which does its own compositing can very well do scaling *today*. All this GL stuff about which we've been hearing lately is an attempt at being able to do the compositing with GL, which is good for many reasons.
Before becoming irate and all UPPERCAPPY, you may want to actually research what are the uses of this developments you describe as eye-candy. At the very least, do not go shouting at developers like that.
Natural, live languages are not spec'ed and frozen, and the meaninings of words and expressions change. That is how they work. Of course, some of that change is due to mistakes (for example, ignorance of that "begging the question" originally meant, and taking a slightly immaginative semi-literal interpretation) but that is not bad. English itself can be described as the result of a long long list of spelling, pronounciation, morphological and grammatical errors and deviations.
Unless you are saying that Adobe uses unpublished information on how to render PDF files, you most certainly do not make much sense. It is not a condition one can reasonably impose on a spec, that it be automatically implementable to perfection by every coder out there. The kind of guarantee you are talking about simply does not make sense.
Hmm, this is not insightful: it is in the very definition of "modeling" that information is lost. If you "model" something in such a way that you do not lose anything, what you do is duplicate, and, quite clearly, you have not simplified your task of understanding it a tiny bit...
I've seen people around here complaining that the models we make of reality are lossy with the same spirit that ID proponents say that evolution is "just a theory". Man that annoys me! (Not saying you were doing this, mind)
Roger Penrose has argued (Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press, 1994, ISBN 0-19-853978-9) essentially that your belief is wrong.
He argues against the notion that the description of the physical world we have now is correct, based on the fact that the description we have is (let's say, for simplicity) computable; being based on an axiomatic system, like you prefer to say, is basically equivalent to that. That our minds cannot be computable, or formal-system-based, is a quite an important point in his argument.
I am not saying Penrose is right; I have not yet been able to decide if I think his argument holds water or not. He presents a very well thought-out argument, with much more detail that what most people will take. You may want to look into it.
An axiom is nothing more than an accepted but unproven truth, ie: an assumption.
This is a very widely held view of what an axiom is, but it is completely different from the usage of the word in current math (and, of course, in any discipline which uses math to model its object of study)
Axioms are simply assumptions, and that is not the same thing as an accepted and unproven truth. It is just an assumption.
When the group theorists posits that "the group operation is associative", she is by no means whatsoever stating that she believes that all operations are associative but she can't prove it.
An axiom is just an assumtion. Mathematicians use axioms instead of just stating the assumption explicitely before every single statement that depends on them only because of convenience and psychological reasons. Thus, there is no difference between:
Axiom 1: Pigs fly.
Theorem 1: The sun revolves around the earth.
Proof. We leave this as an easy excercise for the reader. QED
and
Theorem 1: If pigs fly, then the sun revolves around the earth.
Proof. We leave this as an easy excercise for the reader. QED
Of course, if what you are writing is a list of 500 theorems, putting the assumptions at the top is much more convenient.
The usage of the word axiom you have in mind more or less aplies to what the Greeks had in mind, and other people that came after them. There was a change when non-euclidean geometries were discovered, when algebra became modern algebra (probably, when van den Wearden's book came out).
As long as that "higher math" you speak of is a formal system, incompleteness persists; the statement with which you had a problem will have been resolved, maybe, but there will be others, maybe new ones that appeared because of the enlargement of the theory, that are as problematic as the old one. Read up on \omega-incompleteness.
If this "higher math" is not a formal system, well, then you stopped doing math...
There is no reason why the number of dimensions in the model we make of the physical world has to correspond to the number of dimensions of the physical world. In fact, if you think about it, the world does not have a "number of dimensions": this number 3 that people keep bringing up, or 4, when they have read some relativity and/or seen some SciFi movie, is the number of dimensions of the model of physical space we are most familiar with.
BTW, have you heard of an experiment designed to show that the world is 3- or 4-dimensional?
This is not as ridiculous as it sounds. For example, experiments can be designed to decide whether physical space (excluding time...) is odd- or even-dimensional. Indeed, detailed study of the wave equation, which models electromagnetism "exactly", shows that waves have very, very different behaviour in odd-dimensional spaces and in even-dimensional spaces (for the cognoscenti, the difference comes from the very different shapes of the supports of the fundamental solutions of the wave equation).
Being a math minor should be enough for you to know by now that "dt" is not "really tiny", "aproximately zero", or anything like that: it is not a number, nor a function nor anything.
The notation df/dt is simply a different way of writing "f'(t)" or "t |-> lim_{s\to t}(f(s)-f(t))/(s-t)" It is a very very good notation, because in lots of situations, it does behave like a fraction, and one can use it as a mnemonic tool (as in the rule dx/dt=dx/ds.ds/dt, and what not), but unless you understand that this kind of usage is only justified by the fact that there is a theorem which backs it, you have not understood anything.
In particular, the pieces of the notation "df/dt" have no meaning when used by themselves (at least, at this level of calculus... there are things like exterior differentiation, the grassman calculus and what not; but those are a different game)
One usually sees people (in fact, this tends to be done by physicists...) using "dt" to stand for a "very small quantity", but that is just an heuristic way of doing things, and those reasonings are, formally, of no use; of course, the trained mind will know to limit itself to only use those reasonings that can be justified appropiately by real theorems. One can deduce a formula for the volume of a revolution solit using arguments that begin like "take an element of volume dV of height dh and...", but that is just heuristics.
Infinitesimals do not exist (in standard analysis...), only limit processes, in the same way that infinite sums do not make any sense, but limits of finite sums do.
Slashdot Mirror
And being paid by a company that uses your app somehow puts you out of the community now?
I fail to see the grounds for being sarcastic about the worthy endeavor to use words correctly. Indeed, I for one make sure I understand the meaning of each word I use, independently of the topic being discussed, and I do believe the fact that words be used accoding to their senses to be invaluable while discussing any topic.
Enjoy.
"Exponential" does not mean "a lot", you know... What you are seeing is linear growth.
It is when I am in my "regular user" mode that I displike Windows the most.
Hmph. I mentioned the representation theory of semisimple Lie groups and the fact that it basically "reduces" to "elementary" manipulations of diagrams and tableaux as an example of another theory which "is in the end reduced to simple operations" yet as a theory "is not trivialized"; in the same vein, I mentioned number theory, which in some respects strives (with various levels of success...) to reduce its answers to elementary manipulations with integers.
I was trying to show both that it is not only signal processing which achieves this "reduction" and that this "reduction" by no means renders the theory uninteresting or useless. In fact, these examples suggest the very opposite!
The fact that you did not understand this is only a reflection of the patently poor intelligibility of my post.
Hmm. Did I ever say that the theorem was relevant to the design of audio codecs? In the post I replied to originally, you said
and I did not even mention that point. On the other hand you continued to say:
Of these points, I chose to ignore the first one (which is a truism) and contested the second one.
As for the last line in this last post of yours, well, since I fail to see what you are referring to, I'll ignore it: honestly it's a tad boring, figuring out where you see name-dropping and such.
Those questions, which are fundamental in signal processing of any kind, by no means reduce to finite linear algebra!
You need some kind of smoothness (piecewise continuous differentiability works, for example) in order to prove pointwise convergence.
Saying that Fourier analysis becomes, when you go to the discrete domain, "simple linear algebra" shows very clearly that you do not understand the complexities of discrete fourier analysis. That something is in the end reduced to simple operations does not mean that the subject has been trivialized. Most of number theory "reduces" to finite, discrete computation with integer numbers; the representation theory of semisimple Lie groups (and that of the symetic groups) "reduces" to "mere" handling of Ferrers diagrams and Young tableaux. No one who knows anything about these subjects would ever use the word "simple"...
In any case, if you think that one could do with just the discrete version of Fourier theory, and that somehow the original, 'continuous' theory becomes irrelevant when dealing with iPods and what not, well, you need to go back and read up on those subjects...
I marvel at people that do not realize that man pages are worth a read.
Go RTFM!
Proposing prizes as the motivation for scientists shows that you do not understand how scientific work gets done.
Of course, there are prizes around, offered to people that solve specific unsolved problems (like, say, the Clay Milennium Prize Problems), and, more usually, prizes given out to people for their achievements.
But relevant scientific work is not only the solution of those "prize-worthy" problems like the ones you list, but the every day toil in the quest for information, without which those big problems would never ever be solved. If for some reason we were left only with the geniuses, science would essentially stop.
Rewarding geniality only is blindness, as science is not built upon the work of geniuses. Geniuses do show the way, but the roads are not usually built by them. And, do note, it is quite rare that society can benefit from the mere knowing of where the way is: society needs to get to where the way leads to.
Indeed, every single study made on this subject shows that the correlation is strong. Not only in the US, but everywhere. This Newsweek journalist seems not to have done his homework.
Great idea!
In fact, I'm going to go around spreading the word to my fellow mathematicians right now. Thanks!
What is the "officially provided" and commonly agreed upon way to access a sftp:// URL? One that will interact with the ssh-agent, and look into my keychain for keys, of course. What about fonts:///? What is the standard way of opening a resource located by an http URL? cp(1) seems not to grok
It may be related, I guess, to the fact that open(3) for some reason does not like SMB mounts.The GnomeVFS API may well be difficult to work with (*), but this statement of yours applies to absolutely every API out there.
(*) Actually, most programs rarely need much more than rather small subset of the API, which basically mimics the POSIX file API; hence, your use of the word `impossible' in this context is, at the very least, dubious.
Among the many things a compositing manager can do is precisely that, scaling windows. A window manager which does its own compositing can very well do scaling *today*. All this GL stuff about which we've been hearing lately is an attempt at being able to do the compositing with GL, which is good for many reasons.
Before becoming irate and all UPPERCAPPY, you may want to actually research what are the uses of this developments you describe as eye-candy. At the very least, do not go shouting at developers like that.
Have you read <http://en.wikipedia.org/wiki/Begging_the_question #Modern_usage>?
Natural, live languages are not spec'ed and frozen, and the meaninings of words and expressions change. That is how they work. Of course, some of that change is due to mistakes (for example, ignorance of that "begging the question" originally meant, and taking a slightly immaginative semi-literal interpretation) but that is not bad. English itself can be described as the result of a long long list of spelling, pronounciation, morphological and grammatical errors and deviations.
Unless you are saying that Adobe uses unpublished information on how to render PDF files, you most certainly do not make much sense. It is not a condition one can reasonably impose on a spec, that it be automatically implementable to perfection by every coder out there. The kind of guarantee you are talking about simply does not make sense.
Well, as I said, this is related to Gödel's work on omega-incompleteness. I would hardly classify it as "obvious"!
Hmm, this is not insightful: it is in the very definition of "modeling" that information is lost. If you "model" something in such a way that you do not lose anything, what you do is duplicate, and, quite clearly, you have not simplified your task of understanding it a tiny bit...
I've seen people around here complaining that the models we make of reality are lossy with the same spirit that ID proponents say that evolution is "just a theory". Man that annoys me! (Not saying you were doing this, mind)
Roger Penrose has argued (Roger Penrose, Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University Press, 1994, ISBN 0-19-853978-9) essentially that your belief is wrong.
He argues against the notion that the description of the physical world we have now is correct, based on the fact that the description we have is (let's say, for simplicity) computable; being based on an axiomatic system, like you prefer to say, is basically equivalent to that. That our minds cannot be computable, or formal-system-based, is a quite an important point in his argument.
I am not saying Penrose is right; I have not yet been able to decide if I think his argument holds water or not. He presents a very well thought-out argument, with much more detail that what most people will take. You may want to look into it.
This is a very widely held view of what an axiom is, but it is completely different from the usage of the word in current math (and, of course, in any discipline which uses math to model its object of study)
Axioms are simply assumptions, and that is not the same thing as an accepted and unproven truth. It is just an assumption.
When the group theorists posits that "the group operation is associative", she is by no means whatsoever stating that she believes that all operations are associative but she can't prove it.
An axiom is just an assumtion. Mathematicians use axioms instead of just stating the assumption explicitely before every single statement that depends on them only because of convenience and psychological reasons. Thus, there is no difference between:
and
Of course, if what you are writing is a list of 500 theorems, putting the assumptions at the top is much more convenient.
The usage of the word axiom you have in mind more or less aplies to what the Greeks had in mind, and other people that came after them. There was a change when non-euclidean geometries were discovered, when algebra became modern algebra (probably, when van den Wearden's book came out).
Cheers.
As long as that "higher math" you speak of is a formal system, incompleteness persists; the statement with which you had a problem will have been resolved, maybe, but there will be others, maybe new ones that appeared because of the enlargement of the theory, that are as problematic as the old one. Read up on \omega-incompleteness.
If this "higher math" is not a formal system, well, then you stopped doing math...
BTW, have you heard of an experiment designed to show that the world is 3- or 4-dimensional?
This is not as ridiculous as it sounds. For example, experiments can be designed to decide whether physical space (excluding time...) is odd- or even-dimensional. Indeed, detailed study of the wave equation, which models electromagnetism "exactly", shows that waves have very, very different behaviour in odd-dimensional spaces and in even-dimensional spaces (for the cognoscenti, the difference comes from the very different shapes of the supports of the fundamental solutions of the wave equation).
Being a math minor should be enough for you to know by now that "dt" is not "really tiny", "aproximately zero", or anything like that: it is not a number, nor a function nor anything.
The notation df/dt is simply a different way of writing "f'(t)" or "t |-> lim_{s\to t}(f(s)-f(t))/(s-t)" It is a very very good notation, because in lots of situations, it does behave like a fraction, and one can use it as a mnemonic tool (as in the rule dx/dt=dx/ds.ds/dt, and what not), but unless you understand that this kind of usage is only justified by the fact that there is a theorem which backs it, you have not understood anything. In particular, the pieces of the notation "df/dt" have no meaning when used by themselves (at least, at this level of calculus... there are things like exterior differentiation, the grassman calculus and what not; but those are a different game)
One usually sees people (in fact, this tends to be done by physicists...) using "dt" to stand for a "very small quantity", but that is just an heuristic way of doing things, and those reasonings are, formally, of no use; of course, the trained mind will know to limit itself to only use those reasonings that can be justified appropiately by real theorems. One can deduce a formula for the volume of a revolution solit using arguments that begin like "take an element of volume dV of height dh and...", but that is just heuristics.
Infinitesimals do not exist (in standard analysis...), only limit processes, in the same way that infinite sums do not make any sense, but limits of finite sums do.