A lot of people are willing to speak of the beauty and power of mathematics. The physicist Wigner, for example, wrote an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences."
I, however, often find myself being forced to adopt a contrary view. In many ways mathematics is highly impotent and close to being virtually useless. The best we may be able to say is that mathematics is only slightly better than nothing at all.
The area of differential equations is a good example. Such equations are the only models of the physical world but in most cases no solution, other than brute force simulation, is possible.
The linear harmonic oscillator, y'' + k*y = 0, is perfectly soluble and well characterized. But the natural world is seldom perfectly linear. If we introduce a slight non-linearity to the problem, y'' + k1*y + k2*y^3 = 0, the equation becomes hopelessly intractable. Our ability to characterize this simple reality becomes completely lost. Is this power? Is this beauty?
Another example is the more famous three-body problem. Such a problem is the template for many physical situations but, again, no solution is possible. (Oftentimes the only progress in mathematics is a proof of impossibility.) Only a brute-force approach that is devoid of insight can provide a practical answer.
How about quantum mechanics? For anything beyond hydrogen or one-electron helium, the Schroedinger equation cannot be solved. Only humongous computer power can redeem this pitiful situation.
Many more examples could be given but the basic idea, at least to those who use mathematics, should be clear. Mathematics is only beautiful and powerful in certain limited cases. For everything else it is quite impotent.
You forgot to mention that mathematics, as conceived by the mathematician David Hilbert, is the highest and purest form of science ever conceptualized by the human mind.
rabid_fan <r...@righthere.net> writes: > A lot of people are willing to speak of the beauty and power > of mathematics. The physicist Wigner, for example, wrote > an article titled "The Unreasonable Effectiveness of > Mathematics in the Natural Sciences."
> I, however, often find myself being forced to adopt a > contrary view. In many ways mathematics is highly impotent > and close to being virtually useless. The best we may > be able to say is that mathematics is only slightly better > than nothing at all.
> The area of differential equations is a good example. > Such equations are the only models of the physical world > but in most cases no solution, other than brute force > simulation, is possible.
> The linear harmonic oscillator, y'' + k*y = 0, is > perfectly soluble and well characterized. But the > natural world is seldom perfectly linear. If we > introduce a slight non-linearity to the problem, > y'' + k1*y + k2*y^3 = 0, the equation becomes > hopelessly intractable.
Actually, not so. The solutions can be written in terms of Jacobi elliptic functions. According to Maple:
> Our ability to characterize > this simple reality becomes completely lost. Is this > power? Is this beauty?
Yes, in this case there is power and beauty.
> Another example is the more famous three-body problem. > Such a problem is the template for many physical > situations but, again, no solution is possible. (Oftentimes > the only progress in mathematics is a proof of impossibility.) > Only a brute-force approach that is devoid of insight can > provide a practical answer.
There is more to differential equations than exact solutions and numerical approximations. In particular, there is the qualitative approach. -- Robert Israel isr...@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
On Fri, 20 Nov 2009 04:34:15 +0000, rabid_fan wrote: > A lot of people are willing to speak of the beauty and power > of mathematics. The physicist Wigner, for example, wrote > an article titled "The Unreasonable Effectiveness of > Mathematics in the Natural Sciences."
> I, however, often find myself being forced to adopt a > contrary view. In many ways mathematics is highly impotent > and close to being virtually useless. The best we may > be able to say is that mathematics is only slightly better > than nothing at all.
> The area of differential equations is a good example. > Such equations are the only models of the physical world > but in most cases no solution, other than brute force > simulation, is possible.
> The linear harmonic oscillator, y'' + k*y = 0, is > perfectly soluble and well characterized. But the > natural world is seldom perfectly linear. If we > introduce a slight non-linearity to the problem, > y'' + k1*y + k2*y^3 = 0, the equation becomes > hopelessly intractable. Our ability to characterize > this simple reality becomes completely lost. Is this > power? Is this beauty?
> Another example is the more famous three-body problem. > Such a problem is the template for many physical > situations but, again, no solution is possible. (Oftentimes
> the only progress in mathematics is a proof of impossibility.) > Only a brute-force approach that is devoid of insight can > provide a practical answer.
> How about quantum mechanics? For anything beyond > hydrogen or one-electron helium, the Schroedinger > equation cannot be solved. Only humongous computer > power can redeem this pitiful situation.
> Many more examples could be given but the basic idea, > at least to those who use mathematics, should be clear. > Mathematics is only beautiful and powerful in certain > limited cases. For everything else it is quite impotent.
On Thu, 19 Nov 2009 23:36:54 -0600, Robert Israel wrote:
> Actually, not so. The solutions can be written in terms of Jacobi > elliptic functions.
The elliptic functions are simply a substitute for an inexpressible integral. In other words, we know a solution exists but the solution cannot be expressed in a form that we can directly utilize. We then call this inexpressible solution the elliptic function.
It is a common tactic. The logarithm function is another substituted name for an inexpressible integral.
This is not exactly beauty. This is not exactly power.
More importantly, however, is that the elliptic function gives no real insight into the original problem. In this case, an approximate approach shows that a third harmonic can arise for particular driving frequencies. But the elliptic function does not directly reveal this fact.
> There is more to differential equations than exact solutions and > numerical approximations. In particular, there is the qualitative > approach.
The entire qualitative approach to differential equations was started by Poincaire when he failed to solve the three-body problem. Ideally, an exact solution was desired. Out of frustration was born the qualitative approach which gives insight into the behavior of solutions in general, but for any specific case we must still resort to brute force simulation.
On Thu, 19 Nov 2009 21:23:19 -0800, Shubee wrote: > You forgot to mention that mathematics, as conceived by the > mathematician David Hilbert, is the highest and purest form of science > ever conceptualized by the human mind.
I don't disagree with the sentiment, but Hilbert never really descended from his ivory tower to muddy his hands in the inelegant business of practical computation.
If he had, then maybe his ideas of loftiness and purity would be tempered a bit.
> On Thu, 19 Nov 2009 23:36:54 -0600, Robert Israel wrote:
> > Actually, not so. The solutions can be written in terms of Jacobi > > elliptic functions.
> The elliptic functions are simply a substitute for an > inexpressible integral. In other words, we know a solution > exists but the solution cannot be expressed in a form that we > can directly utilize. We then call this inexpressible solution > the elliptic function.
> It is a common tactic. The logarithm function is another > substituted name for an inexpressible integral.
> This is not exactly beauty. This is not exactly power.
> More importantly, however, is that the elliptic function > gives no real insight into the original problem. In this > case, an approximate approach shows that a third harmonic > can arise for particular driving frequencies. But the elliptic > function does not directly reveal this fact.
> > There is more to differential equations than exact solutions and > > numerical approximations. In particular, there is the qualitative > > approach.
> The entire qualitative approach to differential equations > was started by Poincaire when he failed to solve the three-body > problem. Ideally, an exact solution was desired. Out of frustration > was born the qualitative approach which gives insight into > the behavior of solutions in general, but for any specific > case we must still resort to brute force simulation.
> Again, this is neither real power or beauty.
If you find mathematics so ugly and so useless, pray give it up and direct your interests elsewhere.
On 2009-11-20, Bill Barber <b...@moregood.info> wrote:
> The elliptic functions are simply a substitute for an inexpressible > integral. In other words, we know a solution exists but the > solution cannot be expressed in a form that we can directly utilize.
What functions do you consider that we can "directly utilize"? Obviously not logarithms, as you state later. Do trig functions or noninteger powers count?
As far as I can see, the only difference is in the familiarity of the user with the function, and nothing inherent in the function itself.
On 20 Nov 2009 04:34:15 GMT, rabid_fan <r...@righthere.net> wrote:
>I, however, often find myself being forced to adopt a >contrary view. In many ways mathematics is highly impotent >and close to being virtually useless. The best we may >be able to say is that mathematics is only slightly better >than nothing at all.
Somewhat like a Pythagorean devotee realizing that their mathematics could only deal with the rationals, leaving an infinite number of real world values unaccounted for.
> On 20 Nov 2009 04:34:15 GMT, rabid_fan > <r...@righthere.net> wrote:
> >I, however, often find myself being forced to adopt > a > >contrary view. In many ways mathematics is highly > impotent > >and close to being virtually useless. The best we > may > >be able to say is that mathematics is only slightly > better > >than nothing at all.
> Somewhat like a Pythagorean devotee realizing that > their mathematics > could only deal with the rationals, leaving an > infinite number of real > world values unaccounted for.
Since irrationals are uncountable, one has a continuum of "real world values" unaccounted for. Was the pun intentional? Guess one can say that an irrational lives in the "real world", by that meaning the world of the real numbers. :-) --D Cass
rabid_fan wrote: > I don't disagree with the sentiment, but Hilbert never > really descended from his ivory tower to muddy his hands > in the inelegant business of practical computation.
Is Hilbert's book "Methods of Mathematical Physics" no longer very well known, or is his book not what you mean by "practical computation"?
Regarding the harmonic oscillator and other examples that you said (in another post) rarely occurred in the real world, I thought their importance was as first approximations that one could then build more accurate approximations from, as well as for understanding things. One could argue that the only way to completely incorporate everything into a model is to exactly replicate the system being studied (i.e. have a pile of sand instead of various equations that capture certain aspects of a pile of sand), and this isn't the purpose of a mathematical model. It sounds like you're focused entirely on "getting some kind of answer" and ignoring the process of creating useful conceptual models of reality. Both have their place in the applications of math to our world.
On Fri, 20 Nov 2009 07:29:44 +0000, Tim Little wrote:
> What functions do you consider that we can "directly utilize"? Obviously > not logarithms, as you state later. Do trig functions or noninteger > powers count?
> As far as I can see, the only difference is in the familiarity of the > user with the function, and nothing inherent in the function itself.
Once again, I will state that my objection is only to these Neo-Platonists who insist that mathematics represents some sort of perfect and potent creation that exists independently of the human mind.
How can one argue that defining a function as the solution to an intractable integral is not a "dippy process" and about as far from a perfect Platonic realm as one can get?
In other words, we cannot express the integral and so we just lump the entire inexpressible mess into this shorthand notion of a new function. We create a solution that is not really a solution.
To be sure the function (i.e. the relation between sets as functions are currently defined) exists, but if it cannot be discerned directly then how can we claim that we are using a potent and beautiful method? The function is a "black box" whose inner workings are not completely known to us and from which we can only glimpse certain properties. This is neither beauty or power, but it does represent human mathematics.
> If you find mathematics so ugly and so useless, pray give it up and > direct your interests elsewhere.
No. I will not give up my interest in mathematics.
I will, however, walk away from haughty and prejudiced individuals such as yourself who cannot tolerate the slightest rattling of their entrenched notions.
On Fri, 20 Nov 2009 08:16:23 -0500, John Bailey wrote:
> Somewhat like a Pythagorean devotee realizing that their mathematics > could only deal with the rationals, leaving an infinite number of real > world values unaccounted for.
The Pythagoreans were not mathematicians in the sense that we understand today. They were religious mystics that worshipped numbers as the essence of all existence. Such people admit no controversy or even open speculation. Fortunately, civilization has advanced well beyond that kind of autocratic game.
Or has it?
Many people today, even if they do not know it, espouse the Platonic ideal. These people will admit no speculation or argument -- and these people inhabit the highest places
On Fri, 20 Nov 2009 07:58:53 -0800, Dave L. Renfro wrote:
> Is Hilbert's book "Methods of Mathematical Physics" no longer very well > known, or is his book not what you mean by "practical computation"?
The book of Hilbert's student, Richard Courant, is probably better known (except I don't know it because I can't recall the title.)
But at the risk of being unclear due to being too brief and simplistic, let me offer the following.
Until his dying day, the physicist Paul Dirac maintained that the entire point and purpose of mathematics was to develop the beautiful. He believed that the value of a mathematical result could literally be judged by its beauty. I could be wrong, but such an attitude toward mathematics smacks of pure Platonism.
In contrast, another physicist, Feynmann, referred to the mathematics of quantum electrodynamics, which is an elaboration of the work begun by Dirac, as a "dippy process" due to certain artifices that are necessary to prevent infinite results.
In my view, these are the two basic views that one can have on the practice of mathematics. Some may believe that mathematics is discovering portions of the eternal ideals of Plato. Others may be inclined to perceive it only as a haphazard and "dippy process."
Considering many aspects of the theoretical nature of mathematics, such as the ad hoc definitions of the elliptic functions, and the often inordinate difficulties of practical computation, I find myself leaning toward the idea of "dippiness."
> Regarding the harmonic oscillator and other examples that you said (in > another post) rarely occurred in the real world, I thought their > importance was as first approximations that one could then build more > accurate approximations from, as well as for understanding things. One > could argue that the only way to completely incorporate everything into > a model is to exactly replicate the system being studied (i.e. have a > pile of sand instead of various equations that capture certain aspects > of a pile of sand), and this isn't the purpose of a mathematical model.
Let us consider quantum mechanics and the Schroedinger equation.
(The Schroedinger equation has been long superseded by the Dirac equation, but since the Schroedinger is easier to talk about I'll use it in the example.)
The model of a quantum system is succinctly:
H * psi = E * psi (H is the quantum Hamiltonian)
This model happens to incorporate *everything*. *Everything* about the quantum system falls out of it. That is the triumph of the Hamiltonian.
But, as they say, the devil is in the details. Constructing a Hamiltonian for all but the simplest of systems is amazingly difficult and then solving the resulting equation becomes even more difficult. Only approximations and other "dippy processes" can save the day.
The theoretician may, ensconced within his ivory tower, laud the beauty and potency of the Schroedinger equation, but the experimentalist can only scream: "Eccch!"
Where is Plato when you need him (as a whipping post)?
Can God create a rock so heavy that he can't lift it? (This used to be a legitimate question of scholastic philosophy.)
I don't know, but a mathematician can certainly create beautiful formulas and models that he can't solve.
> It sounds like you're focused entirely on "getting some kind of answer" > and ignoring the process of creating useful conceptual models of > reality. Both have their place in the applications of math to our world.
The purpose of computation is not numbers; it is obtaining insight. When the model admits no solution according to mathematics then no insight is possible.
On Nov 20, 11:05 am, rabid_fan <r...@righthere.net> wrote:
> On Fri, 20 Nov 2009 07:29:44 +0000, Tim Little wrote:
> > What functions do you consider that we can "directly utilize"? Obviously > > not logarithms, as you state later. Do trig functions or noninteger > > powers count?
> > As far as I can see, the only difference is in the familiarity of the > > user with the function, and nothing inherent in the function itself.
> Once again, I will state that my objection is only to these > Neo-Platonists who insist that mathematics represents some sort > of perfect and potent creation that exists independently of the > human mind.
The quote about the "unreasonable effectiveness of mathematics" refers to the surprising effectiveness of mathematics, as a tool, at solving problems--in physics, engineering, economics, biology, chemistry, even linguistics and philosophy--problems which arise directly from human need. The "unreasonable effectiveness of mathematics" isn't a Platonic (or neo-Platonic) idea but rather a very human idea.
In article <pan.2009.11.20.15.18...@righthere.net>,
rabid_fan <r...@righthere.net> wrote: > On Fri, 20 Nov 2009 00:09:21 -0700, Virgil wrote:
> > If you find mathematics so ugly and so useless, pray give it up and > > direct your interests elsewhere.
> You are, not unexpectedly, missing the point. I have no quarrel > with the utility of mathematics and I find it to be a highly > interesting subject.
> My objection is only to these people who promote mathematics as > as an almost mystical object of perfect beauty and power (in the > manner of Plato).
While I, for example, am not turned on by Wagner operas, there are those who are, and my reaction to them is "whatever floats their boat".
I have no doubt that what turns you on will turns off lot of people, so why are you so uptight about being turned off by what turns some others on?
> To me, mathematics seems more and more to be what Feynmann called > a "dippy process."
On Nov 20, 10:24 am, rabid_fan <r...@righthere.net> wrote:
> Many people today, even if they do not know it, espouse the > Platonic ideal. These people will admit no speculation or > argument -- and these people inhabit the highest places
It seems a little ironic; in another thread, you seem to have a very specific, one might say idealized or Platonic, perception of what "universities" and "degrees" are. Here, you have one of what a Platonist (whether aware or not) "is" and will do. And it seems that you reject any argument or comment that might deviate from those perceptions...
In article <pan.2009.11.20.16.13...@righthere.net>,
rabid_fan <r...@righthere.net> wrote: > On Fri, 20 Nov 2009 00:09:21 -0700, Virgil wrote:
> > If you find mathematics so ugly and so useless, pray give it up and > > direct your interests elsewhere.
> No. I will not give up my interest in mathematics.
> I will, however, walk away from haughty and prejudiced individuals > such as yourself who cannot tolerate the slightest rattling > of their entrenched notions.
From what you had previously said, it appeared that you found any contemplation of mathematics to be repulsive and distressing, and I was just suggesting a path that might ameliorate your distress.
But apparently you enjoy that feeling, whatever it may be, so much that you are addicted to it.
rabid_fan <r...@righthere.net> wrote: > On Fri, 20 Nov 2009 08:16:23 -0500, John Bailey wrote:
> > Somewhat like a Pythagorean devotee realizing that their mathematics > > could only deal with the rationals, leaving an infinite number of real > > world values unaccounted for.
> The Pythagoreans were not mathematicians in the sense that we > understand today. They were religious mystics that worshipped > numbers as the essence of all existence. Such people admit no > controversy or even open speculation. Fortunately, civilization > has advanced well beyond that kind of autocratic game.
> Or has it?
> Many people today, even if they do not know it, espouse the > Platonic ideal. These people will admit no speculation or > argument -- and these people inhabit the highest places
Are the consequences of espousing that Platonic ideal any worse than those of espousing, say, creationism. I very much doubt it.
If the worst that mathematics engenders is such espousal, it is in better shape than the rest of the world.
rabid_fan <r...@righthere.net> wrote: > On Fri, 20 Nov 2009 07:58:53 -0800, Dave L. Renfro wrote:
> > Is Hilbert's book "Methods of Mathematical Physics" no longer very well > > known, or is his book not what you mean by "practical computation"?
> The book of Hilbert's student, Richard Courant, is probably > better known (except I don't know it because I can't recall > the title.)
> But at the risk of being unclear due to being too brief > and simplistic, let me offer the following.
> Until his dying day, the physicist Paul Dirac maintained > that the entire point and purpose of mathematics was to > develop the beautiful. He believed that the value of > a mathematical result could literally be judged by its > beauty. I could be wrong, but such an attitude toward > mathematics smacks of pure Platonism.
> In contrast, another physicist, Feynmann, referred to > the mathematics of quantum electrodynamics, which is an > elaboration of the work begun by Dirac, as a "dippy > process" due to certain artifices that are necessary > to prevent infinite results.
> In my view, these are the two basic views that one > can have on the practice of mathematics. Some may > believe that mathematics is discovering portions > of the eternal ideals of Plato. Others may be > inclined to perceive it only as a haphazard > and "dippy process."
> Considering many aspects of the theoretical nature of > mathematics, such as the ad hoc definitions of the elliptic > functions, and the often inordinate difficulties of practical > computation, I find myself leaning toward the idea of "dippiness."
Perhaps it is because you are by nature dippy and it is merely like drawn to like.
> The purpose of computation is not numbers; it is obtaining > insight. When the model admits no solution according to > mathematics then no insight is possible.
That only speaks to your own limitations. Others may well draw insights from what is meaningless to you.
On 20 nov, 11:17, rabid_fan <r...@righthere.net> wrote:
> On Fri, 20 Nov 2009 00:09:21 -0700, Virgil wrote:
> > If you find mathematics so ugly and so useless, pray give it up and > > direct your interests elsewhere.
> You are, not unexpectedly, missing the point. I have no quarrel > with the utility of mathematics and I find it to be a highly > interesting subject.
> My objection is only to these people who promote mathematics as > as an almost mystical object of perfect beauty and power (in the > manner of Plato).
> To me, mathematics seems more and more to be what Feynmann called > a "dippy process."
The fact that mathematics have shown that are Chaos in nature speaks in its favor. If Poincaré have not applied math. no one could know that the three body problem was so complex. Computation is a discipline of mathematics, then what is the problem to apply it to resolve non-linear differential equations?. If any function can be represented by a Fourier Series, why do you say that some math. formulas are swindling? If you would have some familiarity with Number Theory then you did'n speak in that form. You have a point of view Marxist, mine is Platonist. What is the problem? Ludovicus
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