
Re: Is this an exceptionally hard set of questions to answer?
Posted:
Oct 16, 2002 7:48 PM


Alberto Moreira <junkmail@moreira.mv.com> writes:
> Kevin Foltinek <foltinek@math.utexas.edu> said: > >Which model of gravity uses no math? > > Gravity itself is the model
This is probably semantics; I would say that gravity is a phenomenon, that we have given the name "gravity" to the observed phenomenon of things landing at our feet when we let go of them.
> What I call mathematics is what I have met, in broad terms, that > supports my profession: engineering and computers. There may be plenty > of math beyond that, but I hope you admit that such math is basically > of interest to mathematicians
Yes, it is (by definition) of interest to mathematicians, but there is a lot of it that is also of interest to nonmathematicians. (Yes, there is some that is currently not of interest to nonmathematicians, but that does not mean it never will be. Witness the Radon transform, invented by a mathematician for the sole purpose of studying mathematics, and many years later finding applicability in MRI and similar applications.)
The nonmathematicians to whom some area of mathematics is of interest may not be aware that that subject is mathematics, and in many cases would benefit, perhaps greatly, from learning what mathematicians have said about it.
> And I propose that the importance of that math in real life > applications is sort of decreasing, because with the advent of > computers we can resort to simulation and get results while totally > bypassing the mathematical theory behind something.
This reminds me of the argument made that computers and robots will allow everyone to work less because they will do the work for the people. It didn't happen that way, because people found more things to do with the computers. I propose that, even if the mathematical theory could be bypassed, other mathematical theory will be developed to advance this bypassing process or otherwise get more things done.
In any case, I disagree that the mathematical theory can be bypassed.
> I don't care how broad mathematics is, my point is that all that > breadth sits on top of intuitive number sense.
Certainly your notion of "intuitive number sense" historically predated the development of the corresponding mathematics. However, the mathematics has been developed and is more powerful than the intuition, both in the sense of direct applicability (if you have *proved* that something is true, you will trust it more than if your intuition is telling you that it seems to be true) and in the sense of generalizations, new developments, and so on.
> And the applications of > math decrease dramatically as we get away from numbers: graph theory > and group theory are things that most of us won't ever need,
As I mentioned above, you might not *need* it, but there is a good chance that you would *benefit* from it (or something else).
> Classical geometry is very much about numbers, and it is > rooted on "numberizing" our intuitive geometrical sense
Classical geometry is very much *not* about numbers; it's about Euclid's axioms (or variations), axioms about abstract things called "points" and "lines".
> here, again, > the teaching of elementary geometry is better done by strongly > leveraging on the student's intuition, or geometry will just NOT be > learned.
Before the intuition an be leveraged, we need to make sure that it is correct. (One can easily imagine circumstances in which a child has poor intuition, not the least of which are learning disabilities or neurological problems.) Sometimes it may not be present, yet the student might have an aptitude for logic and excel at formal proofs in classical geometry. As you said, every individual is different.
But these differences are relevant to *how* something is taught, not whether it *should* be taught.
> Anything beyond those axioms and where they lead will be of great > interest to the mathematician, but maybe not to the rest of us.
The "where they lead" is a lot of what mathematicians do, and your intuition may not lead you there (and in fact often does not).
What I would really like to see, educationwise, is for students to be exposed first to (more) logic, so that they have the ability to reason and so that they see the importance of nonambiguity, and second to more mathematics in the sense of exploration of the consequences of axioms, modification of axioms, and so on, so that wherever they end up, they will be capable of "pushing the envelope" beyond what they were formally taught(*), they will be capable of asking relevant questions about the assumptions being made, and they will be capable of recognizing that there are assumptions being made and identifying what those assumptions are. (And probably more things  this isn't a wellformed position paper. :>)
(*) By "pushing the envelope", I am referring to the realworld analog of the student who complains "how can this question be on the test, it wasn't in any of our homework".
> Definitions only exist in mathematical space. > [...] > none of them captures the intuitive concept of two > I and most of my students seem to have.
I'd guess that your intuitive concept of "two" is closely related to counting; "one" is somehow "fundamental", and "two" is "the next one". The Peano axioms, or the settheoretic construction of the natural numbers ({}, {{}}, {{},{{}}}, ...), do (I think) capture everything that your intuition is telling you: there is a "first" object, and there is a way to go from a given object to the "next" object.
Perhaps this would be a good time to mention "zero". There was quite a controversy a few hundred years ago; many people had serious problems accepting the notion of "zero", accepting that there could be nothing. I would guess that you, on the other hand, find the concept of "zero" to be very intuitive. (Perhaps not as intuitive as "one", but pretty close.)
Obviously, intuition is to a large extent learned.
> >Probably not, hence the need for the application of mathematics to > >enable them to get the optimization job done. > > Today we have ways of optimizing that by and large bypass math as we > know it. Try genetic algorithms, for example:
I have used these. I was thinking of the GA when I referred to onealgorithmfitsall but runs at a snail's pace. (Snail's pace is obviously relative to the end use  in the context of realtime control, GAs are very often too slow and will continue to be so for many years to come.)
However, the serious practitioners of the art of GA application are aware that, first, you should spend considerable effort tailoring the encoding of your variables to capture the "structure" of the problem, and second, you should find out what is already known about the problem and try to leverage that (perhaps through the encoding of variables, perhaps by playing "intelligent design" with your population to push it in the right direction, perhaps other ways). This requires some understanding of the problem, and usually some understanding of the mathematics of the problem.
> again, coming back to our K12 > problem, I don't see why we should bother with engineering grad school > issues at that level. There's way more to be taught, and little time > for it !
I'm not saying we should teach grad school content to K12 students.
> >Engineering is about the application of science and mathematics, and > >it includes considerations of performance characteristics  "good > >enough" means that you have satisfied all the requirements. Sometimes > >you are fortunate enough to be able to satisfy all the requirements > >with an offtheshelf algorithm, but often you are not. > > Engineering is not just about the application of science and > mathematics. Engineering is about building things that work.
See the part where I wrote "considerations of performance characteristics"? That means "things that work".
> And let > me tell you, I see engineers building things that work all the time, > and they often don't care for the math, and they just as often look at > the math results and discard them in favor of a more conservative > approach.
Many times that more conservative approach doesn't satisfy all of the performance characteristics. (It's really really easy to make an airplane that won't break apart in a highgee turn. It's considerably more difficult to make such an airplane that will actually get off the ground, much less achieve that turn.)
The math that engineers tend to discard first is the undergraduate analog of the gradeschool rote arithmetic drills: the methods of integration, the solution methods for families of ODEs, etc. The math that they don't usually discard, although they might not realize it, is the abstract reasoning skills they developed or enhanced (and the particular details of their education that they actually use every day).
> The issue here is neither computer hardware nor software. The issue > here is communications bandwidth. Don't blame computers for the speed > of your wires !
I'm not blaming computers for the speed of the wires, I'm blaming the wires.
> And this "laws of physics" of yours reminds me the > late Grace Hopper and her "one nanosecond wire". Well, we broke that > barrier long ago, eh ?
Really? We exceeded the speed of light? Do tell more.
> Engineers have a bad habit of redefining the parameters of physical > and mathematical modeling.
One thing I was thinking about was Shannon's communications and information theory. Haven't seen many engineers redefining those "parameters", but what I have seen is people doing mathematics to remove redundancy and irrelevancies from the data ("compression"), increasing the information density of what is being communicated.
> What I call number sense exists in us almost from cradle; it is not > learned, it's an intuition. It can be developed, but some of that is > orthogonal to mathematics.
See my previous comments about "zero", and then may I suggest that you reconsider your thoughts about using intuition during education.
> The only time mathematics tells an engineer that the intuition is > wrong is when it comes to detailed computations.
Wrong. Qualitative behaviour is sometimes counter to intuition, and sometimes in very important ways.
> when math and intuition clash, you bet that what we do > is to go for the high ground: we apply safety enough that it covers > BOTH worlds.
See my previous comments about making the airplane get off the ground.
> It's a poor engineer who takes math models at face value > and doesn't build all sorts of redundancies into his or her stuff.
Again, getting the airplane off the ground. Sometimes you can include the redundancies and still meet performance requirements, sometimes not.
> Kevin, definitions are only required for mathematical modelling. I do > not need to define anything to have the concept of "fall": I just drop > that ball, it falls, I look at my student, and we both KNOW what we're > talking about without even any words being uttered. [...] > > I do not need any definitions to communicate outside mathematical > space. I need commonsense and intuitive communication.
Your example involving the dropping ball and the student involved communication  you were demonstrating the falling phenomenon, most likely introduced by something like "watch this", and followed by something like "that's what we call 'falling'". The communication of the actual concept of falling happened to be accomplished visually rather than via written or spoken language. Without the commentary after the experiment, in which you defined the word "fall", the student would have very little clue what you meant when you said "objects fall".
You need definitions all the time. You just happen to have absorbed a lot of them into what you call "common sense" and "intuition".
> Because, mind you, there's a big, big distance between solving a > problem algorithmically, or mathematically, and solving it inside a > real computer.
Sometimes the distance is big (sometimes mathematics involves nonconstructive techniques  we can prove something exists, but the proof tells us nothing about how we would go about finding or building such a thing); lots of times it is not so big. Often the biggest distance is in constructing the "interfaces" which allow you to follow a mathematical algorithm. (This might involve constructing classes for the relevant types of objects, or it might involve rewriting the algorithm into an equivalent but more computational and less abstract version.)
Of course the efficiency may be a problem  but that requires either faster computers or doing more mathematical analysis to find a better algorithm.
> You can throw math at anything, but that doesn't mean that what you > throw math at is mathematics.
Of course not. What you throw math at is the application of the mathematics. On the other hand, you can do mathematics without realizing that what you are doing is mathematics. (There is no clear boundary between subject areas; things can be more than one subject.)
> There is no such a thing as the "gravitational constant" if we take > its value out of the context. It's merely a fudge factor to make an > equation work  and as such, it changes with the unit system.
It is much more than a "fudge factor"; it seems to be a universal constant of physical laws. Newton's law of gravity, in fact, is the assertion that Fr^2/Mm=G is a universal constant. In this model (which appears to be correct in the nonrelativistic cases), an alien species can measure this constant in terms of their standard length, time, and mass references, and if we know how to translate our length, time, and mass references into theirs, then we can translate our measurement of G into their units, and we will get the same value as they measured.
The constant G does not change with the units. Its numerical value changes with the units.
> >Newton's law of gravity is F=GMm/r^2. There are no units in that > >equation. > > Oh, indeed there are. They're just hidden, we abstract from them to > make the formula more legible and stronger shored on our intuition and > on the physical model.
There are no units in the equation. There are the things that the units measure  length, time, and mass.
If I measure (in whatever units I choose) F, M, m, and r, and compute G, and then I measure M, m, and r for a different system, I can compute F, and I will get the same number as if I measured F. It DOES NOT MATTER what units I use.
A unit is nothing more than an arbitrary object (or equivalent) that we have all agreed to use as a scale factor. The unit is the fudge factor, not the universal constant G.
> If the mass, the force, the distance, and the > constant are not expressed in coherent units, the equality won't > happen.
Right. But it doesn't matter *what* consistent set of units we use, the equality will happen.
> The real equation is something that looks more like > Fu=GvMwmx/(ry)^2, where u, v, w, x and y are "1 something".
If your equation was the real equation, then we would have different equations for different systems of units, for the exact same phenomena. But we don't. (Well, there is that thing about 4pi in electrodynamics, but that's another story.)
In fact, if you look at your "1 something"s, you'll see that w=x, and that v can be expressed in terms of u,w,y, and then you can cancel all the "1 something"s from both sides of your equation, and end up with an equation involving only numerical values; for example, F = 6.67e11 Mm/r^2 .
No, that's not the real equation. The real equation is F=GMm/r^2, where r is a distance, etc. When we introduce our units, i.e., our standard objects, u, w, and y, and we measure something, say r, we write r=Ry, where R is simply a number. (What this means, of course, is that if we make a bunch of copies of the object y, and line them up endtoend, with one end at the location of one of the masses and the other at the location of the other, then we will need exactly R copies of y [with the standard interpretation if R is not an integer].) Similarly, we can measure M=Nw, m=nw, and F=fu. N, n, and f are numbers. Then we can perform a numerical computation, fR^2/Nn , and then interpret this in terms of units, fu R^2 y^2 / (Nw nw) = (fR^2/Nn)(uy^2/w^2) . If we do this again, with the same units (u,w,y), then we will find the same value of the number fR^2/Nn . The universal graviational constant G is not the value of fR^2/Nn, it is (fR^2/Nn)(uy^2/w^2). The value of fR^2/Nn is the numerical value of G in the u,w,y units.
The units are not in the equation. The model of gravity is the equation together with the interpretations of what the symbols F, M, m, and r represent, and the claim that G is a universal constant.
> To us > engineering types that kind of thing is, mind you, intuitive, as it > was already intuitive when I was beginning to learn elementary physics > at 10th grade
A common error made by students is to write down something like: If m = 3 kg and a = 2 m/s^2, then F = ma = (3 kg) (2 m/s^2) = 3*2 = 6 = 6 N (or F = ma = 3*2 = ...) . There are two places where the equals symbol does not belong. Equality is a very strong statement. It takes a long time to eliminate this error, and my experience in speaking with students has been that it takes so long because of intuition like yours  that the units are a part of the equation, or that the equation is strictly numerical.
Kevin.

