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Re: Is this an exceptionally hard set of questions to answer?
Posted:
Oct 14, 2002 2:43 PM
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Alberto Moreira <junkmail@moreira.mv.com> writes:
> [About Newton's law of gravity]
> The math wouldn't exist if the model hadn't
> been crafted in terms of "force", just look at Relativity if you doubt
> it.
The math (calculus) was (sort of re-)invented to describe/model the
phenomenon of force. Einstein's general theory of relativity was a
combination of his thoughts about space and gravity with the
mathematician Riemann's purely mathematical attempts to describe
physical spaces other than Euclidean. His ideas about gravity would
not have gone beyond "gravity is a distortion of space" without
mathematics (and I think it is quite likely that they never would have
gotten that far - they were generalizations of the special theory,
which itself was rather mathematical).
I'd like to see a model of gravity that doesn't use any mathematics
(and that has more predictive power than "things pull on each other").
> Physical quantities like
> force, work, power, current, voltage, they only really exist inside
> the intellectual model we created to explain objective reality: and
> only then, after we establish the model, mathematics comes to
> formalize and quantify the model.
Again, I'd like to see this "intellectual" non-mathematical model.
> Every assertion in physics should be preceded by the caveat
> "everything in nature behaves as if ..." - your mathematical laws
> apply to quantities that only exist inside models.
Yes, but this is irrelevant to whether you can construct useful models
without mathematics.
> My point is, I don't NEED to find a closed form expression for an
> integral, I can use my computers in a way that I BYPASS that closed
> form.
My point is, you are still using mathematics.
> Then, if all of a sudden I don't need the closed form of an
> integral to do any application of mathematics,
The application of mathematics is not limited to the computation of
the closed form of an integral - the fact that you desire the value
(whether exact or approximate) of an integral means that you are
applying mathematics.
In any case, no matter what integration scheme you use, you are still
using at least one closed-form definite integral (perhaps of the
constant function).
> >In all cases, you could appeal to the mathematical definition of
> >definite integral, together with some basic numerical analysis, and
> >obtain a numerical estimate of the definite integral. You can also
> >obtain bounds on your error.
>
> If I divide an area into little squares such that there are n squares
> along the contour of the curve, my error is no larger than n times the
> area of my square
Maybe you need an indefinite integral (you know, integral from zero to
infinity).
Or maybe it's rather difficult to fit the contour of the curve with a
reasonable number of squares, in which case you're basically throwing
brute force computer power at the problem - see below for the problem
with that.
> >Mathematics provides many techniques for finding the maximum; it also
> >provides conditions under which some techniques are better than
> >others, and you'd better understand those if you are going for
> >efficiency.
>
> I will only go for efficiency if I need efficiency, and I only need
> efficiency if my computer isn't fast enough. But few things in today's
> scientific landscape progress faster than machine speed, so, that
> threshold of efficiency keeps raising by leaps and bounds. Why should
> I even bother with efficiency if the difference is between one
> milisecond and one second worth of computation ? One blip on the
> computer screen of a $700 machine ? There's more in life to occupy
> us.
There are many real-time applications which require that difference
between 1 ms and 1 s computation time. For example, optimal control
of airplanes. There are also applications which require that the
computation be performed millions of times. (Do you know how long a
million seconds is?) Moore's law says that we'll have to wait fifteen
years for that 1000-fold improvement in hardware, or we could invest a
couple of hours now and get the time improvement in the algorithm.
> >No. Throwing an off-the-shelf algorithm at something for which it
> >might be horribly inefficient is OK if you want to do it once, but if
> >you're writing software to do this, you'd better have a good
> >justification for your choice of algorithm.
>
> Efficiency is only needed when we're at the threshold of machine
> power,
See above.
> and that today keeps being more and more restricted to
> scientific research and big time development. For the overwhelming
> amount of other applications, efficiency is rarely that much of a
> concern.
A lot of work is being done to increase efficiency in networking,
servers, communications, etc.
> >You evidently have no idea at all what is included in the subject of
> >mathematics, and you evidently are proud of that ignorance.
>
> And, to put it bluntly, I'm not sure I care for it in the context of
> this discussion. We're talking about teaching K12 kids, remember ? Not
> about the ins and outs of mathematics. It's about time we get off that
> tangent and back to the mainstream of the debate.
The debate started when you (more or less) claimed that repetitive
rote arithmetic drills are very useful; this tangent arose because you
did not (and evidently still do not) realize that mathematics is about
far more than arithmetical computation (in fact, arithmetical
computation is arguably not even mathematics), and that mathematics
arises in many areas even though the workers in those areas may not
realize that they are doing or using mathematics. My argument is that
it is therefore better to teach more mathematics than arithmetic. I
am not claiming that we should abandon arithmetic, either, but that
arithmetic drills can and often are overused.
> >Mathematics provides the framework or language for the model.
> >Familiarity with mathematics is an integral part of the ability to
> >construct good models.
>
> Mathematics does no such thing. Mathematics gives us a way to detail a
> model once its basic frame is constructed. Models are first
> constructed in problem space, and only then mathematics kicks in.
I've resisted doing this, but I cannot resist any more. Please define
"problem space".
> Again, I can find correct answers by understanding the model right and
> just replicating it inside a machine.
Your replication in the machine is using mathematics - and if your
model is not mathematical, I doubt you will be able to replicate it in
the machine.
> >If you use the any of the words "function", "variable", "array",
> >"integer", "floating point", or many others, in your computer science
> >classes, then you are using mathematics.
>
> I don't think so. What I may be doing is throwing a mathematical model
> to detail my problem space model. But as it often happens, I can
> create my own model and use words from mathematics to describe things
> that aren't quite the same as what math says. So, for example, I can
> create a data type that is an array where row and column indices
> aren't numbers but strings,
Still using mathematics. In this context, the array can probably be
thought of as a function from the set of pairs of finite strings to
some other set. (Also, read what Knuth has to say about hashing in
TAOCP, and notice how many times "Theorem...Proof..." occurs. Also
notice how many of the journal article citations in that section are
to journals with the word "mathematics" in their title.)
> or other data structures: and go find me a
> good mathematical text on the kinds of objects we deal with in our
> everyday programming practice !
TAOCP is rather mathematical, and it is probably one of the two or
three texts that every computer programmer should own.
> >Rather, to be revolutionary, maybe we should be teaching mathematics
> >and logic.
>
> The computer revolution has created a new balance between using math
> and using simulation and modeling to get results in application space.
> Yes, we should teach math and logic. No, we shouldn't teach only math
> and logic - more and more it's happening that we don't really need the
> good old math to get answers.
Perhaps not, but we need it to formulate questions.
> >OK, please put together two unit cubes and get a rod of length 1.
> >(Notice the word "unit" - it means "side of length 1".)
>
> Easy, I put them together and call the resulting length "one".
I have defined the reference for measurement of length - the unit cube
has sides of length "one". It's rather difficult to call the length
of two adjacent unit cubes "one" without violating logic and
definitions (either that or it's just stupid).
You can put them together and call the resulting length "one flurble",
where you define a flurble as two of the reference cubes, but you
cannot call it "one".
> The
> length of the cube was 1, the length of the new object is also 1. The
> key question lays in problem space: ONE WHAT ?
That was made explicit in my question: the "unit" is the reference
upon which all other measurements are based.
> Quatities, mind you, depend on units, and units are at the mercy of
> the modeler.
The model should not depend on units. Only the measurements and the
computations should see the units.
Kevin.
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