> Kevin Foltinek <email@example.com> said: > > >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). > > The mathematics on its own would be unable to even start the theory. > It's only after the physical framework is established that the math > kicks in.
I suggest you read Riemann's inaugural lecture <http://www.maths.tcd.ie/pub/HistMath/People/Riemann/ Geom/WKCGeom.html> or if you want to go to the library look for Spivak's "Comprehensive Introduction to Differential Geometry", Vol. 2, in which Spivak discusses what Riemann was saying.
That lecture *was* the start of the theory - it described the concept of an abstract non-Euclidean space.
> >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"). > > The model in itself uses no math. The math quantifies the model, but > the math is not the model.
Which model of gravity uses no math?
> >Again, I'd like to see this "intellectual" non-mathematical model. > > Read any physics book. Force, work, power, energy, these are all > model-space entities that predate the math that details it.
I've read many many physics books. I have yet to see one that defines any of those concepts (in a usable way) without the use of mathematics. (Just like "things pull on each other" is not a model of gravity, "force is when you push on something" and "energy is the capacity to do work" are not models of physics.)
> >My point is, you are still using mathematics. > > Maybe what I cal mathematics is not what you call it,
That is a point that I have been trying to make (though perhaps not explicitly enough) - you are using mathematics without realizing it, because mathematics is a very broad subject. I think that what you call mathematics is a rather small sample of all of mathematics.
> or maybe you are > denying that intuitive number sense that is the very stuff that math > attempts to model.
Again, mathematics is much broader than an attempt to model your "intuitive number sense". Much of mathematics does not include numbers (except that it might use them for what are essentially counting purposes); some examples: graph theory, group theory, classical geometry.
> Numbers, shapes, process, these things exist without the math, > they're upstream from math: the math is merely an attempt to > formalize and detail concepts that to many of us are intuitive.
No, that is not what mathematics is. Mathematics is (more or less) the study of the logical consequences of certain sets of axioms. Sometimes these axioms are chosen in an effort to capture our intuition about things, such as numbers or geometry, but often these axioms are then changed so that we can see what other possibilities there are.
The claim that "numbers exist without the math" is somewhat troublesome, because you have not defined the word "number", and any attempt to do so (in a way that makes numbers usable) will (probably) be mathematics. (And you have already stated that "there's no such a thing in real life as a number", so where do these mathless numbers exist?)
> >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. > > You fail to see the point: I don't NEED the concept of integral to > compute the area under or inside a curve, all I have to do is to let a > computer turtle traverse it for me and count squares.
You fail to see the point that letting your computer turtle follow the curve and count squares is (if you get rid of the turtle) 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). > > I'm not even using integrals, I'm just counting. You use integration > as an intellectual model for the computation of areas: I don't.
Yes you do: you know (somehow) that if you want a better estimate of the area, you need smaller and smaller squares; you are using the definition of an integral (without realizing it). But you don't know if your use of smaller and smaller squares will converge to a limit; you are assuming it does. (See my other message about the Navier- Stokes equations.)
> [2 GHz computers] > in fact, I'm going to bet that your on-board aircraft computers aren't > as a rule nearly that fast.
Probably not, hence the need for the application of mathematics to enable them to get the optimization job done.
> Moreover, the difference between optimal > and suboptimal may not be large enough to justify the additional cost > and expenditure of intellectual bandwidth:
I was not talking about optimal versus suboptimal, I was talking about the one-algorithm-fits-all (but runs at a snail's pace so doesn't meet the performance requirements) versus a specialized algorithm. Either of these may generate a suboptimal solution.
> and you quickly reach the > boundary between engineering and science. Engineering isn't about > optimal, it's about good enough, and then we throw all sorts of safety > features on top of it.
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 off-the-shelf algorithm, but often you are not.
> >A lot of work is being done to increase efficiency in networking, > >servers, communications, etc. > > That's not computers but wires.
It's a combination of hardware and software. For a given physical system, the laws of physics say that you can only do so much; when you throw some mathematics into the mix, you can do better than the naive application (example: compression of data to get better effective bandwidth).
> I did claim that drill is > required to develop NUMBER SENSE in individual students. I also > claimed - and I DO claim - that this number sense is UPSTREAM from > mathematics, and that it is a necessary requirement for much > APPLICATION of mathematics to real life problems that I have been > involved with.
I think your definition of "number sense" is much more broad than the usual one, and that if you use the real (broader) definition of "mathematics" that I outlined above, then you will see that some of what you call "number sense" is mathematics. Therefore, you are really claiming that an understanding of mathematics is a necessary requirement for application of mathematics.
> This is like driving a car: a car needs mechanical engineering, but > that doesn't mean I'm doing mechanical engineering when I drive it.
False analogy. The computer programmer is not driving the car; the computer programmer is designing and building the car. The person who plugs in the game cartridge and plays the game that you wrote is driving the car.
> For much real life application, arithmetic is what we need.
For much other real life application, logic and more abstract things than arithmetic are needed (or if not needed, would certainly help).
> More, algebra isn't but arithmetic anyway
No. There is a lot of algebra that is not about numbers; arithmetic is the combination of a certain type of algebraic structure with a standard notational convention.
> It's even worse: in many specialties, not even arithmetic is required, > yet that primeval number sense must be there. Take music, for example, > counting is fundamental, the notion of frequency and interval is > fundamental, and none of that is mathematics: the math rather > FORMALIZES it, and helps DETAIL it, but we can and often do master > music without bothering to learn the math underneath: because we use > INTUITION to replace the need to know math.
It is not formally called "mathematics", but there is still mathematical structure involved. The partitioning of the pitches by octaves; the temporal subdivisions into measures, beats, and further refinement; just because you're not calling them "quotients of affine spaces by discrete group actions" doesn't mean that they're not.
This is why I suggested that what you call "number sense" is really what I'm calling mathematics, though you are apparently less interested in proving theorems than in empirically finding patterns (or developing experience-based intuition).
> The real applicability of math to real life problems comes when our > intuition fails
Sometimes (as I have said previously) the mathematics tells you that your intuition is wrong; there are certainly cases where it would be preferable to learn this before you (perhaps metaphorically) build something and it breaks.
> Problem space: the set of phenomena that occurs in the raw subsystem > we are trying to model or to solve a problem within. Those phenomena > do not originate inside our intellect, neither do they spring out from > the usage of a mathematical model. For example, > > - "Objects fall" is a problem space assertion.
The trouble with this is that you still need to define things. For example, the word "fall" is defined in terms of the words "move", "down", etc. The word "move" is defined in terms of place, direction, or position. I could keep chasing down definitions in the dictionary, and never get anywhere, until I construct some sort of model which abstracts these concepts - in effect, a mathematical model.
(I am being deliberately pedantic with this particular example, to illustrate the point. The same point could be applied to other statements about "problem space".)
I suppose the main point here is that in order to unambiguously communicate about "problem space", you need precise definitions of the words you are using, and this precision of definition is in some sense equivalent to mathematical axioms and definitions.
> There's a lot of people throwing math at computer programming, but > that's an after-effect: computer programming is NOT about > mathematics.
Alan Turing is widely considered the first computer scientist. He was a mathematician. Computer science is historically an after-effect of mathematics. Of course, computer science and computer programming can be very different, and you're right, often computer programming is not about mathematics - it's often about "grunt work" (translating data formats between different systems, for example).
> Because much programming is based on > intuition, not in science: programming, in the end, is an art and a > craft before it is a science.
As is mathematics.
> >TAOCP is rather mathematical, and it is probably one of the two or > >three texts that every computer programmer should own. > > It's now dated, and therefore only of interest to computer scientists.
Read some comp.* newsgroups and see how many times the answer to a question "how do I ..." is "see TAOCP".
> Today we have object orientation, and that has changed the way we see > the world of computers and programming. For a more modern treatment of > algorithms, you can try Sedgewick, for example, and pay good attention > to how the math detail only springs out occasionally: much of that > text is targeted at improving one's programming intuition.
Oddly enough, I was considering mentioning Sedgewick's books ("Algorithms in [language of choice]") as another example of textbooks with significant mathematical content. This is probably another instance of your too-narrow interpretation of the word "mathematics". (Just because it doesn't have an equation in it doesn't mean it's not mathematics.)
> >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). > > In other words: you have restricted your problem space so much that it > ceased to be relevant to the original point, which was, to teach > problem space concepts to children in problem space.
The purpose of Cuisenaire rods is to teach addition and multiplication of small integers.
> IF we fix the standard, if, if, if, if, then maybe your imagery is > valid;
We did, did, did, did, fix the standard - I explicitly said that's what the "unit" in "unit cube" does.
> but I reject the idea that fixing all those ifs for the sake of > a nineteenth century view of mathematical modeling is a good idea, > here and today, as far as teaching our kids goes.
We're really not doing mathematical modelling with Cuisenaire rods. Mathematical modelling can't be done until we've actual learned mathematics.
> >The model should not depend on units. Only the measurements and the > >computations should see the units. > > The model DOES depend on units. Newton's gravitational constant > changes depending on which units we have, measurements can be > falsified if we don't use a coherent set of measurement yardsticks, > and so on.
Newton's gravitational constant does not change depending on the units. Its numerical value changes. 1 inch equals 2.54 cm, even though 1 does not equal 2.54.
It is assumed that measurements are made consistently with respect to some standard, and that measurement units are converted where necessary into compatible units prior to calculation.
Newton's law of gravity is F=GMm/r^2. There are no units in that equation.