What makes for good science and mathematics education?

[Sam H]

i believe that Galileo believed that,

Galileo was neither the first

Plato in Timaeos: "Innocent light-minded men, who think that astronomy can be learnt by looking at the stars without knowledge of mathematics will, in the next life, be birds."

nor the last

Fourier: "The profound study of nature is the most fruitful source of mathematical discoveries."
Gibbs: "Mathematics IS the language of nature."
Einstein: How can " ... mathematics, being after all a product of human thought, [be] so admirably appropriate to the objects of reality?"
Wigner: "The unreasonable efficacy of mathematics in the natural sciences."
Dirac: "Physical laws should have mathematical beauty."

... (no evidence) that the world could be understood perfectly via mathematics.

I am not sure what kind of evidence one could even talk about. And the world can NEVER be understood perfectly. No sane scientist claims this. Furthermore, isn't the enormous success of physics in explaining the world - imperfectly? - after Galileo sufficient evidence?

... maths is perhaps richer than Nature.

You can claim this only if we know Nature COMPLETELY. If we know Nature only partially, as we do, how can we judge if math is richer than it?

Nature (and only Nature) provides rich mathematics, which is applied back to Nature to discover its RICHER parts. There are many examples of this, but I give you one. Differential equations came from Nature. Group theory came from the mind of a 21-year old French mathematician. Sophus Lie combined the two to invent Lie group theory. Lie group theory has been applied to:

1. the classification of elementary particles (eight-fold way) and the PREDICTION of the existence of omega minus;
2. the prediction of quarks;
3. the explanation of the weak nuclear force, its unification with the electromagnetic force, AND the prediction of the Higgs boson;
4. the explanation of the strong nuclear force and the confinement of quarks.

 

[lenny]

Galileo was neither the first

i did not intend to suggest he was the first. only that he might well have believed that mathematics described the world perfectly.

And the world can NEVER be understood perfectly. No sane scientist claims this.

Laplace came close. and many scientists ave considered the possibility that a perfect mathematical description may exist, even while agreeing your point that we could never know we had it.
Mach in his book on "Error" has a few nice comments on this, about 1850 (i am not at home, or i'd offer you the citation). Feynmann had this as one of his two options for the (long term) future of science.

If we know Nature only partially, as we do, how can we judge if math is richer than it?

if we view nature as consistent, and note that maths is happy to admit alternatives, each mathematically valid within itself but incompatible with each other. then why could one not argue that maths was richer? more varied?

i personally prefer to view them as just very different types of things.

Nature (and only Nature) provides rich mathematics, which is applied back to Nature to discover its RICHER parts.

i think you'll find the parenthetical hard to prove. what evidence do you have that "only Nature" can provide mathematics?

Differential equations came from Nature.

setting aside the continuum hypothesis for the moment, woul dyou tell me how you see those differential equations relationship to nature?

do the govern what happens? or merely describe?

 

[Sam H]

Laplace came close.

If I remember correctly, Laplace merely stated that IF we knew the positions and momenta of ALL particles in the universe, we could, IN PRINCIPLE, predict its fate. I don't think that he actually believed in the possibility of doing this, and therefore, that the universe could be understood mathematically perfectly.

if we view nature as consistent, and note that maths is happy to admit alternatives, each mathematically valid within itself but incompatible with each other. then why could one not argue that maths was richer? more varied?

I believe our definitions of "rich" are different. I don't necessarily consider variety "rich." Richness, to me, is breadth, universality, and elegance. For example, I consider differential geometry, which combines analysis, geometry, algebra, and group theory richer than, say, number theory - not to devalue the latter!

i think you'll find the parenthetical hard to prove. what evidence do you have that "only Nature" can provide mathematics?

I agree that the statement is too strong. My intention was to consider Nature as the SOURCE of rich mathematics a la Fourier. For example, the semi-Riemannian geometry started in 1908 when Minkowski, bsed on the special theory of relativity, considered the possibility of metrics which were not positive definite.

In fact, if one brings evolution into this discussion, one has to conclude that the source of ALL mathematics is Nature. I am sure that the brain of homo erectus was incapable of analyzing mathematical arguments!

setting aside the continuum hypothesis for the moment, woul dyou tell me how you see those differential equations relationship to nature?

Newton wrote the first differential equation when he applied his second law to the motion of a single planet around the sun. Partial differential equations came into existence when the eighteenth century physicists/mathematicians applied the second law of motion to fluids. The entire Fourier analysis had its origin in the study of heat.

do the govern what happens? or merely describe?

Obviously, no human product, including differential equations, can "govern" what happens in Nature! They also don't merely describe Nature, they PREDICT what Nature will do.

 

[lenny]

do they govern what happens? or merely describe?

Obviously, no human product, including differential equations, can "govern" what happens in Nature! They also don't merely describe Nature, they PREDICT what Nature will do.

Surely you mean more than mere prediction. A p-random number generator can PREDICT , often not very well, sometimes embarrassing well... What do u mean by predict that males it more than mere description?

 

[Sam H]

Surely you mean more than mere prediction. A p-random number generator can PREDICT , often not very well, sometimes embarrassing well... What do u mean by predict that males it more than mere description?

Einsteins general relativity predicted the bending of starlight; it did not describe it because it didn't exist prior to GR's prediction.

 

[lenny]

Einsteins general relativity predicted the bending of starlight; it did not describe it because it didn't exist prior to GR's prediction.

i am not sure i got the referent for each of those "it"s correctly. are you suggesting tha light was not defected by massive bodies before GR?

what is the "it" in "it didn't exist prior to GR's prediction"?

 

[Sam H]

I think of "description" as something with which we have already had experience. Perhaps "explain" is a better word for it. GR "explained" - or described - the precession of the perihelion of Mercury, but "predicted" the bending of light and the expansion of the universe.

 

[ddt]

I think of "description" as something with which we have already had experience. Perhaps "explain" is a better word for it. GR "explained" - or described - the precession of the perihelion of Mercury, but "predicted" the bending of light and the expansion of the universe.

But the bending of light and expansion of the universe had also been going for approx. 13.7 billion years before Einstein formulated GR.

 

[elgarak]

But the bending of light and expansion of the universe had also been going for approx. 13.7 billion years before Einstein formulated GR.

But no one had ever bothered to look before at the light bending in detail.

Not that this is important.

 

[Sam H]

But the bending of light and expansion of the universe had also been going for approx. 13.7 billion years before Einstein formulated GR.

The difference between the precession of the perihelion of Mercury and the bending of light (and the expansion of the universe) is that the former was known to astronomers for at least a century before GR and it was accounted for by classical gravity except for a tiny portion which GR "explained." The bending of light and the expansion of the universe, however, were completely unknown - and unexpected! That is why the latter are called "predictions" of GR.

 

[ddt]

The difference between the precession of the perihelion of Mercury and the bending of light (and the expansion of the universe) is that the former was known to astronomers for at least a century before GR and it was accounted for by classical gravity except for a tiny portion which GR "explained." The bending of light and the expansion of the universe, however, were completely unknown - and unexpected! That is why the latter are called "predictions" of GR.

This seems to be mainly a semantic quibble now over what is a "prediction" or a mere "description". IMHO, the bending of light is an ongoing phenomenon, so it is described by GR. OTOH, the specific observations that Eddington made of the solar eclipse - one specific instance of that bending of light - were predicted by GR.

 

[Sam H]

This seems to be mainly a semantic quibble now over what is a "prediction" or a mere "description".

It may very well be. But I would like to differentiate between describing - explaining, predicting, ... - what we already know but have no explanation for it, and what we don't know - which obviously must be happening, otherwise we would NEVER know about it - but a theory makes us aware of it. Any suggestions for words - to replace the words "describe" (or explain) and "predict" - that differentiate between these two cases?

 

[Sam H]

IMHO, the bending of light is an ongoing phenomenon, so it is described by GR.

Here is another twist to the whole discussion. While the bending of light is "an ongoing phenomenon" which, therefore, according to ddt could only be described by GR, Higgs boson is not an ongoing phenomenon; it existed only in the first nanosecond after the big bang and disappeared until artificially produced on July 4th, 2012. Should we say that the standard model - before July 4th, 2012 - "described" or "predicted" Higgs boson?