Posted on 03/15/2008 1:58:58 AM PDT by neverdem
However, geometric terms such as "angle" or "bisection" or "plane" refer to physical concepts, just as the number "2" has meaning only because it has physical counterparts. Basic mathematical constants turn out not to be abstract at all. Pi has an undeniable physical connection; and "e" and its associated natural logarithm have a habit of showing up in all manner of physical manifestations. As, for that matter, does "i." The "physical" countours of a plotted fractal (e.g., the Mandelbrot set) is another example.
I think it is safe to say that we don't understand the connection between mathematics and physical reality. But that is not the same as saying there is no connection between them. The evidence suggests otherwise.
However, geometric terms such as "angle" or "bisection" or "plane" refer to physical concepts, just as the number "2" has meaning only because it has physical counterparts.Often, mathematical concepts are inspired by the real world. They are defined, however, in purely abstract terms, without referring to the real world at all. The old Greeks already noticed that over 2000 years ago.
Suppose we have already developed set theory. Now we can define
Basic mathematical constants turn out not to be abstract at all. Pi has an undeniable physical connection; and "e" and its associated natural logarithm have a habit of showing up in all manner of physical manifestations. As, for that matter, does "i."These numbers, which exist in mathematics, reflect logical properties of certain mathematical constructions. As such, they are purely abstract, and remain so, of course, forever. That they also show up again and again in mathematical models of the real world that physicists construct, is indeed very interesting, but it does not mean that these numbers all of a sudden cease to be abstract mathematical concepts. Mathematics does not change just because you apply it to physics.
I didn’t know 6x7 off the top of my head but I did know 6x6=36, so my mind worked it out as 6x6+6 = 36+6 = 42, FWIW.
Explain pi.
Yep, any way is fine except asking the calculator :-)
Explain pi.π is often defined to be the ratio of the circumference of a circle to its diameter. But not always. Another way that is sometimes used is to define first
The fact remains, however, that the ratio of the circumference of a circle to its diameter is no other than pi -- there is clearly a physical aspect to this mathematical stalwart. It was discovered, after all, by measurement, rather than by theoretical means.
The fact that you provided a series form for the computation neglects that fact that it is based on another physical ratio (which is what a cosine is....). As such, the series form is interesting, but is still ultimately tied to the circle, and (physical) angular measurement.
One supposes that Plato's theory of "forms" is relevant here....
Love the image!! You might like this story:
http://bestsciencefictionstories.com/2008/01/22/the-feeling-of-power-by-isaac-asimov/
The fact remains, however, that the ratio of the circumference of a circle to its diameter is no other than pi -- there is clearly a physical aspect to this mathematical stalwart. It was discovered, after all, by measurement, rather than by theoretical means.I am not sure what you mean here, and how it relates to what I wrote.
The fact that you provided a series form for the computation neglects that fact that it is based on another physical ratio (which is what a cosine is....). As such, the series form is interesting, but is still ultimately tied to the circle, and (physical) angular measurement.I did not provide the series for computation. I was explaining that the cosine, and π, are sometimes defined by the power series alone! It is more common, especially when talking to kids, to define the cosine by geometric means, of course. What is absolutely never done in mathematics, however, is to define the cosine by any physical measurements. Of course, Euclidian geometry is sometimes used as an approximate model of the real world. So, you can use the model to predict the results of physical measurements. These numbers may include π, and the cosine, of course. That does not mean, however, that π, or the cosine, is somehow defined by these measurements. It only means that you are using Euclidian geometry as a model. An inaccurate model, by the way, as Einstein has shown.
You have absolutely no notion of pure mathematics, it seems. Have you ever read about the axiomatic method? Or seen how pure mathematics actually works? Apparently not, and I fear that it is impossible to give you an idea of how it works with a few FR postings. So, I guess I'll give up now.
I mean that pi was not originally discovered through any abstract mathematical means; the ancient Greeks discovered it by trying to physically measure the relationship between circumference and diameter.
You have absolutely no notion of pure mathematics, it seems.
Well, I did say that I'm an engineer by trade and inclination -- which means that I don't get too hung up on artificial distinctions such as you're trying to create. I simply accept that there is a real connection between mathematics and physical reality -- and by golly, it works!!!!
http://www.maa.org/devlin/LockhartsLament.pdf
A Mathematician's Lament
by Paul Lockhart
excerpt:
And there you have it. A complete prescription for permanently disabling young minds — a proven cure for curiosity. What have they done to mathematics!
There is such breathtaking depth and heartbreaking beauty in this ancient art form. How ironic that people dismiss mathematics as the antithesis of creativity. They are missing out on an art form older than any book, more profound than any poem, and more abstract than any abstract. And it is school that has done this! What a sad endless cycle of innocent teachers inflicting damage upon innocent students. We could all be having so much more fun.
“Students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.”
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Those who can and want to pass Algebra II are likely to have a higher IQ. The higher the IQ the more likely they will finish college.
When I was in school, Algebra II was the minimum math requirement for college admission. You had to have Alg I,II and geometry, and preferably trig. I didn’t take trig, and added an extra year of language. So, I had two years of French and four of Latin.
You also had to three years of science, no two, probably biology and chemistry, physics was probably optional.
Then, everyone was required to take one year of higher math, based on your achievement tests, in college. I remember because my brother always did ridiculously well on testing, and tested out of all but the highest calculus that was offered, and had to take it. You also had to take two years of a foreign language in college.
The current approach to learning music and learning to play a musical instrument needs to be changed.
All of my children learned to read music and play a musical instrument. Talent had absolutely **NOTHING** to do with it. It was expected that they would do it.
Most children **can** learn to read music and play a musical instrument to (at minimum) the intermediate level. Yes, pure genius can produce a protege but basic proficiency it achievable by nearly all children.
Oh, NO!! Didn’t I just huff-n-puff and rant and rave and slather through this same headline just last week???? I daren’t look at the article, I just have barely cooled down.
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