Plus, human mathematics is not pure. We have met the enemy, and he is us — Pogo
Secondly, I think education should be valuer for its own sake rather than for its practical purposes. Realising one can do maths for maths's sake ought to be part of the syllabus ; one of the pitfalls of maths, and especially of maths for practical purposes, is that of mistaking them for a set of problem-solving techniques, which make understanding maths (and actual further problem solving) harder.
If up to 15 you only teach the maths that are practically useful, not looking into tome abstract details, it actually becomes very hard to do further maths, and since those one has learned are only a disjointed set of algorithms and quick answers, they are fast forgotten, too. Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
(I mean, I an almost see it, I think, but I'm one of those who sees the application, I'm seeing it backwards maybe, from the machine to the parts to the materials to geology, into the atoms, and...out there in the land of the abstract....mathematics!
I almost called this diary "The Joy of Maths"...I keep thinking of applications...."What the numbers meant to Ka Ne Suss was that the thing was about to blow."
I'm intrigued! Don't fight forces, use them R. Buckminster Fuller.
As an example, that first of all demonstrations, that of Pythagorean theorem : how does the figure prove that in a triangle with a 90° angle, a²+b²=n² ? Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
a2+b2+2ab=2ab+n2
so a2+b2=n2 Life should consist in at least fifty percent pure waste of time, and the rest doing what you please.
At no point in the proof there is a nonhomogenoeous polynomial adding a length to an area, for instance. So Ceebs' argument can be written out in words involving areas.
I think that diagrammatic proof of Pythagoras' theorem may have originated in India? We have met the enemy, and he is us — Pogo
I'm still pondering the idea of an "imageless" maths that invokes images (the diagram above), or a graph--something visual at any rate that stands for...the invisible maths behind the image...
So it may be easy, but easy is good (for me) if it helps me concentrate on the underlying aspect, in this case:
When one advances enough, the pleasures of maths become non-visual ; Maths after all is the art of abstract symbol manipulation.
(Thing is, I have pondered this and I am wondering whether maths' claim to be somehow bigger than the universe (maths gives us "the universe" + 1)--I mean the idea that maths "encompasses" the universe as compared to the universe being "bigger than maths"--Heh, I'll have to try and explain this again later, but I mean something like: "What does it mean that we can't beyond a certain exponential--I was thinking about mathematical models of the universe--there is the "empty box" model, we are in it and the sides are an endless distance away. Then there is the "closed form" model, balls, saddles, but always (inevitably) seen from "outside"...heh...I'll post this just to remind myself that I had a thought in there somewhere. Don't fight forces, use them R. Buckminster Fuller.
Notice that before the development of symbolic algebra in the middle ages, elgebra had always a geometric interpretation. Squares were the areas of squares. Linear quantities were the lengths of segments. Cubes were the volumes of actual cubes. Inhomogeneous polynomials (mixing quantities of different degree) didn't often occur.
Mathematics has always been visual, touchy-feely, intuitive, until the formalization in the 19th century. We have met the enemy, and he is us — Pogo
But this one should be really easy. We had to find this out on ourselves, I don't know, maybe as sixth graders.
I mean the idea that maths "encompasses" the universe as compared to the universe being "bigger than maths"
Do you know that Set Theory proves that there is no Universe?
Then there is the "closed form" model, balls, saddles
Saddles are a representation of open ever-expanding hyperbolic universes. *Traitor*, n. A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.
Logic is how you are allowed to manipulate abstract symbols ; the rest of maths is deciding about some abstract symbol, and then playing with them a lot... Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
In my view N only has the meaning we give it through axioms ; axioms which are rules on how to write proofs.
What do you mean by 'meaning' ? :) Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
Axiomatization is not the prerequisite for mathematics, it's the endpoint. We have met the enemy, and he is us — Pogo
Also, is R more concrete than the set of p-adic numbers? is Euclidean geometry less abstract than other geometries ? Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
R is more concrete than the set of p-adic numbers. That is why it was invented centuries earlier.
And while Euclid and his contemporaries had axioms, mathematics had existed before them. The greeks may have invented the axiomatic-deductive method, but they did not invent mathematics. We have met the enemy, and he is us — Pogo
Which geometry is concrete to a blind person ?*
It seems you define concrete as intuitively accessible to the human brain... It makes god a very concrete concept nowadays. Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
It seems you define concrete as intuitively accessible to the human brain...
As for spherical geometry being invented after euclidean geometry, I don't know what came first, but spherical geometry was highly developed by babylonian astronomers while the Babylonian value for pi was still the integer 3. We have met the enemy, and he is us — Pogo
Also, is there mathematical truth independent of thought processes : is logic only a cognitive process ? Colman was contrasting logics with the rest of mathematics. Is logic "true" because it agrees with our thought processes - but many people think without adhering to the laws of logic. Why would logic be different from the rest of maths ? Auferre, trucidare, rapere, falsis nominibus imperium; atque, ubi solitudinem faciunt, pacem appellant.
Concepts of abstract and concrete can depend where you're looking at them from: N can be relatively concrete. In a moment we can consider what is concrete, precisely.
You have a couple of things going on - your intuition about the structures you're dealing with, your visualisation of them (which I don't mean in a way that's easily mappable to visualising real things, but you're using the same part of the mind), the symbolic representations and the available facts about the symbolic representations. So you're working on several levels, and different people enjoy different parts. Generally I think people are guided by intuition to propose things which they then need the symbolic machinery to prove - it's too easy to let your imagination run away with you. Sometimes your intuition is mistaken and the symbolic machinery will help you understand why.
Where Mathematics Comes From - Wikipedia, the free encyclopedia
Human cognition and mathematics Lakoff and Núñez's avowed purpose is to begin laying the foundations for a truly scientific understanding of mathematics, one grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path.
Lakoff and Núñez's avowed purpose is to begin laying the foundations for a truly scientific understanding of mathematics, one grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path.
Now, whether the quadratic equation and planar geometry should be part of general mathematics education is a different story. We have met the enemy, and he is us — Pogo
Y^2 = b^2/(4ac^2) - c/a, where Y = x + b/2a.
That's how it was taught to me in highschool first grade, which was then ninth grade overall, e.g. we were aged 14-15. But planar geometry, if I guess right what that is, came much later, maybe only college, around the same time complex numbers. I'd pull complex numbers ahead. *Traitor*, n. A benighted individual who perceives an illusory distinction between serving his nation and abetting the criminals who govern it.
Regarding planar geometry, historically Complex number - Wikipedia, the free encyclopedia
The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.
you can solve the cubic equation by using trigonometry (or hyperbolic functions) without using complex numbers
Exponentials and logarithms are even easier than the trigonometric functions. For one thing, they are almost arithmetic operations: you just solve for x in e^b=x, or e^x=b, with some peculiar number e, and continuously variable b. Secondly, properties like addition formula exp(a+b)=exp(a)*exp(b) are much simpler. And if you know differential equations, y'=y is simpler than y"+y=0, and y'=1/x is important as well.
Exponential/logarithmic functions might look less exciting than trigonometric functions, but dull looks are deceiving. Everyone who has to pay interest rates must know the exponential function.
You probably know the famous fable of exponential growth: Rice grains on a chessboard:
A courtier presented the Persian king with a beautiful, hand-made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement [demanded] over a million grains on the 21st square, more than a million million on the 41st and there simply was not enough rice in the whole world for the final squares. (From Meadows et al. 1972, p.29 via Porritt 2005)
This tale could be supplemented with the following:
Put a 1g granule of gold on the first square. On the second square, put 1.011 g granule of gold - a 1.1% increase. On the second square, put 1.022 g of gold - still 1.1% more. On the third square, put 1.033 g of gold - another 1.1% increase. If you go on like this, on the last square you put 1.011^63 *1g=1.992 g of gold - almost the double of what is on the first square. In total, there would be about 92.188 g of gold on the chessboard. That is worth about 1629 euro in today's markets (with 17.67 euro per gram). Not too terribly bad so far.
Do we have another chessboard? On the first square of the 2nd board, we put 1.1% more than the 1.992 g on the last square of the 1st board. That is 2.014 g of gold. On the second square of the 2nd board, we put 1.1% more, which is 2.036 g gold, or 2.014 times more than on the 2nd square. If we go on like this, we always put 2.014 times more gold on the Nth square of the 2nd board than on the Nth square of the 1st board. In particular, we would have 4.012 g of gold on the last square of the 2nd board, and the second board would weight 185.67 g of gold.
How many chessborards do we have? Is it 10 in total? After increasing the ammount by 1.1% per next square, the 10th board would contain 2.014^9*92.188g, which is ~50.27 kg (kilograms!) of gold. That is worth 888 thousand euro! All 10 boards would contain ~99.76 kg of gold.
Can we borrow 10 more chessboards? The numbers will be very similar to the original Persian story when we come to the 20th square there, since the numbers 2.014 and 2.0 are about equal.
The moral is that the "interest rate" of exponential growth tells you how fast you double the amount. With 1.1% of interest rate, you double in 63-64 steps. With 1% growth, you double in 69-70 steps. With 4% growth, you double in 17-18 steps. (Compute log(2)/log(1.04).) With 5% growth, you double in 14-15 steps. To compare two interest rates, you should compare the time scales of growth.
Is the exponential function more exciting now?
