| View previous topic :: View next topic |
Boreades
In: finity and beyond
|
|
 |
|
|
Also, as it's mathematics, we may also expect the AEL management to adopt their orthodox position of Careful Ignoral towards the subject.
|
|
 |
|
 |
|
Brian Ambrose
|
|
|
|
|
By the way, I am not mocking for Mick’s disinterest in all things technical. From the age of eight I was forced to listen to teachers learning me French. I wasn’t the slightest interested but they punished me every year until ‘O levels’ when, after an -F result, they believed me. The only way to teach is to make it relevant, interesting or funny; engage mind. When a PT teacher covered for a maths teacher for a term (Mr Chapman in case you remember him), he started with algebra; one Martian plus one Martian is how many Martians? 2m, you get the idea. Algebra became easy and from then on maths was reasonably straightforward. Well, I wasn’t keen on calculus, with its deltas and squiggles, but fortunately I have only rarely found it useful down at the Tesco.
|
|
| |
|
|
|
Boreades
In: finity and beyond
|
|
|
|
Horses for courses, and mental mathematics.
A few months ago, M'Lady instructed me to construct a sight gauge for the side of the heating oil tank. To translate the dipstick depth (the distance from the top of the tank to the top of the oil in the tank) into the volume of litres remaining in the tank.
After a bit of head scratching, a dim & distant memory of O-Level Maths crept back into vision. As the end of the tank was a flat semi-circular shape, the whole volume was a simple derivative of the area of the end. A little Integral Calculus was called for.
| Integral is the representation of the area of a region under a curve. We approximate the actual value of an integral by drawing rectangles. A definite integral of a function can be represented as the area of the region bounded by its graph of the given function between two points in the line. |
By dividing the end-area into horizontal rectangles, each one centimetre wide, the area of each rectangle is the same as the length of the side.This simplifies the integration. My DIY calculus has proved to be accurate within 30 litres, not bad if I say so myself.
|
|
| |
|
|
|
Boreades
In: finity and beyond
|
|
|
|
|
The point of the example is that this kind of mental activity allows one to predict how the real world will behave, with good-enough accuracy. Instead of just guessing.
|
|
| |
|
|
|
Brian Ambrose
|
|
|
|
|
Very smart Borry, but you do know that you can buy a brand new oil tank which speaks the remaining volume, the exact time the oil will run out and automatically re-order it for just £29 from Amazon, right?
|
|
| |
|
|
|
Brian Ambrose
|
|
|
|
The awesome constructs of the human mind
Mathematics is solely a construct of the human a-natural mind - mathematics does not exist in nature (ie in reality). For example, we are told that Pi is a real thing; the ratio of circle to diameter. But the irrational number Pi is just a symbol, part of a mind language called mathematics to make sense of reality. Nature does not need an irrational number to create a circle, it makes it according to a higher rule of reality. This is not a philosophical argument, it demonstrates that mind exists outside of reality, outside of nature, and yet it makes a good approximation as it makes sense of it.
Unlike mathematics, beauty is universally appreciated and has probably been so since mind looked up and smelt the coffee. But is the appreciation of beauty solely a construct of mind, or is nature inherently beautiful and mind merely constructs a ‘language’ to appreciate it? Nobody knows. I do notice that the nature which we can comfortably see is beautiful, but if we get closer it gets very ugly indeed. In any case, mind has a construct to handle beauty.
|
|
| |
|
|
|
Wile E. Coyote
In: Arizona
|
|
|
|
Wiley has never used Kuhn, my feeble conception of paradigms is based around Proofs and Refutations: The Logic of Mathematical Discovery a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics.
| The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs | |
|
| |
|
|
|
Boreades
In: finity and beyond
|
|
|
|
I do have a PDF copy of "Where Mathematics Comes From".
I would willingly share it, but have no idea how that can be done with the existing BB software here.
|
|
| |
|
|
|
Mick Harper
Site Admin
In: London
|
|
|
|
I would not accept 'debating the proof of the Euler characteristic defined for the polyhedron' as paradigmatic. Though the bloke turned up in RevHist so should not be entirely dismissed:
----------
D’Alembert was the pre-eminent thinker of his generation, and Wiki has been kind enough to list the more important of his contributions
D’Alembert criterion
D’Alembert force
D’Alembert’s form of the principle of virtual work
D’Alembert’s formula
D’Alembert’s equation
D’Alembert operator
D’Alembert’s paradox
D’Alembert’s principle
D’Alembert system
D’Alembert–Euler condition
Tree of Diderot and d’Alembert
Cauchy–Riemann equations
Fluid mechanics
Three-body problem
The Preliminary Discourse to Diderot’s Encyclopédie
More than a thousand articles for the Encyclopédie |
I am not personally au fait with much of this, and of course nobody is au fait with the thousand articles he wrote for the Encyclopédie, but my eye has been caught by the d’Alembert–Euler condition because I am au fait with Euler, the great German mathematician. I will therefore confine my remarks to the d’Alembert-Euler condition as an epitome of the oeuvre as a whole
| The d’Alembert-Euler condition is named for Jean le Rond d’Alembert and Leonhard Euler who independently first described its use in the mid-18th century. |
A complication right out of the gate. Not so much the d’Alembert-Euler condition as the d’Alembert condition and the Euler condition. One supposes, if precedent were sought, something akin to the Newton Calculus and the Leibniz Calculus, and we all know what confusion that caused. Can Wiki clear up any confusion?
| It is not to be confused with the Cauchy–Riemann conditions |
You just did! You put it on the same list with only the Tree of Diderot and d’Alembert between them. Pray clarify.
| In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be complex differentiable, that is, holomorphic. |
Duh, we knew that, what we asked for was the connection with d’Alembert and Euler
| This system of equations first appeared in the work of Jean le Rond d’Alembert [d’Alembert 1752] |
And Euler?
| Later, Leonhard Euler connected this system to the analytic functions [Euler 1797] |
I must protest in the strongest possible terms on behalf of our man. This makes it the d’Alembert Condition with Euler bringing up the rear forty-five years later.
Hang on, that can’t be right. D’Alembert and Euler were contemporaries, they both died in 1783, so what exactly is going on? I hope to God we are not back in the world of posthumous monkey business. What happened to his/their equation/equations after his /their death/deaths?
| Cauchy (1814) then used these equations to construct his theory of functions |
Oh no, you’re not catching us with that one. You said they were the Cauchy-Riemann equations. Where’s Riemann got to? Don’t tell us he’s the grad student doing all the work while the boss gets his name on the finished paper. We know all about that one, thank you very much. Just tell us the facts, Wikipedia, and this time no tacking on a few decades just because you feel like it.
| Riemann’s dissertation on the theory of functions appeared in 1851 |
I give up. Maths will have to be left to people with more than a O-level (Grade C).
|
|
| |
|
|
|
Wile E. Coyote
In: Arizona
|
|
|
|
|
Wiley has previously noticed that whenever he has posted on paradigms, Mick has always advised it was total rubbish. I have always chosen to blame Lakatos but, to be fair to Imre, it could be the fault of someone else. I just can't work out who that could possibly be......
|
|
| |
|
|
|
Mick Harper
Site Admin
In: London
|
|
|
|
By AE rules you can't use Imre Lakatos as a stick to beat me with merely by citing him as an authority. After all these years working at the paradigm coalface, I'm an authority too.
As of course are you, or should be. By AE rules you take whatever you have learned from Lakatos and use that to beat me with.
|
|
| |
|
|
|
Wile E. Coyote
In: Arizona
|
|
|
|
| I just can't work out who that could possibly be...... |
To be clearer, that was actually a reference to my own failings.
|
|
| |
|
|
|
Mick Harper
Site Admin
In: London
|
|
|
|
Even so I get the impression you don't regard me as quite on a par with Lakatos. This may of course be the case but it is an example of something we constantly have to fight against.
| We are nobodies who know a lot up against somebodies who are reputed to know a lot. |
PS To be clear in turn, I had never heard of Imre Lakatos.
|
|
| |
|
|
|
Boreades
In: finity and beyond
|
|
|
|
| Mick Harper wrote: | | After all these years working at the paradigm coalface, I'm an authority too. |
Citation required.
|
|
| |
|
|
|
Mick Harper
Site Admin
In: London
|
|
|
|
For what? I can cite you for 'all those years'. As for being an authority, there is no Bumper Book of Authorities, is there? It is true my claim is entirely self-generated but as an authority that seems fair enough.
Unfortunately the world judges by the judges so I fear you will have to judge for yourself. It is a 'I was Lord Kitchener's Valet' situation so take that into account.
|
|
| |
|
|
|
|
The Library Index 
FAQ 
Search
Register
Profile 
Log in to check your private messages 
Log in 
