Mathematics
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Andrew MacLean
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I think so, in fact I think Mathematics can be rather beautiful, but you have to get used to the idea that obvious assumptions are often false!
There is a nobility about maths: a purely logical progression that can produce paradoxes: such as the squre root of -1 (i). This is called an irrational number because it ought not to have any place in a logical system such as Mathematics, but it does!
But as to clusters; if you throw a handful of rice in the air, so that it all lands on a tray, you will not find the grains equally spaced about the surface of the tray. There will be little clusters of them, with some parts of the tray completely empty of rice grains.
See Philippe Caldero, Frederic Chapoton, Ralf Schiffler for a simple exposition of this sort of algebra.
There is a nobility about maths: a purely logical progression that can produce paradoxes: such as the squre root of -1 (i). This is called an irrational number because it ought not to have any place in a logical system such as Mathematics, but it does!
But as to clusters; if you throw a handful of rice in the air, so that it all lands on a tray, you will not find the grains equally spaced about the surface of the tray. There will be little clusters of them, with some parts of the tray completely empty of rice grains.
See Philippe Caldero, Frederic Chapoton, Ralf Schiffler for a simple exposition of this sort of algebra.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.
Andrew MacLean
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Andrew MacLean
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Matthew
I remembver that moment too. Until that time I had always understood that in resolving mathematical formulae you squared and then square rooted your answer so that there were no negative numbers. But i is the square root of negative one, so is i negative?
I puzzled over this for ages and then it came to me: it doesn't matter, it is sufficient for it just to be i. Then all sorts of things in Q begin to make sense.
After all, as well as being rigorously logical, mathematics has also to be rigorously pragmatic. That's wher the boundry between maths and art becomes blurred.
Andrew
I remembver that moment too. Until that time I had always understood that in resolving mathematical formulae you squared and then square rooted your answer so that there were no negative numbers. But i is the square root of negative one, so is i negative?
I puzzled over this for ages and then it came to me: it doesn't matter, it is sufficient for it just to be i. Then all sorts of things in Q begin to make sense.
After all, as well as being rigorously logical, mathematics has also to be rigorously pragmatic. That's wher the boundry between maths and art becomes blurred.
Andrew
Andrew MacLean
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Matthew_
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Andrew,
Despite my denials, I am actually interested in maths and science although I do not clearly share your level of understanding. Are you sure it is logic, though?
I remember in A-Level physic becoming quite agitated because the teacher had told me that one of the assumptions of Newton's Laws are that the earth is perfectly spherical. Of course, the earth is not spherical at all. In fact it is so irregular in shape that they just call it a Geoid which is a fudge by the way. So Newton's laws do not stand, they are based on incorrect assumptions. Sure enough, now we know more, we know that Newton's Laws are at best a loose fit! On the other hand without these scientific fudges, the world today would not be as advanced as it. So its difficult and science is clearly an incredibly important and fundamental movement, but I wonder is there any logic in root-1 being i. Doesn't that just mean,"we don't know what the root of -1 is so lets call it i or t...or fish..or anything really" I mean sqaure roots are works of the human mind they don't exist anywhere else!
I can see your point about the beauty of mathmatics if you believe there is something pure about it and perhaps you are right. Its an immutable language, so it cannot be corrupted, is that what attracts you to it?
But when it comes back to KC and probability, science is going to have move some way yet to understand who gets it and why. There are clearly a whole bunch of factors which again you understand more clearly than I, but it still scratching the surface.
This is a fascinating discussion by the way.
Despite my denials, I am actually interested in maths and science although I do not clearly share your level of understanding. Are you sure it is logic, though?
I remember in A-Level physic becoming quite agitated because the teacher had told me that one of the assumptions of Newton's Laws are that the earth is perfectly spherical. Of course, the earth is not spherical at all. In fact it is so irregular in shape that they just call it a Geoid which is a fudge by the way. So Newton's laws do not stand, they are based on incorrect assumptions. Sure enough, now we know more, we know that Newton's Laws are at best a loose fit! On the other hand without these scientific fudges, the world today would not be as advanced as it. So its difficult and science is clearly an incredibly important and fundamental movement, but I wonder is there any logic in root-1 being i. Doesn't that just mean,"we don't know what the root of -1 is so lets call it i or t...or fish..or anything really" I mean sqaure roots are works of the human mind they don't exist anywhere else!
I can see your point about the beauty of mathmatics if you believe there is something pure about it and perhaps you are right. Its an immutable language, so it cannot be corrupted, is that what attracts you to it?
But when it comes back to KC and probability, science is going to have move some way yet to understand who gets it and why. There are clearly a whole bunch of factors which again you understand more clearly than I, but it still scratching the surface.
This is a fascinating discussion by the way.
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Andrew MacLean
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Absolutely right: the thing about "scientific laws" is that they are always the 'best fit' at the time they are advanced. The task of the scientific community then becomes the rigourous testing of the law until they find a case that does not fit.
When that case is found it is clear that the 'law' was not a law at all, but a staging post in the developing understanding of the world and the way it is.
I am fairly confident that without Newton, Einstein would not have been able to develop his general and special theories of relativity. without Einstein, Edward Lorenz would not have been able to develop his first Chaos Theory, nor Schroedinger his Quantum ... without these building blocks String theory may never have developed.
You are right. I do enjoy mathematics because it is a precise language, but one that contains many paradoxes. i is just one of them; the value of pi is another.
What all this has to do with KC is that both medical ethics and medical advance depends upon the rigorous statistical analysis of data produced by researchers, her this rigour is often absent as one team seeks to advance its claims against all others (you only need to glance through some of the strings on this site to see just exaclty how damaging and dangerous that can be!)
My introduction was through Medical Ethics. At one time I spoke for the Church of Scotland on Medical Ethical issues, and I have served on Ethics committees.
Just as a footnote, I'd be reluctant to dismiss all classical physics. Newton is still pretty reliable on gravity, and his work is still the basis on which people are propelled into space, or into the air in big airoplanes! You see it remains true that although his 'laws' may not be immutable at all, they are still a basis on which predictions can be made about the way things are and will be when forces interact in the real world.
Even Archimedes enables us to predict whether a body will float or sink, and if the King of Sweden had not been so blinded by Huberis and more attentive to Archimedes, then the Vasser may have sailed and the subsequent history of military conquest originating in Europe may have taken a different path altogether.
Andrew
When that case is found it is clear that the 'law' was not a law at all, but a staging post in the developing understanding of the world and the way it is.
I am fairly confident that without Newton, Einstein would not have been able to develop his general and special theories of relativity. without Einstein, Edward Lorenz would not have been able to develop his first Chaos Theory, nor Schroedinger his Quantum ... without these building blocks String theory may never have developed.
You are right. I do enjoy mathematics because it is a precise language, but one that contains many paradoxes. i is just one of them; the value of pi is another.
What all this has to do with KC is that both medical ethics and medical advance depends upon the rigorous statistical analysis of data produced by researchers, her this rigour is often absent as one team seeks to advance its claims against all others (you only need to glance through some of the strings on this site to see just exaclty how damaging and dangerous that can be!)
My introduction was through Medical Ethics. At one time I spoke for the Church of Scotland on Medical Ethical issues, and I have served on Ethics committees.
Just as a footnote, I'd be reluctant to dismiss all classical physics. Newton is still pretty reliable on gravity, and his work is still the basis on which people are propelled into space, or into the air in big airoplanes! You see it remains true that although his 'laws' may not be immutable at all, they are still a basis on which predictions can be made about the way things are and will be when forces interact in the real world.
Even Archimedes enables us to predict whether a body will float or sink, and if the King of Sweden had not been so blinded by Huberis and more attentive to Archimedes, then the Vasser may have sailed and the subsequent history of military conquest originating in Europe may have taken a different path altogether.
Andrew
Andrew MacLean
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Matthew_
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I'm with you on this. I can leave i as a paradox safely parked! As you say the initial scientific discoveries going back Archimedes and beyond are fundamental to what we achieved to this day and the scientific rigour which now exists is fundamental to the future. I don't mean to dis it. I think like everyone else its the people (me included) who take everything on face value. The Starkeys for instance, who were behind a lot of the Austrolopithicus (definitely spelt that wrong) claimed there was no more evidence at the end of all their work, spanning generations than for the sudden arrival of man on the planet. Personally, I am sure their proposition of evolutionary man is based on fact and I have no problem with that, even as a christian. The point it their hypotheses were limited and they said as much. But society as a whole has discarded that bit and decided that everything was just how they said it was. Well it might not have been, its an educated guess and a pretty good one but its still a guess.
Oh by the way, on rigorous testing: Popperian Falsificationism suggests that if you test something often enough you can disprove it, even 2 + 2 = 4. I think that is a real issue with KC because it is so complex it seems to me that more you look into it, the more confusing it becomes. There is plenty of complexity to conjure with!
By the way, is it OK to have this conversation, its only very tenuously KC linked now!
Oh by the way, on rigorous testing: Popperian Falsificationism suggests that if you test something often enough you can disprove it, even 2 + 2 = 4. I think that is a real issue with KC because it is so complex it seems to me that more you look into it, the more confusing it becomes. There is plenty of complexity to conjure with!
By the way, is it OK to have this conversation, its only very tenuously KC linked now!
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rosemary johnson
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'scuse me mentioning this...
i isn't irrational; it's imaginary.
Square root of 2 is irrational, because it can't be expressed as a ratio. That is, there are no two integers x and y such that (x/y) = root 2
(Theancient greeks knew and could prove this - it drove Pythagoras, of triangle fame, potty, IIRR.)
A square root of -1, now: there's really no such thing. There's no real number, whether expressable as a ratio of integers or not, that squares to give -1.
But the mathematicians weren't happy with a definition system in which somethings just didn't exist, so they just had to imagine it. Hence i for imaginary.
And then they started doing maths with this imaginary number and got lots of interesting things coming out.
SOmewhere along the line came fractals - the things that make those very pretty computer graphics drawings.
More technically came a whole branch of mathematics involving what they called complex numbers (with a real part and an imaginary part) to do with things they called complex potentials and related goodies.
This allseemed very esoteric and isolated "good clean pure fun" - until along come some of the folks who'd been doing experiments on fluid dynamics, found this complex maths stuff that the mathemiticans thought they'd imagined with no earthly use, and said "That's just what we need cos it fits our fluid dynamics like a glove!"
Contrary to what many may think (particularly fromt he way I've described it.....) that is actually fluid dynamics as in air flowing round aircraft wings, or water round boats, that makes planes fly, boats sail fast and submarines swim.
or, less exotically, why when you stir a cup of tea, all the tea leaves go in a column up the centre of the cup - which when I was an college doing the hydrodynamics course, I used to know how to do the equations for.
Rosemary
i isn't irrational; it's imaginary.
Square root of 2 is irrational, because it can't be expressed as a ratio. That is, there are no two integers x and y such that (x/y) = root 2
(Theancient greeks knew and could prove this - it drove Pythagoras, of triangle fame, potty, IIRR.)
A square root of -1, now: there's really no such thing. There's no real number, whether expressable as a ratio of integers or not, that squares to give -1.
But the mathematicians weren't happy with a definition system in which somethings just didn't exist, so they just had to imagine it. Hence i for imaginary.
And then they started doing maths with this imaginary number and got lots of interesting things coming out.
SOmewhere along the line came fractals - the things that make those very pretty computer graphics drawings.
More technically came a whole branch of mathematics involving what they called complex numbers (with a real part and an imaginary part) to do with things they called complex potentials and related goodies.
This allseemed very esoteric and isolated "good clean pure fun" - until along come some of the folks who'd been doing experiments on fluid dynamics, found this complex maths stuff that the mathemiticans thought they'd imagined with no earthly use, and said "That's just what we need cos it fits our fluid dynamics like a glove!"
Contrary to what many may think (particularly fromt he way I've described it.....) that is actually fluid dynamics as in air flowing round aircraft wings, or water round boats, that makes planes fly, boats sail fast and submarines swim.
or, less exotically, why when you stir a cup of tea, all the tea leaves go in a column up the centre of the cup - which when I was an college doing the hydrodynamics course, I used to know how to do the equations for.
Rosemary
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Matthew_
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Fractals, isn't that related to chaos, I know you've mentioned in relation to complexity theory. I'm probably really showing my ignorance now as I am not mathematician. I read that chaos students felt one day this would all lead to a "theory of everything".
Hydrodynamics is interesting. I used to be in the submarine service so we get a very rough appreciation for it. Driving a submarine is not like flying a plane underwater like everybody thinks, its more like an airship in fact.
Hydrodynamics is interesting. I used to be in the submarine service so we get a very rough appreciation for it. Driving a submarine is not like flying a plane underwater like everybody thinks, its more like an airship in fact.
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rosemary johnson
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I'm afraid I'm too old to have done chaos theory at college.....!
the general principle behind drwing the fractals pictures is to take a function of a complex number (z = x + iy, which i is root (-1)) and kkeping iterating until the absolute value [root (x^2 + y^2)] goes very large.
That is, draw a graph with rectangular Cartesian axes x and y. For each point (x,y) on this graph, feed the x and y as the initial values into z=x+iy.
Then calculate the value of the function of z, which will have a real ("x") and imaginery ("y") part. Feed these values ack into the equation for the function and re-evaluate.
At each stage, check the value of root (x^2+y^2) has become "large", stop, and put a coloured blob at that point, colour-coded for the number of interations you've done so far.
If you interate and interate and root (x^2+y^2) still isn't "large", then make a black blob and go on to the next.
If you've got an appropriate function and a suitable defiition of "large" you get a picture with one of those black shapes like a heart with a blob on the end with a blob on the end......
It's fairly easy to write a little programme to draw these - well, it must be, because I've tried writing them before!
Can't remember off-hand the functions of z I used, but will try to find the bits of paper I kept if anyone wants a go.
Rosemary
the general principle behind drwing the fractals pictures is to take a function of a complex number (z = x + iy, which i is root (-1)) and kkeping iterating until the absolute value [root (x^2 + y^2)] goes very large.
That is, draw a graph with rectangular Cartesian axes x and y. For each point (x,y) on this graph, feed the x and y as the initial values into z=x+iy.
Then calculate the value of the function of z, which will have a real ("x") and imaginery ("y") part. Feed these values ack into the equation for the function and re-evaluate.
At each stage, check the value of root (x^2+y^2) has become "large", stop, and put a coloured blob at that point, colour-coded for the number of interations you've done so far.
If you interate and interate and root (x^2+y^2) still isn't "large", then make a black blob and go on to the next.
If you've got an appropriate function and a suitable defiition of "large" you get a picture with one of those black shapes like a heart with a blob on the end with a blob on the end......
It's fairly easy to write a little programme to draw these - well, it must be, because I've tried writing them before!
Can't remember off-hand the functions of z I used, but will try to find the bits of paper I kept if anyone wants a go.
Rosemary
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