Ladies and gentlemen....Sir William .. Rowan...
Hamilton! Applause please!
So i awoke wondering why my education in vectors had been soooo lamentable?,Why Cartesian coordinates had so constrained the field of mathematics? Why Descartes had had such an influence on mathematics and science through it, after all it is really a simple reference frame, aand when he introduced it nobody was that bothered.
It slowly dawned that it was the personalities, the camps, the warring groups in the grand game, the nationalities: that in mathematics there was as Hamilton puts it a "mathesis" a way of doing mathematics that was doctrinaire nationalistic, patristic and encrusted with anachronism and tradition.
Because of this english maths suffered a loss for over a century due to the Leibniz Newton farago, important insights were overlooked in the case of Grassman because they were not part of the old boy network, and they discouraged outsiders..didah didah didah... Nothing new then.
Rodrigues and grassman what they would have given to have ccess to the web!
Hamilton is a great Irish figure in distinction to the italian, german and general European figures in our part of the history of maths. Because of the web mathematicians of all ages and abilities can get together and critique create and contribute without the old guard control!
So this new frontier in mathematics won't lead to chaos...because wikipedia has shown that effective democratic controls can be put in place, and minority or special interests can set up their own group, without the need for this centuries long hostility and browbeating.
I know that evolution means it is inevitably going to be involved somewhere along the line,but we invented gods to control this sort of thing, so we should use them!
So Hamilton almost singled out the area of Algebra as the new Messiah for mathematics and the sciences due to the success that had flowed from it due to Bombelli, Cardano,Euler Napier,Gauss, Riemann,Newton,Laagrange and Laplace,, on and on as the simple cartesian coordinate mathesis, method tied together all those in the game of maths,both assisting and frustrating mathematicians in what they wanted to think about, to measure and to manipulate.
Even today the myth of number is used to convey concepts that are not related except algebraically. The emergence of a dynamic applied geometry was masked by a clinging to the number mathesis,myth and method. It is amazing to look back and see how mathematicians and physicists struggled to establish a proportinate measure concept of space, whichGreek and earlier mathematicians had in their geometries, and which was common up to the time of cartesian coordinates and beyond.
Newton did not have to have a vector algebra to deal with vector quantities, or dynamic situations, but he did have to invent a new mathesis for dealing with dynamic situations geometrically. It was based on Descartes, but dynamic not static, that is why he called it fluxions.
Didn't he half get into trouble for it! Berkely Lambasted him later. Fortunately for the shy,autistic Newton he was in a respected position because he was right and bright, and the plague had killed off a lot of other contenders! Still he delayed publishing until asked to by Hooke, because of the criticism and prsonal attack he would be subject to.
In those days it was no joke to be besmirched as Galois indicates. You defended your honour with your life! No reason to disturb the frogs and toads then, by troubling the waters unecessarily!
Descartes methods included small differences called differentails later. Newton studied these extensively and through them and compound interest formulae found the binomial series. With this and the fact that differentials were used to algebraically study tangents through Proportions he was able to develop fluxions as a way of compounding tangents to give a curve solution to a dynamic system.
A differential is thus a "compounding sum" of tangential proportions and gives a curve. But along with tangents areas under curves were being studied again by small differences related to tangents. The small differences of these areas under the tangents could be Aggregated directly and they became integrals and were seen as and shown to be the inverse of the mathesis or method of "tangeation".
The difference between compounding and aggreation was not thought that significant, and yet it is a systematic use of vector addition using the parallelogram rule. It was probably hidden by the infinitesimal numbers or fluxions as Newton called them! These were everywhere evident to Newton because he had developed the binomial series and could see them vanishing away in the limit in the series! But what is overlooked is Newton also regarded them as the result of the parallelogram rule, without which he would have quantities but no direction. Newton needed both quantity or magnitude as they distinguished it then,and direction to trace a curve path by tangential envelope.
So by fusing cartesian coordinat geometry and euclidean geometry with algebra of proportions Newton created fluxions for dynamic systems.Leibniz came to it later but for geometrical purposes and without the binomial series which he did learn from Newton.however he did publish both in differential tangential calculus and integral tangential calculus before Newton, giving no reference to their correspondence or collaboration. That was the basis of the dispute.
So sir William Hamilton was not in a glass bottle when he did his maths degree, and he read and corresponded widely.Whether Grassman and Rodrigues were known to him is question, but in ant case no one claims that he stole their ideas, rather that all mathematicians were looking at how to tackle 3d dynamics. Newton for all his brilliance was constrained to 2d by rhe mathematics of his time, and in any case geometry dealt perfectly well with 3d.
So between Newton and Hamilton a generation of mathematicians enamoured with a more symbolic approach as opposed to a geometric one grew up almost disdaining geometry for its lack of algebraic rigour!
Hamilton puts it succintly, they sought a deeper truth than apparently plane and solid geometry could give, there being no advance in it for thousands of years! Plus no one could make their mark in geometry, that belonged to Euclid! Riemann was giving it a good go though.
It is very simple: Bombeeli showed that there was an algebra that was prior to arithmetic, something that had not been realised until he stated it in his treatise. In addition he had shown that it made sense of the "mene" the √-n . Because of him polynomials were generalised and gauss proved a general theorem using these "imaginaries " as Descartes called them. The imaginaries lead to an explosion in the interest and development and applicability of algebra, in which fiels Hamilton was greatly interested,especially as it applied to geometry and space. Because of their notation and their history Hamilton saw the calculus and the differential geometries as a powerful algebra applying to the real world the secret of imaginaries! But coordinates mucked things up!
Much as Cartesian coordinates werethe greatest unifying simplistic idea, they also constrained the imagination and thinking.
Hamilton studied optics because he hoped to understand how rays (or vectors) behaved in the world. Once he had done that he had the technical framework to be able to dismantle cartesian coordinates. Maybe he studies Moebius as well as
Malus,but regardless he had a geometricl structure linked algebraically to positions and directions in space. from this he sought a deeper connection with the imaginaries which are involved with rotation and therefore reflection equivalents.
Hamilton's theory on
the Algebraic Couples prepared him to be able to do his task. It is to be noted that Hamilton was in a club of Algebraists who were linked to the sciences, along with his friend Graves.
That the imaginaries inspired algebraist is seen in Hamiltons reference tot the doctrine(mathesis) of imaginaries and graves seminal but rejected work.
So i woke to find that these things called vectors by hamilton, both the line and the coordinate system were conceived by him to promote the extraordinary efficacy of Algebra as a valid and useful part of mathematics over arithmetic, and a a gateway into a deeper understanding of the world around us.
Hamilton's
dream has been realised but we are not party to it because some have made it their aim to promote themselves,and to keep us in the dark. Would that we all could see Hamilton's rays of light, and his notion of the vector as unencumberd by coorfinates,freeing the geometry to speak to us of the best form to represent it in.
And now we have computers we can perform vectors as naturally as putting knect pieces, or Zome parts together.
November 21, 2006, 10:00:37 AM
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