Difference between revisions of "The amazing story of Geometry"
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William Rowan Hamilton sits just in the middle of the story. He wanted to find a plausible algebraic model in which to write down Maxwell's equations (involving ''pseudo''-vectorial objects, elements of structures now called ''Clifford-Hamilton alegbras''). Its tentative ''quaternionic geometry'' fails to be the right idea: the natural metric on the space of quaternions fails to model the natural metric on space-time, because of a signature problem (the "physical" space admits isotropic vectors, pointing the direction where light beams propagate: no such vector exists in the quaternionic 4-space). So, pseudovectors seem not to be the right object to describe our universe, but a new formalism to understand the classical motion of bodies, ''analytical mechanics'', stemmed from the work of Hamilton: he used to project the equations of motion of a body on the cotangent bundle of its configuration space, and worked out the formalism in which Quantum Mechanics was written a century after. | William Rowan Hamilton sits just in the middle of the story. He wanted to find a plausible algebraic model in which to write down Maxwell's equations (involving ''pseudo''-vectorial objects, elements of structures now called ''Clifford-Hamilton alegbras''). Its tentative ''quaternionic geometry'' fails to be the right idea: the natural metric on the space of quaternions fails to model the natural metric on space-time, because of a signature problem (the "physical" space admits isotropic vectors, pointing the direction where light beams propagate: no such vector exists in the quaternionic 4-space). So, pseudovectors seem not to be the right object to describe our universe, but a new formalism to understand the classical motion of bodies, ''analytical mechanics'', stemmed from the work of Hamilton: he used to project the equations of motion of a body on the cotangent bundle of its configuration space, and worked out the formalism in which Quantum Mechanics was written a century after. | ||
| − | At the same time, the Italian school in Algebraic Geometry (working in many italian universities like Padua, Rome, Cremona, Florence, Naples, ...) is pursuing a much more subtle task. Italian school has been deeply influenced by the work of Riemann and Emmy Noether: Noether's life and genius could be one of the best "manifesto" of a tentative book about women in Science: from Abstract Algebra to celestial Mechanics, there's no field of pure Mathematics (and engineering, Physics, hydrodynamics...) she didn't touched and radically changed. | + | At the same time, the Italian school in Algebraic Geometry (working in many italian universities like Padua, Rome, Cremona, Florence, Naples, ...) is pursuing a much more subtle task. Italian school has been deeply influenced by the work of Riemann and Emmy Noether: Noether's life and genius could be one of the best "manifesto" of a tentative book about women in Science: from Abstract Algebra to celestial Mechanics, there's no field of pure Mathematics (and engineering, Physics, hydrodynamics...) she didn't touched and radically changed. Anyway she was a woman in the XIX century, so Noether's genius was completely stonewalled by the academics: "Is this a university or a public restroom?". |
| − | + | The profoundness of Noether's idea can be stated in this way: we can associate to any algebraic curve (defined to be the zero-set of a suitable polynomial) a ''ring'', named the ''ring of coordinates'' of the curve. This ring encodes geometric properties translating them into algebraic ones: a suitable family of subsets of the ring (its ''prime ideals'') mimic the behaviour of ''points in a topological space''. ''Primes are points'', she wrote: the set of prime ideals turns out to be a topological space ''homeomorphic to the curve we started from''! | |
| − | Yet to be translated... | + | ==Yet to be translated... |
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Rapidamente si dovette combattere con questo problema: so passare da una curva a un anello, e so che le proprietà geometriche vengono tradotte in certe proprietà algebriche dell'anello stesso. Posso rovesciare questa corrispondenza, e pensare che ad un oggetto algebrico, per quanto bizzarro e selvatico esso sia, corrisponda sempre un oggetto geometrico? Posso estendere il "dizionario" con cui un problema geometrico ha una soluzione algebrica ad uno col quale un problema algebrico si possa "disegnare" e coinvolga un ente geometrico? Pensate a quanto è stato cardinale in Teoria dei Numeri poter passare biiettivamente da un linguaggio all'altro e capirete quanto centrale il problema sia. | Rapidamente si dovette combattere con questo problema: so passare da una curva a un anello, e so che le proprietà geometriche vengono tradotte in certe proprietà algebriche dell'anello stesso. Posso rovesciare questa corrispondenza, e pensare che ad un oggetto algebrico, per quanto bizzarro e selvatico esso sia, corrisponda sempre un oggetto geometrico? Posso estendere il "dizionario" con cui un problema geometrico ha una soluzione algebrica ad uno col quale un problema algebrico si possa "disegnare" e coinvolga un ente geometrico? Pensate a quanto è stato cardinale in Teoria dei Numeri poter passare biiettivamente da un linguaggio all'altro e capirete quanto centrale il problema sia. | ||
Ecco in poche parole il merito di Alexander Grothendieck: dimostrare che questo dizionario esiste, insegnarci a leggerlo, proporre ulteriori generalizzazioni all'idea per cui "l'algebra è geometria passata attraverso lo specchio". Il problema della rappresentazione dello spazio, connesso al problema della soluzione di equazioni, connesso al problema della coordinatizzazione di uno spazio astratto, connesso al problema della logica, altamente non-booleana (ovvero basata su piu valori di verità che non 0-falso e 1-vero), che siamo costretti a trattare studiando la fisica, connesso all'evidente problema ermeneutico che tutti loro portano con sè...'' | Ecco in poche parole il merito di Alexander Grothendieck: dimostrare che questo dizionario esiste, insegnarci a leggerlo, proporre ulteriori generalizzazioni all'idea per cui "l'algebra è geometria passata attraverso lo specchio". Il problema della rappresentazione dello spazio, connesso al problema della soluzione di equazioni, connesso al problema della coordinatizzazione di uno spazio astratto, connesso al problema della logica, altamente non-booleana (ovvero basata su piu valori di verità che non 0-falso e 1-vero), che siamo costretti a trattare studiando la fisica, connesso all'evidente problema ermeneutico che tutti loro portano con sè...'' | ||
Revision as of 04:12, 18 September 2012
The problem of figuration of space is quite an ancient matter. Euclid's Elements first proposed a set of rules Mathematic(ian)s should use to infer "geometrical" properties of a suitable conglomerate of objects (points, lines and polygons in the plane). It's worth to say that formally speaking Euclid's Elements are inconsistent (one of the first theorems in Euclid's book gives a recipe to build an equilateral triangle having prescribed side, but nothing prevents the two circles needed in this construction from having empty intersection); nonetheless his work is worth to be considered the first piece of abstract Mathematics: a set of objects (points and lines) and a bunch of inference rules (the relation of incidence, some formal properties of such relation, mimicing the "natural" behaviour of forms in physical space) is all one needs to do "Geometry".
Jump forward to the IX century and you'll meet (Abū Jaʿfar Muhammad ibn Mūsā) al-Khwārizmī, commonly accepted by mathematical storiography as the father of Algebra. Astronomer, philosopher, mathematician, he combined the work of Diophantus and Brahmagupta to obtain the first set of rules to handle with quadratic equations. He classified "all the different types" of degree-two equation and proposed both an algebraic and a geometrical way of solution: every algebraically solvable problem which lacks a geometrically evident solution, has to be considered incomplete: no "algorithm" (the european word is nothing but an adaption of al-Khwārizmī's name) can stand alone to justify a solving procedure.
Rene Descartes has to be considered the father of analytical geometry. He was the first to state the greatest abstract principle of modern Geometry: a geometric problem can be translated in an algebraic one via coordinates. A precise knowledge of the rules governing the algebraic equations of a subset of the plane totally equals a precise knowledge of the space itself. Numbers are points on a line; a conic is the closed curve arising as the set of zeroes of a degree-two polynomial in x and y; any polynomial of degree n defines a precise curve on the plane: this is made by two simpler (smaller) curves if and only if the polynomial can be factored as a product.
If only Kant knew more Geometry! Space is not a "pure form" of epistemic intuition; it is determined by a more atavic concept, the primeval idea of a coordinate system. Husserl understood Geometry as the obvious science of space, and Algebra as the natural mathematization of time: the idea of an interplay between the two dimensions is quite more ancient than Einstein's theory of relativity.
William Rowan Hamilton sits just in the middle of the story. He wanted to find a plausible algebraic model in which to write down Maxwell's equations (involving pseudo-vectorial objects, elements of structures now called Clifford-Hamilton alegbras). Its tentative quaternionic geometry fails to be the right idea: the natural metric on the space of quaternions fails to model the natural metric on space-time, because of a signature problem (the "physical" space admits isotropic vectors, pointing the direction where light beams propagate: no such vector exists in the quaternionic 4-space). So, pseudovectors seem not to be the right object to describe our universe, but a new formalism to understand the classical motion of bodies, analytical mechanics, stemmed from the work of Hamilton: he used to project the equations of motion of a body on the cotangent bundle of its configuration space, and worked out the formalism in which Quantum Mechanics was written a century after.
At the same time, the Italian school in Algebraic Geometry (working in many italian universities like Padua, Rome, Cremona, Florence, Naples, ...) is pursuing a much more subtle task. Italian school has been deeply influenced by the work of Riemann and Emmy Noether: Noether's life and genius could be one of the best "manifesto" of a tentative book about women in Science: from Abstract Algebra to celestial Mechanics, there's no field of pure Mathematics (and engineering, Physics, hydrodynamics...) she didn't touched and radically changed. Anyway she was a woman in the XIX century, so Noether's genius was completely stonewalled by the academics: "Is this a university or a public restroom?". The profoundness of Noether's idea can be stated in this way: we can associate to any algebraic curve (defined to be the zero-set of a suitable polynomial) a ring, named the ring of coordinates of the curve. This ring encodes geometric properties translating them into algebraic ones: a suitable family of subsets of the ring (its prime ideals) mimic the behaviour of points in a topological space. Primes are points, she wrote: the set of prime ideals turns out to be a topological space homeomorphic to the curve we started from!
==Yet to be translated...
Rapidamente si dovette combattere con questo problema: so passare da una curva a un anello, e so che le proprietà geometriche vengono tradotte in certe proprietà algebriche dell'anello stesso. Posso rovesciare questa corrispondenza, e pensare che ad un oggetto algebrico, per quanto bizzarro e selvatico esso sia, corrisponda sempre un oggetto geometrico? Posso estendere il "dizionario" con cui un problema geometrico ha una soluzione algebrica ad uno col quale un problema algebrico si possa "disegnare" e coinvolga un ente geometrico? Pensate a quanto è stato cardinale in Teoria dei Numeri poter passare biiettivamente da un linguaggio all'altro e capirete quanto centrale il problema sia. Ecco in poche parole il merito di Alexander Grothendieck: dimostrare che questo dizionario esiste, insegnarci a leggerlo, proporre ulteriori generalizzazioni all'idea per cui "l'algebra è geometria passata attraverso lo specchio". Il problema della rappresentazione dello spazio, connesso al problema della soluzione di equazioni, connesso al problema della coordinatizzazione di uno spazio astratto, connesso al problema della logica, altamente non-booleana (ovvero basata su piu valori di verità che non 0-falso e 1-vero), che siamo costretti a trattare studiando la fisica, connesso all'evidente problema ermeneutico che tutti loro portano con sè...