# Burali-forti and boggio to general relativity

Receive emails about upcoming NOVA programs and related content, as well as featured reporting about current events through a science lens. General relativity, the theory of gravity Albert Einstein published years ago, is one of the most successful theories we have.

It has passed every experimental test; every observation from astronomy is consistent with its predictions. Physicists and astronomers have used the theory to understand the behavior of binary pulsars, predict the black holes we now know pepper every galaxy, and obtain deep insights into the structure of the entire universe.

To be more precise: most believe it is incomplete. After all, the other forces of nature are governed by quantum physics; gravity alone has stubbornly resisted a quantum description. Meanwhile, a small but vocal group of researchers thinks that phenomena such as dark matter are actually failures of general relativity, requiring us to look at alternative ideas. The fix is dark matter, particles invisible to light but endowed with gravity.

However, none of our detectors or experiments have ever seen a dark matter particle directly, leading some to doubt that dark matter actually exists. Nevertheless, MOND is successful enough in galaxies to inspire some theorists to try to modify itin hopes of making predictions that more closely match nature. John Moffat was also motivated to modify GR by the problem of dark matter, but is uninterested in reproducing MOND because of its observational failures.

Instead, his modification of general relativity involves allowing the strength of gravity to vary slightly in space and time and changing the way gravity acts over long distances. Farther out still, the force strength drops off in proportion to the mass of the vector field. So far, his theory seems to be able to explain many observed properties of galaxies, galaxy clusters, and other observations.

Few astrophysicists doubt that black holes exist: We know of a large number of very massive, very dense objects in the cosmos, for which the black hole hypothesis is the only one that fits.

The shadow is created as light orbits close to, but not quite in the event horizon; the EHT would see it as a faint ring with a dark interior. GR makes very specific predictions about the shape and size of that ring—which was a dramatic visual effect in the movie Interstellar.

The ultimate arbiter of a theory, after all, is nature.

## Do We Need to Rewrite General Relativity?

If one of the dark matter experiments found particles with the right properties, then the motive to modify GR would diminish; if more and more experiments fail to find dark matter, then researchers are likely to pay more attention to alternative theories, perhaps even ones that are unorthodox or complex. John W. Moffat Find books, papers, and media appearances by Perimeter Institute physicist John Moffat at his personal web site.

Kavli Institute for Theoretical Physics: Dark Matter vs Modified Gravity Caltech physicist Sean Carroll delivers an hour-long lecture on dark matter and the pros and cons of modified gravity theories. Quantum Diaries: How do we know dark matter exists? Support Provided By Learn More. Email Address. Zip Code. Related Tiny Black Holes. Share this article.General relativity GRalso known as the general theory of relativity GTRis the geometric theory of gravitation published by Albert Einstein in and the current description of gravitation in modern physics.

General relativity generalizes special relativity and refines Newton's law of universal gravitationproviding a unified description of gravity as a geometric property of space and timeor spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present.

The relation is specified by the Einstein field equationsa system of partial differential equations. Some predictions of general relativity differ significantly from those of classical physicsespecially concerning the passage of time, the geometry of space, the motion of bodies in free falland the propagation of light.

Examples of such differences include gravitational time dilationgravitational lensingthe gravitational redshift of light, and the gravitational time delay. The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date.

Although general relativity is not the only relativistic theory of gravityit is the simplest theory that is consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity.

Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars.

There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes. For example, microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holesrespectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waveswhich have since been observed directly by the physics collaboration LIGO.

In addition, general relativity is the basis of current cosmological models of a consistently expanding universe. Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.

Soon after publishing the special theory of relativity inEinstein started thinking about how to incorporate gravity into his new relativistic framework. Inbeginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November of what are now known as the Einstein field equations, which form the core of Einstein's general theory of relativity.

The Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But inthe astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the final stages of gravitational collapse, and the objects known today as black holes.

In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant —to match that observational presumption. This is readily described by the expanding cosmological solutions found by Friedmann inwhich do not require a cosmological constant.

During that period, general relativity remained something of a curiosity among physical theories. It was clearly superior to Newtonian gravitybeing consistent with special relativity and accounting for several effects unexplained by the Newtonian theory.

Einstein showed in how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters " fudge factors "[11] and in an expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29,[12] instantly making Einstein famous. Over the years, general relativity has acquired a reputation as a theory of extraordinary beauty.

It juxtaposes fundamental concepts space and time versus matter and motion which had previously been considered as entirely independent.

Chandrasekhar also noted that Einstein's only guides in his search for an exact theory were the principle of equivalence and his sense that a proper description of gravity should be geometrical at its basis, so that there was an "element of revelation" in the manner in which Einstein arrived at his theory. General relativity can be understood by examining its similarities with and departures from classical physics.

The first step is the realization that classical mechanics and Newton's law of gravity admit a geometric description. The combination of this description with the laws of special relativity results in a heuristic derivation of general relativity.

At the base of classical mechanics is the notion that a body 's motion can be described as a combination of free or inertial motion, and deviations from this free motion. Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motionwhich states that the net force acting on a body is equal to that body's inertial mass multiplied by its acceleration.Cesare Burali-Forti attended the University of Pisa, graduating in Immediately after graduating he taught in a school but he moved to Turin in where he was appointed to the Military Academy.

Burali-Forti taught analytic projective geometry at the Military Academy where he continued to teach for the rest of his life. A university teaching position would have been more to Burali-Forti's liking but in this he had difficulties. He was a great believer in vector methods but, at this time, these were not in favour.

It is hard to believe from our present view of mathematics that vector methods would ever be less than welcomed. However, at this time many mathematicians opposed vector methods and unfortunately these views prevailed on the committee that considered Burali-Forti's submission for a doctorate. He was failed on these grounds, never tried again, and as a consequence was never able to teach in a university, although he did give informal lecture courses there. In Burali-Forti gave an informal series of lectures on mathematical logic at the University of Turin.

After the course the lectures were written up as a book and Burali-Forti presented a copy of the book to the Academy of Sciences of Turin in June At the start of the academic session, Burali-Forti became Peano 's assistant at the University of Turin. He was to hold this position until Burali-Forti attended the Congress and presented a paper The postulates for the geometry of Euclid and of Lobachevsky to the Geometry section of the Congress.

Burali-Forti is famed as the first discoverer of a set theory paradox in which was framed in technical terms but in essence reduces to a 'set of all sets' paradox. Cantor was to discover a similar paradox two years later. As well as set theory and vector analysis, Burali-Forti also worked on linear transformations and their applications to differential geometry. Not only was Burali-Forti a prolific writer, with over publications, he was also very interested in how to teach mathematics.

The "Mathesis" Italian Society of Mathematicians, aimed at school teachers of mathematics, was founded in Burali-Forti joined Mathesis in academic year He played a major role in the first congress of the Society which was held in Turin in September Burali-Forti was a close friend of Peano 's but his closest friend and mathematical collaborator was Roberto Marcolongo. Burali-Forti and Marcolongo were called the "vectorial binomial" by their friends. However this collaboration ended when they differed in their views on relativity.

Burali-Forti never understood the theory of relativity and, together with Boggiohe wrote a book which claimed to prove that the theory of relativity was impossible. Kennedy writes in [2]:- Many of his publications were highly polemical, but in his family circle and among friends he was kind and gentle. He loved music, Bach and Beethoven being his favourite composers. He was a member of no academy. Always an independent thinker, he asked that he not be given a religious funeral.The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers for example Peano arithmeticthere are true propositions about the natural numbers that cannot be proved from the axioms.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theoryassuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs.

He also made important contributions to proof theory by clarifying the connections between classical logicintuitionistic logicand modal logic.

In his family, young Kurt was known as Herr Warum "Mr. Why" because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever ; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in his older brother Rudolf born left for Vienna to go to medical school at the University of Vienna.

By that time, he had already mastered university-level mathematics. During this time, he adopted ideas of mathematical realism. Inat the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision.

In it, he established his eponymous completeness theorem regarding the first-order predicate calculus. He was awarded his doctorate inand his thesis accompanied by some additional work was published by the Vienna Academy of Science.

Here he delivered his incompleteness theorems. In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers e.

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalismto find a set of axioms sufficient for all mathematics. In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement.

That is, for any computably enumerable set of axioms for arithmetic that is, a set that can in principle be printed out by an idealized computer with unlimited resourcesthere is a formula that is true of arithmetic, but which is not provable in that system.Recipient s will receive an email with a link to 'The Late Entrance of Relativity into Italian Scientific Community ' and will not need an account to access the content.

Sign In or Create an Account.

### Cesare Burali-Forti

User Tools. Sign In. Article Navigation. Close mobile search navigation Article navigation. Volume 31, Issue 1. This article was originally published in. Previous Article Next Article. Research Article January 01 This Site. Google Scholar. Historical Studies in the Physical and Biological Sciences 31 1 : — Split-Screen Views Icon Views. Guest Access. Get Permissions. Cite Icon Cite. Glick, ed. Michele Biezunski, "Einstein's reception in Paris in ," in Glick ref.

Roberto Maiocchi, Einstein in Italia. La scienza e lafilosofia italiane di fronte alla teoria della relativita Milan, Giuseppe Giuliani and Giulio Marazzini, "The Italian physics com- munity and the crisis of the classical physics: New radiations, quanta and relativity [— ]," Annals of science, 51 Paolantonio Marazzini, Nuove radiazioni, quanti e relativita in Italia: Pavia, Michelangelo De Maria and Giulio Maltese, "I fisici sperimentali italiani e la relativita ," Quaderni di storia della fisica del Giornale di Fisica, 1 Zeeman di Leida," NC, 6 Orso Mario Corbino, "A proposito della interpretazione del fenomeno di Zeeman data dal sig.

Cornu," NC, 7 Augusto Righi, La moderna teoria dei fenomeni fisici radioattivita, ioni, elettroni Bologna,chapt. Lorentz and the electromagnetic view of nature," Isis, 61, on Lewis Pyenson, "The relativity revolution in Germany," inGlick ref. Max Abraham, "Considerazioni critiche sulle radiazioni elettriche," NC, 18 Gli anni tra le due guerre mondiali Milan, Gianni Paoloni, ed. Erwiderung auf eine Bemerkung des Hrn. Einstein," AP, 55, on Pietro Nastasi, ed. Michele La Rosa, Der Aether.

Geschichte einer Hypothese Leipzig, Immediately after graduating he taught in a school but he moved to Turin in where he was appointed to the Military Academy. Burali-Forti taught analytic projective geometry at the Military Academy where he continued to teach for the rest of his life. A university teaching position would have been more to Burali-Forti's liking but in this he had difficulties. He was a great believer in vector methods but, at this time, these were not in favour.

It is hard to believe from our present view of mathematics that vector methods would ever be less than welcomed. However, at this time many mathematicians opposed vector methods and unfortunately these views prevailed on the committee that considered Burali-Forti's submission for a doctorate.

He was failed on these grounds, never tried again, and as a consequence was never able to teach in a university, although he did give informal lecture courses there. In Burali-Forti gave an informal series of lectures on mathematical logic at the University of Turin. After the course the lectures were written up as a book and Burali-Forti presented a copy of the book to the Academy of Sciences of Turin in June At the start of the academic session, Burali-Forti became Peano 's assistant at the University of Turin.

He was to hold this position until Burali-Forti attended the Congress and presented a paper The postulates for the geometry of Euclid and of Lobachevsky to the Geometry section of the Congress.

Burali-Forti is famed as the first discoverer of a set theory paradox in which was framed in technical terms but in essence reduces to a 'set of all sets' paradox.

Cantor was to discover a similar paradox two years later. As well as set theory and vector analysis, Burali-Forti also worked on linear transformations and their applications to differential geometry. Not only was Burali-Forti a prolific writer, with over publications, he was also very interested in how to teach mathematics. The "Mathesis" Italian Society of Mathematicians, aimed at school teachers of mathematics, was founded in Burali-Forti joined Mathesis in academic year He played a major role in the first congress of the Society which was held in Turin in September Burali-Forti was a close friend of Peano 's but his closest friend and mathematical collaborator was Roberto Marcolongo.

Burali-Forti and Marcolongo were called the "vectorial binomial" by their friends. However this collaboration ended when they differed in their views on relativity.

Burali-Forti never understood the theory of relativity and, together with Boggiohe wrote a book which claimed to prove that the theory of relativity was impossible. Kennedy writes in [2]:- Many of his publications were highly polemical, but in his family circle and among friends he was kind and gentle. He loved music, Bach and Beethoven being his favourite composers. He was a member of no academy. Always an independent thinker, he asked that he not be given a religious funeral.He had about publications.

View two larger pictures. Cosimo was Chief Secretary of the Province of Arezzo. He was described as a [ 8 ] After schooling in Arezzo, Cesare attended the Military College in Florence, completing his secondary level studies there in The years he was growing up were difficult years in the new Italy with widespread unrest which eased after Rome became part of Italy in Burali-Forti studied at Pisa from to Documents show that he progressed from the first to the second year of studies in the Faculty of Mathematics and Physics on 13 Novemberand began his third year of studies on 7 Novemberwithout having taken the Physics and the Chemistry examinations.

He moved from the third year to the fourth in November During these years he was taught by several mathematicians, the most influential being those we mentioned above.

Dini held the chair of analysis and higher geometry as well as the chair of infinitesimal analysis. Bianchi had been a student at Pisa but had studied abroad before returning to Pisa in where he taught differential geometry. Bettialthough his interests had been in algebra and topology, was by this time becoming interested in mathematical physics and held the chair of celestial mechanics.

For the first part of Burali-Forti's studies at Pisa, Volterra was a research student but he became Professor of Mechanics in With rapidly developing political events in the new country of Italy, several of Burali-Forti's lecturers were also involved in politics. For example after serving in local government, Dini was elected to the national Italian parliament in as a representative from Pisa.

Among the courses taken by Burali-Forti we mention those on infinitesimal analysis and higher analysis by Diniand courses on rational mechanics, celestial mechanics and mathematical physics taught by Betti. At his final examination he had to answer questions on mathematical physics topics that Betti had discussed with Riemann during his visits to Pisa in the s.

Immediately after graduating Burali-Forti taught at the Technical Institute in Augusta, Sicily, where he was appointed on 29 Augustbut he moved to Turin in when he was successful in the competition for an extraordinary professorship at the Military Academy of Artillery and Engineering.

He could not marry before obtaining a position with an income which would allow him to support a wife and family and this appointment, starting on 1 Septembercertainly meant that he was now financially secure. He also taught at the Sommeiller technical school in Turin until which provided him with further income.

On 29 October ofhe married Gemma Viviani; they had a son Umberto who was born on 9 August His appointment in had been on a temporary basis but on 30 June his position was made permanent.

On 30 October he was promoted to extraordinary professor first-class and then in he became a full professor, holding the chair of projective geometry. He was the only professor at the Military Academy who was not himself a military man. Burali-Forti spent the rest of his career at the Military Academy. Roberto Marcolongo [ 25 ]described him as a lecturer Some of Burali-Forti's colleagues at the Military Academy were particularly important in his developing research interests.

## Paradoxes and Contemporary Logic

The two became good friends and collaborated with each other in mathematical investigations. Through the years Burali-Forti did much to bring Peano 's work to a wider audience. Peano taught mathematics there until as well as having university appointments.