Foundation for matematics: geometry or algebra?

Until the nineteenth century, geometry was regarded by everybody, including mathematicians, as the most reliable branch of knowledge. Analysis got its meaning and its legitimacy from its link with geometry.

In the nineteenth century, two disasters befell. One was the recognition that there’s more than one thinkable geometry. This was a consequence of the discovery of non-Euclidean geometries.

A second disaster was the overtaking of geometrical intuition by analysis. Space-filling curves** and continuous nowhere-differentiable curves** were shocking surprises. They exposed the fallibility of the geometric intuition on which mathematics rested.

The situation was intolerable. Geometry served from the time of Plato as proof that certainty is possible in human knowledge—including religious certainty. Descartes and Spinoza followed the geometrical style in establishing the existence of God. Loss of certainty in geometry threatened loss of all certainty.

Mathematicians of the nineteenth century rose to the challenge. Led by Dedekind and Weierstrass, they replaced geometry with arithmetic as a foundation for mathematics. This required constructing the continuum—the unbroken line segment—from the natural numbers. Dedekind,** Cantor, and Weierstrass found ways to do this. It turned out that no matter how it was done, building the continuum out of the natural numbers required new mathematical entities—infinite sets.

Philosophy of mathematics

 

                                                                                                       https://www.durham.ac.uk/departments/academic/mathematical-sciences/

Philosophy of mathematics should be tested against five kinds of mathematical practice: research, application, teaching, history, computing.

**

The need to check philosophy of mathematics against mathematical research doesn’t require explication. Many important philosophers of mathematics were mathematical researchers: Pascal, Descartes, Leibniz, d’Alembert, Hilbert, Brouwer, Poincaré, Rényi, and Bishop come to mind. Applied mathematics isn’t illegitimate or marginal. Advances in mathematics for science and technology often are inseparable from advances in pure mathematics. Examples: Newton on universal gravitation and the infinitesimal calculus; Gauss on electromagnetism, astronomy, and geodesy (the last inspired that beautiful pure subject—differential geometry); Poincare on celestial mechanics; and von Neumann on quantum mechanics, fluid dynamics, computer design, numerical analysis, and nuclear explosions.

**

G. H. Hardy “famously” boasted: “I have never done anything ‘useful’. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” Nevertheless, the Hardy-Weinberg law of genetics is better known than his profound contributions to analytic number theory. What’s worse, cryptology is making number theory applicable. Hardy’s contribution to that pure field may yet be useful. Twenty years after the war, mathematical purism was revived, influenced by the famous French group “Bourbaki.” That period is over. Today it’s difficult to find a mathematician who’ll say an unkind word about applied math.

**

On the basis of this reduction, philosophers of mathematics generally limit their attention to set theory, logic, and arithmetic. What does this assumption, that all mathematics is fundamentally set theory, do to Euclid, Archimedes, Newton, Leibniz, and Euler? No one dares to say they were thinking in terms of sets, hundreds of years before the set-theoretic reduction was invented. The only way out (implicit, never explicit) is that their own understanding of what they did must be ignored! We know better than they how to explicate their work! That claim obscures history, and obscures the present, which is rooted in history.

An adequate philosophy of mathematics must be compatible with the history of mathematics. It should be capable of shedding light on that history. Why did the Greeks fail to develop mechanics, along the lines that they developed geometry? Why did mathematics lapse in Italy after Galileo, to leap ahead in England, France, and Germany? Why was non-Euclidean geometry not conceived until the nineteenth century, and then independently rediscovered three times? The philosopher of mathematics who is historically conscious can offer such questions to the historian. But if his philosophy makes these questions invisible, then instead of stimulating the history of mathematics, he stultifies it. Computing is a major part of mathematical practice. The use of computing machines in mathematical proof is controversial. An adequate philosophy of mathematics should shed some light on this controversy.

 

Her şey Hatırlandığı Gibi

“Anneme giderken onlara Eukleides’ten söz etmeye başladım. Geometrisinin temelini oluşturan beş aksiyomunu anlattım. Beşincisi üzerine yapılan tartışmalara değindim, konuyla ilgilenmiş birkaç büyük matematikçinin adını andım ve gelip Eukleides’çi olmayan geometriye dayandım çünkü hiçbir şey göründüğü, hatta yaşandığı gibi değil!

Her şey hatırlandığı gibi.”

Ramanujan

A thousand years before the British came, Indians were doing mathematics. Before the seventh century, while the West was still mired in awkward Roman numerals, India had introduced the numerals we use today. The zero, a symbol expressing nothingness, represented a particular triumph; it may go back to as early as the second century B.C. but definitely appeared in a book in the third century and on the wall of a temple near Gwalior, in central India, in the ninth (where it helped specify a flower garden as 270 units long).

Many of India’s contributions to mathematics were spurred by the need to know, based on astronomical factors, the correct times for Vedic ceremonies. Algebra, geometry, and trigonometry were all thereby enriched. Figures like Aryabhata, born in A.D. 476, who established one of the earliest and best values for π, and Brahmagupta, 150 years later, left theorems even now associated with their names.

It was a rich tradition, but one quite different from that of Greece, the cradle of Western mathematics. Whereas the Greeks, especially Euclid, emphasized formal proof, as in the step-by-step process high school students first encounter in geometry, Indian mathematics stressed the results themselves, however obtained. And without that winnowing out of mathematical dross that formal proof achieved, Indian mathematics was wildly uneven; some of it was just plain wrong. One Muslim writer noted in a book about India that Hindu mathematics was “a mixture of pearl shells and sour dates . . . of costly crystal and common pebbles.”

“By the twentieth century, the pearl shells and crystal had long lain buried in the dust of time. For centuries, India had stood its mathematical ground against the rest of the world. But now, that was ancient history; of late it had added little to the world’s mathematical treasure. Only a line of brilliant mathematicians in Kerala, on the subcontinent’s southwest tip, broke the gloom that otherwise extended back to the great Bhaskara of the twelfth century. The birth of the Mathematical Society could not ensure a rebirth. But its founders—hungry to connect with the West, proud of their country’s heritage yet soberly aware that reverence for the past was no substitute for present achievement—surely hoped it did.

It was into this nascent new world that Ramanujan “came out,” as it were, as a mathematician in 1911.”

I teach high-school math. Is my work pointless?

I am a mathematician and I am a math teacher. That does not mean that I am smarter or less sociable than the next fellow; math is just another profession. One thing I am certain of is that being a mathematician is the best profession in the world.

I am sometimes asked – by students, parents or strangers who see me wearing a geeky math T-shirt – what is the point of studying math? The question does not bother me. It is legitimate and non-trivial. What bothers me are the simplistic answers that are often given.

The most obvious rationale is to develop basic numeracy skills – from counting your change, to figuring out what your mortgage rate means, to interpreting some basic statistics in the news. Everybody agrees, but these only require math knowledge accumulated up to Grade 7. I teach high school. Is my work pointless?

Not if you think of math as a gym for the mind. Children develop strength, co-ordination and risk taking by climbing jungle gyms on the playground. Grown-ups stay healthy – both mentally and physically – by staying active. Math – when rightly and regularly done – keeps the mind active. It develops analytical thinking, creativity, pattern recognition, patience and focus.

Of course, the operative words here are “rightly” and “regularly.” If we just load the students with a bag full of tools that they hardly use, there will be little benefit. It is like teaching your soccer team how to do correct throw-ins and how to recognize an offside position – but not spending much time playing games. The “right” way to do mathematics is not to learn many techniques, but to solve many problems using the learned techniques. The word “regularly” is also of essence. Some parents, who know how important thorough and focused practice is when it comes to developing piano or hockey skills, don’t seem to realize that completing math work in the car when driven from the piano lesson to the hockey practice is not a formula for success.

It’s not easy to see an obvious practical use for different branches of mathematics learned in school. To see the benefits, one needs to look beneath the surface.

Take Euclidean geometry, with its proofs of similarity and collinearity. Who cares if we prove beyond any doubt that three points are collinear?

But did you know that former U.S. president Abraham Lincoln was a lawyer by trade who carefully studied Euclid’s Elements, a book that, through geometry, builds the fundamentals of clear and correct reasoning? In an ideal society, not just lawyers and mathematicians should be able to present a point of view in a convincing way: A follows from B, which is a consequence of C.

Algebra is also a target of smart alecs from respected authors (Stephen Leacock comes to mind) to stand-up comedians and even fellow teachers. Yes, I admit, factoring polynomials and completing squares – the stuff of teenage nightmares – hardly comes in handy in our daily lives. I also admit that we math teachers sometimes indulge in too much fancy technicality that may be more interesting to us than it is for our audiences. However, no matter what the sins of the math teachers, algebra is probably one of the most useful inventions of humankind. Airplanes fly because physics and engineering are written in algebraic terms. Computers are living, breathing algebra-based golems. Any science that does not use the language of algebra to some extent, well, is not quite science – it’s opinion.

Paradoxically, one of the greatest practical advantages of learning mathematics is to develop abstract thinking. In a math problem, a bus travelling from A to B is stripped of other details – colour, size, age of driver – so we can apply the same formula when solving a problem about a motorcycle. Abstract thinking is one of the greatest assets of intelligent beings. From the paintings in the Lascaux caves to Salvador Dali’s work, creativity stems from abstract thought. We cannot live our concrete lives without a feel for abstract concepts, from rhythm and comfort to the more complex: morals, beauty and love.

I’ve left the most difficult to understand benefit of studying math to the end: I think that math is beautiful. It is also a collection of some of the most amazing creations of human genius. Can I convince the uninitiated? It would be hard. One first needs to have the patience, respect and desire to learn the language of mathematics, then its idioms, then its quirks and subtleties. Then we can talk beauty.

In class, when we get tired of factoring polynomials, I tell my students that life is all about discovering the beauty that surrounds us – math, music, poetry, the structure of a living organism. You only need to have the patience to learn the proper language to see it.

Many, many years ago, I was graduating from high school and I was considering a career in mathematics. I asked a math professor for advice. “Why do you want to study math?” he asked me. “Because I think it is fun.” He told me it wasn’t, really, that there would be much tedious work and frustration. Then he paused, and added with a smile, “I guess it also has its fun and awesome moments.” I spent my next 40 years doing math and never looked back.

Alexandru Pintilie

http://www.theglobeandmail.com/life/facts-and-arguments/i-teach-high-school-math-is-my-workpointless/article34086848/

Özsaygı ve Delilik

Özsaygı her tarafa birbirinden farklı bin bir tarzda mutluluk yaymıyor mu? Bir tanesi maymun kadar çirkin olduğu halde kendini Niera kadar güzel sanır; öteki kendine bir ikinci Eukleides nazarıyla bakar, çünkü pergel yardımıyla birkaç eğri çizmeyi başarmıştır. Bir üçüncü müziğe, kılıktan yana doğanın en büyük sillesini yemiş eşekten daha fazla yeteneği olmadığı, sesi horozun sesi kadar nahoş ve kısık olduğu halde, Hermogenes kadar iyi şarkı söylediğini sanır. Yukarıdakiler kadar hoş bir delilik türü de hizmetlerinde bulunanların erdem ve hünerleriyle -sanki Tanrı bunları kendilerine bahşetmiş gibi- gururlanan ve övünen kimselerin deliliğidir. Seneca'nın söz ettiği şu mutlu zengin işte böyle idi. Bu adam ne zaman bir masal anlatsa, yanında daima, isimleri kendisine gizlice söyleyecek hizmetkârlar bulundururdu ve hayatı artık bir solumalık nefesten ibaretken en ünlü pehlivanlarla cenge cüret etmeye hazırdı, çünkü yanında bulunan bütün tutsakların gücüne sahip olduğuna inanırdı.

Burada güzel sanatlarla uğraşanlardan söz etmeye gerek var mı? Bunların hepsinde özsaygı o kadar doğaldır ki, dâhilik ününü terk etmektense babadan kalma küçük servetlerinin bütününü terk etmeye razı olmayan belki aralarında bir tane yoktur. Özellikle, aktörler, müzisyenler, hatipler ve şairler böyledirler. Ne kadar az hüner sahibiyseler, o derece kibirli ve gururludurlar. Bununla beraber, bütün deliler, kendilerini alkışlayan başka deliler bulurlar; zira bir şey ne kadar sağduyunun karşıtı ise, o kadar çok hayranı kendine çeker; en fena şey, her zaman çoğunluğu okşayan şeydir.

**

Doğa, özsaygının mutlu armağanlarını yalnız bireylere vermiş değildir.

Genellikle her kavim, her millet, hatta her şehir bunlardan oldukça bol nasibini almıştır, ingilizler, güzel adam, iyi müzisyen ve ziyafetlerinde cömert olmakla övünürler. İskoçyalılar, asaletleri, unvanları, krallarının hanedanı ile olan akrabalıkları ve skolastik tartışmalardaki olağanüstü incelikleri ile iftihar ederler. Fransızlar, nezaket iddiasındadırlar; Parisliler, özellikle Sorbonne'larında en bilimsel teoloji okuluna sahip olmakla gururlanırlar. Edebiyat ve söz söyleme sanatına sadece kendilerinin sahip olduklarına inanan italyanlar, kendilerini dünyanın barbarlık karanlıklarına dalmamış biricik kavmi sanırlar.

Aralarında bu tatlı yanılgıyı en fazla yaşayanlar Romalılardır; eski Romalıların büyüklüğünü sayıklar ve onlardan bir şeyler aldıklarına iyice inanırlar. Venedikliler, asaletlerini düşünmekle, Grekler bilimlerin kurucuları olduklarını düşünüp eski kahramanlarının sıfatlarını kendilerine takmakla mutludurlar. Türkler ve yeryüzünün dörtte üçünü kaplayan şu sayısız barbarlar, doğru dine girmiş olmakla övünürler, boş inanç sahibi alçak kimseler saydıkları Hıristiyanlara yukardan bakarlar. Çok daha mutlu olan Yahudiler meşinlerini tatlı tatlı bekleyerek yaşar ve bu arada daima Musa'nın dinine bağlı kalırlar. İspanyollar dünyanın en büyük askerleri geçinirler; yüksek boylarından gururlanan Almanlar, sihirden anladıklarını, büyük sihirbaz olduklarım iddia ederler.

Non-Euclidean Geometry

Gauss’s final contribution to research on the fifth postulate came shortly before he died, when, already seriously ill, he set the title for the probationary lecture of one of his brightest students, 27-year-old Bernhard Riemann: ‘On the hypotheses that lie at the foundations of geometry’. The cripplingly shy son of a Lutheran pastor, Riemann at first had some kind of breakdown struggling with what he would say, yet his solution to the problem would revolutionize maths. It would later revolutionize physics too, since his innovations were required by Einstein to formulate his general theory of relativity.

Riemann’s lecture, given in 1854, consolidated the paradigm shift in our understanding of geometry resulting from the fall of the parallel postulate by establishing an all-embracing theory that included the Euclidean and non-Euclidean within it. The key concept behind Riemann’s theory was the curvature of space. When a surface has zero curvature, it is flat, or Euclidean, and the results of The Elements all hold. When a surface has positive or negative curvature, it is curved, or non-Euclidean, and the results of The Elements do not hold.

The simplest way to understand curvature, continued Riemann, is by considering the behaviour of triangles. On a surface with zero curvature, the angles of a triangle add up to 180 degrees. On a surface with positive curvature, the angles of a triangle add up to more than 180 degrees. On a surface with negative curvature, the angles of a triangle add up to less than 180 degrees.

A surface with negative curvature is called hyperbolic. So, the surface of a Pringle is hyperbolic. The Pringle, however, is only an hors d’oeuvre in understanding hyperbolic geometry since it has an edge. Show a mathematician an edge and he or she will want to go over it.

Mathematics Does Not Age

Unlike the humanities, which are in a permanent state of reinvention, as new ideas or fashions replace old ones, and unlike applied science, where theories are undergoing continual refinement, mathematics does not age. The theorems of Pythagoras and Euclid are as valid now as they always were—which is why Pythagoras and Euclid are the oldest names we study at school. The GCSE syllabus contains almost no maths beyond what was already known in the mid seventeenth century, and likewise A-level with the mid eighteenth century. (In my degree the most modern maths I studied was from the 1920s.)

Math and Islamic History

In his work on algebra, al-Khwarizmi worked with both what we now call linear equations – that is, equations that involve only units without any squared figures – and quadratic equations, which involve squares and square roots. His advance was to reduce every equation to its simplest possible form by a combination of two processes: al-jabr and al-muqabala.

Al-jabr means ‘completion’ or ‘restoration’ and involves simply taking away all negative terms. Using modern symbols, al-jabr means simplifying. Al-muqabala means ‘balancing’, and involves reducing all the postive terms to their simplest form. 

In developing algebra, al-Khwarizmi built on the work of early mathematicians from India, such as Brahmagupta, and from the Greeks such as Euclid, but it was al-Khwarizmi who turned it into a simple, all-embracing system, which is why he is dubbed the ‘father of algebra’. The very word algebra comes from the title of his book, al-Kitab al-mukhtasar fi hisab al-jabr wa’l muqabala or The Compendious Book on Calculating by Completion and Balancing.

***
Beyond al-Khwarizmi, many other Arabic-speaking scholars explored mathematics. Indeed, it was fundamental to so many things, from calculating tax and inheritance to working out the direction of Mecca, that it is hard to find a scholar who did not at some time or other work in mathematics. But it wasn’t just practical applications that fascinated many of them, and they began to push mathematics to its limits.

In the early 11th century in Cairo, Hassan ibn al-Haitham, for instance, laid many of the foundations for integral calculus, which is used for calculating areas and volumes. Half a century later, the brilliant poet/mathematician Omar Khayyam found solutions to all thirteen possible kinds of cubic equations – that is, equations in which numbers are cubed. He regretted that his solutions could only be worked out geometrically rather than algebraically. ‘We have tried to work these roots by algebra, but we have failed’, he says ruefully. ‘It may be, however, that men who come after us will succeed.’

***
Omar Khayyam is one of the most extraordinary figures in Islamic science, and tales of his mathematical brilliance abound. In 1079, for instance, he calculated the length of the year to 365.24219858156 days. That means that he was out by less than the sixth decimal place – fractions of a second – from the figure we have today of 365.242190, derived with the aid of radio telescopes and atomic clocks. And in a highly theatrical demonstration involving candles and globes, he is said to have proved to an audience that included the Sufi theologian al-Ghazali that the earth rotates on its axis.

***
Trigonometry was first developed in ancient Greece, but it was in early Islam that it became an entire branch of mathematics, as it was aligned to astronomy in the service of faith. Astronomical trigonometry was used to help determine the qibla, the direction of the Ka’bah in Mecca. Modern historians such as David King have discovered that the Ka’bah itself is astronomically inclined. On one side it points towards Canopus, the brightest star in the southern sky. The axis that is perpendicular to its longest side points towards midsummer sunrise.

Mecca’s significance is such that when a deceased person is to be buried, contemporary Islamic tradition determines that his or her body must face Mecca. When the famous call to prayer is announced, it must be done facing Mecca. And when animals are slaughtered, slaughtermen must also turn in the direction of the holy city. Islamic-era astronomers began to compute the direction of Mecca from different cities from around the 9th century. One of the earliest known examples of the use of trigonometry (sines, cosines and tangents) for locating Mecca can be found in the work of the mathematician al-Battani, which, according to David King, was in use until the 19th century.