Einstein, twenty-six years old, only three years away from crude privation, still a patent examiner, published in the Annalen der Physik in 1905 five papers on entirely different subjects. Three of them were among the greatest in the history of physics. One, very simple, gave the quantum explanation of the photoelectric effect—it was this work for which, sixteen years later, he was awarded the Nobel prize. Another dealt with the phenomenon of Brownian motion, the apparently erratic movement of tiny particles suspended in a liquid: Einstein showed that these movements satisfied a clear statistical law. This was like a conjuring trick, easy when explained: before it, decent scientists could still doubt the concrete existence of atoms and molecules: this paper was as near to a direct proof of their concreteness as a theoretician could give. The third paper was the special theory of relativity, which quietly amalgamated space, time, and matter into one fundamental unity. This last paper contains no references and quotes to authority. All of them are written in a style unlike any other theoretical physicist's. They contain very little mathematics. There is a good deal of verbal commentary. The conclusions, the bizarre conclusions, emerge as though with the greatest of ease: the reasoning is unbreakable. It looks as though he had reached the conclusions by pure thought, unaided, without listening to the opinions of others. To a surprisingly large extent, that is precisely what he had done.

Language as putative science. - The significance of language for the evolution of culture lies in this, that mankind set up in language a separate world beside the other world, a place it took to be so firmly set that, standing upon it, it could lift the rest of the world off its hinges and make itself master of it. To the extent that man has for long ages believed in the concepts and names of things as in aeternae veritates he has appropriated to himself that pride by which he raised himself above the animal: he really thought that in language he possessed knowledge of the world. The sculptor of language was not so modest as to believe that he was only giving things designations, he conceived rather that with words he was expressing supreame knowledge of things; language is, in fact, the first stage of occupation with science. Here, too, it is the belief that the truth has been found out of which the mightiest sources of energy have flowed. A great deal later - only now - it dawns on men that in their belief in language they have propagated a tremendous error. Happily, it is too late for the evolution of reason, which depends on this belief, to be put back. - Logic too depends on presuppositions with which nothing in the real world corresponds, for example on the presupposition that there are identical things, that the same thing is identical at different points of time: but this science came into existence through the opposite belief (that such conditions do obtain in the real world). It is the same with mathematics, which would certainly not have come into existence if one had known from the beginning that there was in nature no exactly straight line, no real circle, no absolute magnitude.

5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one.5.5 So a) To what extent might human relationships be expressed in a mathematical or logical formula? And b) If so, what signs might be placed between the integers?Plus and minus, self-evidently; sometimes multiplication, and yes, division. But these sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of zero; whereas an entirely successful one can be represented by both addition and multiplication. But what of most relationships? Do they not require to be expressed in notations which are logically improbable and mathematically insoluble?5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v?B = s - v (*/+) a (to the first)Or a (to the second) + v + a (to the first) x s = b 5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility?5.8 Or is "link" a false metaphor?5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for such breaking? On the links immediately on the other side, or on the whole chain? But what do you mean by "the whole chain"? How far do the limits of responsibility extend?6.0 Or we might try to draw the responsibility more narrowly and apportion it more exactly. And not use equations and integers but instead express matters in the traditional narrative terminology. So, for instance, if...." - Adrian Finn

Fermat žil poměrně stranou od hlavních center matematického života, a byl tedy odkázán na korespondenci. Rozeslal jí hodně a měl při tom zvláštní a pro partnery asi dost rozčilující zvyk - posílal jim hotové matematické věty, ktoré objevil, ale bez jejich odvození, takže bylo na nich, aby si správný postup našli sami (s tím, že z jeho dopisu už vědeli, jak zní dokazované tvrzení). Navíc posílal jen některé ze svých poznatků a další si nechával pro sebe - "výrobní tajemství" si chránit nemusel. Nejspíš mu v té chvíli nepřipadaly tak důležité, aby platil postilionovi, možná počítal s tím, že na jejich vylepšení bude sám dál pracovat, a nechtěl přijít o prvenství. Tyto výsledky pak shromáždil až Fermatův syn Clément Samuel Fermat a vydal je po otcově smrti tiskem. Styk s "tajnůstkářským" Fermatem asi nebyl vždy nejpříjemnější a ne každý na něj měl náturu. Celkem hladce s ním spolupracoval (to znamená dopisoval si s ním) Blaise Pascal, když spolu tvořili počet pravděpodobnosti, naopak Décartes ho nejspíš nesnášel a v korespondenci s přáteli pro něj má výrazy, ktoré prudérnější interpreti překládají jako chvástal - i když tlučhuba by asi bylo přesnější.

He walked straight out of college into the waiting arms of the Navy. They gave him an intelligence test. The first question on the math part had to do with boats on a river: Port Smith is 100 miles upstream of Port Jones. The river flows at 5 miles per hour. The boat goes through water at 10 miles per hour. How long does it take to go from Port Smith to Port Jones? How long to come back?Lawrence immediately saw that it was a trick question. You would have to be some kind of idiot to make the facile assumption that the current would add or subtract 5 miles per hour to or from the speed of the boat. Clearly, 5 miles per hour was nothing more than the average speed. The current would be faster in the middle of the river and slower at the banks. More complicated variations could be expected at bends in the river. Basically it was a question of hydrodynamics, which could be tackled using certain well-known systems of differential equations. Lawrence dove into the problem, rapidly (or so he thought) covering both sides of ten sheets of paper with calculations. Along the way, he realized that one of his assumptions, in combination with the simplified Navier Stokes equations, had led him into an exploration of a particularly interesting family of partial differential equations. Before he knew it, he had proved a new theorem. If that didn't prove his intelligence, what would?Then the time bell rang and the papers were collected. Lawrence managed to hang onto his scratch paper. He took it back to his dorm, typed it up, and mailed it to one of the more approachable math professors at Princeton, who promptly arranged for it to be published in a Parisian mathematics journal.Lawrence received two free, freshly printed copies of the journal a few months later, in San Diego, California, during mail call on board a large ship called the U.S.S. Nevada. The ship had a band, and the Navy had given Lawrence the job of playing the glockenspiel in it, because their testing procedures had proven that he was not intelligent enough to do anything else.

A Puritan twist in our nature makes us think that anything good for us must be twice as good if it's hard to swallow. Learning Greek and Latin used to play the role of character builder, since they were considered to be as exhausting and unrewarding as digging a trench in the morning and filling it up in the afternoon. It was what made a man, or a woman -- or more likely a robot -- of you. Now math serves that purpose in many schools: your task is to try to follow rules that make sense, perhaps, to some higher beings; and in the end to accept your failure with humbled pride. As you limp off with your aching mind and bruised soul, you know that nothing in later life will ever be as difficult.What a perverse fate for one of our kind's greatest triumphs! Think how absurd it would be were music treated this way (for math and music are both excursions into sensuous structure): suffer through playing your scales, and when you're an adult you'll never have to listen to music again. And this is mathematics we're talking about, the language in which, Galileo said, the Book of the World is written. This is mathematics, which reaches down into our deepest intuitions and outward toward the nature of the universe -- mathematics, which explains the atoms as well as the stars in their courses, and lets us see into the ways that rivers and arteries branch. For mathematics itself is the study of connections: how things ideally must and, in fact, do sort together -- beyond, around, and within us. It doesn't just help us to balance our checkbooks; it leads us to see the balances hidden in the tumble of events, and the shapes of those quiet symmetries behind the random clatter of things. At the same time, we come to savor it, like music, wholly for itself. Applied or pure, mathematics gives whoever enjoys it a matchless self-confidence, along with a sense of partaking in truths that follow neither from persuasion nor faith but stand foursquare on their own. This is why it appeals to what we will come back to again and again: our **architectural instinct** -- as deep in us as any of our urges.

Mathematics is not truthBut it is subject that help us to figure out truth.गणित म्हणजे सत्य नव्हे , पण सत्यापर्यंत पोहचवणारे शास्त्र म्हणजे (तर्कशास्त्र) गणित !काय योग्य- अयोग्य, चूक-बरोबर , खरा इतिहास -खोटा इतिहास इत्यादी ठरवण्यासाठी लागणारे मुलभूत निकष देण्याचे कार्य गणिताचे (तर्कशास्त्राचे). गणित हा वैज्ञानिक दृष्टीकोनाचा आधारस्तंभ आहे. गणित म्हणजे केवळ आकडेमोड नाही किंवा पाठांतर केलेल्या सूत्राचा वापर करून विशिष्ट प्रकारचे स्पर्धा परिक्षांतील प्रश्न सोडवणे नव्हे तर गणित म्हणजे काही स्वयंसिध्द मुलभूत संकल्पनांच्या आधारे अनिश्चित प्रश्नांचा अथवा रहस्यांचा तर्काचा वापर करून उलगडा करणे.गणित ही मानवाने विकसित केलेली वैश्विक भाषा आहे.गणित हे शास्त्रांचे शास्त्र आहे.हाती असलेल्या साधनांचा / ज्ञानाचा वापर करून तर्कसुसंगत विचारसरणीच्या साहाय्याने इच्छित ध्येयापर्यंत पोहचणे म्हणजे गणित. ज्याला गणिताचा हा खरा अर्थ उलगडतो व जो केवळ स्पर्धा परिक्षांच्या नव्हे तर आयुष्याच्या अनुषंगाने गणित / तर्कशास्त्र शिकतो, तोच आयुष्याचे गणित सोडवू शकतो.जवळ-जवळ सर्वच क्षेत्रांत गणित डोकावते.उदा. एखाद्या उत्कृष्ट लेखकाचा एखादा लेख निवडा व त्यातील कोणत्याही परिच्छेदाचे निरीक्षण करा, त्यात लगतची वाक्ये आपणांस तर्कशास्त्रीय दृष्टीकोनातून सुसंगतपणे माडलेली आढळतील.गणित विषयाचे सौंदर्य म्हणजे यात इतर विषयांप्रमाणे यात जास्त लक्षात ठेवावे लागत नाही. येथे केवळ मुलभूत नियम लक्षात ठेवावे लागतात व त्यांच्या साहाय्याने प्रश्नाची उकल काढण्यासाठी 'तर्काचा वापर कसा करायचा ?' हे शिकावे लागते.