Gh Hardy Quotes

It is not worth an intelligent man's time to be in the majority. By definition, there are already enough people to do that.

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

Real mathematics must be justified as art if it can be justified at all.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full.

It (proof by contradiction) is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

Exposition, criticism, appreciation, is work for second-rate minds. I

The seriousness of a theorem, of course, does not lie in its consequences, which are merely the evidence for its seriousness.

Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity

I was advised to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.

No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young man's game. … Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work later; … [but] I do not know of a single instance of a major mathematical advance initiated by a man past fifty. … A mathematician may still be competent enough at sixty, but it is useless to expect him to have original ideas.

The beauty of a mathematical theorem depends a great deal on its seriousness, as even in poetry the beauty of a line may depend to some extent on the significance of the ideas which it contains.

Poetry is more valuable than cricket, but Bradman would be a fool if he sacrificed his cricket in order to write second-rate minor poetry (and I suppose that it is unlikely that he could do better).

Ambition is a noble passion which may legitimately take many forms; there was something noble in the ambitions of Attila or Napoleon; but the noblest ambition is that of leaving behind something of permanent value.

The Greeks were the first mathematicians who are still ‘real’ to us to-day. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greeks first spoke a language which modern mathematicians can understand: as Littlewood said to me once, they are not clever schoolboys or ‘scholarship candidates’, but ‘Fellows of another college’. So Greek mathematics is ‘permanent’, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.

I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, 'Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.

I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.

Plenty of mathematicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of the day, he assigned an 80. To Ramanujan he gave 100.

A MAN who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be.

One rather curious conclusion emerges, that pure mathematics is on the whole distinctly more useful than applied. ... For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.

The geometer offers to the physicist a whole set of maps from which to choose. One map, perhaps, will fit the facts better than others, and then the geometry which provides that particular map will be the geometry most important for applied mathematics.

Twentieth-century British mathematician G.H. Hardy also believed that the human function is to "discover or observe" mathematics rather than to invent it. In other words, the abstract landscape of mathematics was there, waiting for mathematical explorers to reveal it.

The best mathematics is serious as well as beautiful—‘important’ if you like, but the word is very ambiguous, and ‘serious’ expresses what I mean much better

Immortality is often ridiculous or cruel: few of us would have chosen to be Og or Ananias or Gallio. Even in mathematics, history sometimes plays strange tricks; Rolle figures in the textbooks of elementary calculus as if he had been a mathematician like Newton; Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before; the names of five worthy Norwegians still stand in Abel’s Life, just for one act of conscientious imbecility, dutifully performed at the expense of their country’s greatest man. But on the whole the history of science is fair, and this is particularly true in mathematics. No other subject has such clear-cut or unanimously accepted standards, and the men who are remembered are almost always the men who merit it. Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of investments.

Regarding mathematics, there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation.

I do not think that G. H. Hardy was talking nonsense when he insisted that the mathematician was discovering rather than creating... The world for me is a necessary system, and in the degree to which the thinker can surrender his thought to that system and follow it, he is in a sense participating in that which is timeless or eternal.

The proof is by reductio ad absurdum, and reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons5. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.

The play is independent of the pages on which it is printed, and ‘pure geometries’ are independent of lecture rooms, or of any other detail of the physical world.

We have concluded that the trivial mathematics is, on the whole, useful, and that the real mathematics, on the whole, is not.

No one should ever be bored. One can be horrified, or disgusted, but one can't be bored.

Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean. G.H. Hardy 23

If I had a statue on a column in London, would I prefer the columns to be so high that the statue was invisible, or low enough for the features to be recognizable? I would choose the first alternative, Dr Snow, presumably, the second.

It is a tiny minority who can do something really well, and the number of men who can do two things well is negligible. If a man has any genuine talent he should be ready to make almost any sacrifice in order to cultivate it to the full

We must guard against a fallacy common among apologists of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls.

It is hardly possible to maintain seriously that the evil done by science is not altogether outweighed by the good. For example, if ten million lives were lost in every war, the net effect of science would still have been to increase the average length of life.

The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. G.H. Hardy

The mathematician is in much more direct contact with reality. This may seem a paradox, since it is the physicist who deals with the subject-matter usually described as 'real' ... A chair may be a collection of whirling electrons, or an idea in the mind of God : each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense. ... neither physicists nor philosophers have ever given any convincing account of what 'physical reality' is, or of how the physicist passes, from the confused mass of fact or sensation with which he starts, to the construction of the objects which he calls 'real'. A mathematician, on the other hand, is working with his own mathematical reality. ... mathematical objects are so much more what they seem. ... 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.

Published mathematical papers often have irritating assertions of the type: “It now follows that…,” or: “It is now obvious that…,” when it doesn't follow, and isn't obvious at all, unless you put in the six hours the author did to supply the missing steps and checking them. There is a story about the English mathematician G.H. Hardy, whom we shall meet later. In the middle of delivering a lecture, Hardy arrived at a point in his argument where he said, “It is now obvious that….” Here he stopped, fell silent, and stood motionless with furrowed brow for a few seconds. Then he walked out of the lecture hall. Twenty minutes later he returned, smiling, and began, “Yes, it is obvious that….” If he

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill. Time may change all this. No one foresaw the applications of matrices and groups and other purely mathematical theories to modern physics, and it may be that some of the 'highbrow' applied mathematics will become 'useful' in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life.

I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated....

Judged by all practical standards, the value of my mathematical life is nil; and outside mathematics it is trivial anyhow. I have just one chance of escaping a verdict of complete triviality, that I may be judged to have created something worth creating. And that I have created something is undeniable: the question is about its value. The case for my life, then, or for that of any one else who has been a mathematician in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any of the other artists, great or small, who have left some kind of memorial behind them.

Είναι μελαγχολική εμπειρία για έναν κατ' επάγγελμα μαθηματικό να βρεθεί στην θέση να γράφει για τα Μαθηματικά. Η λειτουργία ενός μαθηματικού είναι να δημιουργεί, να αποδεικνύει νέα θεωρήματα, να προσθέτει καινούργια πράγματα στα Μαθηματικά· και όχι να μιλά για τα επιτεύγματα του ίδιου ή άλλων μαθηματικών. Οι δημόσιοι άνδρες απεχθάνονται τους εκδότες, οι ζωγράφοι τους κριτικούς τέχνης, και οι γιατροί, οι φυσικοί, ή οι μαθηματικοί τρέφουν συνήθως για διάφορους παρόμοια συναισθήματα. Δεν υπάρχει πιο μεγάλος ψόγος, ή, εν γένει πιο δικαιολογημένος, από αυτόν που έχουν οι άνθρωποι που δημιουργούν για τους ανθρώπουν που αναλύουν. Η παρουσίαση, η κριτική, η εκτίμηση ενός πράγματος, θεωρείται έργο για μυαλά δευτέρας κατηγορίας.

Θα υποθέσω ότι γράφω για αναγνώστες που είναι έμπλεοι, ή ήσαν τέτοιοι κατά το παρελθόν, ενός ορθού πνεύματος φιλοδοξίας. Το πρώτο καθήκον ενός ανθρώπου, και προπάντων ενός νέου, είναι να είναι φιλόδοξος. Η φιλοδοξία είναι ένα ευγενές πάθος το οποίο μπορεί να πάρει πολλές νόμιμες μορφές. Υπήρχε _κάτι_ το ευγενές στη φιλοδοξία του Αττίλα ή του Ναπολέοντα: η ευγενέστερη φιλοδοξία όμως, είναι να αφήνει κανείς πίσω του κάτι με διαχρονική αξία. Εδώ στην επίπεδη αμμουδιά, ανάμεσα σε ξηρά και θάλασσα, τι θα κτίσω ή τι θα γράψω ενάντια στη νύχτα που πέφτει; Πες μου να χαράξω μαγικά σύμβολα να κρατούν το κύμα που σκάει ή να φτιάξω οχυρά που θα ζήσουν πιο πολύ από μένα.

Οι νέοι οφείλουν να είναι επηρμένοι, αλλά οφείλουν και να μην είναι ανόητοι.

Αν η περιέργεια του πνεύματος, η επαγγελματική υπερηφάνεια και η φιλοδοξία είναι τα κύρια κίνητρα για την έρευνα, τότε σίγουρα κανείς δεν έχει πιο καλή ευκαιρία να τα ικανοποιήσει απ' ό,τι ένας μαθηματικός. Το αντικείμενό του είναι το πιο περίεργο απ' όλα - δεν υπάρχει κανένα άλλο στο οποίο η αλήθεια να παίζει τόσο παράξενα παιγνίδια. Το αντικείμενο αυτό έχει την πιο εκλεπτυσμένη και γοητευτική τεχνική, και δίνει ασυναγώνιστες ευκαιρίες για την επίδειξη μιας ανώτερης επαγγελματικής ικανότητας. Τελικά, όπως αποδεικνύει κατά πολλούς τρόπους η Ιστορία, τα μαθηματικά επιτεύγματα, ανεξάρτητα από την εγγενή τους αξία, αντέχουν στο χρόνο περισσότερο απ' όλα τα άλλα. Μπορούμε να το δούμε αυτό, ακόμη και στους πρώιμους πολιτισμούς της Ιστορίας. Ο Βαβυλωνιακός και ο Ασσυριακός πολιτισμός έχουν χαθεί· ο Χαμουραμπί, ο Σαργκόν και ο Ναβουχοδονόσωρ είναι σκέτα ονόματα. Κι όμως, τα βαβυλωνιακά Μαθηματικά είναι ακόμη και σήμερα ενδιαφέροντα, και το βαβυλωνιακό σύστημα αρίθμησης με βάση το 60 χρησιμοποιείται ακόμη στην Αστρονομία. Αλλά φυσικά, η κρίσιμη περίπτωση είναι εκείνη των Ελλήνων. Οι Έλληνες είναι οι πρώτοι μαθηματικοί που εξακολουθούν να είναι «πραγματικοί» και για μας σήμερα. Τα Μαθηματικά της Ανατολής μπορεί να προκαλούν το ενδιαφέρον, αλλά στα ελληνικά βρίσκεται η ουσία του πράγματος. Οι Έλληνες ήταν οι πρώτοι που μίλησαν με μια μαθηματική γλώσσα που μπορούν να την καταλάβουν οι σύγχρονοι μαθηματικοί. Όπως μου είπε κάποτε ο Littlewood, δεν πρόκειται για έξυπνους μαθητές σχολείου ούτε για «υποψήφιους υποτροφίας», αλλά για «Εταίρους από ένα άλλο πανεπιστήμιο». Έτσι, τα ελληνικά μαθηματικά είναι κάτι «μόνιμο», πιο μόνιμο και από την ελληνική Λογοτεχνία. Τον Αρχιμήδη θα τον θυμούνται ακόμη κι όταν ο Αισχύλος θά 'χει ξεχαστεί, επειδή οι γλώσσες πεθαίνουν ενώ οι μαθηματικές ιδέες όχι. Η «αθανασία» μπορεί να είναι μια ανόητη λέξη αλλά, κατά πάσα πιθανότητα, ένας μαθηματικός έχει περισσότερες ευκαιρίες για ό,τι μπορεί αυτή να σημαίνει. Ο μαθηματικός δε χρειάζεται σοβαρά να φοβάται ότι το μέλλον θα τον αδικήσει. Η αθανασία είναι συχνά γελοία ή βάρβαρη: λίγοι από εμάς θα διάλεγαν να είναι ο Ωγ ή ο Ανανίας ή ο Γαλλίων. Ακόμη και στα Μαθηματικά, η ιστορία παίζει καμιά φορά περίεργες φάρσες. Ο Rolle ποζάρει στα βιβλία του Στοιχειώδους Λογισμού σαν να ήταν ένας μαθηματικός του διαμετρήματος του Νεύτωνα. Ο Farey είναι αθάνατος επειδή απέτυχε να κατανοήσει ένα θεώρημα που ο Haros είχε ήδη αποδείξει πριν από 14 χρόνια. Τα ονόματα πέντε άξιων Νορβηγών βρίσκονται ακόμη στον _Βίο_ του Abel, μόνο εξ αιτίας μιας ενέργειας ενσυνείδητης βλακείας που συνετελέσθη, από τυπολατρεία, εις βάρος του μεγαλύτερου άνδρα της χώρας τους. Αλλά, συνολικά, η ιστορία της επιστήμης είναι δίκαιη, και αυτό ισχύει ιδιαίτερα στα Μαθηματικά. Κανένα άλλο αντικείμενο μελέτης δεν έχει τόσο καθαρά οριοθετημένα ή ομόφωνα αποδεκτά υψηλά κριτήρι, και οι μαθηματικοί που θυμόμαστε είναι σχεδόν πάντα αυτοί που το αξίζουν. Η μαθηματική δόξα, αν μπορούσε να εξαγοραστεί, θα ήταν μια από τις πιο υγιείς και σταθερές επενδύσεις.

Η εις άτοπον απαγωγή, που ο Ευκλείδης αγαπούσε τόσο πολύ, είναι ένα από τα ωραιότερα όπλα του μαθηματικού. Είναι πιο όμορφο από οποιοδήποτε σκακιστικό γκαμπί. Ένας σκακιστής μπορεί να θυσιάσει ένα πιόνι, ή ακόμη και ένα κομμάτι, αλλά ο μαθηματικός προσφέρει το ίδιο το παιγνίδι.

Το θεώρημα του Ευκλείδη είναι ζωτικής σημασίας για όλη την δομή της Αριθμητικής. Οι πρώτοι αριθμοί είναι το αρχικό υλικό με το οποίο έχουμε οικοδομήσει την αριθμητική, και το θεώρημα του Ευκλείδη μας εξασφαλίζει ότι έχουμε άφθονο υλικό γι' αυτό το σκοπό.[...] Το Θεώρημα του Ευκλείδη μας λέει ότι έχουμε ένα ικανό απόθεμα υλικού για την κατασκευή μιας συγκροτημένης αριθμητικής των ακεραίων αριθμών. Το Θεώρημα του Πυθαγόρα και οι επεκτάσεις του μας λένε ότι, άπαξ και κατασκευάσθηκε αυτή η αριθμητική, δεν θα αποδειχτεί αρκετή για τις ανάγκες μας, αφού υπάρχουν πολλά μεγέθη τα οποία μας επιβάλλουν την παρουσία τους και τα οποία αυτή η αριθμητική είναι ανήμπορη να μετρήσει: η διαγώνιος του τετραγώνου είναι απλώς το πιο προφανές παράδειγμα. Η μεγάλη σπουδαιότητα αυτής της ανακάλυψης αναγνωρίστηκε αμέσως από τους Έλληνες μαθηματικούς. Είχαν ξεκινήσε με την υπόθεση ότι (σε συμφωνία, υποθέτω, με τις «φυσικές» επιταγές της «κοινής λογικής») όλα τα μεγέθη του ίδιου είδους είναι σύμμετρα (ότι για παράδειγμα, δύο οποιαδήποτε μήκη είναι πολλαπλάσια κάποιας κοινής μονάδας μήκους), και είχαν κατασκευάσει μια θεωρία αναλογιών στηριγμένη σ' αυτή την υπόθεση. Η ανακάλυψη του Πυθαγόρα κατέδειξε το μη στέρεο της θεμελίωσης αυτής και οδήγησε στην κατασκευή της πολύ βαθύτερης θεωρίας του Ευδόξου που εκτίθεται στο πέμπτο βιβλίο των Στοιχείων και που θεωρείται από πολλούς σύγχρονους μαθηματικούς ως το εξέχον επίτευγμα των ελληνικών Μαθηματικών.

Φαίνεται ότι οι μαθηματικές ιδέες είναι κατά κάποιο τρόπο διατεταγμένες κατά στρώματα, με τις ιδέες σε κάθε στρώμα συνδεδεμένες μέσω ενός πλέγματος σχέσεων τόσο μεταξύ τους όσο και με αυτές που βρίσκονται από κάτω και από πάνω τους. Όσο πιο χαμηλό είναι το στρώμα, τόσο πιο βαθειά (και εν γένει δυσκολότερη) είναι η ιδέα. Έτσι, η ιδέα του «άρρητου» είναι βαθύτερη από εκείνη του ακεραίου, και το Θεώρημα του Πυθαγόρα είναι γι' αυτό το λόγο βαθύτερο από εκείνο του Ευκλείδη. Ας συγκεντρώσουμε την προσοχή μας πάνω στις σχέσεις μεταξύ των ακεραίων, ή κάποιας άλλης ομάδας αντικειμένων που ανήκουν σε κάποιο συγκεκριμένο στρώμα. Μ' αυτόν τον τρόπο, μια από αυτές τις σχέσεις μπορεί ενδεχομένως να κατανοηθεί πλήρως: επί παραδείγματι, να αναγνωρίσουμε και να αποδείξουμε μια ιδιότητα των ακεραίων χωρίς καμμία γνώση του περιεχομένου των πιο κάτω στρωμάτων. Έτσι αποδείξαμε το θεώρημα του Ευκλείδη θεωρώντας μόνο ιδιότητες των ακεραίων. Αλλά υπάρχουν επίσης πολλά θεωρήματα για τους ακεραίους που δεν μπορούμε να εκτιμήσουμε όσο πρέπει - και ακόμη λιγότερο να αποδείξουμε - χωρίς να σκάψουμε βαθύτερα και χωρίς να εξετάσουμε τι συμβαίνει πιο κάτω. Εύκολα βρίσκουμε τέτοια παραδείγματα στη θεωρία των πρώτων αριθμών. Το θεώρημα του Ευκλείδη είναι πολύ σημαντικό, αλλά όχι πολύ βαθύ: μπορούμε να αποδείξουμε ότι υπάρχουν άπειροι πρώτοι, χωρίς να χρησιμοποιήσουμε κάποια έννοια βαθύτερη από εκείνη της «διαιρετότητας». Αλλά μόλις μάθουμε την απάντηση στο ερώτημα αυτό για το πλήθος των πρώτων, νέα ερωτήματα γεννώνται αμέσως από μόνα τους. Υπάρχουν άπειροι πρώτοι, αλλά πώς κατανέμονται αυτοί οι άπειροι αριθμοί; Αν δοθεί ένας μεγάλος αριθμός Ν, ας πούμε ο 10^80 ή ο 10^10^10, πόσοι περίπου πρώτοι είναι μικρότεροι από Ν; Όταν κάνουμε αυτές τις ερωτήσεις βρίσκουμε τους εαυτούς μας σε μια αρκετά διαφορετική θέση. Μπορούμε να τις απαντήσουμε, με εκπληκτική μάλλον ακρίβεια, αλλά μόνο μετά από «διάτρηση» σε πολύ μεγάλο βάθος, αφήνοντας για λίγο πάνω από εμάς τους ακεραίους, και χρησιμοποιώντας τα πιο ισχυρά όπλα της σύγχρονης Θεωρίας Συναρτήσεων. Έτσι, το θεώρημα που απαντά στις ερωτήσεις μας (το αποκαλούμενο «Θεώρημα των Πρώτων Αριθμών») είναι πολύ πιο βαθύ από εκείνο του Ευκλείδη ή ακόμη και του Πυθαγόρα.

Τι «καθαρά αισθητικά» ποιοτικά χαρακτηριστικά μπορούμε να διακρίνουμε σε θεωρήματα παρόμοια με του Ευκλείδη και του Πυθαγόρα; Δεν θα διακινδυνέψω τίποτα περισσότερο από μερικές σκόπιες παρατηρήσεις. Και στα δύο θεωρήματα (και σ' αυτά φυσικά περιλαμβάνω και τις αποδείξεις) υπάρχει ένας πολύ υψηλός βαθμός απροσδόκητου σε συνδυασμό με στοιχεία αναπόφευκτου και εξοικονόμησης. Η επιχειρηματολογία τους παίρνει μια παράξενη και εκπλητική μορφή: τα όπλα που χρησιμοποιούνται φαίνονται εντελώς παιδικά εν συγκρίσει με τα αποτελέσματα που είναι μεγάλου βεληνεκούς. Αλλά δεν υπάρχει τρόπος διαφυγής από τα συμπεράσματα. Δεν υπάρχουν επιπλοκές εξ αιτίας λεπτομερειών - μια γραμμή επίθεσης είναι αρκετή σε κάθε περίπτωση. Και αυτό ισχύει επίσης για τις αποδείξεις πολύ πιο δύσκολων θεωρημάτων, που για να εκτιμηθούν πλήρως απαιτείται ένας αρκετά υψηλός βαθμός επαγγελματικής ικανότητας στην πράξη. Δεν θέλουν πολλές «διακυμάνσεις» στην απόδειξη ενός μαθηματικού θεωρήματος: η «απαρίθμηση περιπτώσεων» είναι, πραγματικά, μια από τις πιο πληκτικές μορφές μαθηματικής επιχειρηματολίας. Μια μαθηματική απόδειξη πρέπει να μοιάζει με έναν απλό και ευδιάκριτο αστερισμό και όχι με ένα νεφέλωμα σκορπισμένο στον Γαλαξία μας.

The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.

We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.

he had made such and such a move, then I had such and such a winning combination in mind.’ But the ‘great game’ of chess is primarily psychological, a conflict between one trained intelligence and another, and not a mere collection of small mathematical theorems.

will end with a summary of my conclusions, but putting them in a more personal way. I said at the beginning that anyone who defends his subject will find that he is defending himself;

Formalism has an austere elegance. It appeals to people like G.H. Hardy, Antonin Scalia, and me, who relish that feeling of a nice rigid theory shut tight against contradiction. But it's not easy to hold to principles like this consistently, and it's not clear it's even wise. Even Justice Scalia has occasionally conceded that when the literal words of the law seem to require an absurd judgment, the literal words have to be set aside in favor of a reasonable guess as to what Congress must have meant. In just the same way, no scientist really wants to be bound strictly by the rules of significance, no matter what they say their principles are. When you run two experiments, one testing a clinical treatment that seems theoretically promising and the other testing whether dead salmon respond emotionally to romantic photos, and both experiments succeed with p-values of .03, you don't really want to treat the two hypotheses the same. You want to approach absurd conclusions with an extra coat of skepticism, rules be damned.

İnsanları araştırma yapmaya yönelten pek çok neden vardır; ancak bunlardan üçü diğerlerinden daha önemlidir. Birincisi (ki bu olmadan öbür nedenler işe yaramaz), entelektüel merak, gerçeği öğrenme arzusudur. İkincisi, profesyonel saygınlık, yaptıklarının kendini tatmin etmem endişesidir; ortaya koyduğu eser, yeteneği ile orantılı olmadığı zaman her onurlu zanaatçının duyduğu utanma hissidir. Sonuncusu da başarma hırsı, mevki ve üne kavuşma arzusu, hatta sağlanacak para ve onun getireceği güçtür. A Mathematician's Apology

Matematiksel sonuçlar, içerdikleri değerler ne olursa olsun, diğerlerinin içinde en kalıcı olanlardır.

Matematiğin çok küçük bölümü pratik yarar sağlar; o küçük bölüm de oldukça sıkıcıdır.

I spoke of the 'real' mathematics of Fermat and other great mathematicians, the mathematics which has permanent aesthetic value, as for example the best Greek mathematics has, the mathematics which is eternal because the best of it may, like the best literature, continue to cause intense emotional satisfaction to thousands of people after thousands of years. These men were all primarily pure mathematicians; but I was not thinking only of pure mathematics. I count Maxwell and Einstein, Eddington and Dirac, among 'real' mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as 'useless' as the theory of numbers. It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill.

If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. But science works for evil as well as for good; and both Gauss and lesser mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.

Even a pure mathematician may find his appreciation of this geometry [applied geometry] quickened, since there is no mathematician so pure that he feels no interest at all in the physical world; but, in so far as he succumbs to this temptation, he will be abandoning his purely mathematical position.

A man who is always asking 'Is what I do worthwhile?' and 'Am I the right person to do it?' will always be ineffective himself and a discouragement to others.

Good work is no done by ‘humble’ men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ‘Is what I do worth while?’ and ‘Am I the right person to do it?’ will always be ineffective himself and a discouragement to others.

OLD BRANDY came to mean a taste that was eccentric, esoteric, but just within the bounds of reason.

Good work is no done by ‘humble’ men. It is one of the first duties of a professor, for example, in any subject, to exaggerate a little both the importance of his subject and his own importance in it. A man who is always asking ‘Is what I do worth while?’ and ‘Am I the right person to do it?’ will always be ineffective himself and a discouragement to others.

We are merely explorers of infinity in the pursuit of absolute perfection.

It seems that mathematical ideas are arranged somehow in strata, the ideas in each stratum being linked by a complex of relations both among themselves and with those above and below. The lower the stratum, the deeper (and in general more difficult) the idea. Thus the idea of an ‘irrational’ is deeper than that of an integer; and Pythagoras’s theorem is, for that reason, deeper than Euclid’s.

In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras, and will not end with Einstein, but is the oldest and the youngest of all.

a science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life...

But it was Poincare who wrote that what guided him in his unconscious gropings towards the 'happy combinations' which yield new discoveries was 'the feeling of mathematical beauty, of the harmony of number, of forms, of geometric elegance. This is a true aesthetic feeling that all mathematicians know.' The greatest among mathematicians and scientists, from Kepler to Einstein, made similar confessions. 'Beauty is the first test; there is no permanent place in the world for ugly mathematics', wrote G.H. Hardy in his classic, A Mathematician's Apology. Jacques Hadamard, whose pioneer work on the psychology of invention I have quoted, drew the final conclusion: "The sense of beauty as a "drive" for discovery in our mathematical field, seems to be almost the only one.' And the laconic pronouncement of Dirac, addressed to his fellow-physicists, bears repeating: 'It is more important to have beauty in one's equations than to have them fit experiment.

I am interested in mathematics only as a creative art.

3 A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then.