“Science is a fundamentally optimistic enterprise. More than a cheery disposition, it is the source of a philosophical outlook that we might call ‘optimistical’. It reliably produces fundamental and actionable knowledge about the world. […] But the engines behind science, surprisingly, are ignorance, the unknown, failure, and, perhaps most vexingly, uncertainty. In recent decades, science has undergone a change in perspective and practice — from viewing the universe like a clockwork regimented by laws and formulas to recognizing it as irreducibly complex and uncertain. Perhaps counterintuitively, this has freed science to exploit previously unimaginable possibilities and opportunities. […] And yet socially and societally we remain mired in a 19th century view of deterministic science. We might instead learn to revel in the adventure of navigable uncertainty and take advantage of the creative opportunities of a world where we can confidently say ‘It could be otherwise’. Possibility of this sort is the rarest and purest form of optimism.”

~ Stuart Firestein

“Don’t fear failure — not failure, but low aim is the crime. In great attempts, it is glorious even to fail.”
~ Bruce Lee

“It is almost with reluctance that I am about to speak of a sentiment, which appears to arise from narrow-minded views, or from a certain weak and morbid sentimentality — I allude to the fear entertained by some persons, that nature may by degrees lose a portion of the charm and magic of her power, as we learn more and more how to unveil her secrets, comprehend the mechanism of the movements of the heavenly bodies, and estimate numerically the intensity of natural forces. It is true that, properly speaking, the forces of nature can only exercise a magical power over us, as long as their action is shrouded in mystery and darkness, and does not admit of being classed among the conditions with which experience has made us acquainted. The effect of such a power is, therefore, to excite the imagination, but that, assuredly, is not the faculty of mind we would evoke to preside over the laborious and elaborate observations by which we strive to attain to a knowledge of the greatness and excellence of the laws of the universe. […]

The astronomer who, by the aid of the heliometer or a double-refracting prism, determines the diameter of planetary bodies, who measures patiently, year after year, the meridian altitude and the relative distances of stars, or who seeks a telescopic comet in a group of nebulae, does not feel his imagination more excited — and this is the very guarantee of the precision of his labours — than the botanist who counts the divisions of the calyx, or the number of stamens in a flower, or examines the connected or the separate teeth of the peristoma surrounding the capsule of a moss. Yet the multiplied angular measurements, on the one hand — and the detail of organic relations on the other, alike aid in preparing the way for the attainment of higher views of the laws of the universe. […]

I cannot, therefore, agree with Burke when he says: “It is our ignorance of natural things that causes all our admiration, and chiefly excites our passions”.”

~ “Cosmos”, Alexander von Humboldt.

“Among the colossal mountains of Cundinamarca, of Quito, and of Peru, furrowed by deep ravines, man is enabled to contemplate alike all the families of plants, and all the stars of the firmament. There, at a single glance, the eye surveys majestic palms, humid forests of bambusa, and the varied species of musaeese, while above these forms of tropical vegetation appear oaks, medlars, the sweetbrier, and umbelliferous plants, as in our European homes. There, as the traveller turns his eyes to the vault of heaven, a single glance embraces the constellation of the Southern Cross, the Magellanic clouds, and the guiding stars of the constellation of the Bear, as they circle round the arctic pole. There the depths of the earth and the vaults of heaven display all the richness of their forms and the variety of their phenomena. There the different climates areranged the one above the other, stage by stage, like the vegetable zones, whose succession they limit; and there the observer may readily trace the laws that regulate the diminution of heat, as they stand indelibly inscribed on the rocky walls and abrupt declivities of the Cordilleras.”

~ “Cosmos”, Alexander von Humboldt.

“Nature considered rationally, that is to say, submitted to the process of thought, is a unity in diversity of phenomena; a harmony, blending together all created things, however dissimilar in form and attributes; one great whole animated by the breath of life. The most important result of a rational inquiry into nature is, therefore, to establish the unity and harmony of this stupendous mass of force and matter, to determine with impartial justice what is due to the discoveries of the past and to those of the present, and to analyze the individual parts of natural phenomena without succumbing beneath the weight of the whole.”

“In reflecting upon the different degrees of enjoyment presented to us in the contemplation of nature, we find that the first place must be assigned to a sensation, which is wholly independent of an intimate acquaintance with the physical phenomena presented to our view, or of the peculiar character of the region surrounding us. In the uniform plain bounded only by a distant horizon, where the lowly heather, the cistus, or waving grasses, deck the soil; on the ocean shore, where the waves, softly rippling over the beach, leave a track, green with the weeds of the sea; everywhere, the mind is penetrated by the same sense of the grandeur and vast expanse of nature, revealing to the soul, by a mysterious inspiration, the existence of laws that regulate the forces of the universe. Mere communion with nature, mere contact with the free air, exercise a soothing yet strengthening influence on the wearied spirit, calm the storm of passion, and soften the heart when shaken by sorrow to its inmost depths. Everywhere, in every region of the globe, in every stage of intellectual culture, the same sources of enjoyment are alike vouchsafed to man. The earnest and solemn thoughts awakened by a communion with nature intuitively arise from a presentiment of the order and harmony pervading the whole universe, and from the contrast we draw between the narrow limits of our own existence and the image of infinity revealed on every side, whether we look upwards to the starry vault of heaven, scan the far-stretching plain before us, or seek to trace the dim horizon across the vast expanse of ocean.”

~ “Cosmos”, Alexander von Humboldt.

“It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.”

~ “Citizenship in a Republic”, Theodore Roosevelt.

“Pauca sed matura”
~ Gauss’ motto

“I matematici del XVIII secolo avevano infatti l’impressione di internarsi in un nuovo mondo dello spirito, ed erano bramosi di delimitarne subito i confini.”
~ “Storia e filosofia dell’analisi infinitesimale”, Ludovico Geymonat.

“Se Newton e Leibniz avessero pensato che le funzioni continue possono non avere una derivata, e che anzi ciò costituisce proprio il caso generale, il calcolo differenziale non sarebbe nato; allo stesso modo le idee inesatte di Lagrange sulla possibilità degli sviluppi in serie di Taylor hanno reso immensi servizi… Era indispensabile che all’inizio le cose sembrassero semplici… Senza voler troppo generalizzare, si può dire che l’errore è talvolta utile, e che, nelle epoche veramente creatrici, una verità incompleta o approssimata può essere più feconda che la medesima verità accompagnata dalle restrizioni necessarie.”
~ E. Picard

“Nessuno più di Leibniz ha compreso a fondo il valore scientifico dei simboli: “ai simboli — egli scrive — è da richiedere che essi si prestino alla ricerca; ciò succede principalmente quando essi esprimono in modo conciso e quasi dipingono l’intima natura della cosa, perché essi allora risparmiano mirabilmente lo sforzo del pensiero”. E altrove giunge a dire che tutti i progressi, da lui fatti compiere all matematica, provengono unicamente dall’essere egli riuscito a scoprire simboli idonei a rappresentare le quantità infinitamente piccole e le loro relazioni. Bene dunque conclude il Couturat, affermando: “l’originalità profonda del calcolo infinitesimale consiste nel rappresentare, con dei segni appropriati, nozioni e operazioni che non hanno più nulla di aritmetico e a sottometterli a un algoritmo formale. Proprio qui si ha ciò che costituisce il merito essenziale dell’invenzione di Leibniz, il suo principale vantaggio in confronto al metodo delle flussioni di Newton”. E lo Zeuthen di rincalzo: “non si debbono cercare nuovi risultati nelle formule differenziali di Leibniz. Ciò che conferisce ad esse il loro significato fondamentale per l’intera matematica, è la circostanza di trovarsi legate a un simbolismo che le pone a base di un nuovo calcolo, con cui si possono effetuare le ricerche infinitesimali proprio come, per mezzo del caclolo letterale, si effettuava l’analisi del finito”.”

~ “Storia e filosofia dell’analisi infinitesimale”, Ludovico Geymonat.