🥀On Style in Mathematics

"The figures which excite in us the ideas of beauty seem to be those in which there is uniformity amidst variety. […] What we call beautiful in objects, to speak in the mathematical style, seems to be in compound ratio of uniformity and variety: so that where the uniformity of bodies is equal, the beauty is as the variety; and where the variety is equal, the beauty is as the uniformity." (Francis Hutcheson,"An Inquiry Concerning Beauty, Order, Harmony, Design", 1973)

"Numeracy has two facets-reading and writing, or extracting numerical information and presenting it. The skills of data presentation may at first seem ad hoc and judgemental, a matter of style rather than of technology, but certain aspects can be formalized into explicit rules, the equivalent of elementary syntax." (Andrew Ehrenberg,"Rudiments of Numeracy", Journal of Royal Statistical Society, 1977)

"Moreover, ‘fact’ doesn’t mean ‘absolute certainty’; there ain’t no such animal in an exciting and complex world. The final proofs of logic and mathematics flow deductively from stated premises and achieve certainty only because they are NOT about the empirical world. Evolutionists make no claim for perpetual truth, though creationists often do" (and then attack us falsely for a style of argument that they themselves favor)." (Stephen J Gould, "Evolution as Fact and Theory", Discover, 1981)

"[...] probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"Mathematics is not just a collection of results, often called theorems; it is a style of thinking. Computing is also basically a style of thinking. Similarly, probability is a style of thinking." (Richard W Hamming, "The Art of Probability for Scientists and Engineers", 1991)

"The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof – which we know and can recognize, without the formal advice of the logicians. […] Intuition is glorious, but the heaven of mathematics requires much more. Physics has provided mathematics with many fine suggestions and new initiatives, but mathematics does not need to copy the style of experimental physics. Mathematics rests on proof - and proof is eternal." (Saunders Mac Lane,"Reponses to …", Bulletin of the American Mathematical Society Vol. 30" (2), 1994)

"Allowing more than two players into the game and/or postulating payoff structures in which one player's gain does not necessarily equal the other player's loss brings us much closer to the type of games played in real life. Unfortunately, it's generally the case that the closer you get to the messiness of the real world, the farther you move from the stylized and structured world of mathematics. Game theory is no exception." (John L Casti, "Five Golden Rules", 1995)

"The body of mathematics to which the calculus gives rise embodies a certain swashbuckling style of thinking, at once bold and dramatic, given over to large intellectual gestures and indifferent, in large measure, to any very detailed description of the world. It is a style that has shaped the physical but not the biological sciences, and its success in Newtonian mechanics, general relativity and quantum mechanics is among the miracles of mankind. But the era in thought that the calculus made possible is coming to an end. Everyone feels this is so and everyone is right." (David Berlinski, "A Tour of the Calculus", 1995)

"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103 (7), 1996)

"Physicists have been drawn to elegant mathematical relationships that bind the subject together with economy and style, melding disparate qualities in subtle and harmonious ways. But this is to import a new factor into the argument - questions of aesthetics and taste. We are then on shaky ground indeed. It may be that M theory looks beautiful to its creators, but ugly to N theorists, who think that their theory is the most elegant. But then the O theorists disagree with both groups [...]" (Paul C W Davies, "Cosmic Jackpot: Why Our Universe Is Just Right for Life", 2007)