In an ideal math paper, the abstract and intro make clear the assumptions and conclusions. So a reader who trusts the authors to avoid error can ignore the rest of the paper for the purpose of updating their beliefs. In non-ideal math papers, in contrast, readers are forced to dig deep, as key assumptions are scattered throughout the paper.
Robin Hanson, Review of ‘Semi-informative priors over AI timelines’, December 9, 2020
At one time I had undertaken to write a book on von Neumann’s scientific life. In trying to plan it, I thought of how I, along with many others, had been influenced by him; and how this man, and some others I knew, working in the purely abstract realm of mathematics and theoretical physics had changed aspects of the world as we know it. […] It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.
Stanisław Ulam, Adventures of a Mathematician, New York, 1976, pp. 4-5
Often the best way to make sure you’re being logical is to express your arguments mathematically. Early in this century, the eminent economist Alfred Marshall offered this advice to his colleagues: when confronted with an economic problem, first translate into mathematics, then solve the problem, then translate back into English and burn the mathematics.
Steven Landsburg, More Sex Is Safer Sex: The Unconventional Wisdom of Economics, New York, 2007, p. 174
There was hunger in the cities but not in the food-producing villages, and the peasants hoarded and hid food. One way to get some bread and butter, or maybe a chicken, was to walk to a village not too far from [Odessa], carrying along some silk handkerchiefs, a few pieces of family silver, or even a golden watch, and to exchange these for food. Many enterprising city inhabitants did this, even though it was a dangerous undertaking.
Here is a story told to me by one of my friends who was at that time a young professor of physics in Odessa. His name was Igor Tamm (Nobel Prize laureate in Physics, 1958). Once when he arrived in a neighboring village, at the period when Odessa was occupied by the Reds, and was negotiating with a villager as to how many chickens he could get for half a dozen silver spoons, the village was captured by one of the Makhno bands, who were roaming the country, harassing the Reds. Seeing his city clothes (or what was left of them), the capturers brought him to the Ataman, a bearded fellow in a tall black fur hat with machine-gun cartridge ribbons crossed on his broad chest and a couple of hand grenades hanging on the belt.
“You son-of-a-bitch, you Communistic agitator, undermining our Mother Ukraine! The punishment is death.”
“But no,” answered Tamm, “I am a professor at the University of Odessa and have come here only to get some food.”
“Rubbish!” retorted the leader. “What kind of professor are you?”
“I teach mathematics.”
“Mathematics?” said the Ataman. “All right! Then give me an estimate of the error one makes by cutting off Maclaurin’s series at the nth term. Do this, and you will go free. Fail, and you will be shot!”
Tamm could not believe his ears, since this problem belongs to a rather special branch of higher mathematics. With a shaking hand, and under the muzzle of the gun, he managed to work out the solution and handed it to the Ataman.
“Correct!” said the Ataman. “Now I see that you really are a professor. Go home!”
George Gamow, My World Line: An Informal Autobiography, New York, 1970, pp. 19-20
To Engels’s mind, there was nothing in math that was not already in nature; mathematics was simply a reflection and an explanation of the physical world. As a result, he attempted to crowbar all sorts of mathematical models into his system of dialectics. “Let us take an arbitrary algebraic magnitude, namely a,” begins one passage in Dialectics of Nature. “Let us negate it, then we have -a (minus a). Let us negate this negation by multiplying -a by -a, then we have +a, that is the original positive magnitude, but to a higher degree, namely to the second power.” As the Trotskyist scholar Jean van Heijenoort points out, this is all horribly confused: to take just one example, ”negation” in Engels’s usage can refer to any number of differing mathematical operations. Worse was to come as Engels, playing the reductive philistine, dismissed complex numbers and theoretical mathematics—those parts of theoretical science that went beyond a reflection of natural phenomena—as akin to quackery: “When one has once become accustomed to ascribe to the [square root of] -1 or to the fourth dimension some kind of reality outside of our own heads, it is not a matter of much importance if one goes a step further and also accepts the spirit world of the mediums.”
Tristam Hunt, Marx’s General: The Revolutionary Life of Friedrich Engels, New York, 2009, pp. 286-287
Wherefore in all great works are Clerkes so much desired? Wherefore are Auditors so richly fed? What causeth Geometricians so highly to be enhaunsed? Why are Astronomers so greatly advanced? Because that by number such things they finde, which else would farre excell mans minde.
Robert Recorde, Arithmetic: or, The Ground of Arts, London, 1543, p. 34
[T]here is one purpose at any rate which the real mathematics may serve in war. When the world is mad, a mathematician may find in mathematics an incomparable anodyne. For mathematics is, of all the arts and sciences, the most austere and the most remote, and a mathematician should be of all men the one who can most easily take refuge where, as Bertrand Russell says, “one at least of our nobler impulses can best escape from the dreary exile of the actual world.”
G. H. Hardy, A Mathematician’s Apology, Cambridge, 1940, sect. 28
[An] argument I always hear around the mathematics department [is that] mathematics helps you to think clearly. I have a very low opinion of this self-serving nonsense. In sports there is the concept of the specificity of skills: if you want to improve your racquetball game, don’t practice squash! I believe the same holds true for intellectual skills.
David Edwards, ‘The Math Myth’, EconLog, September 1, 2016
Among the worst of barbarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3…(n-1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4 &c. should be found in mathematical results.
Augustus De Morgan, The Penny Cyclopædia of the Society for the Difusion of Useful Knowledge, London, 1842, vol. 23, p. 444
In ethics, as in mathematics, the appeal to intuition is an epistemology of desperation.
Philip Kitcher, ‘Biology and Ethics’, in David Copp (ed.), The Oxford Handbook of Ethical Theory, Oxford, 2006, p. 176