Quotes & Sayings About Integers

I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had." In the next proof-sheets I read (what now stands), "It was Littlewood who said ... " — John Edensor Littlewood

I have tried, with little success, to get some of my friends to understand my amazement that the abstraction of integers for counting is both possible and useful. Is it not remarkable that 6 sheep plus 7 sheep makes 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, I find it both strange and unexplainable. — Richard Hamming

God made the integers; all the rest is the work of Man. — Leopold Kronecker

My problem is that I have been persecuted by an integer. — George Armitage Miller

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules ... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms. — Kurt Godel

With energy there is this difference, that there are no blocks, so far as we can tell. Also, unlike the case of the blocks, for energy the numbers that come out are not integers. I — Richard Feynman

The trouble with integers is that we have examined only the very small ones. Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way. Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed. Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions. — Ronald Graham

The integers of language are sentences, and their organs are the parts of speech. Linguistic organization, then, consists in the differentiation of the parts of speech and the integration of the sentence. — John Wesley Powell

If you object (as some of us did to Dr. Goris) that Cantor's transfinite numbers aren't really numbers at all but rather sets, then be apprised that what, say, 'P(Infinity to the Infinity +n), really is is a symbol for the number of members in a given set, the same way '3' is a symbol for the number of members in the set {1,2,3}. And since the transfinites are provably distinct and compose an infinite ordered sequence just like the integers,they really are numbers, symbolizable (for now) by Cantor's well-known system of alephs or '(Aleph symbol's). And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers-although, just as with 0, the rules for these operations are very different in the case of (Alephs) and have to be independently established and proved. — David Foster Wallace

God made the integers; all else is the work of man. — Leopold Kronecker

The series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences. — Albert Einstein

He was shaken by an unwelcome insight. Lives did not add as integers. They added as infinities. — Lois McMaster Bujold

Increasing pressure on students to subject themselves to ever more tests, whittling themselves down to rows and rows of tight black integers upon a transcript, all ready to goose-step straight into a computer. — Leah Hager Cohen

When we combine the set of whole numbers with their opposites, including zero, we
obtain a set of numbers we call integers. That — Judith Sowder

Nature does not count nor do integers occur in nature. Man made them all, integers and all the rest, Kronecker to the contrary notwithstanding. — Percy Williams Bridgman

Harriet grinned at Betty Armstrong, hearing the familiar academic wrangle begin. Before ten minutes had passed, somebody had introduced the word "values." An hour later they were still at it. Finally the Bursar was heard to quote: "God made the integers; all else is the work of man." "Oh, bother!" cried the Dean. "Do let's keep mathematics out of it. And physics. I cannot cope with them. — Dorothy L. Sayers

Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc ... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. — Abraham Robinson

All results of the profoundest mathematical investigation must ultimately be expressible in the simple form of properties of the integers. — Leopold Kronecker

God made the integers, man made the rest. — Leopold Kronecker

Integers are the fountainhead of all mathematics. — Hermann Minkowski

[L]ife ceases to be a fraction and becomes an integer. — Harry Emerson Fosdick

In this communication I wish first to show in the simplest case of the hydrogen atom (nonrelativistic and undistorted) that the usual rates for quantization can be replaced by another requirement, in which mention of "whole numbers" no longer occurs. Instead the integers occur in the same natural way as the integers specifying the number of nodes in a vibrating string. The new conception can be generalized, and I believe it touches the deepest meaning of the quantum rules. — Erwin Schrodinger

In they'd come, integers; out they came, squared. — Lauren Groff

Let's zoom in on a particular form of synesthesia as an example. For most of us, February and Wednesday do not have any particular place in space. But some synesthetes experience precise locations in relation to their bodies for numbers, time units, and other concepts involving sequence or ordinality. They can point to the spot where the number 32 is, where December floats, or where the year 1966 lies.8 These objectified three-dimensional sequences are commonly called number forms, although more precisely the phenomenon is called spatial sequence synesthesia.9 The most common types of spatial sequence synesthesia involve days of the week, months of the year, the counting integers, or years grouped by decade. In addition to these common types, researchers have encountered spatial configurations for shoe and clothing sizes, baseball statistics, historical eras, salaries, TV channels, temperature, and more. — David Eagleman

The Good Lord made all the integers; the rest is man's doing. — Leopold Kronecker

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 signs are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total 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 insoluble? — Julian Barnes

The page of my notebook was filled with many messy integrals, but all of a sudden I saw emerge a formula for counting. I had begun to calculate a quantity on the assumption that the result was a real number, but found instead that, in certain units, all the possible answers would be integers. This meant that areas and volumes cannot take any value, but come in multiples of fixed units. — Lee Smolin