Integers Quotes

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

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

5.4 The question of accumulation. If life is a wager, what form does it take? At the racetrack, an accumulator is a bet which rolls on profits from the success of one of the horse to engross the stake on the next one.
5.5 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 sings are limited. Thus an entirely failed relationship might be expressed in terms of both loss/minus and division/ reduction, showing a total of 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 improbable and mathematically insoluble?
5.6 Thus how might you express an accumulation containing the integers b, b, a (to the first), a (to the second), s, v?
B = s - v (*/+) a (to the first)
Or
a (to the second) + v + a (to the first) x s = b
5.7 Or is that the wrong way to put the question and express the accumulation? Is the application of logic to the human condition in and of itself self-defeating? What becomes of a chain of argument when the links are made of different metals, each with a separate frangibility?
5.8 Or is "link" a false metaphor?
5.9 But allowing that is not, if a link breaks, wherein lies the responsibility for ~ Julian Barnes

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

It as mathematical, marriage, not, as one might expect, additional; it was exponential. This one man, nervous in a suite a size too small for his long, lean self, this woman, in a green lace dress cut to the upper thigh, with a white rose behind her ear. Christ, so young. The woman before them was a unitarian minister, and on her buzzed scalp, the grey hairs shone in a swab of sun through the lace in the window. Outside, Poughkeepsie was waking. Behind them, a man in a custodian's uniform cried softly beside a man in pajamas with a Dachshund, their witnesses, a shine in everyone's eye. One could taste the love on the air, or maybe that was sex, or maybe that was all the same then.
'I do,' she said.
'I do,' he said.
They did. They would.
Our children will be so fucking beautiful, he thought, looking at her.
Home, she thought, looking at him.
'You may kiss,' said the officiant.
They did, would.
Now they thanked everyone and laughed, and papers were signed and congratulations offered, and all stood for a moment, unwilling to leave this gentile living room where there was such softness.
The newlyweds thanked everyone again, shyly, and went out the door into the cool morning. They laughed, rosy. In they'd come integers, out they came, squared.

Her life, in the window, the parakeet, scrap of blue midday in the London dusk, ages away from what had been most deeply lived. Day on a rocky beach, creatures in the tide pool. ~ Lauren Groff

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

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

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

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

Integers are the fountainhead of all mathematics. ~ Hermann Minkowski

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

A good example of the archetypal ideas which the archetypes produce are natural numbers or integers. With the aid of the integers the shaping and ordering of our experiences becomes exact. Another example is mathematical group theory. ...important applications of group theory are symmetries which can be found in most different connections both in nature and among the 'artifacts' produced by human beings. Group theory also has important applications in mathematics and mathematical physics. For example, the theory of elementary particles and their interactions can in essential respects be reduced to abstract symmetries.
[The Message of the Atoms: Essays on Wolfgang Pauli and the Unspeakable] ~ Kalervo V. Laurikainen

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

The danger of speaking about life exclusively in terms of problem and solution is that we are thus tempted to overlook the limitations of this detective game and the very existence of the initial arbitrary rules that makes the playing of it possible. The rule is to exclude from the terms of the problem everything that the solution cannot solve. It is diverting and useful to know that, for the chemist, a man is made up of a few pennyworth of salt, sugar, iron, and what not, together with an intolerable deal of water. But we must not assert that 'man is, in fact, nothing but' these things, or suppose that the solution of the pennyworths in the water will produce a complete and final solution of man. . . .

It was said by Kronecker, the mathematician: 'God made the integers; all else is the work of man.' Man can table the integers and arrange them into problems that he can solve in the terms in which they are set. But before the inscrutable mystery of the integers themselves he is helpless, unless he calls upon that tri-unity in himself that is made in the image of God, and can include and create the integers. ~ Dorothy L. Sayers

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

Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. ~ Gregory Bateson

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

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

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

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

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

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

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