Integer Quotes (12 quotes)

A few days afterwards, I went to him [the same actuary referred to in another quote] and very gravely told him that I had discovered the law of human mortality in the Carlisle Table, of which he thought very highly. I told him that the law was involved in this circumstance. Take the table of the expectation of life, choose any age, take its expectation and make the nearest integer a new age, do the same with that, and so on; begin at what age you like, you are sure to end at the place where the age past is equal, or most nearly equal, to the expectation to come. “You don’t mean that this always happens?”—“Try it.” He did try, again and again; and found it as I said. “This is, indeed, a curious thing; this is a discovery!” I might have sent him about trumpeting the law of life: but I contented myself with informing him that the same thing would happen with any table whatsoever in which the first column goes up and the second goes down.

Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. … Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.

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…”. What had happened was that Hardy had received the remark in silence and with poker face, and I wrote it off as a dud.

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.

Integers are the fountainhead of all mathematics.

Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics.

Mssr. Fermat—what have you done?

Your simple conjecture has everyone

Churning out proofs,

Which are nothing but goofs!

Could it be that your statement’s an erudite spoof?

A marginal hoax

That you’ve played on us folks?

But then you’re really not known for your practical jokes.

Or is it then true

That you knew what to do

When n was an integer greater than two?

Oh then why can’t we find

That same proof…are we blind?

You must be reproved, for I’m losing my mind.

Your simple conjecture has everyone

Churning out proofs,

Which are nothing but goofs!

Could it be that your statement’s an erudite spoof?

A marginal hoax

That you’ve played on us folks?

But then you’re really not known for your practical jokes.

Or is it then true

That you knew what to do

When n was an integer greater than two?

Oh then why can’t we find

That same proof…are we blind?

You must be reproved, for I’m losing my mind.

Of all the reciprocals of integers, the one that I best like is 1/0 for it is a titan amongst midgets.

Our knowledge of the external world must always consist of numbers, and our picture of the universe—the synthesis of our knowledge—must necessarily be mathematical in form. All the concrete details of the picture, the apples, the pears and bananas, the ether and atoms and electrons, are mere clothing that we ourselves drape over our mathematical symbols— they do not belong to Nature, but to the parables by which we try to make Nature comprehensible. It was, I think, Kronecker who said that in arithmetic God made the integers and man made the rest; in the same spirit, we may add that in physics God made the mathematics and man made the rest.

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.

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.

This formula [for computing Bernoulli’s numbers] was first given by James Bernoulli…. He gave no general demonstration; but was quite aware of the importance of his theorem, for he boasts that by means of it he calculated

91,409,924,241,424,243,424,241,924,242,500.

*intra semi-quadrantem horæ!*the sum of the 10th powers of the first thousand integers, and found it to be