William Rowan Hamilton Quotes

Recreational mathematics is a type of play which is enjoyable and requires mathematical thinking or skills to engage with; typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts.

The land of easy mathematics where he who works adds up and he who retires subtracts.

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples...

My earliest mathematical memory is my father explaining to me the theorem that three angles in a triangle add up to 180 degrees. The idea that something could be proved to be always true was very appealing to me.

I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission.

There are men of a certain type of mind who are never wearied with gibing at mathematics, at mathematicians, and at mathematical methods of inquiry. It goes almost without saying that these men have themselves little mathematical bent. I believe this to be a general fact; but, as a fact, it does not explain very well their attitude towards mathematicians. The reason seems to lie deeper. How does it come about, for instance, that whilst they are themselves so transparently ignorant of the real nature, meaning, and effects of mathematical investigation, they yet lay down the law in the most confident and self-satisfied manner, telling the mathematician what the nature of his work is (or rather is not), and of its erroneousness and inutility, and so forth? It is quite as if they knew all about it. It reminds one of the professional paradoxers... They, too, write as if they knew all about it. Plainly, then, the anti-mathematician must belong to the same class as the paradoxer, whose characteristic is to be wise in his ignorance, whereas the really wise man is ignorant in his wisdom. ...What is of greater importance is that the anti-mathematicians sometimes do a deal of mischief. For there are many of a neutral frame of mind, little acquainted themselves with mathematical methods, who are sufficiently impressible to be easily taken in by the gibers and to be prejudiced thereby; and, should they possess some mathematical bent, they may be hindered by their prejudice from giving it fair development. We cannot all be Newtons or Laplace's, but that there is an immense amount of moderate mathematical talent lying latent in the average man I regard as a fact; and even the moderate development implied in a working knowledge of simple algebraical equations can, with common-sense to assist, be not only the means of valuable mental discipline, but even be of commercial importance (which goes a long way with some people), should one's occupation be a branch of engineering for example.

It might have been beans, successful hunts, or victories in battle, but, for millennia, people were using maths to describe things - counting them, measuring them, dividing them up.

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

We should not believe... that commensurability is a quality of every magnitude as of all the numbers; and whoever has not investigated this subject, shows a gross and unseemly ignorance of what the Athenian Stranger says in the seventh treatise of the Book of the Laws, [namely], "And besides there is found in every man an ignorance, shameful in its nature and ludicrous, concerning everything which has the dimensions, length, breadth, and depth; and it is clear that mathematics can free them from this ignorance. For I hold that this [ignorance] is a brutish and not a human state, and I am verily ashamed, not for myself only, but for all Greeks, of the opinion of those men who prefer to believe what this whole generation believes, [namely], that commensurability is necessarily a quality of all magnitudes. For everyone of them says: "We conceive that those things are essentially the same, some of which can measure the others in some way or other. But the fact is that only some of them are measured by common measures, whereas others cannot be measured at all".

Mathematics may be a way of developing physically, that is anatomically, new connections in the brain.

Prometheus: ...
List rather to the deeds
I did for mortals; how being fools before,
I made them wise and true in aim of soul. ...
But teaching you the intention of my gifts,
How first beholding, they beheld in vain,
And hearing, heard not, but, like shapes in dreams,
Mixed all things wildly down the tedious time,
Nor knew to build a house against the sun...
But lived like silly ants, beneath the ground
In hollow caves unsunned. There, came to them
No steadfast sign of winter, nor of spring...
But blindly and lawlessly they did all things,
Until I taught them how the stars do rise
And set in mystery, and devised for them
Number, the inducer of philosophies,
The synthesis of Letters, and, beside,
The artificer of all things, Memory,
That sweet Muse-mother.

Maths is the language spoken by all the sciences, taking us to the frontiers of knowledge, from the workings of the Universe to the workings of our minds, which enabled us to dream it all up in the first place.

A marveilous newtrality have these things mathematicall and also a strange participation between things supernaturall, imortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, compounded and divisible.

I've always felt that a teacher can introduce recreational math; and I'm defining it in the very broad sense to include anything that has a spirit of play about it. I don't know of any better way to hook the interests of the students.

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of... intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings. ...We are leaving out of account that most important faculty which distinguishes topics of interest from others; in fact, we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions.