Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.
No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a "mixed number " while 5/2 is an "improper fraction." They're EQUAL for crying out loud. They are the exact same numbers and have the exact same properties. Who uses such words outside of fourth grade?
The probability of an event is the reason we have to believe that it has taken place, or that it will take place.The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible.
Nothing comforted Sabine like long division. That was how she had passed time waiting for Phan and then Parsifal to come back from their tests. She figured the square root of the date while other people knit and read. Sabine blamed much of the world's unhappiness on the advent of calculators.
Mathematics had never had more than a secondary interest for him [her husband, George Boole]; and even logic he cared for chiefly as a means of clearing the ground of doctrines imagined to be proved, by showing that the evidence on which they were supposed to give rest had no tendency to prove them.
{Replying to G. H. Hardy's suggestion that the number of a taxi (1729) was 'dull', showing off his spontaneous mathematical genius}No, it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways, the two ways being 13 + 123 and 93 + 103.
I had been to school most all the time, and could spell, and read, and write just a little, and could say the multiplication table up to six times seven is thirty-five, and I don't reckon I could ever get any further than that if I was to live forever. I don't take no stock in mathematics, anyway.
The language of categories is affectionately known as "abstract nonsense," so named by Norman Steenrod. This term is essentially accurate and not necessarily derogatory: categories refer to "nonsense" in the sense that they are all about the "structure," and not about the "meaning," of what they represent.
The physical method becomes a philosophy when it asserts there is no higher knowledge than the empirical knowledge of scientific phenomena. The mathematical method becomes a philosophy when it asserts that some higher knowledge is needed to explain scientific facts, and that higher knowledge is mathematics.
How is it that there are so many minds that are incapable of understanding mathematics? ... the skeleton of our understanding, ... and actually they are the majority. ... We have here a problem that is not easy of solution, but yet must engage the attention of all who wish to devote themselves to education.
One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
One can be enlightened about proofs as well as theorems. Without enlightenment, one is merely reduced to memorizing proofs. With enlightenment about a proof, its flow becomes clear and it can become an item of astonishing beauty. In addition, the need to memorize disappears because the proof has become part of your soul.
In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world.
The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.
[Regarding mathematics,] there are now few studies more generally recognized, for good reasons or bad, as profitable and praiseworthy. This may be true; indeed it is probable, since the sensational triumphs of Einstein, that stellar astronomy and atomic physics are the only sciences which stand higher in popular estimation.
