[Benjamin Peirce's] lectures were not easy to follow. They were never carefully prepared. The work with which he rapidly covered the blackboard was very illegible, marred with frequent erasures, and not infrequent mistakes (he worked too fast for accuracy). He was always ready to digress from the straight path and explore some sidetrack that had suddenly attracted his attention, but which was likely to have led nowhere when the college bell announced the close of the hour and we filed out, leaving him abstractedly staring at his work, still with chalk and eraser in his hands, entirely oblivious of his departing class.

Believe me, if Archimedes ever had the grand entrance of a girl as pretty as Gloria to look forward to, he would never have spent so much time calculating the value of Pi. He would have been baking her a Pie! If Euclid had ever beheld a vision of loveliness like the one I see walking into my anti-math class, he would have forgotten all the geometry of lines and planes, and concentrated on the sweet simplicity of soft curves. If Pythagoras had ever had a girl look at him the way Gloria's eyes fix in my direction, he would have given up his calculations on the hypotenuse of right triangles and run for the hills to pick a bouquet of wildflowers.

There was yet another disadvantage attaching to the whole of Newton’s physical inquiries, ... the want of an appropriate notation for expressing the conditions of a dynamical problem, and the general principles by which its solution must be obtained. By the labours of LaGrange, the motions of a disturbed planet are reduced with all their complication and variety to a purely mathematical question. It then ceases to be a physical problem; the disturbed and disturbing planet are alike vanished: the ideas of time and force are at an end; the very elements of the orbit have disappeared, or only exist as arbitrary characters in a mathematical formula.

To the average mathematician who merely wants to know his work is securely based, the most appealing choice is to avoid difficulties by means of Hilbert's program. Here one regards mathematics as a formal game and one is only concerned with the question of consistency ... . The Realist position is probably the one which most mathematicians would prefer to take. It is not until he becomes aware of some of the difficulties in set theory that he would even begin to question it. If these difficulties particularly upset him, he will rush to the shelter of Formalism, while his normal position will be somewhere between the two, trying to enjoy the best of two worlds.

I've got a few ideas," (Amy) admitted. "But I don't know where we're going in the long term. I mean - have you ever thought about what this ultimate treasure could be?""Something cool." (Dan)"Oh, that's real helpful. I mean, what could make somebody the most powerful Cahill in history? And why thirty-nine clues?"Dan shrugged. "Thirty-nine is a sweet number. It's thirteen times three. It's also the sum of five prime numbers in a row - 3,5,7,11,13. And if you add the first three powers of three, 3 to the first, 3 to the second, and s to the third, you get thirty-nine."Amy stared at him. "How did you know that?""What do you mean? It's obvious.

I think a strong claim can be made that the process of scientific discovery may be regarded as a form of art. This is best seen in the theoretical aspects of Physical Science. The mathematical theorist builds up on certain assumptions and according to well understood logical rules, step by step, a stately edifice, while his imaginative power brings out clearly the hidden relations between its parts. A well constructed theory is in some respects undoubtedly an artistic production. A fine example is the famous Kinetic Theory of Maxwell. ... The theory of relativity by Einstein, quite apart from any question of its validity, cannot but be regarded as a magnificent work of art.

I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

Another mistaken notion connected with the law of large numbers is the idea that an event is more or less likely to occur because it has or has not happened recently. The idea that the odds of an event with a fixed probability increase or decrease depending on recent occurrences of the event is called the gambler's fallacy. For example, if Kerrich landed, say, 44 heads in the first 100 tosses, the coin would not develop a bias towards the tails in order to catch up! That's what is at the root of such ideas as "her luck has run out" and "He is due." That does not happen. For what it's worth, a good streak doesn't jinx you, and a bad one, unfortunately , does not mean better luck is in store.

Velmi brzy začal projevovat zájem o matematiku - dokonce tak silný, že mu otec, který nechtěl zanedbat jeho filozofické, jazykové a humanitní vzdělání, zakázal v jedenácti letech číst matematické knihy, dokud nedosáhne věku patnácti let. Zákaz odvolal, když syna přistihl, jak kusem uhlí píše na zeď v jakési zapadlé chodbě jejich domu svůj vlastní důkaz, že součtem úhlů v trojuhelníku je vždy přímý úhel. Bylo to asi jediné inteligentní graffiti v dějinách.

I entered Princeton University as a graduate student in 1959, when the Department of Mathematics was housed in the old Fine Hall. This legendary facility was marvellous in stimulating interaction among the graduate students and between the graduate students and the faculty. The faculty offered few formal courses, and essentially none of them were at the beginning graduate level. Instead the students were expected to learn the necessary background material by reading books and papers and by organising seminars among themselves. It was a stimulating environment but not an easy one for a student like me, who had come with only a spotty background. Fortunately I had an excellent group of classmates, and in retrospect I think the "Princeton method" of that period was quite effective.

What a shame," signed the Dodecahedron. "They're so very useful. Why, did you know that if a beaver two feet long with a tail a foot and a half long can build a dam twelve feet high and six feet wide in two days, all you would need to build Boulder Dam is a beaver sixty-eight feet long with a fifty-one-foot tail?""Where would you find a beaver that big?" grumbled the Humbug as his pencil point snapped."I'm sure I don't know," he replied, "but if you did, you'd certainly know what to do with him.""That's absurd," objected Milo, whose head was spinning from all the numbers and questions."That may be true," he acknowledged, "but it's completely accurate, and as long as the answer is right, who cares if the question is wrong? If you want sense, you'll have to make it yourself.

A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science:'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' (equally like to happen).From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.

People enjoy inventing slogans which violate basic arithmetic but which illustrate “deeper” truths, such as “1 and 1 make 1” (for lovers), or “1 plus 1 plus 1 equals 1” (the Trinity). You can easily pick holes in those slogans, showing why, for instance, using the plus-sign is inappropriate in both cases. But such cases proliferate. Two raindrops running down a window-pane merge; does one plus one make one? A cloud breaks up into two clouds -more evidence of the same? It is not at all easy to draw a sharp line between cases where what is happening could be called “addition”, and where some other word is wanted. If you think about the question, you will probably come up with some criterion involving separation of the objects in space, and making sure each one is clearly distinguishable from all the others. But then how could one count ideas? Or the number of gases comprising the atmosphere? Somewhere, if you try to look it up, you can probably fin a statement such as, “There are 17 languages in India, and 462 dialects.” There is something strange about the precise statements like that, when the concepts “language” and “dialect” are themselves fuzzy.

Turing attended Wittgenstein's lectures on the philosophy of mathematics in Cambridge in 1939 and disagreed strongly with a line of argument that Wittgenstein was pursuing which wanted to allow contradictions to exist in mathematical systems. Wittgenstein argues that he can see why people don't like contradictions outside of mathematics but cannot see what harm they do inside mathematics. Turing is exasperated and points out that such contradictions inside mathematics will lead to disasters outside mathematics: bridges will fall down. Only if there are no applications will the consequences of contradictions be innocuous. Turing eventually gave up attending these lectures. His despair is understandable. The inclusion of just one contradiction (like 0 = 1) in an axiomatic system allows any statement about the objects in the system to be proved true (and also proved false). When Bertrand Russel pointed this out in a lecture he was once challenged by a heckler demanding that he show how the questioner could be proved to be the Pope if 2 + 2 = 5. Russel replied immediately that 'if twice 2 is 5, then 4 is 5, subtract 3; then 1 = 2. But you and the Pope are 2; therefore you and the Pope are 1'! A contradictory statement is the ultimate Trojan horse.

Before I walked into the door, the room got shades darker as a cloud did a summersault in front of the sun. I turned my head up to the sky and saw Gauss in the glass smirking down at me. In that moment I was reminded of a story about Gauss. 
When he was in the fifth grade, his teacher wanted some quiet, so he asked his class to add up all the numbers from 1-100. Thinking he had plenty of time to relax, he was shocked that within minutes Gauss had an answer. Gauss had cleverly noticed that the numbers 1 and 100 added up to 101, and 2 and 99 also added up to 101 and on down until you hit 50 and 51. So there are 50 pairs of 101, and a simple multiplication problem by Gauss left his teacher perplexed.
The recollection of this story reminded me about my own fifth grade experience. Thor was the volunteer at my school for the “Math Superstar” program. After each assignment, stars of various colors signifying degrees of excellence were stuck on all the papers handed in. Like the Olympics, gold was the highest honor. 
Wendy, the girl who sat next to me, was baffled that no matter how many wrong answers I got (usually all of them), I consistently had gold stars on my papers. She thought Thor was showing a personal bias towards me, but the truth is that I knew where he kept his boxes of stars, so I simply awarded myself what I thought I deserved. Hey, Gauss, how’s that for clever?