Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does. Possibly philosophers would look on us mathematicians the same way as we look on the technicians, if they dared.
Paul HalmosRelated topics
cliche error experimentation find guesswork hypothesis laboratory mathematics prove reason respect science start theorem trial try way technician factsRelated quotes
Subjective reason ... is inclined to abandon the fight with religion by setting up two different brackets, one for science and philosophy, and one for institutionalized mythology, thus recognizing both of them. For the philosophy of objective reason there is no such way out. Since it hold to the concept of objective truth, it must take a positive or a negative stand with regard to the content of established religion.
Max Horkheimer
At present, when the prevailing forms of society have become hindrances to the free expression of human powers, it is precisely the abstract branches of science, mathematics and theoretical physics, which ... offer a less distorted form of knowledge than other branches of science which are interwoven with the pattern of daily life, and the practicality of which seemingly testifies to their realistic character.
Max Horkheimer
As soon as an opinion becomes common it is sufficient reason for men to abandon it and to uphold the opposite opinion until that in its turn grows old, and they require to distinguish themselves by other things. Thus if they attain their goal in some art or science, we must expect them soon to cast it aside to acquire some fresh fame, and this is partly the reason why the most splendid ages degenerate so quickly, and, scarcely emerged from barbarism, plunge into it again.
Luc de Clapiers, Marquis de Vauvenargues
Science is a magnificent force, but it is not a teacher of morals. It can perfect machinery, but it adds no moral restraints to protect society from the misuse of the machine. It can also build gigantic intellectual ships, but it constructs no moral rudders for the control of storm tossed human vessel. It not only fails to supply the spiritual element needed but some of its unproven hypotheses rob the ship of its compass and thus endangers its cargo.
William Jennings Bryan
Science sometimes improves hypotheses and sometimes disproves them. But proof would be another matter and perhaps never occurs except in the realms of totally abstract tautology. We can sometimes say that if such and such abstract suppositions or postulates are given, then such and such abstract suppositions or postulates are given, then such and such must follow absolutely. But the truth about what can be perceived or arrived at by induction from perception is something else again.
Gregory Bateson