A Quote by Charles Sanders Peirce
This branch of mathematics [Probability] is the only one, I believe, in which good writers frequently get results which are entirely erroneous.
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Evil grows and bears fruit, which is understandable, because it has logic and probability on its side and also, of course, strength. The resistance of tiny kernels of good, to which no one grants the power of causing far-reaching consequences, is entirely mysterious, however. Such seeming nothingness not only lasts but contains within itself enormous energy which is revealed gradually.
In no other branch of mathematics is it so easy for experts to blunder as in probability theory.
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.
One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics.
We shall see that the mathematical treatment of the subject [of electricity] has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.
In abstract mathematics, of course operations alter those particular relations which are involved in the considerations of number and space, and the results of operations are those peculiar results which correspond to the nature of the subjects of operation.
Mathematics is a logical method. . . . Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
Man must have results, real results, in his inner and outer life. I do not mean the results which modern people strive after in their attempts at self-development. These are not results, but only rearrangements of psychic material, a process the Buddhists call 'samsara' and which our Holy Bible calls 'dust'.
If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.
Coming from bad results, you have more tension and you get more into the game, maybe. You never know which is best. I prefer to come from good results. You have more confidence and you believe you are doing things well. But in football everything can change very quickly.
But the other notion is, we also believe that those folks closest on the ground that we're holding accountable for the results can decide, and ought to evaluate which programs get results.
I had ... come to an entirely erroneous conclusion, which shows, my dear Watson, how dangerous it always is to reason from insufficient data.
The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
When a branch of mathematics ceases to interest any but the specialists, it is very near its death, or at any rate dangerously close to a paralysis, from which it can be rescued only by being plunged back into the vivifying source of the science.
Are not all Hypotheses erroneous, in which Light is supposed to consist in Pression or Motion, propagated through a fluid Medium? For in all these Hypotheses the Phaenomena of Light have been hitherto explain'd by supposing that they arise from new Modifications of the Rays; which is an erroneous Supposition.
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