Quotes by Mathematicians - Page 2

It is not what the man of science believes that distinguishes him, but how and why he believes it. His beliefs are tentative, not dogmatic; they are based on evidence, not on authority or intuition.
Bertrand Russell
I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.
Alan Turing
The apex of mathematical achievement occurs when two or more fields which were thought to be entirely unrelated turn out to be closely intertwined. Mathematicians have never decided whether they should feel excited or upset by such events.
Gian-Carlo Rota
We are not the stuff that abides, but patternsthat perpetuate themselves.
Norbert Wiener
Men spend their time in following a ball or a hare it is the pleasure even of kings.
Blaise Pascal
... who can forget the amazement of a child balancing an adult on a see-saw, simply by being placed at the right position. How could this be? Where did all that extra force come from!?The only wonder nowadays is that a physics student is unlikely to produce a satisfactory answer to this question. Perhaps we will be offered a few mumblings about moments, force times distance, laws of the lever perhaps even the "principle of virtual work". But we probably won't get an answer that seems to explain where that extra force comes from; and it is highly unlikely that we will get an answer that begins by establishing principles about rigid bodies, even though the rigidity of the lever is an absolute necessity for it to work.In fact, the whole path from Newton's Laws, which basically concern "point masses", to bodies whose shape and extent are significant, is often rather dubiously traversed, even though elementary physics courses blithely pose such problems of the most diverse sorts.
Michael Spivak
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.
Euclid
One of the most curious consequences of quantum physics is that a particle like an electron can seemingly be in more than one place at the same time until it is observed, at which point there seems to be a random choice made about where the particle is really located. Scientists currently believe that this randomness is genuine, not just caused by a lack of information. Repeat the experiment under the same conditions and you may get a different answer each time.
Marcus du Sautoy
To understand this new frontier, I will have to try to master one of the most difficult and counterintuitive theories ever recorded in the annals of science: quantum physics. Listen to those who have spent their lives immersed in this world and you will have a sense of the challenge we face. After making his groundbreaking discoveries in quantum physics, Werner Heisenberg recalled, "I repeated to myself again and again the question: Can nature possibly be so absurd as it seemed to us in these atomic experiments?" Einstein declared after one discovery, "If it is correct it signifies the end of science." Schrödinger was so shocked by the implications of what he'd cooked up that he admitted, "I do not like it and I am sorry I had anything to do with it." Nevertheless, quantum physics is now one of the most powerful and well-tested pieces of science on the books. Nothing has come close to pushing it off its pedestal as one of the great scientific achievements of the last century. So there is nothing to do but to dive headfirst into this uncertain world. Feynman has some good advice for me as I embark on my quest: "I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?' because you will get 'down the drain,' into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
Marcus du Sautoy
Even if we have a reliable criterion for detecting design, and even if that criterion tells us that biological systems are designed, it seems that determining a biological system to be designed is akin to shrugging our shoulders and saying God did it. The fear is that admitting design as an explanation will stifle scientific inquiry, that scientists will stop investigating difficult problems because they have a sufficient explanation al
William A. Dembski
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
George Pólya
It is our task, both in science and in society at large, to prove the conventional wisdom wrong and to make our unpredictable dreams come true
Freeman Dyson
It is not what the man of science believes that distinguishes him, but how and why he believes it. His beliefs are tentative, not dogmatic; they are based on evidence, not on authority or intuition.
Bertrand Russell
I believe that at the end of the century the use of words and general educated opinion will have altered so much that one will be able to speak of machines thinking without expecting to be contradicted.
Alan Turing
We are not the stuff that abides, but patternsthat perpetuate themselves.
Norbert Wiener
To satisfy our doubts . . . it is necessary that a method should be found by which our beliefs may be determined by nothing human, but by some external permanency -- by something upon which our thinking has no effect. . . . Our external permanency would not be external, in our sense, if it was restricted in its influence to one individual. It must be something which affects, or might affect, every man. And, though these affections are necessarily as various as are individual conditions, yet the method must be such that the ultimate conclusion of every man shall be the same. Such is the method of science. Its fundamental hypothesis, restated in more familiar language, is this: There are Real things, whose characters are entirely independent of our opinions about them; those Reals affect our senses according to regular laws, and, though our sensations are as different as are our relations to the objects, yet, by taking advantage of the laws of perception, we can ascertain by reasoning how things really and truly are; and any man, if he have sufficient experience and he reason enough about it, will be led to the one True conclusion. The new conception here involved is that of Reality.
Charles Sanders Peirce
Thus, by science I mean, first of all, a worldview giving primacy to reason and observation and a methodology aimed at acquiring accurate knowledge of the natural and social world. This methodology is characterized, above all else, by the critical spirit: namely, the commitment to the incessant testing of assertions through observations and/or experiments — the more stringent the tests, the better — and to revising or discarding those theories that fail the test. One corollary of the critical spirit is fallibilism: namely, the understanding that all our empirical knowledge is tentative, incomplete and open to revision in the light of new evidence or cogent new arguments (though, of course, the most well-established aspects of scientific knowledge are unlikely to be discarded entirely).. . . I stress that my use of the term 'science' is not limited to the natural sciences, but includes investigations aimed at acquiring accurate knowledge of factual matters relating to any aspect of the world by using rational empirical methods analogous to those employed in the natural sciences. (Please note the limitation to questions of fact. I intentionally exclude from my purview questions of ethics, aesthetics, ultimate purpose, and so forth.) Thus, 'science' (as I use the term) is routinely practiced not only by physicists, chemists and biologists, but also by historians, detectives, plumbers and indeed all human beings in (some aspects of) our daily lives. (Of course, the fact that we all practice science from time to time does not mean that we all practice it equally well, or that we practice it equally well in all areas of our lives.)
Alan Sokal
The thing I want you especially to understand is this feeling of divine revelation. I feel that this structure was "out there" all along I just couldn't see it. And now I can! This is really what keeps me in the math game-- the chance that I might glimpse some kind of secret underlying truth, some sort of message from the gods.
Paul Lockhart
If, years later, I do use the slit detector to observe which way the electron went, it will mean that many years earlier the electron must have passed through one slit or the other. But if I don't use the "slit detector," then the electron must have passed through both slits. This is, of course, extremely weird. My actions at the beginning of the twenty-first century can change what happened thousands of years ago when the electron began its journey. It seems that just as there are multiple futures, there are also multiple pasts, and my acts of observation in the present can decide what happened in the past. As much as it challenges any hope of ever really knowing the future, quantum physics asks whether I can ever really know the past. It seems that the past is also in a superposition of possibilities that crystallize only once they are observed.
Marcus du Sautoy
Your Excellency, I have no need of this hypothesis.
Pierre-Simon Laplace
Each religion makes scores of purportedly factual assertions about everything from the creation of the universe to the afterlife. But on what grounds can believers presume to know that these assertions are true? The reasons they give are various, but the ultimate justification for most religious people’s beliefs is a simple one: we believe what we believe because our holy scriptures say so. But how, then, do we know that our holy scriptures are factually accurate? Because the scriptures themselves say so. Theologians specialize in weaving elaborate webs of verbiage to avoid saying anything quite so bluntly, but this gem of circular reasoning really is the epistemological bottom line on which all 'faith' is grounded. In the words of Pope John Paul II: 'By the authority of his absolute transcendence, God who makes himself known is also the source of the credibility of what he reveals.' It goes without saying that this begs the question of whether the texts at issue really were authored or inspired by God, and on what grounds one knows this. 'Faith' is not in fact a rejection of reason, but simply a lazy acceptance of bad reasons. 'Faith' is the pseudo-justification that some people trot out when they want to make claims without the necessary evidence.But of course we never apply these lax standards of evidence to the claims made in the other fellow’s holy scriptures: when it comes to religions other than one’s own, religious people are as rational as everyone else. Only our own religion, whatever it may be, seems to merit some special dispensation from the general standards of evidence.And here, it seems to me, is the crux of the conflict between religion and science. Not the religious rejection of specific scientific theories (be it heliocentrism in the 17th century or evolutionary biology today); over time most religions do find some way to make peace with well-established science. Rather, the scientific worldview and the religious worldview come into conflict over a far more fundamental question: namely, what constitutes evidence.Science relies on publicly reproducible sense experience (that is, experiments and observations) combined with rational reflection on those empirical observations. Religious people acknowledge the validity of that method, but then claim to be in the possession of additional methods for obtaining reliable knowledge of factual matters — methods that go beyond the mere assessment of empirical evidence — such as intuition, revelation, or the reliance on sacred texts. But the trouble is this: What good reason do we have to believe that such methods work, in the sense of steering us systematically (even if not invariably) towards true beliefs rather than towards false ones? At least in the domains where we have been able to test these methods — astronomy, geology and history, for instance — they have not proven terribly reliable. Why should we expect them to work any better when we apply them to problems that are even more difficult, such as the fundamental nature of the universe?Last but not least, these non-empirical methods suffer from an insuperable logical problem: What should we do when different people’s intuitions or revelations conflict? How can we know which of the many purportedly sacred texts — whose assertions frequently contradict one another — are in fact sacred?
Alan Sokal
Take care of the sense and the sounds will take care of themselves.
Lewis Carroll