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 Halmos
One of the bad effects of an anti-intellectual philosophy such as that of Bergson, is that it thrives upon the errors and confusions of the intellect. Hence it is led to prefer bad thinking to good, to declare every momentary difficulty insoluble, and to regard every foolish mistake as revealing the bankruptcy of intellect and the triumph of intuition. There are in Bergson's work many allusions to mathematics and science, and to a careless reader these allusions may seem to strengthen his philosophy greatly. As regards science, especially biology and physiology, I am not competent to criticize his interpretations. But as regards mathematics, he has deliberately preferred traditional errors in interpretation to the more modern views which have prevailed among mathematicians for the last eighty years. In this matter, he has followed the example of most philosophers. In the eighteenth and the early nineteenth centuries, the infinitesimal calculus, though well developed as a method, was supported, as regards his foundations, by many fallacies and much confused thinking. Hegel and his followers seized upon these fallacies and confusions, to support them in their attempt to prove all mathematics self-contradictory. Thence the Hegelian account of these matters passed into the current thought of philosophers, where it has remained long after the mathematicians have removed all the difficulties upon which the philosophers rely. And so long as the main object of philosophers is to show that nothing can be learned by patience and detailed thinking, but that we ought rather to worship the prejudices of the ignorant under the title or 'reason' if we are Hegelians, or of 'intuition' if we are Bergsonians, so long philosophers will take care to remain ignorant of what mathematicians have done to remove the errors by which Hegel profited.
Henri-Louis Bergson
Newton and Descartes started to try and prove that God existed in the same way as they would try and prove something in the laboratory or with their mathematics … And when you try and mix science and religion you get bad science and bad religion. The two are doing two different things. ... Science can give you a diagnosis of cancer. It can even cure your disease, but it cannot touch your grief and disappointment, nor can it help you to die well.
Karen Armstrong
There are many men now living who were in the habit of using the age-old expression: 'It is as impossible as flying.' The discoveries in physical science, the triumphs in invention, attest the value of the process of trial and error. In large measure, these advances have been due to experimentation.
Louis Brandeis
Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i.e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.
Hans Reichenbach
Neither Chomsky nor Church would make my top ten list. No, I don’t actually have such a list, but my feeling is that their impact has been more on what is taught than what is done. Part of my unease is that I’m uncertain to what extent ‘Computer Science’ is a science. I feel more comfortable comparing what my colleagues and I do with the activities of architects and engineers than with mathematicians, physicists, or biologists. There are things in software that feel more as if they have been discovered than invented, but most of what [passes] for computer science is purely man-made.
Noam Chomsky
Halmos, Paul
Halonen, Tarja
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