A Quote by Ian Hacking
Philip Kitcher thinks that mathematics is surprisingly like empirical science. Few mathematicians would agree; philosophers too, from Socrates on, have held the opposite opinion.
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In fact, the answer to the question "What is mathematics?" has changed several times during the course of history... It was only in the last twenty years or so that a definition of mathematics emerged on which most mathematicians agree: mathematics is the science of patterns.
Socrates: So even our walks are dangerous here. But you seem to have avoided the most dangerous thing of all. Bertha: What's that? Socrates: Philosophy. Bertha: Oh, we have philosophers here. Socrates: Where are they? Bertha: In the philosophy department. Socrates: Philosophy is not department. Bertha: Well, we have philosophers. Socrates: Are they dangerous? Bertha: Of course not. Socrates: Then they are not true philosophers.
It appears that the solution of the problem of time and space is reserved to philosophers who, like Leibniz, are mathematicians, or to mathematicians who, like Einstein, are philosophers.
So in the end it wasn't Gödel, it wasn't Turing, and it wasn't my results that are making mathematics go into an experimental mathematics direction, in a quasi-empirical direction. The reason why mathematicians are changing their working habits is the computer. I think that this is an excellent joke!
Today, it is not only that our kings do not know mathematics, but our philosophers do not know mathematics and - to go a step further - our mathematicians do not know mathematics.
Socrates claimed famously that one never loses by doing the right thing. Stephen Post and his contributors claim, a little less boldly, that at least the generous will, probably, stay healthy—and, improving on Socrates, they support this claim with careful empirical science, impressive for its comprehensive detail. Here ethics and religion join science and enjoin us to be more caring and healthy. A seminal work, with an urgent message.
Formal logic is mathematics, and there are philosophers like Wittgenstein that are very mathematical, but what they're really doing is mathematics - it's not talking about things that have affected computer science; it's mathematical logic.
Mathematics, the non-empirical science par excellence . . . the science of sciences, delivering the key to those laws of nature and the universe which are concealed by appearances.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
Socrates: Have you noticed on our journey how often the citizens of this new land remind each other it is a free country? Plato: I have, and think it odd they do this.Socrates: How so, Plato?Plato: It is like reminding a baker he is a baker, or a sculptor he is asculptor.Socrates: You mean to say if someone is convinced of their trade, they haveno need to be reminded.Plato: That is correct.Socrates: I agree. If these citizens were convinced of their freedom, they would not need reminders.
Mathematics may be the only exception in the sciences that leaves no room for skepicism. But, if mathematical results are exact as no empirical law can ever be, philosophers have discovered that they are not absolutely novel - instead, they are tautological.
I think philosophers can do things akin to theoretical scientists, in that, having read about empirical data, they too can think of what hypotheses and theories might account for that data. So there's a continuity between philosophy and science in that way.
It has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics. . . required. . . had already been worked out by pure mathematicians for their own amusement. . . . The moral for statesmen would seem to be that, for proper scientific "planning", pure mathematics should be endowed fifty years ahead of scientists.
[I can] scarcely write upon mathematics or mathematicians. Oh for words to express my abomination of the science.
As for mathematicians themselves: don't expect too much help. Most of them are too far removed in their ivory towers to take up such challenges. And anyway, they are not competent. After all, they are just mathematicians-what we need is paramathematicians, like you... It is you who can be the welding force, between mathematicians and stories, in order to achieve the synthesis.
Each generation has its few great mathematicians, and mathematics would not even notice the absence of the others. They are useful as teachers, and their research harms no one, but it is of no importance at all. A mathematician is great or he is nothing.
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