Philosophy, for some people, suggests a neatly ordered garden. Philosophers are like those suburban husbands of popular legend, whose chief pleasure in life is to tend their syllogisms and uproot the occasional outbreak of sloppy reasoning. It is an enclave of repose and delight, a refuge from the hard world of business and politics and from the irritations of domestic life. Here, everything is easy and tractable; everything submits to the firm hand of reason. Having discovered such a paradise, one could almost forgive philosophers for forgetting about the world outside.

There is something magical about this conception of philosophy; it has a whimsical, Alice-in-Wonderland quality to it. It is easy, particularly at Oxford or Cambridge, to succumb to its charm. But it is a betrayal of the discipline. The reason why philosophy cannot detach itself from the world in this way is that it has no subject matter that it can call its own. Historians study history, medics study medicine, but there is no such thing as "philosophy" that philosophers study. The neatly ordered garden does not exist. "Philosophy" is just a word for a certain manner of being confused. Philosophical problems, as Ludwig Wittgenstein once remarked, have the form of "I don't know my way around here". Most of the time, life moves along well-oiled grooves. Problems are encountered and overcome with mechanical ease. Occasionally, however, one encounters a problem that doesn't go away; it nags at you with irritating insistence. This is then called "a philosophical problem". One can stumble into philosophy in the course of any activity - reading novels, doing maths, bringing up children, fighting wars. Philosophy is not something that can be separated from puzzlement in general; it is simply disciplined, self-conscious puzzlement.

Such an inclusive conception of philosophy lies behind this collection of essays edited by Ray Monk and Frederic Raphael, who are both more than philosophers. Monk is an accomplished mathematician and logician, as well as the author of excellent biographies of Wittgenstein and Bertrand Russell. Raphael is a well-known novelist and critic. The 12 philosophers covered in the book include two figures not normally regarded as philosophers at all - Karl Marx and Alan Turing. Marx earns his place in the pantheon by virtue of being what his essayist, Terry Eagleton, calls an "anti-philosopher" - a deflator of philosophy's claim to universal, disinterested truth. Turing, here dealt with in a superb essay by Andrew Hodges, is a more interesting addition. He is best known as the Bletch-ley Park mathematician who cracked the Enigma code and laid the foundations of the computer revolution. But in trying to create a machine that could "think", Turing was forced to answer the more fundamental question: "What is it for a machine - or, indeed, anything else - to think?" Turing is a good example of someone who stumbled into phil- osophy while doing something else. His career is proof of the impossibility of separating philosophy from thought in general.

As well as introducing new figures into the canon, the essayists make an effort to dust down the old classics. David Berman wisely ignores Berkeley's well-known argument that "to be is to be perceived" - something through which every undergraduate student of Berkeley has had to trudge - and concentrates instead on his little-known thought experiments. Many of these would now fall within the province of psychology rather than philosophy, which shows how recent and artificial these departmental boundaries are. John Cottingham demonstrates that Descartes's theory of mind is a lot subtler than the "Cartesian dualism" commonly attributed to him, and the inclusion of Heidegger is a gracious acknowledgement that the old distinction between analytic and Continental philosophy is breaking down. The best contemporary philosophers - Richard Rorty, Alasdair MacIntyre, Charles Taylor - fit neither category. And this convergence is also retrospective. One can now see that Wittgenstein had more in common with Heidegger, despite the difference of idiom, than with his colleague and friend Bertrand Russell.

Mathematics is a constant theme. It has always been one of the chief provocations to philosophical puzzlement. Bernard Williams, in his essay on Plato, recounts a famous scene in the Meno. A slave boy is led - not without a little prompting from Socrates - to understand Pythagoras's famous theorem. How can this be possible, asks Socrates, given that the boy has never even heard of Pythagoras? The answer is that the boy must have known the theorem in a previous life and is now remembering it. Few philosophers have been convinced by this answer, and many have accused Socrates of leading the boy in the demonstration. But, as Williams remarks, this misses the point. Had the question been one in history or geography, Socrates could not have led the boy to the answer. He would have had to tell him straight out. This captures the peculiarity of mathematical knowledge - a peculiarity that philosophers have registered by designating it as a priori know-ledge: that which is not based on experience, but issues from the mind itself. It has fascinated and perplexed philosophers ever since.

The appeal of a priori knowledge lies in its utter certainty. A mathematical theorem, once proven, is unassailable: no possible experience could ever prove it false. Mathematics opens up a world of crystalline purity. Here is the garden of repose and delight that philosophers have always dreamed of. Monk recounts the ecstasy of the 11-year-old Russell on first reading Euclid. Brought up in an atmosphere of bereavement and secrecy, Russell responded with joy to "the exhilarating possibility that some beliefs at least can be provided with absolutely cast-iron foundations". One of the perennial quests of philosophy has been to extend the certainty of the a priori beyond the bounds of mathematics. Plato taught that knowledge, not only of mathematics, but also of justice and beauty, is based on recollection of a previous life. Spinoza, examined here by Roger Scruton, constructed a whole system of ethics in the style of Euclid, complete with axioms, deductions and theorems.

Yet why does this pretension to mathematical certainty fail to convince us? Why do we suspect Plato and Spinoza of simply foisting their own opinions on us? Is there something male about this obsession with certainty? Perhaps philosophy is simply another of those tiresome male fanaticisms, like stamp collecting or motor-racing. All 12 of the "great philosophers" in this book, and all 12 of their commentators, are men. There were some great women philosophers in the 20th century. Yet none of them has had this obsession with the a priori.

Not only have philosophers failed to extend the a priori beyond its citadel of mathematics, but they have failed to defend it even there. As Russell grew older, his childish exhilaration was replaced by doubt. The propositions of Euclid may follow with certainty from the axioms, but what about the axioms themselves? What are they based on? Russell spent the first half of his adult life struggling to supply mathematics with a logical foundation. It was a labour of Sisyphus. The logical notation that he devised for the purpose became uncontrollably baroque. Principia mathematica takes one and a half volumes to prove that one plus one equals two. Russell finally abandoned the enterprise in despair. Under the influence of Wittgenstein, he came to the unhappy conclusion that mathematics is nothing more than a set of tautologies. The whole story, by turns funny and moving, is excellently recounted by Monk.

With the notable exception of Russell, English philosophers have avoided the obsession with mathematics. Descartes, Pascal and Leibniz were all world-class mathematicians as well as philosophers; Locke, Berkeley and Hume were not. Berkeley was an accomplished mathematician, but he held the subject in low regard. Hume was a weak mathematician. "Commentators," writes Anthony Quinton in this book, "almost universally draw a veil over this part of his work." When Russell went up to Cambridge to read mathematics, he was disgusted to find that "the whole subject . . . was taught as a set of clever tricks by which to pile up marks in the Tripos". Russell had to turn to the work of contemporary German and French mathematicians to discover the Pythagorean purity he sought. The separation of philosophy from mathematics is one aspect of the notorious English empiricism, the tendency to eschew grand theorising in favour of the piecemeal collection of facts.

Andrew Hodges's essay on Turing is the last in the book, and forms a kind of epitaph on philosophy's romance with the a priori. Turing was a mathematician by training. But he never conceived of mathematics in the manner of the young Russell, as a revelation of super-sensible beauty and order. A strict materialist, he always thought of mathematics as a technique for getting things done. While Russell wore himself down in the quest for certainty, Turing got on with cracking the German Enigma code and inventing the computer. Russell conceived of mathematical logic as an enquiry into fundamental reality; Turing ended up by transforming it into an adjunct of computer science. Turing's answer to the question "What is it for a machine to think?" is typical of his tough-minded pragmatism. Putting the old metaphysical questions of consciousness to one side, he devised his famous test. If it is possible to have a blind conversation with a machine and to confuse it with a human being, then it is justifiable to say that the machine can think. The question of whether the machine really finds things funny or sad is laid to one side as unanswerable and futile; all that matters for the purposes of the test is that it behaves as if it does. Turing has removed the problem of consciousness from philosophy and handed it over to science.

Turing's own life was a tragedy and Russell's was a success. But the pragmatic spirit of Turing has won over the aristocratic spirit of Russell. The old metaphysical problems of being and consciousness appear increasingly remote and unreal. Why not get on with life and stop puzzling over questions that clearly have no answer? So long as our techniques work, there seems to be little point in inquiring whether they correspond to ultimate reality. So, perhaps philosophy - in the sense of the science as practised by Socrates, Spinoza and Russell - has finally come to an end.

Edward Skidelsky's reviews appear monthly in the NS