Actually, he was also a distinguished concert pianist well after losing the arm, and right through the first half of the century filled the great halls of Europe and the United States. He taught himself to play the piano with his left hand alone, and challenged all the great composers of his time - Richard Strauss, Ravel, Prokofiev, Britten - to write him exclusive left-hand symphonies, cantatas and sonatas. The result was some wonderful music, some entertainingly spiky correspondence with the composers (there were a few problems with Ravel) and some interesting technical advances in the use of the sustain pedal, to the benefit of other composers and pianists.
The thing which really arrested my attention was that when Paul first lost his arm, his brother the philosopher wrote a pessimistic entry in his diary saying that he thought constantly about Paul, so cruelly deprived of his occupation, that philosophy had no answers, and that he only wished there was some answer apart from suicide. There was, indeed, to be more than one suicide in that generation of the Wittgenstein family.
But meanwhile, back home, Paul Wittgenstein was not in fact plotting suicide. He was practising seven hours a day with his remaining hand, deciding which composers to target first and how to tie up the business end by making the deals exclusive. He fully intended to make a triumphant comeback sooner rather than later. And in the end, he did just that. Later on, when the Nazis rose to power and a Jew could no longer perform in public, he showed the same brisk spirit by deciding to leave Austria for the US with the words: "If your house is burning down you forget about your possessions and get out." Or, as his American-born daughter put it in the programme:"If life hands you a lemon, make lemonade."
It may be fanciful, but it feels to me as if the difference between the two brothers connects somehow to their different callings. Ludwig lived in realms of abstract thought and intellectual musing. Confronted by a tragedy, he thought and wondered and repined and generally made everything worse. Paul, while a great artist, had a practical skill. And when half his tools were taken from him, that determined, meticulous practicality led him back to the piano, to mental recovery and to a quirky but genuine musical triumph.
It fits my longtime theory that practical skills - playing a musical instrument, doing a sport well, mastering a craft - are powerful antidepressants. We are, after all, animals. We have bodies made to be active, prehensile fingers, perceptions designed to help us handle ourselves as we move through space. We are not that hackneyed science-fiction horror, a brain in a bottle connected to a keyboard and loudspeaker. We demand action. Even the most cerebral among us tend to seek physical outlets: cooking, gardening, sport, or (although with less benevolent social results) even recreational sex.
But we are careless about ensuring that schoolchildren get that chance. Not only are playing fields ever scarcer, leisure centres expensive, and sports coaching the preserve of the competitive elite, but music lessons are far from universal. Moreover, in school subjects the sense of practical, manual mastery can rapidly wither away. Technology becomes overloaded with presentations and folders and verbal exposition. Cooking has become food-tech, largely a study of commercial production and little to do with rubbing butter into flour or tossing pancakes. Privileged c hildren may master horses, or dinghies, or be taken skiing. But an awful lot of schoolchildren live at desks or in front of screens, and unless they go out joyriding or breaking into building sites, can only sense the exhilaration of dexterity through the medium of the computer-game joystick.
And quite apart from the health implications, I think this has a depressing effect. Ask anybody who has seen a sullen child turned round, elated and improved in every way by music, or steering a little boat, or working alongside grandad at the garage workbench, or tinkering with a car. Physical mastery is part of education, part of life, part of joy. The life of the mind matters too, and someone's got to write the occasional Tractatus, but of the two Wittgensteins, the thinker and the player, I know which one I would feel happier about if I was their mother...