BOOK REVIEW / The house that Ludwig built: 'Ludwig Wittgenstein, Architect' - Paul Wijdeveld: Thames & Hudson, 45 pounds: Hugo Barnacle explores the architectural inventiveness of the only great philosopher to have turned master-builder

PLATO never built his ideal gym. We cannot tell whether it would have had Socratic simplicity or Speer-like monumentalism, whether the plan would have been open or full of niches for assignations, whether the facilities would have been mixed, segregated or men-only, whether the whole place would have been agreeable or sinister.

The Kremlin never asked Sartre to draw up the perfect Gulag. Bacon not only never wrote Shakespeare's plays, he never erected a theatre either. Pascal never put up so much as a betting-shop, nor Nietzsche a telephone booth (for supermen). There is only one building in the world designed by a great philosopher, and it stands on the corner of Kundmanngasse in the IIIrd District of Vienna: a large city mansion, pure and severe looking in white stucco, raised in 1926-9 by Ludwig Wittgenstein for his sister Margaret.

Margaret Stonborough, by then amicably separated from her American millionaire husband, wanted a new house with the character of a suburban villa, in a park or garden setting, but also with the formal grandeur of a traditional palais like the one she and Ludwig grew up in. She originally gave the commission to Paul Engelmann, a friend of Wittgenstein's since the Great War and a pupil of Adolf Loos, one of the deserving artists in various fields to whom Wittgenstein had anonymously given away his inheritance.

After visiting Margaret, Engelmann wrote to Wittgenstein: 'We had long discussions as to whether such an undertaking is still possible in these days. I believe no rather than yes. . .' He did not mean moneywise. Money was no problem at all for Margaret, whose international investments had done rather well out of the war. He was doubtful of achieving a result that would be authentic, truthfully of its time, instead of the neo- Baroque or English country-house pastiche that most such ventures lapsed into.

Furthermore Engelmann had only designed interiors up to then, and, following Loos, he dealt in bourgeois informal space planning, the creation of a relaxing environment to come home to after a hard day's salary-earning. This would not be quite the ticket.

All the same, revising his ideas through ten basic phases, Engelmann eventually came up with a subtle arrangement of interlocked cubes which looked good enough to proceed with. Wittgenstein had just given up schoolteaching - he lost his temper too easily - and for a while wondered whether to go back to Cambridge. Margaret consulted him about the plans, and from then on Engelmann was practically off the project.

Wittgenstein's main contribution was to simplify at all costs, and the costs were huge. He not only deleted the false attic storey and parapet, he forbade all decoration whatsoever. The house has no mouldings, architraves, skirtings or thresholds, so every join and angle had to be finished to exact tolerances unprecedented in the history of domestic architecture. The doorhandles of Wittgenstein's own design fit into their holes in the plain steel doors with an utter precision undisguised by any rings or collars.

Using only his eye and intuition, he altered the size and placing of windows to give the exterior a dynamic aspect that Engelmann's drawings lacked. When Margaret was ready to move in and the builders were packing up he decided the ceiling of the principal reception room needed to be 3cm higher, so work began again. He never was satisfied with the stairwell.

The house, austere and rigorous, with its white walls, iron shutters and naked 200-watt bulbs, ought almost to suggest an insane asylum, but the pictures in Paul Wijdeveld's book display an uncanny, tranquil beauty that a BBC film crew also captured a few years back. Although the site has been spoilt by the building of a tower block in the garden where the ancient trees used to stand, and the house is now home to the Bulgarian Cultural Mission, Wittgenstein's personality and philosophy still cling to the place.

Above all, of course, the house does not say anything, it only shows something, and is thus an eloquent memorial to the man who concluded that what we cannot speak about we must consign to silence.

Wijdeveld is an academic, well capable of a sentence like 'The struggle for symmetry in the breakfast room has been amply described by Jan Turnovsky', but he is thorough, reproducing all of Engelmann's sketches along with fine crayon drawings by Wittgenstein's other sister which picture the house as actually furnished and lived in. (Looks pretty comfy, strange to say.) Diagrams give the exact alignment of every black xylolite floor slab, the placing of every air vent and radiator, every last cupboard.

What is not confronted is the possible implication that philosophy, like building a superb mansion for one's rich sister, is a pursuit for the leisured: noble but finally inconsequential. At least, Wijdeveld does not confront it; Wittgenstein almost certainly did, and that is part of what the house, in its grand integrity, shows us.