Principialism is a non-academic philosophical movement which draws on the content of an unpublished work of Bertrand Russell. Though it is relatively obscure, in academic circles it is widely regarded as part of the lunatic fringe and not taken seriously. Its adherents are often seen as eccentric or harmlessly psychotic, though they are also considered a nuisance by some people in academic life and IT.
The Principialists claim that Russell's lost manuscript was an attempt to construct a logical basis for the foundations of mathematics and that the concept expressed in it can be extended in a number of ways. One is that functional completeness can be achieved with only one operator and that this idea enables the construction of simpler and therefore potentially smaller, faster and more efficient digital machines, and that there are a number of other shortcuts to digial design. There are various factions within the groups which argue for different logical operators for this purpose and a few people have gone so far as to buy large quantities of integrated circuits of a single kind to test their hypotheses. This amounts to an expensive hobby and they are generally only able to produce very primitive devices. They are also seen as quixotic because they lack the resources to develop their ideas to a useful degree should they be useful. It is in this respect that they are seen as a nuisance, because they tend to contact IT companies and academic departments incessantly with their ideas and also sometimes believe that there is a conspiracy to hold back the development of technology for commercial reasons, pursued by large companies such as IBM, Acorn and Sinclair
A second idea is more arcane. They also believe that logic can be extended to express necessity and contingency using multivalent truth values. Some take this further, claiming that possible world semantics involve indexical expressions, and have gone so far as to attempt to construct devices which can detect information from other possible worlds. These people are generally considered misguided or insane even by most of the other Principialists.
Orthodox Views on Principialism
It is generally agreed that mathematics cannot be reduced to logic because a discrepancy in the logical analysis of mathematics would clearly be perceived as an error in the analysis rather than in mathematics, so any logical analysis is superfluous. It is also held that the idea of functional completeness being achieved by a single operator would violate known mathematical laws about the smallest possible sets of logical connectives to achieve this aim.
Concerning the claims regarding possible worlds, the difficulty is seen as the consistency of claiming that there can be mutually isolated universes in any meaningful sense. However, this is taken considerably less seriously than the other claim.
Principialism is often compared to the attempt to construct a perpetual motion machine or square the circle and its adherents are seen as pursuing a hopeless will o' the wisp like creationism or the idea of a flat earth. Whereas a few academics have made similar claims, this is a bad career move and no serious journals have published any principialist work, though a few wealthy enthusiasts have self-published books on the "theory".
There have also been a few psychological and sociological studies of principialists, which have generally concluded that many of them suffer from personality disorders or other psychopathology or are in some way socially deviant.