Among the more important schools we may distinguish roughly:1
1) The logistic school represented by Peano, Russell, and Whitehead, who accept the chrisippian, two-valued, restricted form of the el 'logic' and so may be called the chrisippian school.
2) The axiomatic school, represented by Hilbert and his followers, which may be called the aristotelian school.
3) The 'intuitional' school represented by Brouwer and Weyl who question the 'law of excluded third', and so may be called the non-chrisippian school.
4) The Polish school of: (a) 'intuitional' formalism with Lukasiewicz. Tarski, Lesniewski as representatives, which may be called the non-aristotelian school. Lukasiewicz generalized the A 'logic' to three-valued 'logic' which covers modality. Lukasiewicz and Tarski finally produced a general many-valued 'logic' of which the two-valued represents only a limiting case. Lesniewski produced Protothetic, a still more general 'logical' system, by introducing variable 'funktors', .* (b) The restricted semantic school represented by Chwistek and his pupils, which is characterized mostly by the semantic approach, and by paying special attention to the number of values, establishing the thesis that the older 'freedom from contradictions' depends on one-valued formulations, as discovered by Skarzenski and quoted by Chwistek. This school has already produced new foundations (still elementalistic) for 'logic' and mathematics, and leads to generalized arithmetics and analysis.
5) The average prevalent mathematical technician, who does not realize that he belongs to the numerically large class which may be called the 'christian science' school of mathematics, which proceeds by faith and disregards entirely any problems of the epistemological foundations of their supposed 'scientific' activities.
It should be noticed that all existing mathematical schools accept implicitly, at least, A elementalism and do not challenge identity, a principle which happens to be invariably false to facts and which therefore should be entirely abolished.
The above classification suggests that, in spite of great achievements in the field of mathematical foundations, no school can expect to be convincing or accepted by other schools as long as we all flounder in the A and el ambiguities which prevent any possibility of agreement. It becomes obvious also that when a A and non-el system is formulated it will necessitate a new paradox-free foundation for mathematics and so a new school of mathematics will arise which may be called:
6) The general semantic, non-aristotelian, non-elementalistic school of mathematics. It is premature to give the names of the leading pioneers in this field at present.
♦At present Lukasiewicz and Tarski call their many-valued 'logic' non-chrisippian, but this name does not seem appropriate because these authors generalized both forms of the aristotelian 'logic' to a many-valued 'logic' of which the two^valued becomes only a limiting case. Thus it seems that their many-valued 'logic' is better described by the term non-aristotelian, yet still elementalistic 'logic'.