In the context of mathematical discussions at that time, logicism was an extreme minority project whose failure was as little known to a broad mathematical public as its takeoff had been a few decades earlier.
The paradoxes of set theory discovered/generated at the turn of the century were initially not considered worthy of crisis beyond logicism, especially since these paradoxes could at least be brought under control to such an extent by the additional theory of type hierarchies that the (logicist) theory business could continue.
Nevertheless, as early as 1910, when Alfred Whitehead and Bertrand Russell published their Principa Mathematica (Whitehead/Russell 2008), not everyone was satisfied with the type theory proposed there; in this respect, the paradoxes of set theory remained, if not a crisis trigger, at least a strong incentive to find other ways of dealing with these paradoxes - mostly, such as Ernst Zermelo's axiomatic set theory, as offers to prevent paradoxes in a more noble way (cf. Lau 2005).
That the paradoxes of set theory, despite their emergency treatment by the logician type theory resp. – much more prominently – by the formalistic axiomatization of set theory (which was supposed to prevent paradoxes from occurring by cleverly knitted definitional nets) was evaluated at all as a sign of a "new fundamental crisis of mathematics" (Weyl 1965), and that with increasing gain of attention, was a communicative success of the counter-modernists, who could fix their fundamental always-already rejection of the meanwhile prevailing formalistic paradigm on set theory and its paradoxical entanglements.
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Fuchs und Hoegl, „Die Schrift der Form. Peter Fuchs und Franz Hoegl über George Spencer-Browns Laws of Form“.
Through the works of Gottlob Frege and Bertrand Russell, logic had acquired an unprecedented diversity, and so it was expected that a suitable problem would be found for this impressive solution: the "justifiable" mathematics. As is well known, the reason-theoretical program of logicism created problems already at the beginning, to be fixed at the so-called Russell paradox of "naive" set theory ("Does the set of all sets which do not contain themselves contain themselves?").