bound to fail, as pointless or even meaningless. Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko, although it is not clear that either would endorse the title. More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. Some consider the work of Paul Erdős as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g., via the Erdős number. Reuben Hersh has also promoted the social view of mathematics, calling it a "humanistic" approach, similar to but not quite the same as that associated with Alvin White; one of Hersh's co-authors, Philip J. Davis, has expressed sympathy for the social view as well. Beyond the traditional schools Unreasonable effectiveness Rather than focus on narrow debates about the true nature of mathematical truth, or even on practices unique to mathematicians such as the proof, a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain. Popper's two senses of number statements Realist and constructivist theories are normally taken to be contraries. However, Karl Popper argued that a number statement such as can be taken in two senses. In one sense it is irrefutable and logically true. In the second sense it is factually true and falsifiable. Another way of putting this is to say that a single number statement can express two propositions: one of which can be explained on constructivist lines; the other on realist lines. Philosophy of language Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said, the language of science. Although some mathematicians and philosophers would accept the statement "mathematics is a language" (most consider that the language of mathematics is a part of mathematics to which mathematics cannot be reduced), linguists believe that the implications of such a statement must be considered. For example, the tools of linguistics are not generally applied to the symbol systems of