Editing Philosophy of mathematics (section)

====Popper's two senses of number statements<!--Linked from 'Karl Popper'-->====
Realist and constructivist theories are normally taken to be contraries. However, [[Karl Popper]]<ref>Popper, Karl Raimund (1946) Aristotelian Society Supplementary Volume XX.</ref> argued that a number statement such as {{nowrap|"2 apples + 2 apples {{=}} 4 apples"}} 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.<ref>Gregory, Frank Hutson (1996) "[[s:Arithmetic and Reality: A Development of Popper's Ideas|Arithmetic and Reality: A Development of Popper's Ideas]]". City University of Hong Kong. Republished in Philosophy of Mathematics Education Journal No. 26 (December 2011)</ref>