Mathematics 1 is about the structure of immediate experience and the potentially infinite progression of sequences of such experiences; it involves the creation of truth which has an objective meaning in which its statements that cannot be interpreted as questions about events all of which will occur in a potentially infinite deterministic universe are neither true nor false in any absolute sense.
They may be useful properties 2 that are either true or false relative to a particular formal system.
Hempel C.G. (2001) thought that the truths of mathematics, in contradistinction to the hypotheses of empirical science, require neither factual evidence nor any other justification because they are self-evident.
However, the existence of mathematical conjectures 3 shows that not all mathematical truths can be self-evident and even if self-evidence were attributed only to the basic postulates of mathematics.
Hempel C.G. claims that mathematical judgments as to what may be considered as self-evident are subjective that is they may vary from person to person and certainly cannot constitute an adequate basis for decisions as to the objective validity of mathematical propositions.
While Shapiro 4 perceives that we learn perceptually that individual objects and systems of objects display a variety of patterns and we need to know more about the epistemology of the crucial step from the perspective of places-as-offices, which has no abstract commitments, to that of places-as-objects, which is thus committed.
He views that mathematical objects can be introduced by abstraction on an equivalence relation over some prior class of entities.
Shapiro 5 invokes an epistemological counterpart that, by laying down an implicit definition and convincing ourselves of its coherence, we successfully refer to the structure it defines.
1 --------, 2004, “A philosophy of mathematical truth”, Mountain Math Software, Retrieved 2004
3 Hempel, C.G., 2001, “On the Nature of Mathematical Truth”, Retrieved 2004
4 Shapiro in Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://firstname.lastname@example.org>
5 In Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://email@example.com>