Research On Helmholtz's Thought Of Mathematical Philosophy

Mathematical epistemology involves the relation of thought and existence in philosophy, and because the mathematical objects have something of abstraction and speciality, its method, mathematical deduction, is different from that in natural science. Its speciality of method leads to some epistemological issues: what is the nature of mathematic? Whether is mathematic a science of deduction or empirical or transcendental science? This became a basic question of mathematical philosophy. In his Critique of pure reason, Kant claims that mathematical propositions are synthetic judgments a priori. The origin of geometric axioms are transcendental and he claims in his spatial theory that not only empirical space is the Euclidean and bases on the Euclidean demonstrations, but that we can therefore know a priori that physical space and physical objects are described by Euclid's geometry with apodictic certainty.As a famous physiologist and physicist, Helmholtz challenged Kant's position. He argued for empirical theory rather than transcendental theory. He quoted the mobility concept in his abstruse axiom system and improved the non-Euclidean geometries into a mature mathematical science with mathematical experiment (thought experiment)and rich imaginability. Helmholtz argued that Kant was simply wrong to take specifically Euclidean geometry as the necessary form of our spatial intuition or perception. The new non-Euclidean geometries are not only logically possible, they are also perceptually or intuitively possible as well, in that we can very well imagine what our perceptual experience would be if we lived in a non-Euclidean world. The necessary form of our spatial intuition was therefore the much more general structure common to the three classical geometries of constant curvature (Euclidean, hyperbolic, and elliptic); and this structure was described, accordingly, not by the specific axioms of Euclid, but rather by the much more general principle of"free mobility"permitting arbitrary continuous motions of rigid bodies. The existence of non-Euclidean geometries, and the fact that we might make measurements would yield the conclusion that space is non-Euclidean, then the question of the structure of physical space becomes a matter of empirical investigation.Helmholtz'principle of perception forming shows that the space is not only the form of transcendental intuition, but can be intuited through touch and sight regaining. The system of geometric axioms should be different according the different kinds of space in which creatures live. Therefore, the origin of geometric axioms is empirical rather than transcendental. All transcendental geometric system is short of empirical contents, and only geometric axioms have been combined with certain mechanical principle, can we get a geometric system which has a real nature and can be conformed or refuted by experience, and can we gain propositions which has empirical significance.Non-Euclidean geometries have changed the conventional idea of mathematics, that is, geometric objects are not only the form and relation of general space, it can also be other possible form and relation which analogous to them. Helmholtz'thought of mathematical philosophy shows his philosophical thought of empiricism.