For instance, classical mathematicians by definition rely on Plato's theory of forms as the underlying basis of their mathematical worldview. The Platonist assumes the existence of true, immutable, and universal forms and structures that the mathematician approaches through the language of numbers and equations. For instance, the classical mathematician holds to the Platonic belief in the expansion of pi; to approach the expansion of pi from any other perspective "would require a restructuring of all of mathematical analysis," (p. 414). The paradigm would have to change; a most likely candidate for the restructured mathematical analysis would be constructivism, which relies more exclusively on the number system. The very existence of varied paradigms in mathematics points to the essentially "soft" core underlying all mathematical pursuits.
Finally, the growth of mathematics depends partly on the evolution of technology as well as the evolution of thinking. Paper and pencil has largely given way...
Yet many mathematicians criticize the dependency on computers to generate complex strings or to approximate answers. Computer hardware and software are not infallible. Yet human reason is also fallible (p. 418). Appel & Haken's proof of the four color conjecture relied on computers some mathematicians note that the use of the computer undermined its rigor (p. 425).
Mathematics changes as the overall body of human knowledge changes, via continual questioning and the revisiting of old ideas. Mathematics changes through evolution of technology, the evolution of paradigm, and the temporal evolution of mathematical ideas. In each case, mathematics changes by a continual transformation of soft knowledge such as intuition into hard knowledge such as a proof or theory. An idea need not be proven formally to become working knowledge. In time, however, proof becomes possible as mathematicians can avail themselves of new technologies or ways…
