This paper examines the dynamic nature of mathematical truth and how the discipline evolves over time. Drawing on examples from number theory, Platonist philosophy, and computational proofs, the paper argues that mathematics progresses through the gradual transformation of "soft" knowledge—such as intuition, heuristic evidence, and ideological assumptions—into "hard" knowledge in the form of rigorous proof. The paper also considers the role of technology, particularly computers, in shaping modern mathematical practice, and acknowledges the fallibility of both human reasoning and computational tools. Together, these forces illustrate that mathematical truth, while not dependent on any single mathematician, is shaped by the cumulative, time-bound growth of human knowledge.
"Mathematical truth is time-dependent, although it does not depend on the consciousness of any particular live mathematician" (p. 415). In other words, mathematics grows as the body of human knowledge grows; each generation gleans new wisdom from the environment, experimentation, or personal experience and transmits that knowledge to contemporary and future generations either orally or in writing. Noted mathematicians may have their names printed in textbooks or permanently etched into the theorems they discovered, but the greater body of mathematics grows whether or not momentous discoveries warrant an individual mathematician's fame. One of the primary ways mathematics changes over time is through the transformation of soft sources of information—such as common knowledge, intuition, or hunch—into hard information in the form of proof.
Mathematicians have accepted hunches and other soft sources of information as provisionally "true" even before formal proof has been established. Number theory is especially full of instances in which mathematicians rely fairly well on assumptions without demanding full proof: "in number theory, there may be heuristic evidence so strong that it carries conviction even without rigorous proof" (p. 411). For example, mathematicians do not know for certain whether an infinite number of twin prime pairs exist, and yet the field proceeds as if they do. Mathematicians take some ideas for granted, unless those ideas are proven wrong.
In any case, proofs often take generations or even centuries to materialize. The prime number theorem was first postulated in 1792 by a fifteen-year-old Gauss, but it remained unproven until 1896. Mathematicians rely on soft information that can best be described as working knowledge until hard information becomes available.
Mathematicians also operate within underlying belief systems, biases, and ideological frameworks that may influence the validity, reliability, or provability of their theories. 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 infinite expansion of pi; to approach that expansion from any other perspective "would require a restructuring of all of mathematical analysis" (p. 414).
The paradigm would have to change entirely. A leading candidate for any such 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.
"Computer-assisted proofs raise questions about mathematical rigor"
Mathematics changes as the overall body of human knowledge changes, via continual questioning and the revisiting of old ideas. Mathematics changes through the evolution of technology, the evolution of paradigm, and the temporal evolution of mathematical ideas. In each case, mathematics advances by a continual transformation of soft knowledge—such as intuition—into hard knowledge, such as a proof or theorem. An idea need not be formally proven to become working knowledge. In time, however, proof becomes possible as mathematicians avail themselves of new technologies and new ways of thinking.
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