John von Neumann — "Mathematics is an experimental science, and definitions are its axioms."
Mathematics is an experimental science, and definitions are its axioms.
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More from John von Neumann
"I think that a good deal of the 'mathematical thinking' that goes on in our heads is not mathematics at all, but rather thinking about physical analogies."
"Truth is much too complicated to allow anything but approximations."
"I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more."
"The computer is a universal machine. It can do anything that can be described algorithmically."
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Understanding this quote
What it means
Mathematics isn't purely deductive reasoning built from eternal, fixed truths — it works experimentally, like any science. Mathematicians choose definitions, observe what structures emerge, and revise. Those definitions function as axioms: foundational assumptions that determine everything built on top of them. The creative act in math isn't just proving theorems but choosing what to define and how. A well-chosen definition unlocks entire fields; a poor one leads nowhere useful.
Relevance to John von Neumann
Von Neumann lived this principle. His 1932 axiomatization of quantum mechanics chose definitions that made the theory rigorous. His game theory breakthrough lay in defining strategy, utility, and zero-sum precisely enough to generate theorems. His stored-program computer architecture was itself a definition of what computation is. He understood that mathematical power comes from choosing the right abstractions, not just proving from existing ones. His career was a series of definitional revolutions.
The era
Von Neumann worked through mathematics' foundational crisis — Russell's paradox, Hilbert's formalism, and Gödel's incompleteness theorems had shattered faith in fixed axiomatic truth. His era demanded pragmatic mathematics: WWII weapons design, quantum physics, and early computing all required inventing new mathematical frameworks rapidly. The question became not 'what are the true axioms?' but 'which definitions make this problem tractable?' His view reflected a generation that built math to solve problems, not to uncover eternal truth.
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