First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as thus to discern the particular laws governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general laws that complete theories by bringing a multitude of apparently very diverse phenomena together under a single governing principle.
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
Interpretation
What this quote means
Mathematics is both a rational and creative endeavor, merging logical analysis with intuitive thought to achieve beauty and understanding.
This quote by Richard Courant emphasizes that mathematics is not only a logical discipline but also an artistic one that reflects human creativity and intellect. It highlights how the combination of different elements—such as the interplay between logic and intuition—contributes to the richness and value of mathematical science, suggesting that the synthesis of contrasting forces is essential for understanding its true beauty and utility.
Themes
In practice
Example use cases
This quote can be used in a lecture on the philosophy of mathematics to illustrate its dual nature.
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