# Mathematics, Form and Function

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Mathematics, Form and Function

Mathematics, Form and Function is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.

## Mac Lane's relevance to the philosophy of mathematics

Mac Lane founded (with Samuel Eilenberg) category theory, which enables a unified treatment of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding. Nevertheless, his views—however informal—are a valuable contribution to the philosophy and anthropology of mathematics. His views anticipate, in some respects, the much richer and more detailed account of the cognitive basis of mathematics given by George Lakoff and Rafael E. Núñez in their Where Mathematics Comes From. Lakoff and Núñez argue that mathematics emerges via conceptual metaphors grounded in the human body, its motion through space and time, and in human sense perceptions.

## Mathematics and human activities

Throughout his book, and especially in chapter I.11, Mac Lane informally discusses how mathematics is grounded in more ordinary concrete and abstract human activities. The following table is adapted from one given on p. 35 of Mac Lane (1986). The rows are very roughly ordered from most to least fundamental. For a bullet list that can be compared and contrasted with this table, see section 3 of Where Mathematics Comes From.

 Human Activity Related Mathematical Idea Mathematical Technique Collecting Collection Set; class; multiset; list; family Connecting Cause and effect ordered pair; relation; function; operation " Proximity; connection Topological space; mereotopology Following Successive actions Function composition; transformation group Comparing Enumeration Bijection; cardinal number; order Timing Before & After Linear order Counting Successor Successor function; ordinal number Computing Operations on numbers Addition, multiplication recursively defined; abelian group; rings Looking at objects Symmetry Symmetry group; invariance; isometries Building; shaping Shape; point Sets of points; geometry; pi Rearranging Permutation Bijection; permutation group Selecting; distinguishing Parthood Subset; order; lattice theory; mereology Arguing Proof First-order logic Measuring Distance; extent Rational number; metric space Endless repetition Infinity; Recursion Recursive set; Infinite set Estimating Approximation Real number; real field Moving through space & time: curvature calculus; differential geometry --Without cycling Change Real analysis; transformation group --With cycling Repetition pi; trigonometry; complex number; complex analysis --Both Differential equations; mathematical physics Motion through time alone Growth & decay e; exponential function; natural logarithms; Altering shapes Deformation Differential geometry; topology Observing patterns Abstraction Axiomatic set theory; universal algebra; category theory; morphism Seeking to do better Optimization Operations research; optimal control theory; dynamic programming Choosing; gambling Chance Probability theory; mathematical statistics; measure

Also see the related diagrams appearing on the following pages of Mac Lane (1986): 149, 184, 306, 408, 416, 422-28.

Mac Lane (1986) cites a related monograph by Lars Gårding (1977).

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