Mathematics, Form and Function


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.

Contents

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.[1] 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;[2] 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).

Notes

  1. ^ On the anthropological grounding of mathematics, see White (1947) and Hersh (1997).
  2. ^ Also see the "Basic Metaphor of Infinity" in Lakoff and Núñez (2000), chpt. 8.

References

  • Gårding, Lars, 1977. Encounter with Mathematics. Springer-Verlag.
  • Reuben Hersh, 1997. What Is Mathematics, Really? Oxford Univ. Press.
  • George Lakoff and Rafael E. Núñez, 2000. Where Mathematics Comes From. Basic Books.
  • Mac Lane, Saunders (1986). Mathematics, Form and Function. Springer-Verlag. ISBN 0-387-96217-4. 
  • Leslie White, 1947, "The Locus of Mathematical Reality: An Anthropological Footnote," Philosophy of Science 14: 289-303. Reprinted in Hersh, R. , ed., 2006. 18 Unconventional Essays on the Nature of Mathematics. Springer: 304–19.

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