# Mathematics, Form and Function

Mathematics, Form and Function, a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.

## 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 | Object 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).

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

Mac Lane cofounded category theory with Samuel Eilenberg, 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 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.