Mathematics, Form and Function

Short description: Book on philosophy of mathematics

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.

## Contents

* 1 Mathematics and human activities
* 2 Mac Lane's relevance to the philosophy of mathematics
* 3 See also
* 4 Notes
* 5 References

## 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;[1] 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.[2] 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.

## See also

* 1986 in philosophy

## Notes

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

## 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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