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 ActivityRelated Mathematical IdeaMathematical TechniqueCollectingObject CollectionSet; class; multiset; list; familyConnectingCause and effectordered pair; relation; function; operation Proximity; connectionTopological space; mereotopologyFollowingSuccessive actionsFunction composition; transformation groupComparingEnumerationBijection; cardinal number; orderTimingBefore & AfterLinear orderCountingSuccessorSuccessor function; ordinal numberComputingOperations on numbersAddition, multiplication recursively defined; abelian group; ringsLooking at objectsSymmetrySymmetry group; invariance; isometriesBuilding; shapingShape; pointSets of points; geometry; piRearrangingPermutationBijection; permutation groupSelecting; distinguishingParthoodSubset; order; lattice theory; mereologyArguingProofFirst-order logicMeasuringDistance; extentRational number; metric spaceEndless repetitionInfinity;Also see the "Basic Metaphor of Infinity" in Lakoff and Núñez (2000), chpt. 8. RecursionRecursive set; Infinite setEstimatingApproximationReal number; real fieldMoving through space & time:curvaturecalculus; differential geometry --Without cyclingChangeReal analysis; transformation group --With cyclingRepetitionpi; trigonometry; complex number; complex analysis --Both Differential equations; mathematical physicsMotion through time aloneGrowth & decay e; exponential function; natural logarithms; Altering shapesDeformationDifferential geometry; topologyObserving patternsAbstractionAxiomatic set theory; universal algebra; category theory; morphismSeeking to do betterOptimizationOperations research; optimal control theory; dynamic programmingChoosing; gamblingChanceProbability 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.On the anthropological grounding of mathematics, see White (1947) and Hersh (1997). 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.

Mathematics, Form and Function

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