Mathematics, Form and Function explained

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 Technique
CollectingObject CollectionSet
class; multiset; list; family
ConnectingCause and effectordered pair
"Proximity
connection
Topological space
mereotopology
FollowingSuccessive actionsFunction composition
transformation group
ComparingEnumerationBijection
cardinal number; order
TimingBefore & AfterLinear order
CountingSuccessorSuccessor function
ordinal number
ComputingOperations on numbersAddition, multiplication recursively defined
abelian group; rings
Looking at objectsSymmetrySymmetry group
invariance; isometries
Building; shapingShape
point
Sets of points; geometry; pi
RearrangingPermutationBijection
permutation group
Selecting; distinguishingParthoodSubset
order; lattice theory; mereology
ArguingProofFirst-order logic
MeasuringDistance
extent
Rational number
metric space
Endless repetitionInfinity
[1] Recursion
Recursive set
Infinite set
EstimatingApproximationReal number
real field
Moving through space & time:curvaturecalculus
differential geometry
--Without cyclingChangeReal analysis
--With cyclingRepetitionpi
trigonometry; complex number; complex analysis
--BothDifferential equations
mathematical physics
Motion through time aloneGrowth & decay e
exponential function; natural logarithms;
Altering shapesDeformationDifferential geometry
topology
Observing patternsAbstractionAxiomatic set theory
universal algebra; category theory; morphism
Seeking to do betterOptimizationOperations research
optimal control theory; dynamic programming
Choosing; 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.[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

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