• 107328  Infos
Mathematics Form and Function is an excellent and accessible survey of the whole of mathematics including its origins and deep structure Its author Saunders Mac Lane co-founded the field of category theory which is used to study mathematical structures and relationships between them

See also

  • From Action to Mathematics per Mac Lane

References


Throughout his Mathematics Form and Function and especially in chapter I11 the American mathematician Saunders Mac Lane informally discusses how mathematics is grounded in more ordinary concrete and abstract human activities This entry From Action to Mathematics per Mac Lane sets out a summary of his views on the human grounding of mathematics
Ironically Mac Lane is noted for developing category theory which enables a far-reaching unified treatment of mathematical structures and relationships between them at the cost of breaking away from their cognitive grounding His views however informal are a contribution to the philosophy and anthropology of mathematicsOn mathematics and anthropology see White (1947) and Hersh (1997) which anticipates 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 Where Mathematics Comes From Lakoff and Núñez (2000) argue that mathematics emerges via conceptual metaphors grounded in the human body its motion through space and time and in human sense perceptions
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
CollectingCollectionSet; class; multiset; list; family
ConnectingCause and effectordered pair; relation; function; operation
"Proximity; connectionTopological space; mereotopology
FollowingSuccessive actionsFunction composition; transformation group
ComparingEnumerationBijection; cardinal number; order
TimingBefore & AfterLinear order
CountingSuccessorSuccessor function; ordinal number
ComputingOperations on numbersAddition recursively defined]; abelian group; rings
Looking at objectsSymmetrySymmetry group; invariance; isometries
Building; shapingShape; pointSets of points; geometry; pi
RearrangingPermutationBijection; permutation group
Selecting; distinguishingParthoodSubset; order; lattice theory; mereology
ArguingProofFirst-order logic
MeasuringDistance; extentRational number; metric space
Endless repetitionInfinity;Also see the Basic Metaphor of Infinity of Lakoff and Núñez (2000) chpt 8 RecursionRecursive set; Infinite set
EstimatingApproximationReal number; real field
Moving through space & time:
--Without cyclingChangeReal analysis; transformation group
--With cyclingRepetitionpi; trigonometry; complex number; complex analysis
--BothDifferential equations; mathematical physics
Motion through time aloneGrowth & decaye; exponential function; natural logarithms
Altering shapesDeformationDifferential geometry
Observing patternsAbstractionAxiomatic set theory; universal algebra; category theory; morphism
Seeking to do betterOptimizationOperations research; optimal control theory; dynamic programming
Choosing; gamblingChanceProbability theory; mathematical statistics

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 Gärding (1977)

Footnotes

See also


References

  • Gärding 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
  • Saunders Mac Lane 1986 Mathematics: Form and Function Springer Verlag
  • Leslie White 1947 "The Locus of Mathematical : 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