- From Action to Mathematics per Mac Lane

Throughout his

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

Human Activity | Related Mathematical Idea | Mathematical Technique |

Collecting | 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 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;Also see the Basic Metaphor of Infinity of Lakoff and Núñez (2000) chpt 8 Recursion | Recursive set; Infinite set |

Estimating | Approximation | Real number; real field |

Moving through space & time: | ||

--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 |

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 |

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)

- Conceptual metaphor
- Cognitive science
- Cognitive science of mathematics
- Embodied philosophy
- Foundations of mathematics
- Saunders Mac Lane
- Philosophy of mathematics
*Where Mathematics Comes From*

- 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