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'''''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]]''.
{| class="wikitable" style="text-align: center;"
|-
|'''Human Activity'''
|'''Related Mathematical Idea'''
|'''Mathematical Technique'''
|-
|Collecting
|Object Collection
|[[Set (mathematics)|Set]]; [[class (set theory)|class]]; [[multiset]]; [[list (computing)|list]]; [[indexed family|family]]
|-
|Connecting
|[[causality|Cause and effect]]
|[[ordered pair]]; [[relation (mathematics)|relation]]; [[function (set theory)|function]]; [[operation (mathematics)|operation]]
|-
| "
|[[Proximity space|Proximity]]; [[connectedness|connection]]
|[[Topological space]]; [[mereotopology]]
|-
|Following
|Successive actions
|[[Function composition]]; [[transformation group]]
|-
|Comparing
|Enumeration
|[[Bijection]]; [[cardinal number]]; [[order theory|order]]
|-
|Timing
|Before & After
|[[Linear order]]
|-
|[[Counting]]
|[[Successor ordinal|Successor]]
|[[Successor function]]; [[ordinal number]]
|-
|Computing
|[[operation (mathematics)|Operations]] on [[number]]s
|[[Peano axioms|Addition, multiplication recursively defined]]; [[abelian group]]; [[ring (mathematics)|rings]]
|-
|Looking at objects
|[[Symmetry]]
|[[Symmetry group]]; [[invariant (mathematics)|invariance]]; [[isometries]]
|-
|Building; shaping
|[[Shape]]; [[point (geometry)|point]]
|[[Set (mathematics)|Sets]] of [[point (geometry)|points]]; [[geometry]]; [[pi]]
|-
|Rearranging
|[[Permutation]]
|[[Bijection]]; [[permutation group]]
|-
|Selecting; distinguishing
|[[mereology|Parthood]]
|[[Subset]]; [[order theory|order]]; [[lattice (order)|lattice theory]]; [[mereology]]
|-
|[[argumentation|Arguing]]
|[[Mathematical proof|Proof]]
|[[First-order logic]]
|-
|[[measurement|Measuring]]
|[[Distance]]; extent
|[[Rational number]]; [[metric space]]
|-
|Endless repetition
|[[Infinity]];<ref>Also see the "Basic [[Metaphor]] of [[Infinity]]" in Lakoff and Núñez (2000), chpt. 8.</ref> [[Recursion]]
|[[Recursive set]]; [[Infinite set]]
|-
|Estimating
|[[Approximation]]
|[[Real number]]; [[real closed field|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 (mathematical constant)|e]]; [[exponential function]]; [[natural logarithms]];
|-
|Altering shapes
|[[deformation (engineering)|Deformation]]
|[[Differential geometry]]; [[topology]]
|-
|Observing patterns
|[[Abstraction]]
|[[Axiomatic set theory]]; [[universal algebra]]; [[category theory]]; [[morphism]]
|-
|Seeking to do better
|[[Optimization (mathematics)|Optimization]]
|[[Operations research]]; [[optimal control theory]]; [[dynamic programming]]
|-
|Choosing; [[gambling]]
|[[Probability|Chance]]
|[[Probability theory]]; [[mathematical statistics]]; [[Measure (mathematics)|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 [[unifying theories in mathematics|unified treatment]] of mathematical structures and of the relations among them, at the cost of [[abstract nonsense|breaking away from their cognitive grounding]]. Nevertheless, his views—however informal—are a valuable contribution to the [[philosophy of mathematics|philosophy]] and [[anthropology]] of mathematics.<ref>On the anthropological grounding of mathematics, see White (1947) and Hersh (1997).</ref> His views anticipate, in some respects, the more detailed account of the [[cognitive science of mathematics|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 metaphor]]s grounded in the [[embodied philosophy|human body]], its motion through [[space]] and [[time]], and in human sense perceptions.
== See also ==
* [[1986 in philosophy]]
==Notes==
<references />
==References==
*Gårding,
* [[Reuben Hersh]], 1997. ''What Is Mathematics, Really?'' Oxford Univ. Press.
*[[George Lakoff]] and [[Rafael E. Núñez]], 2000. ''[[Where Mathematics Comes From]]''. Basic Books.
* {{cite book |first=Saunders |last=Mac Lane |title=Mathematics, Form and Function |year=1986 |publisher=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.
[[Category:1986 non-fiction books]]
[[Category:Mathematics books]]
[[Category:
[[Category:Cognitive science literature]]
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