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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 [[United States|American]] mathematician [[Saunders Mac Lane]].

== Mathematics and human activities==

== Relevance to cognitive science of mathematics ==

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]]''.

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. This section sets out a summary of his views on the human grounding of mathematics.

Mac Lane is noted for co-founding the field of [[category theory]], which enables a far-reaching, [[unifying theories in mathematics|unified treatment]] of mathematical structures and relationships between them at the cost of [[abstract nonsense|breaking away from their cognitive grounding]]. Nonetheless, his views&mdash;however informal&mdash;are a valuable contribution to the [[philosophy of mathematics|philosophy]] and [[anthropology]] of mathematics<ref>On mathematics and anthropology, see White (1947) and Hersh (1997).</ref> which anticipates, in some respects, the much richer and more detailed account of the [[cognitive science of mathematics|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 metaphor]]s grounded in the [[embodied philosophy|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]]''.

{| class="wikitable" style="text-align: center;"

Line 16 ⟶ 12:

|-

|Collecting

|[[CollectionObject (mathematics)|Collection]]

|[[Set (mathematics)|Set]]; [[class (set theory)|class]]; [[multiset]]; [[list (computing)|list]]; [[familyindexed (mathematics)family|family]]

|-

|Connecting

Line 24 ⟶ 20:

|-

| "

|[[Proximity space|Proximity]]; [[connectedness|connection]]

|[[Topological space]]; [[mereotopology]]

|-

Line 40 ⟶ 36:

|-

|[[Counting]]

|[[Successor ordinal|Successor]]

|[[Successor function]]; [[ordinal number]]

|-

Line 52 ⟶ 48:

|-

|Building; shaping

|[[Shape]]; [[point (geometry)| point]]

|[[Set (mathematics)|Sets]] of [[point (geometry)|points]]; [[geometry]]; [[pi]]

|-

Line 64 ⟶ 60:

|-

|[[argumentation|Arguing]]

|[[Mathematical proof|Proof]]

|[[First-order logic]]

|-

Line 72 ⟶ 68:

|-

|Endless repetition

|[[Infinity]];<ref>Also see the "Basic [[Metaphor]] of [[Infinity]]" ofin 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

Line 96 ⟶ 93:

|Motion through time alone

|Growth & decay

|[[e (mathematical constant)|e]]; [[exponential function]]; [[natural logarithms]];

|-

|Altering shapes

|[[deformation (engineering)|Deformation]]

|[[Differential geometry]]; [[topology]]

|-

|Observing patterns

Line 111 ⟶ 108:

|-

|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 Gärding[[Lars Gårding]] (1977).

== Mac Lane's relevance to the philosophy of mathematics ==

==See also==

Mac Lane iscofounded noted[[category fortheory]] co-founding the field ofwith [[categorySamuel theoryEilenberg]], which enables a far-reaching, [[unifying theories in mathematics|unified treatment]] of mathematical structures and relationshipsof betweenthe relations among them, at the cost of [[abstract nonsense|breaking away from their cognitive grounding]]. NonethelessNevertheless, his views&mdash;however informal&mdash;are a valuable contribution to the [[philosophy of mathematics|philosophy]] and [[anthropology]] of mathematics.<ref>On mathematicsthe andanthropological anthropologygrounding of mathematics, see White (1947) and Hersh (1997).</ref> whichHis anticipatesviews anticipate, in some respects, the much richer and 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 (2000) 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.

* [[Conceptual metaphor]]

* [[Cognitive science]]

== See also ==

* [[Cognitive science of mathematics]]

* [[Embodied1986 in philosophy]]

* [[Foundations of mathematics]]

* [[Saunders Mac Lane]]

* [[Philosophy of mathematics]]

*''[[Where Mathematics Comes From]]''

==Notes==

<references />

==References==

*GärdingGårding, 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.

* {{cite book |first=Saunders |last=Mac Lane |title=Mathematics, Form and Function |year=1986 |publisher=Springer-Verlag |idisbn=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: 304-19304–19.

[[Category:1986 non-fiction books]]

[[Category:Mathematics books]]

[[Category:Philosophy of mathematics literature]]

* [[Category:Cognitive science literature]]