Mathematics, Form and Function: Difference between revisions

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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]].

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

Mac Lane founded (with [[Samuel Eilenberg]]) [[category theory]], 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&mdash;however informal&mdash;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.

== 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. &nbsp;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;"

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|-

|Collecting

|[[CollectionObject (mathematics)|Collection]]

|[[Set (mathematics)|Set]]; [[class (set theory)|class]]; [[multiset]]; [[list (computing)|list]]; [[Familyindexed (disambiguation)#Mathematicsfamily|family]]

|-

|Connecting

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|-

| "

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

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

|-

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|-

|[[Counting]]

|[[Successor ordinal|Successor]]

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

|-

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|-

|Building; shaping

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

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

|-

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|Estimating

|[[Approximation]]

|[[Real number]]; [[real closed field|real field]]

|-

|Moving through [[space]] & [[time]]:

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|Choosing; [[gambling]]

|[[Probability|Chance]]

|[[Probability theory]]; [[mathematical statistics]]; [[Measure_Measure (mathematics)|measure]]

|}

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Mac Lane (1986) cites a related monograph by [[Lars Gårding]] (1977).

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

Mac Lane founded (withcofounded [[Samuelcategory Eilenbergtheory]]) with [[categorySamuel theoryEilenberg]], 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&mdash;however informal&mdash;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==

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* [[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: 304–19.

[[Category:1986 non-fiction books]]

[[Category:Mathematics books]]

[[Category:Philosophy of mathematics literature]]

[[Category:Cognitive science literature]]