Mathematics, Form and Function: Difference between revisions

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[[Category:Mathematics books]]

[[Category:Mathematics literature]]

{{mergeto|Mathematics, Form and Function|date=July 2008}}

Throughout his ''[[Mathematics, Form and Function]]'', and especially in chapter I.11, the [[United States|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, [[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]]. His views, however informal, are a 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;"

|-

|'''Human Activity'''

|'''Related Mathematical Idea'''

|'''Mathematical Technique'''

|-

|Collecting

|[[Collection (mathematics)|Collection]]

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

|-

|Connecting

|[[causality|Cause and effect]]

|[[ordered pair]]; [[relation (mathematics)|relation]]; [[function (set theory)|function]]; [[operation (mathematics)|operation]]

|-

| "

|[[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]]

|[[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]]s of [[point (geometry)|points]]; [[geometry]]; [[pi]]

|-

|Rearranging

|[[Permutation]]

|[[Bijection]]; [[permutation group]]

|-

|Selecting; distinguishing

|[[mereology|Parthood]]

|[[Subset]]; [[order theory|order]]; [[lattice (order theory)|lattice theory]]; [[mereology]]

|-

|[[argumentation|Arguing]]

|[[Proof]]

|[[First-order logic]]

|-

|[[measurement|Measuring]]

|[[Distance]]; extent

|[[Rational number]]; [[metric space]]

|-

|Endless repetition

|[[Infinity]];<ref>Also see the Basic [[Metaphor]] of [[Infinity]] of Lakoff and Núñez (2000), chpt. 8.</ref> [[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 (mathematical constant)|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).

== Footnotes ==

<references />

==See also==

* [[Conceptual metaphor]]

* [[Cognitive science]]

* [[Cognitive science of mathematics]]

* [[Embodied philosophy]]

* [[Foundations of mathematics]]

* [[Saunders Mac Lane]]

* [[Philosophy of mathematics]]

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

==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 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:Philosophy of mathematics]]