Mathematics, Form and Function: Difference between revisions

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'''''Mathematics, Form and Function''''' 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 ==

== Relevance to psychology of mathematical conception ==

Mac Lane isfounded noted(with for[[Samuel co-founding the field ofEilenberg]]) [[category theory]], 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 views anticipatesanticipate, 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.

== 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. ThisThe sectionfollowing setstable outis aadapted summaryfrom one given on p. 35 of hisMac viewsLane on(1986). theThe humanrows groundingare 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 mathematics''[[Where Mathematics Comes From]]''.

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 72 ⟶ 70:

|-

|Endless repetition

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

|[[Recursive set]]; [[Infinite set]]

|-