''Mathematics, Form and Function'', a book published in 1986 by

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, is a survey of the whole of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ''Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being'' (hereinafter ''WMCF'') is a book by George Lakoff, a cognitive linguistics, cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, ''WMCF' ...

''. 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 Lars Gårding (7 March 1919 – 7 July 2014) was a Swedish mathematician. He made notable contributions to the study of partial differential equations and partial differential operators. He was a professor of mathematics at Lund University in Swe ...

(1977).

Mac Lane's relevance to the philosophy of mathematics

Mac Lane cofounded

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

with Samuel Eilenberg, which enables a unified treatment of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding. Nevertheless, his views—however informal—are a valuable contribution to the

philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

and

anthropology Anthropology is the scientific study of humanity, concerned with human behavior, human biology, cultures, societies, and linguistics, in both the present and past, including past human species. Social anthropology studies patterns of behavi ...

of mathematics.On the anthropological grounding of mathematics, see White (1947) and Hersh (1997). His views anticipate, in some respects, the more detailed account of the cognitive basis of mathematics given by

George Lakoff George Philip Lakoff (; born May 24, 1941) is an American cognitive linguistics, cognitive linguist and philosopher, best known for his thesis that people's lives are significantly influenced by the conceptual metaphors they use to explain comple ...

and

Rafael E. Núñez Rafael E. Núñez is a professor of cognitive science at the University of California, San Diego and a proponent of embodied cognition. He co-authored '' Where Mathematics Comes From'' with George Lakoff. External links Academic home page* Rafa ...

in their ''

''. Lakoff and Núñez argue that mathematics emerges via

conceptual metaphor In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality (e.g. "the pr ...

s grounded in the

human body The human body is the structure of a Human, human being. It is composed of many different types of Cell (biology), cells that together create Tissue (biology), tissues and subsequently organ systems. They ensure homeostasis and the life, viabi ...

, its motion through

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...

and

time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...

, and in human sense perceptions.

Notes

References

*Gårding, Lars, 1977. ''Encounter with Mathematics''. Springer-Verlag. * Reuben Hersh, 1997. ''What Is Mathematics, Really?'' Oxford Univ. Press. *

and

, 2000. ''

''. 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 Leslie Alvin White (January 19, 1900, Salida, Colorado – March 31, 1975, Lone Pine, California) was an American anthropologist known for his advocacy of the theories on cultural evolution, sociocultural evolution, and especially neoevoluti ...

, 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. 1986 non-fiction books Mathematics books Philosophy of mathematics literature Cognitive science literature

Mathematics and human activities

Mac Lane's relevance to the philosophy of mathematics

See also

Notes

References