Abstract
We will begin this chapter with examining the primitive (undefined) notions of “a set” and “an element” and then investigate the basic set-theoretical concepts of subsets, equality of sets, the null set, the union, and intersection of sets. In the final section, we will take a closer look at the language of mathematical proof. At every stage of our close reading of the mathematical narrative, we will be looking for the mental patterns like image schemas (e.g., the container image schema), small spatial stories (actors moving in space, manipulating objects), and conceptual integration.
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- 1.
The canonical today, axiomatic set theory called ZFC, does not include the definition of a set either. The set remains a primary, undefined notion there as well. We should also mention that many important mathematical theorems (e.g., the continuum hypothesis, Suslin hypothesis, diamond principle) were proven to be “independent” of ZFC, which means they can neither be proved nor disproved within this framework. Which of course is one of the reasons some mathematicians contest the claim of the fundamental role of the set theory in modern mathematics.
- 2.
The British National Corpus, version 3 (BNC XML Edition), 2007. Distributed by Bodleian Libraries, University of Oxford, on behalf of the BNC Consortium. URL: http://www.natcorp.ox.ac.uk/, accessed 2017-10-10.
- 3.
A Polish mathematician, Stanisław Leśniewski, was so taken aback by the unintuitive nature of the naive (and ZFC) set theory that he decided to create his own theory, known today as mereology (the study of parts). In his theory, (created c.a. 1914) element and subset are one and the same, which is also an elegant way of avoiding the famous Russell’s paradox. (cf. Sect. 3.8).
- 4.
We will return to the subject in Chap. 5, when we discuss the uniqueness of the identity element in a group.
- 5.
http://www.imdb.com/title/tt3636060/, accessed 2017-10-10.
- 6.
In fact, conceptual blending is always mostly unconscious—“These operations (conceptual blending, JW) -basic, mysterious, powerful, complex, and mostly unconscious-are at the heart of even the simplest possible meaning” (Fauconnier and Turner 2002: 6).
- 7.
Leśniewski anticipates here, for example, the famous Banach-Tarski paradox 11 years before it was discovered (in 1924).
- 8.
Mark Turner (1996: 9) calls the above “mental patterns of parable.”
Bibliography
Fauconnier, M. & M. Turner. (2002). The Way We Think: Conceptual Blending And The Mind’s Hidden Complexities. New York: Basic Books.
Herstein, I. (1975). Topics in Algebra. New York: John Wiley & Sons.
Lakoff, G. & R. Núñez. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
Leśniewski, S. (1913). “Krytyka filozoficznej zasady wyłączonego środka”. Przegląd Filozoficzny, Issue 16. Pages 315–352.
Turner, M. (1996). The Literary Mind. Oxford & New York: Oxford University Press.
Leśniewski, S. (1930). "O podstawach matematyki". Przegląd Filozoficzny, Issue 30. Pages 165-206.
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Woźny, J. (2018). Sets. In: How We Understand Mathematics. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-319-77688-0_3
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