Abstract
In this last stage of our exploration of mathematics, we will analyze three more algebraic structures of increasing complexity—rings, fields, and vector spaces. Herstein begins his chapter on rings in the following way:
As we indicated in Chapter 2, there are certain algebraic systems which serve as the building blocks for the structures comprising the subject which is today called modern algebra. At this stage of the development we have learned something about one of these, namely groups. It is our purpose now to introduce and to study a second such, namely rings. The abstract concept of a group has its origins in the set of mappings, or permutations, of a set onto itself. In contrast, rings stem from another and more familiar source, the set of integers. We shall see that they are patterned after, and are generalizations of, the algebraic aspects of the ordinary integers. (1975: 120)
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Notes
- 1.
One could think that this is where mathematics is weak and fuzzy, a sort of Achilles heel of mathematics. Conversely, I think this is exactly where the strength of mathematics lies. The undefined primitives give mathematics (via small spatial stories and blending) its amazing flexibility and effectiveness. Just as the fuzziness of human categories makes them so much more effective and energy-efficient in communication than the Aristotelian categories (cf. Rosch 1978).
- 2.
A task for the reader: prove that those are the only integer solutions of ab = a + b.
- 3.
Well, the refund will probably not be granted because (2,2) and (0,0) are not “real couples” but just ordered pairs of one unique element (cf. our remarks on uniqueness in Chaps. 3 and 5). “I am really sorry, Sir, but Wholesome, Threesome & co. is an agency for couples, not just one person pretending to be a couple.” On the other hand, 0 may get this refund anyway. It pretended to be a couple of (0,0), true, but was matched with itself by both agencies. Not fair really, because the “fake couple” of (2,2), for example, was matched with a delightful and quite separate character called “4.”
- 4.
- 5.
Republic Act 6955 prohibits the business of organizing or facilitating marriages between Filipinas and foreign men (https://en.wikipedia.org/wiki/Online_dating_service, accessed 2016-12-28).
- 6.
The so-called Lucasian Chair of Mathematics is considered to be one of the most prestigious professorships in the world. Its occupiers over the years were, among others, Sir Isaac Newton, Paul Dirac, and—most recently—Stephen Hawking.
- 7.
A task for the reader: prove that there are 1 × 2 × 3 = 6 possible “brick-changing-places moves.”
- 8.
The full title is of course Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being.
- 9.
http://www.pdst.ie/sites/default/files/Mental%20Maths%20Workshop%201%20Handbook.pdf, accessed 2017-01-03.
- 10.
https://www.nctm.org/Grants-and-Awards/Supporters/John-A_-Van-de-Walle-Biography/, accessed 2017-01-04.
- 11.
Herstein seems to be aware of this disparity when he says, for example, “Very early in our mathematical education—in fact in junior high school or early in high school itself—we are introduced to polynomials. For a seemingly endless amount of time we are drilled, to the point of utter boredom, in factoring them, multiplying them, dividing them, simplifying them. Facility in factoring a quadratic becomes confused with genuine mathematical talent” (Herstein 1975: 153).
- 12.
Emphasis added
- 13.
The definition of a module is almost identical with the definition of a vector space, except for axiom 4, which is absent because a ring, as we remember, does not have to contain a “1” (a multiplication identity element).
- 14.
In the schema of adding objects to a collection or putting objects in a container.
Bibliography
Herstein, I. (1975). Topics in Algebra. New York: John Wiley & Sons.
Johnson, M. (1987). The Body in the Mind. Chicago: University of Chicago Press.
Lakoff, G. & M. Turner. (1989). More Than Cool Reason: A Field Guide to Poetic Metaphor. Chicago: University of Chicago Press.
Lakoff, G. & R. Núñez. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
Rosch, E. H. (1978). “Principles of categorization”. In: E. Rosch & B. Lloyd (eds.), Cognition and Categorization. Pages 27–48. Hillsdale, N.J.: Erlbaum Associates.
Talmy, L. (2000). Toward a Cognitive Semantics. Cambridge: The MIT Press.
Turner, M. (1996). The Literary Mind. Oxford & New York: Oxford University Press.
Van de Walle, J. (2007). Elementary and Middle School Mathematics Teaching Developmentally. Boston: Allyn and Bacon (Pearson).
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Woźny, J. (2018). Rings, Fields, and Vector Spaces. In: How We Understand Mathematics. Mathematics in Mind. Springer, Cham. https://doi.org/10.1007/978-3-319-77688-0_6
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