How We Understand Mathematics: Conceptual Integration in the Language of Mathematical Description

© Springer International Publishing AG, part of Springer Nature 2018

Jacek WoźnyHow We Understand MathematicsMathematics in Mindhttps://doi.org/10.1007/978-3-319-77688-0_1

1. Introduction

Jacek Woźny¹ 

(1)

Institute of English Studies, University of Wrocław, Otmuchów, Poland

1.1 The Effectiveness of Mathematics, Conceptual Integration, and Small Spatial Stories

On July 20, 1969, the lunar module of Apollo 11 landed on the moon. The trajectory of this historic space flight has been calculated by hand by a group of the so-called human computers.¹ It is just an example of the effectiveness of mathematics in modeling (and changing) the world around us. Mathematics continues to be productively applied in engineering, medicine, chemistry, biology, physics, social sciences, communication, and computer science, to name but a few. As Hohol (2011: 143) points out, this fact is often treated by philosophers as an argument for mathematical realism of the Platonian or Aristotelian variety. It is from this perspective that Quine-Putnam’s indispensability argument, Heller’s hypothesis of the mathematical rationality of the world, and Tegmark’s mathematical universe hypothesis have been discussed. Eugene Wigner, a physicist, often quoted in this context, finished his paper titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences in the following way:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. (1960: 14)

James C. Alexander, a professor of mathematics, also sees the unreasonable effectiveness of mathematics as of a mystery but offers the following explanation for it:

It is a mystery to be explored that mathematics, in one sense a formal game based on a sparse foundation, does not become barren, but is ever more fecund. I posit [...] that mathematics incorporates blending (and other cognitive processes) into its formal structure as a manifestation of human creativity melding into the disciplinary culture, and that features of blending, in particular emergent structure, are vital for the fecundity. (Alexander 2011: 3)

I agree with the above solution to the puzzle and have no doubt that it deserves further study. The subject of this book, further explained in the next section, is to prove that conceptual blending (integration), paired with the human ability for story (Turner 2005: 4), accounts for the effectiveness of mathematics. One could add, paraphrasing Wigner, that those two correlated mental features of the human mind make the effectiveness of mathematics reasonable. The conceptual blending theory mentioned by James Alexander in the above quotation is thus introduced by Evans and Green (2006):

Blending Theory was originally developed in order to account for linguistic structure and for the role of language in meaning construction, particularly ‘creative’ aspects of meaning construction like novel metaphors, counterfactuals and so on. However, recent research carried out by a large international community of academics with an interest in Blending Theory has given rise to the view that conceptual blending is central to human thought and imagination, and that evidence for this can be found not only in human language, but also in a wide range of other areas of human activity, such as art, religious thought and practice, and scientific endeavour, to name but a few. Blending Theory has been applied by researchers to phenomena from disciplines as diverse as literary studies, mathematics, music theory, religious studies, the study of the occult, linguistics, cognitive psychology, social psychology, anthropology, computer science and genetics. (401)

Over the last two decades, the importance of conceptual blending and other mental processes in mathematics has been extensively studied by, among others, Lakoff and Núñez (2000), Fauconnier and Turner (2002), Turner (2005), Núñez (2006), Alexander (2011), Turner (2012), and Danesi (2016). Let us just quote two little fragments, starting with the groundbreaking Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and Raphael Nunez.

Blends, metaphorical and nonmetaphorical, occur throughout mathematics. Many of the most important ideas in mathematics are metaphorical conceptual blends (2000: 48)

Mark Turner adds the concept of small spatial story as a vital component of conceptual blending in mathematics:

Our advanced abilities for mathematics are based in part on our prior cognitive ability for story [...] - understanding the world and our agency in it through certain kinds of human-scale conceptual organizations involving agents and actions in space. Another basic human cognitive operation that makes it possible for us to invent mathematical concepts [...] is conceptual integration, also called blending. Story and blending work as a team." (2005: 4)

Considering the already existing, impressive body of the literature on the subject of cognitive exploration of mathematics, we might question the point of adding yet another text to it; however, we have to bear in mind that mathematics is a vast discipline that has been evolving over millennia—there are still vast here be dragons areas on the map. All of the existing studies so far are case studies—usually focusing on a few selected mathematical concepts. For example, the foundational text by Lakoff and Nunez (2000) covers set theory, algebra, and various selected topics like infinity, complex numbers, and Euler’s equation. However, its coverage of algebra is about 10 pages long (110–119), and this is certainly not enough for one of the most important branches of mathematics. The other sources I mentioned above (Fauconnier and Turner 2002; Turner 2005; Núñez 2006; Alexander 2011; Turner 2012; Danesi 2016) are equally selective in their choice of mathematical topics. And this is why a more comprehensive approach, further described in the next section, is called for.

1.2 The Point and Method of the Book

I will prove that the construction of meaning in mathematics relies on the iterative use of basic mental operations of story and blending and demonstrate exactly how those two mental operations are responsible for the effectiveness and fecundity of mathematics. It will be done by analyzing the language, the primary notions, axioms, definitions, and proof in Herstein’s (1975) excellent Topics in Algebra—a classic handbook² addressed to the most gifted sophomores in mathematics at Cornell (8). Possible further effects of this study are making mathematics more accessible (easier to teach and learn) and perhaps demystifying mathematics as a product of the human mind rather than some eternal Platonic ideal.³

The research is systematic in two ways. Firstly, it covers all crucial areas of modern algebra, focusing on the fundamental notions such as set and element, mapping, group, binary operation, homomorphism, ring, and vector space. Secondly, it avoids what Stockwell (2002: 5) calls a trivial way of doing cognitive poetics—treating a literary (mathematical in our case) text only as a source of raw data to apply some acumen of cognitive psychology and cognitive linguistics. I don’t set aside impressionistic reading and imprecise intuition (ibid.). The book’s scrutiny of mathematical narrative is not limited to just spotting the mental patterns mentioned above but goes further to demonstrate how those universal patterns of the way we think influence our understanding of mathematics—the construction of mathematical meaning.

1.3 Who Is the Book Addressed To

The book is addressed to cognitive scientists, cognitive linguists, mathematicians, teachers of mathematics, and anybody interested in explaining the question of how mathematics works and why it works so well in modeling (what we perceive as) the world around us. I could not agree more with Rafael Nunez when he postulates that mathematics education should demystify truth, proof, definitions, and formalisms and that new generations of mathematics teachers, not only should have a good background in education, history, and philosophy, but they should also have some knowledge of cognitive science.⁴ Although our focus is academic-level mathematics, I have been trying not to befuddle the reader with too many advanced level formulas. The book, I very much hope, should be easy to follow by someone with no mathematical or cognitive science grounding. And the next chapter, in which the basic concepts are explained, is designed for that very purpose.

1.4 The Organization of the Book

After introducing our main research tools (basic human cognitive abilities) and presenting an overview of our research area (modern algebra) in the next chapter, we will follow the order of a typical university-level algebra course (in our case, Herstein 1975). We will start with analyzing the set theory and mappings (Chaps. 3 and 4, respectively)—considered to be the foundation of the whole edifice of modern mathematics—and continue along the path of increasing complexity to groups (Chap. 5), rings, fields, and vector spaces (Chap. 6). On each of those stages, we will take a close look at the primary concepts, axioms, definitions, and proof to see the telltale traces of the basic human cognitive patterns of story and conceptual blending.

Bibliography

Alexander, J. (2011). Blending in mathematics. Semiotica, Issue 187. Pages 1–48.Crossref

Bernays, P. (1935). Platonism in Mathematics. Lecture delivered June 18, 1934, in the cycle of Conferences internationales des Sciences mathematiques organized by the University of Geneva. Translated from French by C. D. Parsons. http://​www.​phil.​cmu.​edu/​projects/​bernays/​Pdf/​platonism.​pdf, accessed 2017-11-07.

Danesi, M. (2016). Language and Mathematics: An Interdisciplinary Guide. New York: Mouton de Gruyter.Crossref

Evans, V. & M. Green. (2006). Cognitive Linguistics: An Introduction. Edinburgh: Edinburgh University Press.

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.MATH

Hohol, M. (2011). Matematyczność ucieleśniona. In B. Brożek, J. Mączka, W.P. Grygiel, M. Hohol (eds.), Oblicza racjonalności: Wokół myśli Michała Hellera. Pages 143–166. Kraków: Copernicus Center Press.

Lakoff, G. & R. Núñez. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.MATH

Núñez, R. (2006). Do Real Numbers Really Move?. In R. Hersh (ed.), 18 Unconventional Essays on the Nature of Mathematics. Pages 160–181. New York: Springer.Crossref

Stockwell, P. (2002). Cognitive Poetics: An Introduction. London: Routledge.

Turner, M. (2005). Mathematics and Narrative. Paper presented at the International Conference on Mathematics and Narrative, Mykonos, Greece, 12-15 July 2005. http://​thalesandfriends​.​org/​wp-content/​uploads/​2012/​03/​turner_​paper.​pdf, accessed Nov. 11, 2016.

Turner, M. (2012). Mental Packing and Unpacking in Mathematics. In Mariana Bockarova, Marcel Danesi, and Rafael Núñez (eds.), Semiotic and Cognitive Science Articles on the Nature of Mathematics. Pages 248–267. Munich: Lincom Europa.

Wigner, E. (1960). The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications in Pure and Applied Mathematics, Issue 13(I). Pages 1–14.Crossref

Footnotes

1

Including an African-American NASA mathematician, Katherine G. Johnson, recently made famous by the highly acclaimed film Hidden Figures (2016).

2

Undergraduate modern algebra courses are sometimes referred to as Herstein-level courses.

3

The philosophical reflection on the ontological status of mathematical entities is beyond the scope of this book, but let us just point out that Platonic realism seems to prevail in this respect among mathematicians, Herstein included. The famous Swiss mathematician and philosopher, Paul Bernays (1935: 5), after analyzing the foundational contributions of Dedekind, Cantor, Frege, Poincare, and Hilbert, concluded, 40 years before the first edition of Herstein’s Topics in Algebra, that Platonism reigns today in mathematics.

4

http://​www.​cogsci.​ucsd.​edu/​~nunez/​web/​PME24_​Plenary.​pdf, accessed 12.12.2016.

© Springer International Publishing AG, part of Springer Nature 2018

Jacek WoźnyHow We Understand MathematicsMathematics in Mindhttps://doi.org/10.1007/978-3-319-77688-0_2

2. The Theoretical Framework and the Subject of Study

Jacek Woźny¹ 

(1)

Institute of English Studies, University of Wrocław, Otmuchów, Poland

2.1 Overview

The following sections will introduce the tools of study and the subject to be studied—mental operations of story and conceptual blending and modern algebra.

2.2 Language, Cognition, and Conceptual Integration

2.2.1 Cognitive Linguistics

Cognitive linguistics is a relatively modern discipline based on the assumption that language reflects patterns of human thought, perception, motor system, and bodily interactions with the environment. As Eve Sweetser concisely puts it, Linguistic system is inextricably interwoven with the rest of our physical and cognitive selves (1990: 6). Evans and Green (2006) describe the origin of cognitive linguistics in the following way:

Cognitive linguistics [...] originally emerged in the early 1970s out of dissatisfaction with formal approaches to language. Cognitive linguistics is also firmly rooted in the emergence of modern cognitive science in the 1960s and 1970s, particularly in work relating to human categorisation, and in earlier traditions such as Gestalt psychology. Early research was dominated in the 1970s and 1980s by a relatively small number of scholars. By the early 1990s, there was a growing proliferation of research in this area, and of researchers who identified themselves as ‘cognitive linguists’. In 1989/90, the International Cognitive Linguistics Society was established, together with the journal Cognitive Linguistics. In the words of the eminent cognitive linguist Ronald Langacker (1991: xv), this ‘marked the birth of cognitive linguistics as a broadly grounded, self conscious intellectual movement’. (3)

One of the reasons for the described above, rapid expansion of the discipline was the fact that language, fascinating as it is, does no longer have to be studied for its own sake.

An important reason behind why cognitive linguists study language stems from the assumption that language reflects patterns of thought. Therefore, to study language from this perspective is to study patterns