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Allen Chai Math 4388 Simon Stevin: Better mental representations of math reality lead to better understanding of math reality

Dutch Simon Stevin (1548-1620) is known primarily for two mathematical achievements. With his book De Thiende (“The Art of Tenths”), published in 1585, he became the primary contributor to modern notation (Katz, 2012). This is because modern decimal notation developed from his notation, which historians refer to as “Stevin numbers”. In fact, Stevin numbers can simply be seen as an unwieldly form of modern decimal notation.

For example, here is a random number in modern decimal notation: 18.347. Here is the same number in Stevin numbers: 18⓪3①4②7③. Each circled number represents that exponent of

1 1 1 1 1 . In other words, 18⓪3①4②7③ = (18 × ) + (3 × ) + (4 × ) + (7 × ). 10 100 101 102 103

Modern decimal notation simply takes Stevin numbers, introduces the decimal point, and does away with the circled numbers as superfluous. [It is likely who invented the decimal point, and who popularized its use (“Simon Stevin”, n.d.).]

Stevin’s second claim to mathematical fame was through the same book. He became the primary contributor to the concept of real numbers (Katz, 2012). He wrote passages such as: “there are no absurd, irrational, irregular, inexplicable or surd numbers” (Katz, 2012, p. 110), as well as, “It is a very common thing amongst authors of to treat numbers like √8 and similar ones, which they call absurd, irrational, irregular, inexplicable or surds etc. and which we deny to be the case for number [sic] which turns up” (Katz, 2012, p. 110-111).

Katz further quotes D. Fearnley-Sander with writing: “the modern concept of [...] was essentially achieved by Simon Stevin, around 1600, and was thoroughly assimilated into in the following two centuries” (2012, p. 111), and van der Waerden with writing:

“[Stevin’s] general notion of a real number was accepted, tacitly or explicitly, by all later scientists” (2012, p. 111).

I believe it is no coincidence that the progenitor of modern decimal notation and the progenitor of the concept of real numbers is the same person. Before Stevin, European were fully reliant on various forms of notation to represent , such as Egyptian fractions (Wildberger, 2015). This weakness in notation—in representation of reality— negatively affected our understanding of reality. Fraction notation could only represent rational and irrational numbers as segregated and irreconcilable with one another, as fraction notation could not represent irrational numbers. Hence, our understanding of reality was that rational and irrational numbers are segregated and irreconcilable with one another. Simon Stevin’s notation of Stevin numbers was an important innovation because it was an improved representation of reality (certainly improving our representation of irrational numbers). Finally, there was a notation that represented rational and irrational numbers as unified and harmonious. Hence, our understanding of reality changed, and now we understand rational and irrational numbers as equal citizens of the continuum of numbers that has come to be called the real numbers.

The story of Simon Stevin inventing the precursor form of modern decimal notation and forming the concept of real numbers in the late sixteenth century is one of how a better mental representation of reality (here, a better system of symbols, i.e., notation), resulted in a better understanding of reality (that rational and irrational numbers are unified and harmonious in their relationship, together forming the real numbers). Therefore, I would like to spend the rest of my paper introducing a better representation of reality (here, a taxonomic system), which I hope will bring me a better understanding of reality (how various mathematical objects and their properties relate to one another). I am organizing these objects and their properties under the common categories of math: for example, “” is the grouping of “geometric objects” such as circles and triangles, and their “properties”.

The creator of this taxonomic system is David Tall, Emeritus Professor in Mathematical

Thinking at the University of Warwick, and he has named it the “Three Worlds of Mathematics”.

I offer a succinct introduction based on his book How Humans Learn to Think Mathematically:

Exploring the Three Worlds of Mathematics.

Tall’s taxonomic system is structured on the premise that there are three broad categories of mathematical objects and associated properties. He uses the word embodiment to refer to the family of concrete mathematical objects. These are objects such as geometric shapes and space— anything that intrinsically has a physical referent(s). For example, a line segment, whether drawn on a board and consisting of atoms of chalk, or imagined in the mind and consisting of mental images, still has the intrinsic physical referent of length.

Tall uses the word symbolism to refer to the family of abstract mathematical objects—those objects with no intrinsic physical referent. For example, the object “three” has no length, color, position in space-time; it is devoid of any physical referent. With abstract mathematical objects,

Tall is careful to note that he uses the word “object” broadly. Objects may be what we conceptualize as “things”, as “nouns”, to be acted upon, such as a “number” or “sum”. However, what we conceptualize as “actions”, as “verbs”, performed on objects—mathematical operations/procedures/processes (choose your preferred vocabulary) such as “” or

“integration”—can also be thought of as objects, subject to other operations/procedures/processes. Because objects so easily and flexibly switch between being objects and processes, Hall often uses the term “procept” (Tall, 2013, p. 13). Tall uses the word formalism to refer to a third category of mathematical objects: mathematical proofs. These are based on the objects and properties found in the embodiment and symbolism areas of his taxonomic system. Tall is very strict about what constitutes a formal mathematical proof: All objects and properties from the embodiment and symbolism areas must be defined in terms of sets (i.e., set theory). Axioms must also be defined in terms of sets. The proof must then explicitly show relationships between objects and objects, properties and properties, or objects and properties (all defined in terms of sets), via deduction.

Tall’s complete taxonomic system, in my own words, looks like the following (meant to be read from the bottom up):

Three Worlds of Mathematics Taxonomy

Formal math Proofs of all the below objects and properties relying on axioms, set theory, and deduction. Blended (embodied and Blended (symbolic and formal) formal) Geometric proofs of the Algebraic proofs of the below below objects and properties objects and properties that that rely solely on compass rely solely on and and straightedge. . Embodied math Blended (embodied and Symbolic math Geometry symbolic) Calculus (calculus can be (Objects and their properties) The invention of the seen as algebra concerned Cartesian coordinate system with procepts found beyond This is the geometry of for Euclidean space was a traditional algebra, such as Euclid, devoid of algebra. revolutionary innovation that limits, differentiation, and married geometry and algebra integration) (“Cartesian coordinate Algebra system”, n.d.). Arithmetic (For all three above, objects This section is for all the and their properties) graphs possible when you marry the objects of geometry with the objects of algebra.

Figure 1: Graphic depicting Professor David Tall’s taxonomic system for organizing

mathematical objects and their properties

In conclusion, I want to remark that has been somewhat stressful for me.

Every time we move forward in the historical record toward the present, we are introduced to another field of math (another field of mathematical objects and properties): arithmetic, geometry, calculus, formal mathematics concerned with deductive proof. I never learned their relationships in the actual courses bearing those titles, and I know I have to place them into some mental organizational framework to remove the stress of not being sure of their connections.

Simon Stevin altering his mental representation of reality to better understand reality is what I am attempting to do by studying Tall’s taxonomy of all mathematical objects and properties.

Katz remarked on Tall’s taxonomy, and I looked Tall up. I intend to use this taxonomy to see the relationships amongst math fields (associated objects and properties), unless/until I discover a better one.

References

Cartesian coordinate system. (n.d.) In Wikipedia. Retrieved November 1, 2016, from https://en.wikipedia.org/wiki/Cartesian_coordinate_system

Katz, K.U., & Katz, M.G. (2012). Stevin numbers and reality. Foundations of , 17(2),

109-123.

Simon Stevin. (n.d.) In Wikipedia. Retrieved November 1, 2016, from https://en.wikipedia.org/wiki/Simon_Stevin

Tall, D.O. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge: Cambridge University Press.

Wildberger, N.J. [njwildberger]. (2015, March 8). MathHistory24: Number systems and Stevin's . [Video File]. Retrieved from https://www.youtube.com/watch?v=erFVGPqpRxQ.

List of Figures

Figure 1: Graphic depicting Professor David Tall’s taxonomic system for organizing mathematical objects and their properties