• Paraphrasing the Sufi master Nasreddin: It is important to see where you’ve been to gain perspective on where you are going.
• The Author’s burning desire is for his beloved Living Algorithm to be accepted into the pantheon of Mathematics. Unfortunately, the followers of Mathematics believe her to be one rather than many. In other words they believe that to open the door to the world of Mathematics to allow to enter in
How the Embodied Mind brings Mathematics into Being is the subtitle of Drs. Lakoff and Nunez’s marvelous book. This brief statement is a marvelous articulation of the theme of this work. Let’s deconstruct this short phrase to provide context.
The Embodied Mind is the subject of the sentence, the entity that does the ‘bringing’. ‘A Mind that is Embodied’ is a rephrasing of the two words. This is a direct statement of a modern axiom of cognitive science. Via cross-disciplinary studies cognitive scientists have established beyond a reasonable doubt that abstract thought originates in the neural networks of the brain. Hence our thinking mind is embodied in the relationships between our neurons.
The authors’ momentous book, Where Mathematics Comes From, was written to provide yet more evidence for this well-established theory. To accomplish this task they started from the ground up. If abstract thought originates in the mind, what are the innate structures of the brain that provide the foundation? What cognitive structures are we born with? Understanding these innate structures is crucial for our mathematical deconstruction.
Ingenious experiments provide evidence that humans babies as young as a few days old already exhibit an innate mathematical talent – the ability to subitize. This means that they can differentiate the numbers of objects up to three. Further, babies can add and subtract these numbers. Music provides a great example of how this innate ability influences our symbolic expression. The rhythms of Western music are based almost entirely upon combinations of twos and threes. Even Dave Brubeck’s Take Five, a famous jazz piece that features a five count rhythm, is based upon an alternation of 2 and 3, not a pure 5.
Our innate ability to subitize the world provides one of the mathematical foundations of this discussion. The authors carefully demonstrated that our entire range of arithmetic skills is based, in part, in this innate ability. By blending just one conceptual metaphor with our ability to subitize, they were able to develop the full range of the basic arithmetic operations: addition, subtraction, multiplication, division, and even a limited form of exponentiation. As such, our ability to subitize is associated with the operations of basic arithmetic.
Subitization has to do with whole numbers only. As such in basic arithmetic, these operations are performed upon whole numbers. Division of one whole number by another whole number yields the fractions, the rational numbers. Division by whole numbers provides approximations of irrational and transcendental numbers, but nothing exact. While the entire field of arithmetic deals with the positive real numbers. Basic arithmetic, as we define it, only employs whole numbers.
This basic arithmetical perspective based upon our ability to subitize eventually rose to conquer the scientific-mathematical world. As a testimony to the power of this perspective, witness the wide spread global influence of our binary-based digital computers. All operations become computer code written in 1s and 0s, the most fundamental of whole numbers. These binary operations determine the interactions between other 1’s and 0s. Number, as in the whole numbers of subitization, is the dominant feature of this arithmetical perspective.
The obsession with Number, as in counting numbers, has a few distinct features associated with it. 1) Counting numbers are exact. The number of people in a room is a precise number, no ambiguity. In contrast, the numbers associated with measurements are inherently ambiguous. Acknowledging this basic fact, Engineers provide tolerance limits on their specifications. Counting numbers have none of the ambiguity associated with measurements.
One of the advantages of this exactitude is absolute precision. Computer scientists have employed the abstract precision of counting numbers to provide us with the marvels of Internet. The exact replication that we demand from our computer technology is based in the absolute precision of counting numbers. Mathematicians at the end of the 19th century linked the absolute precision of these subtizable counting numbers with scientific rigor.
As a reminder: computers only employ the counting numbers in their operations. As evidence, witness the computer keyboard. Further, these counting numbers consist of electronic collections of 1s and 0s only. As such, computer code employs the basic math operations on counting numbers to approximate all the complex operations that include negative and irrational results. The computer code is turned into computer applications that allow us to easily employ irrational and negative numbers along with a wide variety of complex operations.
• 2) Discretization is a second feature of the counting numbers associated with subitization. Counting numbers are discrete. They are not continuous. As such, there is a notable lack of connectivity between counting numbers. Counting numbers can only be employed to approximate continuity. Digital CDs and analog records exhibit the innate difference between counting numbers and continuity.
Although similar, we count with ordinal numbers and order with cardinal numbers, such as 1st, 2nd, 3rd. The ability to count can be employed to approximate the continuity of space. In similar fashion, the ability to order numbers can be employed to approximate the continuity of time.
Just as counting numbers are discrete, the members of a set are also discrete. Just as we count things, we can also count the members of a set and get an exact determination. We can also order a set with numbers. As such, traditional set theory can encompass continuous space and time, however as approximations. Check out the music from your CDs and the movies on your DVDs to see how precise these approximations are. Although these imbedded electronic bytes create the illusion of motion through space and time, they are isolated members of a set, with no connectivity whatsoever.
Discretization is linked our ability to subitize. It is also linked with the absolute precision of scientific rigor. This attitude towards rigor was a factor in the development of traditional set theory. In this sense, we could say that our innate ability to subitize the world led to an entire branch of mathematics based upon Number, as in counting number.
Now The Importance of Deconstructing the Real Number Line.