Epistemology and Mathematics

Epistemology in Mathematics, as Cole highlighted, is an intricate philosophical domain that I’ve contemplated extensively over my six decades of life. Mathematics examines patterns and relationships, formulating models based on axioms and deriving implications through logical reasoning. While mathematics is fundamentally a form of applied logic, it also requires significant creativity in creating axiom systems and investigating their properties. It serves as a method for systematically inferring conclusions and exploring the emergent characteristics of established relationships among objects.

Historically, mathematics has been driven by practical motivations. Axiomatic systems were initially conceived as representations of real-world phenomena. Although this was not the original intent behind imaginary numbers, there exists a tangible situation whose geometry closely aligns with the fundamental nature of the imaginary unit i.

As the 20th century progressed, mathematics evolved to be more abstract, having exhausted simpler patterns. This shift was influenced by thinkers like David Hilbert and, most notably, Amelie Noether, who tirelessly pursued her principle of "Begriffliche Mathematik," or conceptual mathematics. In Noether's perspective, this meant rigorously stripping problems of any extraneous elements, honing in on their purest essence, and exploring the most general and abstract versions to their fullest potential. Her influence has left a profound mark on the modern mathematical rigor we adhere to today.

Real-Valued Models of Imaginary Numbers

Now, let’s ground ourselves in a more tangible example! One effective way to engage students and help them appreciate the reality of imaginary numbers is through the geometry of circular motion and rotations.

To illustrate this motion, we can describe our position using a position vector defined by x and y coordinates that change over time t. The constraint here is that the length of the vector remains constant at 1. This means that whatever path our point takes, it must reside on the unit circle.

Next, we can delve deeper into our motion. By calculating the time derivative of the aforementioned position, we arrive at a fascinating conclusion: the velocity vector is always orthogonal to the position vector, indicating that the inner product between the two is zero. While this may initially appear alien, it is logically equivalent to the concept of circular motion. Embracing the unfamiliar can lead to valuable insights.

Now, let’s introduce some matrix geometry, which may seem perplexing at first, but with practice, it becomes intuitive. I can confidently derive this because I recognize that a 2×2 matrix geometrically represents a counterclockwise rotation around the origin by a right angle. This means that the derivative of the position vector (representing velocity) is proportional to the vector itself, rotated 90 degrees. This reinforces our inner product statement.

However, it’s important to note that we’ve slightly altered our initial statement—by taking the derivative, we’ve eliminated the constant 1 that constrained our inner product to the original radius. The revised statement now describes circular motion of any radius centered at the origin, with the additional flexibility for speed, v(t), to vary over time.

Solving the first-order differential equations associated with this motion leads us to:

The first video, Imaginary Numbers Are Real [Part 1: Introduction], explores the fundamental concepts surrounding imaginary numbers and their significance in mathematics.

To understand the nature of the exponential function, we should explore its power series. Functions like sine, cosine, sinh, and cosh exhibit an exceptionally well-behaved series expansion. These are categorized as entire functions, meaning they are complex analytic functions without singularities across the entire finite complex plane. Their Taylor series converges universally, thanks to the factorial in the denominator, which leads to what is known as "hyperconvergence."

Similarly, matrix functions defined by series with growing factorials also exhibit converging behavior. This ensures that the series yields well-defined answers that satisfy the differential equation for any finite z.

Let’s break our findings into real and imaginary parts, since i² evaluates to -1 times the identity matrix. By employing the Taylor series for sine and cosine, we establish that these functions are periodic. This periodicity, associated with uniform circular motion, helps us identify the circle's circumference, which is normalized to 2𝜋. Proving this solely from the sine and cosine series would be challenging.

Next, we observe that our definitions contain only real number matrices, with no complex numbers in sight. The matrix representing i is indeed a real-valued 2x2 matrix. Interestingly, calculating i² reveals that it satisfies the equation i² = -1 (multiplied by the identity matrix). This provides a geometric interpretation where multiplication induces a counterclockwise rotation of the Cartesian plane.

To illustrate further, consider the following representation of an anticlockwise rotation by an angle θ around the origin, combined with a uniform dilation factor r.

The second video, How Imaginary Numbers Make Real Physics Easier to Understand, demonstrates the practical applications of imaginary numbers in physics and their role in simplifying complex calculations.

By analyzing matrices of the form a + bi, where a and b are real numbers, we can confirm that they commute, scale, and multiply in alignment with the abstract properties of complex numbers. This means we have effectively constructed a real-valued geometric model that corresponds to the axiomatic definition of complex numbers discussed earlier.

Quaternion Dreams: Exploring Higher Dimensions

Can we extend this concept to higher dimensions? Indeed, we can! Some argue that higher dimensional fields of numbers cannot exist, but in four dimensions, we encounter quaternions. Quaternion multiplication does not commute, making them a skew field. The complex numbers are the only commutative valuation skew field alongside the reals, while quaternions represent another valuation skew field.

Interestingly, these dimensional possibilities—1, 2, 4, and 8—are the only potential configurations for meaningful number definitions. Quaternions are particularly intriguing, as they can be perceived as three-dimensional rotations combined with a scale factor, both centered on the origin.

I look forward to diving deeper into the fascinating world of quaternions in upcoming discussions. Until then, perhaps we’ll meet at the Party at Infinity, or even the Restaurant at the End of the Universe! Who knows, maybe I’ll share a dance with Donna Summer there!

Reference

Bartel L. van der Waerden, "Hamilton's Discovery of Quaternions", Mathematics Magazine, Nov., 1976, Vol. 49, №5 (Nov., 1976), pp. 227–234