While talking to one of my professors for my Data, Justice, and Society class (Dr. Posner you’re the best!), we discussed how models are only our best approximations of things in the world, and we replace them as we learn more and have to model new things. That being said, as someone who wants to delve further into mathematics, our field creates a lot of frameworks, which are inevitably used in other fields to make models. Today I wanted to talk about some instances where mathematics itself changed its own model, adding in new features like an operating system to solve new problems.

Imaginary Numbers: Unlocking the Secrets of Negative Roots

The concept of imaginary numbers, which allows us to take the square root of negative numbers, has been a cornerstone of mathematics since its introduction in the 16th century. Initially met with skepticism, it wasn’t until the 18th century that mathematicians like Leonhard Euler and Carl Friedrich Gauss fully developed and popularized this idea. Euler’s work was instrumental in applying imaginary numbers to solve equations that were previously unsolvable, while Gauss used them to develop the theory of Gaussian integers, which are complex numbers of the form $a + bi$, where $a$ and $b$ are integers and $i$ is the imaginary unit (the square root of -1).

Historical Background

Imaginary numbers were first explored by Girolamo Cardano, but it was Euler who truly integrated them into mainstream mathematics. The term “imaginary” was initially used to convey a sense of mystery or unreal nature (and Descartes used it as a derogatory term), but as their utility became apparent, they became an indispensable tool in various mathematical disciplines.

Modern Applications

Today, imaginary numbers have numerous practical applications across several fields:

  • Electrical Engineering: Imaginary numbers are crucial for analyzing and designing AC circuits, which involve sinusoidally varying currents and voltages. Techniques like the Fourier transform rely heavily on complex numbers to simplify these analyses.
  • Navigation: In GPS systems, imaginary numbers help correct for errors caused by satellite motion, ensuring accurate location information.
  • Medical Imaging: Techniques such as MRI and CT scans use imaginary numbers to reconstruct body images accurately, compensating for movement during the imaging process.
  • Quantum Mechanics: Imaginary numbers are essential for representing wave functions of particles, allowing predictions about their behavior in complex systems.
  • Statistics: Imaginary numbers aid in analyzing and modeling complex systems by correcting for data movement during analysis.
  • Computer Science: They are used in designing algorithms that solve complex problems efficiently, often leveraging powerful mathematical tools like the Fourier transform.

Cantor Time! Comparing Infinities and the Creation of Set Theory

The concept of infinity has long fascinated mathematicians and philosophers alike. Originally, the Aristotelian philosophical tradition viewed infinity as potential or incomplete. In the late 19th century, Georg Cantor revolutionized our understanding of infinity by developing set theory, which allowed for the comparison of different infinite sets by quantifying them in a sense.. This breakthrough transformed mathematics by showing that not all infinities are equal.

Historical Background

Cantor’s work built upon earlier mathematical foundations, but it was his innovative approach to set theory that truly opened up new possibilities. He introduced the concept of cardinality, which enabled mathematicians to compare the size of infinite sets. This was a radical departure from traditional views, where infinity was often seen as a singular, unknowable entity.

Cantor’s Contributions

Cantor’s most significant contribution was the realization that infinite sets can have different cardinalities. For example, the set of natural numbers (1, 2, 3, …) and the set of real numbers (including fractions and decimals) both contain an infinite number of elements, but they have different sizes. The set of real numbers is “larger” than the set of natural numbers because there is no one-to-one correspondence between them—a concept known as the continuum hypothesis.

Modern Applications

Today, Cantor’s work on infinity has far-reaching implications across various fields:

  • Mathematics: Set theory forms the basis of modern mathematics, influencing areas such as topology and abstract algebra.
  • Computer Science: Understanding different sizes of infinity helps in designing algorithms and data structures that can handle large datasets efficiently.
  • Philosophy: Cantor’s ideas have sparked debates about the nature of infinity and its implications for our understanding of reality.
  • Physics: The concept of infinity is crucial in theories like quantum mechanics and cosmology, where infinite series and sets are used to model complex phenomena.

Removing Geometric Axioms: What Happens When Geometry Goes Non-Euclidean?

For over two millennia, Euclidean geometry, as outlined in Euclid’s Elements, was the cornerstone of geometric understanding. It was based on a set of axioms that seemed intuitively true, including the parallel postulate, which states that through a point not on a line, there is exactly one line parallel to the original line. However, in the 19th century, mathematicians began to challenge this fundamental assumption, leading to the development of non-Euclidean geometries.

Historical Background

Non-Euclidean geometries emerged when mathematicians like Nikolay Lobachevsky and Bernhard Riemann independently explored alternatives to Euclid’s parallel postulate. This led to the creation of hyperbolic and elliptic geometries, where the parallel postulate does not hold. Hyperbolic geometry allows for multiple parallel lines through a point, while elliptic geometry allows for no parallel lines.

Impact of Non-Euclidean Geometries

The shift towards non-Euclidean geometries had profound implications:

  • Mathematical Revolution: It challenged the idea of a single, universally true geometry, showing that different axioms could lead to consistent and useful mathematical systems.
  • Scientific Impact: Non-Euclidean geometries paved the way for Albert Einstein’s theory of general relativity, which describes physical space as non-Euclidean.
  • Philosophical Implications: The development of non-Euclidean geometries clarified the relationship between mathematics, science, and observation, highlighting that mathematical truths are not absolute but depend on the axioms chosen.

Modern Applications

Today, non-Euclidean geometries have numerous applications:

  • Physics: They are crucial in understanding gravitational fields and the curvature of space-time according to Einstein’s general relativity.
  • Computer Science: Non-Euclidean geometries are used in computer graphics and game development to create realistic models of curved spaces.
  • Navigation: GPS systems rely on non-Euclidean geometry to accurately calculate positions on the Earth’s surface, which is not perfectly Euclidean.

Conclusion

The expansion of mathematical possibilities has been a recurring theme throughout history. From the development of imaginary numbers to the creation of non-Euclidean geometries, mathematicians have consistently pushed the boundaries of what is thought to be possible. This process of expansion is not limited to mathematics alone, as it has far-reaching implications for the various fields that rely on it to formulate their models. By challenging traditional assumptions and exploring new perspectives, we can unlock new possibilities and gain a deeper understanding of the world around us.