Section 1.1: The Framework of Mathematics

So, what exactly constitutes mathematics? It is built upon a foundation of logic, along with basic syntactical and semantic elements that enable us to formulate "well-formed mathematical statements." This framework comprises fundamental truths, primitive objects (like lines or sets), and logical rules that guide our reasoning.

For example, if we establish that statement A is true and A implies B, we can conclude that B must also be true (this reasoning method is known as modus ponens, a fun fact to share at parties if you’re looking to bore someone). Mathematics revolves around identifying which statements hold true within this structured environment, with established truths known as theorems, propositions, or lemmas.

However, before we can engage in this process of deriving new truths from established ones, we require a foundation of true statements that do not necessitate proof. These fundamental truths are called axioms, and they should be so self-evident that they do not need validation.

In 300 BC, the Greek mathematician Euclid proposed five axioms that laid the groundwork for what we now term Euclidean geometry. This is the geometry most commonly taught in schools, involving lines, triangles, planes, and other shapes. In this geometry, spaces are typically "flat," and we observe properties like the sum of angles in a triangle equaling 180 degrees.

This method of mathematics, where we assume certain statements to be true and derive all other mathematical concepts from them, remains in use today.

Subsection 1.1.1: Reevaluating the Foundations

Section 1.2: The Foundations in Question

However, this approach raises several challenging questions. First, who decides which axioms we adopt? The choice of axioms significantly influences the type of mathematics that emerges.

For instance, altering some of Euclid's axioms can yield a universe where triangles no longer have a sum of angles equal to 180 degrees, and all lines intersect—this leads us into exotic geometries. By simply modifying one axiom, specifically the parallel postulate, we can derive three distinct geometrical systems.

While I won’t delve into the details of the parallel postulate here, it’s fascinating to note that tweaking these axioms opens up entirely different realms of geometry. One of these is spherical geometry, which you might find familiar, as it describes the Earth’s surface.

Imagine a journey from the North Pole to the equator along a great circle arc, then turning along the equator before heading north again; you would create a triangle with angles summing to more than 180 degrees since both angles at the equator measure 90 degrees. In this geometry, all lines converge, adding to its intrigue.

Alternatively, if we adjust the axioms to allow triangles to have an angle sum of less than 180 degrees, we enter the realm of hyperbolic geometry. This form of geometry can be visualized by thinking of shapes like a saddle.

The takeaway is that merely modifying one axiom out of five can lead to three vastly different geometries. But this principle applies to the entirety of mathematics, not just geometry!

The "Solution"

I wish to believe that 1 + 1 = 2, and that the existence of infinitely many prime numbers remains constant, regardless of the mathematical framework employed. After all, cicadas have utilized prime numbers for survival strategies long before humanity recognized their significance.

I hope these truths hold firm on any given day. Perhaps this viewpoint aligns with Platonism. Many mathematicians and physicists share this sentiment, suggesting that a "true" axiomatic system exists—one that reveals fundamental truths about nature. However, we may never realize such an ideal system.

So, what is the way forward? While Gödel's theorems indicate the limitations of mathematics, my personal recommendation is to embrace the belief in mathematics. Yes, simply have faith in it.

Physicists rely on mathematics to navigate complex theories, while engineers trust mathematical principles to design airplanes that carry loved ones on summer vacations.

That's about the best we can do in this intricate field—admittedly, it feels somewhat humbling.

Some intriguing ventures by mathematicians involve expanding our modern axiomatic system (known as ZFC) by introducing new axioms that do not conflict with established ones. This pursuit aims to validate statements that appear obvious but cannot be directly derived from ZFC. It’s like adding an unfamiliar player to the team—while they may contribute positively, they could also disrupt the established order.

One notable example is the quest to resolve the continuum hypothesis, which posits:

Continuum hypothesis:

There is no set whose cardinality is strictly between that of the integers and the real numbers.

Interestingly, different infinities exist—some are larger than others! The cardinality of integers is less than that of real numbers, and the continuum hypothesis asserts that no infinity lies strictly between these two.

The answer to this hypothesis is independent of ZFC, making it the first statement to carry this label. Consequently, we may incorporate it into our axioms within ZFC. However, we remain unable to ascertain whether ZFC itself is consistent, leaving us uncertain about the validity of this new set of axioms.

This narrative is extensive, but we must conclude at some point. I sincerely hope you continue to regard mathematics as a rigorous discipline. Who would have thought that, in the end, mathematics is about belief?

Thank you for reading.

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This video explores the intricate landscape of mathematics, illustrating the various types and their foundational principles.

An insightful overview of all types of mathematics, condensed into a concise 9-minute format.