The Nature of Mathematics

So, what constitutes mathematics, and how is it structured? At its core, mathematics rests on a foundation of logic, along with basic syntactical and semantic structures that enable us to formulate "well-formed mathematical statements." These include fundamental truths, basic elements (such as lines and sets), and logical rules for reasoning within this framework.

For example, if we know statement A is true and that A implies B, we can deduce that B is also true (a principle known as modus ponens). Mathematics essentially revolves around determining which statements hold true within this system, establishing truths that lead to further truths. Verified statements are called theorems (or propositions, lemmas, etc.).

However, before we can explore the process of deriving new truths from established ones, we require a solid foundation composed of statements that are accepted as true without needing proof. These foundational truths are termed axioms, designed to be so evidently true that they require no further justification.

In 300 BC, the Greek mathematician Euclid introduced five axioms that became the cornerstone of what we now recognize as Euclidean geometry. This is the geometry most students learn in school, featuring flat surfaces and properties such as the sum of angles in a triangle equaling 180 degrees.

This method of assuming certain statements to be true and deriving all of mathematics from them remains in practice today. Yet, this approach carries significant complications.

Shaking the Foundations

First, we must select a set of axioms to start with, but who decides which ones to use? These axioms greatly influence the type of mathematics we develop. For instance, modifying some axioms in Euclidean geometry can lead to a framework where the sum of angles in a triangle is not 180 degrees, and lines intersect in unexpected ways. Notably, simply altering one axiom—the parallel postulate—can yield three distinct geometries.

Without revealing what the parallel postulate entails, it's fascinating to note that tweaking these axioms can produce various geometries. One of them is Euclidean geometry, while another allows triangles to have angle sums greater than 180 degrees—this is spherical geometry.

Spherical geometry, which is employed in measuring the Earth, presents an interesting phenomenon. Consider traveling from the North Pole to the equator in a straight line, then moving along the equator before heading straight north again; you would create a triangle with angles exceeding 180 degrees, as the equatorial angles each measure 90 degrees.

Conversely, if we adjust the axioms so that triangles can have an angle sum of less than 180 degrees, we enter the realm of hyperbolic geometry. This involves performing geometry on shapes resembling a saddle rather than a sphere—though explaining it fully would require more space.

The key takeaway here is that by modifying just one axiom out of the five foundational ones in geometry, we can arrive at three fundamentally different geometrical systems. This principle applies not only to geometry but to all of mathematics.

Choosing the right axioms shapes the type of mathematics we explore, yet there’s a more troubling aspect: the entire edifice of mathematics rests on shaky logical foundations that remain unresolvable. Let me explain.

The axiomatic systems in use today mirror Euclid’s approach, generally employing around nine axioms. However, how can we be certain that these axioms do not contradict one another? The unfortunate truth is, we can't.

To grasp this, we need to understand the concept of consistency. An axiomatic system is termed consistent if there is no statement that can be proven true and false simultaneously within that framework. If it fails this condition, it is labeled inconsistent.

Interestingly, any system capable of proving its own consistency is, in fact, inconsistent. This has been established. Additionally, within any axiomatic system, there will be statements that can neither be proved nor disproved, meaning we will never ascertain their truthfulness.

To sum up: “We will never know if our mathematics is fundamentally sound, and there will always be truths that elude us.”

While we often perceive mathematics as the most certain discipline, we've shown that it is, paradoxically, rife with uncertainty. This revelation poses a significant challenge to humanity's quest for knowledge.

The "Solution"

I wish to believe that 1 + 1 = 2 and that there are infinitely many prime numbers, regardless of the mathematical framework employed. After all, cicadas utilized prime numbers to devise survival strategies millions of years ago, relying on their properties.

I desire these truths to hold steadfastly, whether on a Sunday or a Monday. I suppose one could label me a Platonist.

Many mathematicians and physicists share this sentiment, hinting at a belief in an ideal axiomatic system that yields inherent truths about the nature of reality. Yet, such an idealized system remains out of reach.

So what is the answer? While Gödel’s theorems outline the limitations of mathematics with no clear resolution, my personal advice is to embrace belief in mathematics. Indeed, that is the best we can do. It may feel embarrassing.

Some mathematicians are actively working to expand our contemporary axiomatic framework (known as ZFC) by introducing new axioms that do not conflict with established ones. This effort seeks to prove statements that appear obvious but cannot be derived from ZFC alone, aiming for a "better" version of mathematics.

This endeavor resembles allowing an unfamiliar player onto the field; while they might contribute to our success, they could also lead us astray. One example of this exploration is the quest to address the continuum hypothesis.

The continuum hypothesis asserts: > Continuum hypothesis: > There is no set whose cardinality is strictly between that of the integers and the real numbers.

It turns out that different sizes of infinity exist—some larger than others! The cardinality of the integers is smaller than that of the real numbers, and the continuum hypothesis posits that no infinity lies strictly between these two categories.

The resolution to this issue remains independent of ZFC, marking it as the first statement to receive this intriguing classification. Thus, we could potentially add it to our axioms in ZFC. However, without knowing whether ZFC itself is consistent, we cannot ascertain the consistency of this new set of axioms.

There's much more to this narrative, but we must conclude at some point. I hope you continue to regard mathematics as a rigorous discipline. Who would have imagined that, in the end, mathematics is fundamentally about belief?

Thank you for reading.