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# How Fundamental Truths are Defined in Math

### An in-depth analysis of axiomatic theory in math. Discussing Logic, Proof, Structure, Consistency & Limitations of Axioms

In the month of August, I intend to write articles on some of the foundational concepts of math that are extremely important to understand for anyone seriously interested in physics. Some of those topics include, **understanding axioms, set theory, Gödel’s incompleteness theorems**, etc.,

**We start our journey of understanding the foundations of maths by discussing about Axioms or fundamental truths.**

In the world of mathematics, axioms stand as the bedrock upon which entire mathematical systems are constructed. These fundamental statements, accepted as true without proof, serve as the starting points for logical reasoning and deduction.

In this article, we explore the significance of axioms in mathematics, their role in shaping mathematical theories, and their profound impact on our understanding of the universe.

## The Essence of Axioms

Axioms are the foundations upon which the entire edifice of mathematics rests. **They are concise, self-evident statements that are accepted as true without requiring proof.** At the core of the essence of axioms lies their pivotal role in shaping mathematical theories and their ability to unlock world of mathematical reasoning. I will try to define the essence of what makes and ‘axiom’ an ‘axiom’, by exploring its key characteristics.

### Building Blocks of Mathematical Reasoning

Axioms are are carefully chosen to capture the fundamental properties and relationships between mathematical objects. From these initial truths, mathematicians apply logical deductions to derive new theorems and results.

**Every mathematical statement within a particular system can be traced back to the axioms through a chain of logical steps**, ensuring the integrity and coherence of the entire mathematical structure.

### Self-Evident Intuition

The beauty of axioms lies in their inherent self-evidence. **They are crafted to be intuitively true, requiring no further justification or evidence.** These self-evident truths resonate with our innate understanding of mathematical concepts and allow mathematicians to explore the nature of mathematical structures with confidence.

### Universal Yet Tailored

Axioms possess a dual nature: **they are universal and context-specific**. While the core essence of an axiom remains true across mathematical systems, different branches of mathematics require distinct sets of axioms tailored to their specific areas of study. For instance, the axioms governing Euclidean geometry differ from those of set theory, reflecting the unique properties of each field.

### Independent and Consistent

Axioms must be carefully selected to ensure **they are independent from one another**. Each axiom must stand on its own, without being derivable from a combination of the others. **This independence ensures the coherence and internal consistency of the mathematical system**. When the axioms are logically consistent, they create a reliable and robust foundation for mathematical exploration.

### Empirically Connected

Though axioms are accepted as self-evident truths, their selection is often grounded in **empirical observations and intuitive reasoning**. For instance, Euclidean geometry's parallel postulate is based on the observation that parallel lines never intersect in our physical world. This empirical connection bridges the gap between abstract mathematics and the tangible reality it seeks to describe.

### Dynamic Nature

The essence of axioms is not stagnant but rather dynamic. As mathematics evolves and new discoveries are made, **axioms may be refined, expanded, or replaced** to accommodate emerging insights and understandings.

## Logical Structure and Consistency

Having discussed the essence of what makes an ‘axiom’, let us now discuss how axioms are defined. To define an axiom formally, we need to make sure that they are logically structured and consistent. This ensures the validity and reliability of mathematical truths.

Without quoting the examples of pre-existing and accepted axioms of mathematics, these concepts may sound too abstract. Therefore, I will be quoting three examples to elucidate these principles.

**Example 1: Euclidean Geometry**

Euclidean geometry is one of the oldest and most well-known branches of mathematics, characterised by its study of flat surfaces and the relationships between points, lines, and angles. It’s axioms, derived from the work of ancient Greek mathematician Euclid, form the basis of the entire system. One of Euclid's five postulates, the **"Parallel Postulate,"** states that

Given a line and a point NOT on that line, there exists exactly one parallel line passing through that point.

From this postulate, along with the other axioms, mathematicians deduce a myriad of theorems. For instance, they can prove that the angles of a triangle sum to 180 degrees, or that the opposite sides of a parallelogram are equal. The logical structure of Euclidean geometry ensures that each theorem follows soundly from the chosen axioms, leading to a consistent and reliable mathematical system.

**Example 2: Peano Arithmetic**

Peano arithmetic is a foundational system for natural numbers, named after the Italian mathematician Giuseppe Peano. **It’s axioms define the properties of the natural numbers (1, 2, 3, ...) and the operations of addition and multiplication.** Peano's axioms include statements like

0 is not the successor of any natural number

and

Every natural number n has a unique successor, denoted as S(n).

From these axioms, mathematicians prove various theorems about natural numbers. For instance, they can prove that addition and multiplication are commutative and associative, and they can derive properties of prime numbers.

The logical structure of Peano arithmetic ensures that each theorem is established by applying the rules of logic to the axioms, preserving consistency throughout the system.

### Example 3: Set Theory

Set theory is a foundational branch of mathematics that explores the properties of sets, which are collections of objects. **Its axioms define the fundamental operations and relationships between sets.** One of the axioms, the **"Empty Set Axiom,"** states that

There exists a set with no elements, denoted as ∅ (the empty set)

From the axioms of set theory, mathematicians prove various theorems related to sets. For instance, they can prove properties of union, intersection, and set complement. They can also establish results concerning infinite sets, such as the cardinality of the set of natural numbers.

The logical structure of set theory ensures that each theorem derives logically from the chosen axioms, providing a coherent framework for reasoning about sets and their properties.

## Different Mathematical Domains, Different Axioms

Mathematics is a versatile discipline that encompasses a vast array of subfields, each with its own unique objects of study and properties. The diversity of mathematical domains necessitates tailoring axioms to suit the specific context and goals of each branch. Let's delve into this idea further by using the example of the contrasting axioms of Euclidean vs Hyperbolic Geometry.

**Euclidean Geometry vs. Hyperbolic Geometry**

Euclidean geometry, developed by Euclid over two millennia ago, deals with the properties of flat surfaces like a sheet of paper. Its axioms form the basis of plane geometry. As discussed earlier in the article, one of Euclid's postulates, the "Parallel Postulate," states:

Given a line and a point NOT on that line, there exists **exactly one parallel line** passing through that point.

This axiom is intuitive in the context of the flat Euclidean plane we encounter in our everyday experience.

However, mathematicians in the 19th century discovered that alternative geometries are possible by slightly modifying the parallel postulate. Hyperbolic geometry is one such non-Euclidean geometry, where the parallel postulate is replaced with

Given a line and a point NOT on that line, there exist **infinitely many parallel lines** passing through that point.

In hyperbolic geometry, the notion of parallelism differs from Euclidean geometry. Mathematicians found that various theorems in hyperbolic geometry are distinct from those in Euclidean geometry. For instance, the sum of angles in a hyperbolic triangle is always less than 180 degrees, contrary to the sum being 180 degrees in Euclidean triangles. The adoption of different axioms tailored to each geometry yields diverse mathematical results and structures.

## Axiomatic Foundations and Mathematical Proof

**Mathematical proof is the process of systematically deriving new statements, known as theorems, from the accepted axioms.** As always, let’s use an axiom of math as an example and apply the process of logical deduction to establish its proof.

**The Axiom of Addition for Natural Numbers**

One of the fundamental axioms in Peano arithmetic is the **Axiom of Addition.** It defines the basic operation of addition and establishes the properties of adding natural numbers.

**Axiom**

For every natural number n, n + 1 is also a natural number.

This axiom asserts that if n is a natural number, then adding 1 to n results in another natural number.

**Proof**

To prove the Axiom of Addition, we need to demonstrate that for every natural number n, n + 1 is also a natural number. We will utilise the basic properties of Peano arithmetic to build the proof step by step.

**Step 1: **The base case is *n = 1*. We need to show that *1 + 1 is a natural number*.

By definition, *1* is the successor of *0*, so we can represent *1 as S(0) (where S(n) denotes the successor of n).* Therefore, the statement becomes

*S(0) + 1 is a natural number*

By the definition of addition, *S(0) + 1* is the successor of *(S(0) + 0)*. As *0* is the additive identity (i.e., adding 0 to any number does not change the number), we can rewrite the statement as

*S(0) + 1 is the successor of S(0).*

According to Peano's axioms, the successor of any natural number is also a natural number. Therefore,

*S(0) + 1 is indeed a natural number.*

**Step 2:** The induction step. Assume the axiom holds for some arbitrary natural number *k*, i.e., *k + 1 is a natural number.*

Now, we need to show that

*(k + 1) + 1* *is also a natural number.*

By the definition of addition, *(k + 1) + 1* is the successor of *((k + 1) + 0)*. Since *0* is the additive identity, we can rewrite the statement as *(k + 1) + 1* is the successor of *(k + 1).*

By the assumption from the induction hypothesis, *k + 1* is a natural number. And as previously established, the successor of any natural number is also a natural number. Therefore,

*(k + 1) + 1 is indeed a natural number.*

## The Limitations of Axioms

While axioms play a fundamental role in mathematics, they also have certain limitations that impact the scope and completeness of mathematical systems.

These limitations arise from the nature of axioms as self-evident truths and the reliance on intuitive understanding and empirical observations.

**Subjectivity and Intuition**

Axioms are selected based on their intuitive truth and self-evidence, but **what may be intuitively true to one person might not be the same for another**. For example, in Euclidean geometry, the Parallel Postulate seems self-evident for flat surfaces, but it doesn't hold in non-Euclidean geometries, which leads to different axioms and theorems.

**Incompleteness**

**Axiomatic systems can be incomplete**, meaning there are true mathematical statements that cannot be proven from the given axioms. Kurt Gödel's incompleteness theorems showed that certain mathematical systems will always have statements that are true but unprovable within that system. This highlights the inherent limitations of axioms to capture the entirety of mathematical truth.

**Independence**

Axioms should be independent of each other to ensure a solid foundation for mathematical reasoning. However, establishing the independence of axioms can be challenging, and **there may be hidden connections or dependencies** that could affect the logical structure of the system.

**Context-Dependence**

Different branches of mathematics require different sets of axioms, tailored to the specific context. While this allows for flexibility and specialisation, it also means that **axioms may not provide a unified foundation for all mathematical domains.**

**Consistency**

While mathematicians aim for consistency among axioms, it is possible to have axioms that lead to **contradictions or paradoxes**, making the system inconsistent.

**Necessity of Axioms**

Axioms are necessary for any mathematical theory, but the acceptance of axioms as fundamental truths **requires some level of assumption**. There is no way to prove the axioms themselves within the system they govern.

## How Fundamental Truths are Defined in Math

To summarise, in this article **we covered how fundamental truths or axioms are defined in math.**

We discussed about the **essence of an axiom**, i.e., its key features and characteristics. We ventured into real-world mathematical axioms to demonstrate how **axioms need to be** **logically structured, coherent, and consistent**. We also demonstrated how to derive **the proof or theorem of an axiom** using the example of addition of natural numbers. Finally, we delved into the **limitations and drawbacks of mathematical axioms**.

I hope this was a good primer into mathematical axioms for you. **This article is going to be crucial in understanding the upcoming articles in August.** We will be discussing topics such as **Gödel’s incompleteness theorems** and **Russel’s Paradox**, all of which require a solid understanding of how axioms are defined in mathematics.