School mathematics deals mostly with concrete and pragmatic applications of mathematical ideas. The student who goes through regular schooling usually goes through learning about arithmetic, euclidean geometry, trigonometry, analytic geometry, algebra, statistics and calculus.

It took me a while to come out of the cognitive repression that this system imposes upon the student. Ideas are distilled into calculative apparatus, mechanical rote procedures for solving problems asked in any specific type of examination. No mention is made of the genealogy of ideas, nor of the current status of their development in active mathematical circles.

I do not claim to have beaten the school math system by coming out on top. Nor do I claim to have a complete understanding of the current state of abstract theories such as category theory and it’s derivative group theory, or other fascinating concepts that seem obvious at initial glance, but have nuances that need to be specified in a precise sense by the usage of symbolic manipulation. That is why there is rigour in mathematics. Things are specified accurately to the point of redundancy to avoid any possible misalignment in the interpretation of a mathematical statement once it is accepted by the community.

It is also quite worthwhile to note where these ideas are applied in the present day in fields that use mathematics as a language of expression.

Added: 14th feb, 2026 What follows is an update to this deepdive that I’m working on.

There are many ways in which one can appreciate the power and beauty of mathematics. Over the course of just the last few thousand years, the collective intelligence of humanity has grappled with ideas at various levels of abstraction that offer different points of view. I say just because the timescale is a blink of an eye in the evolution of our species over several million years (which is another blink in the geological record, but that is another story).

As I understand it, what happens is that an abstraction, once invented, becomes a sub-field within the discipline that is studied both in isolation and in consonance (TODO: raise this in some forum, can the word “consonance” be used as an antonym of “isolation”. There does not seem to be a specific word for this, just near antonyms) with other sub-fields of mathematics.

Thus, the abstraction of the numbers, allowed humans to count things, even an infinite number of them. For example, the natural numbers 1,2,3.. extend to infinity. The interesting aspect here is that on the Real number line, there are an infinite number of numbers between each of these pairs of natural numbers, the decimal fractions (rational numbers) and the irrationals. Before Cantor, it was not possible to speak about an infinity of one thing being different from an infinity of another thing.

Cantor apparently proved that the infinity of real numbers is larger than the infinity of natural numbers. This seems logical, given the definitions, but the consequences introduced powerful notions that allowed us to think about infinite quantities in a new light. The proof of this is delightful in itself, you may want to look it up.

Specifically, he introduced the concept of countable and uncountable infinities, the natural numbers form a countable infinity and the reals form an uncountable infinity. (TODO: elaborate further on the consequences, specifically the usage of order in the counting, and the general era in which this was done, as opposed to specific work on categories and posets).

The abstraction of the sets, allowed humans to group related things together, and define operations on how multiple groups of things interact with each other or are differentiated from each other. For example, talking about numbers, the set of even numbers are collectively different from the set of odd numbers, in terms of the elements they contain. Also, multiplying any number by an even number always gives an even result.

In this monograph, we will explore one such point of view in quite detail. Particularly, we will explore the consequences of imposing order on a collection of related entities, in this case, numbers. But, we won’t be going in top-down approach, rather a bottom-up approach, which serves to replicate the way these ideas were originally conceived. We look into why certain ideas needed to come into existence. Also, the tone will be conversational and rigor will be minimal. This is both because I am personally not a trained mathematician who could express these concepts in a much more rigorous way than I possibly could, but I hope the clarity that this brings about to the general reader more than compensates for this.

We start with two ordinary natural numbers. In the context of number theory, we can define the following:

Least Common Multiple (LCM): The smallest number that “contains” both original numbers. Thus, the LCM is the smallest number that is able to be divided by both the original numbers.

Greatest Common Divisor (GCD): The largest number such that both original numbers “contain” this number. Thus, the GCD is the largest number that divides both the original numbers.

Finding the LCM: Take all the prime factors across both the numbers and raise each to the highest power across both the numbers for that prime factor.

This is a Set Union operation, where the union is on the set of prime factors of each number. However, a function, max(power of prime factor in A, power of prime factor in B) is applied to each of the elements (here, prime factors) in the sets.

We are making sure that while all the unique prime factors across both the numbers are accounted for, the contribution of each prime factor to the LCM is the maximum contribution of that prime factor in either one of the two numbers.

A succinct way of putting this: The LCM of two numbers is equivalent to the union of the multiset of their prime factors. (TODO: check the definition of multiset, it must be a set with duplicates)

This leads us to: Connection between Number Theory and Set Theory

Number theory has: The fundamental theorem of arithmetic Every integer n>1 can be uniquely represented as a product of prime powers

$n = p_1^{a_1} \cdot p_2^{a_2} \dots p_k^{a_k}$ Set theory has: If we have two numbers, x and y, we can represent them as sets (specifically multisets) of their prime factors.

On this multiset, the standard set operations of Union and Intersection are equivalent to the definitions of the LCM and GCD in Number Theory.

The LCM as a Union ($\cup$)

if $x = \prod p_i^{a_i}$ and $y = \prod p_i^{b_i}$:

$\text{LCM}(x, y) = \prod p_i^{\max(a_i, b_i)}$ This “max” operation on exponents is the exact numerical equivalent of a Set Union. It ensures that the resulting set of factors is large enough to cover both x and y.

The GCD as an Intersection ($\cap$)

$\text{GCD}(x, y) = \prod p_i^{\min(a_i, b_i)}$ This “min” operation on exponents is the exact numerical equivalent of a Set Intersection. It ensures that the resulting set of factors produces the largest number that divides both x and y.

Concrete example:

Take the numbers 12 and 18:

• $12 = 2^2 \cdot 3^1$ → Prime Set $A = \{2, 2, 3\}$

• $18 = 2^1 \cdot 3^2$ → Prime Set $B = \{2, 3, 3\}$

Operation	Logic	Result
Union (LCM)	$\max(2^2, 2^1)$ and $\max(3^1, 3^2)$	$2^2 \cdot 3^2 = 36$
Intersection (GCD)	$\min(2^2, 2^1)$ and $\min(3^1, 3^2)$	$2^1 \cdot 3^1 = 6$

Consider the identity: $x \cdot y = \text{LCM} \cdot \text{GCD}$

Why this works:

In terms of cardinality, set theory provides the following language:

$|A \cup B| = |A| + |B| - |A \cap B|$

$|A \cap B| = |A| + |B| - |A \cup B|$

Adding these,

$|A \cup B| + |A \cap B| = (|A| + |B| - |A \cap B|) + (|A| + |B| - |A \cup B|)$

which is equivalent to,

$|A \cup B| + |A \cap B| = 2*(|A| + |B|) - |A \cap B| - |A \cup B|$

This is saying, the sum of the cardinalities of union and intersection is the same as removing the cardinality of the union and the cardinality of the intersection from twice the cardinality of the original sum. Let this be our statement #1

Recall that, we are considering the multiset of a number composed of it’s prime factors. That is what A and B represent for this case.

So, “twice the cardinality of the original sum”, here, means twice the number of elements in the multiset of A + B.

In Number theoretic terms, the multiset of A + B is the decomposition of the sum of A and B into prime factors. What happens when we double this set? Are we multiplying by 2? No, each element has to be considered twice.

$n = p_1^{a_1} \cdot p_2^{a_2} \dots p_k^{a_k}$ If this is the multiset of A + B = n, 2*(cardinality of n) implies, we want to double the number of elements in this set. If we multiply the whole expression by 2, we are adding just 1 element to the multiset. So, the power of two that must be used is equal to the number of elements in the multiset currently. What is the power, then? Is it k? k is the count of unique prime factors in the multiset. If you observe carefully, the power of two is the sum of the exponents, a1..ak.

Consider, $a_1 + a_2 + ... + a_k = power(2)$ required for the original problem.

What is the effect of summing the exponents over a common base? We are saying, use a multiple of this base, here 2, “sum” times. The sum here, is nothing but the cardinality of the multiset of the sum of cardinalities of A and B.

From our statement #1,

$|A \cup B| + |A \cap B| = 2*(|A| + |B|) - |A \cap B| - |A \cup B|$

$2*(|A| + |B|)$

becomes,

$2^{|A| + |B|}$

Let this statement be #2 We have,

$\text{LCM}(x, y) = \prod p_i^{\max(a_i, b_i)}$

$\text{GCD}(x, y) = \prod p_i^{\min(a_i, b_i)}$

Both definitions are in terms of products.

Also,

$|A \cup B| = |A| + |B| - |A \cap B|$

$|A \cap B| = |A| + |B| - |A \cup B|$

Both definitions are in terms of sums.

Thus, when we are essentially converting from one language to the other, the addition in set theoretic terms, becomes multiplication in number theoretic terms.

When we multiply numbers, their exponents add up, by definition.

$2*(|A| + |B|) = 2*(|A \cup B| + |A \cap B|)$

Let this statement be #3. From statement #2,

$2*(|A| + |B|) \space is\space equivalent\space to\space 2^{|A| + |B|}$

For the LHS is the language of Sets, and the RHS is the language of Numbers, let this be statement #4. Since the union is defined as equivalent to the LCM, and the intersection as equivalent to the GCD,

In number theoretic terms, we get:

$\text{LCM}(x, y) * \text{GCD}(x, y) = x * y$

which is our Identity, as we need to change the addition to multiplication when moving from sets to numbers.

From statement #3 and statement #4,

$2^{|A| + |B|} = 2^{|A \cup B| + |A \cap B|}$

Where, the cardinalities are still in set theoretic terms, converting to numbers, we get:

$2^{x*y} = 2^{LCM(x,y)*GCD(x,y)}$

Which reduces to our desired identity.

Looking closer at the identity, we can we that the product of LCM and GCD is essentially “counting” each factor in x and y exactly as many times as it appears in the product of x and y.

Further generalizations (connection to Algebra): In mathematics, these Set Operations in which the Union and Intersection are defined by functions, (in this particular case, the maximum and the minimum over the exponents of the prime factor decomposition of numbers), uniquely define a structure known as a Lattice.

Lattice of divisors A distributive lattice of natural numbers ordered by divisibility.

In this context, the operations of LCM and GCD are known by their formal algebraic names: The Join ($\vee$) and The Meet ($\wedge$)

The Join ($\vee$) The Union from set theory, becomes the Join in Lattice theory. In the particular lattice of divisors, the join of two numbers is their LCM.

The Meet ($\wedge$) The Intersection from set theory, becomes the Meet in Lattice theory. In the particular lattice of divisors, the meet of two numbers is their GCD.

Visualizing the Hierarchy: The Hasse Diagram

The Geometric connection

Least Upper Bound The smallest number that both a and b can “reach” through multiplication.

Greatest Lower Bound The largest number that “fits” into both a and b.

To see this “Maximum” and “Minimum” logic in action, mathematicians use a Hasse Diagram. In this diagram, the LCM (Join) sits at the top of a diamond shape, and the GCD (Meet) sits at the bottom. (TODO: add a diagram)

Any two numbers you pick will have a unique “highest point” where their paths meet (the Join/LCM) and a unique “lowest point” where their paths originate (the Meet/GCD).

Informal Term	Set Theory Term	Lattice Theory Term	Number Theory Result
”The Max”	Union ($\cup$)	Join ($\vee$)	Least Common Multiple
”The Min”	Intersection ($\cap$)	Meet ($\wedge$)	Greatest Common Divisor

Example (TODO: add this specific diagram): Here is a Hasse Diagram showing the lattice of divisors for the number 36. This will help you visualize the “Meet” (GCD) and “Join” (LCM) operations we discussed.

How to read this Lattice:

• The “Join” (LCM): If you pick two numbers—say 4 and 6—follow the lines upward until they first meet at a single point. That point is 12, which is their LCM. You are taking the “maximum” of their prime factors.
• The “Meet” (GCD): If you take the same two numbers (4 and 6) and follow the lines downward, they first meet at 2. That is their GCD. You are taking the “minimum” of their shared factors.
• The Top and Bottom: In this specific set, 36 is the “Universal Upper Bound” and 1 is the “Universal Lower Bound.”

Summary: So far, we have elaborated one way of looking at things in a bottom-up manner. Considering that numbers are fundamental, we looked at how Number theory relates to Set theory and have had a taste of generalization of Sets into Lattices. In the following sections, we will look at generalization as applied to Sets in more detail.

Next section brief: Lattice theory is essentially a specialized, “shorthand” version of Category Theory. If Category Theory is the study of mathematical structures and the relationships between them, Lattice Theory is what happens when you apply those rules to Ordered Sets. The relationship between Lattice theory and Category theory is best understood through the concept of a Poset (Partially Ordered Set) viewed as a category.

The consequences of order We tend to apply order in the everyday world around us. Some things are ahead or behind another, above or below another. The pages of this document (when printed) are in order (hopefully), such that a lesser numbered page appears before a higher numbered page when flipping pages in-order from the beginning of the document.

Family trees are an everyday application of order that most of us are familiar with. These “genealogy” trees describe people as belonging to successive generations and map the relations they have with each other and others. We will see later that sometimes, things aren’t so straightforward even in what at first glance seems so. The concept of a partial order applies to such trees, rendering it impossible to mark certain relations within the confines of the tree.

What is an order in mathematical terms? An order allows for specific arrangements of the elements in a container, like in a set. An order allows one to pick from the set, any two elements and compare one or more of their properties. Specifically, there are four distinct possibilities: one is less than the other, one is greater than the other, the two are equal, and the two are incomparable

While the first three might seem obvious, the last one may not be. What does it mean for elements in the same set to be incomparable with each other?

Let us try to go through some examples to better understand what order means.

Consider the price of a stock before and after a stock split. As the price before the split would have been different due to a lower outstanding number of shares, as compared to the price after the split due to a higher outstanding number of shares, it is not possible to compare the two prices as it is. It becomes necessary to transform one of them by a scaling factor such that the two can be compared.

Note the three different usages of the word compare in the sentences above. The first two are general English usage, but the final use is closely aligned with the mathematical definition of comparison.

Why be pedantic about it? Because, in mathematics, comparison, is an order on elements of a specific container. That container defines rules regarding what is included or excluded from itself. If an element is included in a container that has an order, it will be a given that the element obeys the rules of the order. This will become clearer later as we move on to abstract algebraic structures.

Consider some quantity of red apples and some quantity of white cotton. They may be comparable in terms of a property like weight or volume. But in order to do this comparison, one must define the container that contains apples and cotton as elements of itself. Say, we have agricultural products ordered by market price. In what way would they be incomparable? The ordering here is the market price of each element, which exists for all elements. If you want to define some other ordering such as shades of red specified in RGB values, all the cotton will be excluded. Thus, here we see that a distinction can be made from orders applicable to some subset of the container as opposed to all the elements of the container. (TODO: elaborate on this)

A given natural number is either less than, greater than or equal to another natural number. This is a strict order on the natural numbers. For that matter, any discrete collection of entities (a collection of things that can be counted) implies the existence of a strict order on the count. For example, the ordering of the English alphabet as specified by either a dictionary or some other canon of the language. Note that the representation of the alphabet in a computer in some encoding like ASCII or Unicode, relies on the existence of this canonical equivalent. It is not the case that ‘a’ < ‘b’ in the C programming language due to ASCII ordering. But that ASCII ordering is a representation of the English language’s inherent feature that requires this.

Continuing the previous example, consider two different spoken languages that have a written equivalent. It is clear that the alphabets used varies greatly between languages. So, technically, the same comparison “function” cannot be applied to characters when they are from different languages.

Suppose, in a classroom, we are talking about the order of students as a measure of distance from the blackboard. So, the students in each row of desks are at the same distance from the blackboard and a case can be made for each row of students being part of a strict order. But what about students within each row? They are at an equal distance from the blackboard, so they cannot be part of a strict order on the student collection of the class. This is called as a partial order. In a partial order, there are no single greatest and least elements, only a set of maximal elements (back-benchers) and a set of minimal elements (teacher’s pets). These types of sets are called as a Poset, short for partially ordered set.

Consider any game or sport which has rules specifying the outcome as win, lose or draw. The unique intermediate states of this game will be part of the state space of the game tree, from some initial state. It is to be noted here that there may be arbitrary paths from a given initial state to one of the final states. Thus, the set of games that lead to winning outcomes can be considered as a poset of all possible games that could be played, under the same rules. Is one state of a game less than or equal to another? A given intermediate state of a game could be seen as being comparable with another game if they are seen as being on some level of a game tree, the levels being different from the definition an intermediate state. Thus, there could be cases where the same intermediate state is realized on multiple levels. The path length from the root level to the leaf level, given by the height of the tree, or to some intermediate state, given by the an expansion of the tree as the game proceeds, helps in making choices that lead to sooner or later outcomes.

As an aside, we said we are picking elements from a set. The specific mathematical property that enables picking is known as the axiom of choice. (TODO: elaborate)