## Archive for April 3rd, 2022

### What could be the bigger picture while solving the function transformation problems in school textbook?

Posted on: April 3, 2022

Himansu Sahu. Bangalore

May 14, 2021

# Arithmetic

At first, we learned counting 1, 2, 3… Then we learned how to “add”, “subtract” and then to “multiply” and “divide”. It made sense. (To whom?)

When we divide, we sometimes end up in “parts”, rather than “whole” — a very natural way to think about Fractions or Rational Numbers.

The concept of “spending” vs “earning”, “borrowing” vs “lending” made intuitive sense to think about negative numbers.

In the meantime, we also started denoting repeated multiplication of the same number as “exponentiation”. Now, reversing that gives rise to “square-root”, “cube root” and so on.

So, what do we have? We have the “numbers” — Natural numbers, Integers, Rational numbers, Irrational numbers (like what? the complex number i?) Real numbers”, and a bunch of “operations” — add, subtract, multiply, divide, square, square-root, exponent, logarithm, and so on.

We have these “ objects” (numbers) and “operations” on the objects.

# Algebra

We were taught not to worry about a specific number. Let’s call it “x”, representing any number, and see what we can do. Turns out, we can do everything that we did earlier — add, subtract, multiply, divide, and in fact, all of the previous “operations”.

What did we do really? Welcome to the world of “abstraction”!

We have, in fact, abstracted out the “numbers” by an “algebraic symbol”. We continued to learn “algebra”, continued to express the mathematical rules at this abstract level. It turned out to be more powerful because it could represent the mathematical truths for everything that the abstract “x” represented!

In essence, we upgraded to these abstract objects and operations on them.

# Calculus

We didn’t stop there. We could formulate more complex “operations” out of the simple ones (add, subtract, multiple, divide, etc.). We further extended the operations to be rather arbitrary — representing a mapping from a set of inputs (domain) to a set of outputs (range).

Again, what did we really do? We abstracted the “operations” as “function”!

What’s more? Turns out, with these “functions”, we could do a lot more wonders.

Issac Newton and Gottfried Leibniz extended to us the incredible gift of Calculus.

We learned how to do more “operations” on these functions — “Differentiation”, “Integration”, and so forth.

What do we have now?

We have these higher-level abstract objects (including “functions”) and more operations on them!

# Dimensions and Vectors

We realized that combining (or ordering) objects can represent another object — e.g. 3 east and 4 north is, in fact, 5 North-East!

Likewise, the sine of theta on the X-axis and cosine of theta on the Y-axis, is, in fact, 1 at an angle theta. Hence, we formed an “ordered set” of objects and abstracted them as “vectors”.

urns out that we could do all the previous operations, including calculus, on vectors (vector calculus). And, we could define even more involved operations on them, e.g. “gradient”, “divergence”, “curl” etc.

Again, what do we have?

We have even higher-level abstract objects (vectors) and more operations on them!

# Higher Mathematics — Abstract Algebra, Category Theory

As one can imagine, the abstraction does not stop here.

Higher-level mathematics (e.g., Abstract Algebra) takes the hierarchy of abstraction further up. It defines entities, e.g., “Rings”, “Groups”, “Fields”, as sets of super- abstract “objects” — numbers, vectors, matrices, functions, transformations, etc.

It defines abstract operations, which are specified merely by their attributes (“associativity”, “commutativity”, “identity” and “inverse”).

While looking at these higher-level mathematics topics (Abstract Algebra, Topologies, Analysis, and so forth), one can get truly intrigued dwelling at such abstract levels, and wander endlessly, given enough curiosity and luxury of time!

What of the topic in Mathematics called “Category Theory”.

It abstracts everything into a collection of categories and morphisms (“objects” and “relations” between them).

According to Prof. Barry Mazur of Harvard University, Category Theory is a “template” for all of Mathematics.

Depending upon what you feed into the template, the appropriate realm of Mathematics emerges. In her blog, Tai-Danae Bradle, a researcher at X, the Moonshot Factory — formerly Google X, describes Category Theory as the “ultimate vantage point in the landscape of Mathematics”.

It is quite fascinating to note the high level of abstractions, at which these topics are developed and dealt with (with rigor).

As I said earlier, it is almost impossible for me to comprehend the concepts without formal, rigorous training on the subjects. Nevertheless, I can not but draw some parallels here.

# The analogy with Physics

What about the quest for a “Grand Unified Theory” and a “Theory of Everything” in Physics?

The Grand Unified Theory attempts to combine three of the four fundamental forces in the universe (the “electromagnetic”, the “strong”, and the “weak” interactions) into a single coherent theory. The quest for a Theory of Everything is an attempt to unify even gravity into a single universal theory that can explain everything about the universe, in a simple way.

The quest started right during the renaissance period, when Newton formulated gravity, suggesting that the phenomenon which governs the movement of planets around the sun is the same one that holds us onto the surface of the earth.

In the 19th century, when Maxwell came up with the simple set of equations to suggest that electricity and magnetism are just different manifestations of the same underlying principle – electromagnetism, it was another leap forward.

In fact, subsequently, starting from Einstein, during much of the twentieth century, until this day, Physicists have been looking for that one, single, unified theory which can elegantly explain everything about the universe.

short video by Dr. Don Lincoln of Fermilab explains this with amazing lucidity.

# Ultimate Abstract Class

Coming back to the topic of Mathematics, it seems only natural to me that the development and application of modern Mathematics endeavors to push the envelope of abstraction where it truly encapsulates all the rules and principles of mathematics.

To borrow from Computer Science, (and a bit metaphorically speaking), I would call it the ultimate abstract class!

# Mathematics & Computer Science

Speaking of Computer Science, let’s consider the following:

• A very simple “bully” algorithm in Hash table insertion uses a property of “generator of a cyclic group (modulo P, where P is a prime)”. This guarantees that in the event of a hash collision the algorithm will scan all the indices of the table for a free spot, even with a randomized increment of the index.
• Modern Internet cryptography is entirely based on the “incalculable” nature of factorization of the product of large primes — part of fundamental Number Theory.
• I noted in one of my previous articles, how the Father of Computer Science Alan Turing conceptualized the Turing machine and in effect, left the problem-solving aspect of computing to the creativity of the human mind.
• Hence, since the advent of (general purpose) computing machines, Programming Languages have been the substrate on which this creative endeavor is flourishing. The core constructs of programming language, i.e., automata theory, finite state machines are built upon such firm mathematical foundations, e.g. Field Theory.
• Algorithms are at the heart of Computer Science and algorithms are nothing but step-wise instructions for computations (read “operations”) on real-world data (read “objects”).
• It is fascinating to study Lambda Calculus, a formal system of mathematical logic, and how it lends itself to building the very basis of computational constructs — i.e., functions and arguments.

The list goes on… It is indeed eye-opening for me, how the formal underpinnings of the vast discipline of Computer Science are firmly rooted in such rigorous, but beautiful abstractions of higher-level mathematics!

True to its evolution, the field of Computer Science is (again, metaphorically speaking) derived from this ultimate abstract class of Mathematics!!

# Finally, the innate beauty

I urge the reader to watch this amazing piece of Ted Talk (if not already done) by Grant Sanderson (of 3blue1brown) at Berkeley, just to pause and marvel at the beauty of the “Language of the Universe”!

Note: In a recent article of how we were warned as children to learn Not to mix apples with oranges in our operations.

It turned out that all applied Math (physics, chemistry, biology…) is just mixing all kinds of independent variables (orange and apples and bananas…) with all kinds of operations. And we are totally sure that these equations will accurately predict/forecast…outcome of events of nature and in man-made system.

### Blog Stats

• 1,518,661 hits