Algebra: The art of making X disappear like my motivation to solve for it.
Summary: Mathematics and biology share a foundational truth: complexity arises from simple principles. Peano’s axioms define numbers, just as ‘Survival of the Fittest’ shapes life. Our existence is a chance occurrence, not a predetermined goal. Yet, understanding this cosmic connection brings both humility and awe, revealing the profound beauty of existence.
There is profound beauty in accepting that one plus one is two and recognizing that this simple truth lays the foundation for far more complex mathematical structures — ones that are not only abstractly intriguing but also essential in modeling and explaining the workings of the real world.
Giuseppe Peano, an Italian mathematician (August 27, 1858 — April 20, 1932), proposed five axioms about natural numbers that, in their simplicity, seem almost self-evident:
- Zero is a number and serves as the foundation of all numbers.
- Every natural number has a successor, which is also a natural number. If you start at 0, the next number is 1, then 2, then 3, and so on.
- Zero is not the successor of any natural number. In Peano’s system, there is no number that comes before 0 — this framework does not include counting backward.
- Two natural numbers with the same successor must be the same number. If different numbers led to the same successor, the number system would become inconsistent.
- If a rule works for zero and remains valid as you move to each successive number, it holds for all numbers. This is the principle of mathematical induction.
If that sounds complicated, here’s what it means in simpler, everyday terms:
- There’s always a starting point. Imagine a basket of oranges. Even if the basket is empty, that still represents a number — zero oranges.
- You can always add one more orange to the basket. If you add one to an empty basket, you have 1 orange. Add another, and you have 2. This process continues indefinitely.
- Zero is special — it’s where we start. If the basket is empty, that’s 0 oranges. You can’t take oranges from an empty basket and still have oranges. (In this basic system, we don’t consider negative numbers.)
- If you and I are counting oranges and I say “3” while you say “4,” that means I counted up from 2, and you counted up from 3. Since we started from different numbers, we arrived at different results. No two different numbers can lead to the same “next” number — otherwise, counting would break down.
- If something is true at the beginning and remains true step by step, it is true forever. If a rule holds for 0 and continues to hold for each next number, then it holds universally.
Starting from these five axioms, increasingly complex mathematical structures emerge. Each builds upon the previous, leading to interconnected frameworks that underpin much of modern mathematics.
By modifying Peano’s axioms, one can construct alternative mathematical systems. While his original framework defines natural numbers, altering these axioms or introducing new ones gives rise to different number systems and algebraic models.
This brings us to a broader point: the understanding of complex systems — or fundamental questions about our existence — often starts with a few basic principles. Questions such as: How did we come about? Do we have a purpose? If life began again, would we be here?
A few undeniable facts can lead to profound consequences. One simple realization is that in an environment with limited resources and the inherent influence of randomness, if biology were to arise, the emergence of the principle of ‘Survival of the Fittest’ would be inevitable. And once this principle is in place, so many other pieces of the existence puzzle fall into place.
Starting from this, we can deduce that evolution did not have us in mind as an end goal. We are a product of chance. If the process were to start over, it is almost certain that we would not be here.
There is no predetermined purpose for our existence. The principle of survival of the fittest dictates that once self-replicating molecules appear, complexity will evolve — culminating in forms capable of learning from the past and anticipating the future to better compete for limited resources. That, in itself, defines the extent of our existence’s meaning.
Yet, there is profound beauty in understanding these questions through a few simple truths. The intricate details of how we came to may be complex (and not fully understood), but we grasp the fundamental reasons behind our existence.
In that understanding, there is also a deep, almost cosmic connection — a realization that threads link us to the earliest moments of the universe and those extending into the unknown future.
In that understanding, sometimes, we can hear the sublime vibrations that permeate the cosmos and will continue to do so forever.
And in that understanding, we recognize that our existence is a rare and fragile chance occurrence — one that should fill us with both awe and humility.
Ciao, and thanks for reading.
