I would rather have questions that can't be answered than answers that can't be questioned
– Richard P. Feynman
Long ago, before the age of scientific inquiry, people looked to the natural world to understand the workings of the universe. This was the era of Natural Philosophy, a time when thinkers sought to understand the natural world through observation, speculation, and deduction.
Fast forward to the present day, and the picture is very different. We live in a world where science and technology have transformed our lives in countless ways. From smartphones to space exploration, we rely on scientific discoveries to navigate our daily lives. However, it is important to remember that science is not simply a means to an end. It is not just a tool for making our lives easier or more comfortable. At its core, science is about uncovering the truth about the world we live in.
The pursuit of truth in science is a sacred endeavor that cannot be reduced to mere convenience or handouts. This sentiment is echoed in a quote by Walter White: "Is that what you want? This short order cook? You're not flipping hamburgers here, Pal." Science is not a simple matter of following instructions or taking orders. Rather, it requires creativity, ingenuity, and perseverance to uncover the secrets of the natural world.
Scientists are not prophets who hand down divine knowledge from on high, but rather explorers who push the boundaries of our understanding. Ultimately, it is only by following in the footsteps of these explorers that we can truly appreciate and benefit from their discoveries. By questioning, experimenting, and observing, we can unlock the secrets of the natural world and harness its power for the betterment of humanity.
Throughout the course of this project, I have gained valuable insights into what makes a derivation good. These insights can be distilled into two axioms that I believe are crucial for any successful derivation.
There is a common misconception that derivations are just backward proofs. However, the truth is that proofs and derivations are fundamentally different. The purpose of a proof is to show what is true and what is not, without necessarily explaining why. This often involves tedious and meaningless algebraic manipulation that only makes sense in hindsight. On the other hand, derivations not only show whether something is true, but also explain why it is true.
To create a meaningful derivation, each step must be motivated and not simply enforced because it is known to lead to the desired result. In other words, every step must have a clear justification and contribute to the overall understanding of the concept being presented. This ensures that the derivation is not just a sequence of arbitrary steps, but rather a coherent and logical progression of ideas.
In summary, derivations and proofs serve different purposes, and derivations require more than just the ability to manipulate equations. They require a clear understanding of the underlying concepts, as well as the ability to motivate and justify each step in the derivation.
When deriving equations or theorems, it is crucial to aim for the highest level of symmetry possible. Symmetry, in this context, refers to a generalized concept that does not rely on specific examples or cases. By creating a derivation that is symmetric, we can gain a better understanding of the underlying principles and deeper patterns that are at play. This can ultimately lead to more fundamental theorems or equations that have broader applicability and greater significance.
Specific examples or cases can be helpful in understanding a concept, but when they are used in a derivation, it can limit its applicability and make it less symmetric. A derivation based on a specific case may only be applicable to a particular set of conditions, whereas a symmetric derivation will be applicable to a broader range of conditions.
Moreover, the more assumptions that are needed in a derivation, the less symmetric it becomes. When too many assumptions are made, the derivation can become cluttered with unnecessary details, making it difficult to identify the underlying principles and patterns. In contrast, a symmetric derivation is clean and elegant, highlighting the essential elements of the problem at hand.
In summary, when creating a derivation, it is essential to avoid relying on specific examples or cases and aim for the highest level of symmetry possible. A symmetric derivation reveals deeper patterns and underlying principles that are not immediately apparent, leading to more fundamental theorems or equations that have broader applicability and greater significance. By keeping the derivation clean and elegant and avoiding unnecessary assumptions, we can create a more powerful and useful tool for understanding the natural world.