Is pi on a par with the physical constant of nature the fine-structure constant?

I had this chat with ChatGPT 3.5 on 4 Nov 2023.

Patrick Reany

6 November 2023

While I was watching the YouTube video

Ravi Vakil: Algebraic geometry and the ongoing unification of mathematics.

I was inspired to ask the question: Should the irrational number pi be considered a fundamental constant of nature in much the same way as the fine-structure constant is? After all, they're both unitless numbers that have empirical content.

Comment: I asked this question of ChatGPT 3.5, as I was interested what its largely objective and unemotional reasoning might have to bear on it. What I like about ChatGPT is that it is conservative, even skeptical, but openminded. It will now present an assortment of relevant facts that bear on the question to help us decided the matter, if possible. However, the biggest fact against my hypothesis is that there is little if anything in the literature to support my hypothesis, which was prejudicial against it. Understand that I'm not claiming the hypothesis is true, merely that it's worth thinking about.

ChatGPT:

The question of whether π (pi) should be considered a fundamental constant of nature, similar to the fine-structure constant, is a matter of perspective and context. Let's explore this issue:

Definition of Constants:

π (pi) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is a fundamental concept in geometry and trigonometry. The fine-structure constant, denoted by α (alpha), is a dimensionless physical constant that characterizes the strength of the electromagnetic interaction in quantum electrodynamics.

Role in Physics:

π is not a fundamental constant of physics in the same sense as the fine-structure constant. It doesn't directly govern physical interactions or properties of particles. The fine-structure constant, on the other hand, is a fundamental constant in the context of quantum electrodynamics and plays a crucial role in describing electromagnetic interactions among particles.

Empirical Content:

Both π and the fine-structure constant have empirical content in the sense that they arise from observations and experiments. However, they have different roles in describing physical phenomena. π arises naturally in many physical contexts, including those related to circles, waves, and oscillations. It's a constant that relates to the geometry of space. The fine-structure constant emerges from the behavior of electric charge in quantum systems, and it has profound implications for atomic and subatomic physics.

Universality:

The fine-structure constant is considered universal, meaning it has the same value in any part of the universe and at any point in time, assuming the laws of physics remain the same. π is also universal in the mathematical sense, but it doesn't represent a physical constant that is fundamental to the behavior of the physical universe.

In summary, while both π and the fine-structure constant are dimensionless numbers with empirical content, they serve very different roles in physics. π is a mathematical constant related to geometry, whereas the fine-structure constant is a fundamental constant of nature governing the behavior of electromagnetic interactions in the quantum realm. Whether π should be considered a fundamental constant of nature is a matter of interpretation, but it doesn't have the same direct physical significance as constants like α in the context of fundamental physics.

Patrick:

I think that the viewpoint you have promoted would be espoused by most physicists, though I wonder if we are missing something here. Perhaps the dismissal of equal physical importance of pi and alpha by physicists is just the result of a long-standing bias on their part that hasn't been fully analyzed yet. Perhaps physicists are wont to dismiss pi as being "just mathematical" rather than physical. Perhaps we've overlooked something important in this dismissal.

Comment: So, I made an argument along the lines that the reason there is nothing in the literature to support my hypothesis is just due to a cavalier attitude that physicists have developed about the physical status of pi. In other words, perhaps physicists have never seriously considered thinking of pi as a fundamental constant of nature, and thus they may have overlooked something important in doing so. Therefore, what I'm saying, basically, is that the facts warrant that we keep an open mind about this hypothesis. Remember that at one time the geometry of space was considered to play only a passive role in physical processes, until Einstein decided to make use of the overlooked principle of equivalence between constant acceleration and uniform gravity. Who's to say that it's impossible for a physical theory to come out someday that will make pi even more fundamental than alpha, in as much as geometry is more fundamental than electromagnetism.

ChatGPT:

Your viewpoint raises an interesting and thought-provoking question. It is true that the distinction between what is considered "just mathematical" and what is considered "fundamental to the physical universe" can be somewhat subjective, and it's worth exploring whether there might be a more nuanced perspective.

Comment: This is as close as I've ever remembered ChatGPT presenting a human-like response to me. Is this LLM "interested"? Is showing interest imply showing an emotion? Anyway, I give high marks for ChatGPT simulating a rational person. In any case, it goes on to present a persuasive essay to support my hypothesis, only going so far as to suggest that it's worth giving consideration to the hypothesis. I think what ChatGPT realizes is that the literature does not present a good reason to distinguish between "mathematical constants" and "physical constants." Perhaps this is why ChatGPT was able to so easily conclude that the stance on this hypothesis "is a matter of perspective." But now it will present evidence that is designed to weaken this cavalier division between the two.

Foundational Nature: While π might not be traditionally classified as a fundamental physical constant like the fine-structure constant, it does have a foundational nature in various mathematical and physical contexts. It appears in equations describing wave phenomena, oscillations, and circular motion, all of which have direct physical applications.

Unity of Science: Some scholars argue for the unity of science, suggesting that the divide between "pure mathematics" and "applied physics" may not be as clear-cut as we often assume. The relationship between mathematical constants like π and the physical constants of the universe might be more intertwined than we traditionally think.

Philosophical Perspectives: There are also philosophical perspectives on the nature of mathematical and physical reality. Some philosophies, such as mathematical Platonism, argue that mathematical entities have an objective existence and are just as real as physical entities, blurring the line between the mathematical and physical worlds.

In light of these considerations, it's valid to explore whether the distinction between mathematical and physical constants is overly rigid. However, it's important to acknowledge that the fine-structure constant is a dimensionless constant that directly governs physical interactions and behaviors at the quantum level, while π, while deeply important in mathematical and physical contexts, does not have the same direct physical influence. This distinction is a practical one made in the context of the current state of our understanding of the physical universe.

While it's always valuable to challenge existing paradigms and explore new perspectives, it's crucial to do so within the framework of empirical evidence and the current state of scientific knowledge. If new evidence or insights emerge that suggest a different perspective on the role of mathematical constants like π in the physical universe, the scientific community would certainly be open to reevaluating these ideas.

Conclusion:

If we are given to offhand dismissal of consideration of this hypothesis, we may miss an important new principle in the foundations of physics.

As Doctor Who once said:

Keep an open mind --- that's the secret.