Throughout the writings of Plato, one of the cornerstone thinkers of Western Philosophy, we are taught that the driving force of true education is the “pursuit of the Forms”, and that the ideal realm of Forms and Absolutes constitutes the greater reality, “forming”, as it does, the basis for our discernible reality. For Socrates (as we are told by Plato) the ideal realm is the only real realm, it defines our world and allows it to have meaning, but it is completely unknowable using our physical senses. Herein lies a paradox; how can it be that what is true, what is real, according to Plato, is not what we can sense, yet is knowable through observation of our surroundings? (I’ll leave that as a puzzle for the reader, in true Socratic form) Indeed, as discussed in the allegory of the cave, like a person shackled to the floor of a cavern, observing the dancing shadows of objects cast by the wavering light of a fire, we see not an urn, or a pot, or a tree, or even a human, but only an imperfect representation of their ideal forms. Such a concept may seem strange to us in our modern world, even antiquated and naïve, reflecting as it does, perhaps, the inexperienced grasping at truth of our species’ budding rational minds, desperately trying to understand the world around us. In fact, some of Plato’s contemporaries found his ideas absurd or bizarre. Even his student, Aristotle, took issue with the Theory of Forms (as it has come to be known). The fact is, however, that many fields of inquiry in modern times depend in a very real sense on the basic premise of the theory, which, I would venture to say, is this: “Observing and understanding the substantial world around us gives us insight into the abstract rules and instructions that govern it.” There exists several schools of knowledge where this idea is a central driving force: Science and mathematics, and more precisely, as I argue in this paper, physics and its mathematical underpinnings. Perhaps this would seem to stretch his theory too far. Socrates (via Plato) evidently took interest primarily in social issues of his time in Athens, and the bulk of his theory is dedicated to explaining how social behaviors and ideas (such as justice, education, social roles, etc.) can be modeled and understood precisely using the concept of the ideal realm. Why would I choose now to apply this millennia’s old idea to a field of study that is widely regarded as being at the cutting edge of progress and advancement; a pursuit that is usually perceived as being dispassionate and unconcerned with social quibbles?
In High School, most of us learned about a simple formula from our science teachers, Newton’s Second Law of Motion, which states in its simplest form, Force equals Mass times Acceleration, or “F = ma”. This means that the force that any given object or system exerts on any other object or system is a function of its mass multiplied by its acceleration at the moment that it interacts with the second object or system. This process is not only intuitive, it is readily apparent and observable at any time one likes. One can simply take a game of billiards as an example (to use the most clichéd analogy in Newtonian physics). Take a billiard ball and roll it towards another billiard ball on the playing surface. The resulting movement of the second ball in response to the impact of the first can be understood precisely (or perhaps as precisely as Newtonian theory allows, which is to say not so precise in an objective sense, but that’s a topic for another discussion) by knowledge of Newton’s laws. More specifically, the amount of force exerted on the second ball is a direct and necessary product of the mass and acceleration of the first ball. We can see such manifestations of this law by observation of almost any mundane situation. For example, we can observe an automobile accident and we understand that quite a lot of force is imparted to the frame of a car when it impacts a light pole. We intuitively know that the greater the mass of the pole, the greater the mass of the car, and the greater the speed of the car, the more force is applied to the body of the car and the pole. But this paper is not meant to be an exposition on physics, instead I want to bring attention to a key aspect of the previous discussion. Accustomed as we are to the rather dry language of physics, perhaps the reader has missed the subtle clues throughout that demonstrate the inexorable dependence of physical processes on a Platonic understanding of abstract physical rules. How can one, for example, “see” the force exerted on the frame of a car after an accident? Perhaps the reader would respond that one can observe the physical damage and destruction of the body, as well as the resulting injuries incurred by the occupants, but such observations are merely consequential. Other scenarios involving different types of objects exhibit other manifestations of force. One would not say for example that force is the damage of the car frame or injuries of the driver. One would also not say that the speeding car is acceleration or speed, or that mass is 1,500 kilograms (the average mass of an automobile). Instead we observe that the mass of the car, in this instance, is 1,500 kilograms, that 60 miles per hour is the speed of the car, and that the damage done to the car, occupants, and light pole is the result of the force applied to the system as a whole. These parameters all vary in any given situation. Obviously, the mass of different objects varies, and the speed and acceleration of different objects can be different values at any given time. Thus we can say that in any given instance, the measured properties of a system are manifestations of Mass, Acceleration, Force, etc.. The implication is clear, such abstract concepts can be best understood as Forms, ideas that are intangible and yet, strangely, easily measurable. The above scenarios, taken as a whole, can themselves be described as manifestations of the Form of Newton’s Second Law of Motion. I cannot “touch” or “grab” “force equals mass times acceleration”. I cannot point to something and say, “Look, there is ‘force equals mass times acceleration’!” I can only observe and describe events which are instances of the law manifesting itself.
Physics has long been one of my favorite subjects, and one which I have studied my entire life in an informal capacity, indeed, this connection is not a new one to me, it has been with me since I read Roger Penrose’s “The Road To Reality: A Complete Guide to the Laws of the Universe” in which Penrose drew some connections between Mathematical Platonism (The idea that mathematical constructs like numbers and operations have a separate, absolute existence apart from our mental manipulations) and his goal of uniting the General Theory of Relativity with Quantum Mechanics (which do not play well together, and this has long been one of the great enigmas and banes of physics). One of the central aspects of physics which both this book and many others on the philosophy of science demonstrate is that there is a fundamental dichotomy at the heart of the science of physics. There are two general realms of specialization into which all subsets of physics fall: Experimental and Theoretical. Experimental physics is the process of performing experiments, or observing natural processes, and collecting, analyzing, and drawing conclusions from the data. Theoretical physics is an entirely mathematical endeavor, starting with simple and a priori mathematical rules and formulae and following a strict line of logic to arrive at more complex calculations which describe some system or set of systems interacting. These two fields are not at odds, nor are they disconnected from one another, instead, theoretical physics provides hypotheses and suggests lines of inquiry which experimental physics then uses as a guide to the processes which should be observed to either confirm or refute the proposed hypotheses. Theoretical physicists then use these results to either continue down a particular line of reasoning or begin again from scratch. It is truly remarkable to reflect on the fact that our world obeys mathematical laws, and does so to such an exacting standard that we can describe the workings of any physical system with nearly arbitrary precision using math. Thus, it could be said, and has in fact, that the natural world is a manifestation of ideal mathematical laws. Mathematics itself, then, is not some irrelevant creation of our minds, but instead a real, albeit intangible realm, accessible only through our mental processes. No other method can access this realm in the same way, a calculator or computer may do math, but only through our own mental gymnastics can we understand math. In this sense, then, the physical world, including calculators and computers, are described by and follow the prescriptions of math, but the realm itself is only able to be glimpsed through cognition and thought. So our minds can comprehend math, and math in turn describes our physical world, but there is one more link in this chain which will turn it into a perfect example of an impossible causality loop: Our minds are the result of the physical world. Electrochemical processes lie at the heart of the operation of our brains, and if you believe as I do, and as the overwhelming evidence from science suggests, that our conscious minds are a result of the operation of the brain, then you must accept that our minds are a product of the physical world.
So then, we have here the basic outline for a Platonic understanding of our pursuit of natural law (another word for physical law). A realm of absolutes, the Platonic or Mathematical realm, which dictates the ideal form. The Physical realm, which follows the rules of math and, one could say in some sense “imperfectly” mimics the clean Mathematical realm, and the Mental realm, which is created by the Physical and can comprehend the Mathematical, which is itself in turn a result of minds like ours. For without minds to comprehend the math, or observe the physical world it creates, who is to say that math truly exists? And yet without the physical world, or the math it depends upon, there would be no minds to comprehend them in the first place! Included below is a diagram provided by Penrose to demonstrate this idea:
As you can see, each realm is a direct consequence of the proceeding realm, and yet there is no beginning or end, much like the Ouroboros, we have a continual cycle of creation creating itself. Each world mimicking the world that creates it. To use the allegory of the cave, the shadows represent the physical world, the objects that create the shadows are the mathematical world, and the mind of the prisoner, who, let’s say, comes to understand that the shadows represent a higher reality, is the mental world. Physics can thus be described as the process of discovering that the shadows are not a thing in themselves, but a direct consequence of the objects that lie beyond our senses. This is not to say that the physical world does not exist, but without the mathematical realm, the physical realm becomes unintelligible, and an unintelligible existence is, in my estimation, no existence at all.
Unlike the cave’s prisoners, there is no way to really step outside of the cave of our existence, we are confined by the limits of our senses, but our minds give us a unique way to access the realm of absolutes, to comprehend the rules that govern our world and use these rules to predict future states of a system by knowing the parameters of an earlier state of the system. This is the core of physics, without this predictive power, physics, and indeed all of science, becomes pointless. So, physicists are, to me, philosophers in their own right, concerned with the philosophy of existence and substance, and engaged in perhaps the most important endeavor humans can ever embark upon: understanding the world and how it works. For from this all other things flow, the entirety of all experiences and occurrences are dictated by a set of relatively simple physical rules, and if our minds can comprehend the mathematical realm, we can understand the physical realm, and thus we can understand the mental realm, as well.
In Isaac Newton’s time, “physics” was not a word, and people did not describe the discipline as such, instead, they used a term which I think helps to elucidate my main idea: the study was called “Natural Philosophy”. I believe this term is more appropriate then we may think in our modern times, for nothing is as fundamentally philosophical as the study of Nature.