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.