# Boxing Pythagoras

## On Free Will and Determinism

Given that I hold to a very mathematical, materialist view of the cosmos in which all points of time are extant with one another, it should come as no surprise that I tend to favor the idea of Determinism. This is a philosophy which states that, given perfect knowledge of the physics of the cosmos and adequate data, one could accurately predict any future event with 100% certitude. The idea is that everything operates according to determinable mechanisms, and that understanding these mechanisms can lead to an accurate understanding of their outcomes even before those outcomes have occurred. This view has quite often butted heads with philosophers who argue that Determinism eliminates the possibility of Free Will. Free Will, they argue, does not exist if the outcome of a choice can be predicted with certainty or if the future is already set.

As a Compatibilist, I contend that Determinism and Free Will are not mutually exclusive concepts.

In my humblest of opinions, the reason that these two concepts are often pitted against one another is due to a false impression about the implications of Free Will. It is a very common view that Free Will requires that the agent making the choice has the ability to fundamentally alter reality. On this view, the choice is that which actually informs the future and brings potentiality into reality. This view would fairly obviously stand in contradiction to my own view that all of time– past, present, and future– coexists in reality equally.  However, this implication is not actually a requirement of the concept of Free Will.

In order to clarify, let’s start with some definitions. What is Free Will? Free Will is the ability of agents to make choices unconstrained by certain factors. That is to say, the agent making a choice does so according to its own motives. What is a choice? A choice is a selection of one particular outcome from a field of possible outcomes. So, a Free Will agent is one which selects a particular outcome for a situation according to its own motives. There are two things which are noteworthy to our conversation, implicit in these definitions. First is that the choices are unconstrained by certain factors, but that this does not mean they are unconstrained by all factors. Obviously, any choice is going to be constrained in some manner. For example, if a child is offered a choice between a red ball and a blue ball, he cannot choose a green ball. His choice is constrained by possible options, or input. No proponent of Free Will would argue that this constraint removes the child’s liberty of choice. The second noteworthy concept is that these definitions say nothing at all about Free Will’s relationship to time.

If an agent makes a choice according to its own motives, that choice was made freely, whether or not it may have been possible to know how the agent would choose prior to the selection being made, and whether or not the future containing the choice was real prior to the selection being made.

So, then, what does it mean to make a choice according to one’s own motives? This is a concept which I find easiest to describe through mathematical means.  Let’s say we have some function $f(x)$. Whenever we plug in a value for $x$, the function selects an outcome based upon its own properties, and unconstrained by the properties of unrelated functions. So, regardless of the outcome of $g(x)$, the function $f(x)$ will always make its choices according to its own definition or motive. This function is therefore Free, despite the fact that it has a determinate outcome. Similarly, all possible points on the x– and y-axes over which the function is defined exist simultaneously, but this does not mean that $f(x)$ was thereby constrained by some unrelated function. Look at the plots of $f(x)$ and $g(x)$ in Figure 1, below.

Figure 1: The functions f(x) and g(x) are independent of one another. The outcomes selected by f(x) are free from the influence of g(x), and vice versa.

If we were to plug some value $x$ into our function $f(x)$, but the outcome that we received was some value which does not exist on the red curve in Figure 1, we would immediately know that our result was constrained by some outside influence. The function $f(x)$ was not free to produce a selection based solely upon its own motivations. This result might be represented mathematically as $f(x)+c(x)$, where $c(x)$ represents the constraint on our function.

On a Deterministic view, any particular agent could be similarly represented by a mathematical function. So long as the outcome of any choice is truly representative of that function, rather than having been artificially altered by some outside constraint, it is perfectly coherent to claim that this agent operated by Free Will. The results of its choices were solely dependent upon its own motivations, and were not controlled by another entity. The agent’s freedom is not impacted by the fact that it’s choices are knowable, nor by the fact that the dimensions by which it is measured exist together in their entirety. The only factor relevant to the choice’s outcome is the agent, itself.

Contrary to popular sentiment, Determinism and Free Will are not contending philosophies. Neither one contradicts or precludes the other, and the Compatibilist view is perfectly coherent. We are the navigators of our own course through time.

## 19 thoughts on “On Free Will and Determinism”

1. In quantum mechanics, the momentum of a particle is represented as a wave of probabilities. If you constrain the “free will” of a particle by forcing it to fly through a small aperture (confining it’s position probability wave), it compensates by fanning out it’s momentum probability, and in the case of light, where speed is constant, this equates to uncertainty in the target angle after passing through the hole. This is something anyone can do in a garage with a laser pointer and a vise. And so you see that determinism and free will are not only incompatible, but any attempt to know the present with precision results in a less-knowable future.

• Thanks for reading and replying! The implications of quantum mechanics on determinism are certainly interesting, and relevant to the discussion.

However, I do have to say that your conclusion, here, is wholly dependent upon your preferred interpretation of Quantum Mechanics. On the Cophenhagen Interpretation, which is the most commonly taught interpretation of QM, I’d say you have a point. However, many of the other popular interpretations– for example, the de Broglie or Many Worlds or Time Symmetric interpretations– do offer views of QM which are compatible with Deterministic philosophy.

2. Ignostic Atheist on said:

So while compatibilism considers the object in question as an agent as a whole, determinism considers the agent as the sum of his deterministic parts.

Let’s say we have some function f(x). Whenever we plug in a value for x, the function selects an outcome based upon its own properties, and unconstrained by the properties of unrelated functions. So, regardless of the outcome of g(x), the function f(x) will always make its choices according to its own definition or motive. This function is therefore Free, despite the fact that it has a determinate outcome. Similarly, all possible points on the x- and y-axes over which the function is defined exist simultaneously, but this does not mean that f(x) was thereby constrained by some unrelated function. Look at the plots of f(x) and g(x) in Figure 1, below.

The difference is that the function has a steady state, while the individual changes over time. Baby Pythagoras Boxer is not the same person as adult Pythagoras Boxer. He has changed over time by exposure to his external environment. So, f(x) at time t later becomes g(x) at time t+n. To say that your functions are free of external influence would be false. It can only be viewed as true if you take it moment by moment, more along the lines of f(t) = f(t-1)+x. At f(0), the individual is the sum of all prior experience. At f(1), the sum of all prior experience, plus that last moment, which was external, but has been made internal.

But I do agree, determinism and compatibilism don’t strictly compete. The fighting difference is that determinists feel that deep-sixing the notion of free will and replacing it with pro-active incarceration or rehabilitation, and forward thinking social programs, is better than retaining the status quo.

I think there are some fundamental problems with your understanding of functions, here. Firstly, unless the ‘x’ is a temporal parameter, f(x) will be exactly the same at time t as it is at time t’. For example, if f(x)=x^2 (as I used in my example graph in Figure 1) then it doesn’t matter whether it is 5:00 am on New Year’s Day of 1904 or if it is 7:26 pm on Halloween of 2014, the solution to f(2) will always be 4, and the solution to f(16) will always be 256, et cetera. No matter how much time passes, f(x) will never “become” g(x). Either the two are equivalent functions, or else they are not.

Even if a function is defined recursively with a temporal parameter, as in your example of f(t)=f(t-1)+x, all possible solutions to the function coexist alongside one another. I do not have to wait until time t’ in order to evaluate f(t’).

When you say that “the individual changes over time,” you are describing a mathematical function. Functions describe change over one or more measurable dimensions. Time is a measurable dimension. For example, a fairly typical function of time from Physics 101 would be d=vt. The displacement, d, is a function of time, t, at constant velocity, v. Now, it doesn’t matter if our experiment is actually at t=0 or at t=10 or anything in between, the value of d at t=5 will always be 5v. However, 5v is only one solution to the function. It is not the whole function d=vt. All possible values for d coexist, regardless of what time it is at the present. The function does not change. The solution to the function changes.

Similarly, the deterministic function for an agent, as I have postulated in the article, is not meant to be resolved at any particular moment. Rather, it is meant to describe the function of all space-time coordinates representative of our particular entity. Baby Pythagoras Boxer is, indeed, the same person as Adult Pythagoras Boxer– each distinction is simply an evaluation of the function over a particular range of possible coordinates, rather than over the whole range of the function.

• Ignostic Atheist on said:

No matter how much time passes, f(x) will never “become” g(x).

Exactly, and yet people change over time by factors that aren’t internal. Hence, your analogy that a person is f(x) doesn’t work.

Even if a function is defined recursively with a temporal parameter, as in your example of f(t)=f(t-1)+x, all possible solutions to the function coexist alongside one another. I do not have to wait until time t’ in order to evaluate f(t’).

X is arbitrary in this. It would vary moment to moment, and I admit that’s not clear. It could be viewed deterministically as a function of the universe at x, y, z, and t. Basically x is the universe affecting you.

When you say that “the individual changes over time,” you are describing a mathematical function.

Yes, but your conception suggests that the outcome is determined by the consistent properties within the function. I am saying that the properties aren’t consistent, and themselves vary as a function of the environment. It’s kind of like describing what you would do in a fight versus actually being in a fight.

This is a distinction between a compatibilist and determinist. The determinist views the environment as a whole, including the individual. The compatibilist views the individual as a whole, excluding the environment. Both sides admit the (more or less) deterministic nature of reality, but compatibilists put the individual in a bubble, and say, “If you stop looking beyond this point, there is free will.”

Rather, it is meant to describe the function of all space-time coordinates representative of our particular entity. Baby Pythagoras Boxer is, indeed, the same person as Adult Pythagoras Boxer– each distinction is simply an evaluation of the function over a particular range of possible coordinates, rather than over the whole range of the function.

It could be viewed as such, but if you keep the bubble around the Boxer of Pythagoras from infancy to adulthood, you’re slicing off the massive web of connections Pythagoras Puncher has accumulated over time from the surrounding environment.

• I’m going to try to enumerate my replies, for the sake of clarity. Please let me know if I miss anything.

1. People change over time in exactly the same manner as f(x) changes over x. Again, when I say that an agent can be represented mathematically on a deterministic point of view, time is one of the dimensions over which the function is defined.

2. If you intend for the ‘x’ to be variable in the function, then you are actually talking about a different function. Perhaps f(x,t)=f(x,t-1)+x, in which case the function would still be defined for all possible values along the x- and t-axes, at once, regardless of the momentary values of x and t.

3. As someone who gets paid to talk about what I would do in a fight and to literally get into fights every day, I can assure you that this fits precisely with my functional approach. My system of jiu-jitsu is the function. Plug in all the parameters, and the outcome flows naturally from the function.

Similarly, in the case of our agent, anything which is not consistent– a constant in the function– is a parameter of the function. Time, location in space, and outside stimuli would all be parameters in the function describing the agent, on a deterministic system.

4. I’m not sure what you mean by “compatibilist,” as you seem to be implying something wholly disparate from the definition I employed in the article. Perhaps you meant “libertarian incompatibilist?” Generally, a compatibilist IS a determinist, and one who does not see Free Will as being mutually exclusive with determinism. On the other hand, a libertarian incompatibilist asserts that determinism and free will ARE contradictory, and sides with free will against determinism.

5. Again, the function is meant to describe all of these interactions described by “the massive web of connections.” If it does not, it cannot properly be considered a deterministic function of the agent.

• Ignostic Atheist on said:

4. Not sure where you’re getting that from. I’ve explicitly said that the difference between compatibilism and determinism is in how useful they regard the concept of free will. I’m well aware that if you cut short the line of causation, you can regard the individual as the agent. Quite frankly, it’s how I function to get through life as a social animal. But, I recognize that if you take the line of causation far enough, you will leave the confines of that individual and be searching through the environment instead.

If I’m reading you right, you’re saying that f(x) represents the agent and all environmental inputs which have had an effect upon it, all the way back, with the x representing the inputs at any given moment. Not seeing where free will emerges in that notion – you’re still the result of a lifetime’s collection of external influences. That collection is “you” for sure, but it’s still made up of innumerable things which are not of you. Even when you get down to genetics, you might say your sequence of DNA is you, but it’s not. Half of it is your dad’s, half your mom’s. And theirs before them. When you say that f(x) represents the agent as he is now, you are cutting the line of causation between the way the agent is, and what caused him to be so. Again, I recognize that f(x) can be viewed as the agent, but I think that the recursive time function better represents the reality of the individual being an accumulation of environmental effects.

So while I don’t hold people directly responsible for what they do, I do recognize the value of creating incentives to toe the social line by enacting punishments, focusing on rehabilitation over vengeance, and positive social action to aid those who were born into environments that keep them down.

• On Compatibilism and Determinism,
The reason I am confused is because you keep contrasting Compatibilism against Determinism; however, Compatibilism is a subset of Determinism. A Compatibilist is a Determinist. As such, the idea of a “difference between compatibilism and determinism” is particularly confusing. It’s like saying “the difference between a dog and a greyhound.”

On Stimuli,
There is a difference between influencing a selection and controlling a selection. Again, if we are to use the mathematical example of f(x)=x^2, when the function is evaluated for any particular value of x, the function is the only thing responsible for the output value. So, we would not say that the number 2 is responsible for the fact that f(2)=4. However, if we were evaluating f(2) and received the number 1.3145, we would know that some outside constraint forced the function to produce a different output. If an agent can be described as a function, that function’s purpose is to evaluate input and to produce an output. So long as the output is truly representative of that function, it has been chosen freely. If the output is not representative of that function, then it has been constrained by an outside factor.

On Responsibility,
Given that I do assert the existence of Free Will, I absolutely hold people directly responsible for what they do. Given that I am a Determinist, I feel that punishment for wrongdoing– or the threat of such punishment– is just one more input stimulus into the function describing that moral agent. Similarly, social aid is an input stimulus. These influence, but do not control, the agent’s choices.

• Ignostic Atheist on said:

Yeah, I’d just have to say the distinction between influence and control strikes me as so much hand waving. Control is what you call it when you’ve lost track of all the influences which cause an action. Again, you’re creating a bubble around a meat bag and saying, “This is mine. All prior influences which I’ve internalized are no longer effects, but a single massive causal agent, which I will describe as f(x).” But, of course, the next moment will introduce new influence which will change the function that you will be using down the road. So, the function is the only thing responsible the output. Except in the future, when it is the function plus the influences between then and now which have altered its processes. And except for the past, where it is the function minus all that you’ve absorbed since then. None of what makes you, you, can be traced to anything that is you, so to say that you control outcomes is to cut the lines of causation and wrap a bubble around the present arrangement of your atoms.

Over this conversation I’ve been a bit confused as to whether you feel f(x) is the agent, or the agent plus the relevant parts of the universe. If f(x) is just the agent, then constraints (resulting in unexpected results) factor into future calculations, meaning the function going forward can no longer be f(x). If f(x) is the latter, a recursing function that internalizes prior influences, then the constraint is part of x, which means that the result is not actually unexpected. You might also view f(x) as the function over the lifetime of the agent, as opposed to going forward from the present. However, this would still fail to account for breaks in free will by outside constraints, and fail to account for how f(x) adjusts itself to the new course imposed by that break in free will. By your definition, f(x) does not account for these breaks, because the result would be unexpected, and it’s not as though f(x) will just necessarily continue on as before as though this break didn’t happen.

This has gone on long enough that I feel I should remind you: you know I love you, right?

• Thanks for the reminder, but I never doubted it for a moment! The whole purpose of this blog has been to foster just these sorts of conversations, and I’m extremely glad and honored to have your input, here!

I do, indeed, intend the f(x) analogy to apply solely to the agent, and not the agent-plus-relevant-parts-of-the-universe. And, accordingly, I would say that the whole idea of “future calculations” or “going forward” is not really cogent in discussing a function. The function is always just the function, no matter what time it is. Our f(x) never changes to become g(x). Again, using the mathematical example f(x)=x^2, the function will always equal x^2 regardless of the particular value of x involved. Saying “the function going forward can no longer be f(x)” is precisely akin to claiming “the function as x increases can no longer be x^2,” which is obviously incorrect. The function exists over the whole range of time, already.

I know that it can be difficult to think atemporally, given how the concept of time is quite naturally ingrained into our thought and language. However, try to remember that (on a Deterministic, Tenseless Theory of Time point of view) the whole of time is set. The future is not some unrealized potential waiting to occur, nor is the past some ever-increasing collection of bypassed present moments. These are simply particular values along a one dimensional axis. The whole system is static. What we think of as “change” is a particular description about the output of a function between an arbitrary set of points along a particular direction. The idea does not apply to the system, as a whole. “Change” is applicable to a function’s output, but not to the function, itself.

• Ignostic Atheist on said:

Saying “the function going forward can no longer be f(x)” is precisely akin to claiming “the function as x increases can no longer be x^2,” which is obviously incorrect. The function exists over the whole range of time, already.

The point being, f(x) is what happens if free will is not impinged. If free will is impinged, then the result is not what f(x) would otherwise produce. To say that this constraint will not affect the subsequent values of f(x) simply makes no sense, unless x already accounts for the break in free will. In that case, the result is not unexpected.

• I don’t believe that there can be such a thing as a “break in free will.” Either Free Will exists, or else it does not.

Continuing with my f(x)=x^2 example, let’s say that we are handed a graph which we are told was produced using this function. Now, let’s say that the curve of the graph matches exactly with what we would expect from f(x), except for a single point at x=3, where an anomalous point at (3,7) mysteriously exists on our graph. In this case, we can immediately see that x^2 was not free to produce this graph, and some external constraint has been placed upon it. It does not matter that the infinitely many other points have all been graphed precisely according to x^2; the fact that a single point does not conform to our function tells us that our function was not in complete control of this graph.

Similarly, if it is at all possible to break any agent’s Free Will, then that agent was never actually free, in the first place. Either an agent has free will, or an agent’s choices can be constrained by an outside factor. An agent cannot be both free and not free.

• Ignostic Atheist on said:

Free will that cannot be impinged upon is a useless concept. I mean really, what do you get? Everyone is at fault for any effect derived from them, no matter the circumstances which brought them to that point. It seems to me that you have to view the free will used by social society as an illusion just as much as I do, but in the opposite manner.

• I’d say there’s a fair difference between the concept of Free Will and the social concept of punishable offense. An action undertaken by a person might be punishable or it might be justifiable, but that does not remove the fact that the person committed the action of his own agency.

If I understand the direction you’re going, a good example might be a bank teller being held-up by a robber. The robber points the gun at the teller’s head and demands that she put all the cash in her drawer into the robber’s bag. In a Free Will sense, that woman is responsible for her choice to remove cash which does not belong to her and give it to someone else. However, her action would not be considered an illegal one, in a criminal court, due to the robber’s coercion. In a legal sense, the stimulus of the robber’s action justifies the action undertaken by the teller; however, in a philosophical sense, the teller was still free to choose whether she would give the robber the cash or not.

• Ignostic Atheist on said:

And I understand where you are coming from, but I don’t understand how it is in any way meaningful. It’s pretty much a tautology: a person will do what a person will do. In the game of why, you’ve stopped at one.

And never mind that that person is a transitionary collection of atoms and patterns. To view a person as a constant function separate from the rest of the universe is impossible, because you are constantly absorbing material and concepts from the universe (and excreting them back out). I guess the point I’m making is that the concept of a static, independent f(x) person does not empirically jive with reality. Then you can consider the question of dualism: if the material us is nearly entirely replaced every decade, what the hell is this unchanging f(x)? A soul? Is it a Platonic form that will last for eternity without physical reference, or does it end when our consciousness ends (if so, then how is it unchanging)?

You’re talking about something that has no reference in reality. If your flavor of free will is a real concept, then this f(x) must be real in some way. But again, f(x) is pretty much a tautology.

• Whether one agrees with Determinism or not, Free Will tends to mean that a person will do as a person will do. I’m not sure I understand the objection, here.

I don’t think it is quite right to say that I’m referring to a function “separate from the rest of the universe.” The purpose of a function is to describe a particular subset of the universe. When I plot f(x)=x^2 on a Cartesian plane, x^2 is not separate from the plane. Rather, it simply describes a set of points which exist within the plane. In the same way, the function doesn’t “end” (that is, it does not cease to exist) in the parts of the plane which it does not describe. We would not say that x^2 “ends” when y<0. The function I am proposing for an agent would similarly describe a set of space-time coordinates within the 4-Dimensional manifold that is our universe. The function does not cease to exist beyond the range for which it is temporally defined any more than x^2 ceases to exist when y<0.

The individual bits of material out of which a person is composed are not what makes a person who they are. If a person loses an arm in an accident, or undergoes a heart transplant, or has a cerebral hemispherectomy, no one considers the resultant entity an entirely different person. Similarly, the fact that our cells die and are replaced by new cells does not mean that we become different people than we once were. Generally, when we refer to a specific person, we are referring to the experiences and choices which that person has had and has made. That is, we are referring to the manner in which that person acts. This even comes out colloquially, in our speech. When someone's behavior changes radically and suddenly, you'll often hear their loved ones exclaim, "It's like you are a whole new person!" In that sense, a Deterministic function would do a much better job of describing a person than does their material constituency.

Interestingly, this does share some things in common with the classical idea of a "soul." Neither a function nor a soul is a physical entity. Both could be considered eternal and outside the influence of time. However, the classical conception of a soul ascribes some sort of sentience and intelligence to the entity which persists after the death of the physical body. A mathematical function, on the other hand, does not have sentience, nor does it have intelligence. A function cannot transmigrate into a new physical body, nor can a function go to Heaven or Hell. A function simply exists.

3. Not being any kind of mathematician, your argument about f(x) and g(x) and fancy graphs are lost on me. Is there any way to translate that into English?

• Basically, I am saying that regardless of whether the outcome of a choice is determinate or not, that choice may still be made by an agent unconstrained by outside factors.

Similarly, regardless of whether or not the future is actual or potential, the choice may still be made by an agent unconstrained by outside factors.

As such, Determinism and Free Will are compatible concepts.

4. I too agree that they are compatible. I don’t truly understand all the terms, but I am “determined” to “choose” things in a way that “determines” other things I desire.