What is determinism?

The question mark in the title is not rhetorical. As a result of the recent discussion of free will on these pages, I’ve come to realize that I don’t really have a very clear understanding of this very basic concept — which is a big problem because my argument (that free will is inconsistent with both determinism and indeterminism) doesn’t mean anything if I can’t say what determinism is.

Here are some disjointed notes, which I hope will eventually congeal into something useful.


The problem is that determinism and indeterminism are theories about what could happen, but the only data we have is about what actually did happen. (Actually, we don’t even have that — it has to be inferred — but let’s simplify and say we do.) It’s like looking at a single game of chess, having no previous knowledge of the rules of the game, and trying to figure out what moves the players could have made at each point in the game and whether or not they could have made different moves than they did in fact make. Of course, a single game contains only a very limited subset of all possible chess moves, and any given game has infinitely many different sets of rules which are equally consistent with it. How can you distinguish moves which are forbidden by the rules of chess from moves which are legal but which the players happened not to make in this particular game? If neither player happened to castle or to capture en passant, how could you infer that such moves were possible? If in this particular game, no pawn advanced two spaces except on its first move and no queen moved diagonally except on its first move, how could you conclude that this is a rule for the pawns but only a coincidence for the queens?


We can conceptualize a universe as an ordered series of numbers, with each value representing a particular state of the universe and consecutive members of the series representing consecutive points in time. (Yes, we are simplifying rather drastically by making time finitely divisible and ignoring relativity, but you have to start somewhere.) Which such series can be called deterministic?

A starting definition might be that a series is deterministic if it is possible to derive the whole series by applying an algorithm to a subset thereof.

The series <1, 2, 4, 8, 16, 32, 64, 128> is deterministic by this definition. Choose any one member of the series, apply the algorithm, and you can derive the whole series.

The series <5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34> is also deterministic. Here, if you know any two members of the series, you can derive all the others by a simple algorithm.

But what about <92, 75, 30, 92, 46, 87, 69, 89, 2, 48, 630>? The first ten numbers were given by a random number generator and follow no pattern; the 11th and final number is the sum of the others.  We would intuitively hesitate to call it a deterministic series, but it differs from the previous two only in degree; if you know all but one of the members, the remaining member (whichever it may be) can be derived by an algorithm. Are there perhaps degrees of determinism — the less you have to know in order to derive the whole series, the more deterministic it is?

And what about <92, 75, 30, 92, 46, 87, 69, 89, 2, 48>? Yes, that’s the same series of random number as before, minus the checksum. There’s no pattern to be found. But couldn’t we still devise a complicated ad hoc algorithm by which the whole series could be derived from one or a few of its members? Isn’t that always possible? Of course, then all the information would be in the algorithm, not in the members from which we are supposedly deriving the set — but where do we draw the line between “real” determinism and this bogus ad hoc variety?

It’s beginning to look as if determinism is a question of data compression: if lossless compression is possible, the series is deterministic. Unfortunately, I don’t know anything about data compression.

For an infinite series of numbers, the question of determinism is clearer. Since the ad hoc algorithm for an infinite series of random numbers would itself have to be infinitely long, we can say that an infinite series is deterministic if it can be derived from a finite subset of its members by a finite algorithm.


In the examples of deterministic series given above, the whole series can be derived from any adequate subset, whether it comes from the beginning, middle, or end of the series.

But in the real universe, people (trained physicists excluded) generally believe that determinism is directional, that it runs in only one temporal direction. Which direction that is depends on who you ask and how you phrase the question. Most people will tell you that causes precede effects, that the past determines the future in a way that the future does not determine the past. Most people will also tell you what seems like the exact opposite: that present evidence can be used to reconstruct the past with a much higher degree of certainty than that with which it can be used to predict the future. Most people — and I am most definitely including myself in this group — obviously don’t know what the hell they’re talking about when it comes to determinism.

Conway’s Game of Life is a good example of a system with unidirectional determinism. Given a snapshot, a simple algorithm can tell you what will come next, yielding perfectly accurate predictions of the arbitrarily distant future — but nothing can tell you what came before.


Filed under Philosophy

12 responses to “What is determinism?

  1. Luther

    Well, I know a bit about this, so let me see if I can help. I’ll address compression, a mathematical definition of causality, and the notion of inputs.

    Compression is the process of re-posing data in a new basis wherein a datum you expect to see is smaller than a datum you do not expect to see. In other words, we codify expectations in a codec and transmit only each datum’s deviation from expectation.

    If our expectations are perfect then we compress the entire universe down to zero and are left with only the codec, no data. I think this is a common notion of “deterministic.” Another may be “Is there an algorithm guaranteed to compress the entire past and future?”, which (via the pigeonhole principle) is the same question as “are there any states that, over infinite time, are less common than others?” If there is anything approaching a law anywhere in the universe, the answer is an unambiguous yes.

    Codecs need not themselves be finite; they need only produce any finite prefix of the output in a finite amount of time. We have no accepted model of algorithmic size.

    Let me suggest a different formalism, common to the fields of operator theory, dynamic systems, and differential equations. You postulated the universe as a vector of numeric states; let’s add to that a time-invariant operator and an unmodelable input vector.

    The operator is a function f(s, i) that outputs the difference between consecutive states: the operator applied to (92, 75, 30, 92, 46, 87, 69, 89, 2, 48) would map to (-17, -45, 62, -46, 41, -18, 20, -87, 46, ??). I got these by simple subtraction (the discrete equivalent to taking a derivative).

    A system is considered to be causal if there exists some version of the operator that, for sufficiently large n, need consider only the first n states (and inputs) in order to determine the nth element of the output. For example, the series (1, 2, 4, 8, 16, 32, 64, …) is causal, for n = 1: the output is simply the same as the input. The random sequence you proposed was likely generated by a computerized pseudo-random number generator, probably the Mersenne twister which is causal for n = 624.

    As with codecs, the size of the operator is not well defined.

    With the arguable exception of a few inevitability arguments in time travel stories, I have never met a fiction, philosophy, or religion that did not assume the universe was causal.

    Now for inputs. Suppose I modify the n = 1 sequence (1, 2, 4, 8, 16, 32, 64, …)’s operator to take unmodelable input that is either 0 or 1, which gets added to the output. My sequence might now be (1, 2, 5, 10, 21, 43, 86, …). Virtually all engineering applications assume such an unmodeled input, generally to cover portions of the real world they decided not to model, such as wind.

    I think the most probable meaning of “deterministic” is “casual without inputs.”

    Quantum physics offers a proof (by contradiction, and hence applicable only to those who believe in classical logic, not intuitionistic or constructivist logicians) that inputs exist (and lots of them!). Every quantum event requires an impossible-to-be-modeled-within-the-universe input at each and every quantum event. They call these inputs “probability” and, since they’ve proven they cannot be individually modeled, consider only the trends in their behavior, not details nor source.

    I believe most free will believers would model free will as an input to the universe. If they mean this literally, there is by definition nothing more that can be derived about free will because it lies outside the universe itself.

  2. Luther, thank you. Yours is precisely the sort of well-informed comment I had hoped to receive.

    Some of your specific examples didn’t get posted, though (“the operator would map to .” and “My sequence might now be .”) I’m guessing you used angle-brackets, which were mistaken for html tags and deleted. Could you try posting them again? (Update: Luther has now done so, and I have edited his original comment accordingly.)

    It’ll take me some time to fully digest this, since I don’t have the mathematical training you do, but the model of free will you suggest doesn’t seem to me to be true free will. If an input comes from outside the universe, then it doesn’t come from the person who is supposedly making the decision. It’s like saying a chessman has free will because its moves are determined by input from outside the chessboard.

    (And, no, I’m not trying to pass off pseudo-random numbers as random. We’re strictly on the up-and-up around here, and my numbers are random as advertised, generated by random.org using atmospheric noise.)

  3. Luther

    A re-try on the examples:

    the operator applied to (92, 75, 30, 92, 46, 87, 69, 89, 2, 48) would map to (-17, -45, 62, -46, 41, -18, 20, -87, 46, ??). I got these by simple subtraction (the discrete equivalent to taking a derivative).

    For example, the series (1, 2, 4, 8, 16, 32, 64, …) is causal, for n = 1: the output is simply the same as the input.

    Suppose I modify the n = 1 sequence (1, 2, 4, 8, 16, 32, 64, …)’s operator to take unmodelable input that is either 0 or 1, which gets added to the output. My sequence might now be (1, 2, 5, 10, 21, 43, 86, …).

    Per the free will item, if I understand you are saying free will can’t be part of the state because the state is acted upon, not acting, and if it isn’t part of the state then the state doesn’t have free will. But it seems to me that your chess piece does have free will, a will named WmJas. That said will is not part of the chess universe doesn’t seem to make much difference in it’s being free will.

  4. It seems to me that in terms of your formalism every conceivable series of states is causal. Take your example, where the nth element times two plus the input yields the (n+1)th element. If we remove the stipulation that the input must be either 0 or 1, this operator can be made to yield any sequence we please, including any given sequence of random numbers.

    It seems, further, that any finite series of states can be considered not only causal but deterministic (defined as “causal without inputs”), since all the work which was being done by the inputs can also be done by a sufficiently complicated ad hoc operator.

    Have I misunderstood something?

    Regarding the chessman’s free will, I see what you’re getting at, but I don’t think it quite works. If the inputs controlling the chessman were coming from a simple, perfectly predictable computer program, we wouldn’t say it had free will. The chessman would have free will only if the source of the inputs itself had free will — which would mean that it, too, would have to receive inputs from outside its universe. If the chess universe requires inputs from the physical universe, and the physical universe requires inputs from the spirit world, does the spirit world also require inputs from yet another universe? Either we have an infinite regress of universes, or else things have to terminate somewhere in either predictability or randomness, neither of which is the same as free will.

  5. Luther

    The presence of causality is always a postulate. There always exists a system that can generate any arbitrarily-long finite sequence of arbitrary outputs, but said system cannot necessarily produce future outputs correctly. Hence mathematical causality is a property of a model of a system (as all mathematics works only on models, not reality per se), which (per Occam’s razor) is preferred to be the simplest model consistent with observation; simple is taken to mean both mathematical simplicity of the operator and minimized influence of the input.

    As for the chess-playing-program argument against input-based free will, you are implicitly utilizing the axiom that inputs must come from a setting admitting the same laws of mathematics and logic as the present universe: that is, that they are not inputs to the mathematical system for they are subject to the same laws as the present system. Quantum physics tells us that at least the input that selects locations of quantum wave collapse cannot be modelled by any extant branch of mathematics or logic beyond an aggregate probability function; the existence of any such in-universe model would invalidate quantum physics and any reasonably similar theory as an explanation for many observed phenomena.

    things have to terminate somewhere in either predictability or randomness

    That assumes the excluded middle “for all X, predictable(X) or random(X) = true”. Why? Both why only these two, but also why either of these two? Why do you accept either deterministic systems as being extant or randomness as being extant? What does random mean? If random means only “cannot be predicted by any deterministic analysis of past observation” then what does determine it, or is it “undetermined”, and if undetermined what does that mean? Also, what does predictable mean? Do we have to be able to compute a mathematical model of events for them to count as predictable, or just check the math is not violated, or just have faith that math exists even if we can’t compute it?

    Allowing that I am likely wrong, I see you saying “predictable or not predictable” and questioning the meaning of predictability in this post. But I also see you defining “not predictable” as incapable of having meaning/freedom/etc, which only follows logically if said attributes are demonstrated or defined to be subsets of predictability.

  6. i’ve left a ( comprehensive ) nonsensical rant at : http://transamoebae.blogspot.com/2011/02/determinism-fate-or-phat.html

    But basically; as mentioned in the forward of the above rave,
    i suspect that you understand Determinism perfectly well,
    But, like too many philosophers, find D- too abhorrent to accept.

    The Correct Conclusion: Ignore it.
    Reality is Wrong.

  7. Dad

    I’m sorry to be jumping into the discussion so late, but I only just noticed this entry. Here are some of my thoughts on the subject.

    It seems to me that free will has nothing to do with being predictable or unpredictable. It is possible to have free will and be predictable at the same time. The fact that I predictably choose not to drink poison in no way implies that I have no choice in the matter.

    Nor does it have anything to do with being influenced by something else. If you hold a gun to my head and tell me hand over my money, I am still free to do whatever I want, in spite of the fact that I may rather conveniently choose to do exactly what you so eloquently suggest.

    Nor does it have anything to do with the possibility of doing something else. In fact the whole idea that I “could have” decided to do otherwise seems to me to be a meaningless counterfactual which only serves to cloud the issue.

    It also has nothing to do with what happened before. In a given situation I either possess the power to freely choose or I do not, and this is completely independent of how that situation or I came to be.

    So what does it have to do with? In my opinion, free will cannot be meaningfully discussed separate from consciousness. The reason I believe a refrigerator has no free will is not because I can predict its behavior but because I don’t think it is conscious. It isn’t free to do what it wants simply because there is no ‘it’ to want anything. The same applies to an arbitrarily complex computer program. Does a dog have a will? That depends on whether it is conscious or not. And consciousness happens to be something completely outside of the materialistic model of the universe–something it can neither define nor detect, let alone explain or predict.

    To have a will, we must be conscious, possessing desires, and have the ability to act in accordance with those desires. But there is more to free will than this. Free will as defined by the gospel requires the additional ability to choose to some degree in opposition to our desires. A dog is conscious and has desires and a will, but it is compelled to choose in accordance with its desires. We however are not. People are to a greater or lesser extent free “to act for themselves and not to be acted upon”, and therein lies the purpose of our lives here on earth. The act of choosing affects not only what we do but what we become. Each decision we make in accordance with our base selfish desires strengthens those desires and weakens our ability to choose against them in the future, until we are eventually made slaves to them–having a will but being compelled to use it in accordance with our desires. “Now this is what is meant by the chains of hell.”

    Righteousness refers to our making those choices which strengthen our ability to choose and gain more and more mastery over our desires, until eventually we are completely free. “If ye continue in my word, then are ye my disciples indeed; And ye shall know the truth, and the truth shall make you free.” “If the Son therefore shall make you free, ye shall be free indeed.”

  8. Dad, thanks for your comments. It seems that we are in agreement that the traditional idea of free will, with its oxymoronic concept of non-deterministic causation, is garbage, and that free will is more usefully defined as the ability to do what you want — or, since we want to avoid the slippery idea of “ability,” with its counterfactual implications about what one “could have done,” free will means actually doing what you want to do.

    I don’t think people ever really choose contrary to their desires — not contrary to all their desires, anyway. It often happens that we desire several different things, and that choosing to satisfy any one of them would preclude satisfying the others, and in such cases we may act contrary to desire A — but only in order to satisfy desire B. I don’t want to go to work today, but I will anyway, because my desire to stay home and work on other things is overridden by my desire to keep my job and my reputation. This kind of thing is consistent with free will, but when people really act contrary to all their desires (for example, when someone with Tourette syndrome shouts out curse words), they are not exhibiting free will.

    By the way, dogs also have the ability to suppress one desire in order to serve another. I have seen a man put a dog biscuit on his dog’s snout and command the dog to sit still — which the dog did for several minutes, despite obviously wanting very much to eat the biscuit. Finally its master gave it permission to eat the treat, and it did so. The dog’s desire to please its master was strong enough to override its desire to eat the biscuit. Is that really any different from what we do when our nobler desires keep our baser ones in subjection?

  9. Dad

    Although we don’t have terminology for it, but I think we experience two kinds of wants. I’m referring to the difference between what I want and what my passions want. I know that smoking is killing me and I want to stop, and yet at the same time I don’t want to. I think there is a difference between being a helpless slave to a passion (and thus having no free will in that matter) and being able to resist the passion and yet choosing to give in to it. That difference is free will. The distinction between the two states is obviously subjective (as all of consciousness is), but I think it is real nonetheless and central to our purpose on earth.

  10. Earlier you said

    Nor does it have anything to do with the possibility of doing something else. In fact the whole idea that I “could have” decided to do otherwise seems to me to be a meaningless counterfactual which only serves to cloud the issue.

    But this idea of “being able to resist a passion and yet choosing to give in to it” depends on this “could have” concept which you had dismissed as meaningless. Apparently we still haven’t arrived at a coherent understanding of free will.

  11. Dad

    Your point is well taken. That is what I meant by my statement that it was subjective, just as consciousness is. When you put a gun to my head, it does not deprive me of my free will. I am still able to oppose your commands even if I choose not to. By this do not mean something vague about probability or alternate universes.

    This whole subject reminds me of the definition Ronald Reagan once gave of an economist as someone who sees something work in practice and wonders if it would work in theory. I am indeed conscious and possess free will. I know it, and what’s more, you know it. Not by definitions or proofs but by common experiences of the most direct and undeniable kind. More direct and undeniable than any microscope or any proof. And any system of reasoning that denies it is deeply and irredeemably flawed, for it denies reality. It is simply a smoke screen created in a vain attempt to rid myself of the glorious and terrifying knowledge that I am indeed responsible for how I act.

  12. Well, in a sense Reagan’s economist is asking a valid question — so long as his purpose is to test his theory, not (as Reagan was obviously implying) to decide whether or not to ignore the data of experience. If all you know is that something works in practice (that is, in the current situation), then you have no way of knowing whether it would still work if the situation were changed. “X works because of Y, and if Y were different it wouldn’t work” is much more useful knowledge than “X works now, so just to be safe we’d better not change anything.”

    My motive for trying to figure out free will is not to determine whether or not I have it (like you I assume that I do, even though it’s not entirely clear to me what that means), but to determine whether or not my model of reality is consistent with it. Many of my ideas about the world — that there is no God, that the soul is dependent on the brain, that the passage of time is an illusion — have been challenged by people who say, “If that were true, there could be no free will.” To evaluate such challenges, I need to know more than just the bare fact that we have free will; I need to know what it is, why we have it, and which possible worlds are consistent with it. The same goes for consciousness. Of course we have it, and so a statement like, “If there were no spirits, there could be no consciousness” is an attempt at a reductio ad absurdum; to know whether or not the reductio is valid, I need to have some idea of what consciousness is and what makes it possible for it to exist.

    Consciousness and free will are not quite on the same footing, though. My subjective knowledge that I have consciousness is sufficient, since consciousness is a subjective thing. Free will, though, seems to be a claim about something more than subjective experience; it involves some kind of proposition about causation and possibility. If it seems to you that you are conscious, then you are conscious; the seeming is inseparable from the fact. The same cannot be said of free will.

    What would be different if I were not conscious? The answer is immediately obvious, though I can’t express it in scientific terms. What would be different if I didn’t have free will? That one isn’t so obvious or easy to visualize. (When I try to imagine it, I picture myself watching my body do various things and thinking, “Hey! Why is my body doing this? This isn’t what I want to do!”)

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