Tag Archives: self-reference

The past isn’t your master (part 3 of 5ish)

The law of causality, I believe, like much that passes muster among philosophers, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm.

Bertrand Russell, Selected Papers

Scientific determinism isn’t sufficient for causality, as we pointed out last time. That’s because the laws of nature we actually have allow us to derive the past states from a complete description of the present, every bit as much as they allow us to derive the future from the present. Consider some very simple possible universes called checkerboard worlds (modeled after Carroll [2010: 127]), shown in the following figures.  They have one dimension of time, which will be displayed along the vertical, and one of space, along the horizontal.  To establish conventions, say that time t increases from t=1 at the bottom to t=10 at the top row, and position increases from x=1 at left to x=10 at right.  Discrete spacetime regions can have either of two states, portrayed as white or gray.  Our job as scientists is to come up with a most elegant statement of the rules, or patterns, embodied in the universe.  We can further imagine that after hazarding a hypothesis, more of the universe (more time and/or space) will be revealed to test it. Our first checkerboard universe is C, below.

state(t,x) = state(t-1,x-1) (or state(t,x) = state(t+1,x+1)

The law of nature of C is simple, and shown in the caption. Note that it can be stated as a rule for deriving later-time descriptions from earlier ones, or as a rule for deriving earlier descriptions from later ones. Let’s call that kind of law of nature bidirectional in time. That doesn’t mean that the direction of time make no difference. If we flip universe C in time, around the middle of the diagram, we get C-prime:

state(t,x) = state(t-1,x+1) (or state(t,x) = state(t+1,x-1))

Note that the law is different, in that the state at position x matches the previous state at position x+1 in C’, not the position at x-1. In three spatial dimensions, if we flip all spatial directions at once, physicists call that parity reversal.  Universe C has Parity-and-Time (PT) reversible laws.  The best accounts of the laws of physics of our actual universe,  general relativity and the Schrödinger equation of quantum mechanics, are Charge-Parity-and-Time (CPT)-reversible.

It’s worth staring at picture C’ and asking:  does row 8 make row 9 have gray cells at positions 4 and 8?  Or does row 9 make row 8 have gray cells at 5 and 9?  Or does the whole metaphor of physical states bossing each other around lose its grip here?

Let’s look at a checkerboard universe that doesn’t have bidirectional laws. This one has four possible states for any spacetime location, which will be represented as white, light gray, dark gray, or black.

irreversible dynamics and information loss

A black pattern moves to the left (lower x) with time, but ends if it collides with a dark gray column. Similarly, a light gray pattern ends if it collides with a dark gray column. And a dark gray square can appear “out of nowhere” (that is, when no other dark gray square is within +/- 1 spatial location) – at least when the local neighborhood contains only white squares. It looks like universe M might contain at least one probabilistic law: something like “if x-1 through x+1 are white squares at t-1, then the probability of dark gray at t,x is 0.01”. If not for this last feature, universe M would have been unidirectionally deterministic. Information about the past would be lost, but the future could still be derived from the present. But with the addition of the probabilistic law, universe M loses information in both temporal directions.

M is for macroscopic. The world we can observe with our unaided senses, and which we are accustomed to acting on and caring about, is like M. In the human-scaled world the same effect can come about from multiple causes, and different results can issue from the same situation. On the latter point, to vary our earlier example, a cup of cool water could be caused by leaving a cup of cool water in place, and no heat exchange taking place. Or it could be caused by leaving a cup of hot water with some ice in it for a long time, instead.  So, tracing from the same state (same in macroscopic terms) and going backward in time, different possibilities appear, which all converge on the same present state. And likewise forward in time: the weather, and human decisions, are notoriously unpredictable. We can offer Alice the choice of lunch destinations, and sometimes she picks Mexican and other times Korean, without any observable difference (such as how long it’s been since her last visit to these places) to explain it. Her futures diverge from the same macroscopic present state.

It’s the convergence of macroscopic causality – the fact that the cause (ice water left in the room) guarantees the effect (cool water) but not vice versa – that leads to the idea that the cause is master, and the effect its slave. From our everyday experience, we infer that there is a power differential which is explained precisely by time-order: the past has power over the present, which has power over the future, in a one-way chain of relationships. And that’s a reasonable inference to a simple and elegant theory which neatly fits the macroscopic data – but it’s wrong, as our best theories of microphysical processes reveal. It’s lower entropy, not being in the past as such, which explains why different past states converge on the same macroscopic present state. (More on that next time.)

When someone tells us that the divergence from the present state into multiple future states is an illusion, but neglects to tell us (or notice) that the convergence from past to present states is also an illusion, they’re telling half truths. Half truths can be misleading. This one is.

The Future-Rewound Argument

“I can prove you don’t have free will,” says your strangest friend. “If you had free will, that would mean you’d be able to do something different, even keeping all facts about the future beyond that act fixed. So for example, you think it was up to you where you took your last winter vacation. But if we hold fixed the fact that you were in New York on December 31, and rewind the movie of the universe from there, we find that necessarily, you arranged to travel to New York earlier. So in fact, you had to arrange to travel to New York.”

That’s a crazy argument, which no one would make. The crazy is easy to spot, it’s the idea of keeping all facts about the future beyond that act fixed. You shouldn’t keep those facts fixed, because they are not independent of your decision. But in a universe of bidirectional-in-time fundamental laws, the complete (microscopically detailed) past state of the universe is also not independent of your decision.

But many future facts, like being in New York, are not only not independent of your decision, they are caused by your decision. Causation involves an additional condition: a one-way relationship from cause to effect. Maybe it’s only this one-way causation from your decision to the future that frees you from the need to “keep fixed” those future facts? No: a symmetric relationship will suffice. We can see this in the next two thought experiments.

Self-Referential Pie Chart

inspired by Randall Munroe, XKCD comic https://xkcd.com/688/ (hat tip: Ismael 2012)

The pie chart makes two statements, both of which are true, and it has two pie slices of different size. The size of the reddish slice doesn’t cause the corresponding statement to be true in an asymmetric way: the reddish slice being the size it is, is identical with the truth of the corresponding statement. Once we have a convention that the color of a slice stands for the portion of the chart which is that color, we don’t even have to leave room for a statement written in the same color-range.

Readers are invited to make their own self-referential charts. Be careful in sizing portions and choosing colors: you wouldn’t want to get it wrong! (In the original XKCD comic, Randall Munroe makes his pie chart only partially and indirectly self-referential. So he actually did need to be careful. I call self-nerdsniping.)

The same logic of self-reference that frees us, in this chart-making, from the need to match an independent reality, also applies to our relation to the past under time-symmetric determinism.

Betting on the Past

In my pocket (says Bob) I have a slip of paper on which is written a proposition P. You must choose between two bets. Bet 1 is a bet on P at 10:1 for a stake of one dollar. Bet 2 is a bet on P at 1:10 for a stake of ten dollars. […] Before you choose whether to take Bet 1 or Bet 2 I should tell you what P is. It is the proposition that the past state of the world was such as [to correspond, according to laws of nature, to your action to] take Bet 2.

Ahmed 2014, p. 120

If we take Bet 2, we pocket a dollar. Our usual ignorance about which present states relate to microscopic past details has been removed, thanks to the logic of self-reference. Instead of trying to control a specific macroscopic past event – which is how we usually think about “affecting the past” – here we refer to a present macroscopic event, our taking Bet 2, and work backwards to refer to a widely scattered set of past events, down to the microscopic details.

It’s vital that proposition P is about you. If Bob’s proposition is about what Alice will take when offered this bet, it no longer makes sense to take Bet 2 unless you are Alice. What is up to you depends on who/what you are, including where you stand in the universe. This shouldn’t be surprising.

You might wonder how Bob is supposed to know that the past state of the world was such as to lawfully correspond to your taking Bet 2. But that is easy, if we suppose that Bob is a scientific determinist. He will take our choice as sufficient evidence. Does this mean that we only care about the future, i.e. Bob’s reaction? No: we are honest bettors, who want to take Bob’s money, but only by genuinely satisfying the winning conditions of the bet.

Let’s go back to those counterfactuals. What would happen if we had taken Bet 1? We would have lost a dollar, that’s what. The experiment can be repeated as many times as you like: it would support the hypotheses that “if you had taken Bet 1, you’d lose money” and “if you had taken Bet 2, you’d win money”. Normally, experiments don’t have logically foreseeable results like this; we normally need to know what the specific laws of nature are, not just that they are deterministic. But apart from that, these experiments support “would” statements just as other experiments support more ordinary statements such as “if I dropped this cup, it would fall.” (Remember, even well grounded scientific laws and meta-statements (like determinism) about laws can be supported or undermined by experiments.)

If we can show in more detail that there’s no need to posit an objective Power Hierarchy of Time in which earlier times rule over later ones, then counterfactuals can reach into the past just as easily as into the future. This doesn’t mean that we can change the past, as if some particular past time could be one way – Kennedy was assassinated on Nov 22, 1963 – and then be another way – Kennedy retired after two full terms. Nor does it mean that we can affect the past, if “affect” is a causal verb. It just means that if we had taken Bet 1, we would have lost a dollar. The fact that the consequent follows from the antecedent plus the natural laws, seems like a sufficient reason to accept that counterfactual.

It’s worth noting that even under time-symmetric determinism, the proponent of the Consequence Argument has one last-ditch option to preserve premise

(2) The distant past state of the universe is such that, for every action A you could take, if you did A, that past state would still obtain.

They can assert as an additional premise that there is only one action you could take: the one you did take. So in other words, they simply assume you have no options – they don’t need no stinking reason! This eliminates the Consequence Argument in favor of a Consequence Assertion. There is nothing to recommend such an assertion.

Next up: we show in more detail that there’s no need to posit an objective Power Hierarchy of Time in which earlier times rule over later ones. Hint: entropy explains the irreversible behavior of macroscopic processes. Second hint: the scientific and engineering approach to understanding systems explains how we understand causality.


Ahmed, Arif 2014. Causal Decision Theory and the Fixity of the Past, The British Journal for the Philosophy of Science 65/4: 665–85. https://doi.org/10.1093/bjps/axt021

Sean M Carroll, From Eternity to Here: the Quest for the Ultimate Theory of Time. New York: Penguin, 2010.

Ismael, Jenann. Decision and the Open Future, in The Future of the Philosophy of Time, ed. Adrian Bardon. New York: Routledge, 2012.

Munroe, Randall, xkcd: Self-Description. https://xkcd.com/688/

Pearl, Judea. Causality: Models, Reasoning, and Inference. New York: Cambridge University Press, 2000.