Friday, November 13, 2020

The Eternal Society Paradox Argument: Symbolic Logic Approaches

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In August 2020 the Maverick Christian YouTube channel featured an interview on the Eternal Society paradox, something I’ve also talked about in article explaining why the past cannot be infinite. Here I’m going to dive into a symbolic logic approach to arguing for the premises of the Eternal Society paradox argument. This article is for logic nerds and less layperson-friendly than most of my other blog articles.

Defining the Predicates

The domain of discourse are the years of the past.
  • Py = the Eternal Society did the chant in a year prior to y.
  • Cy = the chant was done in year y.
  • Fy = the coin (indeterministic, probabilistically independent) was flipped in year y.
  • Hy = the coin came up heads in year y.
The propositional variables are as follows:
  • I = the past is infinite and beginningless.
  • L = the coin came up heads last year.
  • V = the coin came up heads every year.
  • D = (I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)])
    • In English: the past is infinite and for every year y: if a flipped coin came up heads in year y and the society did not do the chant in a prior year, then the society does the chant in year y, otherwise they do not do the chant in year y.
  • A = D ∧ ∀y[Fy]
    • In English: the Eternal Society engages in the Annual Coin-Flipping Tradition.
  • E = the Eternal Society (roughly, a society that has existed throughout the infinite, beginningless past and in each year, they can do what we humans can do in contemporary society, e.g. in any year they can flip coins and do chants) obtains.
  • S1 = Scenario S1 obtains (A is true and the coin comes up heads for the first time last year).
  • S2 = Scenario S2 obtains (A is true and the coin comes up heads each year of the infinite past).

Argument 1: If ◊S1 then ◊S2

Where a possible world is the way the world is or could have been like, the modal operator □ is such that □P means that P is necessarily true (true in all possible worlds), and ◊P means that P is possibly true (true in at least one possible world).

Here is an argument that if Scenario S1 is possible then Scenario S2 is possible. The justification for premise (2) below is that in the Annual Coin-Flipping Tradition the coin-flips are probabilistically independent and so any particular permutation of coin clips is possible, including a permutation where the coin came up heads each year it was flipped.
  1. □(S1 → A)
  2. □(A → ◊(A ∧ V))
  3. □(S2 ↔ (A ∧ V))

  1. ◊S1 conditional proof assumption
    1. ¬◊S2 indirect proof assumption
      1. □¬S2 5, equivalence
        1. ¬S2 6, T-reiteration
        2. S2 ↔ (A ∧ V) 3, T-reiteration
        3. ¬(A ∧ V) 7, 8, biconditional elimination
      1. □¬(A ∧ V), 7-9, necessity intro
        1. □¬(A ∧ V) 10, S4-reiteration
        2. ¬◊(A ∧ V) 11, equivalence
        3. A ↔ ◊(A ∧ V) 2, T-reiteration
        4. ¬A 11, 12, biconditional elimination
        5. S1 → A 1, T-reiteration
        6. ¬S1 14, 15, modus tollens
      1. □¬S1 11-16 necessity intro
      2. ¬◊S1 17, equivalence
      3. ◊S1 ∧ ¬◊S1 5, 18, conjunction introduction
    1. ◊S25-19, indirect proof
  1. ◊S1 → ◊S24-20, conditional proof

Sunday, August 9, 2020

Mathematical Argument for God Debunked?

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Stephen Woodford has a YouTube channel called Rationality Rules and he posted a video titled Craig's Mathematical Argument for the Existence of God DEBUNKED in which Woodford is himself responding to a Reasonable Faith video explaining that argument. In this article I’ll explain the argument (something like this is one of the reasons I retained my belief in God in moments of doubt) and respond to some of what Woodford said.

The mathematical argument for God’s existence

To get a better idea behind the mathematical argument for God’s existence I’m going to kind of use a computer analogy. Consider these two conceivable universes:
  1. A universe akin to a hard drive that has its ones and zeros randomly set; a chaotic jumbled mess, disorderly unpredictable behavior at every moment.
  2. A universe akin to a complex computer program with sophisticated mathematical algorithms directing behavior; a universe with consistent mathematical patterns ubiquitously imprinted into nature via physical laws such that it makes physics almost ludicrously successful in precisely predicting behavior (examples: relativity and quantum mechanics). The physical laws are akin to a program’s mathematical algorithms in that behavior is directed in an orderly and predictable fashion.
To get a clearer idea of what I mean by universe (2), consider this equation for the relation between mass, velocity, and kinetic energy.

Where m0 is the rest mass (roughly, the mass it has at zero velocity), c is the speed of light, and v is the velocity of the mass. Or for our purposes (well below the speed of light) it gets pretty close to this:

So for example the approximate kinetic energy values for the following mass and velocity would be the following:

K.E. (joules)mass (kg)velocity (m/s)

This provides a sort of mathematical elegance and robust consistency for the universe’s behavior. But we can conceive the kinetic energy relation being more like a randomized hard drive, where the kinetic energy values for various pairs of mass and velocity are assigned haphazardly with no meaningful pattern rather than fitting some neatly ordered equation:

K.E. (joules)mass (kg)velocity (m/s)

With the universe also yielding different kinetic energy values for the same mass/velocity pairs for different locations. We can conceive the relation being even more like a randomized hard drive in that the relation changes unpredictably from moment to moment. This is all still describable with math just like a randomized hard drive is with its randomly set ones and zeros, but this doesn’t have the same type of robustly consistent mathematical elegance as in the case where this mathematical algorithm is ubiquitously imprinted into the universe:

As you may have guessed, our universe operates like universe (2). Physics has been extraordinarily effective in predicting accurate and precise behavior thanks to the mathematical algorithms ubiquitously imprinted into nature. (The Reasonable Faith video describes this quite well at around 0:39 to 2:00, which among other things notes how scientists used math to pinpoint the location of a previously undiscovered planet, and Peter Higgs using math to predict an elementary particle which scientists found after exerting billions of dollars and millions of work-hours.) Conceivably, this scientific use of math didn’t have to be nearly as stunningly effective as we observe. So why is it?

For theists the answer is simple: the universe has this remarkable mathematical order because it was designed. For the atheist, the only viable option for this type of mathematical applicability is that it’s just a happy coincidence. But a happy coincidence of this magnitude strikes some people as...too coincidental to be very plausible.

The aforementioned Reasonable Faith video presents this mathematical argument for the existence of God (around 4:21 to 4:38):
  1. If God does not exist, the applicability of mathematics is just a happy coincidence.
  2. But the applicability of mathematics is not just a happy coincidence.
  3. Therefore, God exists.

Woodford’s Response

As I suggested earlier, Stephen Woodford of Rationality Rules responded to the Reasonable Faith video. For sake of time I’m not going to discuss everything Woodford says, instead focusing mostly on the argument from the universe’s mathematical order, but I would like to respond to couple somewhat off topic things.

Regarding providing an alternative explanation for the effectiveness of mathematics Woodford’s video (around 8:23 to 11:04) shows clips of scholars some of which include the following:
Sabine Hossenfelder: I don’t think it’s all that unreasonable that mathematics is effective in the natural sciences, because what is mathematics about? It is a way to describe patterns, to describe regularities, and that’s exactly what we do in the natural sciences.

Steven Weinberg: I don’t think mathematics can ever be regarded as an explanation in itself of anything, and this has not always been—well, understand, perhaps it’s even still controversial—physical theories aren’t eth way they are because of principles of mathematics. Principles of mathematics are the language in which we state our physical principles, and they are the way—the intellectual tools we use for calculating the consequences of those principles, but nothing is the way it is because of some mathematical principles.

George Lakoof: It’s [mathematics] not in the world The world is as it is. Let’s take a very simple case. Take a spiral nebula. The logarithmic spiral is not in the nebula, it’s in your understanding of the nebula. The marvelous thing about mathematics is that we can create mathematics with our brains that fiat phenomena in the world remarkably. It is not a miracle that that’s the case because we have the capacity to see and understand the world, to categorize it in terms of what our brains do, and then we can create a mathematics out of that in a systematic way using what our brains allow us.
None of that really answers the question at hand. For example, yes we describe regularities in the natural sciences, but conceivably these precise mathematical regularities didn’t have to exist, and their existence is exactly what is to be explained in the first place. This is no more an explanation for the consistent mathematical patterns in the universe than saying that the reason opium causes sleepiness is because of its dormitive powers, where “dormitive powers” just means it has the power to cause sleepiness. In philosophy this type of pseudo-explanation is called a “dormitive principle” where one reiterates the thing to be explained in different words, which potentially gives the illusion of an explanation where none existed. The rest of the clips, while they may say true or plausible stuff, also don’t answer why the universe has the remarkable mathematical structure it has, because again, the universe conceivably didn’t have to be this way (think back to the kinetic energy example, where the values for kinetic energy for given a mass/velocity pair could conceivably have varied from location to location or from one moment to the next).

Woodford said he has an explanation, but what is it? At around 11:50 to 12:02 in response to why the universe has such a stunningly elegant mathematical structure:
At the risk of sounding like a broken record, it’s because the laws of the universe are robustly consistent.
We kind of have a dormitive principle here. The Reasonable Faith video referenced Eugene Wigner’s paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences which had this:
It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities.
Yes, the laws of the universe are robustly consistent, so much so that we actually have mathematical algorithms ubiquitously imprinted into the universe in which physics is almost ludicrously successful in making accurate and precise predictions, but that is exactly what is to be explained. Reiterating the thing to be explained in different words is a non-answer; it’s the equivalent of, “Because I said so.”


Ultimately the only viable alternative to design for why the universe behaves more a hard drive imprinted with algorithms, rather than a randomized hard drive with ones and zeros assigned haphazardly, is that it’s just a happy coincidence. A proposed explanation that is actually just a dormitive principle is a non-answer, stalls progress, and rots the mind. To be fair Woodford does say this in his video he might be missing something (15:54 to 15:57). He is, but to be fair to Woodford again, I don’t think the argument from mathematics was argued as strongly or as clearly as it could have been in a number of cases, including the video Woodford responded to. I think the argument from the universe’s mathematical order becomes clearer when you contrast our universe with the way physical reality conceivably could have been like, that is, it could conceivably have been more like a randomized hard drive and a lot less like software running elegant mathematical algorithms directing everything in a more orderly fashion.

Wednesday, July 1, 2020

Why the Past Cannot be Infinite

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Relevance to Theism

A finite past bears relevance to the kalam cosmological argument (KCA) which goes like this:
  1. Anything that begins to exist has a cause.
  2. The universe begins to exist.
  3. Therefore, the universe has a cause.
Further arguments are given to show that the cause of the universe is (among other things) a transcendent personal cause. If we have adequate grounds for thinking the universe has a transcendent personal cause, this gives at least some evidence for the truth of theism. I’ve justified premise (1) in my anything that begins to exist has a cause article. In this article I’ll argue for premise (2) by arguing for a finite past.

Would a finite past mean that not even God is sempiternal (i.e. having existed for a beginningless, infinite duration)? Yes it would. One idea is that God is timeless sans creation. Note that if spacetime itself began to exist and our spacetime universe had a cause, that cause would have to transcend space and time. Whether you want to call this spacetime-transcending cause supernatural or not, such a cause would have to be something beyond the physical laws as we know them today. The fact that there is some sort of (at least) de facto supernatural cause beyond space and time creating the universe would seem to make atheism less plausible.

An Infinite Traversal

Some philosophy lingo: a potential infinite is a collection that grows towards infinity without limit but never actually gets there. For example, if you started counting “one, two, three…” at a rate of one number per second and continued indefinitely, the number you’re at would grow larger and larger without limit but you’d never actually arrive at “infinity.” An actual infinite is a collection that really is infinite, such as the set of all positive whole numbers.

Traversing an actual infinite region at a finite rate seems impossible. Suppose for example there were a road that starts at a particular location and is infinitely long. Someone named Jill Walker starts at the beginning of the road and walks at a rate of one meter per second. Will she ever traverse an infinite region? She will not; the distance (and time!) she traverses is a potential infinite only. What if she were given infinite time? The problem is that traversing an actual infinite amount of time can never happen. Even if she is given unlimited time she will never traverse an actual infinite amount of time or an actual infinite distance; both will be a potential infinite only.

A similar problem occurs with a beginningless past: for a beginningless, infinite past to exist an actual infinite amount of time would need to be traversed, which is impossible, and thus we never would have arrived at the present moment. Another way to look at it: imagine if we viewed a universe with an infinite past and rewound it, traversing it at the same rate as time normally goes but backwards. Could we traverse the entirety of the infinite past? The infinite past would be impossible to completely traverse even given unlimited time. Similarly, going the other direction would be impossible because it requires an infinite traversal and we never would have arrived at the present moment (or at any moment, since any moment in the infinite past has an infinite amount of time before it).

The Eternal Society Paradox

There are also various paradoxes one can make with an infinite past, an example of which is the Eternal Society paradox. Roughly (in the paper the Eternal Society paradox was published), an Eternal Society is a society that has existed for a beginningless, infinite duration of time and has the abilities of ordinary human beings in each year of its existence; e.g. in each year people in the society can flip coins, write books, sing songs, and pass on information possessed in the current year to the next year. Because of the society’s extremely modest abilities, it seems like an Eternal Society would be possible if an infinite past were possible (note that by “possible” in this article I’ll be referring to metaphysical possibility, as opposed to e.g. physical possibility).

Now imagine the Eternal Society has the following Annual Coin Flipping Tradition: each year they flip a coin and if it comes up heads, they all get together to do a particular chant but only if they have never done the chant before. If the coin does not come up heads they do not do the chant for that year.

The coin flips are probabilistically independent events, so any particular infinite permutation of coin flips is equally unlikely but also equally possible. Consider scenario S1 in which the coin came up heads for the first time last year. The Eternal Society gets together to do the chant for the first time. This seems like it would be possible if an infinite past were possible (an eternal society with the ability of ordinary humans, by which I mean the society has the ability of ordinary humans in each year of its existence, could surely do something like this), but this scenario is provably not possible.

Again, the coin flips are probabilistically independent events, so if scenario S1 were possible, then another scenario, that we can call scenario S2, would be possible: the coin came up heads each year of the infinite past. If the coin came up heads each year, did the Eternal Society ever do the chant? They would have had to have done the chant some year, because they would have done the chant last year if they hadn’t done it yet (since the coin came up heads last year). And yet any year you point to, there is a prior year in which they would have done the chant if they had not done the chant before. So they had to have done the chant (since the coin came up heads last year), yet they could not have done the chant (there is no year they could have done it), and so this scenario creates a logical contradiction.

Although scenario S1 is not directly self-contradictory, scenario S1 is impossible because it implies the possibility of a logical contradiction. The Eternal Society argument against an infinite past goes like this:
  1. If an infinite past were possible, an Eternal Society would be possible.
  2. If an Eternal Society were possible, then scenario S1 would be possible.
  3. If S1 would be possible, then S2 would be possible.
  4. S2 is not possible.
  5. Therefore, an infinite past is not possible.
One could deny premise (4) particularly since that seems to be the most vulnerable premise, but as the Eternal Society paradox paper says, “Surely there is something metaphysically suspicious about an infinite past if an eternal society with the abilities of ordinary humans can actualize a logical contradiction.” The idea that an infinite past is possible but an Eternal Society is not possible strikes me as overly ad hoc due to the Eternal Society’s extremely modest abilities (the abilities of ordinary humans in each year of its existence).


While there is also scientific evidence favoring a finite past, philosophical arguments seem to provide a strong case for temporal finitism (the view that the past is finite). For a beginningless, infinite past to exist an actual infinite amount of time would need to be traversed, which is impossible, and thus we never would have arrived at the present moment. Moreover, the Eternal Society paradox shows that an eternal society with the abilities of ordinary humans would have been able to create a logical contradiction, which strongly suggests that an infinite past is metaphysically impossible.