Thursday, December 24, 2020

An Objection to the Eternal Society Paradox

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Intro



The Eternal Society Paradox is one in which an eternal society with the abilities of ordinary human beings in each year of its existence uses its modest abilities to create a logical contradiction, thereby casting doubt on the metaphysical possibility of an infinite past. If the past is finite, this in turn can be used as part of an argument for the universe having a transcendent personal cause. Somebody wrote an article that among other things critiqued the Eternal Society Paradox argument against an infinite past…sort of. The original link is here but in case something disastrous happens, the archived link is here.

Background



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. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise 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 for the Eternal Society practicing the aforementioned Annual Coin Flipping Tradition. 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 for the Eternal Society engaging in the Annual Coin Flipping Tradition. 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.
The Eternal Society Paradox Argument Against an Infinite Past is a deductively valid argument—the conclusion (line 5) follows logically and inescapably from the premises (lines 1-4). A sound argument is a valid argument with all true premises, so the only way the argument can fail to be sound is with a false premise.

One could deny premise (1) 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).

The Rebuttal



When using the phrase “Eternal Society Paradox” the author seems to have in mind specifically scenario S2. From the article:
…the solution [to the paradox] is straightforward: The Eternal Society Paradox is presupposing a logical contradiction.
How is this a solution? The fact that the Eternal Society Paradox (in scenario S2) entails a logical contradiction is part of the point; it’s not a solution to simply to concede part of the claim.
It presupposes a first and a last element to a supposedly infinite series, so the Eternal Society Paradox commits the First-and-Last Fallacy.
Simply calling something a fallacy doesn’t make it so. The “first-and-last fallacy” is described as follows:
The First-and-Last Fallacy occurs if and only if a person envisions a supposedly infinite series as having both a first and a last element.
I didn’t envision it, and neither did scenario S2. Indeed, part of the reason there’s a contradiction is that the scenario (if anything) envisions that there is no first element.

Another problem with the objection is that the Eternal Society Paradox Argument is logically valid, so if the argument is unsound, which premise is false? This objection doesn’t actually attack any premise of the argument! This sort of objection is actually a red herring (for more on red herrings, see my red herring video).

Conclusion



When considering an objection against a logically valid argument, consider whether the objection attacks the truth or justification of any premise of the argument; if it doesn’t, it might be a red herring. The magic of logic is such that if the premises of a logically valid argument are true, then the conclusion follows inescapably regardless of what else might be true.

Friday, November 13, 2020

The Eternal Society Paradox Argument: Symbolic Logic Approaches

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Introduction



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. I will, however, have at least rough English translations of symbolic logic for each of the premises right underneath the symbolic logic language. For those enterprising individuals who aren’t familiar with symbolic logic but want to try to understand it, I’ll explain some of the symbols in the next section.

Some Symbolic Logic



Some of the logic I will use (including the various names for the rules of logic) can be found in this introductory logic page. I won’t go into the various rules of logic here but I will explain what some symbols mean. First there are symbols of basic propositional logic:

Type of
connective
EnglishSymbolic
Logic
When it’s true/false
Conjunctionp and qp ∧ qTrue if both are true; otherwise false
Disjunctionp or qp ∨ qFalse if both are false; otherwise true
ConditionalIf p, then qp → qFalse if p is true and q is false; otherwise true
Biconditionalp, if and only if qp ↔ qTrue if both have the same truth-value (i.e. both are true or both are false); otherwise false
NegationNot-p¬pTrue if p is false; false if p is true


To give an example of predicate logic, consider the following symbolization key:

B(x) = x is a Bachelor.
U(x) = x is Unmarried.


The letters B and M in these examples are predicates which say something about the element they are predicating. Sometimes parentheses aren’t used; e.g. Bx being used to mean “x is a bachelor.” The symbol; ∀ means “For All” or “For Any” such that the following basically means “All bachelors are unmarried:”

universal quantification
 
In English In Symbolic Logic
For any x: [if x is B, then x is U] ∀x[B(x) → U(x)]


There’s also the existential quantifier ∃ that denotes the existence of something.

existential quantification
 
In English In Symbolic Logic
There exists an x: [x is B and x is not U] ∃x[B(x) ∧ ¬U(x)]


Note that ∀x¬[Fx] = ¬∃[Fx].

In philosophy, a possible world is a complete description of the way reality is or could have been like. On possible world semantics, a proposition is possibly true if and only if it is true in at least one possible world, and a proposition is necessarily true if and only if it is true in all possible worlds.

EnglishSymbolic
Logic
When it’s true/false
p is possible◊pTrue if p is true in at least on possible world; otherwise false
p is necessary□pTrue if p is true in all possible worlds; otherwise false
p is not possible¬◊pTrue if p is false in all on possible worlds; otherwise true


Note that ¬◊p is equivalent to □¬p. Indeed, the ◊ and □ operators can even be defined in terms of each other; e.g. one could define ◊ as “true in at least one possible world” and then define □ as this:
□p = ¬◊¬p


Eternal Society Paradox



Roughly, 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. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise 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 for the Eternal Society practicing the aforementioned Annual Coin Flipping Tradition. 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 for the Eternal Society engaging in the Annual Coin Flipping Tradition. 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.
As mentioned in the paper, premises (1)-(3) can be understood as material conditionals, even though there is a sense in which I think the subjunctive mood is appropriate. Some people have disputed these premises, and to argue for them I’ll use symbolic logic.

Defining the Predicates and Propositional Variables



With the domain of discourse being the years of the past, the predicates are defined as follows.
  • Py = There exists a year y’ prior to year y in which the Eternal Society did the chant in year y’.
  • Cy = the chant is done in year y.
  • By = there exists a year before year y.
  • PBy = Where y’ represents the year before y, Py’ is true (they did the chant in a year prior to y’).
  • CBy = The chant is done in the year before y.
  • Fy = the coin (indeterministic, probabilistically independent) is flipped in year y.
  • Hy = the coin comes up heads in year y.
The propositional variables are defined as follows:
  • I = the past is infinite and beginningless such that I entails ∀y[By].
  • L = the flipped coin came up heads for the first time last year.
  • V = ∀y[Hy].
    • In English: the flipped 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 = ∀y[Fy] ∧ D = ∀y[Fy] ∧ (I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)])
    • 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 = A ∧ L = ∀y[Fy] ∧ D ∧ L
    • In English: Scenario S1 obtains.
    • Note that it follows from this definition that S1 entails A.
  • S2 = A ∧ V = ∀y[Fy] ∧ (I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) ∧ ∀y[Hy]
    • In English: 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 (7) 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 for the Annual Coin-Flipping Tradition, including a permutation where the coin came up heads each year it was flipped.
  1. □(S1 → A)
    • Scenario S1 entails that the Annual Coin-Flipping Tradition obtains.
  2. □(A → ◊(A ∧ V))
    • Necessarily: if the Annual Coin-Flipping Tradition obtains then heads coming up each year is a possible outcome.
  3. □(S2 ↔ (A ∧ V))
    • Necessarily: the Annual Coin-Flipping Tradition with the coin coming up heads each year obtains if and only if scenario S2 obtains. (Recall that scenario S2 is defined to be this way.

  1. ◊S1 conditional proof assumption
    1. ¬◊S2 indirect proof assumption
      1. □¬S2 10, equivalence
        1. ¬S2 10, T-reiteration
        2. S2 ↔ (A ∧ V) 8, T-reiteration
        3. ¬(A ∧ V) 12, 13, biconditional elimination
      1. □¬(A ∧ V) 12-14, necessity intro
        1. □¬(A ∧ V) 15, S4-reiteration
        2. ¬◊(A ∧ V) 16, equivalence
        3. A ↔ ◊(A ∧ V) 7, T-reiteration
        4. ¬A 16, 17, biconditional elimination
        5. S1 → A 6, T-reiteration
        6. ¬S1 19, 20, modus tollens
      1. □¬S1 16-21 necessity intro
      2. ¬◊S1 22, equivalence
      3. ◊S1 ∧ ¬◊S1 9, 23, conjunction introduction
    1. ◊S210-24, indirect proof
  1. ◊S1 → ◊S29-25, conditional proof


Argument 2: If ◊I then ◊E, and if ◊E then ◊S1



Next is an argument for the idea that if an infinite past is possible, then an Eternal Society is possible and S1 is possible. Premises (2) and (3) can be derived rigorously from lines (27)-(35) below:
  1. □(E → I)
    • Necessarily: an Eternal Society existing implies the past is infinite.
  2. □(I → (I ∧ ∀y[Py ∨ ¬Py]))
    • Necessarily: if the past is infinite, then (the past is infinite and for any year y, either the chant was done prior to year y or it is not the case that the chant was done prior to year y).
  3. □(I → (I ∧ ∀y[Fy → (Hy ∨ ¬Hy]))
    • Necessarily: if the past is infinite, then (the past is infinite and for any year y, if the coin was flipped in y then either the coin landed heads in y or it is not the case that it landed heads in y.
  4. {◊I ∧ □(I → (I ∧ ∀y[Py ∨ ¬Py]))□(I → (I ∧ ∀[Fy → (Hy ∨ ¬Hy]))} → ◊E
    • If [the infinite past is possible and necessarily: (if the past is infinite, then for any year y either the chant was done in a prior year or it is not the case the chant was done in a prior year) and necessarily (If the past is infinite, then for any year y if the coin was flipped then either it came up heads or it didn’t in year y)] then the Eternal Society is possible.
  5. ◊E → ◊(E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)])
    • If the Eternal Society is possible, then it is possible for that Eternal Society do the following in each year y: if there is a flipped coin that comes up heads and the chant was done in a prior year they do the chant, otherwise they don’t.
  6. □{D ↔ (I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)])}
    • Roughly, the letter D is defined as a placeholder letter to represent the following: the past is infinite and if for any year y there exists the flipped coin that comes up heads and the chant was not done in a prior year, then the chant is done in year y, otherwise the chant is not done in y.
  7. □(D ∧ E → ◊(D ∧ ∀y[Fy]))
    • Necessarily: if the Eternal Society exists and D is true for that society (e.g. they do the chant depending on inter alia whether the coin came up heads), then it is possible for the Eternal Society to also actually flip a coin each year with D being true (thereby engaging in the Annual Coin-Flipping Tradition).
  8. ◊(D ∧ ∀y[Fy]) → ◊(D ∧ ∀y[Fy] ∧ L)
    • If the Annual Coin-Flipping Tradition is possible (the probabilistically independent coin is flipped each year etc.), then it is possible for the coin to have come up heads for the first time last year.
  9. □((D ∧ ∀y[Fy] ∧ L) ↔ S1)
    • Roughly: S1 is defined to be the scenario in which the Annual Coin-Flipping Tradition is done and the coin comes up heads for the first time last year.

  1. ◊I conditional proof assumption
    1. ◊I ∧ □(I → (I ∧ ∀y[Fy → (Hy ∨ ¬Hy])) 28, 36, conjunction introduction
    2. ◊I ∧ □(I → (I ∧ ∀y[Fy → (Hy ∨ ¬Hy])) ∧ □(I → (I ∧ ∀y[Fy → (Hy ∨ ¬Hy]) 37, 29, conjunction introduction
    3. ◊E 30, 38, modus ponens
  1. ◊I → ◊E 36-39, conditional proof
  2. □¬(D ∧ E) conditional proof assumption
       □
    1. ¬(D ∧ E) 41, T-reiteration
    2. E → I 27, T-reiteration
    3. D ↔ (I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) 32, T-reiteration
    4. E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)] indirect proof assumption
      1. E 45, conjunction elimination
      2. I 43, 46, modus tollens
      3. (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)] 45, conjunction elimination
      4. I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)] 47, 48, conjunction introduction
      5. D 44, 49, biconditional elimination
      6. D ∧ E 46, 49, conjunction introduction
      7. (D ∧ E) ∧ ¬(D ∧ E) 42, 51, conjunction introduction
    1. ¬(E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) 45-52, indirect proof
    1. □¬(E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) 42-53, necessity intro
  1. □¬(D ∧ E) → □¬(E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) 41-54, conditional proof
  2. ¬◊(D ∧ E) → ¬◊(E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) 55, equivalence
  3. ◊(E ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) → ◊(D ∧ E) 56, transposition
  4. ◊E → ◊(D ∧ E) 28, 52, hypothetical syllogism
  5. □¬(D ∧ ∀y[Fy]) conditional proof assumption
       □
    1. □¬(D ∧ ∀y[Fy]) 59, S4-reiteration
    2. ¬◊(D ∧ ∀y[Fy]) 60, equivalence
    3. (D ∧ E) → ◊(D ∧ ∀y[Fy]) 33, T-reiteration
    4. ¬(D ∧ E) 61, 62, modus tollens
    1. □¬(D ∧ E) 65-68, necessity intro
  1. □¬(D ∧ ∀y[Fy]) → □¬(D ∧ E) 59-64, conditional proof
  2. ¬◊(D ∧ ∀y[Fy]) → ¬◊(D ∧ E) 65, equivalence
  3. ◊(D ∧ E) → ◊(D ∧ ∀y[Fy]) 66, transposition
  4. ◊E → ◊(D ∧ ∀y[Fy]) 58, 67, hypothetical syllogism
  5. ◊E → ◊(D ∧ ∀y[Fy] ∧ L) 34, 68, hypothetical syllogism
  6. □¬S1 conditional proof assumption
       □
    1. ¬S1 70, T-reiteration
    2. (D ∧ ∀y[Fy] ∧ L) ↔ S1 35, T-reiteration
    3. ¬(D ∧ ∀y[Fy] ∧ L) 71, 72 biconditional elimination
    1. □¬(D ∧ ∀y[Fy] ∧ L) 71-73, necessity intro
  1. □¬S1 → □¬(D ∧ ∀y[Fy] ∧ L) 70-74, conditional proof
  2. ¬◊S1 → ¬◊(D ∧ ∀y[Fy] ∧ L) 75, equivalence
  3. ◊(D ∧ ∀y[Fy] ∧ L) → ◊S1 76, transposition
  4. ◊E → ◊S1 69, 77 hypothetical syllogism
Line (40) matches up with premise (1) (“If an infinite past is possible, then an eternal society is possible”) and line (78) matches up with premise (2) (“If an Eternal Society were possible, then scenario S1 would be possible”).


Argument 3: ¬◊S2



Odd as it may seem, I’ve even seen some question premise (4), even though theoretically it should be an uncontroversial premise. In addition to the definition of scenario S2 applied in premise (81) I make use of several necessary truths that apply to this situation, such as the beginningless, infinite past entailing that each year has a year before it (line (79)); that where y’ is the year right before some arbitrary year y, if the chant was done in y’ then the chant was done in a year prior to y (line (84)); and that if there exists a year y where the chant was done in a year prior to y, then there exists a year in which the chant was done (line 85).
  1. □(I → ∀y[By])
    • The past being beginningless and infinite entails that for every y there exists a year before y.
  2. □(I → (∃y[Cy] ∨ ∃y[¬Cy]))
    • The past being beginningless and infinite entails that either there exists a year in which the chant is done or there exists a year in which it is not the case that the chant was done.
  3. □(S2 → (∀y[Fy] ∧ I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) ∧ ∀y[Hy]))
    • Roughly, scenario S2 is defined such that: the Annual Coin-Flipping Tradition in which the coin comes up heads each year. (Note that scenario S2 is in fact defined this way.)
  4. □(∀y[¬Py → Cy] → ∀y[By → (¬PBy → CBy)])
    • Necessarily: if (for every year y, if the chant is not done in a year prior to y then the chant is done in y) then (if there exists year y’ before y, if the chant is not done in a year prior to y’ then the chant is done in y’)
  5. □(∀y[PBy → Py])
    • Necessarily: for any year y if there exists a year y’ right before y and the chant was done prior to year y’ then the chant was done prior to year y.
  6. □(∀y[CBy → Py])
    • Necessarily: for any year y if there exists a year y’ right before y and the chant was done in y’, then the chant was done prior to year y.
  7. □(∃y[Py] → ∃y[Cy])
    • Necessarily: if there exists a year y in which the chant was done in a year prior to y, then there exists a year in which the chant was done.

   □
  1. I → ∀y[By] 79, T-reiteration
  2. I → (∃y[Cy] ∨ ∃y[¬Cy]) 80, T-reiteration
  3. S2 ↔ (∀y[Fy] ∧ I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) ∧ ∀y[Hy] 81, T-reiteration
  4. ∀y[¬Py → Cy] → ∀y[By → (¬PBy → CBy)] 82, T-reiteration
  5. ∀y[PBy → Py] 83, T-reiteration
  6. ∀y[CBy → Py] 84, T-reiteration
  7. ∃y[Py] → ∃y[Cy] 85, T-reiteration
  8. S2 indirect proof assumption
    1. (∀y[Fy] ∧ I ∧ (∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) ∧ ∀y[Hy] 88, 93, modus ponens
    2. ∀y[Fy] 94, conjunction elimination
    3. I 94, conjunction elimination
    4. ∀y[((Hy ∧ ¬Py) → Cy) ∧ (¬(Hy ∧ ¬Py) → ¬Cy)]) 94, conjunction elimination
    5. ∀y[Hy] 94, conjunction elimination
    6. ((Ha ∧ ¬Pa) → Ca) ∧ (¬(Ha ∧ ¬Pa) → ¬Ca) 97, universal instantiation
    7. (Ha ∧ ¬Pa) → Ca 99, conjunction elimination
    8. ¬(Ha ∧ ¬Pa) → ¬Ca 99, conjunction elimination
    9. ∀y[(Hy ∧ ¬Py) → Cy] 100, universal generalization
    10. ∀y[¬(Hy ∧ ¬Py) → ¬Cy] 101, universal generalization
    11. ∀y[By] 86, 96, modus ponens
    12. ¬Pb conditional proof assumption
      1. Hb 98, universal instantiation
      2. Hb ∧ ¬Pb 105, 106 conjunction introduction
      3. (Hb ∧ ¬Pb) → Cb 102, universal instantiation
      4. Cb 107, 108, modus ponens
    1. ¬Pb → Cb 105-109, conditional proof
    2. ∀y[¬Py → Cy] 110, universal generalization
    3. Pc conditional proof assumption
      1. ¬¬Pc 112, double negation
      2. ¬Hc ∨ ¬¬Pc 113, disjunction addition
      3. ¬(Hc ∧ ¬Pc) 114, De Morgan’s law
      4. ¬(Hc ∧ ¬Pc) → ¬Cc 103, universal instantiation
      5. ¬Cc 115, 116, modus ponens
    1. Pc → ¬Cc 112-117, conditional proof
    2. ∀y[Py → ¬Cy] 111, universal generalization
    3. ∀y[By → (¬PBy → CBy)] 89, 111, modus ponens
    4. Bd 104, universal instantiation
    5. Bd → (¬PBd → CBd) 120, universal instantiation
    6. ¬PBd → CBd 121, 122, modus ponens
    7. CBd → Pd 91, universal instantiation
    8. PBd → Pd 90, universal instantiation
    9. ¬Pd indirect proof assumption
      1. ¬PBd 125, 125, modus tollens
      2. ¬CBd 124, 126, modus tollens
      3. ¬¬PBd 123, 128, modus tollens
      4. ¬PBd ∧ ¬¬PBd 127, 129, conjunction introduction
    1. Pd 126-130, indirect proof
    2. Pd → ¬Cd 119, universal instantiation
    3. ¬Cd 131, 132, modus ponens
    4. ∀y¬[Cy] 133, universal generalization
    5. ¬∃y[Cy] 134, quantifier negation
    6. ∃y[Cy] ∨ ∃y[¬Cy] 87, 96, modus ponens
    7. ∃y[¬Cy] 135, 136, disjunctive syllogism
      1. ¬Ce 137, existential instantiation
      2. ¬Pe → Ce 111, universal instantiation
      3. ¬¬Pe 138, 139, modus ponens
      4. Pe 140, double negation
      5. ∃y[Py] 141, existential introduction
    1. ∃y[Py] 137-142, existential instantiation
    2. ¬∃y[Py] 92, 135, modus tollens
    3. ∃y[Py] ∧ ¬∃y[Py] 143, 144, conjunction introduction
  1. ¬S2 93-145, indirect proof
  1. □¬S2 86-146, necessity intro
  2. ¬◊S2 147, equivalence
And there you have it; proof for ¬◊S2.

Sunday, August 9, 2020

Mathematical Argument for God Debunked?

Home  >  Philosophy  >  Atheism/Theism

Introduction



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)
923
1843
2763
3683


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)
7923
2043
1363
2483


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 the 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 Lakoff: 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.”

Conclusion



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.