Causal finitism holds that the causal history of anything is finite. On purely formal grounds (and assuming the Axiom of Choice--or at least Dependent Choice), it turns out that there are exactly two ways that a world could violate causal finitism:
- The world contains an infinite regress.
- Some effect is caused by infinitely many causes.
But perhaps the last observation can be turned into an argument for causal finitism. For if it is possible to have infinitely many objects working together causally, it should be possible to haven an infinite Newtonian universe. But it would be strange to suppose that some but not all infinite arrangements of physical objects are compossible with the Newtonian laws. After all, we can imagine asking: "What would happen if angels shuffled stuff?" So it should be possible to suppose a universe that has nothing in the half to the left of me, but in the half of the universe to my right is an infinite number of objects arranged in a uniform density in space. If that happened, I would experience an infinite force to the right (think of the gravitational force of a solid ball of uniform density at the surface: the Newtonian law makes the force be proportional to the ball's radius as the cube-dependence of the mass beats out the inverse-square-dependence), and accelerate infinitely to the right. That's impossible.
15 comments:
Sure you can suppose that angels shuffle stuff, but in your example you suppose that they shuffled an infinite amount of stuff. That is enough to explain the impossibility without saying that the Newtonian universe is impossible.
Hi Dr. Pruss,
You've done a lot of work on the intersection of infinity and probability. I've been thinking a lot about this paradox––do you have any thoughts? It is a variant I have constructed, inspired by your infinite fair lotteries.
Probabilities get washed when in an infinite sequence. For example, imagine an infinite past. Now, imagine that two men have each been flipping a coin once every day, infinitely into the past. You will choose one of the two men. I tell you that you will be murdered if the man you choose has, in his lifetime, landed heads even once. So, the probability you will live is infinitesimally small (for you to live the entire infinite sequence of coin flips must be tails).
But, I let you in on a catch. One of the men stopped flipping his coin in the year 1300. The other stopped flipping his in 2020. So, if you select the man who stopped in 1300, that's 262800 less coin flips to worry about! And, if you select the man who stopped in 2020, the probability that it makes no difference and you still live is 1/[4.81793009364255366254971341160591836971377319401090945834 × 10^79110]! Wow! You should really hope you choose the man who stopped in 1300. I now offer to tell you which man is which. Should you accept my offer?
But, wait a second. The probability that the man who stopped in 1300 has landed all tails is infinitesimally small. And the probability that the man who stopped in 2020 has landed all tails is ALSO infinitesimally small. The probability that either man has landed all tails is exactly the same. So, you shouldn't care for my information at all. In fact, if I offered to tell you the identity of both men, or give you a donut, you should take the donut every time.
But that is VERY strange. What's stranger is that if I replaced one of the men with another man who has been rolling a 19287493749837498374893-sided die every day, and the only condition under which you live is if he has rolled a "1" every single day, you should not prefer to choose the man with the coin. Your probability of living is exactly the same in either case! Do you think such a paradox further supports causal finitism?
Hallo Ibrahim Dagher,
How exactly is your paradox supporting causal finitism?
You could derive the same paradox in an infinite universe with infinite identical coins and dices rolling at the same time and asking us to bet on the person, who just flipped tails with an infinite amount of identical coins at the same time.
It appears to me, that you are trying to make a hasty generalisation from an example, which has not much to do with an infinite causal history at all and which has rather much more to do with infinity itself.
It is also very dubious how Alexander R. Pruss mentions causal finitism in this blogpost of his. He mentions that here, if that would have been properly derived and if it would be a fact of reality even though his derivation of causal finitism is also nothing more than a hasty generalisation from dubious examples like yours, Ibrahim.
Sure there are contradictions with infinities and infinite causal chains, but from those few examples it is wrong to conclude, that each infinity or infinite causal chains are impossible since not any infinity or infinite causal chains contains a contradiction. It is as false to conclude causal finitism from those few examples as it is false to conclude, that marriages and bachelors aren't supposed to exist because of the impossibility of the existence of married bachelors.
The infinite causal chain of eggs and chickens (... egg, chicken, another egg, another chicken,...) doesn't suppose to be possible cause of the impossibility of the Grim Reaper Paradox and its "conclusion"/hasty generalisation of causal finitism, even though that infinite causal chain containing eggs and chickens doesn't even contain a single paradox or a contradiction?
I don't think so.
Best regards
Zsolt Nagy
Hi Zsolt,
Very interesting. I agree with you in that the following inference is dubious at best:
(1) Some infinite causal chains are contradictory/paradoxical.
(2) Therefore, all infinite causal chains are impossible.
But I wonder if the paradoxes that Pruss uses [and maybe even the one I mentioned above] could be used in a different inference. This inference, I think you would agree, is valid:
(1) If it is possible to have an infinite causal chain, then [insert a specific causal chain] would be possible.
(2) But, [] causal chain is NOT possible.
(3) So, it is not possible to have an infinite causal chain.
In other words, the inference I am trying to support is not so much that because a certain causal chain is contradictory/impossible, then all of them are. I think what I would argue is that if it were possible for an infinite causal chain to exist, then we should expect it to be possible that some given scenario is possible. So, for example, imagine an infinite past. IF this is a genuine possibility, then we should also expect my man-flipping-a-coin scenario to be possible, because all we're really adding to the infinite past is a man and a coin! Note that an infinite past, by itself, is not contradictory. But what I'm arguing is that if this infinite past WERE possible, some other states of affairs would be possible. But that state of affairs is the contradictory one (ie, not possible).
I'm still thinking a lot about these arguments. Any further thoughts Zsolt?
Best,
Ibrahim
Ibrahim,
It's an interesting story, but I am not convinced. Let's modify your story to suppose a completely finite but extremely long past. Then the donut would still matter more to you than the difference between the two tiny, but non-infinitesimal, probabilities.
Hallo Ibrahim,
That argument of yours doesn't seem to be valid since
"(1) If it is possible to have an infinite causal chain, then [insert a specific causal chain] would be possible."
and
"(2) But, [] causal chain is (contradictory/paradoxical. Hence, it is) NOT possible."
So, it is not possible to have some infinite causal chain
and NOT-"an" infinite causal chain.
There is a big logical difference there.
If you want to do an Universal Modus Tollens, then please do that properly and correctly. Otherwise please don't make any hasty generalizations masqueraded as a "normal modus tollens"!
Thank you!
My suggestion and thought would be for you not to think too much and too hard on causal finitism, but instead invest some more quality time on learning proper logic and the proper usage of quantifiers.
Best regards,
Zsolt
Hi Zsolt,
It seems you may have misinterpreted the syllogism. Let me write it out formally.
(1) [Possibly, there exists an X such that X has an infinite causal history] -> [Possibly, there exists an X such that X has (the paradoxical scenario in question) as a causal history]
(2) ~[Possibly, there exists an X such that X has (the paradoxical scenario in question) as a causal history]
(3) ~[Possibly, there exists an X such that X has an infinite causal history]
Best,
Ibrahim
Dr. Pruss,
That might be true (although maybe it is always correct to take a higher probability of saving one's own life over a donut). However, the infinite case has a startling feature over any finite case. I can revise the story and place the two men at *any* finite difference in time, and the probabilities will always be the same. Plausibly, in the finite case, there will come a point where the difference in years between the two men is so great that one should take the chance of saving one's own life over the donut. Even if the difference in two small, non-infinitesimal probabilities is always the same. But that is not even a possibility in the infinite case.
"although maybe it is always correct to take a higher probability of saving one's own life over a donut": If that were true, it would be wrong to cross the street to buy a donut, since crossing the street has a tiny a probability of resulting in death.
"Plausibly, in the finite case, there will come a point where the difference in years between the two men is so great that one should take the chance of saving one's own life over the donut."
Not really. If each number of years is sufficiently large that the probability of death is negligible -- say, much less than the probability of dying by a meteor strike -- then the fact one of these probabilities is many times larger than the other doesn't really matter.
By the way, arguments like your (1)-(3) are very much my way of proceeding in my paradoxes book.
Hallo Ibrahim,
Since the infinite causal chain containing eggs and chickens apparently doesn't contain any contradictions or paradoxes and since contradictions and paradoxes are generally not possible, hence, any such causal history containing contradictions and paradoxes are not possible, therefore it is the case, that [Possibly, there exists an X such that X has an infinite causal history (apparently)] AND NOT[Possibly, there exists an X such that X has (the paradoxical scenario in question) as a causal history], which is exactly the negation of [Possibly, there exists an X such that X has an infinite causal history] → [Possibly, there exists an X such that X has (the paradoxical scenario in question) as a causal history].
So your premise (1) is apparently not correct.
Since your argument in that form is valid (conserving truth-values), therefore, your conclusion is also apparently not correct.
Do you know, why those kinds of proofs and generalizations in mathematics with examples do work? Because the examples are not specific examples, but general examples. If there is no counterexample, then the generalizations from those examples are correct. But if you find even one counterexample, then the generalisation is simply wrong.
Take any triangle, make two copies from it. You can always arrange them in such a way, that the three interior angles α, β and γ add up to a straight angle - to 180°.
Since you can do that with any triangle, therefore the sum of the three interior angles α, β and γ of any triangle will always add up to 180°.
If a person would find a counterexample or there would be any counterexamples, then that generalization and previous statement would be false.
If you want to try and find counterexamples for that, then please be my guest with this.
Sure, there are some contradictory and paradoxical causal histories or infinite causal chains, but not all infinite causal chains are contradictory and paradoxical and if an infinite causal chain doesn't contain any contradictions or paradoxes, then why should that infinite causal chain be impossible? Cause of Causal finitism and causal finitism is true because of some specific examples, but not general examples with infinite causal chains?
hahahahhahhahhahhahahhahhahhahahhahhahhahahhahhaha...
Since the examples for causal finitism are so specific, there are exactly that: specific examples and NOT general examples.
Causal finitism simply isn't derivable from ANY example of infinite causal chain - counterexample: infinite causal chain only containing eggs and chickens.
If you are capable of deriving and concluding causal finitism from the infinite causal chain containing eggs and chickens example, then congratulations on proving causal finitism.
If you are not capable of doing that, then also congratulations. Congratulations to wasting your time with other specific and not general examples of infinite causal chains.
Best regards,
Zsolt
Hi Dr. Pruss,
Good points. I didn't think of the fact that probabilities can be made negligibly small even in finite cases.
But something still nags me about the fact that in an infinite past, the probabilities are the *exact same*. Maybe I can modify the situation as follows?
Two men, Bob and Sam. have each been flipping a coin every day infinitely into the past. Bob has stopped in 2020. Same will never stop, flipping indefinitely into the future. Same rules apply: if the man you choose has ever landed a "heads", you will die. Still, even in this case, it will not matter which man you choose. But this seems odd, given that if I choose Bob, I will take the risk of dying once. But, if I choose Sam, every day I will have to check and worry about how I have a 50% chance of dying.
And there is no finite case that can achieve the same result, given that for any finite number of past years there will come a point where the difference between Bob and Sam's flipping is no longer non-negligible.
Would this solve your concerns? Or does this run into problems about presentism vs eternalism?
And yes! I formulated (1)-(3) from your work (as well as Josh Rasmussen's).
Hallo Ibrahim,
I don't like life-death scenarios. So how about this one?
There are four persons: Alice, Bob, Charlie and Dora, who have decided to flip fair identical coins simultaneously with each other. Further each person has decided on their own, how many coins he or she likes to flip.
(A) Alice has decided to flip as many coins as there are natural numbers.
(B) Bob has decided to flip as many coins as there are odd numbers.
(C) Charlie has decided to flip as many coins as there are even numbers.
(D) Since Dore adores and favors prime numbers so much, Dora has decided to flip as many coins as there are prime numbers.
And how life usually goes, of course there are going to be bets made based upon the outcomes of that event and the outcomes of those coinflips, but in this case we are only allowed to make a specific kind of bet.
We are allowed to bet something on one person from those four persons, Alice, Bob, Charlie and Dora, and on the outcome of the betted person achieving with his or her coinflips "No heads at all - only tails".
If we bet something on person x and person x achieves with his or her coin flips the outcome (O) "No heads at all - only tails", then we win the bet and we get something based upon the thing, we have bet.
If we bet something on person x and person x doesn't achieve with his or her coin flips the outcome (O) "No heads at all - only tails", then we lose the bet and we lose the thing, we have bet.
So then on which person do we want to bet something, if we want to make a bet at all?
Should we bet on Alice, Bob, Charlie or Dora?
Should we go with the obvious answer or with the least obvious one?
Best regards,
Zsolt
Dora's friend, Eve, likes the idea of flipping coins simultaneously. So she joins the four other persons doing that.
(E) Since Eve also favors some primes. *But for some unknown reason she specifically dislikes exactly 262800 primes of all primes. So Eve has decided to flip as many coins as she is favoring - all primes except exactly 262800 primes.
If I made a bet, then now should I change my bet to Eve?
(Let's just suppose, that we can change bets till the coins have been flipped simultaneously by each person at once.)
Alice gets wind of the bets about the outcome of those coin flips.
She wants to capitalize on that. So she does actually this:
(A') Alice declares to the public, that "She has decided to flip as many coins as there are natural numbers". But actually she is going to flip coins for each second prime of all primes. So as many as there are 2,5,11,17,... prime numbers.
She thinks, that even though she is unlikely to win that bet, she now has at least the best chance to win considering all five persons doing the event of simultaneously flipping coins at once with each other. So she instructs her friend, Xenu, to make a bet on her splitting the win, if they win at all.
Sure, Alice is immoral while doing that, but is that rational?
Is her thought of herself now having the better chances of winning that bet rational at all?
I don't think so.
*Note, that there is an artificial reason for the number 262800.
It's the same number as the number of one man doing, that amount of coin flips more or less than the other man in your first hypothetical scenario and example.
Maaan those or these kinds of scenarios and hypothetical examples must be headbreaking, if one has not a single clue about cardinalities and how those cardinalities are in relation with probabilities in these kinds of scenarios and hypothetical examples.
What does this all have to do with causal finitism or infinite causal chains?
I guess, that these and your kinds of examples have nothing to do with that.
What's the sample space Ω(2) of tossing the same coin twice? It's this:
Ω(2)={(H|H),(H|T),(T|H),(T|T)}
with H: The coin flip results in "heads".
and T: The coin flip results in "tails".
and (outcome of first coin toss|outcome of second coin toss)
What's the sample space Ω'(2) of tossing two identical coins simultaneously?
It's this: Ω'(2)={(H|H),(H|T),(T|H),(T|T)}
with (outcome of the first coin|outcome of the second coin)
So we get basically this:
Ω(2)=Ω'(2)={(H|H),(H|T),(T|H),(T|T)}
Considering the same probability distributions for those cases, then the probability spaces will be also the same. Questioning anything about the probability spaces of those cases will result in basically the same answers and conclusions.
What's the sample space Ω(3) of tossing the same coin trice? It's this:
Ω(3)={(H|H|H),(H|H|T),(H|T|H),(T|H|H),(H|T|T),(T|H|T),(T|T|H),(T|T|T)}
and (outcome of first coin toss|outcome of second coin toss|outcome of third coin toss)
What's the sample space Ω'(3) of tossing three identical coins simultaneously?
It's this:
Ω'(3)={(H|H|H),(H|H|T),(H|T|H),(T|H|H),(H|T|T),(T|H|T),(T|T|H),(T|T|T)}
with (outcome of the first coin|outcome of the second coin|outcome of the second coin)
So we get basically this:
Ω(3)=Ω'(3)={(H|H|H),(H|H|T),(H|T|H),(T|H|H),(H|T|T),(T|H|T),(T|T|H),(T|T|T)}
Considering the same probability distributions for those cases, then the probability spaces will be also the same. Questioning anything about the probability spaces of those cases will result in basically the same answers and conclusions.
We then get this per induction:
Ω(n)=Ω'(n) for any n∈ℕ.
and also the probability space being the same for any n∈ℕ.
From this it's not difficult to argue for n→∞ or ad infinitum resulting in:
Ω(∞)=Ω'(∞)
or to be more precise
Ω(ℵ0)=Ω'(ℵ0),
if we identify n with the amount of tosses with the same coin or if we identify n with the amount of identical coins flipped simultaneously in the case of n→∞ or ad infinitum - if we identify n→∞ or ad infinitum with the cardinality of ℕ; namely with ℵ0.
This is not just true for the sample spaces, but the probability spaces are also basically the same in the case of n→∞ or ad infinitum considering the same probability distributions in both cases of tossing the same coin infinitely many times or flipping an infinite amount of identical coins simultaneously.
So questioning anything about the probability spaces of those cases will result in basically the same answers and conclusions.
So, Ibrahim, if you've got a "problem" or any "problems" and examples with tossing the same coin infinitely many times (distributed over any time intervals or over any amount of time) while questioning anything about its probability space, then the same "problem" or "problems" and examples can be found with tossing an infinite amount of identical coins simultaneously.
If so, then how would those "problems" and examples of yours be ever considerable supporting causal finitism, being against the possibility of any (proper) infinite causal chains or being against the possibility of the past being infinite?
In my opinion if the same "problems" and examples can be stated without any considerations of any amount of time and if specially those and your kinds of "problems" and examples have been already solved by number theory, then your examples have no merit to even remotely substantiate the impossibility of an infinite past.
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