Why maximize expected value of an unbounded utility function if it’s irrational? What other reasons do you have to do it over alternatives? Biting the bullet of St Petersburg doesn’t just mean accepting the lottery, it also means in principle paying to avoid learning information, and choosing options that are strictly dominated by others, so predictably losing. Or you have to think ahead and make commitments you’ll predictably later want to break. Maybe such cases won’t come up in practice, though.
Also, if you’re biting the bullet on expectational total hedonistic utilitarianism, infinities will dominate everything, and you should ignore anything that doesn’t have infinite EV.
I think the problem is a bit worse than this?
If your decision procedure is “maximize the EV of an unbounded utility function,” you basically cannot make any decisions. After all, for any action you could take, there is an extremely low but still nonzero chance that the action is infinitely good, and a similarly low-but-nonzero chance that it is infinitely bad. Infinity minus infinity is undefined. So all actions have an undefined expected value.
I agree that all actions would have undefined EV (and a chance of positive infinity and a chance of negative infinity) under the standard extended real numbers. However, increasing the probability of positive infinity and decreasing the probability of negative infinity would extend expectationalism in that case, following from extended rationality axioms (without continuity) and still make sense.
You could also consider different ways of doing arithmetic with infinities to avoid things usually being undefined.
I see now my reply just above misinterpreted of what you said, sorry. If I understand correctly, you were referring to what you mentioned here:
All options maximize expected utility (EU), since the expected utility will be undefined (or infinite) regardless. There’s always a nonzero chance you will end up choosing the right religion and be rewarded infinitely and a nonzero chance you will end up choosing badly and be punished infinitely, so the EU is +infinity + (-infinity) = undefined. (I got this from a paper, but I forget which one; maybe something by Alan Hájek.)
In response to 1, you might say that we should maximize the probability of +infinity and minimize the probability of -infinity before considering finite values. This could be justified through the application of plausible rationality axioms directly, in particular the independence axiom. This could reduce to EU maximization with some prior steps where we ignore equiprobable parts of distributions with the same values. However, infinite and unbounded values violate the continuity axiom. Furthermore, if we’re allowing infinities, especially as limits of aggregates of finite values like an eternity of positive welfare, then it would be suspicious to not allow unbounded finite values at least in principle. Unbounded finite values can lead to violations of the sure-thing principle, as well as vulnerability to Dutch books and money pumps (e.g. see here, here and my reply, and here). If the bases for allowing and treating infinities this way require the violation of some plausible requirements of rationality or require ad hoc and suspicious beliefs about what kinds of values are possible (infinity is possible but finite values must be bounded), then it’s at least not obvious that we’re normatively required to treat infinities this way. Some other decision theory might be preferable, or we can allow decision-theoretic uncertainty.
The 1st point is not a problem for me. For the reasons described in Ellis 2018, I do not think there are infinities.
As for the 2nd point, the definition of unbounded utilities Paul Christiano uses here and here involves “an infinite sequence of outcomes”. This point is also not a worry for me, as I do not think there are infinite sequences in the real world.
Similarly, I think zeros only exist in the sense of representing arbitralily small, but non-null values.
Do you just mean that you shouldn’t use 0 as a probability (maybe only for an event in a countable probability space)? I agree with that, which is called Cromwell’s rule.
(Or, are you saying zero can never accurately describe anything? Like the number of apples in my hand, or the number of dollars you have in a Swiss bank account? Or, based on your own claim, the number of infinite sequences that exist? The probability that “the number of things that exist and match definition X is 0” is in fact 0, for any X?)
I would say 0 can be used to describe abstract concepts, but I do not think it can be observed in the real world. All measurements have a finite sensitivity, so measuring zero only means the variable of interest is smaller than the sensitivity of the measurement. For example, if a termometer of sensitivity 0.5 K, and range from 0 K to 300 K indicates 0 K, we can only say the temperature is lower than 0.5 K (we cannot say it is 0).
I agree 0 should not be used for real probabilities. Abstractly, we can use 0 to describe something impossible. For example, if X is a uniform distribution ranging from 0 to 1, the probability of X being between −2 and −1 is 0.
If I say I have 0 apples in my hands, I just mean 0 is the integer number which most accurately describes the vague concept of the number of apples in my hands. It is not indended to be exactly 0. For example, I may have forgotten to account for my 2 bites, which imply I only have 0.9 apples in my hands. Or I may only consider I have 0.5 apples in my hands because I am only holding the apple with one hand (i.e. 50 % of my 2 hands). Or maybe having refers to who bought the apples, and I only contributed to 50 % of the cost of the apple. In general, it looks like human language does not translate perfectly to exact numbers.
Why maximize expected value of an unbounded utility function if it’s irrational?
Why would that be irrational? Intuitively, if one thinks maximising expected value is fine for non-tiny probabilities and non-astronomical values, the reasoning should extend to tiny probabilities and astronomical values.
What other reasons do you have to do it over alternatives?
When I say I have a credence of 1 on expectational total hedonistic utilitarianism (ETHU), I mean I can assume it to be exactly 1 in practice, and therefore consider true everything which follows from it without considering other reasons. I worry this sounds dismissive and overconfident. To be clear, my credences are rarely this close to 1, and I am very uncertain about what actions one should do in the real world. I just think the uncertainty is empirical (including uncertainty about the real-world heuristics which correlate with maximising expected total hedonic utility). Since most people have lower credences than me on ETHU, I guess I am understanding it in a more general way than the one described in the literature.
Biting the bullet of St Petersburg doesn’t just mean accepting the lottery, it also means in principle paying to avoid learning information, and choosing options that are strictly dominated by others, so predictably losing.
To clarify, by “biting the bullet of the St. Petersburg paradox”, I meant I am willing to maximise expected value under all and any conditions. I do not know what this implies in terms of accepting of rejecting the St. Petersburg Paradox:
If it involves money instead of utility, the expected value is finite (assuming utility increases with the logarithm of money), and one should not keep gambling forever.
In practice, there are physical limits to how much money/utility one can get (the universe has finite resources), so it only applies in its original form to thought experiments.
Also, if you’re biting the bullet on expectational total hedonistic utilitarianism, infinities will dominate everything, and you should ignore anything that doesn’t have infinite EV.
I think getting infite expected value would violate our current understanding of physics. Even if our current understanding is wrong, and it is possible to produce infinite value, actions producing infinite expected value may not be available. For example, when we assume the utility of an action can be modelled as a normal distribution, we are allowing for the possibility of negative and positive infinite utility. However, the expected value of the action is still finite (and equal to the mean of the distribution).
Moreover, if we had actions with infinite expected utility, we may still be able to decide which one is better as long as resources are finite. To illustrate, we can imagine 2 actions A and B with the following expected utilities:
E_A = (E_max—E)^-1.
E_B = (E_max—E)^-2.
E and E_max are the energy used and available to perform the actions. As E tends to E_max:
E_A → +inf.
E_B → +inf.
E_A/E_B → 0.
So, although the expected utility of both actions tends to infinity, we can still say B would be better than A.
In general, I do not understand why infinites are said to be problematic. Intuitively, I would expect indeterminations of the type inf/inf or inf—inf can be resolved analysing the generating functions. I may well be missing something.
Also, if you’re biting the bullet on expectational total hedonistic utilitarianism, infinities will dominate everything, and you should ignore anything that doesn’t have infinite EV.
As I said, I do not think the possibility of infinite value implies there are actions with infinite expected value, and, even if these exist, there would still be ones which are better than others.
when we assume the utility of an action can be modelled as a normal distribution, we are allowing for the possibility of negative and positive infinite utility. However, the expected value of the action is still finite (and equal to the mean of the distribution).
The first sentence here is not true. The formula below is the PDF of a normal distribution:
f(x)=1σ√2πexp(−12(x−μσ)2)
The limit of f(x) as x approaches either ∞ or −∞ is zero.
Moreover, if the first sentence I quoted from your comment were true, there would be no way for the second sentence to be true. This is the definition of expected utility:
∑outcomesU(outcome)P(outcome)
Where U(outcome) is the utility of an outcome and P(outcome) is its probability.
If you have an unbounded utility function, and you have any probability greater than zero (say, 10−101010) that the outcome of your action has infinitely positive utility, and a similarly nonzero probability (say, 10−10101010) that it has infinitely negative utility, then the formula for expected utility simplifies to
The limit of f(x) as x approaches either ∞ or −∞ is zero.
By “possibility of negative and positive infinite utility”, I meant there is a non-null probability of a negative or positive utility with arbitrarily large magnitude. I think infinite is often used as meaning arbitrarily large, but I see now that Michael was not using it that way. Sorry for my confusion, and thanks for clarifying!
Moreover, if the first sentence I quoted from your comment were true, there would be no way for the second sentence to be true.
I agree. In the 1st sentence, “infinite” was supposed to mean “arbitrarily large” (in which case the 2nd sentence would be true).
I shared some links upthread to arguments that expected utility maximization with an unbounded utility function is irrational. It can make you choose infinitely many of options that are definitely worse together, or without even dealing with infinitely many choices, make you averse to information and choose finite sequences of options that are stochastically dominated. All of this seems decision-theoretically irrational, and preventing such behaviour makes some of the main and strongest arguments for expected utility maximization, but with a bounded utility function.
I don’t think you should assume current physics is correct with 100% probability (e.g. we could always be wrong, and we’ve been wrong before), and even if it is, there are ways to get infinities or unbounded expected values, e.g. evidential decision theory and correlations with other agents in a spatially infinite universe, possibly quantum tunneling (or so I’ve heard).
On your specific approach for infinities, note that, in principle, the limits of ratios can be undefined even if the ratios are bounded (and even never approach 0). So you need to handle such cases. I think there are definitely some infinite cases you can extend to, but you typically need to pick an order according to which to sum things, which seems especially arbitrary and hard to do if you’re handling cases of creation of new universes, especially infinite universes. The results can be sensitive to which basically arbitrary order you choose. Other decisions theories and utility functions also need to deal with cases that involve physical infinities, although they can sometimes (and maybe usually) be ignored, while only infinities matter in practice on the natural extensions of the view you’re defending.
In general, I do not understand why infinites are said to be problematic. Intuitively, I would expect indeterminations of the type inf/inf or inf—inf can be resolved analysing the generating functions. I may well be missing something.
I was missing that by “infinity” you literally meant infinity, whereas I interpreted it as arbitralily large, but finite. I have now checked in more detail the links, and see how infinities in the sense of ∞ can lead to problems. I will have to think more about this...
Ah ok, I was talking about both arbitrarily large but finite (unbounded) values and infinities as two separate issues, but both are related to fanaticism. Unbounded utilities (especially in cases with infinite or undefined expected values) seem irrational, while actual infinite utilities are more just technical problems that are hard to solve non-arbitrarily. The links I shared are mostly about unbounded utilities, but this one discusses infinities:
https://forum.effectivealtruism.org/posts/qcqTJEfhsCDAxXzNf/what-reason-is-there-not-to-accept-pascal-s-wager?commentId=Ydbz56hhEwxg9aPh8
The definition of unbounded utilities Paul Christiano uses here and here involves “an infinite sequence of outcomes”. I do not think infinite sequences exist in the real world, so I also think unbounded utilities are irrational.
I don’t think this is a valid inference, since there are other ways to define unbounded utilities, e.g. directly with an unbounded real-valued utility function, and the definitions don’t require infinite sequences to actually exist in the real world. However, I suspect all ways of showing unbounded utilities are irrational require infinite sequences, e.g. even St. Petersburg’s lottery is defined with an infinite sequence.
Also, I don’t think you should assign probability 1 to unbounded sequences not existing. In fact, I think some infinite sequences are more likely than not to actually exist, because the universe is probably unbounded in spatial extent, and there are infinitely many agents and moral patients in the universe in infinitely many different locations (although perhaps they’re all “copies” of finitely many different individuals). And for any proposed time bound for our future, there’s also nonzero chance that there will be moral patients past it.
However, I suspect all ways of showing unbounded utilities are irrational require infinite sequences, e.g. even St. Petersburg’s lottery is defined with an infinite sequence.
I got that impression too.
In fact, I think some infinite sequences are more likely than not to actually exist, because the universe is probably unbounded in spatial extent, and there are infinitely many agents and moral patients in the universe in infinitely many different locations (although perhaps they’re all “copies” of finitely many different individuals).
According to this article from Toby Ord (see Figure 15), “under the most widely accepted cosmological modell (ΛCDM)”:
“The part of the universe we can causally affect” (affectable universe) has a radius of 16.5 Gly.
“The part of the universe which can ever have any kind of causal connectedness to our location” has a radius of 125.8 Gly.
There are (abstract) models under which the universe is infinite (see section “What if ΛCDM is wrong?”):
“A useful way of categorising the possibilities concerns the value of an unknown parameter, w. This is the parameter in the ‘equation of state’ for a perfect fluid, and is equal to its pressure divided by its energy density”.
“Relativistic matter has w = 1⁄3. ΛCDM models dark energy as a cosmological constant, which corresponds to w = –1”.
“Our current best estimates of w are consistent with ΛCDM: putting it to within about 10% of –1, but the other models cannot yet be excluded”.
“If dark energy is better modelled by a value of w between –1 and –1/3, then expansion won’t become exponential, but will still continue to accelerate, leading to roughly similar results — in particular that only a finite number of galaxies are ever affectable”.
“If w were below –1, then the scale factor would grow faster than an exponential. (...) Furthermore, the scale factor would reach infinity in a finite time, meaning that by a particular year the proper distance between any pair of particles would become infinite. Presumably this moment would mark the end of time. This scenario is known as the ‘Big Rip’”.
“If w were between –1/3 and 0, then the scale factor would merely grow sub-linearly, making it easier to travel between distant galaxies and removing the finite limit on the number of reachable galaxies”.
Based on the 3rd point, one may naively say w follows a uniform distribution between −1.1 and −0.9. Consequently, there is a 50 % chance of w being:
Lower than −1, leading to a Big Rip. I think this only means the size of the universe tends to infinity, not that it actually reaches infinity, as I do not expect physical laws to generalise until infinity (which would also be impossible to test, as infinities are indistinguishable from very large numbers from an experimental point of view, given the limited range of all measurements).
Between −1 and −1/3, being compatible with ΛCDM. This would mean the affectable universe is finite.
Ya, I think the part of the universe we can causally affect is very likely bounded/finite, but that could be wrong, e.g. the models could be wrong. Furthermore, the whole universe (including the parts we very probably can’t causally affect) seems fairly likely to be infinite/unbounded, and we can possibly affect parts of the universe acausally, e.g. evidential cooperation or via correlated agents out there, and I actually think this is quite likely (maybe more likely than not). There are also different normative ways of interpreting the many worlds interpretation of QM that could give you infinities.
Someone who bites the bullet on risk-neutral EV maximizing total utilitarianism should wager in favour of acts with infinite impacts, no matter how unlikely, e.g. even if it requires our understanding of physics to be wrong.
Ya, I think the part of the universe we can causally affect is very likely bounded/finite, but that could be wrong, e.g. the models could be wrong.
The models are certainly wrong to some extent, but that does not mean we should assign a non-null probability to the universe being infinite. I think we can conceive of many impossibilities. For example, I can imagine 1 = 0 being true, or both A > B and A < B being true, but these relations are still false.
It is also impossible to show that 1 = 0 is false. Likewise, it is impossible to show the universe in infite, because infinities are not measurable (because all measurement have a finite range). So there is a sense in which the universe being finite is similar to the axioms of math.
Furthermore, the whole universe (including the parts we very probably can’t causally affect) seems fairly likely to be infinite/unbounded, and we can possibly affect parts of the universe acausally, e.g. evidential cooperation or via correlated agents out there, and I actually think this is quite likely (maybe more likely than not).
Someone who bites the bullet on risk-neutral EV maximizing total utilitarianism should wager in favour of acts with infinite impacts, no matter how unlikely, e.g. even if it requires our understanding of physics to be wrong.
Unless causal expectational total hedonistic utilitarianism in a finite affectable universe is true, which I think is the case.
I don’t think you can (non-dogmatically) justify assigning 0 probability to any of these claims, which you need to do to justifiably prevent possible infinities from dominating. That seems way too overconfident. An infinite universe (temporally or spatially) is not a logical impossibility. Nor is acausal influence.
Some considerations:
The analogy with math isn’t enough, and the argument also cuts both ways: you can never prove with certainty that the universe is finite, either. And you should just be skeptical that a loose analogy with math could justify 100% confidence in the claim that the universe is finite, if that’s what you intended.
You may be able to gather indirect evidence (although not decisive proof) for the universe being infinite, like we do for other phenomena, like black holes, dark matter and dark energy. For example, the flatter the universe seems to be globally, I think the more likely it is to be infinite (although even a flat universe could be finite).
Multiple smart people knowledgeable on this topic have thought much more about the issues than you (or me) and have concluded in favour of infinities. Giving their views any weight means assigning nonzero probability to such infinities. Not giving their views any weight would seem arrogant. (Of course, we should also give “only finite impacts” positive weight, but that gets dominated by the infinite possibilities under your risk neutral expected value maximizing total utilitarianism.) See also https://forum.effectivealtruism.org/posts/WKPd79PESRGZHQ5GY/in-defence-of-epistemic-modesty
If you could provide a persuasive argument against these infinities that non-dogmatically allows us to dismiss them with 100% certainty, that would be a huge achievement. Since no one seems to have done this so far (or everyone who disagrees after hearing the argument failed to understand it or was so biased they couldn’t agree, which seems unlikely, or the argument hasn’t been read by others), it’s probably very hard to do, so you should be skeptical of any argument claiming to do so, including any you make yourself.
An infinite universe (temporally or spatially) is not a logical impossibility.
I would say infinity is a logical impossibility. During this thread, I was mostly arguing from intuition. Now that I think about it, my intuition was probably being informed by this episose of the Clearer Thinking Podcast with Joscha Bach, who is also sceptical of infinities.
Meanwhile, I have just found The Case Against Infinity from Kip Sewell. I have read the Introduction, and it really seems to be arguing for something similar to my (quite uninformed) view. Here are the 1st and last paragraphs:
Gazing into the sublime immensity of the starry night sky, pondering the awesome depths of the past and future, many hold that the Universe must be “infinite” in space and time. Now infinity is certainly a profound notion, holding a powerful emotional appeal across cultures. But while infinity is widely believed in, it has simply been taken for granted that infinity is a logically coherent concept. Such an assumption is mistaken. I will argue that infinity is in fact a logical absurdity (that is, a self-contradictory notion) like a square circle or a four-sided triangle. And since logical absurdities cannot refer to anything that actually exists, it follows that there is nothing infinite. I will show why we must conclude that the Universe cannot, therefore, be infinite in either space or time.
(...)
In addition, the logical failure of the traditional notion of infinity, and the necessity of replacing infinity as a mathematical value with indefiniteness, carries even more serious implications for physics and cosmology: If infinity as a mathematical value is logically absurd, and if logical absurdities refer to nothing that can actually exist, then infinity as a mathematical value refers to nothing that really exists—at least, not according to the traditional definition of infinity. As a result, measures of space and time cannot really be “infinite” in the usual sense of the term. Thus in showing the selfcontradictions involved with the traditional notion of infinity, I will also be presenting reasons why cosmology and physics must hold that neither space nor time can be infinite—no matter how indefinite the vastness of space or time may be, the Universe as a whole must still be finite.
Not sure whether I will understand it, but I will certainly have a go at reading the rest!
This seems to be arguing against standard mathematics. Even if you thought mathematical (not just physical) infinity was probably a logical impossibility, assigning 100% to its impossibility means dismissing the views of the vast majority of mathematicians, which seems epistemically arrogant.
If the author found a formal contradiction in the standard axioms of set theory (due to the axiom of infinity) or another standard use of infinity, that would falsify the foundations of mathematics, they would become famous, and mathematicians would be freaking out. It would be like solving P vs NP. Instead, the paper is 14 years old, not published in any academic journal, and almost no one is talking about it. So, the author very probably hasn’t found anything as strong as a formal contradiction. The notion of ‘absurdity’ they’re using could be informal (possibly like the way we use ‘paradox’, but many paradoxes have resolutions and aren’t genuine contradictions) and could just reflect their own subjective intuitions and possibly biases. Or, they’ve made a deductive error. Or, most charitably, they’ve introduced their own (probably controversial) premises, but to arrive at 100% confidence in the impossibility of infinity, they would need 100% confidence in some of their own premises. I’m not sure the author themself would even go that far, since that would be epistemically arrogant.
EDIT: I may have been uncareful switching between arguments. The main claim I want to defend is that infinities and infinite impacts can’t justifiably be assigned 0% probability. I do think some infinities are pretty likely and that infinity is very probably logically possible/coherent, but those are stronger claims than I need to justify not assigning 0% probability to infinite impact. Pointing out arguments for those positions supports the claim that 0% probability to infinite impacts is too strong, even if those arguments turn out to be wrong.
EDIT2: Maybe I’ve misunderstood and they don’t mean infinity is logically impossible even in mathematics, just only physically. Still, I think they’re probably wrong, and that’s not the main point here anyway: whatever argument they give wouldn’t justify assigning 0 probability to infinities and infinite impacts.
(I don’t think I will engage further with this thread.)
This seems to be arguing against standard mathematics.
Yes and no:
The answer depends on what mathematics is about. Insofar as mathematics is simply an intellectual game, then the logical breakdown of infinitudes does not amount to much; mathematicians can continue to play with infinities in classical mathematics and infinite sets in transfinite mathematics just as gamers can continue role-playing in fictional worlds. On the other hand, insofar as mathematical operations are used in attempts to reveal the nature of the Universe, then the logical incoherence of infinitudes means that they reveal nothing of nature since logical contradictions cannot refer to real circumstances.
Kip argues:
The answer to revising both transfinite mathematics and classical mathematics is to replace the traditional use of infinitude with an alternative concept—indefinitude.
(...)
If a value is “indefinite,” then that means either (A) it would be found to be the terminating value in a set of values if the set could be measured or counted, but is either too minute or too vast to be measured or counted in actual practice or (B) it is finite in the sense of being currently undefined as the next value, or sequence of values, beyond the highest or lowest value that can be actually computed or invented for a series.
(...)
What will be called “the indefinite” shall refer to any set that has so many members that the highest value(s) in the set is indefinite according to A.
However:
Functionally speaking then [reference to example], nothing is lost by replacing infinity with indefiniteness—values that were traditionally thought to go on “infinitely” could just as well be thought of as going on “indefinitely.”
I think the crux of the disagreement is described here (emphasis added by me):
Many mathematicians have assumed that the infinite really is a property of nature [and one of the axioms of set theory], existing in a Platonic world on its own apart from minds attempting to conceive it, and that Cantor’s transfinite mathematics teaches us how this property of nature operates. But that is a big assumption, and one which we are not intellectually compelled to make. It could equally be assumed that infinite sets are mathematical inventions, in which case the rules for manipulating infinite sets do not indicate “how the infinite works” as if those rules are merely descriptions of an infinite set’s behavior that mathematicians discover like the principles of atomic motion in a condensed gas are discovered by physicists. That is, it could just as well be supposed that the infinite is an invented idea that refers to no existing natural property at all. If that position is correct, the rules for calculating infinite sets are only “discovered” in the same way that a new strategy in chess is discovered. This kind of “discovery” is actually a form of invention; it is the invention of new rules of inference for manipulating concepts in a pattern coherent with the rules previously established for those concepts.
If this is correct, then the rules of transfinite mathematics are not really the discovery of some phenomenon independent of human activity, but simply the invention of a system of inference. Further, because the traditional view of infinite sets contains self-contradictions, that system of inference has no coherent application to understanding reality in terms of measurement. Infinite sets and transfinite mathematics are better interpreted as elements of a mathematical game rather than a means for understanding the quantitative aspects of nature. Transfinite mathematics is therefore actually misleading about the nature of real sets of things. That is, the “infinite sets” of transfinite mathematics not only do not refer to real sets in nature, but actually lead us astray in understanding the quantifiable aspects of reality. Consequently, Cantor’s math ought to be rejected as a tool for investigating reality even if it is saved as a kind of academic game.
In other words:
Claiming that infinite sets of objects can exist because the rules for calculating them remain consistent as long as you don’t allow subtraction and division is like saying square circles can exist because we could create a formula that allows them to be used, provided some qualifications are put in place that don’t allow us to expose the contradictions resulting from attempting to calculate a round square or a circle with corners.
(...) The result would be that square circles are not any less self-contradictory; all that we would end up proving in the construction of such a formula is that a coherent game of square circle calculation can be made as long as the rules are limited in an ad hoc fashion so that the illogic of square circles is not allowed to be exposed by taking the concept to its, well, logical end. Transfinite mathematics is in the same boat—the game is coherent only because we won’t allow logic to proceed down its natural path so that the self-contradictory nature of its subject would be exposed. This allows the idea of computable infinities in the real world to retain the illusion of being logically coherent.
Regarding:
Maybe I’ve misunderstood and they don’t mean infinity is logically impossible even in mathematics, just only physically.
Kip rejects the existence of infinities in both physics and math. The real world does not allow for contradiction, so infinities have to be rejected in physics. In math, it can exist, but Kip argues that it is better to revise it to the extent math is supposed to decribe the real world (see quotations above).
Bach makes a basic error or assumption that’s widely rejected in math:
That there is any set of all sets. The notion is contradictory for more basic reasons like Russell’s paradox, so we use the “class of all sets” and define/construct sets so that there is no set of all sets. Proper classes are treated pretty differently from sets in many cases. Classes are collections of sets only. People don’t use the class of all sets to represent anything in the physical world, either, and I’d say that it probably can’t be used to represent anything physical, but that’s not a problem for infinities in general. There’s no class of all classes under standard set theory, since that would need to contain proper classes.
Even if we used the class of all sets to try to fix the argument, the power set operation has no natural extension to it in standard set theory. It would have to be the class of all subclasses of the class of all sets, which doesn’t exist under standard set theory because it would contain proper classes, but even if it did exist, that object would be different from the class of all sets, so there need not be any contradiction with them having different sizes. (I’d guess the class of all subclasses of the class of all sets would be strictly bigger by the same argument that the power set of a set is bigger than the set, under some set theory where that’s defined naturally and extends standard set theory.)
Sewell assumes subtraction with infinite cardinals should be well-defined like it is for finite numbers without (good) argument, but this is widely rejected. Also, there are ways to represent infinities so that the specific operations discussed are well-defined, e.g. representing the objects as sets and using set operations (unions, differences, partitioning) instead of arithmetic operations on numbers (addition, subtraction, division). N—N = 0 this way and N—N has no other value, where “-” means set difference and N is the set of natural numbers. Subtracting the even numbers (or odd numbers) from the natural numbers would be represented differently on the left-hand side, so that giving a different result isn’t a problem. EDIT: I think he quotes some similar arguments, but doesn’t really respond to them (or probably doesn’t respond well).
They seem to be arguing against strawmen. They don’t seem to understand the basics of standard axiomatic set theory well enough, and they wouldn’t make such bad arguments if they did. I would recommend you study axiomatic set theory if you’re still tempted to dismiss the logical possibility of infinity, or just accept that it’s likely to be logically possible by deferring to those who understand axiomatic set theory, because probably almost all of them accept its logical possibility.
(Again, I don’t intend to engage further, but I guess I’m bad at keeping that kind of promise.)
Understands the notion of infinity does not lead to contradictions in math. As you noted, infinity is one of the axioms of ZMC set theory, which is widely followed in math. So no wonder infinity is true (by definition) for most mathematicians!
Argues that math should be about the real world, so we should not be defining ad hoc rules which have no parallell in physical reality.
As an analogy (adapted from one used by William Craig), we can suppose I have 2 bags with infinite marbles. One contains marbles numbered with the even numbers, and the other marbles numbered with the odd numbers, so they have the same infinity of marbles. If I:
Give both bags to you, I will keep no bags, and therefore will have zero marbles. So inf—inf = 0.
Give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles. So inf—inf = inf.
This leads to 0 = inf, which is contradictory.
I appreciate one can say I have cheated by:
Using the same type of subtraction in both situations (indicated by “-”), whereas I should have used different symbols to describe the different types of subtractions.
Assuming I could perform the operations inf—inf, which is an indeterminate form.
However, as far as I can tell, reality only allows for one type of subtraction. If I have 3 apples in my hands (or x $ in a Swiss bank account ;)), and give you 2 apple, I will keep 1 apple. This is the motivation for 3 − 2 = 1.
In Sewell’s words:
“In classical mathematics the operation of subtraction on natural numbers yields definite answers, and so instances of subtraction can be grounded in real world examples of removal. The act of “removing” a subset of objects from a set of objects is just an instance of applying mathematical subtraction or division to physical collections in the real world”.
“There is nothing in transfinite mathematics implying that mathematical operations on infinite sets cannot be applied to logically possible infinite collections in the real world. So, if we are able to consistently subtract or divide infinite sets in transfinite mathematics, we should then without contradiction be able to carry out the removal of infinite subsets from infinite sets of real objects as well. Subtracting and dividing infinite sets should show what would happen in the real world if we could go about “removing” infinite subsets from infinite sets of physical objects. On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to “remove” an infinite subset of real objects from an infinite set of them. Such a removal would then not be able to be performed in the real world, which does not permit logically contradictory states of affairs to occur. The application of inverse operations in transfinite mathematics to real world instances of removing infinite subsets then, is actually a test of the logical validity of infinite sets. If the math breaks down as we’ve seen, so does the logic of infinite sets in the real world”.
“On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to “remove” an infinite subset of real objects from an infinite set of them.”
This doesn’t follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
Plenty of apparently real things aren’t well-defined unless you specify them in enough detail, but that doesn’t make them nonsense. For example, the weight of a bag after removing an object whose weight is unknown. Or, the center of mass of two objects, knowing only their respective centers of mass (and distance between them).
There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities). It’s of course possible our universe has no infinities and arithmetic is always well-defined when representing any real objects in it, but there’s no decisive proof for either, and hence no decisive proof for the impossibility of infinity. It doesn’t follow by necessity from the finite case.
There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities).
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/morality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
This doesn’t follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
To clarify the contradiction I mentioned above, if n denotes the cardinality operator, v the disjunction operator, ^ the conjunction operator, O the set of odd numbers, E the set of even numbers, ES the empty set, n(ES) = 0, and n(O) = n(E) = inf:
If I give both bags to you, I will keep no bags, and therefore will have zero marbles:
A1: n((O v E)\(O v E)) = n(O v E) - n((O v E) ^ (O v E)) = n(O v E) - n(O v E) = inf—inf.
B1: n((O v E)\(O v E)) = n(ES) = 0.
C1: A1^ B1 ⇒ inf—inf = 0.
If I give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles:
A2: n((O v E)\O) = n(O v E) - n((O v E) ^ O) = n(O v E) - n(O) = inf—inf.
B2: n((O v E)\O) = n((O v E)\E) = n(O) = inf.
C2: A2 ^ B2 ⇒ inf—inf = inf.
So there is a contradiction:
D: C1 ^ C2 ⇒ 0 = inf.
Since, 0 = inf is false, one of the following is false:
The relationship R ⇔ n(X\Y) = n(X) - n(X ^ Y), which I used above, exists in the real world.
Infinites exist in the real world.
I guess you would be inclined towards putting non-null weight into each one of these points being false. However, R essentially means the whole is the sum of its parts, which I cannot see being false in the real world. So I reject the existence of infinites in the real world.
I have now finished reading The Case Against Infinity, and really liked it! I think this paragraph summarises it well:
The infinite is therefore something that is complete, thus divisible, and yet limitless, therefore indivisible. Since being a divisible-yet-indivisible thing is a contradiction in terms, so too the traditional notion of the infinite is self-contradictory.
Why maximize expected value of an unbounded utility function if it’s irrational? What other reasons do you have to do it over alternatives? Biting the bullet of St Petersburg doesn’t just mean accepting the lottery, it also means in principle paying to avoid learning information, and choosing options that are strictly dominated by others, so predictably losing. Or you have to think ahead and make commitments you’ll predictably later want to break. Maybe such cases won’t come up in practice, though.
Also, if you’re biting the bullet on expectational total hedonistic utilitarianism, infinities will dominate everything, and you should ignore anything that doesn’t have infinite EV. See also: https://forum.effectivealtruism.org/posts/qcqTJEfhsCDAxXzNf/what-reason-is-there-not-to-accept-pascal-s-wager?commentId=Ydbz56hhEwxg9aPh8
I think the problem is a bit worse than this?
If your decision procedure is “maximize the EV of an unbounded utility function,” you basically cannot make any decisions. After all, for any action you could take, there is an extremely low but still nonzero chance that the action is infinitely good, and a similarly low-but-nonzero chance that it is infinitely bad. Infinity minus infinity is undefined. So all actions have an undefined expected value.
I agree that all actions would have undefined EV (and a chance of positive infinity and a chance of negative infinity) under the standard extended real numbers. However, increasing the probability of positive infinity and decreasing the probability of negative infinity would extend expectationalism in that case, following from extended rationality axioms (without continuity) and still make sense.
You could also consider different ways of doing arithmetic with infinities to avoid things usually being undefined.
I agree the possibility of infinities does not imply actions will have undefined expected values. My comment here illustrates this.
I see now my reply just above misinterpreted of what you said, sorry. If I understand correctly, you were referring to what you mentioned here:
The 1st point is not a problem for me. For the reasons described in Ellis 2018, I do not think there are infinities.
As for the 2nd point, the definition of unbounded utilities Paul Christiano uses here and here involves “an infinite sequence of outcomes”. This point is also not a worry for me, as I do not think there are infinite sequences in the real world.
Similarly, I think zeros only exist in the sense of representing arbitralily small, but non-null values.
Do you just mean that you shouldn’t use 0 as a probability (maybe only for an event in a countable probability space)? I agree with that, which is called Cromwell’s rule.
(Or, are you saying zero can never accurately describe anything? Like the number of apples in my hand, or the number of dollars you have in a Swiss bank account? Or, based on your own claim, the number of infinite sequences that exist? The probability that “the number of things that exist and match definition X is 0” is in fact 0, for any X?)
I argue for infinite sequences in my other reply.
I would say 0 can be used to describe abstract concepts, but I do not think it can be observed in the real world. All measurements have a finite sensitivity, so measuring zero only means the variable of interest is smaller than the sensitivity of the measurement. For example, if a termometer of sensitivity 0.5 K, and range from 0 K to 300 K indicates 0 K, we can only say the temperature is lower than 0.5 K (we cannot say it is 0).
I agree 0 should not be used for real probabilities. Abstractly, we can use 0 to describe something impossible. For example, if X is a uniform distribution ranging from 0 to 1, the probability of X being between −2 and −1 is 0.
If I say I have 0 apples in my hands, I just mean 0 is the integer number which most accurately describes the vague concept of the number of apples in my hands. It is not indended to be exactly 0. For example, I may have forgotten to account for my 2 bites, which imply I only have 0.9 apples in my hands. Or I may only consider I have 0.5 apples in my hands because I am only holding the apple with one hand (i.e. 50 % of my 2 hands). Or maybe having refers to who bought the apples, and I only contributed to 50 % of the cost of the apple. In general, it looks like human language does not translate perfectly to exact numbers.
Thanks for challenging my assumptions!
Why would that be irrational? Intuitively, if one thinks maximising expected value is fine for non-tiny probabilities and non-astronomical values, the reasoning should extend to tiny probabilities and astronomical values.
When I say I have a credence of 1 on expectational total hedonistic utilitarianism (ETHU), I mean I can assume it to be exactly 1 in practice, and therefore consider true everything which follows from it without considering other reasons. I worry this sounds dismissive and overconfident. To be clear, my credences are rarely this close to 1, and I am very uncertain about what actions one should do in the real world. I just think the uncertainty is empirical (including uncertainty about the real-world heuristics which correlate with maximising expected total hedonic utility). Since most people have lower credences than me on ETHU, I guess I am understanding it in a more general way than the one described in the literature.
To clarify, by “biting the bullet of the St. Petersburg paradox”, I meant I am willing to maximise expected value under all and any conditions. I do not know what this implies in terms of accepting of rejecting the St. Petersburg Paradox:
If it involves money instead of utility, the expected value is finite (assuming utility increases with the logarithm of money), and one should not keep gambling forever.
In practice, there are physical limits to how much money/utility one can get (the universe has finite resources), so it only applies in its original form to thought experiments.
I think getting infite expected value would violate our current understanding of physics. Even if our current understanding is wrong, and it is possible to produce infinite value, actions producing infinite expected value may not be available. For example, when we assume the utility of an action can be modelled as a normal distribution, we are allowing for the possibility of negative and positive infinite utility. However, the expected value of the action is still finite (and equal to the mean of the distribution).
Moreover, if we had actions with infinite expected utility, we may still be able to decide which one is better as long as resources are finite. To illustrate, we can imagine 2 actions A and B with the following expected utilities:
E_A = (E_max—E)^-1.
E_B = (E_max—E)^-2.
E and E_max are the energy used and available to perform the actions. As E tends to E_max:
E_A → +inf.
E_B → +inf.
E_A/E_B → 0.
So, although the expected utility of both actions tends to infinity, we can still say B would be better than A.
In general, I do not understand why infinites are said to be problematic. Intuitively, I would expect indeterminations of the type inf/inf or inf—inf can be resolved analysing the generating functions. I may well be missing something.
As I said, I do not think the possibility of infinite value implies there are actions with infinite expected value, and, even if these exist, there would still be ones which are better than others.
The first sentence here is not true. The formula below is the PDF of a normal distribution:
f(x)=1σ√2πexp(−12(x−μσ)2)
The limit of f(x) as x approaches either ∞ or −∞ is zero.
Moreover, if the first sentence I quoted from your comment were true, there would be no way for the second sentence to be true. This is the definition of expected utility:
∑outcomesU(outcome)P(outcome)
Where U(outcome) is the utility of an outcome and P(outcome) is its probability.
If you have an unbounded utility function, and you have any probability greater than zero (say, 10−101010) that the outcome of your action has infinitely positive utility, and a similarly nonzero probability (say, 10−10101010) that it has infinitely negative utility, then the formula for expected utility simplifies to
∞⋅10−101010−∞⋅10−10101010=∞−∞
which is undefined.
Hi Fermi,
By “possibility of negative and positive infinite utility”, I meant there is a non-null probability of a negative or positive utility with arbitrarily large magnitude. I think infinite is often used as meaning arbitrarily large, but I see now that Michael was not using it that way. Sorry for my confusion, and thanks for clarifying!
I agree. In the 1st sentence, “infinite” was supposed to mean “arbitrarily large” (in which case the 2nd sentence would be true).
I shared some links upthread to arguments that expected utility maximization with an unbounded utility function is irrational. It can make you choose infinitely many of options that are definitely worse together, or without even dealing with infinitely many choices, make you averse to information and choose finite sequences of options that are stochastically dominated. All of this seems decision-theoretically irrational, and preventing such behaviour makes some of the main and strongest arguments for expected utility maximization, but with a bounded utility function.
I don’t think you should assume current physics is correct with 100% probability (e.g. we could always be wrong, and we’ve been wrong before), and even if it is, there are ways to get infinities or unbounded expected values, e.g. evidential decision theory and correlations with other agents in a spatially infinite universe, possibly quantum tunneling (or so I’ve heard).
On your specific approach for infinities, note that, in principle, the limits of ratios can be undefined even if the ratios are bounded (and even never approach 0). So you need to handle such cases. I think there are definitely some infinite cases you can extend to, but you typically need to pick an order according to which to sum things, which seems especially arbitrary and hard to do if you’re handling cases of creation of new universes, especially infinite universes. The results can be sensitive to which basically arbitrary order you choose. Other decisions theories and utility functions also need to deal with cases that involve physical infinities, although they can sometimes (and maybe usually) be ignored, while only infinities matter in practice on the natural extensions of the view you’re defending.
Thanks for taking the time to clarify, Michael!
I said:
I was missing that by “infinity” you literally meant infinity, whereas I interpreted it as arbitralily large, but finite. I have now checked in more detail the links, and see how infinities in the sense of ∞ can lead to problems. I will have to think more about this...
Ah ok, I was talking about both arbitrarily large but finite (unbounded) values and infinities as two separate issues, but both are related to fanaticism. Unbounded utilities (especially in cases with infinite or undefined expected values) seem irrational, while actual infinite utilities are more just technical problems that are hard to solve non-arbitrarily. The links I shared are mostly about unbounded utilities, but this one discusses infinities: https://forum.effectivealtruism.org/posts/qcqTJEfhsCDAxXzNf/what-reason-is-there-not-to-accept-pascal-s-wager?commentId=Ydbz56hhEwxg9aPh8
The definition of unbounded utilities Paul Christiano uses here and here involves “an infinite sequence of outcomes”. I do not think infinite sequences exist in the real world, so I also think unbounded utilities are irrational.
I don’t think this is a valid inference, since there are other ways to define unbounded utilities, e.g. directly with an unbounded real-valued utility function, and the definitions don’t require infinite sequences to actually exist in the real world. However, I suspect all ways of showing unbounded utilities are irrational require infinite sequences, e.g. even St. Petersburg’s lottery is defined with an infinite sequence.
Also, I don’t think you should assign probability 1 to unbounded sequences not existing. In fact, I think some infinite sequences are more likely than not to actually exist, because the universe is probably unbounded in spatial extent, and there are infinitely many agents and moral patients in the universe in infinitely many different locations (although perhaps they’re all “copies” of finitely many different individuals). And for any proposed time bound for our future, there’s also nonzero chance that there will be moral patients past it.
I got that impression too.
According to this article from Toby Ord (see Figure 15), “under the most widely
accepted cosmological modell (ΛCDM)”:
“The part of the universe we can causally affect” (affectable universe) has a radius of 16.5 Gly.
“The part of the universe which can ever have any kind of causal connectedness to our location” has a radius of 125.8 Gly.
There are (abstract) models under which the universe is infinite (see section “What if ΛCDM is wrong?”):
“A useful way of categorising the possibilities concerns the value of an unknown parameter, w. This is the parameter in the ‘equation of state’ for a perfect fluid, and is equal to its pressure divided by its energy density”.
“Relativistic matter has w = 1⁄3. ΛCDM models dark energy as a cosmological constant, which corresponds to w = –1”.
“Our current best estimates of w are consistent with ΛCDM: putting it to within about 10% of –1, but the other models cannot yet be excluded”.
“If dark energy is better modelled by a value of w between –1 and –1/3, then expansion won’t become exponential, but will still continue to accelerate, leading to roughly similar results — in particular that only a finite number of galaxies are ever affectable”.
“If w were below –1, then the scale factor would grow faster than an exponential. (...) Furthermore, the scale factor would reach infinity in a finite time, meaning that by a particular year the proper distance between any pair of particles would become infinite. Presumably this moment would mark the end of time. This scenario is known as the ‘Big Rip’”.
“If w were between –1/3 and 0, then the scale factor would merely grow sub-linearly, making it easier to travel between distant galaxies and removing the finite limit on the number of reachable galaxies”.
Based on the 3rd point, one may naively say w follows a uniform distribution between −1.1 and −0.9. Consequently, there is a 50 % chance of w being:
Lower than −1, leading to a Big Rip. I think this only means the size of the universe tends to infinity, not that it actually reaches infinity, as I do not expect physical laws to generalise until infinity (which would also be impossible to test, as infinities are indistinguishable from very large numbers from an experimental point of view, given the limited range of all measurements).
Between −1 and −1/3, being compatible with ΛCDM. This would mean the affectable universe is finite.
Ya, I think the part of the universe we can causally affect is very likely bounded/finite, but that could be wrong, e.g. the models could be wrong. Furthermore, the whole universe (including the parts we very probably can’t causally affect) seems fairly likely to be infinite/unbounded, and we can possibly affect parts of the universe acausally, e.g. evidential cooperation or via correlated agents out there, and I actually think this is quite likely (maybe more likely than not). There are also different normative ways of interpreting the many worlds interpretation of QM that could give you infinities.
Someone who bites the bullet on risk-neutral EV maximizing total utilitarianism should wager in favour of acts with infinite impacts, no matter how unlikely, e.g. even if it requires our understanding of physics to be wrong.
The models are certainly wrong to some extent, but that does not mean we should assign a non-null probability to the universe being infinite. I think we can conceive of many impossibilities. For example, I can imagine 1 = 0 being true, or both A > B and A < B being true, but these relations are still false.
It is also impossible to show that 1 = 0 is false. Likewise, it is impossible to show the universe in infite, because infinities are not measurable (because all measurement have a finite range). So there is a sense in which the universe being finite is similar to the axioms of math.
To clarify, I think the universe is finite, but unbounded, i.e. that it has a finite size, but no edges/boundaries.
How much of these is still relevant if one puts null weight on evidential decision theory (EDT)?
Unless causal expectational total hedonistic utilitarianism in a finite affectable universe is true, which I think is the case.
I don’t think you can (non-dogmatically) justify assigning 0 probability to any of these claims, which you need to do to justifiably prevent possible infinities from dominating. That seems way too overconfident. An infinite universe (temporally or spatially) is not a logical impossibility. Nor is acausal influence.
Some considerations:
The analogy with math isn’t enough, and the argument also cuts both ways: you can never prove with certainty that the universe is finite, either. And you should just be skeptical that a loose analogy with math could justify 100% confidence in the claim that the universe is finite, if that’s what you intended.
You may be able to gather indirect evidence (although not decisive proof) for the universe being infinite, like we do for other phenomena, like black holes, dark matter and dark energy. For example, the flatter the universe seems to be globally, I think the more likely it is to be infinite (although even a flat universe could be finite).
Multiple smart people knowledgeable on this topic have thought much more about the issues than you (or me) and have concluded in favour of infinities. Giving their views any weight means assigning nonzero probability to such infinities. Not giving their views any weight would seem arrogant. (Of course, we should also give “only finite impacts” positive weight, but that gets dominated by the infinite possibilities under your risk neutral expected value maximizing total utilitarianism.) See also https://forum.effectivealtruism.org/posts/WKPd79PESRGZHQ5GY/in-defence-of-epistemic-modesty
If you could provide a persuasive argument against these infinities that non-dogmatically allows us to dismiss them with 100% certainty, that would be a huge achievement. Since no one seems to have done this so far (or everyone who disagrees after hearing the argument failed to understand it or was so biased they couldn’t agree, which seems unlikely, or the argument hasn’t been read by others), it’s probably very hard to do, so you should be skeptical of any argument claiming to do so, including any you make yourself.
I would say infinity is a logical impossibility. During this thread, I was mostly arguing from intuition. Now that I think about it, my intuition was probably being informed by this episose of the Clearer Thinking Podcast with Joscha Bach, who is also sceptical of infinities.
Meanwhile, I have just found The Case Against Infinity from Kip Sewell. I have read the Introduction, and it really seems to be arguing for something similar to my (quite uninformed) view. Here are the 1st and last paragraphs:
Not sure whether I will understand it, but I will certainly have a go at reading the rest!
This seems to be arguing against standard mathematics. Even if you thought mathematical (not just physical) infinity was probably a logical impossibility, assigning 100% to its impossibility means dismissing the views of the vast majority of mathematicians, which seems epistemically arrogant.
If the author found a formal contradiction in the standard axioms of set theory (due to the axiom of infinity) or another standard use of infinity, that would falsify the foundations of mathematics, they would become famous, and mathematicians would be freaking out. It would be like solving P vs NP. Instead, the paper is 14 years old, not published in any academic journal, and almost no one is talking about it. So, the author very probably hasn’t found anything as strong as a formal contradiction. The notion of ‘absurdity’ they’re using could be informal (possibly like the way we use ‘paradox’, but many paradoxes have resolutions and aren’t genuine contradictions) and could just reflect their own subjective intuitions and possibly biases. Or, they’ve made a deductive error. Or, most charitably, they’ve introduced their own (probably controversial) premises, but to arrive at 100% confidence in the impossibility of infinity, they would need 100% confidence in some of their own premises. I’m not sure the author themself would even go that far, since that would be epistemically arrogant.
EDIT: I may have been uncareful switching between arguments. The main claim I want to defend is that infinities and infinite impacts can’t justifiably be assigned 0% probability. I do think some infinities are pretty likely and that infinity is very probably logically possible/coherent, but those are stronger claims than I need to justify not assigning 0% probability to infinite impact. Pointing out arguments for those positions supports the claim that 0% probability to infinite impacts is too strong, even if those arguments turn out to be wrong.
EDIT2: Maybe I’ve misunderstood and they don’t mean infinity is logically impossible even in mathematics, just only physically. Still, I think they’re probably wrong, and that’s not the main point here anyway: whatever argument they give wouldn’t justify assigning 0 probability to infinities and infinite impacts.
(I don’t think I will engage further with this thread.)
Yes and no:
Kip argues:
However:
I think the crux of the disagreement is described here (emphasis added by me):
In other words:
Regarding:
Kip rejects the existence of infinities in both physics and math. The real world does not allow for contradiction, so infinities have to be rejected in physics. In math, it can exist, but Kip argues that it is better to revise it to the extent math is supposed to decribe the real world (see quotations above).
Bach makes a basic error or assumption that’s widely rejected in math:
That there is any set of all sets. The notion is contradictory for more basic reasons like Russell’s paradox, so we use the “class of all sets” and define/construct sets so that there is no set of all sets. Proper classes are treated pretty differently from sets in many cases. Classes are collections of sets only. People don’t use the class of all sets to represent anything in the physical world, either, and I’d say that it probably can’t be used to represent anything physical, but that’s not a problem for infinities in general. There’s no class of all classes under standard set theory, since that would need to contain proper classes.
Even if we used the class of all sets to try to fix the argument, the power set operation has no natural extension to it in standard set theory. It would have to be the class of all subclasses of the class of all sets, which doesn’t exist under standard set theory because it would contain proper classes, but even if it did exist, that object would be different from the class of all sets, so there need not be any contradiction with them having different sizes. (I’d guess the class of all subclasses of the class of all sets would be strictly bigger by the same argument that the power set of a set is bigger than the set, under some set theory where that’s defined naturally and extends standard set theory.)
See this page for definitions and some discussion: https://en.wikipedia.org/wiki/Class_(set_theory)
Sewell assumes subtraction with infinite cardinals should be well-defined like it is for finite numbers without (good) argument, but this is widely rejected. Also, there are ways to represent infinities so that the specific operations discussed are well-defined, e.g. representing the objects as sets and using set operations (unions, differences, partitioning) instead of arithmetic operations on numbers (addition, subtraction, division). N—N = 0 this way and N—N has no other value, where “-” means set difference and N is the set of natural numbers. Subtracting the even numbers (or odd numbers) from the natural numbers would be represented differently on the left-hand side, so that giving a different result isn’t a problem. EDIT: I think he quotes some similar arguments, but doesn’t really respond to them (or probably doesn’t respond well).
They seem to be arguing against strawmen. They don’t seem to understand the basics of standard axiomatic set theory well enough, and they wouldn’t make such bad arguments if they did. I would recommend you study axiomatic set theory if you’re still tempted to dismiss the logical possibility of infinity, or just accept that it’s likely to be logically possible by deferring to those who understand axiomatic set theory, because probably almost all of them accept its logical possibility.
(Again, I don’t intend to engage further, but I guess I’m bad at keeping that kind of promise.)
Long story short, Sewell:
Understands the notion of infinity does not lead to contradictions in math. As you noted, infinity is one of the axioms of ZMC set theory, which is widely followed in math. So no wonder infinity is true (by definition) for most mathematicians!
Argues that math should be about the real world, so we should not be defining ad hoc rules which have no parallell in physical reality.
As an analogy (adapted from one used by William Craig), we can suppose I have 2 bags with infinite marbles. One contains marbles numbered with the even numbers, and the other marbles numbered with the odd numbers, so they have the same infinity of marbles. If I:
Give both bags to you, I will keep no bags, and therefore will have zero marbles. So inf—inf = 0.
Give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles. So inf—inf = inf.
This leads to 0 = inf, which is contradictory.
I appreciate one can say I have cheated by:
Using the same type of subtraction in both situations (indicated by “-”), whereas I should have used different symbols to describe the different types of subtractions.
Assuming I could perform the operations inf—inf, which is an indeterminate form.
However, as far as I can tell, reality only allows for one type of subtraction. If I have 3 apples in my hands (or x $ in a Swiss bank account ;)), and give you 2 apple, I will keep 1 apple. This is the motivation for 3 − 2 = 1.
In Sewell’s words:
“In classical mathematics the operation of subtraction on natural numbers yields definite answers, and so instances of subtraction can be grounded in real world examples of removal. The act of “removing” a subset of objects from a set of objects is just an instance of applying mathematical subtraction or division to physical collections in the real world”.
“There is nothing in transfinite mathematics implying that mathematical operations on infinite sets cannot be applied to logically possible infinite collections in the real world. So, if we are able to consistently subtract or divide infinite sets in transfinite mathematics, we should then without contradiction be able to carry out the removal of infinite subsets from infinite sets of real objects as well. Subtracting and dividing infinite sets should show what would happen in the real world if we could go about “removing” infinite subsets from infinite sets of physical objects. On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to “remove” an infinite subset of real objects from an infinite set of them. Such a removal would then not be able to be performed in the real world, which does not permit logically contradictory states of affairs to occur. The application of inverse operations in transfinite mathematics to real world instances of removing infinite subsets then, is actually a test of the logical validity of infinite sets. If the math breaks down as we’ve seen, so does the logic of infinite sets in the real world”.
My reply here has some further context.
Sure, I trust your decisions regarding your time. Thanks for the discussion!
“On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to “remove” an infinite subset of real objects from an infinite set of them.”
This doesn’t follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
Plenty of apparently real things aren’t well-defined unless you specify them in enough detail, but that doesn’t make them nonsense. For example, the weight of a bag after removing an object whose weight is unknown. Or, the center of mass of two objects, knowing only their respective centers of mass (and distance between them).
There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities). It’s of course possible our universe has no infinities and arithmetic is always well-defined when representing any real objects in it, but there’s no decisive proof for either, and hence no decisive proof for the impossibility of infinity. It doesn’t follow by necessity from the finite case.
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/morality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
To clarify the contradiction I mentioned above, if n denotes the cardinality operator, v the disjunction operator, ^ the conjunction operator, O the set of odd numbers, E the set of even numbers, ES the empty set, n(ES) = 0, and n(O) = n(E) = inf:
If I give both bags to you, I will keep no bags, and therefore will have zero marbles:
A1: n((O v E)\(O v E)) = n(O v E) - n((O v E) ^ (O v E)) = n(O v E) - n(O v E) = inf—inf.
B1: n((O v E)\(O v E)) = n(ES) = 0.
C1: A1^ B1 ⇒ inf—inf = 0.
If I give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles:
A2: n((O v E)\O) = n(O v E) - n((O v E) ^ O) = n(O v E) - n(O) = inf—inf.
B2: n((O v E)\O) = n((O v E)\E) = n(O) = inf.
C2: A2 ^ B2 ⇒ inf—inf = inf.
So there is a contradiction:
D: C1 ^ C2 ⇒ 0 = inf.
Since, 0 = inf is false, one of the following is false:
The relationship R ⇔ n(X\Y) = n(X) - n(X ^ Y), which I used above, exists in the real world.
Infinites exist in the real world.
I guess you would be inclined towards putting non-null weight into each one of these points being false. However, R essentially means the whole is the sum of its parts, which I cannot see being false in the real world. So I reject the existence of infinites in the real world.
I have now finished reading The Case Against Infinity, and really liked it! I think this paragraph summarises it well: