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.
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: