Boxing Pythagoras

Philosophy from the mind of a fighter

WLC doesn’t understand Infinity, Part 1

One of the topics which William Lane Craig often discusses is a question which has been argued in the Philosophy of Mathematics for at least 2300 years. Can an infinite number of things actually exist? Dr. Craig asserts that such actual infinites cannot exist. This is actually a topic which I have discussed before, on this blog, but Dr. Craig attempts to tackle the question quite differently than does Dr. Wildberger. Interestingly, Dr. Wildberger is a mathematician, and most of my objections to his argument pointed out his unfamiliarity with philosophy; while Dr. Craig, on the other hand, is a philosopher, and most of my objections to his argument will point out his unfamiliarity with mathematics.

Dr. Craig has discussed the topic of actual infinities in a number of different places, but I will be referring to his Excursus on Natural Theology, Part 9, for our discussion today. These are the same arguments which I have generally seen Dr. Craig present in his other work, but this happens to be the most recent exploration of the topic from WLC which is available to us.

Unfortunately, just as he has done many times before (see here and here, for example), William Lane Craig demonstrates that he has a rather poor grasp of the mathematics he’s attempting to discuss.

Early on in the Excursus, Dr. Craig is attempting to explain the difference between an actual infinite and a potential infinite. These are concepts drawn largely from Aristotle’s work, Physics, which were quite central to the discussion of infinite quantities in centuries past. An actual infinity refers to a complete set of entities which has a quantity greater than any finite number. A potential infinity refers to a process or algorithm which can be repeatedly iterated without end. It is at this point in his discussion that Dr. Craig makes the first of many claims about mathematics which are simply not true. He says:

This [potential infinity] is the type of infinity which is used in Calculus, in mathematics, where you have infinite limits.

Yes, the concept of potential infinity is utilized in Calculus, but Dr. Craig seems to be implying that this is done to the exclusion of the actual infinite. This is absolutely not the case. Quite the contrary, the entire purpose of Calculus is to perform calculations over actually infinite sets of entities.

Let’s take a very, very basic example. Let’s say that I am presented with the integral \int_1^\infty \frac {1}{x^2} dx, and asked to evaluate the expression. This is a very simple problem which most students will encounter when they are first introduced to Integral Calculus. Now, let’s take a look at what this expression represents. In Figure 1, below, I have graphed the function \frac{1}{x^2}, for x\geq 1, as the dark purple curve. The shaded region underneath the curve is the value which we are attempting to calculate with the integral \int_1^\infty \frac {1}{x^2} dx. We are trying to determine the area of this region.

Figure 1: Graphing our function

Figure 1: Graphing our function

To perform this calculation, we absolutely do make use of a potential infinite, as Dr. Craig alleges. We take the integral and evaluate it as follows:

\int_1^\infty \frac{1}{x^2} dx = \lim_{b\rightarrow \infty} (-\frac{1}{b}) |_1^\infty = \lim_{b\rightarrow \infty} (-\frac{1}{b})+1=1

We can iterate the limit \lim_{b\rightarrow \infty} (-\frac{1}{b}) repeatedly, over and over, in order to notice that it trends towards a particular value. There is no boundary to the number b which we can utilize in this iteration– it can become bigger and bigger and bigger, with no greatest possible value. This is what Dr. Craig means by a potential infinity.

However, the reason he is wrong (or, at the very least, misleading) to claim that this is “the type of infinity which is used in Calculus” is because the whole purpose of this type of calculation is to allow us to evaluate an actually infinite quantity. In the case of this particular integral, we are adding up an actually infinite collection of infinitesimally narrow areas underneath the curve in order to determine the whole area underneath the curve.

It is precisely the fact that Calculus makes use of actual infinites which made it (and the 17th Century advancements upon which it was built) controversial amongst mathematicians and philosophers when it was originally developed. There’s a wonderful book on precisely this subject, written by Amir Alexander, called Infinitesimal: How a dangerous mathematical theory shaped the modern world. I highly recommend it to anyone interested in Math History.

Moving on, Dr. Craig then begins to discuss the concept of actual infinity. Here, again, he makes a claim which demonstrates that he really doesn’t understand the mathematics which he is attempting to discuss.

This type of infinity [actual infinity] is symbolized by the Hebrew letter, \aleph (Aleph)…

The Aleph symbol, in mathematics, certainly does represent something infinite. However, it is not a symbol which is meant to represent “actual infinity” as a general concept. Rather, the Aleph is a very specific symbol with a very specific meaning. An Aleph describes the cardinality of an actually infinite set. This is a very important distinction, and one which will illuminate a number of the errors which Dr. Craig will make during his discussion of Infinity.

So, Aleph is a number. If you were to ask, “What is the number of elements in the set of Natural numbers?” the answer would be \aleph_0. That is the number of members in the set of Natural numbers.

This is one such error. As I just mentioned, Aleph is the cardinality of a set. It does not refer, necessarily, to the quantity of elements in that set. So, then, what does “cardinality” mean? Cardinality is a description of the algorithm which one might utilize in order to count or list the elements of a particular set. A set is said to have a cardinality of \aleph_0 if the elements of that set can be listed in a one-to-one correspondence with the elements of the set of Natural numbers. While this is sometimes referred to as the “size” of the set, this is not at all the same thing as a “quantity,” in the usual sense of the word.

Soon after this, Dr. Craig makes mention of something which is very important, and very true, but something which he will (unfortunately) either forget or ignore in his subsequent points. He says,

The lemniscate (\infty), or the potential infinity, isn’t a number. It’s not a number.

This is an extremely important point, because it underlies the vast majority of the problems and misunderstandings which most people have when they discuss infinities. Arithmetic operations can only be performed on numbers. You cannot, for example, answer the questions, “What is 5+Red?” or “How much is 7\times Salty?” or “What do you get when you divide Astronomy by John Wilkes Booth?” These are entirely nonsensical, because arithmetic operations can only be performed on numbers. In exactly the same way, one cannot perform arithmetic operations on \infty, because \infty is not a number. Keep this in mind, for later on.

Dr. Craig is then asked about the Aleph numbers, by someone in his class. The questioner wants to know in what way the Alephs can actually be considered numbers, presuming that arithmetic operations cannot be performed upon them (similar to the manner in which we cannot perform arithmetic with the lemniscate). However, WLC corrects this questioner by saying,

You can do transfinite arithmetic using these numbers. You can do multiplication, and you can do addition, and you can do exponentiation (like (\aleph_0)^2, for example). So, this is a number that can be manipulated in arithmetic, in this way. What is interesting (and this will become significant when we talk about whether actual infinites can really exist) is that you can’t do inverse operations, like subtraction and division, with them.

Now, Dr. Craig is certainly correct to point out that \aleph_0 can be utilized in mathematical operations. However, he introduces a bit of an equivocation, at this point, which he never corrects throughout the whole of the discussion. The operations of transfinite arithmetic are not the same operations as utilized in basic arithmetic. Transfinite Addition, Multiplication, and Exponentiation are very different from the Addition, Multiplication, and Exponentiation which children learn in school. It is absolutely fallacious to conflate the two, and yet Dr. Craig will do so many times during his discussion.

After this, Dr. Craig goes on to acknowledge that modern mathematics utilizes actually infinite sets all the time, and discusses the topic without issue. He explicitly notes that mathematicians have developed logically consistent and sound systems for discussing such actual infinites. Dr. Craig raises these points because one common objection to the claim that “actual infinites cannot exist” is that modern mathematics invalidates the sorts of arguments classically used in support of the claim. He responds to this objection by saying,

But is that really the case? Modern Set Theory shows that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way, without contradicting yourself (as I said in response to John’s question). Now, all this does is succeed in setting up a certain universe of discourse for talking consistently about actual infinites, but it does absolutely nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist. If Ghazali is right [about the impossibility of the actual infinite], this universe of discourse may be regarded simply as a fictional realm, rather like the world of Sherlock Holmes, in the Arthur Conan Doyle novels; not something that exists in the real world.

Of course, Dr. Craig neglects to mention that if Ghazali is wrong, then this universe of discourse might be perfectly applicable to the real world. So that leaves us with a rather important question: why should I think that this discourse does not apply to the real world? Well, one way (and, likely, the best way) to show that Modern Set Theory is inapplicable to the real world would be to attack the axioms upon which it has been built. This is the strategy employed by those very few mathematicians in the world (like the aforementioned Dr. Norman Wildberger) who reject the idea of actual infinites. However, this is not the tack taken by Dr. Craig. He doesn’t make any attempt to refute the axioms of Set Theory. In fact, Dr. Craig never even mentions any of the specific axioms of Set Theory.

The only discussion which Dr. Craig proffers in order to support his claim that actual infinites cannot exist is the example of Hilbert’s Hotel. Unfortunately, while he does an adequate job of explaining just what Hilbert’s Hotel is, and the different situations which can arrive in the thought experiment, he doesn’t actually give us very good reasons to think Hilbert’s Hotel refutes the idea of actually existing infinite sets. He simply shows that the thought experiment illustrates counter-intuitive properties of infinite sets, as if that acts as some sort of refutation.

Furthermore, there actually does exist a rather glaring problem with the manner in which Dr. Craig describes the entire thought experiment. Remember that earlier in the article, we explicitly noted that \infty is not a number, and Dr. Craig acknowledged in his discussion that there are some infinite numbers which are larger than other infinite numbers. However, he seems to have completely forgotten this fact as he starts to discuss the Hotel. To simply tell us that the hotel contains “an infinite number” of rooms gives us very little information. What infinite number are we talking about? There are an infinite number of infinite numbers, and some have very different properties from others!

Dr. Craig does not clarify, and seems content with his description of the Hotel as it stands. However, a few small clarifications can alter the whole thought experiment significantly. For example, suppose the Hotel has a countably infinite number of vacant rooms, and an uncountably infinite number of guests arrive. Well, despite the fact that the Hotel currently has an infinite number of empty rooms, it cannot possibly accommodate all of the guests which are arriving (yet another counter-intuitive property)!

This, combined with Dr. Craig’s confusion about cardinality as opposed to quantity, is what leads to his next misunderstanding.

What would happen if people started checking out of the Hotel? Let’s suppose that the people in the odd numbered rooms check out: 1, 2, 3– uh, I mean 1, 3, 5, 7, and so forth. All the odd numbered guests check out. How many guests are left? Well, all the even numbered guests. So, an infinite number of guests are still in the hotel, even though an equal number has already checked out and left the hotel.

Once again, Dr. Craig is ambiguous in his description of these infinite numbers, and does not seem to realize that the simple fact that two numbers are both infinite does not imply that they are equal. Let me reformulate Dr. Craig’s statement, for a moment, to illustrate the problem. In this example, we won’t even talk about an infinite hotel. Let’s say our hotel is finite, with an even number of rooms which are all filled up. Now, let’s say an even number of guests check out of the hotel, leaving an even number of rooms still occupied. So, an even number of guests are still in the hotel, even though an even number of guests has already checked out of the hotel. Would you think that there is any absurdity in that statement? Of course not, because not all even numbers are the same number.

Now, it’s worth noting that all of the mathematics which Dr. Craig has mentioned in his discussion, thus far, is around 100 years old or older. Cantorian Set Theory, Hilbert’s Hotel, transfinite arithmetic– these are things which were developed in the late 19th and early 20th Centuries. And yet, Dr. Craig refers to them as if they are the boundary of modern mathematics with infinities. He does not show any indicator that he is even aware of the fact that infinite mathematics has progressed quite a bit farther since the time of Cantor and Hilbert. Particularly, there is a field of study (now more than 50 years old) which greatly illuminates the problems in WLC’s understanding. Our eminent philosopher seems entirely ignorant of the last half-century’s work with Hyperreal numbers.

You see, despite Dr. Craig’s earlier protestations that inverse operations like Subtraction and Division cannot be performed on infinite numbers, there actually do exist systems in which those operators are defined. The Hyperreal number system is one example. And, unlike the Addition, Multiplication, and Exponentiation operators of transfinite arithmetic, the Hyperreal operators do exactly what you would expect those operators to do based on your childhood math classes.

It’s very easy to now reformulate Dr. Craig’s discussion of people checking out of Hilbert’s Hotel using Hyperreal numbers. Let’s say that the Hotel is currently filled to capacity, and that the Hotel has a positive, countably infinite, Hyperreal number of rooms. Let’s call that number H. We can even define H exactly– let’s say that H is the limit of the sequence (1,2,3,4,5,...). Now, let’s say that all of the occupants in odd numbered rooms check out of the Hotel. Since this amounts to half of the Hotel’s occupants, we can quite easily see that \frac {H}{2}=(\frac{1}{2},1,\frac{3}{2},2,\frac{5}{2},...) guests have checked out, and that another \frac{H}{2} guests yet remain. So the original number of guests is infinite, and the number of guests which check out is also infinite, but these are not equal infinite numbers. Meanwhile, the number of guests which have checked out and the number of guests which remain are equal infinite numbers.

There is absolutely no absurdity in this. There is no inconsistency. The mathematics is every bit as sound as it would be if H had been a finite number.

But now let’s suppose instead that all of the guests in the rooms 3, 4, 5, 6, 7, out to infinity, check out. How many guests are left now? Well, if there’s a room 0, just three are left. And yet, the same number of guests checked out this time as when all of the odd numbered guests left. So you subtract identical quantities from identical quantities and you get non-identical results, which is absurd.

Dr. Craig’s errors persist into this quote. Again, the cardinality of the sets is equal, but the quantity of elements is not. So, again putting this in terms of Hyperreal numbers, we started with H guests, and in Scenario 1, we saw that \frac{H}{2} guests checked out of the Hotel. Now, in Scenario 2, all but three of the guests check out of the hotel. Despite Dr. Craig’s claims, this is not an identical quantity to that which left in Scenario 1. Instead, the quantity of guests leaving in Scenario 2 is H-3=(-2,-1,0,1,2,...). Again, this is still an infinite number, but it is a different infinite number than the others we have mentioned. Contrary to WLC’s assertion, H \neq \frac{H}{2} \neq (H-3), and no absurdity is shown, at all.

Keep in mind that the Hyperreal number system is not just some crazy, esoteric, and untested construction. Quite the contrary, it has been extensively explored by mathematicians for a very long time. In fact, it has been more than half a century since Abraham Robinson proved that the Hyperreal number system is logically consistent if and only if the Real number system is, as well. As a consequence, if you believe that numbers like \sqrt 2 and \pi are applicable to the real world, you’ll be very hard-pressed, indeed, to give a convincing reason to think that Hyperreal numbers are not.

William Lane Craig is woefully ill-equipped to pass judgment upon the metaphysical possibility of an actual infinite. He seems entirely ignorant of nearly a century’s worth of work, on the subject, and he greatly misunderstands those bits of infinite mathematics with which he is aware. The mathematics which deals with the infinite is logically consistent, and does not produce the absurdities which Dr. Craig claims are produced. He does absolutely nothing to discuss, let alone refute, the axioms upon which infinite mathematics are based. He gives us absolutely no good reason to reject the idea that infinite mathematics might be applicable to the real world.

William Lane Craig simply doesn’t know what he’s talking about, when it comes to Infinity.


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23 thoughts on “WLC doesn’t understand Infinity, Part 1

  1. I am wondering what sort of tangle he gets into when faced with complex numbers.

    • At a guess, he would probably argue similarly to what he claimed about Set Theory. That is to say that he would likely acknowledge the math to be logically consistent and valid, but he would still argue that it is only a convenient fiction rather than being reflective of reality.

      Ironically, WLC also loves to argue that Mathematics are so unreasonably applicable to the real world that God must exist.

      https://boxingpythagoras.com/2014/04/21/mathematics-natural-or-supernatural/

      I honestly don’t know how he can possibly rationalize claiming that Set Theory (the foundation of modern mathematics) is inapplicable to the real world while simultaneously claiming that mathematics is unreasonably applicable to the real world.

  2. I have trouble accepting the existence of actual infinities. Given your acceptance, what metaphysical status do you then assign to infinities – or even mathematics as a whole? Do they exist in a physical sense (as in the way the number 10 can be represented \ derived from my fingers), or only in a Platonic sense?

    • I’m a Nominalist, so I don’t believe that numbers “exist” in the Platonic sense. As such, I believe all numbers to simply be descriptors of entities. Whether we’re talking about 2 or 4/7 or -3 or π or 6+5i or some infinite Hyperreal, I see no reason to think these numbers cannot be applicable to the real world.

      • OK, so before I say too much I should add the disclaimer that I am far from being a professional mathematician or philosopher, so I may simply need to be educated. That aside, when I look at those other classes of number (natural, integer, rational, irrational and complex) I can see how they are either derived from empirical observation, or derived from those derivations. For example, the observation of separate entities gives us natural numbers. Then observing the countable relationship between naturals gives us integers (e.g., -3 = three fewer). Then another relationship (ratio) gives us rationals and in some cases irrationals (e.g., circumference over diameter). Complex numbers are derived from applying the same methods that work for naturals to all integers and thus allows us to describe things with those relations mathematically. But when it comes to infinities, I don’t see how you get to infinity without just popping one in arbitrarily. I don’t see where the empirical chain can lead to an actual infinite. Is there an explanation of which I am unaware?

        • There certainly is! The book which I recommend in the article, Infinitesimal by Amir Alexander, does a wonderful job of explaining the manner in which the exploration of actual infinites arose. In a nutshell, it’s a consequence of greater investigation of Geometry, and the relationship between points, lines, planes, and spaces.

          Another great source would be the Epilogue from Keisler’s Elementary Calculus, which is freely available here.

          https://www.math.wisc.edu/~keisler/calc.html

          That discusses the mathematical basis for discussing the Hyperreals.

  3. Hey, so I read the epilogue and didn’t see the empirically justified derivation, so then I looked at Ch. 1 of the book and all I see is the “bold step of introducing a new kind of number”. Can you try explaining the empirical link that leads to accepting the reality of infinites (and infinitesimals)? To be clear, I’m not doubting their value as tools. I’m just questioning whether anything in reality can be an actual infinite.

    • Sure! My favorite example is a sort of variation on Zeno’s Dichotomy Paradox.

      Imagine any two different locations in the real world. We’ll call them positions A and B. How many unique positions exist between A and B?

      Now, I’m sure that we can agree that every possible position between A and B exists just as much as A and B, themselves. So, whatever number we assign to the quantity of those positions is certainly reflective of an actual, completed quantity of entities in the real world.

      So, if all the positions between A and B exist, that means the position halfway between A and B exists. We needn’t actually stand in that position in order that it might exist. Similarly, the position 0.546% of the way between A and B actually exists, as does the position π% of the way between them. In general, any position x% of the way between A and B (where x is a Real number greater than 0 and less than 100) actually exists.

      So, what is the total quantity of these unique, individual positions? Whatever the specific number of that quantity might be, it certainly is not finite. It is, in fact, uncountably infinite. Therefore, infinite numbers would be applicable to the real world.

      • My understanding is that many leading theories quantize space-time at something around the Planck length. Assuming that held up, would there be another reason to accept an actual infinite?

  4. I think, if Travis and I share a similar mental wrangling, the actual infinite, even in the dichotomy paradox (which, fortunately, is one I’m familiar with), seems to become merely a mathematical game or invention the more you follow it through. Of course I understand the relevance, the very real and practical appreciation, that between my office and the train station there are recognisable points: the post-box at half-way; the fruiterer a quarter of the way, the eight paving stones between the door and the fruiterer etc.; and, on an everyday basis, these are factors and distances that we acknowledge, but the more you explore the distances in-between each, the infinite points, as you rightly point out, the more and more it enters the mathematical realm (distance between particles, the distances between the particles and those points, etc.) and leaves reality– the reality of the non-mathematician, in any case. There’s almost a similarity to Sorites’ paradox to the extent you start to enter the absurd.

    Hopefully Travis will correct me if I’m misinterpreting his dilemma and simply projecting my own thoughts on the subject but I think this is what he was grasping at by referring to the empirical, physical infinite. I don’t think there’s an unconscious or conscious acknowledgement on a particle level for the everyday man *without* inventing the mathematics. Do you see what I’m grasping at even if not expressing very well?

    Very much enjoyed this post, by the way; despite much of the maths being lost on me.

    • It seems we’re generally on the same page, though I might go a step farther. I lean toward the perspective that mathematics is nothing more than a conceptual tool for describing reality. In most cases the mathematics of the infinite \ infinitesimal does this very well, but it is equally possible in those cases to substitute the infinite with a ridiculously large \ small number compared to all other values in the equation and get, what is for all practical real-world purposes, the same result. However, sometimes the infinite leads to completely bizarre results – as in BP’s earlier post – and this triggers in me a concern that perhaps something unreal is going on. Couple this with non-algorithmic conceptual issues regarding infinites and I can’t help but wonder whether it is in fact a “useful fiction” which can sometimes reveal itself as unrepresentative of reality when pushed to the limit (pun intended). I grant that I am something less than an amateur mathematician and may be completely wrong on this, but I have yet to encounter a convincing case for the actual reality of the infinite \ infinitesimal.

      Having said all that, allow me to offer a possible rejoinder: I have heard it said (from Pigliucci, I believe) that the equations which describe phase transitions require an actual infinite and do not accurately reflect reality otherwise. This is a domain that I have not pursued and would likely not fully understand, but I believe that there may be something to it. Take a look at section 2.2.2 of “Mathematics and the Natural Sciences” on Google Books to get a taste of this. It’s possible that if I was better equipped to dig in and fully understand this argument then I might be convinced of the reality of the actual infinite

      But then I see in there a discussion of regularization and renormalization and I’m led to recall the comment thread here, discussing the claim that 1 + 2 + 3 + … = -1/12, where a quantum physicist seems to show how that result is not a product of actual infinities but rather a product of a particular regularization technique. Again, it’s all way over my head but as far as I can grasp the topic, it appears that quantum physics does not rely on actual infinites, and since everything above the quantum level is apparently reducible to the quantum level, everything else would not rely on actual infinites either.

  5. I lean toward the perspective that mathematics is nothing more than a conceptual tool for describing reality.

    I completely agree with you here, and have presented exactly this position in my article, Mathematics: Natural or Supernatural.

    However, sometimes the infinite leads to completely bizarre results – as in BP’s earlier post – and this triggers in me a concern that perhaps something unreal is going on

    The fact that something seems counterintuitive or bizarre is a fairly poor gauge of its veracity. Special Relativity, General Relativity, and Quantum Mechanics all introduced mathematics with incredibly bizarre consequences which have, nonetheless, shown to be extremely accurate in describing the workings of the real world.

    Take a look at section 2.2.2 of “Mathematics and the Natural Sciences” on Google Books to get a taste of this.

    This is actually a wonderful source, and I thank you for sharing it.

    I’ll also note that if you think the Real numbers (like √2 or π) are applicable to the real world, it is exceedingly difficult to defend the idea that the Hyperreals are not so applicable. The only manner in which to rigorously define the very concept of the Real numbers is by use of infinite sets.

    Again, it’s all way over my head but as far as I can grasp the topic, it appears that quantum physics does not rely on actual infinites, and since everything above the quantum level is apparently reducible to the quantum level, everything else would not rely on actual infinites either.

    It isn’t actually true that quantum physics does not rely on actual infinites. The wave function, itself, is not quantized. It is a distribution function over a range of Real numbers; and anytime you have a range of Real numbers from a to b (where a<b) you have an uncountably infinite set of numbers.

    It’s also not apparent that everything above the quantum level is reducible to the quantum level. Though there are some rather ingenious attempts at solving the issue, quite possibly the most important problem currently unresolved in physics is that Gravity cannot yet be reduced to quantum scales. It’s possible that it can be. However, it’s equally possible that there do exist fields which do not quantize. And even if quantum gravity can be solved, it is still not evident that actual infinites cannot possibly be applicable to the real world.

    Now, I’m all for skepticism. By all means, you are free to doubt the applicability of actual infinites to the real world. In fact, I actively encourage such thought! However, that’s not what William Lane Craig is doing. WLC is claiming that it is metaphysically impossible for actual infinites to be applicable to the real world. And that claim comes with a rather heavy burden of proof which he has certainly failed to satisfy.

    Even if it is true that there cannot be any actual infinites in the real world, WLC has not even come close to proving that case.

    • The fact that something seems counterintuitive or bizarre is a fairly poor gauge of its veracity.

      Actually, I would suggest that in practice the counterintuitive or bizarreness of something is usually a very good gauge of its veracity and is all we really have in the absence of evidence to the contrary. I would of course agree, however, that it is far from perfect and should be vetoed in the face of evidence, as we have with relativity and quantum mechanics.

      I’ll also note that if you think the Real numbers (like √2 or π) are applicable to the real world, it is exceedingly difficult to defend the idea that the Hyperreals are not so applicable.

      You’re correct to note that the rejection of actual infinitesimals would entail rejection of the actual existence of irrationals and I may have alluded to their acceptance in a prior comment without really thinking it through. That said, I think there would be little to no consequence to their non-existence. Perhaps there is no such thing as a “perfect circle” as defined by π. Perhaps π is just the average value we would get if we were able to measure C/2r in units of the minimal quanta without violating Heisenberg’s uncertainty principle. I don’t see that this would cause a problem for us. After all, every single computational use of π today (i.e., values which don’t include π themselves) ends up actually using a rational value, which means that everything that has ever been engineered by using π ended up actually using something that was not exactly π. Conversely, the practical consequences of accepting or denying infinites looms large. So I think there is a pragmatically distinct difference in the ramifications for the existence of irrationals versus infinities. So the issue for me comes down to the following:
      1) Are there results that the use of infinities \ infinitesimals produces which are demonstrably better than the results we could obtain through finite procedures? I’m not aware of any clear reason to answer ‘yes’, though the phase transition example may be such a case, pending further scrutiny.
      2) Are there results that the use of infinities \ infinitesimals produces which are problematic and which would not obtain through finite procedures? It seems so, but I concede that I could be wrong about that.

      These two answers, taken together, cause me to slightly favor not accepting the existence of an actual infinite – though that is highly tentative and probably better described as “I don’t know”.

      It isn’t actually true that quantum physics does not rely on actual infinites. The wave function, itself, is not quantized. It is a distribution function over a range of Real numbers; and anytime you have a range of Real numbers from a to b (where a<b) you have an uncountably infinite set of numbers.

      I don’t know enough about the derivation of the wave function to answer this, but is it perhaps a possible infinite rather than an actual infinite? Or is it perhaps infinite only because the calculus used to arrive at it assumed infinites as a matter of convention or precedence? I also don’t know how the wave function is actually utilized – is renormalization perhaps used in practice?

      even if quantum gravity can be solved, it is still not evident that actual infinites cannot possibly be applicable to the real world.

      Agreed, and in the absence of a quantum gravity solution, it is still not evident that actual infinities are applicable to the real world.

      Even if it is true that there cannot be any actual infinites in the real world, WLC has not even come close to proving that case.

      Agreed, but his argument could be viewed as probabilistic rather than deductive and then it wouldn’t matter. I suspect that it’s more just a matter of convention that it is formed as a deductive argument rather than a probabilistic argument.

      Thanks for engaging on this topic. It’s both mentally exhausting and infinitely interesting.

      • Actually, I would suggest that in practice the counterintuitive or bizarreness of something is usually a very good gauge of its veracity

        I would wholeheartedly disagree with you, here, for two very important reasons. Firstly, “counterintuitiveness” and “bizarreness” are extremely subjective terms. What may be bizarre to one person might be perfectly intuitive to another.

        The second reason, and the more important one, is that the fact that a person might find something to be bizarre is absolutely no indicator of veracity. It is entirely possible for people to hold that true things are bizarre. Bizarreness does not, in any way, imply falsehood. It may give us sufficient reason to be skeptical, but it does not give good reason to conclude that the bizarre proposition is therefore false.

        That said, I think there would be little to no consequence to their non-existence. Perhaps there is no such thing as a “perfect circle” as defined by π. Perhaps π is just the average value we would get if we were able to measure C/2r in units of the minimal quanta without violating Heisenberg’s uncertainty principle.

        You can drop the “perfect” qualifier. If there are no actual infinites, there are no actual circles, at all (as an aside, a circle is not defined by π; quite the reverse, in fact). But there’s even more bizarre consequences, as well. It might still be possible for a square to exist, in the absence of actual infinites, but it would not be possible for the diagonal of a square to exist. The diagonal of a rectangle whose length and width are in a ratio of 3:4 could possibly exist, but the diagonal of a rectangle whose length and width are in a ratio of 1:2 could not exist.

        If you are averse to bizarreness, then you probably don’t want to try excising actual infinites from mathematics.

        Are there results that the use of infinities \ infinitesimals produces which are demonstrably better than the results we could obtain through finite procedures? I’m not aware of any clear reason to answer ‘yes’

        Yes. It was for this very reason that Isaac Newton invented his Calculus. The results of using infinitesimal calculus to predict the motion of the planets are far, far better than the results we had obtained through finite procedures. Calculations of compound interest, population growth, economics, radioactive decay, and generally anything involving Real numbers or Calculus perform demonstrably better than finite procedures.

        Are there results that the use of infinities \ infinitesimals produces which are problematic and which would not obtain through finite procedures? It seems so, but I concede that I could be wrong about that.

        In general, if a particular problem can be solved using both finite and infinite methods, the results of those two methods will be consistent. I know of no cases in which a problem can be legitimately solved by two methods which yield conflicting answers.

        I don’t know enough about the derivation of the wave function to answer this, but is it perhaps a possible infinite rather than an actual infinite?

        The most common form for the wave function is e^[i(kxωt)]. That e is the base of the natural logarithm. It is an irrational, transcendental number defined by an infinite limit. There’s no way to coherently define e without completed infinite sets.

        Agreed, but his argument could be viewed as probabilistic rather than deductive and then it wouldn’t matter

        He doesn’t do anything to establish a probabilistic argument, either, though. The parts of his dialectic which aren’t blatantly wrong do nothing to show that actual infinities are less probable than the contrary.

        Thanks for engaging on this topic. It’s both mentally exhausting and infinitely interesting.

        • I’m going skip the first part because it’s apt to go way off topic.

          If you are averse to bizarreness, then you probably don’t want to try excising actual infinites from mathematics.

          First, I’m not trying to excise anything from mathematics. There’s no doubt that infinites are valuable mathematical tools. The question is whether they exist in reality. Second, those examples (circle and diagonal)) are purely mathematical constructs. They tell us nothing about the existence of real entities.

          Yes. It was for this very reason that Isaac Newton invented his Calculus. The results of using infinitesimal calculus to predict the motion of the planets are far, far better than the results we had obtained through finite procedures. Calculations of compound interest, population growth, economics, radioactive decay, and generally anything involving Real numbers or Calculus perform demonstrably better than finite procedures.

          Yes, algorithms can use infinites to facilitate easier calculations. But let’s go back to the possibility that reality is fundamentally quantized. Assuming that is true, do you think that anything you have mentioned here would actually be any different than what we see now? We are effectively doing everything in finite maths whenever we don’t report observations in units of irrationals, which makes sense because I don’t even know what it would mean for an observation to include an irrational.

          Aside from the types of cases already mentioned (i.e., phase transitions), one could argue that infinite maths are “more real” than finite maths operating on a fundamental quantum unit because otherwise we wouldn’t see that our observations are reliably centered around the results predicted by using infinites \ irrationals, but I think this assumes that the fundamental quantum unit maths would not be stochastic and thus biased in some way. And quantum theory is fundamentally stochastic.

          I know of no cases in which a problem can be legitimately solved by two methods which yield conflicting answers.

          Yeah, that isn’t quite what I meant and it’s my fault for the way I worded it. I’m thinking more of things like Zeno’s paradox, Hilbert’s hotel and other issues that don’t arise unless there are actual infinites.

          The most common form for the wave function is e^[i(kx–ωt)]… It is an irrational, transcendental number defined by an infinite limit.

          Right, so it’s an outcome of having used infinites in the mathematical derivation. That doesn’t imply the actual existence of the infinite.

          • Second, those examples (circle and diagonal)) are purely mathematical constructs. They tell us nothing about the existence of real entities.

            This seems a fairly curious position, to me. Let’s assume, for a moment, that space is discrete, and quantized at the Planck length. If I choose an arbitrary position in space which actually exists, which we’ll call A, then it seems obvious that I should be able to find another actual position, B, which is exactly 10 Planck lengths away from A. I should similarly be able to find a position, C, which is exactly 10 Planck lengths away from B such that the angle ABC measures precisely 90-degrees; and another position, D, exactly 10 Planck lengths from C such that angle BCD measures precisely 90-degrees. I have now defined a square in real space, ABCD. However, it would be impossible to travel a straight path from A to C, and similarly from B to D.

            The square in this example would not simply be a conceptual construct. We are discussing an actual, physical square. The square would exist in reality, but its diagonal would not.

            Yes, algorithms can use infinites to facilitate easier calculations. But let’s go back to the possibility that reality is fundamentally quantized. Assuming that is true, do you think that anything you have mentioned here would actually be any different than what we see now?

            If the whole of reality is fundamentally discrete, it becomes exceedingly curious why mathematics designed to describe a continuous reality does such a wonderful job of describing the real world.

            Imagine, for a moment, setting a brick on the top step of a flight of stairs. You would expect that the brick stay motionless, right? Well, what if, instead of remaining motionless, that brick glided over the contours of the stairway and ended up at the bottom in such a manner that it’s position at any point during the traversal could be very accurately predicted by assuming the brick was sliding down an icy slope. That’s the conundrum which discrete space would present to us: why does reality seem to behave as if it is continuous if it actually is not continuous?

            And quantum theory is fundamentally stochastic.

            This is not necessarily true. On the Copenhagen interpretation (as well as some others), QM is fundamentally stochastic. This, however, is not the case on all possible interpretations of QM, including a number which are extremely popularly held amongst eminent quantum physicists (such as Everett’s Many Worlds). Personally, I tend to prefer Two-State Vector Formalism, which is certainly not stochastic.

            I’m thinking more of things like Zeno’s paradox, Hilbert’s hotel and other issues that don’t arise unless there are actual infinites.

            Zeno’s paradox is only a paradox if you insist that it is possible to divide a real distance a potentially infinite number of times while denying that actual infinites exist. If you reject potential infinity along with actual infinity, there is no paradox. If you do not reject actual infinites, there is no paradox.

            Hilbert’s hotel doesn’t actually present any paradox, at all. It’s a curious thought experiment, but there is nothing logically inconsistent or absurd in the Grand Hotel, despite WLC’s protestations to the contrary.

            Right, so it’s an outcome of having used infinites in the mathematical derivation. That doesn’t imply the actual existence of the infinite.

            This, also, is a fairly curious position to hold.

            Imagine, for a moment, someone came to you and denied that any number other than the number 1 was actually applicable to the real world. The Naturals, the Rationals, the Reals, et cetera, are all just convenient fictions, according to this person, and the only number which actually describes reality is 1.

            “But wait,” you reply, “I have here two coins in my hand! Obviously, ‘two’ is a real number!” But your friend simply responds that you don’t actually have two coins. Instead, you have one coin and one coin. In fact, he says, everything described by the convenient fiction of mathematics is simply referring to things which are 1, or the absence of things which are 1.

            It doesn’t matter that the convenient fiction seems to very accurately describe the way things interact with one another. It’s simply the outcome of a mathematical derivation. That doesn’t imply the existence of “two” or “three” or “eight-thousand six-hundred fifty-one.”

            If a number can only be defined by use of infinities, and that number is almost certainly applicable to the real world, it seems quite peculiar indeed to insist that actual infinities still cannot be applicable to the real world.

  6. Let’s assume, for a moment, that space is discrete … The square in this example would not simply be a conceptual construct. We are discussing an actual, physical square. The square would exist in reality, but its diagonal would not.

    It seems to me that you have superimposed continuous-space assumptions. The physical manifestation of a “straight line” is the shortest distance in space between two points. The path from A to C and from B to D is straight if it traverses the fewest number of quantum units. We can’t superimpose a continuous spacial dimension on top of that and then use it to say that the quantized space is invalid. For this thought experiment we have already granted the assumption that space is discrete.

    If the whole of reality is fundamentally discrete, it becomes exceedingly curious why mathematics designed to describe a continuous reality does such a wonderful job of describing the real world.

    This is a valid point and one I’ve been pondering throughout our discussion. It’s the reason I brought up the stochastic nature of quantum theory. And you’re correct to note that some interpretations are not stochastic. That said, I think that if a stochastic interpretation is correct then it explains this observation: the consistent physical agreement with results predicted using irrationals and infinitesimals represent cases where the probability space “surrounds” that value, such that we can’t detect a bias toward the rational values which correspond with the quantized results that actually obtain in any one case. But then there’s also the matter of scale…

    Imagine, for a moment, setting a brick on the top step of a flight of stairs … That’s the conundrum which discrete space would present to us: why does reality seem to behave as if it is continuous if it actually is not continuous?

    We’re talking about levels of precision which far exceed our capabilities. That’s a bit like looking at the edge of a razor under an electron microscope and then translating that our perceptual level and concluding that it’s too dull to cut anything. If the physical realization of pi was actually rational and ended at the 1020th digit, with stochastic variation, I don’t see that we would ever be able to tell the difference between that world and the world where the irrational value actually applied.

    Imagine, for a moment, someone came to you and denied that any number other than the number 1 was actually applicable to the real world … If a number can only be defined by use of infinities, and that number is almost certainly applicable to the real world, it seems quite peculiar indeed to insist that actual infinities still cannot be applicable to the real world.

    I don’t see that this analogy maps to the question of the infinite. There is no obvious parallel for “I have here two coins in my hand”. Regardless, if the closing sentence is the key point then this is pretty much the same point as above. And, as noted, I think that’s a valid point, but I also don’t think that it’s obviously true that finite maths (as described above) could not account for those observations.

    Thanks again for the dialogue. I’m probably further into the “I don’t know” camp than when I started, but I’m not an infinite realist yet.

    Also thought I’d share a couple interesting links that I ran across in the last couple days:
    – A free MITx course on “Paradox and Infinity”
    An article where several different parties defend the rejection of infinites

  7. I confess that one of those curious little things my mind just can’t grasp is the distinction between a potential and an actual infinity.

    I understand the reasonings people give in distinguishing them, mind you; it’s just that when I try to think of an infinity and whether it would be regarded as potential or infinite, I can’t convince myself that I must be right. There’s something about the distinction drawn that I’m just not building an intuitive feeling for.

    • It also took me some time to really wrap my head around the distinction. I tried to figure the best succinct way of defining the two concepts that I could, when I wrote this article. After trying out a few different wordings, the phrasing that I utilized was:

      An actual infinity refers to a complete set of entities which has a quantity greater than any finite number. A potential infinity refers to a process or algorithm which can be repeatedly iterated without end.

      So, for example, the set of all points in a line segment (assuming continuity) would be an actual infinite, as it refers to a complete set.

      A potential infinity, on the other hand, implies an iterated process. For example, if I started counting off the Natural numbers, the process would be potentially infinite because there is no upper limit to the steps in this algorithm, and no single individual iteration would ever complete the set.

      So, an actual infinite refers to the properties of a collection, while a potential infinite refers to the properties of a process.

      • That does feel like it may be the sort of explanation I can internalize. At the least I can see distinguishing between a process for generating a set and the set itself.

        It still feels outside the tradition of infinities that I’m used to from calculus/real analysis/mathematical logic as an undergraduate, though.

  8. 1301 on said:

    Do you actually understand the mathematics behind the hyperreals (namely the ultrafilter on N used in the ultraproduct of R) when you claim that exponentiation can be defined (presumably with the usual properties) on hyperreals???

    • I’m not sure what you’re attempting to imply, here. Surely, you aren’t trying to say that exponentiation is completely undefined on *R, nor that the operation behaves entirely differently than it does on R.

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