Infinitesimals: Another argument against actual infinite sets

by common_law5 min read26th Jan 201331 comments


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My argument from the incoherence of actually existing infinitesimals has the following structure:

1. Infinitesimal quantities can’t exist;

2. If actual infinities can exist, actual infinitesimals must exist;

3. Therefore, actual infinities can’t exist.

Although Cantor, who invented the mathematics of transfinite numbers, rejected infinitesimals, mathematicians have continued to develop analyses based on them, as mathematically legitimate as are transfinite numbers, but few philosophers try to justify actual infinitesimals, which have some of the characteristics of zero and some characteristics of real numbers. When you add an infinitesimal to a real number, it’s like adding zero. But when you multiply an infinitesimal by infinity, you sometimes get a finite quantity: the points on a line are of infinitesimal dimension, in that they occupy no space (as if they were zero duration), yet compose lines finite in extent.

Few advocate actual infinitesimals because an actually existing infinitesimal is indistinguishable from zero. For however small a quantity you choose, it’s obvious that you can make it yet smaller. The role of zero as a boundary accounts for why it’s obvious you can always reduce a quantity. If I deny you can, you reply that since you can reduce it to zero and the function is continuous, you necessarily can reduce any given quantity—precluding actual infinities. When I raise the same argument about an infinite set, you can’t reply that you can always make the set bigger; if I say add an element, you reply that the sets are still the same size (cardinality). The boundary imposed by zero is counterpoint for infinitesimals to the openness of infinity, but the ability to demonstrate actual-infinitesimals’ incoherence suggests that infinity is similarly infirm.

Can more be said to establish that the conclusion about actual infinitesimal quantities also applies to actual infinite quantities? Consider again the points on a 3-inch line segment. If there are infinitely many, then each must be infinitesimal. Since there are no actual infinitesimals, there are no actual infinities of points.

But this conclusion depends on the actual infinity being embedded in a finite quantity—although, as will be seen, rejecting bounded infinities alone travels metaphysical mileage. For boundless infinities, consider the number of quarks in a supposed universe of infinitely many. Form the ratio between the number of quarks in our galaxy and the infinite number of quarks in the universe. The ratio isn’t zero because infinitely many galaxies would still form a null proportion to the universal total; it’s not any real number because many of them would then add up to more than the total universe. This ratio must be infinitesimal. Since infinitesimals don’t exist, neither do unbounded infinities (hence, infinite quantities in general, their being either bounded or unbounded).


Infinitesimals and Zeno’s paradox

Rejecting actually existing infinities is what really resolves Zeno’s paradox, and it resolves it by way of finding that infinitesimals don’t exist. Zeno’s paradox, perhaps the most intriguing logical puzzle in philosophy, purports to show that motion is impossible. In the version I’ll use, the paradox analyzes my walk from the middle of the room to the wall as decomposable into an infinite series of walks, each reducing the remaining distance by one-half. The paradox posits that completing an infinite series is self-contradictory: infinite means uncompletable. I can never reach the wall, but the same logic applies to any distance; hence, motion is proven impossible.

The standard view holds that the invention of the integral calculus completely resolved the paradox by refuting the premise that an infinite series can’t be completed. Mathematically, the infinite series of times actually does sum to a finite value, which equals the time required to walk the distance; Zeno’s deficiency is pronounced to be that the mathematics of infinite series was yet to be invented. But the answer only shows that (apparent) motion is mathematically tractable; it doesn’t show how it can occur. Mathematical tractability is at the expense of logical rigor because it is achieved by ignoring the distinction between exclusive and inclusive limits. When I stroll to the wall, the wall represents an inclusive limit—I actually reach the wall. When I integrate the series created by adding half the remaining distance, I only approach the limit equated with the wall. Calculus can be developed in terms of infinitesimals, and in those terms, the series comes infinitesimally close to the limit, and in this context, we treat the infinitesimal as if it were zero. As we’ve seen, actual infinity and infinitesimals are inseparable, certainly where, as here, the actual infinity is bounded. The calculus solves the paradox only if actual infinitesimals exist—but they don’t.

Zeno’s misdirection can now be reconceived as—while correctly denying the existence of actual infinities—falsely affirming the existence of its counterpart, the infinitesimal. The paradox assumes that while I’m uninterruptedly walking to the wall, I occupy a series of infinitesimally small points in space and time, such that I am at a point at a specific time the same way as if I were had stopped.

Although the objection to analyzing motion in Zeno’s manner was apparently raised as early as Aristotle, the calculus seems to have obscured the metaphysical project more than illuminating it. Logician Graham Priest (Beyond the Limits of Thought (2003)) argues that Zeno’s paradox shows that actual infinities can exist by the following thought experiment. Priest asks that you imagine that rather than walking continuously to the wall, I stop for two seconds at each halfway point. Priest claims the series would then complete, but his argument shows that he doesn’t understand that the paradox depends on the stopping points being infinitesimal. Despite the early recognition that (what we now call) infinitesimals are at the root of the paradox, philosophers today don’t always grasp the correct metaphysical analysis.

Distinguishing actual and potential infinities

Recognizing that infinitesimals are mathematical fictions solidifies the distinction between actual and potential infinity. The reason that mathematical infinities are not just consistent but are useful is that potential infinities can exist. Zeno’s paradox conceives motion as an actual infinity of sub-trips, but, in reality, all that can be shown is that the sub-trips are potentially infinite. There’s no limit to how many times you can subdivide the path, but traversing it doesn’t automatically subdivide it infinitely, which result would require that there be infinitesimal quantities. This understanding reinforces the point about dubious physical theories that posit an infinity of worlds. It’s been argued that the many-worlds interpretation of quantum mechanics, which invokes an uncountable infinity of worlds, doesn’t require actual infinity any more than does the existence of a line segment, which can be decomposed into uncountably many segments, but this plurality of worlds does not avoid actual infinity. We exist in one of those worlds. Many worlds, unlike infinitesimals and the conceptual line segments employing them, must be conceived as actually existing



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common_law, you appear to be the same person as metaphysicist. It appears that you have been upvoting metaphysicist's articles.

Is there a reason why you are using two identities here?

Karma requirements for posting? User metaphysicist's karma is 0.

If I remember correctly, this post was at +3ish shortly after being posted. Coincidence ...

This reads like really bad armchair philosophy. You make a bunch of statements about infinities and infinitesimals without any regards for what this actually means, experience wise. Then you bring up Zeno's paradox, which may have been intriguing in the 5th century BCE but was solved nicely by classical physics. You blindly thrash about in math land with no regard for rigor, then conclude with a statement that again has no relation to actual experience.

I'm bad at expressing myself through writing, but this post is really bad.

I downvoted common_law's post, because of some clear-cut math errors, which I pointed out. I'm downvoting your comment because it's not saying anything constructive.

There's nothing wrong with what common_law was trying to do here, which is to show that infinite sets shouldn't be part of our ontology. Experience can't be the sole arbiter of which model of reality is best; there is also parsimony. Whether infinite quantities are actually real, is no less worthy of discussion than whether MWI is actually real, or merely a computational convenience. I only agree with you that the math lacked rigor. This is discussion, so I don't see a problem with posting things that need to be corrected, but I had to downvote the post because it might have confused someone who didn't notice the errors.

There are arguments which are wrong because they lack rigor, but in my opinion this isn't one of them. The main problem is asking a question about "actual existence" of abstract objects without clear understanding what such "actual existence" would represent. I can imagine a rigorous version of this post where "actual existence" was given a rigorous definition, but I doubt it would convince me about anything (as I remain unconvinced that e.g. modal logic is a useful epistemological tool although it can be formalised).

Note that (one of) the apparent motivation(s) of all recent anti-infinity posts is rejection of many-world interpretation of QM, i.e. it is unlikely that the author is aiming at constructing a neat rigorous theory.

I feel like you're saying "these things don't act like numbers, therefore they can't exist." Like this sentence:

When you add an infinitesimal to a real number, it’s like adding zero. But when you multiply an infinitesimal by infinity, you sometimes get a finite quantity.

You don't go into what rules infinitesimals play by. You immediately try to make them play by number-rules:

Few advocate actual infinitesimals because an actually existing infinitesimal is indistinguishable from zero.

Really? When was the last time you integrated zero and got a positive number?

e.g. "if x exists, and y exists, then the ratio of x to y must exist"

Really? When was the last time you integrated zero and got a positive number

It is allowed in some systems, to pick a random real number from the interval [0,1]. The probability for the each of them is 0, yet the probability for the whole interval is 1.

A way to pick a random number from this interval is tossing a fair coin countably many times. The head gives you 1 and the tail gives you 0 in the binary representation. Every toss takes half the time as the previous one, so you finish this construction in a finite time. So called supertasks are allowed sometimes.

No, the probability density function for a uniform distribution on [0,1] is what you are integrating, and that is non-zero.

Is it? How probable is 1/2, for example?

[-][anonymous]9y 7

That's not what a probability density function is.

Still. How probable is 1/2 in the above process of coin toss?

1/2=.1000000... in the binary presentation, means one head and all tails.

[-][anonymous]9y 2
Or also .0111..., one tail and all heads.

Any individual number has probability 0, but the probability density is the probability that you'll get a number between x and dx, divided by dx, in the limit as dx approaches 0.

Any individual real number has the zero probability, but at least one of them - is bound to happen.

One may or may not consider sub intervals. It is a side question. Just as rational numbers, or algebraic numbers on this interval. Every sub-interval has the probability equal of its length what is always nonzero. All rational numbers have the probability 0, for example.

The coin flipping trick will miss plenty of numbers, like one third - and those that are left have a small but non-infenitesimal probability.

Edit: whoops, my bad, read "countable" as "finite".

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in the binary representation

It is obvious you're not overly familiar with mathematics, but I can't help but notice how despite writing "since infinitesimals don't exist" several times in your post, you fail to give any kind of proof for this assumption. I hope you don't consider such a ludicrous line as "few advocate actual infinitesimals because an actually existing infinitesimal is indistinguishable from zero" your proof.

You also fail to define "existence", and the post you link to does little else than go to great lengths to avoid this issue. There's a difference between 2∈ℕ, 2∈ℝ, 2∈ℂ and so on; and a meaningful question isn't whether 2∈ℕ "exists" but whether ℕ is a good enough approximation of some parts of reality. Likewise, your question whether infinitesimals exist should be rephrased as whether ℚ (or ℝ or anything similar) is a good enough approximation of some parts of reality. Assuming the hypothesis of quantization this is not the case for anything made of mass/energy, but I see no reason why the same should be said of anything that is not, like time or space.

Your argument is, at best, incomplete.

Infinitesimals in the old sense are no longer with us. For a century and a half.

But they can be redefined as the hyperreal numbers

Or the surreal numbers, or the dual numbers, or about a billion other systems of infinitesimals that mathematicians use.

  1. Infinitesimal quantities can’t exist;

Taboo exist; we can certainly do math on them and get useful results, so ???

If you modify your claim to "infinitesimals can never actually be constructed/written down in reality", I think you'll get a lot further. But of course that's much less shocking.

I agree with premise (1), that there is no reason to think of infinitesimal quantities as actually part of the universe. I don't agree with premise (2), that actual infinities imply actual infinitesimals. If you could convince me of (2), I would probably reject (1) rather than accept (3). Since an argument for (2) would be a good argument against (1), given that our universe does seem to have actual infinities.

the points on a line are of infinitesimal dimension ... yet compose lines finite in extent.

No. Points have zero dimension. "Infinitesimal" is something else. There are no infinitesimal numbers on the real line (or in the complex plane, for that matter), and no subinterval of the real line has infinitesimal length, so we would have to extend the number system if we wanted to think of infinitesimals as numbers.

When I raise the same argument about an infinite set, you can't reply that you can always make the set bigger; if I say add an element, you reply that the sets are still the same size (cardinality).

But there is a way to use an infinite set to construct a larger infinite set: - the power set. I don't understand the rest of this paragraph.

Consider again the points on a 3-inch line segement. If there are infinitely many, then each must be infinitesimal.

Again, single points have zero length, not infinitesimal length. Note, though, that there are ways to partition a finite line segment into infinitely many finite line segmets, including the partition that Zeno proposed: 1/2 + 1/4 + 1/8 + ... In an integration, we (conceptually) break up the domain into infinitely many infinitesimally wide intervals, but this is just an intuition. None of the formal definitions of integrals I've seen actually say anything about an infinitesimally wide interval.

The series comes infinitesimally close to the limit, and in this context, we treat the infinitesimal as if it were zero.

Actually, we don't have to treat an infinitesimal as zero, we just have to treat zero as zero. If I move along a meter stick at one meter per second, then according to Zeno's construction, I traverse half the distance in 1/2 second, 3/4 of the distance in 3/4 of a second, and so on. As you say, after one second, I have traversed every point on the meter stick except the very last point, because the union of the closed intervals [0,1/2], [1/2,3/4], [3/4,7/8], ... is the half-open interval [0,1). So how much longer does it take me to traverse that last point? Zero seconds, because a single point has zero length. There is no contradiction, and no need to use infinitesimals.

There are no intinifesimals in Zeno's paradox. Each step has a strictly finite length.

There are also no infinitesimals on the line segment. Each point has a size of exactly 0.

Of course, you cannot calculate the length of the segment based on this as L = point size number of points = 0 infinity, because infinity is not a number. You can't pass to the limit of infinity before doing the multiplication unless you know that all multipliers are finite, which is not the case here. What you can do here is calculate N * (L/N) and pass to the limit N -> infinity, in which case both your multipliers are always finite.

Always do the calculation first and then pass to the limit (unless you know that in your specific case it doesn't matter--there are theorems for that, of course).

OP, I'm sorry you got slammed with downvotes. I hope you stick around. (I didn't actually try to follow the argument so this isn't a comment on the discussion itself, I just want to extend condolences because that many downvotes are never fun)