Tag: Infinity

The Better Solution to Zeno’s Paradox of Motion

This is about Zeno’s “dichotomy” paradox of motion.

As described by Wikipedia:

“Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on. […] This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.”

Since I’m not necessarily a mathematical Platonist, I won’t maintain that there “is” an infinite number of points to a number line; they don’t exist until you construct them individually (and then they only exists as abstractions/ideas). The paradox presumes distance should be idealized as a number line in which an infinitude of infinitessimally small distance-intervals exists already. Even if mathematics were Platonic, it would still be questionable whether an ideal number line characterizes real, physical distances.

Now, the standard calculus answer does work—that as time intervals and distance intervals diminish, they both approach the same limit (being zero) simultaneously. But that answer deals with infinities, as well as relies on mathematical equations, which are not process-oriented, and I believe that infinities are unnecessary—since distance intervals don’t exist until you create them—and that, as such, the problem could best be understood as a process—particularly a thought process, given that the problem is a thought experiment. (Yes, motion does actually happen in real life, but there’s obviously no paradox in motion intrinsically, or it wouldn’t happen; thus to contemplate the paradox, per se, is to contemplate the thought experiment as a thought process.)

So here is my deconstruction of the paradox as a thought process.

Here’s what we do: we take one step in this thought process—we imagine the thing to move half-way (say, 4m). Then we take another step in the thought process—we imagine the thing to move half-way again (2m). Then we imagine it moving again (1m), again (.5m), and so on and so on. The problem here is that, while the distance traversed diminishes for every successive iteration of the thought-loop, the time it takes to think it through, doesn’t. Thus, it appears that, while time (being easily conflated with thinking-time) carries on indefinitely, the distances traveled get shorter and shorter incommensurately with the time intervals associated with them. If one could reduce one’s thought-process time by half for each successive iteration as the distance intervals diminish by half, then perhaps one would see things differently.

The proof that this mental effect is what’s responsible for the appearance of a paradox would be that we wouldn’t formulate the argument as so: the object must travel for half the time it takes to get there, then half the remaining time, then half the remaining time again, etc., and therefore it can never get there since it takes an infinite number of steps. That formulation is symmetrical to the distance-based one and carries the same implication, but somehow it just wouldn’t work—­it doesn’t have the same poignancy as the distance-based one.

One other point that comes to mind is that the idea alone that infinitessimal distance intervals don’t exist until you imagine them resolves Zeno’s paradox because there is no infinitude of steps; there is only the circular thought process, which carries on for exactly as long as one carries it on. In the same vein, contrary to popular belief, the value pi is not “infinite”…it’s a prescription for constructing a procession of digits of indefinite length. (Actually, it’s just a value. What was described is the algorithm for expressing that value in decimal form.) The actual value of pi can actually be expressed more succinctly, and precisely, as follows:

(Maybe there’s a more concise way to express that in summation notation, IDK.) Admittedly, this is actually another prescription for iterating in a loop; it just isn’t about creating digits, per se. The more iterations you step through, the closer your approximation is to the limit л. To use this as a prescription for generating accurate digits of pi, you would have to (a) convert your value to decimal (or another base, if you preferred), and (b) know how many digits are accurate at a given iteration and discard the rest. In any case, the above

formula itself is not an approximation, and taken as a whole, it is not infinite; that is, it does not contain an infinite amount of information, or it couldn’t be shown on your screen.

Why the Physical Universe Can’t Be Infinite

I just realized I’ve said more on the subject already here, but in this short essay I’ll take a different (and more trivial) approach.

Astrophyicists seem to say that it’s unknown whether the universe is finite or infinite, such as in this answer: https://www.quora.com/Infinity/Is-the-universe-finite-or-infinite-1/answer/Frank-Heile. But if infinity is not a number and not the same type of value as any finite amount, and no amount of counting or addition can get from a finite amount to infinity, then isn’t it impossible for the universe to be infinite? If the universe is infinite, then infinity is the quantity of mass-energy or information in the universe, yet we are able to measure and observe finite parts of the physical universe, and these two types of quantities are incompatible—they can’t exist on the same scalarity.

In order for the universe to be infinite, it needs to be at least theoretically possible to observe an infinite amount of it, because by the very epistemic nature of existence it makes no sense to say that something exists if we can’t even in principle be affected by it (I explain why this is the case in my essay linked to above). But, since infinity and the finite cannot exist on the same scalarity, it’s impossible to start at a viewpoint of some finite part of the world and keep expanding its breadth until finally you reach a viewpoint where you can observe an infinite slice of the universe.

Also—and I think this is actually closely related to the above points—the axiom of choice has to be metaphysically assumed in order to be able to find yourself observing any particular limited part of the universe if the universe is infinite, and the axiom of choice is problematic in constructivist mathematics, and I’ve shown that mathematical Platonism, which is basically everything that’s not constructivism, is silly here.

However, if there is a multiverse, the universes contained therein don’t necessarily have a spatial (or temporal?) relationship to each other, so its infinitude wouldn’t have to exist on the same scale as our finitude, so it’s entirely possible that the multiverse is infinite. (Except that I contradict myself because if we can’t ever observe an infinite multiverse infinitely then it can’t exist by my definition of “exists.” Oh well.)

Similarly if there is a spiritual reality beyond the physical universe, it could be infinite because measurement on such a plane is subjective and malleable by mind anyway, so the concept of consistency across all scales of measurement breaks down.

Or something. Idk. Nevermind.

Is the Universe Infinite?

As with a lot of simple yet deep philosophical questions and statements, the question is basically nonsensical but appears to make sense because of our tendency to be duped by language. To a certain extent, we tend to think that grammatically correct sentences must makes sense. And that’s to say nothing of the semantic problems raised by the verb “to be” (see Alfred Korzybski).

The universe is neither finite nor infinite.

What does the term “infinite” mean exactly? Basically, it’s a mathematical term that means that a value is so large that any finite value is smaller than it. Of course, the problem with this definition is that to have an actual value it must be finite—otherwise you have a formula for creating values.

For example, any actual whole number must be finite, but the number of whole numbers that “exist” is said to be infinite. Of course, you can’t possibly ever represent, count or observe every possible real number. Not even if you had all the time in the universe. Not even if you had an “infinite” amount of time, whatever that might mean. Because no matter how many numbers you’ve counted, you can always count more, by definition.

So  the set of all whole numbers is “infinitely” large simply because you can execute an algorithm (however you want—by hand, on a computer, in your mind, whatever) to generate more successive (or non-successive, if you prefer) whole numbers for as long as you want. The algorithm itself does not contain all the whole numbers and is not infinite in content, so how can you execute it for as long as you want? The answer is that the algorithm essentially runs in a loop.

In the case of generating successive whole numbers, the algorithm could look something like this:

  1. Start with some number. If you think about it, this number is actually nothing other than a sequence of digits
  2. Copy the contents of the current number to the next number
  3. Start working on the last digit of this number
  4. If the current digit we’re working on is 0, change it to 1. If it’s 1, change it to 2, etc.
  5. If it’s 9, change it to 0, change the working digit to the one before the current one, and go to step 4. If you can’t do this because we’re on the first digit, then prepend a 1 to the entire sequence and change every subsequent element to 0
  6. Go to step 2

..Or something like that. Whatever. The point is that all infinite sets or infinite values (such as the size of an infinite set) actually boil down to algorithms for generating those things that run in loops. If you’re wondering about infinite sets other than the number line, in general any infinite set is ultimately a prescription for finding new elements that belong in the set indefinitely, or at least until you stop.

So when we ask, “Is the universe infinite?”, we’re basically asking if the universe can be generated by a mathematical algorithm in a loop. And even if it could be—which it obviously can’t, because that would create a universe so regular and ordered that it would be uninteresting, not this one—that would only make the universe as big as the time God or whoever spent executing that algorithm. And that’s to say nothing of the fact that mathematical algorithms deal with numbers only, and numbers are purely quantitative and abstract and can’t possibly generate quality or substance. (That’s why the universe can’t fundamentally be made up of math, but I digress.)

I said earlier that the universe neither finite nor infinite. So why is it not finite? Because it’s unlimited. Just like the infinite contents of an infinite set don’t actually exist anywhere, because you can’t define infinite existence except as a looping algorithm or some kind of paradox, the universe doesn’t exist in an “infinite” sense. But neither does it have any boundaries to its existence. The more you look, the more you find, forever.

How can this be true and the universe not be infinite? The answer is that existence itself is relative. If you think about it, in order for something to be said to “exist,” it must be able to affect you in some way. If it can’t affect you, then you have no way of knowing it’s “there” and therefore you can’t rightly posit that it exists.

The concept of “existence” is a tricky one. Emmanuel Kant said, in response to the ontological proof of God’s existence, said that “existence is not a predicate.” While his reasoning surrounding this statement was valid, the statement alone isn’t exactly true. Existence is a predicate, it’s just not a normal one. If existence weren’t a predicate, why would we say that a unicorn—or anything else—is either “existent” or “non-existent”? That’s exactly how predicates work.

You could say that the unicorn that’s non-existent can’t have any predicates because it doesn’t even exist, but if you think about it, all objects we can possibly think or talk about are mental objects; they exist primarily in the mind. They may or may not “point” to objects outside of us.

How do we know if a mental object points to something outside of us? Presumably, we can’t directly know of anything that exists outside of our minds. We only infer as a result of sensation. So how do we know the chair exists even while we’re not sensing it? If we expect that, when we will our muscles to contract in certain ways we call “walking into the dining room,” we will see a chair with specific properties there, then we say that that chair “exists” and that our concept of the chair therefore points to something outside of us. But insofar as we can think of or talk about the chair, it exists in our minds.

We don’t even know if reality outside of our minds (if there is such a thing) is made of objects, or if it’s just some continuous field that wouldn’t even look like objects if we could have a “view from nowhere” (or, to be more epistemologically coherent, at least a “more objective” viewpoint). Indeed, “the chair” is just an arbitrary collection of atoms that we separate as “a chair.”

Let’s say the chair is made of wood and, due to attrition, some wood particles on the bottom of the chair’s legs get scraped onto the floor. Exactly which particles belong to the chair, and which belong to the floor? Where does the chair end and the floor begin? What if a child marked the chair 3 years ago with a magic marker? Are those ink particles now part of the chair, or not? If you break apart the chair with a hammer piece by piece, or burn it to the ground, at what point during the process does it cease to be a chair? Etc.

Since any two things we can possibly compare and contrast to each other (presumably using thought) must necessarily be ideas, the schism between the ideational (that in our minds) non-ideational (that outside of us) must necessarily be the biggest possible schism we can imagine—or, arguably, bigger than any schism we can possibly imagine.

So, back to the existence of the chair. To say that it exists is necessarily merely to say that we expect to perceive particular sensations in response to willing (what we think are) our muscles to do certain things. (We don’t know for certain that we have muscles, but we know for certain what we’re willing since that’s a part of our mind and therefore is directly known.)

If you posit something extant that can’t possibly affect us, any possible description of that thing is equally valid, since none of it is provable/demonstrable or falsifiable.

So, to validly posit that something “exists” must imply positing that it can potentially affect us in some way. If we will X, we expect to sense Y, hence Z exists. E.g., if we will walking to the dining room, we expect to have the visual sensation of a brown geometric form whose shape is determined by our perspective, hence the wooden chair exists. Of course, there are a million other ways we could less directly test its existence, and we can guess they’d all work because reality seems to be self-consistent, but that’s beside the point.

The reason existence is relative is that not every object that exists in the multiverse, according to some kind of fully objective view from nowhere, is potentially available to us at any given time. Most of it isn’t most of the time. Most of it will never be. But anything is experientially available to some entity at any given time (probably some entity you don’t have access to on a certain level), and on the most ultimate level, all entities are one, so the fact that it’s available to them and not to you is a relative fact.

From the perspective of this view from nowhere, every possible experience exists. I said/implied earlier that there’s no such coherent thing as a view from nowhere, which is exactly why we can’t say, based on this view that every possible experience exists “somewhere,” that the universe is infinite. The best we can say is that it’s unlimited or unbounded because your viewpoint constantly changes and therefore the breadth of objects that become extant to you constantly changes. (TBH, in actuality I contradict myself by saying we can’t have a view from nowhere and then saying that in view from nowhere every possible experience exists. But oh well—”I am large, I contain multitudes.” :P)

I don’t know whether the separation between what’s existent to us and what’s not is discretized/bounded according to finite universes within an unlimited multiverse, in which everything in our particular universe is existent to us at once but in the big picture we have access to more than just this universe, or if it’s more of a continuum. Maybe what’s existent to us is everything in our past light cone.

By the way, in case it seems odd that I would appeal to epistemology or the subjective in defining the meaning of “exists,” I will justify it here. First, trying to define “exists” purely ontologically/objectively is problematic and fraught with paradox. Explaining it epistemologically is much more cogent and tidy. Second, our epistemology comes before our knowledge of external reality both in chronology and in logical primacy, so it’s the natural place to ground such a deep concept as “existence.”

Third, to state that something exists is necessarily to talk about a concept, because what we’re talking about isn’t a thing in reality in order for us to be talking about reality when we talk about it. And it can’t be that things that don’t exist are fundamentally concepts while things that do exist aren’t fundamentally concepts, because then you could never decide whether a thing exists or not. Its existence status couldn’t be subject to change. Therefore it has to be that both things we talk about that exist and things we talk about that don’t exist are fundamentally concepts. So it makes sense that “existence” is a conceptual quality/category we ascribe to those concepts.

QED.