# Is infinite divisibility of Something the same concept as Nothing?

There must be some kind of proof for that.

I have always be intrigued by the notion that if something is endlessly divisible then that would mean that it is nothing indeed. (An example? Matter which turns out to show a discrete nature -until now- and current theories show to prove that space and time are discrete too).

I guess we can see the same idea lurking around when thinking about an algebraic system or some knowledge system which must state its axioms first in order to build it. And here enter Gödel proving that in a mathematical system like this it is impossible to prove all truth within. (Yes, the best example would be 'Geometry' with its axioms for 'point' and 'line' which cannot be defined from simpler concepts).

So, in summary:

1. Is it true that endless divisibility of something proves that is nothing? [This is general question with divisibility meaning a kind of 'break' operation]

2. Assuming that there exist these basic axioms or atoms of something: Are we destined to live with the fact that those axioms are impenetrable knowledge?

• "Is it true" questions get tricky in the presence of zeros and infinities. In particular, the interpretation of the phrase that you intend begins to matter greatly. Do you intend that question to be interpreted using the language of modern mathematics, which has strict handling of the laws of arithmetic, zero, and limits, or did you have another interpretation to be used? Also, you use the term "impenetrable logic." Are you referring to concepts such as a formal system that can define its own semantics, or do you have another interpretation you are looking for. – Cort Ammon Oct 31 '15 at 19:26
• The main obstacle to infinite divisibility is when one does not have an adequate way to reason about infinitely divided things. Once you have that, it's no longer an issue. – user6559 Oct 31 '15 at 19:34
• I guess my question was naively conceived because I didn't intend to mean a 'division' in the mathematical sense. It was supposed to mean more of a 'break and see' operation. :-) – fante Nov 3 '15 at 12:46
• Reverse the question. If you multiply something an infinite number of times, what do you get? – user18800 Jan 9 '16 at 3:58
• @fante What does "break and see" mean? – xiota Aug 29 '19 at 2:20

The answer is: No, endless division does not lead to nothing. It leads to "not nothing." This is one answer, anyway, to this a wonderful old question, going back at least to Zeno and "resolved," for practical purposes at least, by the "limit" in Calculus.

The uneasy relations between "zero" and "infinity" led the Greeks to abhor and avoid both. They are unobservables that simply bring in endless problems of incoherence and the ever-lurking "infinite regress," the quicksand of dialectic. By contrast, the atom of Democritus was, unlike ours, definitively "something" bordered by definitive "void."

What you get at the "very bottom" of infinite divisibility became both a mathematical and theological debate when Newton and Leibniz formalized the "infinitesimal" in calculus. Was the least possible "something" the same as "nothing"? All of physics depends on the answer being... no. It is a "not nothing," which Berkeley memorably characterized as "the ghosts of departed quantities."

In physics, Wheeler addressed the problem by coining the phrase "it from bit." This means that "information" is the fundamental unit, and is minimally defined as a "bit" or a "something/nothing" an "either/or" a "(0/1)" And it is precisely because of this irreducible instability or "possibility" that we have "something" and motion, rather than just "nothing."

This is, to the horror of hard-headed physicists, not unlike what Hegel suggests at the beginning of his Logic by deriving "becoming" from the irreducible overlap and interplay of the meanings of "being" and "nothing." To say something "is" is to say "nothing" about it. Thus, the idea of "being" itself gives rise to the idea of "nothing." Which now itself has a "being." And because these bare ideas collapse into one another they give rise to "becoming"... or transformation, motion, change.

So, the problem is not that such knowledge is "impenetrable," but that it is all too penetrable. One arrives by analytic divisibility not at the "atom" or at "nothing," but at an irreducible instability. The answer is that infinite divisibility leads to something that is "not nothing" and is also the generative power of "nothingness" or "negation." Which Sartre, incidentally, equates with us. For after all, there is always "something else" which is doing this endless dividing.

• "What you get at the "very bottom" of infinite divisibility became both a mathematical and theological debate when Newton and Leibniz formalized the "infinitesimal" in calculus." Yes and no. We can endlessly divide a rational or real number without ever getting to the question of infinitesimals. An infinitesimal has a much stronger property than mere divisibility; that of being less than 1/n for any positive integer n. – user4894 Nov 4 '15 at 22:26
• @ Nelson Alexander, Thanx for your encyclopedic answer :-). On the hand, don't you think that the notion of 'infinitesimal' is a clear statement that 'in the end' there are just 'discrete' entities 'making sense of all'. I mean 'calculus' gets wrong the moment you make 'dx' (from dy/dx) equal zero instead of the usual 'finite but near zero' concept. – fante Nov 6 '15 at 12:42

There are several positions on this:

1. Atomism doesn't require infinite divisibility; it ends in something larger than nothing - the atom.

2. Analytically, the real number line, concieved as a set of points has infinite divisibility and ends in a point.

3. Synthetically, it could be argued that the real line is not made of points, but of lines of various sizes - ie topology - and potential infinite divisibility is still possible - but one never ends with a point; for no points were granted - but perhaps the infinitesimal line.

It matters how we concieve of something: the symbol '1' concieved purely as an integer cannot be divided; but as a real number, can be; it can be argued here that the same symbol is actually referring to different concepts - though we tend to tie them together, as there is a certain relationship between the concepts.

• Sorry to comment on somethin so old, but option 2 is not really true. Analytically, if by that you mean in terms of Real and Complex Analysis, infinite divisibility involves iteration, so it is still countable, and that can't get you down to a point. – user9166 Sep 30 '19 at 21:02
• @jobermark: I'm talking about analytic in opposition to synthetic and in this sense its about understanding the real line as being made up of points. The point you're making, which relies on the notion of analyticity in complex analysis (holomorphicity or power series) is not really relevant here. – Mozibur Ullah Oct 2 '19 at 5:44

I don't know of any such proof, nor should we look for one. The history of philosophy is full of bad "theories" that arose because philosophers tried to reason their way to reality rather than making observations. We should let science answer this question.

However, we can look to mathematics for at least some insight. When we do, we see support for both views.

On the one hand, the opposite seems true. Infinite divisibility is the property of something, not nothing. For instance, 1 is not 0, yet 1 is infinitely divisible. Further, nothing is not divisible at all. One of 0's key properties is ∀x <> 0 : 0/x = 0.

On the other hand, one foundation for numbers has them built on nothing -- the empty set. For instance, 0 is the empty set, 1 is the set whose only member is the empty set, and so on. So here, at the most basic level, something is made out of nothing.

• I understand that anything divided by zero is "infinity" and that zero divided by anything is zero. I simply don't know enough to know how "divided" is defined. But isn't here a sense in which zero can be divided into 1 and -1? In that sense, "nothing" exists as "something" and its "removal" or subtraction. I believe there is something like this in Kant's demonstration of space as the intuition of "positionally" without content. – Nelson Alexander Oct 31 '15 at 18:32
• @NelsonAlexander good point; division isn't well defined. If it means division in the mathematical sense, then no. If it means division in the sense of partitioning, then sure. Whether -1 would have any mapping in "reality" is another interesting discussion :) – R. Barzell Nov 2 '15 at 15:35

1) Is the flipside of Zeno's paradox. Time is the primary thing we imagine would be infinitely divisible. But by Zeno's logic, if time really is infinitely divisible, motion is impossible. He decided motion was the illusion, you would choose instead to see infinite divisibility of a real thing as the illusion.

The problem here is, how would we imagine granular time? What would keep us from splitting it? This is the last holdout of Kant's Antinomy of Atoms. If time is not infinitely divisible, we have no natural way of relating to the minimal quantum of time. Yet, if it is infinitely divisible, other problems ensue. The strength of our intuition prevents us from truly accepting either alternative.

I agree (with Kant) that this is a logically unsolvable problem. We just assume our way around it, because we like to model space and time by the real numbers, to which we have built in the notion of either infinitesimals or limit points, and the math serves us well.

2) Seems to be contraindicated by our recent experience with atomic particles and quantum mechanics. The things we have found to be indivisible all generate fields, which we can study from the outside, even though we cannot go further within the quanta of matter. They also change places with one another according to intricate sets of rules.

We have learned a lot about the nature of the smallest particles of matter from astrophysics and from huge particle-colliders, and we keep learning more. If there are atoms of space or time, there will likely be impressive macroscopic effects that will be the key to making sense of them, and we will not need or want to look inside in order to feel like we understand them. (Popular candidates for such forces include dark matter and the universal expansion constant, which are both confusing and entertaining astrophysicists immensely to date.)

In the real numbers, every element is 'endlessly divisible' by 2. For example 10: 10, 5, 2.5, 1.25, 0.675, ... That doesn't mean that 10 is actually nothing, or that 10 cows is the same as 0 cows.

To be honest, I don't get the analogy with geometry. A point or a line can be defined from similar concepts. A point in n dimensions is an n-tuple (n1, n2, ..., nn). A line in n dimensions is a function from N to the set of n-1-tuples. A field in n dimensions is a function from N2 to the set of n-2-tuples, etc.

• I guess my question was not intended to be about a mathematical operation (division) but about 'partition' as R. Barzell suggest. The example about 'plane geometry' maybe wasn't a good one. But consider the struggle for Euclid trying to define those concepts who left them vaguely defined. – fante Nov 3 '15 at 12:25
• @fante the set of natural numbers 0,1,2,3,... is endlessly partitionable, because there are infinitely many singletons {0},{1},{2},... that could all be partitions. That doesn't mean the set of natural numbers is empty. – user2953 Nov 3 '15 at 12:28
• You are right, forgive my naive mathematical expressions. :-) – fante Nov 3 '15 at 12:40