# What's the difference between infinite divisibility and infinte extension?

I understand that most people would say that time, such as the time it takes to write this question, is infinitely divisible. But I'm guessing it's unusual to suppose that makes time infinitely extendable, at least as it is happening.

What's the difference between infinite divisibility and infinite extension?

• A geometrical straight line has infinite extension (lenght) and is infinitely divisible. A line segment is only infinitely divisible. – Mauro ALLEGRANZA Dec 20 '18 at 7:44
• I think infinite extension is spatial and refers to the universe. I suppose the reason time is infinitely divisible could be because the universe is eternal. – Bread Dec 20 '18 at 7:45
• Sounds to me like the two terms refers to the notion that something can become infinitely small(er), e.g. by shrinking or subtracting from it, or infinitely large(r), e.g. by expanding or adding to it. – user34765 Dec 20 '18 at 9:54
• Same as the difference between ordinary divisibility and extension: I can divide a segment in two, or I can extend it to twice its size. Infinity does not add anything new. – Conifold Dec 20 '18 at 20:30
• yeah @MauroALLEGRANZA kinda silly of me sorry are there genuine paradoxes about infinitesimals? can times arrive if they are infinitely small? – user35983 Dec 20 '18 at 23:17

My first thought is that you raise a good point but the issue is tricky. Infinite divisibility cannot apply to a line segment since a segment is divided from the whole so is 'finitely divisible' (or may be divided). For me 'infinitely divisible' means 'not divisible'. No knife would be thin enough to cut a segment from an infinitely divisible continuum.

This is a conceptual problem. A continuum has no parts so cannot be extended. Thus Hermann Weyl proposes that the infinitely divisible number line of mathematics is a fiction. Others argue from the necessarily fictional character of this conception of the number line to the unreality of space-time. Weyl notes that time and space are only ever experienced as 'here' and 'now', such that extension is not an empirical phenomenon.

I'd suggest that you're half-right and that infinite divisibility implies either infinite extension or no extension. Or, to put it another way, infinite divisibility makes no sense when applied to a supposedly real extended object, just as extension itself makes no sense on close analysis.

There are some difficult issues here that take us back to Zeno and Dedekind and there will be a wide range of opinions about all this. I feel that extension is a much understudied phenomena in scholastic philosophy and is an issue more or less swept under the carpet.

• A line segment is infinitely divisible. If you take any two points, you can find a third point that's between them. Obviously, we don't have infinite divisibility of anything physical (we can't divide a quark or lepton, as far as we can tell), but spacetime might be infinitely divisible (it is as far as we can tell). You might call infinite divisibility a fiction, but only in the sense that all of mathematics is fiction. – David Thornley Dec 20 '18 at 23:41
• @DavidThornley - Yes, I'd agree that infinite divisibility is a fiction, thus also an infinitely divisible number line. My point here was that a line segment cannot be defined for an infinitely divisible line since there is no way to identify the limits of the segment. This may be debatable but it seems correct to me right now. Even a notional point on the line would have to be an ill-defined region. – PeterJ Dec 21 '18 at 13:34

Here is the question:

What's the difference between infinite divisibility and infinite extension?

Since we are talking about "divisibility" we need an algebraic structure that has order and a division operation defined. One such structures is a field. Here is how Wikipedia defines a field:

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.

The rational and real fields are also totally ordered. Here is Wikipedia's definition:

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.

That is, given any two elements from one of these domains there are only two possibilities:

1. They are the same.
2. One is bigger than the other.

So take two different elements from the domain of rational or real numbers and subtract the smaller from the larger. Call this a line segment. The length of the line segment is another rational or real number. We will use this length to show infinite divisibility.

Since the domain is a field, we can divide that line segment by 2, a number in both the rationals and reals. Add that result to the smaller number to get another rational or real number which is different from the smaller one. Those two numbers are the endpoints of a smaller line segment.

We can repeat this process indefinitely. How do we know this? Assume we cannot. Then we have reached, after some finite process, a line segment with two end points that are different. The assumption is we cannot take those two different numbers, subtract them and divide the result by 2, but that would violate the fact that they are members of an algebraic field in which we can always subtract them and given any member of the domain divide it by 2. Given that the assumption that we cannot indefinitely divide the line segment leads to a contradiction, we assert that we actually can indefinitely divide the line segment.

Define "indefinitely" as "infinitely" and one has infinite divisibility.

Infinite extension doesn't require an algebraic field. We don't need divisibility although we do need addition. Given a member of a domain we should always be able to add 1 to that member indefinitely. The algebraic structures with such properties are called ordered rings. Here is Wikipedia's definition of it:

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.

Take a number from the ordered ring and add 1 to it. This gives us a new number. One can indefinitely repeat that process as well.

That covers the description of infinite divisibility and infinite extension for algebraic structures. The OP has a more basic question about time:

I understand that most people would say that time, such as the time it takes to write this question, is infinitely divisible. But I'm guessing it's unusual to suppose that makes time infinitely extendable, at least as it is happening.

There are different ways to look at time. See Bradley Dowden's survey of the subject. The basic question is whether time can be modeled using fields and rings such as the rational or real numbers. If it cannot then the definitions of infinite divisibility and infinite extension given by those algebraic structures would not be relevant to time. If it can, then the fact that an algebraic field is also a ring would imply that if time is infinitely divisible, then it is also infinitely extendable.

Bradley Dowden, "Time" Internet Encyclopedia of Philosophy https://www.iep.utm.edu/time/#H7

"Field (mathematics)" Wikipedia https://en.wikipedia.org/wiki/Field_(mathematics)

"Ring (mathematics)" Wikipedia https://en.wikipedia.org/wiki/Ring_(mathematics)

"Total order" Wikipedia https://en.wikipedia.org/wiki/Total_order

• +0.7 I think the answer goes in the right direction, but some points should be changed: (1) You are introducing division in a field, but then interpret dividing as the arithmetic mean, which actually does not make sense in a general field. (2) The introduction of line segments is not necessary, and neither is the order. Just take an element in the field and divide, and divide, and divide... (3) Defining "infinitely" as "indefinitely" is a bad move. There needs to be a distinction between the process and result. Dividing can be repeated as often as you want, but a limit does not need to exist. – Jishin Noben Jan 19 at 15:33
• (4) One should mention that nothing of this has to do with there being infinitely many elements. (5) The physical "object" time could very well be considered to be an algebraic field. Time is melted with space into spacetime. Clocks measure the minkowski distance, which lies in the field (6) The characterisation of spacetime is a local one! The spacetime manifold might be closed, and there is no addition and "infinite extendibility". This manifold might be bounded, so that a priori the values of the minkowski distance between events might be bounded. – Jishin Noben Jan 19 at 15:33

Infinite extension and infinite division both implicate infinity in their notion but in opposite ways.

Whereas the first expands, the latter contracts.

For example, take the unit interval; and then double it, and then double it again; and so on ... infinitely(!) this is infinite extension.

It's opposite is simply halving the interval, and then halving it again; and so on ... to infinity; and this is infinite division.

Phrased like this, we can see immediately that they are opposites.

Notably Aristotle denied the physical actualisation of both. Mathematically, since we deal with ideal quantities, we can do both; and both infinite division and infinite extension is familiar there as the construction of the real number line.