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I am wondering about a system with division defined for non-numbers. From what I have encountered so far, it seems division is only applied to numbers. For example:

4 * 6 = 24

You can then take the resulting 24, and get back one of the original values, given that you provide half of the puzzle.

24 / 6 = 4
24 / 4 = 6

Division is essentially taking the top number and breaking it into chunks the size of the bottom number. It then tells you how many chunks we have. But that's the thing, it tells you how many, which is a number.

I am wondering if there's any way to conceive of division without using numbers. For example:

day / night = ?
person a / person b = ?

If I am dividing a person by another person, maybe that means I'm seeing how much that persons body size fits into the other person. But then that gives me a number again!

The thing is, you can easily add and subtract non-numbers. For example:

water + coloring = colored water
plate + food = a plate of food
block a + block b = stacked blocks

But with division, not only can I not think of an example that makes sense, in the case I can clunkily divide, it gives a number like:

person a / person b = ratio of body sizes

Wondering if there is any way to define division such that it works on non-numbers. Not even sure what you would call "divide" at this point, I haven't thought of any clear metaphor or anything for it.

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    Stacked blocks - block = ? It is unknown how many blocks are stacked. At first it is better to start with multiplication, since division is merely the inverse operation. In fact I'd argue your +'es for NaN's should be turned to set unions.
    – rus9384
    Jun 6, 2018 at 23:37
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    I'm not full grasping the question about philosophy here, but divide means "split into equal parts." Surely, a cake can be divided, and while the division will have a mathematical ratio, the cake itself is no number.
    – virmaior
    Jun 6, 2018 at 23:40
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    In mathematics division is generalized far beyond numbers, but you should as about that on Math SE. You can also look at mereology, part of philosophy that studies relations between parts and wholes. It is unclear what you want beyond a loose association or what it is to be used for.
    – Conifold
    Jun 6, 2018 at 23:57
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    Haha nice, that makes sense. But that is cake /, with no denominator.
    – Lance
    Jun 7, 2018 at 6:15
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    As often in (bad) philosophy you are playing with the ambiguity of words: "division" can be read as a physical operation, in which case we use number (as usual) to count or measure the result. In mathematics division is an operation involving mathematical objects: numbers, polynomials, matrices and can be further generalized to more abstract math objects.
    – Mauro ALLEGRANZA
    Jun 7, 2018 at 6:59

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You could say for example that if you divide the world's population by the amount of food in the world then you get no hunger. But these aren't really mathematical operations without numbers.

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Yes, there are examples. Here's one: how colors work.

If you take the RED color on a piece of paper and divide it into it's components, you get the magenta and yellow colors. No numbers involved. Same goes for any color (dividing them into their original parts gets you to CMY, the colors printers use in their cartridges).

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    In general I must say one is dividing wholes to the parts. At the same time for colors that does not work, like it does not work for different quants in physics (quarks, leptons, etc.). They are not divided, they are recombined or so. Except for white/black color, maybe.
    – rus9384
    Jun 7, 2018 at 8:16
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    I'm talking about physically getting a paint (any color) an chemically treat it so you can separate it's parts. All solid colors can be reduced to only 3 colors (CMY) just like any color as a ray of light can be obtained by only using 3 colors (RGB).
    – Overmind
    Jun 7, 2018 at 8:34
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    You presuppose RGB/CMY models while these are created only for convenience. It's not hard to "divide" green beam to light blue and yellow. Using the same methods that allow to divide yellow beam to red and green.
    – rus9384
    Jun 7, 2018 at 8:38
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    +1. Nice approach.
    – Geoffrey Thomas ♦
    Jun 7, 2018 at 9:17
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    @ rus9384 there is nothing else you can extract from a 100% green beam (525nm). The light frequencies are a totally different matter compared to paints on paper.
    – Overmind
    Jun 7, 2018 at 10:15
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Most people when they think about division in relation to mathematics it's generally with numbers: 8/4 or 9/3.

Numbers have been around for a long time and everyone is familiar with them. However, modern mathematics, which on fact is not so modern - it's been going on for some time now - divides alls sorts of objects: Rings divided into rings, groups divided by groups, manifolds divided into manifolds by the actions of groups ... one doesn't have to understand the jargon to understand that mathematicians have generalised the idea of division into many different directions. Nevertheless all of them, at root, is merely dividing something that you or I may divide. For example cutting a pear into two halves. Or even taking apart a table into its four legs and table top.

This isn't like dividing a stick into two halves, where after you have divided them the two halves look like what you began with - just smaller. The two halves of a pear, do not each look like a pear; and the leg of a table is certainly not like a table.

But the modern mathematician will say, the two halves of pears have a reflective symmetry; the four legs of the table have an identity symmetry; and the table top when turned upside down looks exactly the same as it did when it was the right way up - so we've added a symmetry there that wasn't there before.

Division then, is dividing according to some kind of law; here the law is for further understanding what is under examination. That's the prerogative of mathematics when understood as a science. Put like this, we see that division has no neccessary correlation with numbers ...

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