Discrete quantities and dimensions

Question

This question popped into my head yesterday when my friend told me about a game which he claimed to be "$2.5$-d". Of course, I knew it was just an expression, since it wasn't really $2.5$-d; the camera sometimes rotated around the avatar, but the movements themselves were $2$ dimensional. But this leads me to ask: can a non-integral dimension exist?

I think this is math-based (and not theoretical-physics based, say) because it's talking about quantities. We all know from elementary school that "you can't have $5.6$ apples" or "you can't go on $2.1$ trips". But if we define what these objects actually are, we can make sense of these non-integral quantities. For example, if we define an "apple" as "a juicy fruit weighing exactly one kilogram", then we can have $5.6$ apples. This may or may not tie in with intuition, which depends on the object.

A more general question is: can every object be defined in a manner such that non-discrete quantities (like $3.4$ or $\frac{12}5$) of said object are allowed?

Answer 1

There are objects called fractals, many of which are considered to have non-integer Hausdorff dimension. For example, the Sierpinski triangle has dimension approximately 1.58 according to this definition. The Menger sponge has dimension slightly over 2.72.

The 2010 US Census found an average of 2.58 persons per household. It must be hard to be the last 0.58 person. (That's an old joke.)

Probabilistically, it makes perfect sense to expect a fractional number of objects that can only exist in integer quantities. Once you actually observe the quantity, it will be an integer, but until it is observed it makes sense to use the expected value in various ways that the observed value could be used, even if the expected value is not an integer.

Answer 2

In short, no. That's the issue. If you are in the math world, I would have to say that a physical object cannot exist purely in the form of math, but rather is governed by the laws of physics, since it has a physical existence. From a mathematical point of view, the cardinality of a set is by nature an integer.

Any object that is irreducibly complex cannot be defined as fractional, since the definition of the object would cease to hold once the initial object is divided. Think about a human. A human is either dead or alive (please, no Princess Bride references ;)). Cut one in half, and it is dead.