I understand that "structure" of data is totally dependent on Boolean Algebra, but:


  Why is data considered to be a discrete mathematical entity rather than a continuous one? 


Related to this:


  What are the drawbacks, or invariants, that are violated in structuring data as a continuous entity in $r$ dimensions? 


I am not an expert in the field as I am an undergrad math student, so I'd really appreciate it if someone would explain this to me like I'm five.

Answer


  why was the data considered to be a discrete mathematical entity rather than a continuous one


This was not a choice; it is theoretically and practically impossible to represent continuous, concrete values in a digital computer, or actually in any kind of calculation.

Note that "discrete" does not mean "integer" or something like that. "discrete" is the opposite of "continuous". This means, to have a computer that is truly able to store non-discrete things, you would need to be able to store two numbers a and b where abs(a-b) < ε for any arbitrarily small value of ε. Sure, you can go as deep as you want (by using more and more storage space), but every (physical) computer always has an upper bound. No matter what you do, you can never make a (physical) computer that stores arbitrarily finely resolved numbers.

Even if you are able to represent numbers by mathematical constructs (for example π), this does not change anything. If you store a graph or whatever that represents a mathematical formula, then this is just as discrete as anything else. 

Addendum

The rest is just a little perspective beyond the field of computer science. As the comments have shown, the physical topic is not undisputed, and as you can see I have formulated my next paragraph in a way that is rather uncommitted to whether it's true or not. Take it more as a motivation that the concept of "continuum" is not a trivial one. The answer given above does not depend on whether space is discrete or not.

Note that all of this is not so much a problem of computers, but a problem with the meaning of "continuous". For example, not everyone even agrees, or did agree in the past, that the Universe is continuous (e.g., Does the Planck scale imply that spacetime is discrete?). For some things (e.g., energy states of electrons and many other features in Quantum (sic) Mechanics) we even know that the Universe is not continuous; for others (e.g., position...) the jury is still out (at least regarding the interpretation of research results...). (Notwithstanding the problem that even if it is continuous, we could not measure to arbitrary precision => Heisenberg etc.).

In mathematics, studying the continuum (i.e., the reals) opens up a lot of fascinating aspects, like measure theory, which makes it utterly impossible to actually store a "continuous" kind of number/data.
