I have a question for 0/0 being undefined and wonder if anybody has a refutation.

Your question and ideas touch on both mathematics and philosophy (also metaphysics and the foundations of math)! I try explain as much detail as possible. (Don't get bored in the middle!)

In mathematics, $\frac{0}{0} $ is considered undefined. This is not because it is simply impossible or contradictory, but rather because it does not have a single consistent result.

$\frac{0}{0}$ = 1 then:

$1 \times 0 = 0$ which holds true

$\frac{0}{0} = 5$ then:

$5 \times 0 = 0$ which holds true

Division is defined as the inverse operation of multiplication. So, you can see that it could equal any number, since multiplying any number by 0 still results in 0.

So, $\frac{0}{0}$ is left undefined in math, because it fails to have a unique solution, it leads to indeterminacy, not necessarily logical contradiction. In math, we want operations to be well-defined, meaning they produce a single, unambiguous result. Since $\frac{0}{0}$ could be anything as shown above, it violates this principle.

Now, does 0 represent nothing? Philosophically, it is debated whether 0 should represent absolute nothingness. You can view it in different perspectives. For example, zero as a symbol in mathematics, represents the absence of a quantity, not necessarily the metaphysical concept of nothing. In this sense, it is a placeholder or a marker for when a quantity is missing (e.g., no apples in a basket). However, it is not synonymous with pure non-existence in a metaphysical sense.

Another example is in the concept of set theory. In set theory, zero can be understood as the cardinality of the empty set, a set that contains no elements. Here, zero still has a precise meaning without invoking absolute nothingness, it simply represents the quantity of elements in an empty set.

So, again zero in mathematics is more of a concept that allows for operations and reasoning within certain formal systems, it is not necessarily tied to philosophical notions of non-being.

Zero is a symbol used within a mathematical framework to denote a specific concept. It does not need to correspond to metaphysical nothing any more than the number 1 corresponds to a specific object.

In formal mathematical systems, operations like division by zero are undefined not because of metaphysical concerns but because they violate mathematical consistency. This does not imply a metaphysical contradiction, just a breakdown in the structure of arithmetic as currently defined.

In mathematics, nothing may mean no quantity, while in philosophy, it may mean the absence of any being or existence. These are not the same concept, and confusing them can lead to misunderstandings.

If we think of nothing as literally having no properties, assigning a location or identity to nothing would indeed create a contradiction because nothing cannot be something.

However, when mathematicians work with zero, they are generally not thinking of nothing in an absolute metaphysical sense. Zero in mathematics is more of a concept that allows for operations and reasoning within certain formal systems it is not necessarily tied to philosophical notions of non-being.

The debate over zero and division by zero has a long history, and modern mathematics resolves it by treating $\frac{0}{0}$ as undefined.

Also, calling it "impossible" might suggest there's some inherent contradiction in the concept of dividing nothing by nothing. But the issue is not a logical impossibility in the same way that, say proving 1=2 would be.

Hope this helps!