If we take the meaning of multiplication seriously then:
$$3 \times 4 = 3 + 3 + 3 + 3$$
That is to say, we take the multiplicand and add itself to it the same number of times as the multiplier. (Above $3$ is the multiplicand and $4$ is the multiplier). Hence,
$$4 \times 3 = 4 + 4 + 4$$
So let's say that we take $0 \times 4$, this would be equivalent to:
$0 + 0 + 0 + 0$ which is still zero. However, what would $4 \times 0$ be? That would mean that you take $4$ and add $4$ to it zero times, which would still give you $4$. It seems to me that the result of $4\times 0$ is a matter of deciding what $4$, $0$, multiplication and addition means and then being consistent.
Edit
We prove things in math not by replacing mathematical terms with their definitions and then using statement logic and seeing if the negation leads to a contradiction, it would be nice if we did do that, but no one can do that now, instead we are stuck with simply using the formulas over and over again and if they give us the results we want then we keep using them. For example, with the quadratic equation we can plug the values into the equation and see if we get the result which matches the other side of the equation. If we want to measure the volume of a complex shape we can pour liquid into a straight cylinder where we know the area and pour the liquid into the complex container to prove our results. The multiplication by zero however has very limited applicability. In fact it seems to have no applicability. So the only way to test the if 4*0=0 is to use statement logic and employ a consistent usage of the terms involved. So what I'm asking is if anyone knows of any applications of 4*0 to the real world?
It is very popular to model concepts and procedures in science and engineering using sparse matrices today. A sparse matrix is a stencil of numbers which contains almost only the value $0$ and very few (maybe just one in thousand or one in a million non-zero values). When computing things with sparse matrices very many of the operations will be additions with $0$ and multiplications with $0$. So if the computer knows it can skip doing these operations (because they make no difference) then we can do the computations much faster!