What’s wrong with the modern engineering view that dimensions can be multiplied and divided?

In modern engineering, dimensions are multiplied and divided to verify that equations are dimensionally homogeneous.

However, dimensions cannot be multiplied and divided, for the same reason that pigs cannot be multiplied by trees, and elephants cannot be divided by airplanes. We can write (pigs x trees) and (elephants/airplanes), but these expressions are obviously nonsense—the multiplication and division they call for cannot in fact be accomplished.

If elephants could be divided by airplanes, it would be possible to answer the question:

How many airplanes are in an elephant?

And if miles could be divided by hours, it would be possible to answer the question:

How many hours are in a mile?

These questions make no sense because airplanes and elephants and hours and miles are things, and mathematical operations cannot be performed on things. Mathematical operations can be performed only on numbers. Since dimensions are not numbers, mathematical operations cannot be performed on them.

In summary, the modern engineering view that dimensions can be multiplied and divided is irrational because dimensions cannot be multiplied and divided.

(It should be noted that Fourier was the first to express the view that dimensions could rationally be multiplied and divided. For thousands of years prior to Fourier, the scientific view was that dimensions could not rationally be multiplied and divided.)