There's a simple mathematical argument to thwart your prof. Imagine an equation of two or more variables (such as "capacity"). For example the "capacity" of a storage unit is an equation relating height, width, and depth to volume (3 independent variables).
Any such equation can only be represented by a single number ("capacity" or "productivity") if all variables are dependent (and therefore, there is only one independent variable in the equation).
So the assertion your professor is making is that the "capacity" of a team is always exactly dependent on hours worked per day, and any other proposed dimension of capacity (such as years experience, field of study, languages spoken, cases won, relationships with judges) are dependent on "hours worked per day". If he agrees any one of those variables affects capacity, but does not depend on "hours worked per day", then a single number can never reduce the dimensionality of the output (you need at minimum 2 numbers to represent two independent variables, you can never "collapse" the data).
> Imagine an equation of two or more variables (such as "capacity"). For example the "capacity" of a storage unit is an equation relating height, width, and depth to volume (3 independent variables).
> Any such equation can only be represented by a single number ("capacity" or "productivity") if all variables are dependent (and therefore, there is only one independent variable in the equation).
This doesn't seem right. You can have storage units with varied combinations of height, width, and depth, sure. But whether that matters depends on what you want to use them to store. An example of an approach that doesn't work would be storing unboxed fragile antique dollhouses. They have weird shapes, so you can't fill the floor area, and you can't stack them, so adding height to the storage unit doesn't add any capacity.
Except that of course you wouldn't just toss them into a garage and call it a day. (They'd break!) You'd keep them in boxes. Those pack and stack perfectly. Suddenly volume is what matters again, and increasing the width, length, or height of the unit by 10% will increase the amount you can store by about 10%.
This is even more obvious if you're storing water or oxygen. Fluids take the shape you give them. Your unit might have length, width, and height (though it really shouldn't... you want to store fluids in cylinders), but the only thing that matters for how much water you can put in there is volume.
The Knapsack problem is arguably a counter argument to measuring storage space by volume.
However, air freight is a much more direct one. You have 2 largely independent measurements for weight and volume with either being the limiting metric for each load.
Yes, in air freight weight and volume are independently significant.
But I'm not saying that all multidimensional data can be losslessly reduced to a one-dimensional value. That would be crazy! the point of my comment is that it isn't true that -- as the parent comment asserted -- it is impossible to usefully report multidimensional data with a one-dimensional value. To the contrary, it is quite possible that the multidimensional data adds zero value over the one-dimensional summary.
Some multidimensional data can easily be losslessly reduced to a one-dimensional value. We can easily make a much stronger claim -- all one-dimensional values are "reductions" of other, multi-dimensional characterizations of the same data. But they're not all useless! The number of dimensions you use to describe data is an editorial choice, mostly unrelated to the raw facts.
You can as long as the size of your box is significantly smaller than the size of your warehouse. This is in fact the general case.
It's even truer if you're considering larger expansions; increasing the length of your warehouse by 200% will mean you can store 200% more stuff regardless of how awkward the original fit was.
Any such equation can only be represented by a single number ("capacity" or "productivity") if all variables are dependent (and therefore, there is only one independent variable in the equation).
So the assertion your professor is making is that the "capacity" of a team is always exactly dependent on hours worked per day, and any other proposed dimension of capacity (such as years experience, field of study, languages spoken, cases won, relationships with judges) are dependent on "hours worked per day". If he agrees any one of those variables affects capacity, but does not depend on "hours worked per day", then a single number can never reduce the dimensionality of the output (you need at minimum 2 numbers to represent two independent variables, you can never "collapse" the data).