What’s math but a second-hand emotion?

– Tina Turner, but not quite

Over on Twitter, Ruth Malan said:

She was promptly told to stop the pseudo-mathematics.

Let’s look at this in detail:

From just looking at the tweet, we can’t tell whether this claim has any merit to it — no matter whether it’s expressed as a mathematical equation or as an English sentence. If we wanted to investigate, we’d have to go to the source (directly or indirectly) and learn about the context in which Kurt Lewin made tis claim. Because, y’know, *it depends*…it always does and we probably want to know *what it depends upon*.

If we can say *“Behavior is a function of a person interacting with the environment”* perfectly well in English, what’s the benefit of expressing this as the formula *“B = f(P, E)”*?

Brevity and precision, I think. These help us see what we’re actually saying. In other words, I think it helps us see the *structure* and perhaps the *shape* or *form* of our statements. (Better words, anyone?)

At first sight, we might consider the English sentence and the equation to be equivalent. But are they?

The equation makes it clear that behaviour is considered to be a function of *only* the person and the environment — no other variables are in play. With the information we have for now, we can’t know whether this is true or not, or even whether this was what the author of that English sentence meant to say.

But we brought this into the open and can now reason about it. And argue. And agree. Or disagree.

We could have done this by analysing the English sentence, but I think the equation helps us seeing this issue in a way that the sentence does not.

We don’t know which aspects of the person or the environment are relevant here. We also don’t know what that function **is**, i.e. how exactly the interaction between the person and the environment determines the person’s behaviour. In other words, we don’t know the definition of that function.

But neither the sentence nor the equation imply that we know this. We shouldn’t be impressed or intimidated just because someone expresses a statement as a mathematical equation, and we shouldn’t jump to the conclusion that the underlying situation is fully understood. Instead, we could think about what the equation really *means* and how this meaning can improve our understanding of that situation.

For a function to be useful, we don’t necessarily have to know its definition — at least up to a point. For example, when analysing a system we might determine that some outcome (call it *z*) depends on two variables (call them *x* and *y*):

z = f(x, y)

We might be able to determine the values of z for certain values of x and y from analysing our system. We might be able to determine certain relations between the outcomes and the input variables: for example, we might be able to determine that z goes down when x goes up and when y goes down.

Or better still, we might postulate this from our knowledge about this system and seek to validate this empirically.

Still, we might not know the exact definition of that function…but expressing this relationship as a function can further our understanding and our conversations.

What’s more, we don’t always have to know a function’s definition in order to work with it.

Given

f(x) = g(h(x))

and assuming a couple of math-y conditions, we know (if I remember correctly) that f’s first derivation is

f'(x) = g'(h(x)) * h'(x)

even without knowing the definition of g and h.

Now, this might not be a perfect example (after all, we know enough about the definition of f), but it seems to illustrate my point.

Coming back to Lewin, we could ask deeper questions about that statement. And I think I will and read up on this. Consider the equation once more:

B = f(P, E)

Form. Shape. Structure. A person’s behaviour is a function of that person’s interaction with the environment. If that’s true of *a* person — say me — it’s true for all other person’s — say you, too.

So my behaviour is a function of my interaction with the environment, where part of that environment is your behaviour, which in turn is a function of the interaction between you and the environment.

Things are getting complicated quickly, and a mathematician might be able to help us put that into an equation…which in turn might help us to better see the structure of these relationships.

Mathematicians can help us express better what we’re trying to say and help us understand better what we’re actually saying. And they might help us reason about our statements — and separate what we can safely conclude from our knowledge from what we cannot safely conclude. In other words, they can help us put more rigour in our statements and our thinking.

Now that would be helpful and, I’m sure, much appreciated.