The built-in function max in Python is broken and this article explains why, drawing parallels with other programming and mathematics concepts.

max is broken

The default values

Did you ever notice that in Python, you have:

sum([]) == 0
all([]) == True
any([]) == False
math.prod([]) == 1

Notice that the functions sum, all, any, and math.prod all have one thing in common: they take a list of things and reduce it into a single result. In all four cases above, if we pass the function an empty list we get a default value back.

However, max is also a reduce function and max doesn't have a nice default value. Instead, it throws an error:

max([])  # ValueError

But why? Isn't there a suitable default value for max?

From a mathematical point of view, there is a suitable default value for max, so the fact that the function raises an error instead of returning it is wrong and that is why I claim that max is “broken”. For max to be mathematically correct, the default value for max should be negative infinity, which you can get with float("-inf"):

max([]) == float("-inf")

So, why does Python raise an error instead of being mathematically correct? I'm guessing Python goes with the “practicality beats purity” approach and prefers to raise an error instead of returning infinity values to users, given that infinity can be quite an exotic thing... Especially if we consider that max should return negative infinity and min should return infinity, which are different things.

(Right, I forgot to tell you but min is also broken!)

But why “should” the return values be (negative) infinity? That's because all of these reduction functions (sum, any, all, math.prod, max, and min) should return the identity element for their respective operations.

• sum performs addition and 0 is the identity element for addition.
• any performs the Boolean operation “or” and False is the identity element for “or”.
• all performs the Boolean operation “and” and True is the identity element for “and”.
• math.prod performs multiplication and 1 is the identity element for multiplication.

Likewise, the identity element for the max operation is float("-inf") and the identity element for the min operation is float("inf").

How can you know that float("-inf") is the identity value for the operation max? Try to come up with a numerical value for x such that max(float("-inf"), x) is different from x. I bet you can't, and that's why float("-inf") is the identity value for the operation max; because max(float("-inf"), x) == x for any number x.

Do you get what I'm saying? Feel free to leave a comment below if you don't!

Fixing max

To fix max in a mathematical sense you'll need to set its default parameter to float("-inf"). The version below uses functools.partial to freeze the parameter default:

from functools import partial

pedantic_max = partial(max, default=float("-inf"))

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References

• “The power of reduce”, https://mathspp.com/blog/pydonts/the-power-of-reduce [last accessed 04-01-2024];
• Twitter/X post and discussion therein, https://twitter.com/mathsppblog/status/1742918859383079192 [last accessed 04-01-2024];