1/0 = 0

It's a value type, so you can't really have NULL. I don't see the issue with those examples; if you have no samples, there's nothing to mislabel. An average blood pressure of 0 across 0 patients is only going to harm 0 patients.

Holy moly that is not at all what that paper says! It specifically argues that certain equational properties of a given total language continue to hold in the total fragment of a given partial language. It is an embedding theorem combined with lifting properties, not a license to perform _any_ reasoning at all regarding the non-total portion of the language it considers!

> I don't see the issue with those examples

Let me elaborate then.

Average credit card balance predicts probability of default. Joe who maxed out his credit card is higher risk than Jane who pays back her entire credit card balance every month. Now we have Jack with no credit card. We predict that Jack is low risk because his average credit card balance is zero.

Second example, defibrillator that monitors blood pressure and shocks the patient when his heart stops (blood pressure drops to zero). We attach the defibrillator to a patient, turn it on, and it shocks the patient immediately because there are no blood pressure measurements yet and therefore it thinks the average is zero.

Third example, a thermostat that turns on the freezer if average temperature is above a set threshold. It never turns on because average (of zero observations) is already zero.

Of course you can hard-code handling of those special cases, but if you did not, failing would be preferable to continuing to work incorrectly.

That's interesting. In my career, I've rarely encounted division by zero. I've encountered over and underflow considerably more. Different experiences, different points of view.

Consider the following two functions, that we can define for any field F:

• Our first function f(x, y) is defined for all x in F, and all nonzero y in F. That is, the domain of this function f is (F × F\{0}). The definition of this function f is:

f(x, y) = x times the multiplicative inverse of y

(Note that in the definition of f(x,y) we could say "if y≠0", but whether we say it or not the meaning is the same, because the domain of f already requires that y≠0.)

• Our second function g(x, y) is defined for all x in F, and all y in F. That is, the domain of this function g is (F × F). To define this function g, we pick a constant C in F, say C=0 or C=1 (or any C∈F, e.g. C=17 if 17 is an element of our field). And having picked C, the definition of this function g is:

• g(x, y) = x times the multiplicative inverse of y, if y ≠ 0, and C otherwise [that is, g(x, 0) = C].

Note that this function is defined for all (x, y) in F×F, and when y≠0 it agrees with f, i.e. g(x,y) = f(x,y) when y≠0.

Would you agree that both of these are functions, on different domains?

Next, we have the notation x/y. To assign meaning to this notation (and to the word “division”), there are two conventions we could adopt:

• Convention 1: When we say "x/y", we will mean f(x,y) (as defined above) — that is, x * y^{-1}.

• Convention 2: When we say "x/y", we will mean g(x,y) (as defined above).

The point of the post is that we can well adopt Convention 2: with such a definition of "division", all the properties that were true of f (on the domain F×F\{0}) continue to be true of g, except that g is defined on a wider domain.

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Now, maybe Convention 2 offends you. Maybe you think there is something very sacred about Convention 1. In all your posts in this thread, you seem to be insisting that "division" or "/" necessarily have to mean Convention 1, and the provided justification for preferring Convention 1 seems circular to me — here are some of your relevant comments, with my comments in [square brackets]:

> Since there is no multiplicative inverse of 0, division by 0 is undefined behavior [This seems to be saying: Because of Convention 1, we cannot adopt Convention 2.]

> Since y = x/0, it follows that the product of y and 0 is equal to x, because division is the inverse of multiplication [Here, your reasoning is "because we adopt Convention 1"]

> in algebraic fields division by x is equivalent to multiplication by 1/x. This is precisely why you cannot have a field that admits division by 0: because 0 has no multiplicative inverse. [Again, you're stating that Convention 1 has to hold, by fiat.]

> If F is a field with elements x, y, then the quotient x/y is equivalent to the product x(1/y). If 0 has no multiplicative inverse, there is no division by 0. The two concepts are one and the same [This is just stating repeatedly that we have to adopt Convention 1.]

> A multiplicative inverse is a division. [This is merely insisting that Convention 1 has to be adopted, not Convention 2]

> divisor cannot exist unless it is a multiplicative inverse [Again, stating Convention 1.]

> you can't look at the field axioms, observe that 0 has no multiplicative inverse, then proceed to define a special, one-off division rule that doesn't involve multiplicative inverses for that one element. [Why not?] ...you've introduced a division rule which is just a syntactical sugar [yes the same is true of Convention 1; what's the problem?]

All these comments, which emphatically insist on Convention 1, seem to ignore the point of the article, which is that Convention 2 has no more mathematical problems than Convention 1, because the function g is no less a “valid” function than the function f.

In mathematics when we use words like “obvious” or insist that something is true because it just has to be true, that's usually a hint that we may need to step back and consider whether what we're saying is really mathematically justified. What we have here is a case of multiple valid definitions that we can adopt, and there's no mathematical reason for not adopting one over the other. (There's a non-mathematical reason, namely “it breaks convention”, but the entire point is that we can break this convention.)