NaN is Not a Math
NaN is Not a Math
What is NaN
NaN is a mechanism defined by the IEEE 754 standard for indicating and signaling Floating-Point Arithmetic exceptional conditions. From the standard it says:
1.1 Scope: This standard specifies formats and methods for floating-point arithmetic in computer systems—standard and extended functions with single, double, extended, and extendable precision—and recommends formats for data interchange. Exception conditions are defined and standard handling of these conditions is specified.
This is from the actual standard document and outlines what the standard says it applies to. The purpose of
NaN in the standard is to handle situations where some floating point arithmetic operation doesn't make sense, has a failure mode, or similar situation that "should...afford retrospective diagnostic information inherited from invalid or unavailable data and results."
NaN when you attempt something like these lines of code:
> 0 / 0 NaN
NaN for many things that shouldn't use it, and how there are no "signaling NaNs" or diagnostic information, so trapping the error where it matters is far too difficult. There's also the issue that
NaN is not math at all, but just part of some standard.
1. IEEE 754 is Not Math
NaN is "math" right? As in, established via a proof that has been peer reviewed in a journal of mathematics such that it is considered a piece of the field of mathematics? I can find the paper that establishes
NaN on the same footing as, say, Principia Mathematica or Gödel's Incompleteness Theorem? Right? That exists?
No? Well then,
NaN is not math.
NaN is they think something being "math" makes it impossible to question. "Math NaN" is an unassailable proof that once established can never be refuted, disproved, undone, countered, or hated. Worse yet, hating
NaN as if it's on the same validity level as Gödel's Theorem, and you're an idiot for not knowing this.
This is a Fallacy of False Equivalence.
NaN is not the same as Gödel's Theorem just because they both have numbers in them. In fact,
NaN is based on a standard that is specifically designed to break real floating point arithmetic so that it can work in a discrete CPU that can't efficiently do symbolic operations. If you divide 1/3 in a computer it has to make some compromises to give you some kind of close-enough answer. But, a "close enough" answer is not what mathematics is all about. Real mathematics--the kind based on real peer reviewed proofs--is based on being as logically exact as humanly possible.
IEEE Floating Point math is not perfectly exact, and that means it's also not "math". It's a bastardization of math as a practical compromise to make 1/3 work in a computer.
The Math and Bailey
Now, when you bring up this clearly obvious point that
NaN--and IEEE Floating Point--are not math you'll get a reply that attempts to move the discussion to a more reasonable claim. They'll say, "Well it's not math per-se but it is defined in a standard that we all use so..."
This rhetorical trick is called a motte-and-bailey. A "motte-and-bailey fallacy" is where someone advances a controversial position (the bailey), and when challenged, runs to a defensible narrowing of that position (the motte). Once they defend the easily defended safe motte position they come out claiming they defended the controversial bailey position.
NaN is irrefutable because it is mathematically the same as Gödel's Theorem." When you challenge this position, because
NaN is not math at all, they'll retreat to a more narrow claim that it's "Part of the IEEE Floating Point standard." It's easy to defend that
NaN. That's an uncontroversial stating of a simple fact. Once they defend that position they'll then claim that the original claim of
NaN being just as proven as Gödel's Theorem is still valid, even though all they did was make the uncontroversial claim that you can find
Yet, until someone produces a proof, published in a prestigious journal of mathematics, that establishes IEEE math as equal to Gödel's Theorem then, nope,
NaN is not math.
2. Where's the Signal?
Alright, let's say the motte-and-bailey is working. Someone is retreating to the "It's in the IEEE standard" claim as a way to defend
The IEEE 754-2008 standard requires two types of
qNaN and signaling
NaN value and the program continues running. A signaling
sNaN, only the quiet ones.
NaN with the claim that it's "IEEE Standard" is invalidated as well. If the "
It's Standard or It's Not
Standards are documents created by companies to help integrate and sell their products. These standards are not sources of truth or forces of nature that can't be questioned. You can totally question standards, but more importantly, because these documents are not universal truths they should be applied sparingly. If you take the IEEE 754 standard and attempt to apply it to things not covered in the document then you're stepping outside of the standard and it once again cannot be used as a defense.
NaN around like it's (not a) candy. All kinds of operations not covered in the IEEE 754 standard return
NaN. Parsing integers isn't in the standard, and yet:
> parseInt("what") NaN
NaN from operations not...in the standard.
3. Where's the Diagnostics?
The standard mentions that a
NaN should include diagnostic information that lets someone figure out what went wrong:
Quiet NaNs should, by means left to the implementer’s discretion, afford retrospective diagnostic information inherited from invalid or unavailable data and results. To facilitate propagation of diagnostic information contained in NaNs, as much of that information as possible should be preserved in NaN results of operations.
NaN? Not much. You can test for it with
isNaN, but you can't get useful information like, where it happened, what the operands were used, or anything else. You just get a random
NaN somewhere and have to root out where it went.
NaN to be conforming (and useful) it should include the source location where it was created, as well as information on the two operands. This could be defended by referring to the "by means left to the implementer's discretion", but this doesn't say "at the implementer's discretion". It says, "by means left to", which means diagnostic information is required, but the standard won't say how it should done.
Not only would source location and operand information make
NaN as well as be backward compatible with existing code. We could have additional functions on
Number that throw exceptions, get diagnostic information, etc.
Finally, people have problems with
NaN is not math, and not IEEE 754 compliant, then at a minimum you should realize that the way
NaN causes people problems, so the reply should at least be more sympathetic and educational than, "LOL learn math dood."
NaN useful...as required by the standard.
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