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Laymen's thirst for overly simple solutions and \(0^0\)

Most people seem to be irrational, in fact obsessively irrational, and that's manifested even in mundane discussions. OK, ask how much is the zeroth power of zero:\[

0^0 = ?

\] Great. In calculus where the arguments are continuous, there is still a general way to calculate a power:\[

a^b = \exp(b \log a).

\] The logarithm is natural. Apply it to \(0^0\) and you get:\[

0^0 = \exp(0\log 0) = \exp(0\times (-\infty)) = \exp({\rm ind}) = {\rm ind}.

\] Like \(0/0\), it's a classic indeterminate form. Well, yes, this \(0\times (-\infty)\) in the exponent is "more likely" to be "morally" zero (because the minus infinity may be modest "minus nine" times the logarithm of ten when the base is one billionth, close enough to zero; and the exponent is therefore one) but in principle, the product of the zero and minus infinity may be anything.




There are other ways to show that these two expressions are equally indeterminate. In fact, it's straightforward to prove that they are equal. Use the general identity\[

a^{x-y} = \frac{a^x}{a^y}

\] and use it to see\[

0^0=0^{5-5} = \frac{0^5}{0^5} = \frac{0}{0}.

\] OK, so the "zeroth power of zero" is equal to "zero over zero". If the latter is indeterminate, so must be the former – or you must admit that \(a^{x-y}=a^x/a^y\) doesn't always work (in particular, one side may be well-defined and the other is not). This argument has the opposite impact than the argument that the binomial expansion of \((x+y)^n\) \[

(x+y)^n = \sum_{k=0}^n \frac{n!}{k!(n-k)!} x^k y^{n-k}

\] "should" be valid even for \(y=0\), and \(0^0=1\) is then naturally "needed" because this \(0^0\) appears in the \(k=n\) term.

However, the claim that \(0^0\) is "indeterminate" means that there are subtleties and there may be reasons to justify different answers – so the very existence of these "morally contradictory" arguments is evidence supporting the "indeterminate" answer! If you need to show that \(0^0\) has a unique value, you need to suppress or censor all the inconvenient arguments – and an honest thinker just cannot do that.

Calculators etc. give you an "error" or "Indeterminate" in order to force you to discuss the special case separately – in order to prevent you from "knowing" a unique answer that turns out to be wrong!




Now, as you can see e.g. on Wikipedia (zeroth power of zero), combinatorial people – what really matters is that they are people who imagine that the exponents such as the "second" zero are integers or rational – like to "define" \(0^0=1\). Well, they give some precise meaning to the exponentiation or to the symbol of the power – and strictly speaking, it is a slightly different meaning than the meaning assigned by the continuous people.

One may give rationalizations for this choice. The value "one" is consistent with some considerations. Anything to the zeroth power is equal to one – zeroth power of something means "zero contribution" to products multiplicatively, and you may choose this logic to be valid even if the base is zero.

OK, I obviously prefer the answer from the calculus and continuous numbers because it's more general. The expression is an indeterminate form and calculators dealing with continuous numbers should better return an ill-defined result. And be sure that they do.

Five years ago, there was a Quora question What is the zeroth power of zero? There are some 100 answers. Some of them say basically what I did, others prefer to say that it is one. One answer is by MIT student who set that "Donald Knuth set things straight" in 1992 and "now we assign the useful value one".

Oh, really? Who is "we"?

Donald Knuth, a typical representative of the "discrete mathematics" culture, just gave another rationalization for the claim that \(0^0=1\), like tons of people before him and after him – in fact, this simple question has been debated for centuries. So he shows that it's compatible with some combinatorial procedures to define the power as one.

What I find terribly characteristic and irrational about a majority of the laymen's attitude to all analogous mathematical or scientific questions is:

  1. they believe that the truth is dictated by authorities or votes
  2. they think that proofs may suddenly disappear
  3. they think that the answers must be simplified when some people "vote"
  4. they think that if an identity works in 90% or 97% of cases, it works in general
On Monday, someone asked the same question again, a bit differently:
Why does Wolfram Alpha and many other calculators including TIs and Casios say that \(0^0\) is undefined, when it's been unanimously accepted by the mathematical community to be \(1\) (except in limit form)?
You see the "logic" and the combative tone here: the consensus science has spoken. The arrogant MIT student from has picked an answer and called himself "we" so it must be true. Stephen Wolfram is a heretic, and so is the CEO of Casio and Texas Instruments. Burn them at stake and make sure that all those programs return the correct result \(0^0=1\) that was agreed upon in the Gathering of the Holy Inquisition. Or these programs must be banned and deleted from all computers.

Now, there is a huge percentage of questions on Quora that ask "Why X is true?" where reasonable people consider "X" to be false, or they at least know "X" to be debated and questioned. This is obviously one of them. The claim that the "mathematical community has unanimously accepted that \(0^0=1\)" is clearly a lie – exactly the same kind of a lie as if someone claims that scientists have achieved a consensus that there's an ongoing dangerous climate change.

Even if Stephen Wolfram and his employees were the only exceptions, Wolfram and many folks around him are members of the scientific community to some extent. If I write \(0^0\) to Mathematica, even the newest version of it, it tells me "Indeterminate". This observation is really enough to prove that the claim that "it has been unanimously accepted" is simply a lie.

Needless to say, it's not just Wolfram and employees – and producers of calculators – that keep on insisting it's an indeterminate expression. Everyone who deals with continuous numbers (especially at the place of the exponent) still knows that the expression shouldn't be assigned a particular finite value. The answer that the expression is indeterminate is still the answer required at calculus courses for undergraduate freshmen. The indeterminate answer is still a basic textbook material. All the arguments above – and others – that it is an indeterminate form are still valid. It's really rather obvious that they will always be considered valid by folks who are competent in certain things.

The author of the question has probably read the answers to the previous question sloppily and decided that all the answers are compatible with his personal preference – the result is one. It is not really true because there are many "subtle" answers similar to mine but he didn't want to see them. So his idea about the "scientific" approach is to sloppily look at a list of some random opinions by some random people on an Internet server, distort the content of this list – and then act. The "action" is the most popular step among such people, of course. (That's also true for the "action" on climate change – these people simply love to throw trillions of other people's dollars to the trash bin.) The Wolfram Language and the calculators have to be invaded and made obedient because the consensus has spoken.

The idea that the consensus should be listened to in this way, and even used to eradicate "minority opinions" is unfortunately extremely widespread. You know, no competent scientist is deciding about the correct answers to mathematical and scientific questions by "measuring the opinions of others or a hypothetical consensus". Who decides in this way is pretty much by definition incompetent. It's always like that.

You know, people like climate skeptics often point out that consensus isn't science and the consensus science is an oxymoron. Some of the laymen sometimes repeat this proposition but I still think that most of them don't really believe what they're saying. Or, when it comes to any other question where they may "prefer" another answer, they just don't apply the rule. Most people are still mobs from the Middle Ages who burn witches after some of them scream some emotional, irrational, obviously not very refined and intelligent clichés.

I am sorry but the screaming and consensus votes only impress idiots. There are several billion idiots in the world but they're much less important for the civilization than the people who are careful. The statement that \(0^0\) is an indeterminate form and one must be careful not to mindlessly assign a single finite value to the expression because it could lead to some contradictions is a fact and it will always be a fact. It's likely that producers of calculators will prefer to keep the indeterminate or erroneous answer, too, although I can't be certain how people are going to behave.

Would the world be simpler for you if \(0^0=1\) were the only allowed answer everywhere? Well, it could be simpler for you but whether the world is simple for you isn't important. What is important are the proofs and mathematical arguments. And they imply that \(0^0\) is subtle, if I put it informally yet generally.

It wasn't just the single troll who wanted to burn Stephen Wolfram and Casio, TI CEOs at stake. Another guy whom I debated there – an engineering student at L.A. – was also feeling extremely uncomfortable about the very existence of some "different" answers – meaning different from \(0^0=1\). Quite generally, I think it's a pretty bad sign if a STEM student feels uncomfortable about arguments that vaguely implied something else than other arguments – because this situation is pretty much omnipresent. Many questions are subtle and many objects – like the powers in this example – may be defined in slightly different ways.

Those subtleties don't prove any inconsistency of mathematics – because what the different groups of people call the "exponentiation" or "power" are simply different things – and if you tried to "suppress" some obviously valid arguments because you feel bad about the ambiguities, you're not thinking fairly and rationally. Also, if you think that even "vaguely similar questions" must have the same answers, it's too bad because the details often matter. The devil is often in the details.

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