Things possibly not to bother writing

[sniffnoy]

So lately I seem to have encountered a number of internet arguments over whether or not 0⁰=1. It seems to me to be the new "Is .9 repeating equal to 1?". A page explaining why 0^0=1 even somehow ended up on the front page of Hacker News. (I'll confess to having upvoted it, in the hopes that people would come to understand why 0^0=1, but I have to admit I probably just promoted pointless argument instead.)

Unfortunately it seems to be a bit harder to convince people that 0^0=1 than it is to convince people that .999...=1. The thing is, you can present any number of good reasons why 0^0 really should be 1, but people will respond with reasons why it shouldn't that are wrong but for which it can take quite a bit of time to point out the error. That's really the thing -- the people who claim it shouldn't be 1 are simply going to ignore all your reasons why it should be unless you specifically counter their arguments first.

So I was thinking maybe I should write a long, comprehensive explanation "Yes, 0^0 really is 1" to put up on http://www-personal.umich.edu/~haltman/ that I could link people to whenever the topic came up. It would actually directly address the various arguments I've heard about why it shouldn't be 1 before explaining why it should be, and hopefully stand a slightly better chance at convincing people than the other attempts I've seen.

On the other hand, people are goddamned stubborn and this would probably have very little effect. I feel like if I actually were to, after writing it, post it to Hacker News or Reddit, it would just stir up more pointless argument. Largely from people who didn't actually read it. :P

So: Thoughts? Should I bother doing this or not?

-Harry

Comments

[joshuazelinsky.livejournal.com]

Unlike .9999...=1, this is really a matter of definitional convention. In contrast, .9999...=1 is a theorem that follows from our definitions of real numbers.

For a lot of purposes it makes sense to set 0^0=1, and certainly more than it does for setting it to any other value. But, this really is an issue of convenient convention more than a consequence of our axiomatic systems.

[sniffnoy.livejournal.com]

Well, .999...=1 follows from the definition of a decimal expansion. The hard part, I think, is convincing people that that's how decimal expansions should be defined (that they should represent a limit).

The thing is I don't think I've seen *any* compelling argument for not considering 0^0 to be 1; the ones I keep seeing on the internet are just full of badness.

[joshuazelinsky.livejournal.com]

Well, consider f(z)=z^z, for z in C. Then z=0 is an essential singularity. So there's no natural extension to z=0. I agree that 0^0 if it gets any definition should be 1. It extends the most properties we want and makes a lot of theorems not need to have silly corner cases. But the distinction between this and the .999...=1 case should be clear.

I agree that it is difficult to convince people that decimal expansions should represent limits. Part of the problem there is that people work with decimals well before they ever learn a rigorous notion of limits (if they ever do). So they expect decimals to be grounded on something they are already used to. I suspect that there's also more reaction to .999...=1 than there is to the 0^0 issue because .999...=1 looks weirder. People don't necessarily have good intuition for exponents, but they have developed an intuition for decimal numbers and in that intuition numbers that look different are different.

[sniffnoy.livejournal.com]

Well, consider f(z)=z^z, for z in C. Then z=0 is an essential singularity. So there's no natural extension to z=0.

What, do you think I haven't anticipated people saying that? :) Yeah a lot of what I see is essentially that but dressed up in a bunch of nonsense. (Though for z^z for z in C to make sense you'll need to pick some sort of principal values.) My response: Yup, there's no choice to make it analytic or continuous. Sometimes things are discontinuous. And exponentiation of whole numbers is a more fundamental notion than that of real numbers -- the latter is an extension of the former -- so if at one point it doesn't extend nicely, well, that's OK, we still have the original. Ordinarily we don't define things at singularities, but usually there's no reason to; here we already have a value provided us.

Presumably someone might reply "Why not just use different conventions for real numbers and for whole numbers?" To which I say, using different conventions make sense if the definitions *disagree*, but here one defines it and the other doesn't, so the sensible thing to do is combine them and define it. Also, they get mixed together a lot and the boundary isn't always so clear so that would be a real headache.

So basically the idea is to write that but much longer. :P

[joshuazelinsky.livejournal.com]

That's a decent argument. Another point that helps your case is that lim x^x =1 (as x goes to 0) so the most natural limit if one does want to be in a continuous case is 1. This is a very weak argument since other limits don't go to 1, but a lot of them do. In fact one looks at f(x)^g(x) with f and g both going to 0 as x goes to 0, and f and g are reasonably close to each other (say f/g is bounded above and below by some constant) then the limit is still 1. So, if one forced a real value to make sense one would probably pick 1 also.

[grenadier32.livejournal.com]

If you want it to attract attention, it has to go viral. To make it go viral, you have to write it charmingly/amusingly/funnily/well. That can be the difference between a rehash everyone's seen before and a genuine advance in the discourse.

[sniffnoy.livejournal.com]

Well, I was planning to explicitly rebut the arguments that it should be left undefined, rather than just making the argument that it should be 1, which would not be a rehash, but yeah, like you say, that leaves the problem of getting people to actually see it. I have to admit, I'm at a loss as to how to make the topic much amusing. I expect using lots of profanity is probably counterproductive? :P

[grenadier32.livejournal.com]

Using lots of profanity will attract attention, but that's distinct from charismatic writing. It can be a way of writing charismatically, but it alone won't do it.

I think your best bet is to be funny. Remember the Tau Manifesto?

Also, the obvious course of action is to submit the post to all the usual places (and, in case you're unsure, there's nothing wrong with self-submitting). Is there a space like Hacker News, but for math people? Besides HN itself?

[sniffnoy.livejournal.com]

Math people already know this stuff. :P I don't expect everyone to necessarily agree with my stronger claim that yes 0^0=1 is the right definition *always*, but anyone who's done much math will already recognize that it's the right definition in most cases.

I never actually read the Tau Manifesto because I already agreed with it. :P

Honestly my main concern is just getting past the load of crap that is "It's not undefined, it's an indeterminate form!" If people can just recognize that that's a bunch of nonsense, then hopefully they argue about it sensibly...

[grenadier32.livejournal.com]

I suggest you read the Tau Manifesto and model your writing style after it. It's written in a beautifully lighthearted and succinct tone that I found easily approachable despite not having a strong math background (hell, it helped me grok trig better than years of HS math classes ever did... though that's also because of the pedagogical disaster that is pi ;)).

[sniffnoy.livejournal.com]

So, what I've done: I actually went ahead and wrote a draft. Which I probably won't actually bother to change too much because hey, it's the internet. :P I think I'll only go ahead and actually post it anywhere if I actually see the argument come again, though.