Regarding Nullity

[drplokta]

I'm referring to the BBC's article here, which unfortunately was written by a journalist and not by a mathematician.

The BBC won't tell you, but the original paper is here.

I can't bear to read any more comments on Slashdot, Wikipedia or Digg from people who don't know as much maths as they think they do, so here is what's actually going on. Do not pay any attention to any other commentary on this subject that does not include the words "field" or "ring" in their mathematical senses.

The "transreal numbers" defined in this paper may very well be internally consistent, although it's not been peer-reviewed and it would be too much work to check it out thoroughly. However, it doesn't really matter, because by the author's own axioms A8 and A18, "nullity", "infinity" and "-infinity" do not have additive or multiplicative inverses, which means that the transreal numbers aren't a field (hell, they're not even a ring), which means that everything you ever thought you knew about arithmetic will have to be rigorously revalidated, and a lot of it will probably turn out not to work any more. This is a high price to pay for being able to work with the result of 0/0 arithmetically instead of just treating it as undefined, and so the transreal numbers are effectively useless.

[Edit: bohemiancoast points out that one of the other authors on the paper I reference above looks strangely familiar.]

Comments

[purpletigron.livejournal.com]

I'm tempted to start a 'Campaign for Real References in Journalism', but that would be CamRRJ, which sounds like a cross between Jack Sparrow and Monty Python.

[zeit.livejournal.com]

Um, yeah, OK...

Regarding journalism, some rag (I forget which) published the astronomers' "favourite" astronomy image the other day - the Sombrero Galaxy - which, according to the journo, is "over twenty million miles away!" I wondered why they don't check basic facts instead of confusing miles with light years but then I realised that it IS factually accurate - the Sombrero galaxy IS more than twenty million miles away. So I've no grounds for complaint, really.

[zeit.livejournal.com]

And this from a computer scientist:

"Imagine you're landing on an aeroplane and the automatic pilot's working. If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead."

Wouldn't you like to know what his exception handling is like?

[drplokta]

The killer is that he's given zero a multiplicative inverse (sort of) but only by defining three new numbers that you not only can't divide by, you can't subtract them either. So your exception handling gets a lot more complicated, not simpler.

[bellinghman.livejournal.com]

Bad Science talks about it today: http://www.badscience.net/?p=335

[dalmeny]

I'm glad I didn't come across this before reading your post. It's saved me some aggravation.

[redbird]

Not only does the whole "now you can divide by zero" thing set my bullshit detectors ringing, but I am dubious of anything that needs 18 (or more?) axioms: Euclidean and Riemannian geometry each get by with five, and standard arithmetic, propositional calculus, and other useful things are in the same area (maybe as many as six or seven, I'm working from memory here).

[damienw]

I had a perspex machine once.

[the-gardener.livejournal.com]

Be sure to follow the link at the bottom of Goldacre's post to the Rout Transcript:

http://www.austlii.edu.au/au/other/hca/transcripts/2002/C4/1.html

No, it does because the dividing and multiplying by zero, the set that they are adhering to, enables me - it causes things to cease to exist. Now, I have proven everything is on nothing so if everything is on nothing and you multiply it by zero, then the entire universe and the world does not exist. I have proven it conclusively.

Priceless!

[bellinghman.livejournal.com]

It's reasonable to provide an extra axiom when you run into places that an existing numerical system doesn't go. Hence zero, negative numbers, real numbers, complex numbers ... each of those is a case where the previous system was insufficient.

But 18?

*coughcoughcoughcoughcoughcough*

[minnehaha.livejournal.com]

Um, okay.

B

[minnehaha.livejournal.com]

Yeah. I'm not really impressed here.

B

Surreal Numbers

[minnehaha.livejournal.com]

I'm not sure I'm remembering correctly, but I think Conway solved the divide-by-zero problem more cleanly with surreal numbers.

B

Re: Surreal Numbers

[drplokta]

No, the surreal numbers are a field, so you can't divide by zero. It's infinite and infinitesimal numbers that you can deal with in the surreals.

Re: Surreal Numbers

[minnehaha.livejournal.com]

Right. Thanks.

B