Using Mathematical Induction to &#8220;disprove&#8221; the foundation of&nbsp;Calculus!

November 13, 2008 at 1:05 am

I don’t think it’s a parody. The author has been sprinkling comments here and there around the math blogosphere under the pseudonym “notedscholar”, and may be trying to elicit a reaction from people, but the tone of the comments suggests to me that he or she (I’d bet he) is honestly confused. If he’s just trolling around for laughs, he’s being awfully deadpan about it.

But keep tongue firmly in cheek, just in case!

Anonymous

I’m pretty sure it’s a smart person playing for laughs. In places I think there are actually some good points being made. I’ve also always hated those pictures in pop science articles illustrating the curvature of spacetime because while they might be useful to mathematicians they’re useless to their intended audience because they do actually get interpreted exactly as described. And another post makes a nice little point about the semantics of the verb ‘to see’.

November 13, 2008 at 4:10 pm

Shubhendu Trivedi

@charles
That site reminds me of the onion. *laughs*
One facet of “Onionesque Reality” IF it is something serious. LOL.

Anonymous
Good points are always to be carefully analyzed and respected. If not, we are reduced to being in an inertia of xyz thought, the biggest impediment to good research.

I have not been following blogs other than my marked ones of late (and they don’t include many math blogs), so it would be difficult to say about “noted scholars” actual comments. I am only commenting on the basis of this post.

November 13, 2008 at 5:26 pm

November 13, 2008 at 7:13 pm

I won’t outright dismiss the notion he’s playing for laughs, but the problem is that of the few things I’ve read of his, I see nothing particularly funny or witty about them. Sorry, he looks like a garden-variety nut to me. Might he be the Andy Kaufman of the math blogosphere? You decide.

Getting back to the mathematics, there are of course ordered fields in which one could say something like $0.999... \neq 1$ . Here’s one: consider functions $f: \mathbb{N} \to \mathbb{R}$ where $f$ is any function you can build from polynomials, the exponential function, and the logarithmic function, using the four basic arithmetic operations and composition. [I’ll allow piecewise definitions, provided there are only finitely many pieces.] Define an equivalence relation on such functions where $f \sim g$ if $f(n) = g(n)$ for all sufficiently large $n$ . The equivalence class of a function $f$ is called the “germ” of the function (“at infinity”); let’s denote it by $\left[f\right]$ . Then define $\left[f\right] < \left[g\right]$ to mean $f(n) < g(n)$ for all sufficiently large $n$ .

The ring of germs is actually an ordered field; it’s called a Hardy field (after G.H. Hardy), and it has infinitesimals like the germ of $f(n) = 1/n$ built right in. We can then go on and define “numbers” such as the germ of $f(n) = 1 - 10^{-n}$ , which is less than the germ of the constant function $g(n) = 1$ . I have a feeling that most crankish articles (such as the subject of this post) devoted to this “controversy” are fumblingly trying to express something like what I have just stated rigorously, but at the same time they never deal with some of the other consequences, such as the fact that the germ of $1 - 10^{-(n+1)}$ is greater than the germ of $1 - 10^{-n}$ . That is (to abuse terminology), that the “limit” of

.99, .999, .9999, …

is greater than the “limit” of

.9, .99, .999, …

as a variation of notedscholar’s “proof” would seem to show.

Really, I think one has to have some exposure to the rigorous definition of the standard reals (as forming a complete ordered field) to cope adequately with these matters, but that’s asking just a little more than most cranks would be prepared to handle.

Henry Wilton

I’m with you Todd. I don’t see what’s funny about misunderstanding induction.

That said, being British I consistently fail to get American irony. I don’t see what’s funny about the Geico adverts either.

November 13, 2008 at 10:09 pm

Johan Richter

It’s a satire, I am 99% certain. He calls Dawkins a noted Biblical scholar in one post and Noam Chomsky a mainstream scholar of the natural sciences in another.

And note how almost everything he writes manages to be obvisously and laughably wrong, yet not completely incoherent or insanse?

November 13, 2008 at 10:21 pm

November 13, 2008 at 10:43 pm

Henry, which series?

Henry Wilton

John – are you asking which series of Geico adverts? I find the mockney lizard particularly incomprehensible, but the caveman-themed ones also seem long past their sell-by date.

I’d forgotten about the “so-and-so is not an actor, so to help tell her story, we hired an actor” ones, which I do like. The Little Richard one is particularly good.

November 13, 2008 at 10:54 pm

November 13, 2008 at 11:08 pm

Well that’s not a lizard. It’s a gecko. Originally he was in an ad by accident, wen people kept calling “gecko” instead of “GEICO”.

As for what’s funny, they’re not supposed to be so much funny as memorable. And I’d say that they’ve succeeded.

November 13, 2008 at 11:18 pm

Okay, Johan — I didn’t see those, and I agree that’s pretty convincing evidence!

Henry Wilton

Well that’s not a lizard. It’s a gecko.

John, geckoes are lizards. And yes, I inferred the hilarious backstory.

As for what’s funny, they’re not supposed to be so much funny as memorable. And I’d say that they’ve succeeded.

I suppose they have. But it seems obvious that the David Attenborough spoof in the most recent ones is trying very hard to be funny.

November 15, 2008 at 10:42 pm

November 15, 2008 at 11:25 pm

I must say….. I appreciate the attention and discussion of my work.

But….. somehow I think some people here don’t take it very seriously.

As for “satire,” – not so sure how that word is being used. There certainly is humor in the blog – I am after all making fun of the scientific community. And furthermore my username (“noted scholar”) is a bit tongue and cheek regarding how supposedly it matters where an academic is from… but that’s as far as it goes.

Vishal Lama

notedscholar,

Drawing attention to your work was not the aim I had in mind when I wrote the post. It was to point out the flaw in mathematical reasoning in your “using mathematical induction to disprove the foundation of calculus” argument. It is a somewhat subtle flaw; indeed, a student learning mathematical induction for the first time can easily fall into such a “trap”, arguing incorrectly (as you did) that from the premise “a property $P(n)$ true for all $n \in \mathbb{N}$ “, we can derive the conclusion that “ $P(n)$ is true for some infinite n“. There is no infinite number sitting inside the set of naturals $\mathbb{N}$ even though the cardinality of that set is not finite! To be sure, it is certainly possible to extend the set of naturals to incorporate an “infinite number” (say, $\infty$ ), but then we can’t use mathematical induction for our original purpose anymore. That’s the moral of the story.

November 15, 2008 at 11:35 pm

November 16, 2008 at 12:33 am

notedscholar: Some of the readers of topologicalmusings just about had me convinced a day or two ago that your blog was a parody or elaborate prank, but right now I’m undecided.

Got a couple of questions for you:

“Felis sum et ad furandum veni.” It’s been decades since I last studied Latin, but my best guess at a (free) translation would be “I am a cat and I have come [edit: or perhaps ‘came’] to steal.” But the “furandum” looks like a gerund, so “ad furandum” looks like it would translate more literally to “toward stealing”. Could you shed some light on all this?

Second, assuming for the moment that you’re for real and that I’m not about to be blogospherically Punk’d, would you tell us what would be the decimal representation of the difference between the numbers 0.999… and 1? [You agree that all real numbers admit decimal representations?]

November 16, 2008 at 12:49 am

Vishal Lama:

I think your objection is interesting. I appreciate that it at least takes seriously my argument! Perhaps I will soon revisit the concept on my blog.

Todd Trimble:

Your translation is basically correct. I have to admit the subtitle isn’t really related to the blog – it’s just a reference to something amusing I found once, but in English. Not of major concern.

I don’t think I want to “punk” you. Your question is difficult because it has to employ two of my controversial ideas. And I think the probability of two controversial ideas being true is less than just one! But the answer would have something to do with the Penultimate (P). So 1 – 0.999… would perhaps be best represented by 0.000…1, where the notation “…” is identified with P decimal places.

November 17, 2008 at 6:29 pm

Hm, interesting. How would 10 times this difference be represented?

November 17, 2008 at 7:21 pm

Well, how would 10 times an infinitely small difference be represented? Symbolically with notation – not with an actual number. So it would start out as something like 10(0.000…1). The answer would be interesting, because it would involve P-1 decimal places.

November 26, 2008 at 8:10 pm

Well, you might want to think carefully just what you mean by “P” and “P-1”. It certainly looks to me as if P is your term for the first infinite ordinal (if you happen to what ordinals are). Is that true? Do you then know what you mean by P-1?

It can be fun to play Games with Numbers as you seem to be trying to do; the Field is far richer than you can imagine. But generations of mathematicians have thought very carefully about how the standard real numbers work at a fundamental level, and I assure you that whatever you are trying to invent, with flights of fancy about the Pth decimal place and so on, it is not the system of standard real numbers. And until you understand why not, you will not be taken seriously. [I’m not saying your efforts to invent an internally consistent system are doomed to failure — just that you are inventing something else besides what are conventionally called the standard real numbers.]

You have your own sandbox to play in — have fun! But for now, further comments on your plaything are closed — life is short, and people are busy with other things. Best wishes, etc.

December 5, 2008 at 11:12 pm

I hope you guys post something new soon!

December 24, 2008 at 5:35 pm

You haven’t made a post in forever! I have been waiting. This is now an abnormally long time before posting for you both!

JHanson

I’m certain notedscholar is a parodist. In a recent post, he argued that because you can’t step on a bear and kill it therefore bears can’t become extinct. And because dinosaurs are even bigger than bears and thus even harder to make extinct, therefore dinosaurs could not have gone extinct. I cannot accept that such a stupendously dumb argument could be made by a non-parodist.

December 25, 2008 at 12:43 am

December 26, 2008 at 2:01 pm

JHanson: not only a parodist, but a personally-appointed leveller of personal attacks at Todd, Vishal, and myself. Consider, for example, his recent comment at Rigorous Trivialities.

December 26, 2008 at 6:36 pm

I honestly think the evidence (for whether he’s a parodist) can be read either way. In the post you mention, JHanson, I mean, yeah, the argument about stepping on a bear is of course really stupid, but then he goes on to say that the conclusion that dinosaurs were not extinctified (his coinage) is also clearly implausible.

One reason it’s hard for me to read him as a parodist is because I don’t find the presumptive parody at all amusing or entertaining — and normally I enjoy that kind of thing. If you ask me, he’s more of an annoyance than anything else. That I imagine is the way most bloggers regard him.