*NOTE: "D. Gordon" is not David Gordon who is affiliated with the Mises Institute.*

There has been some pretty aggressive back and forth in the comment section of my two posts referencing Gene Callahan's statement on probability and infinity (post 1 and post 2) ,but noteworthy among the comments is this from David G, who in four short sentences, takes down a Murphy comment and a Callahan comment:

Bob [Murphy], my point is that Gene's statement is meaningless. In the same way, "gobbledegook fluoresces" is meaningless rather than false because "gobbledegook" is a meaningless word.

Yes, I am saying that P(X=0) is not well-defined for a continuous distribution and so is not 0.

Gene, if you are so sure that my statement was false and have access to great experts in the field of probability, then by all means provide a nontrivial definition of P(X=x) for continuous distributions (by nontrivial, I mean that you don't simply declare P(X=x)=0, but that you can derive that conclusion from the definition). You can muster whatever authority you wish, but if you can't give a definition, then my point stands. If it turns out that there is such a definition, then I will stand corrected, but until you produce one, I would ask that you be just a bit more courteous in your tone.

David Gordon - a consummate gentleman (and scholar).

ReplyDeleteI'm lucky enough to be attending his (online) courses at Mises Academy!!

Picture this: Sitting on a boat, anchored in the Bahamas, sipping a rum-something, kicking back and attending a class in real time with world class instructors!!!

Mises Academy rocks!!!

NB: no compensation to me from them for the endorsement

Capitalizing on so many of the fruits of the free market simultaneously. Jefferey Tucker will be proud of you.

DeleteA probability density function tells you the amount of probability at any given point, many points are non-zero...

ReplyDeleteYou can see why David G is wrong at http://en.wikipedia.org/wiki/Probability_distribution#Continuous_probability_distribution

ReplyDeleteIt states how P(a<X<b) is defined. A basic calculus definition shows that this means that P(X=x)=0 for any x.

Wikipedia doesn't show how D. Gordon is wrong... From the wikipedia entry:

DeleteIntuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable.

What Dr. Gordon has been saying is that Callahan has been ambiguous in the original example at to what "X" means such that Callahan hasn't defined for us whether this a continuous distribution.

I would agree with most people, and I posted originally, that if you bound a set AND are tossing a non-zero point dart, the distribution must be discrete.

But why would you assume that, rather than read Callahan charitably? Basically, that means you are assuming he is wrong.

DeleteHowever, if I gave this example to anyone in my math department, they would interpret it the same way I have (as referring to a continuous distribution and being overall correct).

After all, it's a thought experiment, not an attempt at modeling darts.

I've got a better suggestion: give it to the English department and ask them if Callahan's argument was well-composed. Do it without the wikipedia entry and see if they can figure it out.

DeleteBrilliant David G! Rand used to call these Callahanesque gimmicks - "arbitrary statements" that bear no relation to reality..

ReplyDeleteI'm not a math guy but just for fun I will point out that "gobbledegook" is in fact not a meaningless word. "gobbledegook fluoresces" is not without meaning either, in fact the phrase would be useful to use as a means to describe that which occurs when Wenzel deconstructs krugmans copiously spewed "gobbledegook".

ReplyDeleteJust sayin'

Just FYI folks, there's nothing (yet) to suggest that David G and David Gordon are the same person. The post title initially confused me on that point as well.

ReplyDeleteA few points, everyone, and then I think I must retire from this fascinating discussion:

ReplyDelete(1) On the other thread, David Gordon made clear (thank goodness) that he wasn't "David G."

(2) This whole thing has absolutely nothing to do with Gene Callahan. He was repeating a literally

textbooktreatment of these issues. If you want to roll your eyes at such high-falutin' nonsense, OK, but you are mad at mathematicians, not Gene Callahan.(3) There were plenty of decent objections raised against Callahan's initial post, in particular one by David Gordon. Maybe the mathematicians are not being careful enough in the philosophical implications of their approach, and for sure they rely on "real world" illustrations or analogies that actually don't work. Fair enough, but again, your beef is with mathematicians, not with Gene who just repeated verbatim how they would discuss this type of thing.

(4) The one guy on the other thread who was demonstrably wrong in his critique was "David G" (not David Gordon). He wasn't simply accusing Gene of being sloppy for not specifying whether it was a continuous or discrete event space, as some people are trying to claim. No, David G. flat-out said (see the "takedown" that has Wenzel so smitten in this post itself, above) that for a continuous distribution, the probability that any particular number will occur is ill-defined. He is simply wrong.

Bob Murphy,

DeleteI have to disagree with you that this has nothing to do with Callahan. Callahan's statement starts out with the first two sentences that say:

" 'There's a 0% chance that X will happen.' :

"Most people interpret that to be equivalent to, 'X will never happen.' "

He then implies that those who make the above interpretation are wrong. NO, he is wrong. Nowhere in the above does he state that he is discussing am imaginary construct around the concept of infinity.

I think the lesson we all should learn is to not pay so much attention to Gene Callahan.

DeleteI'm not the David G. referred to in this post.

ReplyDelete