When I accepted Mr. O’Neill’s challenge

**The North Pole has never been ice-free; not once in the history of the earth. **

… and I can prove it. I’ll wager you $100 to be given to the other’s favorite charity.

I said I would give hm “opportunity to address my criticisms” of his proof. On July 26th I asked Mr. O’Neill the following…

You have submitted a few short comments after my refutation. Shall I take these comments as your response to my refutation? If so, then I am ready to proceed.

Mr. O’Neill responded with…

If for some reason you are waiting for me to tell you to proceed in your rebuttal – then please proceed.

Mr. O’Neill’s opportunity to respond to my refutation of his “proof” is now over.

Readers can consider the aggregate of his comments dated July 22nd to July 27th to be his address of my criticisms.

## Conclusion

I will give my conclusion before I rebut Mr. O’Neill’s comments, since most people will not want to read the boring details.

Mr. O’Neill has not addressed my refutation of his “proof.” Instead he has engaged in a sophist and sophomoric game centered around his claimed inability to associate a scalar value with a point. I suspect that he will never concede defeat and pay the required $100 to Save The Children. I will wait a week in the weak hope that his conscience will get the better of him. After that, I will pay Save The Children the $100 myself to prevent them from being stiffed by Mr. O’Neill. I will post a receipt when that time comes.

Mr. O’Neill has posted two other comments that remain in my moderation queue. One of them is a 1500 word treatise on my supposed moral, mental and/or character deficiencies. The other is a whiny diatribe about how I treated him unfairly (boo hoo) by pointing out that a journal article he cited actually supported my view. Both of these comments will receive special treatment and be released in their entirety at some future date.

In the mean time, I do not feel obligated to provide a forum for the unending dribble of sophistry coming from Mr. O’Neill. New comments from Mr. O’Neill will go to my moderation queue, and unless they end up in my special treatment page for Mr. O’Neill, then in all likelihood will end up being deleted. Mr. O’Neill has overstayed his welcome

I may still have a little bit of fun with his “proof” in some later posts though.

If you are interested in boring details, have a strong cup of coffee and read on…

## Therefore I proceed

I ended my refutation of Mr. O’Neill’s “proof” with the following…

O’Neill needs to do all of the three following things: he must prove my paleontological & geological evidence is wrong; he must show that his “proof” does not lead to bizarre consequences; he must show that Li/Lt is “undefined” (as he claimed in his proof) as opposed to “indeterminate.”

Mr. O’Neill flippantly dismissed my paleontological & geological evidence with the statement “The geology/paleontology stuff is just irrelevant.” Sorry Mr. O”Neill, your dismissal does not counter my refutation of your proof. On that basis alone I have already won the wager.

However, I will play along with his clumsy sophistry for the moment

**First sophistry (equivocation)**

Mr. O’Neill couches his “proof” in the language of math and physics. He defines the North Pole as a “point” in the mathematical sense. He also defines two properties of this point when he says “Li is L’s ice covered area and Lt is L’s total area.” He creates a metric for the ice covered fraction at L by taking the mathematical ratio of Li and Lt when he says “To satisfy the definition of ‘ice-free’ Li/Lt must be < .15.” He later simply asserts that “for any point L the quotient for Li/Lt is always undefined.”

Why is “the quotient for Li/Lt is always undefined?” He doesn’t explain this in his “proof.” In some of his comments which he says can serve as a rebuttal to my refutation, he says that Lt is simply undefined because L has no area. He explains that area is not a property of a point, and therefore Lt is undefined.

But wait, didn’t he personally define Lt as an AREA when he said “Lt is L’s total AREA.” So, he personally defines Lt as an area, uses it in his “proof,” and then claims Lt is undefined to conclude the validity of his “proof.” This is an extraordinary case of equivocation (Lt is defined as an AREA early in the “proof” and claimed to be undefined later in the “proof.”).

The absurdity of his “proof” would be clear to all when the mask of equivocation is lifted. Suppose his “proof” said “Li is L’s ice-covered area and Lt is *undefined*” and “To satisfy the definition of ‘ice-free’ Li/Lt must be < .15.”

O’Neill can’t just deliberately create what he feels is a bogus metric (Li/Lt with Lt undefined) and then claim that since his metric is bogus the thing being quantified must be undefined.

**Second sophistry (non-sequitur)**

O’Neill claims that since his metric (Li/Lt) cannot quantify the thing he wants to quantify (surface density of ice), then the thing he wants to quantify must be undefined. Why? Does the supposed failure of his metric prove that all other approaches will also fail? This is a non-sequitur.

**Third, and most important sophistry**

By calling the North Pole a “point” he thinks that he has removed all scalar properties associated that point. That is, he deliberately tries to confuse the difference between the properties of a point with the properties associated with a point.

The only properties of a point are the n coordinates that define it in an n-dimensional space. However, there are an infinite number of properties that can be associated with a point. So, for example, while temperature is not a property of a point, it can be associated with a point. When we speak of the temperature at a point, we are not talking about a property of the point, but rather a property associated with the point. Similarly, and more importantly for this discussion, the surface density of ice is not a property of a point, it is a property associated with a point. The usefulness of any coordinate system is zero without the ability to associate properties with points.

A *scalar* is a quantity that can be described by a single number (either dimensionless, or in terms of some physical dimension). A scalar field is an n-dimensional space with a scalar value associated with every point in a that space. Temperature as a function of position and surface density of ice as a function of position are simple examples of scalar fields. The concept of a scalar field is intuitive to most people, but is summed up nicely here…

In mathematics and physics, a scalar field associates a single number (or scalar) to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the **air pressure**.

Note to Mr. O’Neill: **air pressure** is force per unit AREA. By your reasoning, air pressure cannot be defined at a point. But, in fact it is always defined at a point.

Mr. O’Neill’s sophistry denies both the intuitive and mathematically rigorous concepts of a scalar field. Why? Because he denies the possibility of associating a scalar property (such as surface density or temperature) with a point simply because that scalar property is not a property of the point. He seems to see some clever trap or paradox in his own blind spot of understanding.

**Fourth sophistry**

Mr. O’Neill has a choice: he can either say that his proof requires a mathematical foundation, or it does not. O’Neill couches his “proof” in the language of mathematics, with definitions of variables (albeit equivocating definitions) and mathematical ratios. Why blow smoke with all the math when he could have simply argued that “the concept of a surface density of ice at a point is simply undefined and therefore cannot have ever been less than 0.15. QED” He couldn’t make this simple argument because he cannot back it up – the concept of the surface density of ice associated with a point is easily defined.

Mr. O’Neill’s attempts to brush aside the use of L’Hopital’s rule to resolve an indeterminacy of the surface density of ice at a point. In a statement that stands as a monument to his mathematical ignorance, Mr. O’Neill says…

Parenthetically, and irrelevant as far as I can see, I suspect that L’Hopital’s rule would not apply since the denominator is a constant zero – I suppose the numerator is as well. So, even if we accepted your 0/0 equation, would L’Hopital’s rule apply since we’re dealing with two constants and not a converging series?

While L’Hopital’s rule can be a useful tool to judge the convergence of a series, it is by no means used exclusively or even mostly for that purpose. It is typically used to find the value of f(x)/g(x) at a value of x that yields an indeterminate form (such as 0/0 or infinity divided by infinity). Mr. O’Neill’s argument against the use of L’Hopital’s rule in this instance can best be summed up as the fallacy of “invincible ignorance” (seriously, see the Philosophical Society).” Mr. O’Neill, go back and take a first year calculus class, then re-read my refutation of your “proof.”

**Fifth sophistry (tu quoque)**

I am afraid this gets to the heart of the matter for Mr. O’Neill. As I mentioned in my refutation of his “proof,” Mr. O’Neill seeks to claim some sort of moral victory by saying that in order for me to win the wager I must retract some dishonest or misleading claim that the North Pole is a “point” in the mathematical sense. Of course, I never made such a claim, Mr. O’Neill’s ramblings about my pictures of submarines at the North Pole notwithstanding.

When another commenter (Charlie A, July 19, 2010 at 2:17 pm) mockingly criticized Mr. O’Neill’s “proof” by saying “Perhaps, just perhaps, there is a logical fallacy in the proof” O’Neill responded (July 19, 2010 at 11:20 pm) with the following …

“There’s no logical fallacy. What there is is a lesson: It’s **stupid** to define the North Pole as s point when discussing ice-free conditions in the Arctic – otherwise we get this sort of nonsense.”

When I accepted Mr. O’Neill’s wager, I defined the term North Pole as

**North Pole.** This means the *area* at which the axis of rotation exits the current Northern Hemisphere. It is not the magnetic pole. The North Pole, for the purposes of this wager, does not change with a magnetic reversal. The North Pole is not required to include the entire Arctic Ocean or the entire Arctic Basin (features that have not even existed through the whole “history of the [E]arth”).

Mr. O’Neill twists himself into a logical pretzel trying to show that this definition implies I have defined the North Pole as a point in the mathematical sense. He comments (2010/07/22 at 4:58 pm)

Someone said,

…the area at which the axis of rotation exits the current Northern Hemisphere ..

A line intersecting a sphere would be …. a point?

Get it? I said “area” in the colloquial sense, and Mr. O’Neill claims I said “point” in the mathematical sense. Oh well, it doesn’t really make any difference. Mr. O’Neill’s naive arguments about undefined values and the inapplicability of L’Hopital’s rule fail even if the North Pole is defined as a point the the mathematical sense.

In other words, Mr. O’Neill justifies making what he admits is a “stupid” argument because he claims I had already made the same argument (which, of course, I have not). A pathetic case of tu quoque.