Refutation of Kevin O’Neill’s “Proof” that “The North Pole has never been ice-free”July 22, 2010
Never argue with a fool, onlookers may not be able to tell the difference.
I sure hope I don’t prove Twain to be right.
Tom Moriarty (ClimateSanity)
Mr. O’Neill’s “proof”
Refutation of Kevin O’Neill’s Proof
I will approach Kevin O’Neill’s “semi-formal proof” along three lines.
1. Paleontology & Geology
I will go for the low hanging fruit and consider only the Paleocene-Eocene thermal Maximum (PETM).
a. Leonid Polyak, et. al., History of sea ice in the Arctic, Quaternary Science Reviews, 29, 1757-, 1757-1778, 2010
b. James Zachos, et. al., Trends, Rhythms, and Aberrations in Global Climate 65 Ma to Present, Science, 29, 686-693, 2001
c. Appy Sluijs, Subtropical Arctic Ocean temperatures during the Palaeocene/Eocene thermal maximum, Nature, 441/1, 610-613, 2006
2. O’Neill’s proof yields bizarre consequences
It could equally prove that the entire surface of the Earth has always been 100% ice-covered, which is obviously incorrect.
This is the clincher. I will bring Mr. O’Neill up to date with 17th century mathematics with the use of L’Hopital’s rule on an indeterminate ratio.
Paleontology & Geology
Let’s consider the low hanging fruit of the Paleocene-Eocene Thermal Maximum (PETM).
Mr. O’Neill likes to use “History of sea ice in the Arctic” (Leonid Polyak, et. al., History of sea ice in the Arctic, Quaternary Science Reviews, 29, 1757-, 1757-1778, 2010) as a source. I wonder if he missed this part about the Paleocene-Eocene Thermal Maximum…
“Drilling results confirmed that about 50 Ma [million years ago], during the Paleocene-Eocene thermal Maximum (PETM), the Arctic Ocean was considerably warmer than it is today, with summer temperatures estimated as high as 24C [75 degrees F] and freshwater subtropical aquatic ferns growing in abundance (Moran et. al., 2006). This warm environment is consistent with forests of enormous Metasequoia that stood at the same time on shores of the Arctic Ocean – such as on Ellesmere Island across low-lying delta floodplains riddled with lakes and swamps”
No Northern Hemisphere Polar Ice Sheets during PETM
Trends, Rhythms, and Aberrations in Global Climate 65 Ma to Present (James Zachos, et. al., Science, 29, 686-693, 2001) has much to say about polar ice. For example…
Earth’s climate system has experienced continuous change, drifting from extremes of expansive warmth with ice-free poles, to extremes of cold with massive continental ice-sheets and polar ice caps. [emphasis added]
Sachos’ figure number two sums up the polar ice sheets for the last 50 million years.
No Ice at North Pole During PETM
In “Subtropical Arctic Ocean temperatures during the Palaeocene/Eocene thermal maximum” (Appy Sluijs, Nature, 441/1, 610-613, 2006), Sluijs make the following very direct statement …
“Here we identify the Palaeocene/Eocene thermal maximum in a marine sedimentary sequence obtained during the Arctic Coring Expedition. We show that sea surface temperatures near the North Pole increased from ,18 °C to over 23 ° C during this event. Such warm values imply the absence of ice” [emphasis added]
So, the paleontological and geological evidence for an ice-free North Pole during Paleocene-Eocene thermal Maximum (PETM) is abundant. The burden of proof is on Mr. O’Neill. He must show this paleontological and geological evidence to be incorrect.
The bizarre consequences of O’Neill’s proof
Mr. O’Neill’s “proof” that the “North Pole has never been ice-free; not once in the history of the Earth” may be applied to any and all points on the Earth. It can also be used to show that all points have had 100% ice coverage for all time. This is an obviously bogus result which shatters Mr. O’Neill’s “proof.”
Mathematical Refutation via L’Hopital’s Rule
Refer to Mr. O’Neill’s proof at the top of this post.
His conclusion completely relies on his assertion that “Li/Lt is always undefined.” In fact, Mr. O’Neill is wrong. The ratio Li/Lt is “indeterminate,” not “undefined.” This is an extremely important mathematical distinction. A ratio where the denominator is zero, but the numerator is not zero is undefined – that is, there is no proper solution. But if both the numerator and denominator are zero, as in Mr. O’Neill’s “proof,” the result is indeterminate, and there may very well be a proper solution. One of the tools used to resolve the value of an indeterminate ratio is “L’Hopital’s Rule” (also commonly referred to as “L’Hospital’s rule”). Here is a high school Calculus I level explanation with examples using some simple functions.
Here is a very simple application of L’Hopital’s Rule to the question at hand:
Consider an area, Lt, and a subset of that area, Li, that is covered in ice. The fraction of area covered by ice is Li / Lt. O’Neill’s incorrect mathematical argument is that if Lt goes to zero (i.e. Lt represents a single point) then Li / Lt is undefined. Instead, it is indeterminate and can be resolved with L’Hoptial’s rule.
Lt can be given by…
…where dA is the differential area.
Li is the part of Lt that is covered by ice and can be given by some function, f(A), such that…
Given Lt and Li, the fraction of the area covered by ice is…
What happens when the area goes to zero? By a particularly simple application of L’Hospital’s rule…
There are an infinite number of possible choices for f(A) such that it is less than 0.15 (15%) at the chosen point. This result not only has the advantage of being mathematically correct (and it does not ignore several hundred years of mathematical thought), but has the satisfying effect of not leading to bizarre results.
The burden of proof is still on Mr. O’Neill. He must show that The ratio Li/Lt is “undefined” as opposed to “indeterminate.” That is, he must show that when the area goes to zero Li/Lt goes to some non-zero value divided by zero (“undefined’), as opposed to zero divided by zero (“indeterminate”).
A moral victory for Kevin O’Neill?
or, the point of Kevin O’Neill’s “point”
The Paleontological and Mathematical part of my refutation sufficiently ruin Mr. O’Neill’s “proof.” But I suspect that Mr. O’Neill will attempt to claim some kind of moral victory. He will try to latch on to this 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.
O’Neill’s mistaken notion
Mr. O’Neill has claimed ad nauseum in his comments that I (Tom Moriarty) have said the North Pole is a “point” in the mathematical sense. He seems to be obsessive on this issue.
On July 12, 2010 at 5:18 pm, he said…
Now, you and I both know what ice free means in regards to the Arctic. You changed ice free to open water and claimed victory. BUT even if we accept your pointilist definition, then you have no proof the North Pole was ever ice free. Every picture of the North Pole has ice in it. And not one of those pictures shows the North Pole as being in one of the areas of open water. If the pole is a point – then point it out in the pictures. Otherwise you have no proof.
On 2010/07/19 at 11:20 pm referring to me he said…
“It’s stupid to define the North Pole as s point when discussing ice-free conditions in the Arctic.”
On 2010/07/19 at 5:00 am referring to me he said…
“…if one accepts the colloquial definition of ‘North Pole’ and does not insist that the North Pole is a ‘point’.”
I never said the North Pole was a “point”
I have searched for the word “point” in posts and comments looking for place where I may have said that the North Pole is a point. I can’t find it. Mr. O’Neill must be referring to the following comment that I made July 2, 2010 at 5:49 pm where I said…
The point is the difference between the “North Pole,” which is a dot on the map, and the “Arctic Basin,” which is 4 million square kilometers. It is not a big deal for the “North Pole” to be ice free, but a very big deal for the “Arctic Basin” to be ice free.
“Dot on the map” is a colloquialism that does not imply any mathematical rigor. For example, here is a travel guide called DotOnTheMap.com. If you say your home town is a dot on the map, are you saying that your home town has no spatial extent?
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.”
The burden of proof is on Kevin O’Neill. He cannot support that burden. He loses the wager.