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.    

Mark Twain    

I sure hope I don’t prove Twain to be right.    

Tom Moriarty (ClimateSanity)    


Mr. O’Neill’s “proof”

Kevin O'Neill's proof that "The North Pole has never been ice free; not once in the history of the earth."

Kevin O'Neill's proof that "The North Pole has never been ice free; not once in the history of the earth."

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.  

3. Mathematical…

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).     

The setting. 

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.        

Note the onset of the Northern Hemisphere's polar ice sheets in the late Miocene, and their absence in the Paleocene and Eocene. Sachos' original caption says "The vertical bars provide a rough qualitative representation of ice volume in each hemisphere..."

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 / L 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.  

For example…  

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.


  1. Tom,

    In my proof I state that L(sub i) / L(sub t) is undefined.

    In your rebuttal you say that 0/0 is indeterminate.

    The area of a point does NOT equal 0. If it did it would be defined. Each of the terms is undefined. Undefined divided by undefined = undefined.

    The proof is trivial. It was never stated to be otherwise. I have proven the case given the definitions of the North Pole as a point and the calculation for ‘ice-free < 15%' – that is all I'm required to do.

    Someone said,

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

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

  2. 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?

    I’ve never had anything beyond H.S. math – and that 30+ years ago – so I’d have to ask someone more knowledgeable than I. Irrelevant since both L(sub i) and L(sub t) are undefined, but an interesting question.

  3. It can also be used to show that all points have had 100% ice coverage for all time.

    How can this be used to prove that all points have had 100% ice coverage for all time? I can prove that all points are NOT ice-covered, which makes sense; a point can neither be considered ice-free nor ice-covered since both calculations require a defined area, but you’re saying you can prove

    L(sub i) / L(sub t) = 1 for all points

    I don’t believe you can do that. In fact, I’m pretty sure you can’t.

  4. I like Twain, but I’m also fond of Voltaire:

    I have never made but one prayer to God, a very short one: ‘O Lord, make my enemies ridiculous.’ And God granted it.

  5. I must retract some dishonest or misleading claim that the North Pole is a “point” in the mathematical sense.

    Does “free of ice” mean the same thing as “open water”? No, it doesn’t.

    Did Professor Barber ever say the North Pole had never seen open water? No, he didn’t.

    What then was the purpose of submarine pictures? Umm, it was a semantic trick. “free of ice” –> “open water” THAT is the misleading change of terminology you made. Then to justify it you said the North Pole is just a “dot” – i.e., point.

    If the North Pole is to be calculated for ice-free conditions like every other area of the Arctic then we’d need to look at a grid cell area 13 or 25 or 50 km on a side – depending on the satellite telemetry being used. Of course then you wouldn’t have any cute pictures to show because no human has probably ever witnessed an ice-free North Pole – and surely no human with a camera.

    Oh, and Professor Barber’s statement about it happening for the first time in history would be true.

    Now, you could have asked Professor Barber exactly what he meant, but that would have required courtesy, common-sense, and a willingness to engage in dialogue – not a hot air rant complete with pictures of submarines.

    In the end I guess it all works out. The Fisher House gets your check for $100 and I learned that a Master’s Degree in Physics doesn’t always compare favorably to paying attention in 10th grade Geometry class 35 years ago.

    Though if you still think it follows from my proof that:

    It can also be used to show that all points have had 100% ice coverage for all time.

    I’ll offer you double or nuthin’ 🙂

  6. Kevin,

    Thought I would repeat this comment in a more current post where more people would see it.

    I’m surprised you haven’t left a link in these comments to your latest entry over at DailyKos.

    You told your readers about your bet with climatesanity and posted your proof. But the comments weren’t exactly to your liking, were they?

    For the most part, your own Daily Kos readers said you are wrong.

    justalittlebitcrazy said of your proof…

    “That’s such a lame proof since it uses a cheap trick. Sorry.

    Besides, if you go far enough I’m sure the North Pole has been ice-free. Earth began as hot mass of rock with barely any atmosphere or water.”

    Duncan Idaho said
    “The North Pole was almost certainly ice-free during parts of the Mesozoic Era, when dinosaurs lived well north of the Arctic Circle. And CO2 levels were even higher than today.”

    JMMcDonald, who seems to understand some physics and math said
    “When trying to define a value that doesn’t make sense on a point, the usual trick is to define the value on successively smaller intervals around that point, then show that the sequence of values converges as the intervals shrink.

    In many cases, the values don’t converge, or converge to different values depending how you shrink the intervals.

    But physical properties such as ice-coverage tend to be continuous and well-behaved, leading to an unambiguous answer.

    On the whole, I’d say you lose, since there is a standard way (as I allude to above) to assign an unambiguous “ice-coverage” percentage to each point, and when the Earth was covered with magma that value was zero everywhere, but the devil would be in the details of the bet.”

    pragprogess said of your proof that
    “You can’t win an honest bet with a weasel argument like that, can you?

    Isn’t it also true that there were no glaciers on Earth at all for most of its existence (4/5ths)?”

    Ray in TX said
    “Your use of the word “point” as a geometrical term in Step 5 is inconsistent with its use in the Assumption, where it is a geographical term.

    Therefore, your proof falls apart at that point.

    ba-dum BUM”

    Ariseatex said
    ” Sorry man

    You can’t make an outlandish statement like “The North Pole has never been ice-free; not once in the history of the earth” and be able to prove it. All the other side needs to prove it is one instance, and the full statement is wrong.

    The Earth was in a highly-convectively-active, molten state for a billion years following its formation. The North Pole wasn’t frozen then. Tom Moriarty may be a denialist, but he’s not a creationist; he knows this. He accepted the bet because he wanted to trap you.”

    Heres one of my favotites, from a guy who says he is predisposed against “denialists.”
    “Science is the world of Applied Mathematics; utilizing mathematical knowledge to report on the state of the universe. Proofs such as you’re proposing are Theoretical Mathematics; you can stay in the realm of semantics all you want, but the bridge to concrete science takes a physical definition, not a semantic definition.

    (Full disclosure: Atmospheric Scientist; I can’t stand denialists but there’s no use trying to refute them with bad science either.)”

    Jethrock said
    “The North Pole is not a point… without getting into plate tectonics there is a phenomenon called “true polar wander” where the poles actually change.

    Also there were periods where there was no land at the North Pole… and it is believed that there were periods where there were no extensive ice caps… if any ice at all in the Triassic and Jurassic. When there is with no evidence of glaciation.

    I don’t even need to go back as far as a molten Earth.

    So if you are talking about a limited period of time or “recorded history”… that’s one thing… if you are talking “the history of the Earth” like your bet implies. Well… that’s a dumb bet.”

    There were 34 comments, most of them by you, trying to defend yourself aqainst DailyKos people who recognise our weasel words for what they were.

    For some odd reason, you cut of the comments to your post after only five hours. Why is that?


  7. Kevin,

    By the way, have you posted ClimateSanity’s refutation of you proof over at DailyKos yet?


    No, I don’t think you have. Why not?


  8. Kevin,

    One more thing. Those criticisms of your proof at DailyKos were without even having the benefit of ClimateSanity’s refutation.

    I wonder what they would have said after reading his refutation? It probably wouldn’t have been any better for you, would it?


  9. Dave,

    Are you following this at all? Tom has failed to rebut this proof.

    Not only that, but I’ve taken part of his “rebuttal” and offered him another bet that his rebuttal is flat out wrong as well. If you agree with Tom that it follows from my proof that:

    It can also be used to show that all points have had 100% ice coverage for all time.

    Then join in and put your money where your mouth is 🙂

    When all is said and done I’ll likely write another post at DailyKos, but as you know I don’t post there very often. I’ve written more here in a month than I’ve written there in 6 years.

    • Kevin,

      When you “likely write another post” will it include climatesanity’s refutation verbatim?
      If not, why?


      • I’d probably just link to it. That’s what hyperlinks are for.

        Besides, his first attempt at a refutation was either irrelevant or wrong — and he took up a lot of space in doing so.

        The geology/paleontology stuff is just irrelevant. The “bizarre consequences” are flat out wrong (and I’ve offered either of you to prove his assertion). The application of L’Hopital is incorrect since he took the area of a point, a 0-dimensional undefined quantity, and gave it a 2-dimensional value = 0.

        Now, if he miraculously comes up with a beautiful and elegant refutation I would post it in it’s entirety just to share the beauty of it. Unfortunately, I don’t foresee that happening.

  10. Tom Moriarty says:

    It can also be used to show that all points have had 100% ice coverage for all time.

    In other words:

    A. The North Pole is a point
    B. A location is ‘ice-covered’ if it is covered by > = 15% ice

    Every point on Earth has been 100% ice-covered at all times.

    I’ll start the meter running to see who has their proof up first … but I won’t hold my breath.

  11. Tom,

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

    It should be obvious that only the mathematical is necessary; my challenge was always consistent and obvious in its mathematical intent. There are only two items you need to answer.

    How does L’Hopital’s rule apply when the area of a point is not equal to zero, but undefined?.

    Proof for your assertion of “bizarre consequences” that follow from my proof. I do not believe you can prove your assertion. If you can, provide it.

    Anything else is just wasted verbiage.

    P.S. – Just for fun, you might explain how a line intersecting a sphere doesn’t result in a point 🙂

  12. Tom,

    If a point has area = 0, does it not follow that its length and width = 0?

    If a point has length and width = 0, does it not follow that a line (or line segment) has length = 0?

    If all lines and line segments have length = 0, isn’t there something “bizarre” happening?

    In other words, a point’s length, width, and area are undefined for a reason. To define them as equalling zero has truly “bizarre consequences”.

  13. As a newcomer to the website… this argument is idiocy.

    Tom, I think you unfortunately got entangled with a fool who is laughably trying to get out of his wager by arguing his “point”, but ends up looking like one of those geeky eighth graders who loves to appear smart to themselves, but have not a modicum of social smarts to realize they look like a fool to the rest of mankind.

    Even a fifth grader would not conceive of the “North Pole” as a mathematical point, smaller than the width of a neutron, but an “area” of variable geographical substantiality.

    I’ll move on to more relavent posts on this website now, and cry the loss of 15 minutes of my life reading Kevin’s drivel. (And I don’t even know either of you, so I have no dog in this fight but sane thinking).

  14. As a mathematician, I note that point 2, the bizarre consequences of the proof, is in fact a logical demonstration that the proof is invalid. At no point in the proof does any step rely upon either L being the north pole or Li being the ice area. Thus, if valid, the proof would demonstrate that no place P on earth has ever been X-free, for any proposition X. EG, the equator has never been ice-free; or the north pole ever lava-free. Resort to paleogeography and L’Hospital’s Rule is unnecessary. The argument is blatantly silly.

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