Sometimes, things just don’t add up. Take this retraction notice, from the March 2011 issue of Applied Mathematics Letters:
This article has been retracted at the request of the editor as the authors have falsified mathematical findings and have made unsubstantiated claims regarding Euclid’s parallel postulate (Appl. Math. Lett. 23 (2010) 1137–1139. doi:10.1016/j.aml.2010.05.003). This article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and apologies are offered to readers of the journal that this was not detected during the submission process.
There’s actually only one author, an M. Sivasubramanian, of Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamil Nadu, India, but we’re not PhDs in math, so we figured we were missing something important. Fortunately, Ben Steinberg, a high school friend of one of ours — Ivan’s — is a bit of a math rock star and a professor at Carleton University. We asked him for his take:
I am not sure which is more amusing, the article or the retraction. I cannot understand how this could possibly be published anywhere. It seems like a practical joke to test whether articles are actually refereed. Not a single statement in the author’s “proof” makes sense.
First, he draws a sphere as a circle so he can talk about a perimeter for dropping perpendiculars to opposite sides, when of course on a sphere this makes no sense. He also clearly has no idea what is meant by the geodesic line containing two points in spherical geometry since his lines are not great circles (see the Wikipedia entry).
In fact, he may have hyperbolic geometry and spherical geometry confused. In hyperbolic geometry, one draws a circle and geodesic lines hit the perimeter at right angles.
We admit, we often confuse those geometries ourselves. You should see us trying to draw circles at Retraction Watch HQ. But we digress. It gets better:
And then he says Einstein proved some result about shining light using spherical geometry and the proof applies here, although he in no way explains why Einstein’s light source theorem has any relevance (nor does he cite the Einstein result to which he is referring).
Einstein? What does he know about math? Steinberg continues, thankfully turning to a simile in medicine that we can actually grasp:
Basically the paper is pure rubbish and it is hard to believe it is not a hoax. It’s like I am writing to say that some medicine that everybody knows works well on humans does not work on people at all because I can show it doesn’t work on pigs using experiments of Jonas Salk when he studied some completely different medicine on rats. It is just so incoherent that it is funny.
And then the discussion at the end with its…spirituality is too much.
Oh, did we forget to mention the spirituality? This is the paper’s conclusion:
Gödel’s incompleteness theorems put an upper limit to scientists in knowing the ultimate reality of the nature. The theorems express and explain that mathematics cannot solve everything. The problems will remain and remain for ever. Taking this logical fact into account both possible and impossible dominate mathematics. The one side of the coin states that a particular statement is valid but the other side demands and deduces that the same statement is invalid. This is a peculiar truth. Both science and spirituality came from space. Science is based on equations and experiments whereas spirituality relies on beliefs. The spirituality promises that everything in this universe was created in pairs but in opposites. For example, origin and end, man and women, light and dark, day and night, sorrow and pleasure, loss and gain, God and devil, ugly and beauty, good and bad and so on. Similarly, possible and impossible are consistent in mathematics.
That certainly isn’t anything like the math textbooks we remember. But we have learned that science and spirituality both came from space.
How can a serious mathematics journal accept such a paper? Also he doesn’t really understand what Godel’s theorem says about the consistency of mathematics either. It looks like he has an axe to grind with math.
Anyway, the paper should not be retracted for falsifying mathematical results and unsubstantiated claims about Euclid’s postulates. It should be retracted for unsubstantiated claims about spherical geometry that are “proved” via completely vacuous arguments that show the author’s lack of understanding of non-Euclidean geometry.
We think Ben has said it all. Neither the paper’s author, nor the journal’s editor, have responded to our requests for comment. We’ll update if we hear anything that, um, adds to the discussion.
Please see an update with news of another retraction in Applied Mathematical Letters, this one of a paper questioning the second law of thermodynamics.