A mathematical joke asks, “What do you get when you cross a mountain-climber with a mosquito?” Answer: “Nothing: you can’t cross a scalar with a vector.”
Non-mathematically minded readers may not get the ‘joke’ until later in this article. But when you do, you may feel it’s the most expensive ‘joke’ told, and it’s being played on you and me.
If you’ve ever followed the heated debate about man-made global warming you will know the cornerstone of that science is the so-called “greenhouse gas effect” (GHE). It is purported that rising human emissions of carbon dioxide (CO2), one of those so-called GHE gases, is dangerously adding to climate change. The chosen remedy of western governments: we must all pay more taxes, cut back our industrial emissions and invest in various questionable alternative energy schemes to avert a planetary crisis.
To this end many a (government) climatologist or Greenpeace activist will regale you with the glib assertion that the GHE makes our planet “33 degrees warmer than it would otherwise be.” But where does this “33 degrees” number come from and is it scientifically valid? Contrary to media hype this number is not “an observation” it is the product of a 30-year-old calculation from a team of researchers led by NASA’s Dr. James E. Hansen. It is a ubiquitous claim that the number “proves” the GHE is real. [1,2]
Putting the Numbers into Context
This thermometer illustration depicts the numbers in degrees Celsius and Fahrenheit. But it is easier to follow this article’s analysis if we instead apply the Kelvin temperatures (in purple) The lower value (minus 18°C) thus becomes 255°K and this is what Hansen raised by 33 degrees to 288°K (15°C).
Now you may be thinking as you let out a yawn,“Oh boy, he’s about to spout math.” But before your finger presses ‘delete’ consider this: what you are about to read has enormous ramifications for our industrialized society and serious implications for you and your family’s personal tax liabilities for decades to come. If it can be demonstrated that Hansen’s “33 degrees” is the product of a bungled calculation then, at minimum, this puts the onus back on climatologists to explain the errors and re-think their “theory” before our economies are exposed to deeper economic travails.
Two Different Number Concepts: One Bungled Meaning
Our story begins in 1981 when Dr. Hansen led a team of researchers who wanted to pin down some simple and iconic numbers. Their quest was to prove to the wider scientific community that carbon dioxide and certain other very efficient infrared absorbing (and emitting) gases make Earth’s atmosphere warmer than it would otherwise be. Lamentably, these scientists chose to ignore the superb energy emitting qualities of these gases.
No rigorous scientific testing was involved – the numbers were obtained from known values. Firstly, Hansen’s team took a measure of average temperatures at the ground (a scalar) and, secondly, they chose a temperature for infrared radiation as it passes out of the top of the atmosphere, (a vector). Both these two numbers are reasonable in themselves. However, in both mathematics and physics vectors and scalars each describe quantities and each is very distinct from the other being differently obtained and proving separate values. A scalar operates in one dimension, a vector in three dimensions.
Now this is where the ‘joke’ comes into play and we need to remember the old saying: “You can’t mix apples with oranges.” Hansen’s team took the 288°K scalar number (the ‘apple’) with a one-dimensional basis and put it alongside the 255°K vector number (the ‘orange’), the product of a flow field in three dimensions.“So what?” you may say. Well, remember what was stated at the top of this article? Hansen had *forgotten* that “you can’t cross a scalar with a vector.” Again, please remember this is an axiomatic principle from Physics 101: “vector and scalar quantities cannot be added together.”