“”...what you've just said is one of the most insanely idiotic things I have ever heard. At no point in your rambling, incoherent response were you even close to anything that could be considered a rational thought. Everyone in this room is now dumber for having listened to it. I award you no points, and may God have mercy on your soul.
|—James Downey, Billy Madison|
Not even wrong refers to any statement, argument or explanation that can be neither correct nor incorrect, because it fails to meet the criteria by which correctness and incorrectness are determined.
The phrase implies that not only is someone not making a valid point in a discussion, but they don't even understand the nature of the discussion itself, or the things that need to be understood in order to participate.
The phrase apparently originates with physicist Wolfgang Pauli, who used the phrase (in the form "Das ist nicht nur nicht richtig, es ist nicht einmal falsch!" -- "That is not only not right, it is not even wrong!") to describe an unclear research paper. Pauli was known for his detestation of sloppy writing and arguments, and for his somewhat "colourful" objections to such things. The term has since gathered some popularity, amongst those involved in refuting pseudoscience, to reference the difficulties faced in dealing with some of the more out-of-this-world arguments. Examples include so-called creationist escape hatches, which are a series of irrefutable (but equally unprovable) claims that defy correction with conventional logic. It also applies to science stoppers.
A correct argument or explanation is easy to spot; it may look like this:
“”2 + 2 = 4
A wrong argument has an incorrect conclusion, but is presented in such a way that we can evaluate it, like so:
“”2 + 2 = 6
The above two examples can be shown to be right and wrong; they at least make enough sense for us to spot where the error is. Something that is not even wrong is usually so far out of the ballpark or so far from reality that it is, quite simply, flabbergastingly irrelevant. For example:
“”2 + zebra ÷ glockenspiel = homeopathy works!
This is far more than just an argument leading to a wrong conclusion. The premises aren't even related to the conclusion or are themselves completely nonsensical. In a way, a "not even wrong" argument is often an extreme non sequitur - such as the Homeopaths claiming that (purported and unverified) observations of neutrinos breaking the speed of light meant that all science was wrong and therefore homeopathy works. The premises, their arrangements, the conclusion, all are so divorced from facts and logic that even attempting to rationally engage with it gives it too much credit.
Most commonly, not even wrong arguments arise because of a lack of understanding in the person making the argument. For instance, a conspiracy theorist talking about people slipping "ionised radio waves" into someone's coffee - no one who knows anything on the subject could even think of a way that only one or two small misconceptions could lead to that, it really is not even wrong. Either through willful ignorance, or at best by being out of date with current research, they don't know enough about the subject to know what is needed to form a sensible argument. This can be caused by the Dunning-Kruger effect, where someone makes a not even wrong argument but lacks the meta-cognitive ability to recognise that they don't know enough even to make a wrong argument, never mind a right one.
"Not even wrong" applies especially when the premises of an argument are known to be false (based on observational evidence or similar), or when they are used to describe theories which cannot possibly be falsifiable or provide meaningful predictions. For example, any physical theory based on the existence of the aether, and any biological idea based on evolution by Lamarckian inheritance would be classed as not even wrong. In this case, it is a type of informal logical fallacy (an error in the content of an argument) as using incorrect or non-applicable premises will always yield an incorrect answer. As the premises are wrong the conclusion is certainly incorrect, but the conclusion is at least correct based on said premises. The term "not even wrong" therefore describes this situation. The less strict usage of the fallacy to describe people who have no idea what they're talking about on any known level - whether it be Dewey Larson's model of the atom or Gene Ray's cubic time - is usually the manifestation of this in practice.
For a clear but silly example, one can say that "this chair is made of hard, solid wood, therefore I can sit on it". This is perfectly true, except in the case where the chair is made of jelly. The conclusion is correct based on the premise, but the premise is clearly not applicable. Therefore the conclusion is not even wrong.
“”It's as if someone announced his revolutionary discovery that P=NP implies N=1, and then critics soberly replied that, no, the equation P=NP can also be solved by P=0.
The webcomic xkcd parodied the idea by suggesting that someone thought they had managed to disprove special relativity using a "racecar on a train" thought experiment. The blog Science and Math Defeated seems to be the real personification of what this comic strip is about; however, no one is 100% sure if it is real or not.
Examples in real life often involve "skeptics" arguing with established scientists. In climategate, for example, where a thousand e-mails from leading climate change researchers were hacked and released publicly, most critics seemed to not understand the basic meaning behind some of the emails (climate science and atmospheric chemistry are complex disciplines), instead preferring to quote mine a few emails out of context and misunderstanding the use of the word "trick". Climate science is one of the areas where "not even wrong" arguments are common, owing to the complexity of the system under study.
A further example was in the Lenski affair, where conservative website owner Andrew Schlafly demanded that Professor Richard Lenski "release the data" on his long term evolution experiment. This stemmed from Schlafly's misinterpretation of the research paper in question (if indeed he actually read it – as Lenski himself observed, Schlafly almost certainly hadn't at the time of the e-mail exchange). The paper did in fact mention all the relevant data one would need to appraise the experiment. Schlafly's further criticisms regarding the statistics reported in the paper were certainly in the field of "not even wrong," owing to his complete misunderstanding of basic practice when applying statistics as well as ignorance of the computer programs used - confusing the professional software package Statistics 101 with an amateur website.
(And, for those who aren't familiar with the context of the quote above, P and N are not arithmetical constants in a multiplication equation, but rather P and NP are the full names for two very important sets in computational theory. ĤΨ = EΨ is another common equation to suffer from this for people viewing it for the first time.)
 Wronger than wrong
There is one step further than not even wrong and that is wronger than wrong. The phrase was coined by Isaac Asimov in The Relativity of Wrong, (expanded and popularised by Michael Shermer, who called it "Asimov's Axiom".) Wronger than wrong describes any idea that equates errors that clearly aren't equal. The example originally given is that a belief in a flat earth is wrong, but a belief in a spherical earth is also wrong (as it's actually a distorted spheroid) - however, saying that a spherical and flat earth are equally wrong, is more wrong than both those errors combined. Blurring concepts into the same category of "wrong" or "improbable", despite their obvious difference in the magnitude of how "wrong" and "improbable" they are, is an example of the continuum fallacy (the "fallacy of gray").
This phrase has important implications regarding the nature of scientific theories and aptly describes how the scientific method builds up knowledge and understanding - theories may change and adapt, but calling them outright wrong is not necessarily the right way to go about it. One reviewer of Asimov's The Relativity of Wrong asserted that such thinking was a great tool for "arming oneself against the inevitable anti-science attack that one often hears — [that] theories are always preliminary and science really doesn't 'know' anything".