Mathematical fallacy

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Mathematical_fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof.

For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and are not applicable in the cases that are the exceptions to the rules.

The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry, the five colour theorem of graph theory). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.

Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.

Howlers

${\displaystyle \begin{array}{l}{\displaystyle \frac{d}{dx}}{\displaystyle \frac{1}{x}}\\ ={\displaystyle \frac{d}{d}}{\displaystyle \frac{1}{x^2}}\\ ={\displaystyle \frac{\cancel{d}}{\cancel{d}}}{\displaystyle \frac{1}{x^2}}\\ =-{\displaystyle \frac{1}{x^2}}\end{array}}$

Anomalous cancellation in calculus

Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a howler. The following is an example of a howler involving anomalous cancellation: $${\displaystyle \frac{16}{64}=\frac{1\cancel{6}}{\cancel{6}4}=\frac{1}{4}.}$$

Here, although the conclusion ⁠16/64⁠ = ⁠1/4⁠ is correct, there is a fallacious, invalid cancellation in the middle step. Another classical example of a howler is proving the Cayley–Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial with the matrix.

Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Edwin Maxwell. Outside the field of mathematics the term howler has various meanings, generally less specific.

Division by zero

The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number.

1. Let a and b be equal, nonzero quantities

   ${\displaystyle a=b}$

2. Multiply by a

   ${\displaystyle a^2=ab}$

3. Subtract b²

   ${\displaystyle a^2-b^2=ab-b^2}$

4. Factor both sides: the left factors as a difference of squares, the right is factored by extracting b from both terms

   ${\displaystyle (a-b)(a+b)=b(a-b)}$

5. Divide out (a − b)

   ${\displaystyle a+b=b}$

6. Use the fact that a = b

   ${\displaystyle b+b=b}$

7. Combine like terms on the left

   ${\displaystyle 2b=b}$

8. Divide by the non-zero b

   ${\displaystyle 2=1}$

Q.E.D.^[6]

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zero since a = b. Since division by zero is undefined, the argument is invalid.

Analysis

Mathematical analysis as the mathematical study of change and limits can lead to mathematical fallacies — if the properties of integrals and differentials are ignored. For instance, a naïve use of integration by parts can be used to give a false proof that 0 = 1. Letting u = ⁠1/log x⁠ and dv = ⁠dx/x⁠,

${\displaystyle \int \frac{1}{x\log x}dx=1+\int \frac{1}{x\log x}dx}$

after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits a and b.

${\displaystyle {\int }_a^b\frac{1}{x\log x}dx=1{|}_a^b+{\int }_a^b\frac{1}{x\log x}dx=0+{\int }_a^b\frac{1}{x\log x}dx={\int }_a^b\frac{1}{x\log x}dx}$

Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

Multivalued functions

Main article: Multivalued function

Many functions do not have a unique inverse. For instance, while squaring a number gives a unique value, there are two possible square roots of a positive number. The square root is multivalued. One value can be chosen by convention as the principal value; in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for nth roots.

Positive and negative roots

Care must be taken when taking the square root of both sides of an equality. Failing to do so results in a "proof" of 5 = 4.

Proof:

Start from

${\displaystyle -20=-20}$

Write this as

${\displaystyle 25-45=16-36}$

Rewrite as

${\displaystyle 5^2-5\times 9=4^2-4\times 9}$

Add ⁠81/4⁠ on both sides:

${\displaystyle 5^2-5\times 9+\frac{81}{4}=4^2-4\times 9+\frac{81}{4}}$

These are perfect squares:

${\displaystyle {\left(5-\frac{9}{2}\right)}^2={\left(4-\frac{9}{2}\right)}^2}$

Take the square root of both sides:

${\displaystyle 5-\frac{9}{2}=4-\frac{9}{2}}$

Add ⁠9/2⁠ on both sides:

${\displaystyle 5=4}$

Q.E.D.

The fallacy is in the second to last line, where the square root of both sides is taken: a² = b² only implies a = b if a and b have the same sign, which is not the case here. In this case, it implies that a = –b, so the equation should read

${\displaystyle 5-\frac{9}{2}=-\left(4-\frac{9}{2}\right)}$

which, by adding ⁠9/2⁠ on both sides, correctly reduces to 5 = 5.

Another example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity

${\displaystyle {\cos }^{2}x=1-{\sin }^{2}x}$

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

${\displaystyle \cos x=\sqrt{1-{\sin }^{2}x}}$

Evaluating this when x = π , we get that

${\displaystyle -1=\sqrt{1-0}}$

or

${\displaystyle -1=1}$

which is incorrect.

The error in each of these examples fundamentally lies in the fact that any equation of the form

${\displaystyle x^2=a^2}$

where ${\displaystyle a\ne 0}$, has two solutions:

${\displaystyle x=\pm a}$

and it is essential to check which of these solutions is relevant to the problem at hand. In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive. In particular, when x is set to π, the second equation is rendered invalid.

Square roots of negative numbers

Invalid proofs utilizing powers and roots are often of the following kind:

${\displaystyle 1=\sqrt{1}=\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}=i\cdot i=-1.}$

The fallacy is that the rule ${\displaystyle \sqrt{xy}=\sqrt{x}\sqrt{y}}$ is generally valid only if at least one of ${\displaystyle x}$ and ${\displaystyle y}$ is non-negative (when dealing with real numbers), which is not the case here.

Alternatively, imaginary roots are obfuscated in the following:

${\displaystyle i=\sqrt{-1}={\left(-1\right)}^{\frac{2}{4}}={\left({\left(-1\right)}^2\right)}^{\frac{1}{4}}=1^{\frac{1}{4}}=1}$

The error here lies in the incorrect usage of multiple-valued functions. ${\displaystyle (-1{)}^{\frac{1}{2}}}$ has two values ${\displaystyle i}$ and ${\displaystyle -i}$ without a prior choice of branch, while ${\displaystyle \sqrt{-1}}$ only denotes the principal value ${\displaystyle i}$.  Similarly, ${\displaystyle 1^{\frac{1}{4}}}$ has four different values ${\displaystyle 1}$, ${\displaystyle i}$, ${\displaystyle -1}$, and ${\displaystyle -i}$, of which only ${\displaystyle i}$ is equal to the left side of the first equality.

Complex exponents

When a number is raised to a complex power, the result is not uniquely defined (see Exponentiation § Failure of power and logarithm identities). If this property is not recognized, then errors such as the following can result:

${\displaystyle \begin{array}{rl}{e}^{2\pi i}& =1\\ {\left(e^{2\pi i}\right)}^i& =1^i\\ e^{-2\pi }& =1\end{array}}$

The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being ${\displaystyle \{e^{2\pi n}|n\in \mathbb{Z}\}.}$

Geometry

Many mathematical fallacies in geometry arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so.

In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.

Fallacy of the isosceles triangle

The fallacy of the isosceles triangle, from (Maxwell 1959, Chapter II, § 1), purports to show that every triangle is isosceles, meaning that two sides of the triangle are congruent. This fallacy was known to Lewis Carroll and may have been discovered by him. It was published in 1899.

Given a triangle △ABC, prove that AB = AC:

1. Draw a line bisecting ∠A.
2. Draw the perpendicular bisector of segment BC, which bisects BC at a point D.
3. Let these two lines meet at a point O.
4. Draw line OR perpendicular to AB, line OQ perpendicular to AC.
5. Draw lines OB and OC.
6. By AAS, △RAO ≅ △QAO (∠ORA = ∠OQA = 90°; ∠RAO = ∠QAO; AO = AO (common side)).
7. By RHS, △ROB ≅ △QOC (∠BRO = ∠CQO = 90°; BO = OC (hypotenuse); RO = OQ (leg)).
8. Thus, AR = AQ, RB = QC, and AB = AR + RB = AQ + QC = AC.

Q.E.D.

As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.

The error in the proof is the assumption in the diagram that the point O is inside the triangle. In fact, O always lies on the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide). Furthermore, it can be shown that, if AB is longer than AC, then R will lie within AB, while Q will lie outside of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts). Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.

Proof by induction

There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing that if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this, it can be shown to be true for all cases. The following "proof" shows that all horses are the same colour.

1. Let us say that any group of N horses is all of the same colour.
2. If we remove a horse from the group, we have a group of N − 1 horses of the same colour. If we add another horse, we have another group of N horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of N horses.
3. Thus we have constructed two groups of N horses all of the same colour, with N − 1 horses in common. Since these two groups have some horses in common, the two groups must be of the same colour as each other.
4. Therefore, combining all the horses used, we have a group of N + 1 horses of the same colour.
5. Thus if any N horses are all the same colour, any N + 1 horses are the same colour.
6. This is clearly true for N = 1 (i.e., one horse is a group where all the horses are the same colour). Thus, by induction, N horses are the same colour for any positive integer N, and so all horses are the same colour.

The fallacy in this proof arises in line 3. For N = 1, the two groups of horses have N − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of N + 1 = 2 horses is not necessarily all of the same colour. The implication "every N horses are of the same colour, then N + 1 horses are of the same colour" works for any N > 1, but fails to be true when N = 1. The basis case is correct, but the induction step has a fundamental flaw.

2 + 2 = 5

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/2_%2B_2_%3D_5#See_also

Two Plus Two Make Five (1895), by Alphonse Allais, is a collection of absurdist short stories about anti-intellectualism as politics.

2 + 2 = 5 or two plus two equals five is a mathematical falsehood which is used as an example of a simple logical error that is obvious to anyone familiar with basic arithmetic.

The phrase has been used in various contexts since 1728, and is best known from the 1949 dystopian novel Nineteen Eighty-Four by George Orwell.

As a theme and as a subject in the arts, the anti-intellectual slogan 2 + 2 = 5 pre-dates Orwell and has produced literature, such as Deux et deux font cinq (Two and Two Make Five), written in 1895 by Alphonse Allais, which is a collection of absurdist short stories; and the 1920 imagist art manifesto 2 × 2 = 5 by the poet Vadim Shershenevich.

Self-evident truth and self-evident falsehood

Main article: Self-evidence

In establishing the mundane reality of the self-evident truth of 2 + 2 = 4, in De Neutralibus et Mediis Libellus (1652) Johann Wigand said: "That twice two are four; a man may not lawfully make a doubt of it, because that manner of knowledge is grauen [graven] into mannes [man's] nature."

In the comedy-of-manners play Dom Juan, or The Feast with the Statue (1665), by Molière, the libertine protagonist, Dom Juan, is asked in what values he believes, and answers that he believes "two plus two equals four".

In the 18th century, the self-evident falsehood of 2 + 2 = 5 was attested in the Cyclopædia, or an Universal Dictionary of Arts and Sciences (1728), by Ephraim Chambers: "Thus, a Proposition would be absurd, that should affirm, that two and two make five; or that should deny 'em to make four." In 1779, Samuel Johnson likewise said that "You may have a reason why two and two should make five, but they will still make but four."

In the 19th century, in a personal letter to his future wife, Anabella Milbanke, Lord Byron said: "I know that two and two make four—& should be glad to prove it, too, if I could—though I must say if, by any sort of process, I could convert 2 & 2 into five, it would give me much greater pleasure."

In Gilbert and Sullivan's Princess Ida (1884), the Princess comments that "The narrow-minded pedant still believes/That two and two make four! Why, we can prove,/We women—household drudges as we are –/That two and two make five—or three—or seven;/Or five-and-twenty, if the case demands!"

Politics, literature, propaganda

France

In the political pamphlet "What is the Third Estate?" (1789), Emmanuel-Joseph Sieyès provided the humanist bases for the French Revolution (1789–1799).

In the late 18th century, in the pamphlet What is the Third Estate? (1789), about the legalistic denial of political rights to the common-folk majority of France, Emmanuel-Joseph Sieyès, said: "Consequently, if it be claimed that, under the French constitution, 200,000 individuals, out of 26 million citizens, constitute two-thirds of the common will, only one comment is possible: It is a claim that two and two make five."

Using the illogic of "two and two make five", Sieyès mocked the demagoguery of the Estates-General for assigning disproportionate voting power to the political minorities of France—the Clergy (First Estate) and the French nobility (Second Estate)—in relation to the Third Estate, the numeric and political majority of the citizens of France.

In the 19th century, in the novel Séraphîta (1834), about the nature of androgyny, Honoré de Balzac said:

Thus, you will never find, in all Nature, two identical objects; in the natural order, therefore, two and two can never make four, for, to attain that result, we must combine units that are exactly alike, and you know that it is impossible to find two leaves alike on the same tree, or two identical individuals in the same species of tree. That axiom of your numeration, false in visible nature, is false likewise in the invisible universe of your abstractions, where the same variety is found in your ideas, which are the objects of the visible world extended by their interrelations; indeed, the differences are more striking there than elsewhere.

In the pamphlet "Napoléon le Petit" (1852), about the limitations of the Second French Empire (1852–1870), such as majority political support for the monarchist coup d'Ḗtat, which installed Napoleon III (r. 1852–1870), and the French peoples' discarding from national politics the liberal values that informed the anti-monarchist Revolution, Victor Hugo said: "Now, get seven million, five hundred thousand votes to declare that two-and-two-make-five, that the straight line is the longest road, that the whole is less than its part; get it declared by eight millions, by ten millions, by a hundred millions of votes, you will not have advanced a step."

In The Plague (1947), French philosopher Albert Camus declared that times came in history when those who dared to say that 2 + 2 = 4 rather than 2 + 2 = 5 were put to death.

Russia

Soviet propaganda: The "Arithmetic of an Alternative Plan: 2 + 2 plus the Enthusiasm of the Workers = 5" exhorts the workers of the Soviet Union to realise five years of production in four years' time (Iakov Guminer, 1931).

In the late 19th century, the Russian press used the phrase 2 + 2 = 5 to describe the moral confusion of social decline at the turn of a century, because political violence characterised much of the ideological conflict among proponents of humanist democracy and defenders of tsarist autocracy in Russia. In The Reaction in Germany (1842), Mikhail Bakunin said that the political compromises of the French Positivists, at the start of the July Revolution (1830), confirmed their middle-of-the-road mediocrity: "The Left says, 2 times 2 are 4; the Right, 2 times 2 are 6; and the Juste-milieu says, 2 times 2 are 5".

In Notes from Underground (1864), by Feodor Dostoevsky, the anonymous protagonist accepts the falsehood of "two plus two equals five", and considers the implications (ontological and epistemological) of rejecting the truth of "two times two makes four", and proposed that the intellectualism of free will—Man's inherent capability to choose or to reject logic and illogic—is the cognitive ability that makes humanity human: "I admit that twice two makes four is an excellent thing, but, if we are to give everything its due, twice two makes five is sometimes a very charming thing, too."

In the literary vignette "Prayer" (1881), Ivan Turgenev said that: "Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: 'Great God, grant that twice two be not four'." In God and the State (1882), Bakunin dismissed deism: "Imagine a philosophical vinegar sauce of the most opposed systems, a mixture of Fathers of the Church, scholastic philosophers, Descartes and Pascal, Kant and Scottish psychologists, all this a superstructure on the divine and innate ideas of Plato, and covered up with a layer of Hegelian immanence, accompanied, of course, by an ignorance, as contemptuous as it is complete, of natural science, and proving, just as two times two make five, the existence of a personal god." Moreover, the slogan "two plus two equals five", is the title of the collection of absurdist short stories Deux et deux font cinq (Two and Two Make Five, 1895), by Alphonse Allais; and the title of the imagist art manifesto 2 x 2 = 5 (1920), by the poet Vadim Shershenevich.

In 1931, the artist Yakov Guminer [ru] supported Stalin's shortened production schedule for the economy of the Soviet Union with a propaganda poster that announced the "Arithmetic of an Alternative Plan: 2 + 2 plus the Enthusiasm of the Workers = 5" after Stalin's announcement, in 1930, that the first five-year plan (1928–1933) instead would be completed in 1932, in four years' time.

George Orwell

The anti-Nazi propagandist George Orwell, at the BBC during the Second World War (1939–1945)

See also: Doublespeak

In Orwell's Nineteen Eighty-Four, it appears as a possible statement of Ingsoc (English Socialism). The Party (i.e. a political party) slogan "War Is Peace, Freedom Is Slavery, Ignorance Is Strength" is a dogma which the Party expects the citizens of Oceania to accept as true. Writing in his secret diary in the year 1984, the protagonist Winston Smith ponders if the Inner Party might declare "two plus two equals five" as fact, as well as whether or not belief in such a consensus reality substantiates the lie. About the falsity of "two plus two equals five", in the Ministry of Love, the interrogator O'Brien tells the thought criminal Smith that control over physical reality is unimportant to the Party, provided the citizens of Oceania subordinate their real-world perceptions to the political will of the Party; and that, by way of doublethink: "Sometimes, Winston. [Sometimes it is four fingers.] Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once".

George Orwell used the idea of 2 + 2 = 5 in an essay of January 1939 in The Adelphi; "Review of Power: A New Social Analysis by Bertrand Russell":

It is quite possible that we are descending into an age in which two plus two will make five when the Leader says so.

In propaganda work for the BBC (British Broadcasting Corporation) during the Second World War (1939–1945), Orwell applied the illogic of 2 + 2 = 5 to counter the reality-denying psychology of Nazi propaganda, which he addressed in the essay "Looking Back on the Spanish War" (1943), indicating that:

Nazi theory, indeed, specifically denies that such a thing as "the truth" exists. There is, for instance, no such thing as "Science". There is only "German Science", "Jewish Science", etc. The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such and such an event, "It never happened"—well, it never happened. If he says that "two and two are five"—well, two and two are five. This prospect frightens me much more than bombs—and, after our experiences of the last few years [the Blitz, 1940–41] that is not a frivolous statement.

In addressing Nazi anti-intellectualism, Orwell's reference might have been Hermann Göring's hyperbolic praise of Adolf Hitler: "If the Führer wants it, two and two makes five!" In the political novel Nineteen Eighty-Four (1949), concerning the Party's philosophy of government for Oceania, Orwell said:

In the end, the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?

The 1951 British edition of the text, published by Secker & Warburg, erroneously omitted the "5", thus rendering it simply as "2 + 2 =". This error, likely the result of a typesetting mistake, remained in all further editions of the text until the 1987 edition, whereafter a correction was made based on Orwell's original typescript. This misprint did not exist in the American editions of the text, with British students of the text in the meanwhile misinterpreting Orwell's original intentions.

Contemporary usage

Political graffiti in Havana, Cuba questioning government policy perceived to be "2+2=5"

In The Cult of the Amateur: How Today's Internet is Killing Our Culture (2007), the media critic Andrew Keen uses the slogan "two plus two equals five" to criticise the Wikipedia policy allowing any user to edit the encyclopedia — that the enthusiasm of the amateur for user generated content, peer production, and Web 2.0 technology leads to an encyclopedia of common knowledge, and not an encyclopedia of expert knowledge; that the "wisdom of the crowd" will distort what society considers to be the truth.

In 2020, a social media debate followed biostatistics PhD student Kareem Carr's statement, "If someone says 2+2=5, the correct response is, 'What are your definitions and axioms?' not a rant about the decline of Western civilization".

Room-temperature superconductor

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Room-temperature_superconductor

A room-temperature superconductor is a hypothetical material capable of displaying superconductivity above 0 °C (273 K; 32 °F), operating temperatures which are commonly encountered in everyday settings. As of 2023, the material with the highest accepted superconducting temperature was highly pressurized lanthanum decahydride, whose transition temperature is approximately 250 K (−23 °C) at 200 GPa.

At standard atmospheric pressure, cuprates currently hold the temperature record, manifesting superconductivity at temperatures as high as 138 K (−135 °C). Over time, researchers have consistently encountered superconductivity at temperatures previously considered unexpected or impossible, challenging the notion that achieving superconductivity at room temperature was infeasible. The concept of "near-room temperature" transient effects has been a subject of discussion since the early 1950s.

Reports

Since the discovery of high-temperature superconductors ("high" being temperatures above 77 K (−196.2 °C; −321.1 °F), the boiling point of liquid nitrogen), several materials have been claimed, although not confirmed, to be room-temperature superconductors.

Corroborated studies

In 2014, an article published in Nature suggested that some materials, notably YBCO (yttrium barium copper oxide), could be made to briefly superconduct at room temperature using infrared laser pulses.

In 2015, an article published in Nature by researchers of the Otto Hahn Institute suggested that under certain conditions such as extreme pressure H
₂S transitioned to a superconductive form H
₃S at 150 GPa (around 1.5 million times atmospheric pressure) in a diamond anvil cell. The critical temperature is 203 K (−70 °C) which would be the highest T_c ever recorded and their research suggests that other hydrogen compounds could superconduct at up to 260 K (−13 °C).

Also in 2018, researchers noted a possible superconducting phase at 260 K (−13 °C) in lanthanum decahydride (LaH
₁₀) at elevated (200 GPa) pressure. In 2019, the material with the highest accepted superconducting temperature was highly pressurized lanthanum decahydride, whose transition temperature is approximately 250 K (−23 °C).

Though not room temperature, a rare earth 'infinite layer' nickelate was recently discovered that superconducted at the unheard of (for nickelates) temperature of 44K at ambient pressure. This material is stable in air unlike cuprates, and other nickelates may have even higher critical temperatures. The current theory is that these materials leverage very unusual physics including pair density waves (PDW) that may not be as sensitive to the normal pitfalls of high temperature superconductors like low critical current.

Uncorroborated studies

In 1993 and 1997, Michel Laguës and his team published evidence of room temperature superconductivity observed on Molecular Beam Epitaxy (MBE) deposited ultrathin nanostructures of bismuth strontium calcium copper oxide (BSCCO, pronounced bisko, Bi₂Sr₂Ca_n−1Cu_nO_2n+4+x). These compounds exhibit extremely low resistivities orders of magnitude below that of copper, strongly non-linear I(V) characteristics and hysteretic I(V) behavior.

In 2000, while extracting electrons from diamond during ion implantation work, South African physicist Johan Prins claimed to have observed a phenomenon that he explained as room-temperature superconductivity within a phase formed on the surface of oxygen-doped type IIa diamonds in a 10⁻⁶ mbar vacuum.

In 2003, a group of researchers published results on high-temperature superconductivity in palladium hydride (PdH_x: x > 1) and an explanation in 2004. In 2007, the same group published results suggesting a superconducting transition temperature of 260 K, with transition temperature increasing as the density of hydrogen inside the palladium lattice increases. This has not been corroborated by other groups.

In March 2021, an announcement reported superconductivity in a layered yttrium-palladium-hydron material at 262 K and a pressure of 187 GPa. Palladium may act as a hydrogen migration catalyst in the material.

On 31 December 2023, "Global Room-Temperature Superconductivity in Graphite" was published in the journal Advanced Quantum Technologies, claiming to demonstrate superconductivity at room temperature and ambient pressure in highly oriented pyrolytic graphite with dense arrays of nearly parallel line defects.

Retracted or unreliable studies

A magnet levitating above a superconductor (at −200 °C) that is exhibiting the Meissner effect.

In 2012, an Advanced Materials article claimed superconducting behavior of graphite powder after treatment with pure water at temperatures as high as 300 K and above. So far, the authors have not been able to demonstrate the occurrence of a clear Meissner phase and the vanishing of the material's resistance.

In 2018, Dev Kumar Thapa and Anshu Pandey from the Solid State and Structural Chemistry Unit of the Indian Institute of Science, Bangalore claimed the observation of superconductivity at ambient pressure and room temperature in films and pellets of a nanostructured material that is composed of silver particles embedded in a gold matrix. Due to similar noise patterns of supposedly independent plots and the publication's lack of peer review, the results have been called into question. Although the researchers repeated their findings in a later paper in 2019, this claim is yet to be verified and confirmed.

Since 2016, a team led by Ranga P. Dias has produced a number of retracted or challenged papers in this field. In 2016 they claimed observation of solid metallic hydrogen in 2016. In October 2020, they reported room-temperature superconductivity at 288 K (at 15 °C) in a carbonaceous sulfur hydride at 267 GPa, triggered into crystallisation via green laser. This was retracted in 2022 after flaws in their statistical methods were identified and led to questioning of other data. In 2023 he reported superconductivity at 294 K and 1 GPa in nitrogen-doped lutetium hydride, in a paper widely met with skepticism about its methods and data. Later in 2023 he was found to have plagiarized parts of his dissertation from someone else's thesis, and to have fabricated data in a paper on manganese disulfide, which was retracted. The lutetium hydride paper was also retracted. The first attempts to replicate those results failed.

On July 23, 2023, a Korean team claimed that Cu-doped lead apatite, which they named LK-99, was superconducting up to 370 K, though they had not observed this fully. They posted two preprints to arXiv, published a paper in a journal, and submitted a patent application. The reported observations were received with skepticism by experts due to the lack of clear signatures of superconductivity. The story was widely discussed on social media, leading to a large number of attempted replications, none of which had more than qualified success. By mid-August, a series of papers from major labs provided significant evidence that LK-99 was not a superconductor, finding resistivity much higher than copper, and explaining observed effects such as magnetic response and resistance drops in terms of impurities and ferromagnetism in the material.

Theories

Metallic hydrogen and phonon-mediated pairing

Theoretical work by British physicist Neil Ashcroft predicted that solid metallic hydrogen at extremely high pressure (~500 GPa) should become superconducting at approximately room temperature, due to its extremely high speed of sound and expected strong coupling between the conduction electrons and the lattice-vibration phonons.

A team at Harvard University has claimed to make metallic hydrogen and reports a pressure of 495 GPa. Though the exact critical temperature has not yet been determined, weak signs of a possible Meissner effect and changes in magnetic susceptibility at 250 K may have appeared in early magnetometer tests on an original now-lost sample. A French team is working with doughnut shapes rather than planar at the diamond culette tips.

Organic polymers and exciton-mediated pairing

In 1964, William A. Little proposed the possibility of high-temperature superconductivity in organic polymers.

Other hydrides

In 2004, Ashcroft returned to his idea and suggested that hydrogen-rich compounds can become metallic and superconducting at lower pressures than hydrogen. More specifically, he proposed a novel way to pre-compress hydrogen chemically by examining IVa hydrides.

In 2014–2015, conventional superconductivity was observed in a sulfur hydride system (H
₂S or H
₃S) at 190 K to 203 K at pressures of up to 200 GPa.

In 2016, research suggested a link between palladium hydride containing small impurities of sulfur nanoparticles as a plausible explanation for the anomalous transient resistance drops seen during some experiments, and hydrogen absorption by cuprates was suggested in light of the 2015 results in H
₂S as a plausible explanation for transient resistance drops or "USO" noticed in the 1990s by Chu et al. during research after the discovery of YBCO.

It has been predicted that ScH
₁₂ (scandium dodecahydride) would exhibit superconductivity at room temperature – T_c between 333 K (60 °C) and 398 K (125 °C) – under a pressure expected not to exceed 100 GPa.

Some research efforts are currently moving towards ternary superhydrides, where it has been predicted that Li
₂MgH
₁₆ (dilithium magnesium hexadecahydride) would have a T_c of 473 K (200 °C) at 250 GPa.

Spin coupling

It is also possible that if the bipolaron explanation is correct, a normally semiconducting material can transition under some conditions into a superconductor if a critical level of alternating spin coupling in a single plane within the lattice is exceeded; this may have been documented in very early experiments from 1986. The best analogy here would be anisotropic magnetoresistance, but in this case the outcome is a drop to zero rather than a decrease within a very narrow temperature range for the compounds tested similar to "re-entrant superconductivity".

In 2018, support was found for electrons having anomalous 3/2 spin states in YPtBi. Though YPtBi is a relatively low temperature superconductor, this does suggest another approach to creating superconductors.

"Quantum bipolarons" could describe how a material might superconduct at up to nearly room temperature.