# Do mathematical objects disappear?

I am asking this question starting from two orders of considerations.

Firstly, we can witness, considering the historical development of several sciences, that certain physical entities "disappeared": it is the case of luminiferous aether with the surge of Einstein's relativity, or the case of the celestial spheres that disappeared in the passage from the pre-copernica universe to the Copernican universe.

Secondly, if we consider the evolution of mathematics, we assist to quite an opposite phenomenon: new mathematical objects are often invented and enrich already existing ontologies (let us consider the growth of the number system with negative, imaginary, etc., or the surge of non-euclidean geometries).

But can anyone think of examples of mathematical objects that have, on the contrary, disappeared (have been abandoned or dismissed by mathematicians)?

• Could you please define "mathematical objects"? Could you please define "disappear"? By "disappear" do you mean "fall into disuse, to the point that living mathematicians practically forget about the existence of this part of mathematics"? That certainly happens. Dec 22 '15 at 14:22
• They may not disappear, but functions have been known to vanish.... Dec 22 '15 at 14:50
• Thank you for your precisions. I would say that I agree to define "disappear" as "fall into disuse, to the point that living mathematicians practically forget about the existence of this part of mathematics". Dec 22 '15 at 14:57
• In the days before computers, there were algorithms aimed at performing computations by hand with the aid of numerical tables. I remember learning about mantissas and surds in school. Apart from historical interest, those algorithms have fallen into disuse. Of course numerical analysis is essential for computation, but that is different than algorithms for computation by hand. Dec 22 '15 at 15:22
• This would be better for hsm.stackexchange.com . Dec 22 '15 at 16:22

I certainly can't think of examples similar to your physics examples of concepts that were just wrong so effectively became extinct.

Two extremes which are present in mathematics are

1) Things which become too simple to have their name retain prominence as commonly known terminology.

For example: For Aristotle, Square numbers and Oblong numbers were perhaps similarly important. Today the first concept is vibrant while second concept is fairly unfamiliar. Similarly the concept of singly and doubly even integers is fairly unfamiliar being subsumed as the case $p=2$ of the $p$-adic order of an integer (or rational number.)

AND

2) Things which are too complex for the tools of their time and so hibernate for a while

As an example, I feel compelled to quote the stirring first paragraph of The Invariant Theory of Binary Forms

Like the Arabian phoenix rising out of its ashes, the theory of invariants, pronounced dead at the turn of the century, is once again at the forefront of mathematics. During its long eclipse, the language of modern algebra was developed, a sharp tool now at long last being applied to the very purpose for which it was intended. More recently, the artillery of combinatorics began to be aimed at the problems of invariant theory bequeathed to us by the nineteenth century, abandoned at the time because of insufficient combinatorial thrust.

• Can we begin a list of mixed metaphors in mathematical writing? "More recently, the artillery of combinatorics began to be aimed at the problems of invariant theory bequeathed to us by the nineteenth century ..." This is just what I hope for those things bequeathed to me: aim artillery at them! Dec 22 '15 at 15:59
• By the way, I love Rota. I run a math writing seminar, and I always read through his fabulous advice during the seminar. But that sentence made me shiver. Dec 22 '15 at 16:00
• So-called oblong numbers are just twice the triangle numbers, which was studies by at least Gauss (NUM=$\Delta+\Delta+\Delta$). Dec 22 '15 at 16:12
• One certainly hears of doubly even codes. I also like evenly even and oddly even. But is oddly oddly even 8n+2 or 8n+6? Dec 22 '15 at 19:15
• @JasonStarr : Some people like the gun targets they've inherited. Dec 24 '15 at 3:33

I never see references to the haversine (used in spherical trigonometry, itself an obsolescent, if not obsolete, subject). Also, versine and coversine (and their variants) are obscure (and unnecessary) trigonometric functions, and have been for at least fifty years.

• Perhaps the exsecant and excosecant also belong here. Dec 26 '15 at 21:44
• You could also include prosthaphaeresis, replaced by logarithms. Jul 26 '17 at 4:55

It's certainly possible for mathematical terminology to disappear. For instance, we used to have specialized words for different powers of numbers. We still use square and cube in this way, but for higher powers we would now say "[number]th power" rather than say the zenzizenzizenzic for the eighth power.

• Roman numerals have also been in disuse lately... Dec 23 '15 at 0:09
• In disuse in mathematics, yes. In everyday life, not so much. Dec 23 '15 at 2:08
• I saw a talk on Painlevé VI only a month ago. Dec 23 '15 at 13:54
• @DavidEppstein: Well, at least the NFL chickened out on "Super Bowl L." Dec 26 '15 at 23:18

I believe that the closest analogues to your physical examples arise when mathematical texts are, at the time they are written, regarded as being precise, but are later regarded as being insufficiently precise or rigorous, so that the nouns in the older text are seen as requiring reformulation in "modern, rigorous terms" before they are accepted as directly and unambiguously referring to some mathematical entity.

For example, near the beginning of Euclid's Elements we read, "A surface is that which has length and breadth only." Today, many mathematicians would instinctively react, "Say what?" Before they can make sense of Euclid's definition of a surface, they need to reformulate it in modern language. Now, once they do so, they will deny that Euclid's surfaces have "disappeared," because according to the standard platonistic philosophy that forms the basis of the working mathematician's conception of the world, the universe of mathematical objects is an eternally existing thing, and it is impermissible to declare that objects in that universe cease to exist. Rather, we say instead that Euclid used language in an old-fashioned manner, and that we need to translate his text into modern language before we can reliably determine the referents of his nouns. Once the referent of the noun is established, then it lives forever and cannot disappear; the only things that can "disappear" are ways of speaking that are later judged to be not clear enough to unambiguously describe eternally existing realities in the mathematical universe.

If, however, one takes a less platonistic view of mathematics, then one might simply say that Euclid's surfaces (and other concepts) in their original form have disappeared, but that the essential mathematical content of the Euclidean theory has been preserved. In fact, if the modern reformulation is done carefully, much of the older text can be preserved verbatim. There will, however, be some instances where the older terminology and sentences lack a one-to-one correspondence with the modern version; older concepts that are judged to be superfluous or vague are just dropped. In this sense some things "disappear," and it's not just that certain words have fallen into disuse, but the concepts that they were supposed to refer to are deemed insufficiently precise or rigorous.

As others have mentioned, infinitesimals are another example of this kind of "disappearance." In mathematics proper (as opposed to science and engineering), infinitesimals as they were conceived in classical calculus no longer exist in mathematics, in the sense that the concept that people used to have is no longer considered to be rigorous or precise enough. Obviously, the essential mathematical content has been preserved in modern treatments (whether "standard" or "nonstandard" analysis), and some theorems can be preserved verbatim when the words are suitably redefined. But there has been a definite shift in the underlying concepts, even though "no matter has been created or destroyed" in the platonistic universe of all mathematical objects.

A similar story can be told in other cases where there has been a change in "rigor." Algebraic geometry may be another example where so-called classical Italian algebraic geometry was later regarded by some as lacking in rigor.

• Following up on Tim's example of infinitesimals, I would opine that "horn angles" as such have ceased to play a role in modern mathematics. But if I'm wrong I hope someone will correct me! Dec 23 '15 at 14:46
• Timothy, to add to your fine answer I would mention that "infinitesimals as they were conceived in classical calculus no longer exist in mathematics" only in the sense that our set-theoretic ontology since about 1870 is a definite break with the mathematics that preceded that date. As far as the procedures (as opposed to ontology) of classical infinitesimal analysis are concerned, they are very much alive today. There are several studies that explore this line of investigation. Dec 23 '15 at 17:54
• I once gave a lecture on linear algebra. In a section about some elementary geometry, I read to them a translation of Euclid's axiom. This was the only point where the entirely audience was silent and listening. When I told them a bit later that we had to interpret these axioms in modern language, they were rather disappointed (with the interpretation, not with Euclid). Dec 23 '15 at 19:35

Hopefully at some point, GRH will be proved, and then the Siegel zeros of Zeta functions will disappear. There are currently thousands of papers on them.

In Mathematics everyone believed that Hilbert's second problem had an affirmative answer but it was (later) shown that this is not entirely true and I suppose that the consistency of arithmetic has now been "disappeared"

• Good example, although I would say that what has disappeared is the assumption that the basic axioms for mathematics can be proved to be consistent. Dec 24 '15 at 16:03

Before trying to make parallels, you must understand how these other concepts "disappeared".

The point of science is to create theoretical models which allow us to do some reasoning. The goal of natural sciences, such as physics and biology, is to explain how entities in the world around us function. A physics model is only useful when it either

• does not contradict any observations
• contradicts observations in known ways and is usable for answering questions on a desired level of detail. For example, a gravitational model of the Milky Way can assume that each planet is a sphere, while it is known that planets are flattened spheres.

Such models are considered "right" in physics, while the ones which contradict observations are considered "wrong".

Unlike physics, mathematics does not have the primary goal of explaining natural phenomena. While its theoretical models can be applied to entities found in nature, for example arithmetic, they are considered interesting for their own sake even when they don't describe anything we have observed, see for example the Banach Tarski theorem.

Thus, the mechanism which makes some theories in natural sciences wrong (ether in physics, Lamarckian inheritance in biology) does not apply to mathematics. It is impossible for postulated mathematical entities to disappear the same way, because in mathematics, an entity is already a part of a useful model when it is described, not only when it has been found to correspond to an entity in reality.

As others have noted, it is possible for mathematical entities to fall out of fashion. But while they are seen as old and unnecessary baroque, they are not seen as a direct misconception, the way it is with the disappeared physics entities.

From A. A. Ivanov (The Monster Group and Majorana Involutions):

[$\dots$] John Conway suggested calling the extensions of $^2E_6(2)$ the Baby Monster, the double extension the Middle Monster, and the triple extension the Super Monster. Fischer was not in favour of this terminology being used externally. When it was shown that $^2E_6(2)$ can only be extended twice and therefore that the Super Monster does not exist, the prefix Middle was dropped and the name Monster, as we know it, emerged. [$\dots$]

Consequently, the Super Monster is an example of a mathematical object that was named but later disappeared in a puff of nonexistence.

• It sounds like the Super Monster just became the Monster and it was the Middle Monster that disappeared... Sep 5 '17 at 21:19

Screws and Screw Theory.

I own a copy of the book A Treatise on the Theory of Screws by Robert Ball published around 1900. The algebra of screws, and the related twists and wrenches seem to have played an important role in the study of rigid body dynamics at the time. But that terminology seems to play very little role in any writing on that subject that I've seen more recently. Except in around 2000 the terminology picked up again because of applications to robotics and to rigid body simulation (that I learnt about via simulations in the video games and visual effects industry).

Mathematical objects sometimes disappear when proven either non-existing or isomorphic to more well-known objects (such as "Maschler spaces").

• But what are real concrete examples of both? Dec 22 '15 at 23:39
• @FedorPetrov: there are a few examples right here, of which IMO the most relevant to this question is the theory of finite skew fields prior to Wedderburn's theorem. Dec 23 '15 at 0:06

One of my favorites: the Oxford English Dictionary defines aliquot as "Contained in a larger number a certain number of times without leaving any remainder; forming an exact divisor", and the term is in use in elementary number theory. But the antonym, according to the OED, is aliquant: "Contained in a larger number but not dividing it exactly", for example,

1695 W. Alingham Geom. Epitomiz'd "An Aliquant part is a lesser Number in respect of a greater, when it doth not measure it exactly, as 3 is an aliquant part of 7."

There are cases like this ...

Walter Feit and John Thompson wrote a 250-page paper on nonabelian simple groups of odd order, and in the end conclude that there are none.

Perhaps mathematical objects can be said to "disappear" when they were first thought to be consistent, but were later proven to be inconsistent. An example of this is excessively hypercompact cardinals.

Newton's fluxions were abandonned after Leibnitz notion of infinitesimally small numbers proved to be more manageable.

• I don't think this is an accurate depiction of the history. Newton's notation and Leibniz's notation were isomorphic to each other. Newton's $o$ was equivalent to Leibniz's $dt$. (Newton had a convention about omitting the $o$ in certain contexts, which confused things somewhat. See Boyer, p. 201 ) Also, Leibniz did not wholeheartedly endorse the idea that his notation was a notation for infinitesimals. He sometimes preferred to describe differentials in the same way we would today describe differential forms (Boyer, p. 210). Leibniz notation won because it was more expressive. Dec 22 '15 at 16:33
• The notion of "infinitesimally small numbers" itself disappeared, eventually. It is now studied in nonstandard analysis, which as the name suggests is different from the "standard" approach used in calculus and real analysis. Dec 23 '15 at 0:57
• @CarlMummert: Infinitesimals never disappeared from engineering and the sciences. Dec 23 '15 at 1:28
• Or from algebraic geometry! Dec 23 '15 at 3:24

Quaternions haven't disappeared, but they are much less commonly used than they used to be. At one time they were commonly used for the tasks that we would today accomplish using vector calculus. Historian Michael J. Crowe depicts quaternions and vector calculus as being in a kind of Darwinian struggle, which quaternions lost.

• I've got to disagree on that. Quaternions are commonly used by software engineers who deal with 3D computer graphics, animation, robotics, image processing, and such. Refer to Quaternions and Rotation Sequences for applications. Dec 22 '15 at 18:07
• Quaternionic geometry, and the quaternionic hyperbolic plane in particular, remain important in modern geometry, specifically in the study of semisimple Lie groups and lattices therein. The latest such reference I can find is a 2009 paper by Kim and Pansu. An older and very significant one is a 1989 paper by Pansu. Dec 22 '15 at 20:18
• Also in number theory (e.g. endomorphisms of supersingular elliptic curves), representations of finite groups (e.g. the (Frobenius-)Schur indicator). Dec 22 '15 at 23:12
• Actually, just last year I needed the following theorem. Let $K$ be either the reals, the complexes, or the (real) quaternions. Let $A$ be an $n$-by-$n$ matrix whose entries are in $K$ such that the associated $K$-quadratic form $x\mapsto {}^t\overline x A x$ is positive semi-definite. Then there is a matrix $B$ with entries in $K$ such that $A=B^2$. We teach this in linear algebra when $K=\mathbb R$ or $\mathbb C$, but I really did also need it for the quaternions. (The reason suggested in Noam's answer. If $X$ is an abelian variety, then $\text{End}(X)\otimes\mathbb R$ is a product ... Dec 23 '15 at 1:26
• ... of matrix algebras $\text{Mat}_n(K)$ with $K$ being $\mathbb R$, $\mathbb C$, or $\mathbb H$.) Anyway, quaternions still come up quite naturally. Dec 23 '15 at 1:28

Mathematical objects do not disappear, except possibly in the rare cases when something was named and then proved that it does not exist. Mathematical objects come into and out of fashion from time to time, but they almost always come back. Terminology changes of course, but the things remain, whether they have special names or not.

Let me give some simple examples from trigonometry. Of course trigonometry is not a fashionable research subject anymore, so some terminology is forgotten.
David Handelman gives "haversine" as an example. The term is forgotten by mathematicians. But there exists a useful "haversine formula" in astronomy, and unlike aether in physics, the thing itself did not disappear.

Ancient Greeks did not have our trigonometric functions using $\mathrm{chd}\, x$, the chord corresponding to angle $x$ instead. As an independent trigonometric function it disappeared. But the notion of the chord did not disappear. Sometimes this function is indeed handy in trigonometric calculation. When necessary it is resurrected:-)

http://arxiv.org/abs/math/0009251

This does indeed happen. One very clear example is that of a naive set, i.e. a set conceived of as a collection of every object that has some particular property, without any further restrictions on its definition. This was in common currency before Russell's paradox came to light. (Russell's paradox is this: consider the set of all sets that do not contain themselves. Is this set a member of itself or not?) Afterwards it was replaced by the much more refined notion of sets as per ZF set theory (or ZFC), in which the notion of "the set of all sets that do not contain themselves" cannot be expressed.

This example suggests a major difference between mathematics and physics. In physics, a theory can simply be wrong, in the sense of not corresponding at all to reality. For example, the luminiferous ether was not the correct explanation for light waves, and when special relativity came along it had to be discarded entirely. However, in mathematics the only way to be wrong is to be inconsistent. (If a mathematical theory fails to correspond to reality it might lose some applications, but if it's interesting in its own right people will continue to work on it.) As in the case of naive set theory, an inconsistent theory can often be 'patched up' and made consistent in such a way that the core results are preserved. So in physics, the 'disappearance' of a theory can be a rather catastrophic event that affects the whole field, whereas in mathematics it's more likely to take the form of a small change, and be of interest only to a small group of researchers.

• I think naive sets are very much still in play in mathematics - most mathematicians work most of the time without a specific background set theory, under the (almost always correct) assumption that nothing bad will happen. Dec 24 '15 at 6:02
• That's covered in my second paragraph. "An inconsistent theory can often be 'patched up' and made consistent in such a way that the core results are preserved ... in mathematics [the disappearance of a theory is] more likely to take the form of a small change, and be of interest only to a small group of researchers." Naive sets disappeared as an object of study for formal logicians, but the change was so small it barely affected anyone else on a day to day basis. Dec 24 '15 at 6:13
• That's fair - I still don't really think this counts as a "disappearance," especially because even in logic we still consider naive sets (e.g. in paraconsistent set theories, or when talking about possible choices of axioms for set theory). (Btw I was not the downvoter.) Dec 24 '15 at 6:29
• Naive sets are now often called "classes". Dec 26 '15 at 18:43
• @AsafKaragila I would rather say that classes are a different way in which the inconsistent notion of a naïve set can be "patched up" to make it consistent. That they are not the same can be seen from the fact that you can consider the class of sets that don't contain themselves, but you can't consider the class of classes that don't contain themselves, because a class cannot be a member of a class. Dec 27 '15 at 3:34

I would nominate the first apotome of a medial.

It's a great question! The question is a bit "vague" - because you really don't specify "forgotten by who"? Let's assume that you mean "forgotten by the main stream math community at a given time" (because - really once a result is documented in print - it never is really forgotten - only the number of people aware to its existence can vary - there is probably some threshold that makes a result "common knowledge" - where "common" refer to communal).

I cannot think of examples - of "objects" (as in - definitions) that have appeared and disappeared - but I can think of examples of a few results (Theorems) that have been discovered - generally forgotten - and later rediscovered! There are a few examples like that in number theory - some results even quite fundamental. For instance results discovered by Eular - forgotten - later rediscovered in 19-th century - forgotten - later rediscovered circa 1940's - later rediscovered circa 1990's - and every time with different understanding and insight on their contents - so it happens.

Also there is an interesting description of such a process in an Appendix of the very classical book by Gelfand, Kapranov & Zelevinsky (GKZ) on the concept of determinants of complexes required for the computation of discriminants - which was apparently already discovered by Cayley in the early days of modern algerba! The way they write about it is really quite an interesting and recommended read (their modern exposition relies on heavy use of homological algebra). So that might be another example for a historical process of the kind you are asking for.

Yes, definitely. Sometimes the construction become obsolete. For example, in Mumphord's little red book we read about pre-scheme, but essentially nowhere in modern mathematics are "pre-"schemes studied.

Also, there are object which are just too non-mainstream at the time of their conception and never really catch on. Sadly they become forgotten, even though maybe they are ingeritly very interesting and unique objects. For example there is the concept of a Cosmos, which is already forgotten by many mathematicians due to it's inaccessibility (and non-mainstream-ness).

• There seems to be some terminological confusion here. A prescheme in the sense of Mumford (or EGA) is exactly what we call a scheme nowadays. Jan 21 '16 at 11:54
• Why do you say the concept of cosmos is 'inaccessible'? To me 'inaccessible' suggests that it is too abstract or difficult or recondite for most people to grasp. I think the concept of cosmos is quite simple in fact, and also very handy and general. There are also many people who do enriched category theory, a quite common technique in mathematics, who haven't "forgotten". Jan 21 '16 at 13:16