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Abstract nonsense
Abstract nonsense
Abstract nonsense
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In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are nonderogatory terms used by mathematicians to describe long, theoretical parts of a proof they skip over when readers are expected to be familiar with them.[1] These terms are mainly used for abstract methods related to category theory and homological algebra. More generally, "abstract nonsense" may refer to a proof that relies on category-theoretic methods, or even to the study of category theory itself.

Background

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Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content. As a result, mathematical proofs that rely on category-theoretic ideas often seem out-of-context, somewhat akin to a non sequitur. Authors sometimes dub these proofs "abstract nonsense" as a light-hearted way of alerting readers to their abstract nature. Labeling an argument "abstract nonsense" is usually not intended to be derogatory,[2][1] and is instead used jokingly,[3] in a self-deprecating way,[4] affectionately,[5] or even as a compliment to the generality of the argument. Alexander Grothendieck was critical of this notion, and stated that:

The introduction of the cipher 0 or the group concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps... [6]

Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of classifying spaces and universal properties, use of the Yoneda lemma, natural transformations between functors, and diagram chasing.[7]

When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression "Such and such is true by abstract nonsense" rather than provide an elaborate explanation of particulars.[1] For example, one might say that "By abstract nonsense, products are unique up to isomorphism when they exist", instead of arguing about how these isomorphisms can be derived from the universal property that defines the product. This allows one to skip proof details that can be considered trivial or not providing much insight, focusing instead on genuinely innovative parts of a larger proof.

History

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The term predates the foundation of category theory as a subject itself. Referring to a joint paper with Samuel Eilenberg that introduced the notion of a "category" in 1942, Saunders Mac Lane wrote the subject was 'then called "general abstract nonsense"'.[3] The term is often used to describe the application of category theory and its techniques to less abstract domains.[8][9]

The term is believed to have been coined by the mathematician Norman Steenrod,[10][4][5] himself one of the developers of the categorical point of view.

Notes and references

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Abstract nonsense

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Abstract nonsense is a colloquial term in mathematics, particularly associated with , referring to highly abstract proof techniques and conceptual frameworks that emphasize formal relationships—such as those involving commutative diagrams, universal properties, functors, and natural transformations—over the concrete details of specific mathematical structures. Coined by topologist Norman Steenrod in the 1940s or 1950s, the phrase originally described the general methods emerging in and , as developed by figures like , , , and Steenrod himself. These techniques, often involving diagram chasing or arguments by duality, allow mathematicians to establish results that hold across diverse categories without recomputing them for each particular case. The term gained prominence alongside the formalization of in Eilenberg and Mac Lane's seminal 1945 paper, which introduced categories, functors, and natural transformations to organize algebraic concepts in . Steenrod's usage highlighted the power of this abstraction to unify disparate areas of , enabling proofs that are "trivially trivial" once the categorical structure is recognized, as quipped by mathematician Peter Freyd. Key examples include the , first articulated by in 1957 (though predating in oral tradition), which equates natural transformations with elements via representable functors, and the theory of introduced by Daniel Kan in 1958, which formalizes universal constructions like limits and colimits. Despite initial resistance—sometimes viewing it as overly abstract or detached from computation—abstract nonsense has become a foundational tool in modern , facilitating advances in fields like , , and even through concepts like monads. As noted in 1974, these methods support the systematic accumulation of mathematical knowledge by revealing deep analogies across disciplines. Today, it underscores category theory's role as a "" for expressing general principles, with influential texts like Emily Riehl's in Context (2016) demonstrating its practical elegance in theorem-proving.

Overview and Definition

Core Meaning

Abstract nonsense is an informal term used in to describe a style of reasoning and proof technique that emphasizes general abstract structures and relationships, particularly through the framework of , rather than detailed concrete computations or specific examples. This approach typically involves the use of categorical diagrams, functors, and natural transformations to establish results that hold universally across mathematical contexts, allowing proofs to proceed by verifying commutativity in arrow diagrams or invoking universal properties without delving into coordinate systems or explicit element-wise calculations. Key characteristics of abstract nonsense include a focus on structural invariances and morphisms—often represented as arrows in commutative diagrams—over intrinsic properties of objects, enabling elegant derivations that highlight isomorphisms, adjunctions, and other categorical constructs. This method avoids laborious verifications by appealing to the formal consistency of categorical axioms, such as those ensuring that functors preserve certain relations, thereby trivializing what might otherwise require extensive case-by-case analysis. For instance, diagram chasing techniques, a hallmark of this style, systematically traverse commutative squares or longer sequences to deduce equalities or isomorphisms. In contrast to concrete mathematics, which prioritizes explicit constructions and numerical or set-theoretic details, abstract nonsense elevates generality, treating specific instances as manifestations of broader categorical patterns and thereby unifying disparate areas of under shared abstract principles. The term was first used informally by Norman Steenrod during his algebraic topology seminars in the 1940s, affectionately referring to these systematic, diagram-based developments of relations in linear algebra and beyond. serves as the primary framework underpinning this approach.

Philosophical Underpinnings

The Bourbaki group profoundly influenced mathematical by conceptualizing as the study of abstract structures—systems defined by relational properties and operations—rather than isolated objects with intrinsic qualities. This perspective shifted focus from concrete entities to the axiomatic frameworks governing their interactions, promoting a unified view where diverse mathematical domains are analyzed through shared structural lenses. Saunders Mac Lane, along with Samuel Eilenberg, championed and promulgated the term "abstract nonsense"—originally coined by Norman Steenrod—to describe the formal, diagrammatic methods of that uncover profound analogies across mathematical fields, thereby revealing underlying unities in seemingly disparate areas. In his view, this approach transcends specific constructions, enabling mathematicians to identify isomorphic patterns and transfer insights between , , and beyond. The philosophy of abstract nonsense sparks debate on optimal abstraction levels, positing that high-level categorical tools like diagram chasing allow proofs to proceed through commutative diagrams, circumventing tedious case-by-case verifications in favor of general structural arguments. Proponents argue this elevates mathematical reasoning by emphasizing relational invariance over particular instances, though critics contend it risks obscuring concrete intuitions essential for discovery. Linked to formalism, abstract nonsense serves as a rigorous mechanism for achieving generality, axiomatizing constructions via universal properties to minimize dependence on and ensure proofs hold across equivalent categories. This formalist bent underscores category theory's role in bolstering mathematical precision, where diagram-based deductions provide a systematic alternative to element-wise manipulations.

Historical Context

Origins in Early 20th Century

The roots of abstract nonsense trace back to the early , particularly through Emmy Noether's pioneering efforts in during the 1920s. Noether shifted the focus from concrete computations and explicit solutions to the structural properties of algebraic objects, emphasizing ideals and modules as key abstractions. In her seminal 1921 paper, "The theory of ideals in ring domains," she developed the general theory of ideals in commutative rings with unity, introducing concepts like Noetherian rings where every ideal is finitely generated, and proved the theorem for such ideals. This axiomatic approach, detailed in her works from 1920 to 1926, prioritized universal properties over specific representations, laying groundwork for later abstract methods by unifying disparate algebraic structures without reliance on coordinates or explicit forms. In the early 1930s, contributed to this abstract turn through his research in field theory, valuation theory, and arithmetic , which emphasized structural invariants over computational details. Concurrently, mathematicians like and advanced ideas in group and , developing cohomology groups for abstract groups without explicit coordinate-based computations; Hurewicz's 1935 work linked higher groups to homology, providing algebraic invariants for topological spaces in a coordinate-free manner. These efforts marked a departure from classical methods, fostering a more general perspective on algebraic and topological phenomena. The 1930s also witnessed a pivotal transition from classical —focused on explicit polynomial invariants under group actions, as in Hilbert's finiteness theorem—to modern , where invariants were derived from chain complexes and exact sequences. Pioneered by Noether's 1925 observation that Betti numbers represent homology groups and formalized by mathematicians like Leopold Mayer in 1929 through chain complexes, this shift integrated algebraic tools into , yielding coordinate-independent invariants like cohomology classes. This pre-formal era set the stage for later systematization, as seen in the Bourbaki group's post-war axiomatic formalization of similar abstract structures.

Evolution in Post-War Mathematics

In the post-war period, the term "abstract nonsense" emerged as a colloquial descriptor for the diagram-chasing techniques in and , coined by Norman Steenrod during his seminars at in the 1940s. These seminars emphasized proofs relying on commutative diagrams and universal properties rather than explicit constructions, fostering a shift toward more abstract, structural reasoning in . Steenrod's approach highlighted the power of categorical abstractions to unify disparate results, laying groundwork for broader adoption in mathematical discourse. The foundational ideas of , central to abstract nonsense, were formalized in a landmark 1945 paper by and , which introduced categories, functors, and natural transformations as tools for capturing equivalences in algebraic structures. This work provided the theoretical roots for Mac Lane's later book (1971), which systematized these concepts for practical use across . Through publications and collaborations, such as those in the Transactions of the , these ideas disseminated rapidly, influencing post-war research in and . In France, the Bourbaki collective integrated categorical methods into their structuralist framework during the 1950s and 1960s, embedding them in volumes like Algèbre (starting 1950s editions) and influencing the mathematical canon through rigorous, axiomatic treatments. Bourbaki's seminars and texts, such as the 1957 chapter on structures in Théorie des ensembles, adopted functors and universal properties to unify algebraic theories, shaping global curricula and emphasizing abstraction over concrete examples. This integration promoted abstract nonsense as a standard tool in European mathematical education, extending its reach beyond topology to core algebraic pedagogy. By the 1960s, abstract nonsense spread to through Grothendieck's functorial methods, particularly in his development of schemes and toposes via categorical constructions in works like (EGA, 1960 onward). Grothendieck's approach, building on his Tôhoku paper introducing abelian categories and derived functors, reframed geometric objects as functors between categories, enabling powerful generalizations in sheaf theory and . These innovations, disseminated through IHÉS seminars and publications, solidified abstract nonsense as essential for modern , with applications in fields like .

Foundational Concepts

Category Theory Essentials

provides the foundational framework for abstract nonsense by abstracting mathematical structures into objects and relationships that emphasize universal properties over specific implementations. A category consists of a class of objects and, for each pair of objects AA and BB, a set of morphisms from AA to BB, denoted Hom(A,B)\text{Hom}(A, B), satisfying certain axioms. The composition of morphisms must be associative, meaning that for morphisms f:A→Bf: A \to B, g:B→Cg: B \to C, and h:C→Dh: C \to D, the equation (h∘g)∘f=h∘(g∘f)(h \circ g) \circ f = h \circ (g \circ f) holds whenever defined. Additionally, for every object AA, there exists an identity morphism idA:A→A\text{id}_A: A \to A such that idA∘f=f\text{id}_A \circ f = f and g∘idB=gg \circ \text{id}_B = g for any compatible morphisms ff and gg. These concepts were introduced by and in their seminal 1945 paper. Functors are structure-preserving maps between categories, enabling comparisons across different mathematical domains. A covariant FF from category C\mathcal{C} to category D\mathcal{D} assigns to each object AA in C\mathcal{C} an object F(A)F(A) in D\mathcal{D}, and to each f:A→Bf: A \to B in C\mathcal{C} a F(f):F(A)→F(B)F(f): F(A) \to F(B) in D\mathcal{D}, such that F(idA)=idF(A)F(\text{id}_A) = \text{id}_{F(A)} and F(g∘f)=F(g)∘F(f)F(g \circ f) = F(g) \circ F(f) for compatible morphisms. Contravariant functors reverse the direction of morphisms, mapping f:A→Bf: A \to B to F(f):F(B)→F(A)F(f): F(B) \to F(A), while still preserving identities and composition (with reversal). Examples include the from groups to sets, which sends a group to its underlying set and a to its action on elements. Natural transformations allow for the comparison of functors, providing a higher-level abstraction in category theory. Given two functors F,G:C→DF, G: \mathcal{C} \to \mathcal{D}, a natural transformation η:F⇒G\eta: F \Rightarrow G assigns to each object AA in C\mathcal{C} a morphism ηA:F(A)→G(A)\eta_A: F(A) \to G(A) in D\mathcal{D}, such that for every morphism f:A→Bf: A \to B in C\mathcal{C}, the following diagram commutes: F(A)→ηAG(A)F(f)↓↓G(f)F(B)→ηBG(B)\begin{CD} F(A) @>{\eta_A}>> G(A) \\ @V{F(f)}VV @VV{G(f)}V \\ F(B) @>>{\eta_B}> G(B) \end{CD}
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