Recently I have "finished" a 13-year on and off research on the history of the mathematical notion of equivalence. At the end of which, I learned that we owe the nowadays rather elementary process of "Definition by abstraction" (taking equivalence classes as new objects") not to a person, a mathematical theory, a domain
of study, but to the acceptance of what Timothy Gowers (2002, p, 18) calls “the
abstract method in mathematics”, that " a mathematical object is what it does.”
The following comment on the definition of cardinal numbers, written by Hausdorff about a century ago, shows the influence of this attitude on our understanding of equivalence.
This formal explanation says what the cardinal numbers are supposed
to do, not what they are. More precise definitions have been
attempted but they are unsatisfactory and unnecessary. Relations
between cardinal number are merely a more convenient way of expressing
relations between sets; we must leave the determination of the
“essence” of the cardinal number to philosophy. (Hausdorff 1914, pp.
28-29)
I am not sure "the abstract method" to be a common name for this attitude/philosophy. To be honest, I've believed in it since my undergraduate days without having a name for it. Has it ever had any name? What is its story? Why and how did it start and spread? Who were the main advocates? The main antagonists? Asking a single MO-style question: what is the history or a history of the abstract method in mathematics? I would be more than happy to have some references for study.
Best Answer
Repeating the many questions in the OP.
To structure this answer, I'll repeat all the questions of the opening poster. What they emphasize to be their "single MO-style question" is
Needless to say, (OP.history) is too broad to 'answer', except perhaps by pertinent references, of which I know some:
which seems more-or-less exactly what the opening poster is asking for. To facilitate the decision whether to buy the book, here is the beginning of the table of contents:
An August 2017 review, by a professional philosopher (P. A. Ebert), of Mancosu's treatise is here. Excerpt from Ebert's review:
[emphases added; it was new to me that Grassmann held a view that 'abstractum' is, so to speak, a 'type of its own', not only an equivalence class; I did not try to track down a reference for this in the original German]
There are other valuable approximations to 'monograph-on-abstraction-in-mathematics', some mentioned below.
In the OP one also finds the following questions:
I consider (OP.story) to be subsumed in (OP.history).
Ad (OP.history).
This is too broad to thoroughly answer; Paolo Mancosu's 2016 book would be my recommendation of a starting point to the professional literature. Here are some further attempts of mine.
Euclid
According to the usual narrative:
For this, the "abstract method" was at the very least a means to an end. In particular,
Practical considerations of compression and teachability seem to have been at least as important a motivating factor for Euclid as philosophical view s like 'an-object-is-what-an-object-does'.
You probably need not be told that Euclid was far from the logical precision and parsimony that we are used to. This needs little explanation here. Euclid used some impredicative definitions(impredicative), and, Euclid used model-dependent logical inferences (called 'contentual inferences' by some philosophers of mathematics), which made some of Euclid's proof lack logical necessity.
While Euclid mostly defines things by what they 'are' or 'have', sometimes Euclid's definitions are 'behavioristic' (to use Prouté's interesting borrowing from non-mathematical sciences), e.g. the 23rd definition:
Here, Euclid defines 'parallel line segments' by what they (don't) do: if one "produces" them ('produce' is an example of an unclear and 'model-dependent' notion in Euclid), then they nevertheless don't ever meet. That's relevant to the "a mathematical object is what it does.” in the OP.
Then, for centuries, throughout the 'Middle Ages' (needless to say, my knowledge is 'Eurocentric'; I know sadly little about e.g. Chinese or Indian history of abstraction in mathematics; one could e.g. also have a look into the 'golden' 'Kerala' period) Euclid remained an iconic example of abstraction and rigor. What one should mention in addition is the centuries-long European tradition of scholasticism, which did much to preserve and spread logic (though famously they never seem to have invented quantifiers; there are jokes about that which I will keep out of this answer.)
Examples of Euclid's spread through the ages are too numerous to survey; I confine myself to a randomly-chosen unusual example: in Theodor Storm's novella Der Schimmelreiter(Schimmel), Euclid's language-independent 'transmittability' figures prominently; I only summarize: one evening, the adolescent protagonist has the effrontery to question the veracity of something in a school book, whereupon his father says something like "Enough. It's like that. Look into Euclid who will tell you why." Now that has nothing to do with abstraction, but the story then goes on to describe that the protagonist manages to understand a Dutch copy of Euclid, which is the only one available in the household, and
Thinkers relevant to abstraction in mathematics
I am not a historian, but think that a serious and thorough history of 'abstract method', even only insofar as mathematics is concerned (let alone architecture or fine arts) would have to treat at least the following (there are at least three glaring omissions, which I neither have time nor knowledge enough to correct; I draw a line where I perceive an 'abstract method' in the modern sense to come into being):
William of Ockham 1287–1347 (important for 'abstract method' because of work on nominalism, efficiency of reasoning and empty terms in syllogistics )
Isaac Newton 1642-1727 (while an accomplished experimentalist, alchemist and exegist, all of which rather non-abstract, Newton tried to write 'more-geometrico' in his Principia; see also keywords like '(spooky) action at a distance' or Hypotheses non fingo; this may sound like 'I don't use the axiomatic method' but it means the opposite: Newton declined to use the 'constructivist'/'mechanist' continental Cartesianism (for which he was criticised on the 'continent') and **he worked only with what gravity was known to do, not with a model of what gravitation 'is'.
Immanuel Kant 1724-1804 (much scoffed-at, disliked by Cantor, yet important for the history of mathematics; tried to argue axiomatically)
Gottfried Wilhelm Leibniz 1646-1716 (famously unsystematic and eclectic; only posthumously 'systematized' by Wolff; more 'mechanistic' than Newton, but undoubtedly important for 'abstraction in mathematics'; keywords 'identity of indiscernibles'; 'Leibnizian identity' (important in categorical logic; cf. e.g. p. 315 in Jacobs: Categorical Logic and Type Theory Elsevier 1999; also, one can view his idea of the Pre-established harmony as a 'causality-is-what-causality-does'-definition)
Georg Wilhelm Friedrich Hegel 1770-1831 (even more scoffed-at than Kant; can be seen to be trying to use language more abstractly than his contemporaries; category-theoretic interpretations of 'Hegelisms' exist, cf. work of Lawvere; was stuck with the tools available at the time)
Moritz Pasch 1843-1930 (seen by some historians of mathematics as progenitor of the modern axiomatic method)
Gottlob Frege 1848-1925 (seen to as the inventor of quantifiers; his 'Begriffsschrift' important for the history of abstraction in mathematics; note that **quantified variables can be viewed as 'typed' variables, the type telling how the variable 'behaves')
Giuseppe Peano 1858–1932 (tried to 'capture' the idea of the natural numbers abstractly/axiomatically by what they do, and unwittingly ended up with an axiomatic system which (0) is widely-believed-but-not-known to be consistent, (1) of necessity (by Completeness Theorem and Incompleteness Theorem combined) admits of non-standard models.)
David Hilbert 1862-1943 (vociferously defended the axiomatic/synthetic method; in particular defended the usefulness of ideal propositions, which can be 'adjoined' to a system and need not even have 'descriptional content', and 'are what they do')
An example from [Hilbert: Über das Unendliche, Mathematische Annalen 95 (1926), pp. 161–190], which could be given a heading like
Hilbert on "a mathematical object is what it does [for us]."
[my translation:]
Georg Cantor 1845–1918 (not afraid to treat the idea of infinity abstractly and free-of-the-concepts-of-time, and whose work was essential for the later work of Hausdorff, which the opening poster cited in the OP)
John von Neumann 1903-1957 (made important 'axiomatizing' contributions both to set theory and to the mathematics of quantum mechanics, and whose abstract conception of a practicable electronic computer, in particular, his abstract idea of stored program computer may come be seen as one of the most transformational abstractions ever; the idea to treat the program and the data 'on the same footing' is a perfectly abstract and mathematical idea, can be seen to meet Gower's "object is what is does"-definition, and was apparently totally new at the time von Neumann thought of it, and had stunning practical consequences; it is a relevant example of the "mathematical object is what it does" Definition: the mathematical object is the 'program/algorithm/rule', and von Neumann noticed that all what counts is what the program does, and this is data; the program need not 'be' somehow 'machine-like' in appearance, and realized as cogs-and-levers)
Alexander Grothendieck 1928-2014 (almost synonymous with an abstract 'behavioristic' approach to mathematics, and who in particular used category-theoretic methods to rebuild algebraic geometry)
Paul Cohen 1934-2007 (found, in a sense by a method of 'adjunction of generic elements/ideal propositions' which would have delighted Hilbert, models of set theory which, very roughly speaking, do the right things so as to satify the laws of set theor,y but in which the continuum hypothesis is false)
William Lawvere 1937- (one of the most abstract mathematical thinkers; abstracted away inessentials from old ideas like 'implication', 'quantification', 'substitution', 'theory'; Lawvere also concerned about history, philosophy, expository writing and teaching, and sometimes writes things like "[...] category theory does not rest content with mere classification in the spirit of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the mutability of mathematically precise structures (by morphisms) which is the essential content of category theory." [emphasis added]; so you might find it rewarding to look into Lawvere's writings; he has a website with downloadable publications)
Vladimir Voevodsky 1966-2017 (Soulé's 2002 laudatio says "Vladimir Voevodsky is an amazing mathematician. He has demonstrated an exceptional talent for creating new abstract theories, about which he proved highly nontrivial theorems."; found the available formal tools available inadequate for what he needed to do, and set to work on a more abstract treatment of 'equality'; further keywords "isomorphic types are equal", "univalence axiom")
The Homotopy Type Theory program ~1998- (revisited the notorious question of what the meaning of 'is' is)
Further remarks and comments on the question which the opening poster emphasizes to be their 'MO-style question'
I think that what you seem to be primarily be asking for simply does not exist yet, at any rate not in a convenient compressed structured format, let alone in one book, except perhaps Mancosu's 2016 book. I am not a historian though; if you really need to know what there is out there (especially in the way of PhD theses, possibly unpublished), then you could perhaps try asking experts, e.g.
The Logic at Harvard website
Department of the History of Science of Harvard University
Writing a "History of the abstract method in mathematics", thoroughly and expertly, would be possible, but would require years of full-time work, and a professional mathematical historian to tell it. One especially shudders at the many concepts which one would have to tame by appropriate methods. I know little about the methods of historiography.
As far as I can tell, such a comprehensive "History of the abstract method in mathematics" has not yet been written. There are valuable 'approximations':
If you accept a treatise restricted to modern axiomatic set theory, then the book [Penelope Maddy: Defending the Axioms. Oxford University Press. 2011] is something like a history/philosophy of the "abstract method"$|_{\text{set theory}}$
Among the several gems written by Saunders Mac Lane is another approximation:
Koreuber's book starts with very general chapter "Begriffliche Mathematik" Koreuber's book also contains detailed historical and philosophical research on Emmy Noether's influence on abstract mathematics, apparently also taking as yet unpublished material into account.
An anecdote, remembered from memory
I am sure I remember having read some 'minutes' of some meeting of mathematicians, in German, though I cannot find a reference, during which
I am taking the above from memory and do not find a reference (a relevant comment providing a reference would be appreciated).
This is historically significant to the opening poster's question, since it shows (if I am not misremembering that Zermelo-anecdote) that
(0) the term 'abstract method' was a recognizable collocation, at least in the early twentieth century,
(1) it shows that 'abstract method' seems once to have been perceived as pejorative, while now, with good reason (like e.g. the notorious collocation 'sterile pursuit') has come to be used appreciatively
Ad (name).
Apart from 'abstract method', which, as the Zermelo-anecdote seems to prove, is a usual technical term, further relevant and usual technical termsfor your question are
Note, however, that 'structuralism' has a slightly different shade of meaning: it is strongly associated with Bourbaki's great series of books, and, more specifically, with the technical notion of structure on a set. (Which hopefully needs not explaining here.)
But it is close to what you are asking, and often used with a view towards 'mutual relationships between ideas', and
e.g. in C. McLarty's very recommendable article
which is an advanced meta-mathematical research article on questions which can be seen to be relevant to the "abstract method".
Moreover,
Ad (start.and.spread).
There isnt'any simple answer to the start-and-spread-question. For start, look e.g. into Euclid.
Much of the wisdom of antiquity, which was often quite abstract, and Euclid in particular, spread/survived thanks to the good offices of both Arab scholars and European monks, who by their devotion to translating (or even merely transcribing verbatim) the works of the past, kept learning alive during centuries of economic/cultural/hygienic misery.
So regarding the "spread"-part of your question applied to pre-modern times, a one-sentence answer could be: European monasteries and Arab scholars; a pictorial answer to the how-did-abstraction-spread-question could be the following public-domain picture, which I take from Wikipedia, and which shows a European and an Arab 'handling'/'spreading' 'pure abstractions':
Of course, if you are only asking about the recent past, i.e. roughly since 1850, say, then I think the obvious sober answer is
How should it have 'spread' otherwise?
Ad (advocates.and.antagonists).
The 'canonical' examples undoubtably are
I currently won't have time to summarize that.
In the latter, we roughly have
Also,
The technical differences between the two sub-trends of the "abstract method", in a nutshell, is: Pasch advocated a notion of a set of 'points' on which an (order) relation is define, while Hilbert advocated (cf. the famous but perhaps apocryphal 'table, chain, beermug'-anecdote) a purely synthetic approach, with terms which can be interchanged without impairing the usefulness of the system.
Also,
Ad (OP.study)
If you allow taking 'history of the abstract method in mathematics'='history of category theorem' (which I am well aware is too narrow an interpretation; you seem to be asking rather for a history of the axiomatic method, which is a larger, or even separate topic), then the following is an interesting, cogent, one-volume, recommendable item among the many "references for study"
Krömer's thesis is a 400+p. professional history and philosophy (the thesis also self-consciously discusses what the difference between the two terms is, and views itself rather as a 'philosophy of category theory' than a 'history of category theory') of category theory (with an emphasis on topos theory and algebraic).
Incidentally,
I am not sure whether this is what you are looking for, in particular since it is only available in German, but Krömer's thesis definitely is very relevant to the definition
that you give of "abstract method": Krömer's thesis speaks much about concepts like 'Verhalten', 'Werkzeug', and contains philosophical/historical discussions; an illustrative example of the subtle philosophical discussions to be found in Krömer's thesis is the first paragraph on p. 153 of op. cit.
Krömer's thesis is only about an episode in the history of "the abstract method", but a long, important and influential episode; if you have access to modern infrastructure, you can read it with the help of a search engine, piecemeal.
Footnotes.
Alphabetic order.
(analytic statement) An analytic statement about something says something 'about the inside of the thing'. An example is the statement "a prime is a number which 'has' exactly two divisors." Of course, taking analytic statements to be definitions was and is useful in mathematics, but it is not permitted in a strict "object-is-what-object-does"/"terms-substitutable-by-chair-table-beermug" style. In that sense, Euclid is only partially relevant. Euclid's system is somewhat 'hybrid', and uses several 'model-dependent' and 'semantic' notions. But Euclid should figure in every history of abstraction in mathematics.
(impredicative) It is debatable which definitions of Euclid are technically 'impredicative' in a modern sense, but roughly speaking Euclid's definitions, are partly 'impredicative'; e.g., Euclid's very first definition has always sounded to me like the circular 'A point is that which has no points.' (This is a caricature of Euclid's slightly less circular actual definition; I am misrepresenting 'to make a point'.)
(legendary) 'legendary' is not to say 'non-historical'; there seems to be plausible evidence that there once lived some individual roughly like the 'Euclid' we imagine today; but there are also some theories that 'Euclid' was a collective nom-de-plume, like 'Bouraki' or 'Boto-von-Querenburg'; it seems more likely than not that there was 'one' Euclid.
(Schimmel)'Schimmel' is German for 'mould' and it is also a metonym for 'white horse'.