# History of the abstract method in mathematics

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.

• I guess you know the story of accepting that 1 is a number, not just "unit". It might be the first example of "a mathematical object is what it does". Am I wrong? – Anton Petrunin Oct 9 '17 at 2:33
• I just know that story only up to the chapter that people had difficulty grasping 1 as a number. Strangely, I never read the next chapter. However, I see how it is related to my question now upon your mentioning. – Amir Asghari Oct 9 '17 at 6:15
• Perhaps this attitude. of Hausdorff towards the definition of equivalence is reflected in the use of "operational definitions" for terms in physics. – Tom Copeland Oct 9 '17 at 13:17
• Certainly "taking equivalence classes as new objects" is prehistoric. Presumably Hausdorff was talking about infinite cardinals, but let's consider the concept of finite cardinality (AKA "number"). The moment someone took in the notion of, say, "3-ness", they were implicitly considering the equivalence class of all the sets of 3 things they could see around them. Sometime after (but still lost to history), someone wrote a symbol for "3-ness"! Even if later abstractions were farther from everyday experience, they were no more profound. – Nathan Reading Oct 10 '17 at 12:16
• @NathanReading, @Joël makes a similar point below. – LSpice Oct 10 '17 at 12:58

# 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

(OP.history) "what is the history or a history of the abstract method in mathematics?"

Needless to say, (OP.history) is too broad to 'answer', except perhaps by pertinent references, of which I know some:

• closest to a single professional treatise expressly dedicated to "history of the abstract method in mathematics" (to quote the OP) seems the very recent book

Paolo Mancosu: Abstraction and Infinity. Oxford University Press. 2016

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:

[...] Mancosu achieves much more than offering a mere history of abstraction principles. For example, based on his historical overview he distinguishes three options of how to interpret the abstractum of an abstraction principle . 1. It is a representative of the relevant equivalence class, i.e. [it is] one of its objects (characteristic of its use in 19th century number theory) 2. The abstractum of an element in the domain is the equivalence class which contains that element. 3. The abstracta are sui generis and neither coincide with an element of the domain nor with an equivalence class (Grassmann's, as well as the Neo-Logicist conception).

[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:

(OP.story) What is its story?

(OP.name) Has it ever had any name?

(OP.study) "I would be more than happy to have some references for study."

I consider (OP.story) to be subsumed in (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:

the wonderful history of abstraction of course begins with the legendary(legendary) figure of Euclid of Alexandria taking on the Herculean task of systematizing the scattered geometric wisdom of his time.

For this, the "abstract method" was at the very least a means to an end. In particular,

this is only partly relevant to the opening poster's question since Euclid-definitions mostly define objects by analytic statements(analytic statement), only rarely (such as in the 23rd definition) does one ever get an "object-is-what-object-does"-definition

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:

Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. (emphasis added)

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

this is relevant to the topics of 'abstraction in mathematics' and "how did it spread": the protagonist in the novella doesn't even read Dutch, but figures out what Euclid meant; this wouldn't have been possible had Euclid written in a non-abstract, heavily culture-dependent style. Storm is getting at the approximate 'abstract'/'structuralist'/chair-table-beermug'-quality of Euclid (though this was only really achieved by Hilbert).

## 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:]

Let us remember that we are mathematicians, and, as such, have already found ourselves in a similarly awkward situation, and let us remember how the ingenious method of ideal elements has liberated us from the quandary. Some shining role-models for the application of this method I have shown to you at the beginning of my talk. [The article is the published version of a talk Hilbert gave on June 4, 1925 in the German city of Münster, at a meeting of mathematicians.] Just like $$i=\sqrt{-1}$$ was introduced in order to preserve, and in their simplest[-possible] form, the laws of algebra, e.g. the laws about existence and number of roots of an equation; just like the introduction of ideal factors [Hilbert, I think, makes a reference to the intruction of what today is simply called 'ideals' (in a ring), by Dedekind.] happened in order to preserve, among the algebraic integers also, the simple laws of divisibility, like e.g. the introduction of a notion of 'common divisor'[Hilbert omits 'greatest' in the term 'greatest common divisor'.] of the numbers $$2$$ and $$1+\sqrt{-5}$$, while an actual divisor does not exist; just in the same fashion we have to adjoin to the finit[istic] propositions the ideal propositions, in order to preserve the easy formal rules of usual Aristotelian logic. [The word "erhalten" is a German verb which can mean either "to preserve" or "to obtain (something which was not there before)"; Hilbert here means the former meaning: to preserve Aristotelian logic, while doing modern mathematics; he proposes to do so by using ideal objects/propostions, of which the only thing which counts is what they do] And it is a strange coincidence that the modes-of-inference which were so passionately challenged by Kronecker are the exact 'companion piece'[The word "Seitenstück" is an old-fashioned German word, used in drama/literature/music, and means more or less the same as 'companion piece'; Hilbert is basically saying that, behind features of Dedekind's work which Kronecker praised, there are hidden set-theoretic background assumptions, which Kronecker seems not to have noticed, and would would have disapproved of, had he noticed it; Dedekind calculates with infinite sets as if they were numbers, and this is another relevant example of what the opening poster is asking for: the infinite sets 'just do' as numbers, hence 'are' numbers according to the "abstract method"; this example is also relevant to the opening poster's question about "taking equivalence classes as new objects"] to what Kronecker admired so enthusiastically about Kummer number-theoretically-speaking, and which Kronecker praised as a mathematical achievement of the highest quality.

How do we arrive at the ideal propositions? It is noteworthy, and is at any rate a state of affairs which is favorable and supportive [of my views] that we [...]

• 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.

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:

Saunders Mac Lane: Concepts and Categories in Perspective. In: *A century of Mathematics in America, Part I. American Mathematical Society, History of Mathematics 1: 323-365

• If you read German (or are lucky enough to have modern infrastructure available and patient enough to put up with piecemeal machine-translation), then there is another approximation (restricted to a neighborhood of Emmy Noether):

[Mechthild Koreuber: Emmy Noether, die Noether-Schule und die moderne Algebra Springer-Verlag 2015]

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.

• This could not be a thesis-topic the field of 'history of mathematics': the topic is much too broad.

## 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

Ernst Zermelo is said to have made a proposal to limit the use (I don't remember: use in what?) of the term 'abstract method'. The minutes of the meeting duly record this and also note that that Zermelo gave the reason that he felt that the term 'abstract method' was used pejoratively by more senior mathematicians of his time, and that he deplores that (being himself involved in axiomatization), and wants to limit this by avoiding the term 'abstract'. I don't remember more.

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

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

structuralism, structuralist method, etc

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

it is used in collocations like 'categorical structuralism',

e.g. in C. McLarty's very recommendable article

Colin McLarty: Exploring Categorical Structuralism. Philosophia Mathematica, Volume 12, Issue 1, 1 February 2004, Pages 37–53

which is an advanced meta-mathematical research article on questions which can be seen to be relevant to the "abstract method".

Moreover,

in an unsual, yet pertinent, allusion to the non-mathematical concept of 'behaviorism', Alain Prouté in his recommendable 400plus-page lecture notes A. Prouté: Introduction à la Logique Catégorique. Dernière mise à jour de ce texte : 1er novembre 2016, calls this "mehod" by the name behaviorisme. Again: this is not a usual technical term in the meta-mathematical literature. I am mentioning it here since it is relevant tot he opening poster's question, and since Prouté's manuscript a recommendable resource (which should, I think, also be translated into English).

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

it spread through the usual learned mathematical journals and the mathematical conferences which started to exist since the late 19th century

How should it have 'spread' otherwise?

The 'canonical' examples undoubtably are

• the Cantor-Kronecker-feud
• the Hilbert-Brouwer-feud.

I currently won't have time to summarize that.

In the latter, we roughly have

Hilbert: most famous advocate of (even non-contentually-)formalistic/structuralist/naively-optimistic view on mathematics ('no ignorabimus', 'Wir müssen wissen. Wir werden wissen.')

Brouwer: most famous advocate of an anti-logical, anti-formalist, (very valuable in certain respects; has received a new lease of life in categorical logic); Brouwer in particular was dissatisfied with syntactic consistency and first-order logic as the 'least common denominator' for what 'correctness' should mean.

Also,

Osvald Veblen, in his 1903 'A system of axioms for geometry', does the following: as if it were a matter of course to do so, without much explanation, Veblen subdivides the relevant "trend of developments" [p. 344] into two subtrends: I quote: "the then [...] inaugurated by PASCH and continued by PEANO", "rather than that of HILBERT or PIERI." [emphasis added]

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,

• of course, the OP's use of 'abstract method' brings to mind the term 'axiomatic method'. Whether to discuss this is relevant I am not so sure. It is a very common technical term of course. One should note that

The modern 'axiomatic method' goes farther back than David Hilbert's 'Grundlagen der Geometrie' (which is certainly the most famous first 'application' of the 'axiomatic method'), namely to Moritz Pasch who by professional historians of mathematics is seem to be considered something like the 'progenitor of the modern axiomatic method'. (Euclid is not, since Euclid seems to have set great store by the 'self-evidence' of the axioms.) Pasch must be treated in every serious "story" (to quote the OP) of the "abstract method". He in particularly heavily influenced Hilbert.

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"

Ralf Krömer: Die Kategorientheorie: ihre mathematischen Leistungen, ihre erkenntnistheoretischen Implikationen. Inauguraldissertation. Universität des Saarlandes 2004

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,

it might be of interest to some other readers round here that Krömer's thesis has more than twenty pages on Tohoku alone.

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

to the acceptance of what Timothy Gowers (2002, p, 18) calls “the abstract method in mathematics”, that " a mathematical object is what it does.”

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'.

• Your answer has encouraged me to have a deeper look at Essays on the Foundations of Mathematics by Moritz Pasch since Pasch has played a recognizable role on what he calls "Implicit Definition" (Definition by abstraction?). As soon as I move to another point made in your answer (there are too many) I let you know :) – Amir Asghari Oct 9 '17 at 10:46
• Dear Peter, please let me read and digest your current answer before adding anything to it for a while :) – Amir Asghari Oct 9 '17 at 17:28
• There you go with the tiny type sizes again. – Gerry Myerson Oct 9 '17 at 22:15
• Wow, amazingly the "three options" you have mentioned is more or less the conclusion of my research as well, though I have used quite a different language. I'll send my papers to you, the published one, that is more about the history of equivalence as a relation and the under review one that is more about equivalence classes. It is in this second paper that I nearly came to the conclusion that you have mentioned. – Amir Asghari Oct 15 '17 at 21:32

The question is interesting, but presupposes that we understand and accept Amir's conclusion, namely that "we owe Definition by abstraction (taking equivalence classes as new objects) to the the abstract method in mathematics, that a mathematical object is what it does.” I do not fully understand it, and as I understand it now, I am not convinced.

"Taking equivalence classes as new objects" seems essentially as old as human thought or at least human use of language. A word never denotes a "primitive" object, always an equivalence class, and a baby between one and two spends a lot of his time actively learning how to manipulate equivalence classes. By the way, Grothendieck says something similar somewhere in Recoltes et Semailles (this is not intended as an argument by authority, however it may seem :-) about the word "maman" (French version of the cross-language word "ma" or "mama"), in general one of the first word that a baby utters, at first used for what it does (calling Maman for cuddling, fooding, etc.) but which soon becomes a name for the class of all persons that are in the same relation to another baby (then another person or even animal) as the baby's own mother is to itself. (Grothendieck's point by the way is that the "method of abstraction" is universal and that he just pushed it maybe a little further than others)

That seems different, and in a complicated relationship, with the idea that "a mathematical object is what it does", for which I think the proper name is functionalism.

• Let me separate two points of your answer. The first point is that I have assumed that the development and acceptance of "definition by abstraction" has a close tie with development and acceptance of "the abstract method". I cannot comment on this point more since it was the result of my failure to attribute the "first" equivalence to any of the candidates I had: Euclid, Gauss, Dedekind, Pasch, Cantor, Frege, and Russell. – Amir Asghari Oct 9 '17 at 16:44
• As something that "is not intended as an argument by authority" :) let me quote Hermann Weyl: "In looking at a flower I can mentally isolate the abstract feature of colour as such. This act of abstraction would here be primary while the statement that two flowers have the same colour 'red' would be based on it; whereas in mathematical abstraction, it is the equality which is primary, while the feature with regard to which there is equality comes second and is derived from the equality relation. – Amir Asghari Oct 9 '17 at 16:56
• Please read this comment before the second comment :) But then, there is your second point that basically implies there is no first equivalence since it is "as old as human thought or at least human use of language". With this point, I strongly disagree. Consider that in all your examples, you have first abstracted a concept, formed it, named it, and then saw the equivalence of the objects under it. In the definition by abstraction, you are moving in the opposite direction. – Amir Asghari Oct 9 '17 at 17:03

It sounds like you were talking about formalism as a philosophy of mathematics https://en.wikipedia.org/wiki/Formalism_%28philosophy_of_mathematics%29 To me, abstraction would be a different matter, and is as old as mathematics itself -- the very concept of (natural) numbers is a form of abstraction.

Formalism may be most identified with Hilbert, though Dedekind's construction of real numbers may be a prominent precursor.

• I don't follow this: where does the opening poster "talk[..] about formalism as a philosophy of mathematics"? It seems almost wrong to conflate the topic of 'definition by abstraction' (which is an existing collocation and topic in the philosophical literature) with 'formalism' (which of course is also an existing topic in the research literature, but usually tends to refer much more to propositions and proofs than to definitions). I'm not saying this were wrong, yet it seems to me that 'definition and abstraction' and 'formalism-versus-intuitionism' are quite separately discussed topics. – Peter Heinig Oct 16 '17 at 9:26
• Pardon my ignorance. I see that I'm using the word formalism in a broader sense than usual (as in formal proofs), but include "formal definitions". I was led to think this way when asked about what formal linear combination meant, and I came to realize that all (or most) modern definitions are "formal", whatever that means. I was responding to the bold phrases in OP. Perhaps I meant formalism as opposed to realism, not intuitionism. Functionalism may be a better term. – liuyao Oct 17 '17 at 14:50