Abstract
The traditional approach to concept formation via abstraction is tantamount to removing properties and making the corresponding concept less rich. In other words, the more abstract a concept is, the less content it has. This approach to abstraction does not, however, provide an adequate model for concept formation in mathematics. We need to replace the traditional account with one that is, on one hand, true to mathematical practice and the mathematical experience and is, on the other hand, compatible with insights from cognitive science. This chapter adds to the existing literature by homing in on a question that lies at the intersection of the two criteria just mentioned; it is the question of why abstract concepts are perceived not only as more powerful but also richer, not poorer, in content.
I wish to thank two anonymous referees for their careful reading and many helpful observations and remarks.
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Notes
- 1.
The ideas go back mostly to Aristotle who used them to show how we arrive at mathematical ideas (aphairesis) and general concepts (epagoge); see Bäck (2014) for a recent systematic exposition. Boethius merged the Aristotelian ideas about aphairesis and epagoge and rendered them into a single Latin term: abstractio. It has been a staple ever since; see Weinberg (1973) for an overview. It were Arnauld and Nicole who introduced the language of intension (la compréhension) and extension (l’étendue) in their Logic of Port Royal; see (Arnauld and Pierre, 1662, pt. 1, ch. vi). See Lovejoy (1936) for the great chain of being.
- 2.
See (Frege, 1879, pp. 49, 53); emphasis in the original.
- 3.
We owe the last pair of contrasts to Boniface (2007).
- 4.
What got me started to think about embeddings in the present context was an inspiring talk by Irina Starikova on Gromov’s work (on hyperbolic groups and Cayley graphs, see Starikova (2012)).
- 5.
The historical record does not seem to be entirely clear; see (Smoryński, 2008, ch. 4). While Wantzel (1837) announced the results first, Petersen first proved it in the modern vein using the fact that for polynomials over \(\mathbb Q\) to have roots their degree must be a power of 2; see (Peterson, 1878, pp. 159, 166f.). (Note that we use letters such as “\(\mathbb Q\)” indiscriminately for both the set and the set equipped with some algebra.)
- 6.
- 7.
- 8.
Clearly, the last paragraph appeals to and makes sense only if a certain view of mathematics is adopted; for example, a view (we unfortunately cannot argue here) that emphasizes the importance of explanation.
- 9.
For an overview of the Huzita–Hatori axioms, their history, and algebra, see Lee (2017).
- 10.
There is more to the plausibility of our claim than just one counterfactual story. I see it in the context of the growing awareness that culture and natural environment might lead to marked differences in our basic mathematical competencies. The here relevant buzz words include ethnomathematics, WEIRD people, neuroanthropology, and the encultured brain; see, e.g., Rosa et al. (2017), Henrich et al. (2010), Northoff (2010), or Lende and Downey (2012). Some of the challenges this open empirical question poses for mathematical cognition were recently formulated in Beller et al. (2018). I expect that the outcomes will support our claim.
- 11.
See, e.g., Hinman (2005, ch. 2.6).
- 12.
See https://mathscinet.ams.org/msc/msc2010.html. While not every mention of “algebraic,” “topological,” or “category/-ical” in the MSC (to name some of the usual suspects) indicates a case of embedding, their ubiquity still corroborates our point.
- 13.
See (Klein, 1895, p. 3): “This new method of attack is rendered necessary because elementary geometry possesses no general method, no algorithm, as do the last two sciences [sc. algebra and higher analysis]” (emphasis in the original).
- 14.
The closure property was striking enough for Dedekind at the time to make it—“so closed and complete in itself” (in sich so abgeschlossen und vollständig)—the feature that defines a field; see (Dedekind, 1871, p. 424).
- 15.
- 16.
- 17.
One referee wondered whether the case study is too long. I do not believe it is, we simply exercise due diligence. Making a historical argument requires that we look at all the evidence available to make it less likely that our claims are the result of biased readings of selected sources.
- 18.
- 19.
For the record we have to mention three more proposals that were made at the time. We mostly ignore them, however, and do so for two reasons. First, they offer little else than their definition; second, they had no noticeable impact at the time. This renders them mostly useless for our goal of learning about epistemic practices.
1. Charles Méray (1835–1911), a French mathematician in Dijon, was the first to publish what later became known as Cantor’s definition of the reals (see Méray, 1870). So what we call Cantor’s definition should really be labeled Méray-Cantor. Although he subsequently used his definition in two textbooks (Méray, 1872; Méray, 1894), it went unnoticed at the time. It seems it was only three decades later that Jules Molk drew attention to it when he translated and heavily revised Pringsheim’s article (Pringsheim, 1898) on irrational numbers and convergence for the French version of Klein’s Enzyklopädie der mathematischen Wissenschaften; see (Pringsheim and Molk, 1904, § 6). In the same year, 1904, Tannery published a revised edition of his textbook and also acknowledged the priority of Méray (Tannery, 1904, p. vii); but he did so only after Méray had called him out for the oversight (Méray, 1894, p. xxiii). While Méray waxed poetic when it came to criticizing the sad state of analysis in France of the time, he never said much about his definition beyond stating it. When and how Méray’s priority became known more widely, I have no tried to determine; but see Dugac (1970) for some general background.
2. Weierstrass never published the theory of irrational numbers he presented in his lecture course on analytic function theory that he started to offer on a fairly regular basis in the fall of 1861 (see Weierstrass, 1903, pp. 355–360). Thus, lacking ipsissima verba, scholarly caution prevents us from including his theory to our discussion below. For later references, however, we need to know what transpired at the time. We have a somewhat incoherent account by Kossak (1872) based on lectures given in the winter term 1865–1866 (incidentally, this is a lecture that Cantor, too, might have attended) and a thorough reconstruction by Pincherle (1880) based on lectures he attended in the academic year 1877–1878. While both are mentioned by Biermann (1887), his reconstruction of Weierstrass’ theory seems to owe Pincherle everything. We should mention that, in private correspondence, Weierstrass criticized Kossak for having “butchered” (verhunzt) his theory (Mittag-Leffler, 1910, p. 12), while his student Schwarz called it “defaced” (entstellt; Dugac, 1973, p. 144). Biermann’s account met with even more anger; Weierstrass would have intervened with the publisher had he known what was brewing (Dugac, 1973, pp. 142f.). It seems that Dantscher (1908), who published his version as late as 1908—dating back to a lecture Weierstrass gave in 1872!—was given almost no attention at the time. Due in particular to Dugac (1973), and later supplemented by Ullrich (1989), we now have more reliable information, which is sufficient to vet as accurate the little that we need on Weierstrass below.
Weierstrass was the first to use sequences; or, to be more precise, sequences of convergent partial sums (i.e., series). He defined an irrational number as an infinite series of positive and negative rational numbers (q n) such that the partial sum of every finite subseries \((q_{n_k})\) is bounded by a number b: \(\sum _{i=1}^k q_{n_i}<b\). Each q n was a unit fraction, i.e., the fraction of a chosen base unit, say, a: q n = 1∕a, and b a multiple of that base unit: b = m ⋅ a; to show convergence of the sums, Weierstrass employed the notion of absolute convergence.
3. Giulio Ascoli (1843–1896) proposed to use what we now call the Nested Interval Theorem. Let {I n}n ∈ ω, with I n = [a n, b n], be a sequence of closed intervals I 1 ⊇ I 2 ⊇… ⊇ I n ⊇…, and assume limn I n = 0. Then there is exactly one point p such that: p ∈⋂n ∈ ω I n and thereby defined by {I n}n ∈ ω. (See Ascoli, 1895, p. 1065; we here interpret Ascoli’s “∞” to mean “ω.”) While his approach is now one of the usual suspects for defining the completeness of \(\mathbb R\), it does not seem to have garnered much interest at the time.
- 20.
The last step was even more pronounced in Méray (in his book from 1894), Weierstrass (lectures notes taken by Hurwitz in 1878, see (Dugac, 1973, App. I)), and later in Pringsheim (1916): they first established that the respective properties and laws hold for sequences that define rational numbers; then, everything else being equal, what other than prejudice could be the reason for discarding the remaining sequences that define irrational numbers?
- 21.
It is not entirely clear from the text itself how much exactly Heine owes to Weierstrass or to Cantor, respectively. Heine himself stated that his only contribution was to bring oral communications of Weierstrass into a publishable, organized form and that discussions with Cantor had left their mark on how he presented the part on irrational numbers (bedeutenden Einfluss auf die Gestaltung), a part, he noted, that had been finished some time ago (seit längerer Zeit vollendet). Furthermore, he remarked that Cantor’s definition was a felicitous improvement of Weierstrassian ideas he had used earlier (glückliche Fortbildung der ursprünglichen Einführungsart; p. 173). So it might seem that Heine had replaced Weierstrassian series with Cantorian sequences (modulo some adjustments) in an otherwise completed part of the manuscript. This is not, however, how Cantor remembered it after Heine’s death; he claimed (Cantor, 1887, p. 385) that Heine’s entire approach was due to him.
- 22.
All quotations are from (Heine, 1872, p. 172).
- 23.
“Die Frage, was eine Zahl sei, beantworte ich […] nicht dadurch[,] dass ich die Zahl begrifflich definire, die irrationalen etwa gar als Grenze einführe, deren Existenz eine Voraussetzung wäre. lch stelle mich […] auf den rein formalen Standpunkt, indem ich gewisse greifbare Zeichen Zahlen nenne, so dass die Existenz dieser Zahlen also nicht in Frage steht.” ((Heine, 1872, p. 173))
- 24.
We should not keep quiet about a certain ambiguity in the original, German text; it concerns the word Reihe that I translated as “sequence” although today it means “series.” This, and the observation that Cantor was a student of Weierstrass (who had employed series (see footnote 20), prompted some to understand Cantor’s definition in terms of convergent series (see, e.g., Spalt, 1991, p. 357). And indeed, either reading was adopted by mathematicians during Cantor’s lifetime and with reference to him: Stolz used series (Stolz, 1885, pp. iv, 97ff.), Bachmann sequences (Bachmann 1892, pp. iv, 4–9). To the present author, however, it is clear that Cantor meant sequences. This is how he used the term Reihe in his work on Fourier analysis when he distinguished between a sequence (Reihe) and its sum (Summe; e.g., (Cantor, 1932, p. 73); later he used the compound word Reihensumme for series (ibid., pp. 105, 131, 180, 190) and reserved the word Folge (the present-day word for sequence) for transfinite well-orderings (ibid., p. 147, 444). Moreover, Cantor not only portrayed Weierstrass’ definition as very different from his own, but also said this was identical to what Lipschitz had done later (Cantor, 1883, pp. 184ff.); and Lipschitz used sequences, not series (Lipschitz, 1877, § 15).
That Cantor would point to Lipschitz is not without irony, though. We know from the correspondence between Lipschitz and Dedekind that Lipschitz doubted that introducing cuts or proving completeness were even called for. He believed we can find it all in Euclid and the “fundamental property of a line without which no one can conceive a line” (Grundeigenschaft einer Linie […], ohne die kein Mensch sich eine Linie vorstellen kann; ibid., p. 475). Accordingly, while Lipschitz freely mentioned predecessors in his book, he makes no references at all to Cantor or Dedekind when he defines the irrational numbers but rather maintains that “we owe the [Ancient] Greeks the study of such sequences of fractions that converge towards a limit. They elevated the limit associated with such sequences […] to a concept in its own right and realized how to extend the [basic] operations [of arithmetic] to these concepts” (Die Betrachtung solcher Folgen von Brüchen die sich einer Grenze nähern, verdanken wir den Griechen. Sie haben die in einer solchen Folge von Brüchen zugehörige Grenze zu einem selbstständigen Begriff erhoben, und erkannt, wie die Operationen […] auf diese Begriffe auszudehnen seien, Lipschitz, 1877, pp. 36f.). Thus, we must assume that all arguments by Dedekind to the contrary were wasted on him and that Lipschitz believed he can do without Dedekind or Cantor, while at the same time Cantor saw him as executing his own program.
- 25.
It is not entirely clear from the text itself what exactly Dedekind had in mind (not that it would matter, both formulations are equivalent, see (Propp, 2013, pp. 395f.), but we should still address it). The most straightforward reading suggests: every bounded monotone increasing sequence converges toward a limit. But his proof in § 7 suggests reading it as the existence of suprema. Here is an interpolated version of what he wrote: “Every magnitude which grows continually but not beyond all limits [= every set of numbers bounded above], must certainly approach a limiting value [= converges towards a limit or has a least upper bound].” (Jede Größe, welche beständig, aber nicht über alle Grenzen wächst, [muss] sich gewiß einem Grenzwerth nähern. (Dedekind, 1872, pp. 9, 29))
- 26.
“[D]enn ich bin außer Stande, irgend einen Beweis für seine Richtigkeit beizubringen, und Niemand ist dazu im Stande. Die Annahme dieser Eigenschaft der Linie ist nichts anderes als ein Axiom …” (ibid., p. 18)
- 27.
“Mit vagen Reden über den ununterbrochenen Zusammenhang in den kleinsten Theilen ist natürlich Nichts erreicht; es kommt darauf an, ein präcises Merkmal der Stetigkeit anzugeben, welches als Basis für wirkliche Deductionen gebraucht werden kann.” (ibid.)
- 28.
“Hat überhaupt der Raum eine reale Existenz, 18|19 so braucht er doch nicht nothwendig stetig zu sein; unzähliger seiner Eigenschaften würden dieselben bleiben, wenn er auch unstetig wäre.” (ibid., pp. 18f.)
- 29.
Méray displayed similar reservations when he called sequences with rational limits “effective” and those with irrational limits “ideal.” (Méray, 1894, § 50) For the sake of a convenience, however, he wanted to keep the language simple and assigned those ideal limits, although they do not really qualify, number signs that were previously reserved for actually existing quantities (un nombre ou une quantité incommensurable, et qu’on représente par le même signe que si elle existait réellement).
- 30.
“ein Axiom, durch welches wir erst der Linie ihre Stetigkeit zuerkennen, durch welches wir die Stetigkeit in die Linie hineindenken. […18|19 …] Und wüßten wir gewiß, daß der Raum unstetig wäre, so könnnte uns doch wieder Nichts hindern, falls es uns beliebte, ihn durch Ausfüllung seiner Lücken in Gedanken zu einem stetigen zu machen; diese Ausfüllung würde aber in einer Schöpfung von neuen Punct-Individuen bestehen und dem obigen Principe gemäß auszuführen sein” (Dedekind, 1872, pp. 18f.)
- 31.
“es wird daher unumgänglich nothwendig, das Instrument R, welches durch die Schöpfung der rationalen Zahlen construiert war, wesentlich zu verfeinern durch eine Schöpfung von neuen Zahlen der Art, daß das Gebiet der Zahlen dieselbe Stetigkeit gewinnt, wie die gerade Linie.” (ibid., p. 16; emphasis in the original)
- 32.
“Die Annahme dieser Eigenschaft der Linie ist nichts als ein Axiom, durch welches wir erst der Linie ihre Stetigkeit zuerkennen, durch welches wir die Stetigkeit in die Linie hineindenken.” (ibid., p. 18)
- 33.
“Stetigkeit sowohl als Dichtigkeit sind Eigenschaften, die der Natur der Sache nach unserer Sinneswahrnehmung unzugänglich sind […] wie sehr sie uns auch im Wesen unserer Anschauung zu liegen scheinen. Es lassen sich aber wohl reine Begriffssysteme konstruieren, denen …Dichtigkeit und Stetigkeit zukommen.” (Weber, 1895, p. 5) See also ibid., p. 9, where Weber states that measurability (and, consequently, the Archimedean Axiom) is not a given fact but is imposed by the thinking observer (durch den denkenden Beobachter hineingelegt). Some may want to object that Weber, as a close friend of Dedekind, cannot count as an independent source. I would disagree; Weber was an eminent mathematician and therefore had to have an independent mind.
- 34.
“Daß solche Anknüpfungen an nicht arithmetische Vorstellungen die nächste Veranlassung zur Erweiterung des Zahlbegriffs gegeben haben, mag im Allgemeinen zugegeben werden […]; aber hierin liegt gewiß kein Grund, diese fremdartigen Betrachtungen selbst in die Arithmetik […] aufzunehmen” (Dedekind, 1872, p. 17).
- 35.
Russell made a similar remark: “In the past, the definition of irrationals was commonly effected by geometrical considerations. This procedure was, however, highly illogical; for if the application of numbers to space is to yield anything but tautologies, the numbers applied must be independently defined.” (Russell, 1903, p. 278)
- 36.
We refrain from calling it conceptual engineering for that would run counter to the deeply felt conviction among mathematicians that finding the right concept is not an engineering task but more like artistic production. If we agree that what is peculiar to solving a mathematical constraint-based design problem is the fact that it happens entirely in the realm of concepts, then “conceptual artist” (a phrase no longer at our disposal to use) would be more fitting than “conceptual engineer.”
- 37.
Nothing of this is as straightforward as it seems, though; see McCarty (1995).
- 38.
“Der Vorgang bei der korrekten Bildung von Begriffen ist m. E. überall derselbe: man setzt ein eigenschaftsloses Ding, das zuerst nichts anderes ist als ein Name oder ein Zeichen A, und gibt demselben ordnungsmäßig verschiedene, selbst unendlich viele verständliche Prädikate […] die einander nicht widersprechen dürfen. […] ist man hiermit vollständig zu Ende, so […] tritt [der Begriff] fertig ins Dasein” (Cantor, 1883, p. 207, footnotes 7+8).
- 39.
See Tait (1996) for a measured account of Frege’s criticism.
- 40.
- 41.
“What is the meaning of a number sign just by itself? Obviously, a ratio among quantities.” (Was bedeutet nun dabei ein Zahlzeichen allein [sc. without unit of measurement]? Offenbar ein Grössenverhältnis; Frege, 1903, § 73).
- 42.
See Illigens (1889), Illigens (1890). Knowing they were walking on thin ice, the editors added a footnote saying that they wanted to stimulate the discussion of an important question from as many different viewpoints as possible without, however, endorsing the particular views expressed by the author. (“[Wir] wünschen dazu beizutragen, dass die wichtige Frage der Irrationalzahlen unter möglichst vielseitigen Gesichtspunkten zur wissenschaftlichen Diskussion gelangt […aber ohne] Verwantwortung für den besonderen Inhalt des betr. Aufsatzes [zu übernehmen].” (Math. Ann. 33 (1889), p. 155, footnote)) For completeness’ sake, we have to mention that Klein, as editor of the journal, got Cantor involved beforehand and asked for the latter’s opinion (Cantor, 1991, p. 262); for more information on Cantor and Illigens, see (Tapp, 2005, chs 2.4, 11.5).
Since no one else is providing the information (cf. ibid., footnotes 41 and 383 on pp. 73, 353, resp.), we do: Eberhard Heinrich Illigens (October 8, 1858–June 23, 1931) spent most of his life in the small town of Beckum, Westphalia, serving as the manager of the local savings bank his father had founded. Initially, he had studied Catholic theology, which had brought him into contact with a correspondent of Cantor, the Catholic philosopher Gutberlet. But Illigens could not finish his studies; his father’s poor health required him to take over as manager of the bank. (The information was kindly provided by Illigens’ grandson. Despite numerous attempts I was unable to confirm this information from independent historical sources except for his date of birth and death, graciously confirmed by Andrea Langner, Kreisarchiv Warendorf.)
- 43.
- 44.
I plan to address this question more fully in another paper after additional case studies have provided more data.
- 45.
Our reason for saying the discussion was cut short is that it took about three generations to decide a similar ballgame in topology, namely, the question of what serves better as the most basic concept: open sets or neighborhoods. But Phase 1 (to be explained momentarily) did not last that long.
For the sake of clarity, here is how I see the development during the 80 years from 1870 to 1950; I distinguish three phases. During Phase 1, the late nineteenth century, not all textbook authors included a treatment of the irrational or real numbers (e.g., Bertrand, Duhamel, and Hermite did not), but those who did adopted one of the three approaches: Biermann (1887) followed Weierstrass (who suspected, however, that his name was used solely to promote sales (see Dugac, 1973, p. 143)); Capelli (1895), Capelli (1897), Dini (1878), Jordan (1893), Pasch (1882), Peano (1884), Ricci (1893), Ricci (1897), and Weber (1895) built on Dedekind; Tannery switched from Méray to Dedekind with the second edition (Tannery, 1904); (Thomae, 1880, p. iii) explicitly adopted Heine’s “formal stance,” while Bachmann (1892) was sympathetic to it; Cantor got employed, without a detour via Heine, by Lipschitz (1877) (but see footnote 24), Stolz (1885), Dini (1892) as well as by Pringsheim (1916) (who gets included to Phase 1 since his lectures still breathe the spirit of the late nineteenth century). It is this phase that I believe was cut short since the tie between Dedekind and Méray–Cantor was never broken, say, by a decisive discussion of their respective theoretical advantages or by exhibiting compelling benefits in the practice of teaching and research. Rather, the question was superseded and made obsolete by the rapid development that followed. The next phase, Phase 2, the early twentieth century, was heralded by Hilbert’s proposal (Hilbert, 1900) to postulate the real numbers by axioms, and not by “filling the gaps.” And while his ideas took some time to sink in, those who authored textbooks in analysis at the time were anxious to present new results in the framework of set theory and abstract spaces (e.g., Lebesgue measure and integral); they had bigger fish to fry than irrational numbers. Borel (1898) did not define irrational numbers (instead, he discussed Liouville’s approximation method), while Carathéodory (1918, § 18) and Hahn (1921, pp. 30f.) took as their starting point the existence of suprema. (Carathéodory, to be fair, prefixed this with a short discussion of the Peano axioms for the natural numbers.) Ten years later, Hahn (1932) did not bother at all about number systems; the real numbers got introduced as a metric space (ibid., § 25.3). According to Landau, all this was the prevailing approach of the day and caused him to write and publish his well-known book (Landau, 1927). Phase 3, the 1950s and beyond, when mathematicians resumed their regular jobs after prolonged war efforts, Bourbaki ruled the day and in their wake the axiomatic method. Consequently, the first textbook in the new spirit, Dieudonné (1960), introduced the reals as an Archimedean ordered field. The reviewer at the time could not agree more: “These axioms can, of course, be proved […] through the Dedekind or Cantor procedures. Although such proofs have great logical interest, they have no bearing whatsoever on Analysis and teachers should not burden students with them […]. This is the right attitude shared by this text.” (Nachbin, 1961, p. 247) And with translations into 13 languages the arguably most influential textbook in analysis, the Principles of Mathematical Analysis by Walter Rudin (1953) (soon to be dubbed “Baby Rudin”), followed suit in its second edition.
The reason that, in respect to phases 1 and 2, we almost exclusively quoted French, German, and Italian authors is that, as far as we know, this specific development took place within that tri-country community. And we can confirm this for England in regard to which Hardy wrote that before the publication of Young (1906) and Hobson (1907), “real function theory was practically unknown [in Great Britain]” (Hardy, 1934, p. 245).
- 46.
- 47.
The quote and the story that goes with it are reported in (Zukav, 1979, p. 208, footnote)
- 48.
My plan was to exactly do that in the present paper. But I ran out of space. So it has to be left to a follow-up paper. But not just the sample size, also the methodology raises concerns. I described what I take to be aspects of the mathematical experience and defended certain claims. In so doing, I did not base my claims on empirical studies. Rather, I continued the tradition of modern epistemology that, to a considerable extent, has been philosophical psychology that, in the absence of relevant empirical studies, appeals to experiences believed to be familiar to everyone (in our case: everyone who received a college-level education in mathematics). While this is shaky methodology, it seems legitimate as a first step; in the end, however, we cannot shirk empirical studies to (dis)confirm what we found in the first step. But I also used historical case studies to gather data about epistemic practices among mathematicians. This was based on the unproven assumption that these practices have not undergone a fundamental change in the last 150 years and therefore still tell us something about mathematics today.
- 49.
Cf. Frege (1879, p. vif.): “[i]t is a task of philosophy to break the power of words over the human mind, by uncovering illusions that through the use of language often almost unavoidably arise […]”
References
Anonymous (1797) Encyclopædia Britannica, or, A Dictionary of Arts, Sciences, and Miscellaneous Literature; …, 18 vols, 3rd ed. Edinburgh: Bell & Macfarquhar (31797).
Alcock, Lara (2013). How to Study as a Mathematics Major, Oxford: Oxford UP (2013).
Arnauld, Antoine & Pierre Nicole (1662). La Logique, ou l’art de penser, in: Œuvres de messire Antonine Arnauld, I–XLI, ed. G. D. Bellegrade & J. Hautefage, Paris: Darnay (1775–1783; repr. Bruxelles: Culture & Civilisation (1964–1967)), vol. xli, pp. 99–416; first Paris: Desprez (1662).
Ascoli, Giulio (1895). “I fondamenti dell’algebra,” in: Rendiconti di Reale Istituto Lombardo di scienze e lettere, Serie II, XXVIII (1895), pp. 1060–1071.
Bachmann, Paul (1892). Vorlesungen über die Natur der Irrationlazahlen, Leipzig: Teubner (1892).
Bäck, Allan (2014). Aristotle’s Theory of Abstraction (= New Synthese Historical Library; 73), Cham: Springer (2014).
Beachy, John A. & Blair, William D. (2006) Abstract Algebra, Long Grove: Waveland P (32006).
Beller, Sieghard et al. (2018). “The Cultural Challenge in Mathematical Cognition,” in: Journal of Numerical Cognition 4:2 (2018), pp. 448–463.
Biermann, Otto (1887). Theorie der analytischen Funktionen, Leipzig: Teubner (1887).
Boniface, Jacqueline (2007). “The Concept of Number from Gauss to Kronecker,” in: Goldstein C., Schappacher N., Schwermer J. (eds), The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Berlin: Springer (2007), pp. 314–342.
Borel, Émile (1898). Leçons sur la théorie des fonctions, Paris: Gauthiers-Villars (1898).
Bottazini, Umberto & Gray, Jeremy (2013). Hidden Harmony—Geometric Fantasies. The Rise of Complex Function Theory (= Sources and Studies in the History of Mathematics and Physical Sciences), New York: Springer (2013).
Brandom, Robert B. (1994). Making It Explicit: Reasoning, Representing, and Discursive Commitment, Cambridge, MA: Harvard UP (1994).
_________ (2000). Articulating Reasons: An Introduction to Inferentialism, Cambridge, MA; Harvard UP (2000).
Cantor, Georg (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. Ernst Zermelo and a biography by Adolf Fraenkel, Berlin: Springer (1932).
_________ (1872). “Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen,” in Cantor (1932), pp. 92–102; first in: Mathematische Annalen 5 (1872), pp. 123–132.
_________ (1883). “Grundlagen einer allgemeinen Mannigfaltigkeitslehre, Nr. 5,” in Cantor (1932), pp. 165–209; first as “Ueber unendliche, lineare Punktmannichfaltigkeiten,” in: Mathematische Annalen, 21 (1883), pp. 545–591.
_________ (1887/8). “Mitteilungen zur Lehre vom Transfiniten,” in: Cantor (1932), pp. 378–439; first in: Zeitschrift für Philosophie und philosophische Kritik, NF 91 (1887), pp. 81–125; 92 (1888), pp. 240–265.
_________ (1991). Georg Cantor. Briefe, ed. Herbert Meschkowski, Winfried Nilson, Berlin: Springer (1991).
Capelli, Alfredo (1895). Lezioni di algebra complentare, ad uso degli aspiranti alla licenza universitaria in scienze fisiche e matematiche, Naples: Pellerano (1895).
_________ (1897). “Sulla introduzione dei numeri irrazionali col metodo delle classi contigue,” in: Giornale di Matematiche, XXXV (1897), pp. 209–234.
Carathéodory, Constantin (1918). Vorlesungen über reelle Funktionen, Leipzig: Teubner (1918).
Carothers, Neal L. (2000). Real Analysis, Cambridge: Cambridge UP (2000).
Cauchy, Augustin-Louis (1882f.). Œuvres complètes, vols I:1–12, II:1–15, Paris: Gauthier-Villar (1882–1974).
_________ (1821). Course d’analyse de l’Ecole Royale Polytechnique. I: Analyse algébrique, in: Cauchy (1882f.), II:3; first Paris: Debure (1821).
_________ (1823). Le Résumé des Leçons données à I’Ecole royale Polytechnique sur le Calcul infinitésimal, in: Cauchy (1882f.), II:4, pp. 5–261; first Paris: Debure (1823).
Cook, Roy T. (ed) (2007). The Arché Papers on the Mathematics of Abstraction (= Western Ontario Series in Philosophy of Science; 71), Dordrecht: Springer (2007).
Cox, David A. (2012) Galois Theory (= Pure and Applied Mathematics; 106), Hoboken, NJ: Wiley (2012).
Dantscher, Victor von (1908). Vorlesungen über die Weierstrasssche Theorie der Irrationalzahlen, Leipzig: Teubner (1908).
Dawson, Jr., John W. (2015). Why Prove it Again? Alternative Proofs in Mathematical Practice, Basel: Birkhäuser (2015).
Dedekind, Richard (1871). “Ueber die Composition der binären quadratischen Formen” (= Supplement X), in: Peter G. Lejeune Dirichlet, Vorlesungen über Zahlentheorie, posthum ed. R. Dedekind, Braunschweig: Vieweg (21871), pp. 380–497.
_________ (1872). Stetigkeit und irrationale Zahlen, Braunschweig: Vieweg (1872); Engl. tr by Wooster W. Beman, as “Continuity and irrational numbers,” in: Essays on the Theory of Numbers, Chicago: Open Court (1901), pp. 1–27.
Dieudonné, Jean (1960). Foundations of Modern Analysis, Volume 1 (= Pure and Applied Mathematics; 10), New York: Academic P (1960, 21969, rev. 1972, 1974, 1976, 1977, 1978).
Dini, Ulisse (1878). Fondamenti per la teorica delle funzioni di variabili reali, Pisa: Nistri (1878).
_________ (1892). Grundlagen für eine Theorie der Functionen einer veränderlichen reellen Grösse, rev. tr. of Dini (1878) by Jacob Lüroth & Adolf Schepp, Leipzig: Teubner (1892).
Du Bois-Reymond, Paul (1882). Die Allgemeine Functionentheorie. Erster Theil: Metaphysik und Theorie der mathematischen Grundbegriffe: Grösse, Grenze, Argument und Function, Tübingen: Laupp (1882).
Dugac, Pierre (1970). “Charles Méray (1835–1911) et la notion de limite,” in: Revue d’histoire des sciences et de leurs applications, 23:4 (1970), pp. 333–350.
_________ (1973). “Eléments d’analyse de Karl Weierstrass,” in: Archive for History of Exact Sciences, 10:1/2 (April 1973), pp. 41–176.
Ebert, Philip A. & Rossberg, Marcus (eds) (2017). Abstractionism. Essays in Philosophy of Mathematics, Oxford: Oxford UP (2017).
Edwards, Harold E. (1995). “Kronecker on the foundations of mathematics,” in: Hintikka (1995), pp. 45–52.
Ehrlich, Philip (2006). “The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes,” in: Archive for History of Exact Sciences, 60:1 (January 2006), pp. 1–121.
_________ (2018). Contemporary infinitesimalist theories of continua and their late 19th- and early 20th-century forerunners. arXiv:1808.03345v3 (12/27/18); 73 pp.
Fitzpatrick, Richard (ed.) (2007) Euclid’s Elements of Geometry, the Greek text of J. L. Heiberg […] ed. & tr. by R. Fitzpatrick, (2007, rev. 2008); URL: http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
Frege, Gottlob (1879). Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle: Nebert (1879); Engl. tr. of Part 1 (§§ 1–12) in: The Frege Reader, ed. Michael Beaney, Oxford, Blackwell (1997), pp. 48–78.
_________ (1903). Grundgesetze der Arithmetik, begriffsschriftlich abgeleitet, Bd. 2, Jena: Pohle (1903); Engl. tr. P. Ebert and M. Rossberg, Basic Laws of Arithmetic: Derived using concept-script, Oxford: Oxford UP (2013).
Gonzalez-Velasco, Enrique A. (1992). “Connections in Mathematical Analysis: The Case of Fourier Series,” in: American Mathematical Monthly, 99:5 (May 1992), pp. 427–441.
Grabiner, Judith V. (1981). The Origins of Cauchy’s Rigorous Calculus, Cambridge, MA: MIT P (1981).
Hahn, Hans (1921). Theorie der reellen Funktionen. Erster Band, Berlin: Springer (1921).
_________ (1932). Reelle Funktionen. Bd. 1: Punktfunktionen (= Mathematik und ihre Anwendungen; 13), Leipzig: Akademische Verlagsgesellschaft (1932).
Hankel, Hermann (1867). Vorlesungen über die complexen Zahlen und ihre Functionen. I. Theil: Theorie der complexen Zahlensysteme, insbesondere der gemeinen imaginären Zahlen und der Hamiltonischen Quaternionen nebst ihrer geometrischen Darstellung, Leipzig: Voss (1867).
Hardy, Godfrey H. (1934). “Ernest William Hobson, 1856–1933,” In: Obituary Notices of Fellows of the Royal Society, 1:3 (1934), pp. 236–249.
Hawkins, Thomas (1979). Lebesgue’s Theory of Integration. Its Origins and Development, Providence, RI: AMS (2002); first Madison: University of Wisconsin Press (1970); New York: Chelsea (21979).
Heath, Thomas L. (ed.) (1908). The Thirteen Books of Euclid’s Elements, tr. from the text of Heiberg, with introduction and commentary by T. L. Heath, Cambridge; Cambridge UP (1908, 21926); repr. New York: Dover (1956).
Heine, Eduard (1870). “Ueber trigonometrische Reihen,” in: Journal für die reine und angewandte Mathematik (= Crelle’s/Borchardt’s Journal), 71 (1872), pp. 353–365.
Heine, Eduard (1872). “Die Elemente der Functionenlehre,” in: Journal für die reine und angewandte Mathematik (= Crelle’s/Borchardt’s Journal), 74 (1872), pp. 172–188.
Henrich, Joseph & Heine, Steven J. & Norenzayan, Ara (2010). “The weirdest people in the world?,” in: Behavioral and brain sciences, 33:2–3 (June 2010), pp. 61–135.
Hinman, Peter G. (2005). Fundamentals of Mathematical Logic, Wellesley, MA: Peters (2005).
Hilbert, David (1900). “Über den Zahlbegriff,” in: Jahresbericht der Deutschen Mathematiker-Vereinigung, 8 (1900), pp. 180–183.
Hintikka, Jaakko (ed.) (1995). From Dedekind to Gödel. Essays on the Development of the Foundations of Mathematics (= Synthese Library; 251), Dordrecht: Kluwer (1995).
Hobson, Ernest W. (1907). The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Cambridge: Cambridge UP (1907).
Illigens, Eberhard H. (1889). “Zur Weierstrass’-Cantor’schen Theorie der Irrationalzahlen,” in: Mathematische Annalen, XXXIII (1889), pp. 155–160.
_________ (1890). “Zur Definition der Irrationalzahlen,” in: Mathematische Annalen, XXXV (1890), pp. 151–455.
Jordan, Camille (1893). Cours d’analyse de l’École Polytechnique, t. 1: Calcul différentiel, Paris: Gauthioer-Villars (1882, 21893).
Klein, Felix (1895). Vorträge über ausgewählte Fragen der Elementargeometrie, red. by F. Tägert (= Festschrift zu der Pfingsten 1895 in Göttingen stattfinden dritten Versammlung des Vereins zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts), Leipzig: Teubner (1895); cited according to Engl. tr. by W. W. Beman & D. E. Smith, Famous problems of elementary geometry, Boston: Ginn (1897).
Knauer, Ullrich (2011). Algebraic Graph Theory: Morphisms, Monoids and Matrices (= De Gruyter Studies in Mathematics; 41), Berlin: de Gruyter (2011).
Kossak, Ernst (1872). Die Elemente der Arithmetik (= Programmabhandlung des Werderschen Gymnasiums, Berlin), Berlin: Nicolai (1872).
Kronecker, Leopold (1886). “Ueber einige Anwendungen der Modulsysteme auf elementare algebraische Fragen,” in: Werke, ed. Kurt Hensel, Bd. 3.1, Leipzig: Teubner (1899), pp. 145–208; first in: Journal für die reine und angewandte Mathematik (= Crelle’s/Borchardt’s Journal), 99 (1886), pp. 329–371.
Landau, Edmund (1927). Grundlagen der Analysis. (Das Rechnen mit den ganzen, rationalen, irrationalen, komplexen Zahlen) Ergänzung zu den Lehrbüchern der Differential- und Integralrechnung, Leipzig: Akademische Verlagsgesellschaft (1927); numerous reprints.
Lee, Haw Y. (2017). Origami-constructible numbers, Athens, GA: U of Georgia (2017); url: https://getd.libs.uga.edu/pdfs/lee_hwa-young_201712_ma.pdf.
Lende, Daniel H. & Downey, Greg (eds) (2012). The Encultured Brain: An Introduction to Neuroanthropology, Cambridge, MA: MIT P (2012).
Lipschitz, Rudolf (1877). Lehrbuch der Analysis. Bd. 1: Grundlagen der Analysis, Bonn: Cohen (1877).
Lovejoy, Arthur O. (1936). The Great Chain of Being: A Study of the History of an Idea, Cambridge: Harvard UP; many reprints.
Marquis, Jean-Pierre (2013). “Mathematical Abstraction, Conceptual Variation and Identity,” in: Schroeder-Heister, Peter et al., Logic, Methodology and Philosophy of Science. Proceedings of the 14th International Congress (Nancy). Logic and Science Facing the New Technologies, London: College Publications (2014), pp. 299–322.
_________ (2016). “Stairway to Heaven: The Abstract Method and Levels of Abstraction in Mathematics,” in: The Mathematical Intelligencer 38:3 (2016),pp. 41–51.
McCarty, D. Charles (1995). “The mysteries of Richard Dedekind,” in: Hintikka (1995), pp. 53–96.
Méray, Charles (1870). “Remarques sur la nature des quantités définies par la condition de servir de limites à des variable données,” in: Revue sociétés des savantes. Sciences mathématiques, physique er naturelle, Second Series IV (Janvier–Juin 1869; 1870), pp. 280–289.
_________ (1872). Nouveau précis d’analyse infinitésimale, Paris: Savy (1872).
_________ (1894). Leçons nouvelles sur l’analyse infinitésimale et ses applications géométriques. Première partie: Principes généraux, Paris: Gauthier-Villars (1894).
Mittag-Leffler, Gösta (1910). “Sur les fondements arithmétiques de la théorie des fonctions d’après Weierstrass,” in: Compte rendu du congrès des mathematiciens, tenu à Stockholm, 22–25 Septembre 1909, ed. Gösta Mitag-Leffler & Ivar Fredholm, Leipzig: Teubner (1910), pp. 10–31.
Nachbin, Leopoldo (1961). “Review of Dieudonné (1960),” in: Bulletin of the American Mathematical Society, 67:3 (1961), pp. 246–250.
Northoff, Georg (2010). “Humans, brains, and their environment: marriage between neuroscience and anthropology?,” in: Neuron, 65 (March 2010), pp. 748–751.
Pasch, Moritz (1882). Einführung in die Differential- und Integralrechnung, Leipzig: Teubner (1882).
Peano, Guiseppe (1884). Calcolo differenziale e principii di calcolo integrale, pubblicato con aggiunte, Torino: Bocca (1884).
_________ (1899). “Sui numeri irrazionali,” in: Revue de mathématiques (Rivista di matematica), VI:4 (1899), pp. 126–140.
Peterson, Julius (1878). Theorie der algebraischen Gleichungen, Kopenhagen: Höst (1878).
Pincherle, Salvatore (1880). “Saggio di una introduzione alla Teoria delle funzioni analitiche secondo i principii del Prof. C. Weierstrass,” in: Giornale di Matematiche XVIII (1880), pp. 178–254, 317–357.
Pringsheim, Alfred (1898). “Irrationalzahlen und konvergente unendliche Prozesse,” in: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 1: Arithmetik und Algebra, ed. Wilhelm F. Meyer, Leipzig: Teubner (1898–1904), pp. 47–146.
_________ (1916). Vorlesungen über Zahlen- und Funktionenlehre. Bd. 1: Zahlenlehre (= Teubner’s Sammlung von Lehrbüchern auf dem Gebiete der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen; XL:I.1), Leipzig: Teubner (1916, 21923).
____ & Molk, Jules (1904). “Nombres irrationnels et notion de limite,” tr. and rev. of Pringsheim (1898) by J. Molk, in: Encyclopédie des sciences mathématiqus pures et appliquées, publiée …, ed. J. Molk, t. 1: Arithmétique, Paris: Gauthier-Villars (1904), pp. 133–208.
Propp, James (2013). “Real analysis in reverse,” in: American Mathematical Monthly, 120:5 (May 2013), pp. 392–408.
Rav, Yehuda (1999). “Why do we prove theorems?” in: Philosophia Mathematica, 7 (1999), pp. 5–41.
Ricci, Gregorico (1893). “Saggio di una teoria dei numeri reali. Secondo il concetto di Dedekind,” in: Atti del R[eale] Instituto Veneto di Scienze, Lettere ed Arti, Ser. 7:4, LI (1893), pp. 233–281.
_________ (1897). “Della teoria dei numeri reali. Secondo il concetto di Dedekind,” in: Giornale di Matematiche, XXXV (1897), pp. 22–74.
Rosa, Milton & Shirley, Lawrence & Gavarrete, Maria Elena & Alangui, Wilfredo V. (eds) (2017). Ethnomathematics and its Diverse Approaches for Mathematics Education (= ICME 13 Monographs), Cham: Springer (2017).
Rota, Gian-Carlo (2008). “Introduction,” in Davis, Philip J. & Hersh, Reuben (1981). The Mathematical Experience, Boston: Brikhäuser (1981); Study Edition, with Elena A. Marchisotto, Boston: Brikhäuser (1995), pp. xxi–xxiii.
Rudin, Walter (1953). Principles of Mathematical Analysis (= International Series in Pure and Applied Mathematics), New York: McGraw-Hill (1953, 21964, 31976).
Russell, Bertrand (1903). The Principles of Mathematics, Vol. 1, Cambridge: Cambridge UP (1903).
_________ (1919). Introduction to Mathematical Philosophy, London: Allen & Unwin (1919).
Smoryński, Craig (2008). History of Mathematics. A Supplement, New York: Springer (2008).
Spalt, Detlef D. (1991). “Die mathematischen und philosophischen Grundlagen des Weierstraßschen Zahlbegriffs zwischen Bolzano und Cantor,” in: Archive for History of Exact Sciences, 41:4 (1991), pp. 311–362.
Starikova, Irina (2012). “From Practice to New Concepts: Geometric Properties of Groups,” in: Philosophia Scientiæ, 16:1 (2012), pp. 129–151.
Stolz, Otto (1885). Vorlesungen über allgemeine Arithmetik. Nach den neueren Ansichten, Bd. 1: Allgemeines und Arithmetik der reellen Zahlen, Leipzig: Teubner (1885).
Tait, William W. (1996) “Frege versus Cantor and Dedekind: On the concept of number,” in: Frege, Russell, Wittgenstein: Essays in Early Analytic Philosophy (in honor of Leonard Linsky), ed. W. W. Tait, Lasalle: Open Court (1996), pp. 213–248; repr. in: Frege: Importance and Legacy, ed. M. Schirn, Berlin: de Gruyter (1996), pp. 70–113.
Tannery, Jules (1904). Introduction à la théorie des fonctions d’une variable, t. 1, Paris: Hermann (1886, 21904).
_________ (1908). “Review of Dantscher (1908),” in: Bulletin des sciences mathématiques, Ser. II, XXXII (1908), pp. 101–105.
Tapp, Christian (2005). Kardinalität und Kardinäle. Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit (= Boethius: Texte und Abhandlungen zur Geschichte der Mathematik und der Naturwissenschaften; 53), Wiesbaden: Steiner (2005).
Thomae, C. Johannes (1880). Elementare Theorie der analytischen Functionen einer complexen Veränderlichen, Halle: Nebert (1880).
Thomassen, Carsten (1992). “The Jordan-Schonflies theorem and the classification of surfaces,” in: American Mathematical Monthly, 99:2 (Feb., 1992), pp. 116–131.
Ullrich, Peter (1989). “Weierstraß’ Vorlesung zur ‘Einleitung in die Theorie der analytischen Funktionen’,” in: Archive for History of Exact Sciences, 40:2 (1989), pp. 143–172.
Van Vleck, Edward B. (1914). “The influence of Fourier’s series upon the development of mathematics,” in: Science, NS, 39:995 (Jan. 23, 1914), pp. 113–124.
Volkert, Klaus Th. (1986). Die Krise der Anschauung. Eine Studie zu formalen und heuristischen Verfahren in der Mathematik seit 1850 (= Studien zur Wissenschafts-, Sozial- und Bildungsgeschichte der Mathematik; 3), Göttingen: Vandenhoeck & Ruprecht (1986).
Wantzel, Pierre L. (1837). “Recherches sur les moyens de reconnaître si un Probléme de Géométrie peut se résoudre avec la règle et le compas,” in: Journal de mathématiques pures et appliquées 10 (1837), pp. 366–372.
Weinberg, Julius (1973). “Abstraction in the formation of concepts,” in: Dictionary of the History of Ideas. Studies of Selected Pivotal Ideas, I–IV, ed. by Philip P. Wiener, New York: Scribners (1973), vol. 1, pp. 1–9.
Weber, Heinrich (1895). Lehrbuch der Algebra, Bd. 1, Braunschweig: Vieweg (1895, 21912).
Weierstrass, Karl (1903). Mathematische Werke. Bd. 3: Abhandlungen III, Berlin: Mayer & Müller (1903).
Whitney, Hassler (1932). “Non-Separable and Planar Graphs,” in: Transactions of the American Mathematical Society, 34:2 (April 1932), pp. 339–362.
Yates, Robert C. (1942). The Trisection Problem, Baton Rouge, LA: Franklin P (1942).
Young, William H. & Chisholm Young, Grace (1906). The Theory of Sets of Points, Cambridge: Cambridge UP (1906).
Zukav, Gary (1979). The Dancing Wu Li Masters. An Overview of the New Physics, New York: Bantam (1980); first ed. New York: Morrow (1979).
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Buldt, B. (2022). Abstraction by Embedding and Constraint-Based Design. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-95201-3_8
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