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nLab > Latest Changes: "Cohesive Toposes and Cantor's 'lauter Einsen'"

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2013
    • (edited Jan 8th 2014)

    I gave Lawvere’s Cohesive Toposes and Cantor’s lauter Einsen a category:reference-entry.

    The article makes an interesting claim: that Cantor’s original use of terminology was distorted by its editors to become what we now take to be the standard meaning in set theory. But that instead Cantor really meant “cohesive types” when saying “Menge” and used “Kardinale” really for what we call the underlying set of a cohesive type.

    The article also contains the proposal that adjoint modalities capture Hegel’s “unity of opposites”.

    Notice that where the English translations of Science of Logic say “the One”, the original has “Das Eins”, which might just as well be translated with “The Unit”. In view of this and looking through Hegel’s piece on discreteness and repulsion, I think it is clear that Hegel’s “Einsen” is precisely Cantor’s “Einsen” as recalled by Lawvere. Namely: copies of the unit type.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeNov 3rd 2013
    • (edited Nov 3rd 2013)

    Interesting!

    Regarding that we have so few philosophy entries, please add category: philosophy at the bottom of any new entries on this subject. Entry on Hegel’s work could also be linked from philosophy.

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeNov 14th 2013

    Your conjecture about an etymological connection between ‘lauter’ and ‘lot of’ doesn't seem to work:

    • German ‘lauter’ derives from the proto-Germanic root ‘khlutra’ (meaning ‹pure›) (source);
    • English ‘lot’ derives from the proto-Germanic root ‘khlutom’ (meaning ‹lottery›) (source).

    In particular, the meaning ‹a large amount› for English ‘lot’ dates only from the 19th century.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 14th 2013

    My attempt to save an edit to this page seems to have crashed the Lab.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2013
    • (edited Nov 14th 2013)

    Thanks for actually checking this hypothesis!

    But I think there is a problem with a double meaning of “lauter”:

    it is indeed also a somewhat out-dated word for “pure” (kept in modern German mainly in “Läuterung”, which is something like spiritial purification), but this is different from the “lauter” as in “lauter Einsen”, which really means “a lot of”.

    Of course this doesn’t mean my little conjecture is right. But for checking it one would somehow have to find the source of “lauter” with that other meaning.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeNov 14th 2013

    it is indeed also a somewhat out-dated word for “pure” (kept in modern German mainly in “Läuterung”, which is something like spiritial purification), but this is different from the “lauter” as in “lauter Einsen”, which really means “a lot of”.

    So you think that the homonym is of a different etymology (hence not coming there after a sequence of semantic shifts) ?

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2014

    A couple of questions:

    • p.7 Is the distinction between objective and subjective cohesion interesting? Maybe something here corresponds to metaphysical and epistemic modalities (“As far as I know, it might be the case…”)

    • p. 11 Do we see the almost ’Galosian’ search for intermediate categories between the cohesive and base in the (∞,1)(\infty, 1) case? We discussed somewhere the possibility of factorization of general cohesion into infinitesimal and non-infinitesimal parts.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2014

    concerning p. 7, I suppose I see what the vague idea is, but I am not sure what useful formalization it might allude to

    concerning p. 11, yes, an example would be synthetic differential cohesion sitting over infinitesimal cohesion sitting over sets.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 8th 2014

    Re p.11, I suppose what he’s pointing to there is the same idea as the one I mentioned over here about dimensions and levels of determinateness. He’s picturing reflexive directed graphs as 1-dimensional and sitting between kardinale and mengen.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2014
    • (edited Jan 8th 2014)

    Yes, I think so, too.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 2nd 2015

    If I wanted to read some of Cantor’s original writing (hopefully in translation to English), to decide whether I believe Lawvere’s claim about “Mengen” being a precursor of cohesion, where ought I to look? Lawvere doesn’t include any indication other than “page 283 of a copy of Cantor’s works”.

    • CommentRowNumber12.
    • CommentAuthorThomas Holder
    • CommentTimeFeb 2nd 2015
    • (edited Feb 2nd 2015)

    The edition Lawvere refers to is Ernst Zermelo (ed.) : Georg Cantor-Gesammelte Abhandlungen mathematischen und philosophischen Inhalts , Springer Berlin 1932 (it’s still in print). The text that starts on page 282 is ’Beiträge zur Begründung der transfiniten Mengenlehre’ , Math. Ann. 46 (1895) pp.481-512. That should be available in an English translation or you might try for the scan of the German article at the Göttinger digitalisierungszentrum .

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 3rd 2015

    English version here: https://archive.org/details/contributionstof00cant for those that want it.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015

    Thanks, Thomas and David, I have added that to the entry.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015
    • (edited Feb 3rd 2015)

    The relevant bit is probably this here (pp. 85 of the above English translation)

    The conception of Power or Cardinal Number

    By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) MM of definite and separate objects of our intuition or our thought. These objects are called the “elements” of MM. In signs we express this thus :

    (i) M={m}M = \{m\}.

    We denote the uniting of many aggregates MM, NN, PP, ⋯\cdots, which have no common elements, into a single aggregate by

    (2) (M,N,P,⋯)(M, N, P, \cdots).

    The elements of this aggregate are, therefore, the elements of MM, of NN, of PP, …, taken together. We will call by the name “part” or “partial aggregate ” of an aggregate M any other aggregate M 1M_1 whose elements are also elements of MM. If M 2M_2 is a part of M 1M_1 and M 1M_1 is a part of MM, then M 2M_2 is a part of MM.

    Every aggregate MM has a definite “power”, which we will also call its “cardinal number”.

    We will call by the name “power” or “cardinal number” of MM the general concept which, by means of our active faculty of thought, arises from the aggregate MM when we make abstraction of the nature of its various elements mm and of the order in which they are given.

    [482] We denote the result of this double act of abstraction, the cardinal -number or power of M, by (3) M¯¯\overline{\overline{M}}

    Since every single element mm, if we abstract from its nature, becomes a “unit,” the cardinal number MM is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate MM. We say that two aggregates MM and NN are “equivalent,” in signs

    (4) M∼NM \sim N or N∼MN \sim M

    if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. To every part M 1M_1 of MM there corresponds, then, a definite equivalent part N 1N_1 of NN, and inversely.

    [[ …]]

    Every aggregate is equivalent to itself

    (5) M∼MM \sim M

    If two aggregates are equivalent to a third, they are equivalent to one another ; that is to say :

    (6) from M∼PM \sim P and N∼PN \sim P follows M∼NM \sim N.

    Of fundamental importance is the theorem that two aggregates MM and NN have the same cardinal number if, and only if, they are equivalent : thus,

    (7) from M∼NM \sim N we get M¯¯=N¯¯\overline{\overline{M}} = \overline{\overline{N}},

    and

    (8) from M¯¯=N¯¯\overline{\overline{M}} = \overline{\overline{N}} we get M∼NM \sim N.

    Thus the equivalence of aggregates forms the necessary and sufficient condition for the equality of their cardinal numbers.

    [483] In fact, according to the above definition of power, the cardinal number MM remains unaltered if in the place of each of one or many or even all elements mm of MM other things are substituted. If, now, M∼NM \sim N, there is a law of co-ordination by means of which MM and NN are uniquely and reciprocally referred to one another; and by it to the element mm of MM corresponds the element nn of NN. Then we can imagine, in the place of every element mm of MM, the corresponding element nn of NN substituted, and, in this way, MM transforms into NN without alteration of cardinal number. Consequently

    M¯¯=N¯¯\overline{\overline{M}} = \overline{\overline{N}}.

    The converse of the theorem results from the remark that between the elements of MM and the different units of its cardinal number MM a reciprocally univocal (or bi-univocal) relation of correspondence subsists. For, as we saw, M¯¯\overline{\overline{M}} grows, so to speak, out of MM in such a way that from every element ww of MM a special unit of MM arises. Thus we can say that

    (9) M∼M¯¯M \sim \overline{\overline{M}}

    In the same way N∼N¯¯N \sim \overline{\overline{N}}. If then M¯¯=N¯¯\overline{\overline{M}} = \overline{\overline{N}}, we have, by (6), M∼NM \sim N.

    We will mention the following theorem, which results immediately from the conception of equival ence. If MM, NN, PP, … are aggregates which have no common elements, M′M', N′N', P′P', … are also aggregates with the same property, and if

    M∼M′M \sim M', N∼N′N \sim N', P∼P′P \sim P', …,

    then we always have

    (M,N,P,...)∼(M′,N′,P′,⋯)(M, N, P, . . .) \sim (M', N', P', \cdots).

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015

    I’d say this remains a little inconclusive. On the one hand it is true that Cantor says that a cardinal is an “aggregate composed of units”, the units being one for each element of the original aggregate, and he clearly suggests that constructing this is an actual process “of our active faculty of thought” and not just a trivial renaming. But then when in (9) he has M∼M¯¯M \sim \overline{\overline{M}} either we read “∼\sim” as being equivalence after applying that process, or else there is no distinction after all.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015

    Just for entertainment, check out the scan of the German original here. Next to “(4) M∼NM \sim N oder N∼MN \sim M” somebody has written by hand the word “univalenz”.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2015

    Thanks everyone! I agree that it is inconclusive. The passage

    We will call by the name “power” or “cardinal number” of MM the general concept which, by means of our active faculty of thought, arises from the aggregate MM when we make abstraction of the nature of its various elements mm and of the order in which they are given.

    suggests to me that when forming the cardinal of an aggregate we discard only the nature of its elements and their ordering, rather than any “cohesion” that might exist between them. Although I suppose one might generously try to read “the nature of its elements” as including relations between its elements. On the other hand, the passage that Urs quoted

    the cardinal number MM is a definite aggregate composed of units

    suggests that a cardinal is a special kind of aggregate — a “definite” one that is “composed of units” — with the implication that the original aggregate might not have had these properties.

    Probably the “truth”, insofar as there is any, is that Cantor’s conception was not precise enough from a modern perspective to say one way or the other. I expect that when thinking of aggregates, he had particular geometric ones in mind, where the elements do come with “cohesive” relationships to each other, but that he didn’t carefully separate in his mind the elements of those aggregates from the relationships between those elements.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015

    Let’s see. Lawvere argues his point actually by reference to further authors. He says on p. 2:

    Now a very interesting point which emerges from a study of Cantor’s works is that he himself cites the origin of the word ’Mächtigkeit’ in the work of the great Swiss geometer Jakob Steiner, who apparently used this term to signify isomorphism in a different category, namely, the category of algebraic spaces.

    […]

    Cantor himself asserts that he lifted this concept of isomorphism from its geometric context in order to arrive at his—necessarily more abstract—concept of isomorphism.

    Next is a somewhat mysterious remark:

    This fact is not to be found in any book I have seen (although it is emphasized in a recent paper of Colin McLarty who noticed it independently).

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 3rd 2015

    That seems to me to be mainly an argument that the categorical notion of isomorphism was noticed as an informal analogy before it was made precise — which is not at all surprising — and not anything relating to cohesion. Although it’s certainly interesting if the isomorphisms in the “most basic” category of sets (bijections) were originally defined by analogy with previously studied isomorphisms in some more structured category.

    • CommentRowNumber21.
    • CommentAuthorSridharRamesh
    • CommentTimeFeb 3rd 2015
    • (edited Feb 3rd 2015)

    Extremely minor nit: (7) and (8) in post 15 are missing the double overbars on M and N in “M = N”. [This confused me for a bit, which is the only reason I bring it up]

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeFeb 3rd 2015
    • (edited Feb 3rd 2015)

    Thanks! Fixed now.

    By the way, I have created a category:reference entry for this: Beiträge zur Begründung der transfiniten Mengenlehre.

    • CommentRowNumber23.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 4th 2015
    • (edited Feb 4th 2015)

    when in (9) he has M∼M¯¯M \sim \overline{\overline{M}} either we read “∼\sim” as being equivalence after applying that process,

    It should probably be noted that ∼\sim is implicitly ignoring the structure of the Mengen involved, namely each Menge comes with an order etc. Just as one distinguishes between isomorphisms of ordinals as sets and as ordinals. The real content of ∼\sim is given by (7) and (8), and these can in fact give a definition of ∼\sim.

    from every element ww of MM a special unit of MM arises

    this is none other than M→M¯¯M \to \overline{\overline{M}}, which we know is an isomorphism after mapping back to Kardinalen, if we accept that Cantor thought that M¯¯=(M¯¯)¯¯\overline{\overline{M}}=\overline{\overline{\left(\overline{\overline{M}}\right)}}

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 4th 2015

    Actually, I think the really remarkable thing about this passage is that Cantor clearly thinks of the cardinal number M¯¯\overline{\overline{M}} as itself being an aggregate like MM (so that, in particular, we can say things like M∼M¯¯M \sim \overline{\overline{M}}), and yet it also satisfies (7) that equivalent aggregates have equal cardinal numbers — not just equivalent but equal.

    As we know now, there are various ways to define “cardinal number” precisely that have this last property. For instance, the cardinal number of MM could be the set of all sets bijective to MM (perhaps bounded in rank to make it a set) — but then it is not itself a set bijective to MM. The modern set-theorist’s definition is as the smallest von Neumann ordinal bijective to MM — but this is clearly not what Cantor had in mind, because he speaks of the cardinal number as abstracting away from the order in which the elements of MM are given, so that his cardinal numbers are not themselves ordered.

    Or, of course, we could invoke univalence. (-: I think one could make an argument that this is relatively close to what Cantor had in mind — not that he actually thought of univalence as such, but that he thought of the elements of his cardinal numbers as sufficiently “featureless” that two collections of the same number of such elements would actually be the same collection. Which of course implies that he did not think of the elements of an arbitrary aggregate as similarly featureless. I suppose one might take that as an argument for Lawvere’s reading.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 4th 2015

    Of course, logically there is something circular about saying that a “discrete” type is one that is “composed of definite units” meaning “a coproduct of copies of 1”, since then one has to ask “how many copies of 1?” and the only correct answer is that it’s a coproduct indexed by a discrete type. Every type XX is the coproduct of XX-many copies of 1. (-:

    • CommentRowNumber26.
    • CommentAuthorThomas Holder
    • CommentTimeFeb 4th 2015

    For a rough idea of what is going on here I think it is helpful to read Cantor against a ’philosophy’ of number (and compare this e.g. with some of the accounts reviewed in Frege’s booklet ’Grundlagen der Arithmetik’ from 1884) because the Cantorian enterprise is essentially the quest to extrapolate counting from finite aggregates to ω\omega and beyond: to make a respectable number out of ∞\infty.

    So let’s say we think the aggregate of a military parade M=⟨a,b,c⟩M=\langle a, b, c\rangle. We have to chop this up in order to know what we count -soldiers, platoons or rows, hence by first abstraction M¯={a,b,c}\overline{M}=\{a,b,c\}. Now the next basic ingredient of counting is that we ’count as units’ and so we end up with M¯¯={1,1,1}\overline{\overline{M}}=\{1,1,1\}. Of course the problem with this cardinal of lauter Einsen is that it collapses to {1}\{1\} that was the contradiction pointed out by Zermelo: either a and b are definite and different or they are abstract units and coincide!

    Here is where Lawvere comes in by objectifying the contradiction {a,b,c}=M¯¯={1}\{a,b,c\}=\overline{\overline{M}}=\{1\} in the concept of cardinal by the adjoint cylinder. We see the deeper issue is that the idea of number is inherently contradictory: beingdifferent⊣countingasonebeing different\dashv counting as one. This should be explicit in Hegel somewhere, underlies Frege’s acid criticism of other people’s numbers and the ’unit’ idea is probably implicitly based on the intuition that counting is ’summing up’: ∫χ M\int\chi _M.

    So a more balanced view would be to credit Cantor with the idea of a structureless set (Myhill’s observation in Lawvere’s lectures on ETCS) which are arrived at by abstraction from structured sets (Münchhausen foundations à la Lawvere) and credit the ’fallen guy’ of the story -Zermelo with the observation that the concept of structural sets is inherently contradictory and credit Lawvere with the idea that such conceptual contradictions can be productive in mathematics.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2015
    • (edited Feb 4th 2015)

    We see the deeper issue is that the idea of number is inherently contradictory: beingdifferent⊣countingasonebeing different\dashv counting as one. This should be explicit in Hegel somewhere,

    Yes, this is in and around the section Repulsion and Attraction.

    I think this is naturally thought of as being the ∫⊣♭\int \dashv \flat-adjunction: think in a 1-topos first, not to overburden the discussion: then for XX a connected cohesive set, in ∫X≃*\int X \simeq \ast it has collapsed to a single One by all the points collapsing under their cohesive attraction. Conversely, in ♭X\flat X all the points have repelled each other against their cohesive attraction and now form just an aggregate of many separate Ones.

    Moreover, the unity of these two opposites Hegel says is Quality, and that makes perfect sense. Given e.g. a Cartesian space ℝ n\mathbb{R}^n then the opposition between

    1. the single One ʃℝ nʃ \mathbb{R}^n obtained by having all its points collapse under the attraction of its cohesion;

    2. the many Ones ♭ℝ n\flat \mathbb{R}^n obtained by having the points of ℝ n\mathbb{R}^n repel each other against their cohesive attraction

    exhibits exactly the cohesive quality of ℝ n\mathbb{R}^n, the extra quality it carries that distinguishes it from the bare set ♭ℝ n\flat \mathbb{R}^n of its underlying points, as well as from the bare contractible homotopy type ∫X\int X obtained from it.

    On the other hand, the further adjunction ♭⊣♯\flat \dashv \sharp clearly (almost verbatim) matches Hegel’s unity of discreteness and continuity. This unity Hegel says is Quantity and it seems to me that in view of Cantor’s text viewed through Lawvere’s eyes as above this matches nicely with reading “Quantity” as “Cardinal”: given ℝ n\mathbb{R}^n then both ♭ℝ n\flat \mathbb{R}^n and ♯ℝ n\sharp \mathbb{R}^n are the incarnations of its underlying bare set, hence of the underlying quantity.

    When Hegel then finally says that, in turn, the unity of this Quality and Quantity, hence the full adjoint triple ∫⊣♭⊣♯\int\dashv \flat \dashv \sharp is gauge measure, then that’s pretty curious. Because the differential cohomology hexagon says precisely this: this adjoint triple is exactly the axiomatiztion of gauge fields.

    In conclusion, I see this stage of the process neatly formalized as follows:

    Ansichsein Attraktion ʃ ⊣QualitaetEtwas ♭ Repulsion Sein−fuer−Anderes ⊥ Eichmass ⊥ Endlichkeit Dasein Diskretion ♭ ⊣Quantitaet ♯ Kontinuitaet \array{ && && &&\stackrel{Ansichsein}{}&\stackrel{Attraktion}{}& ʃ &\stackrel{Etwas}{\stackrel{Qualitaet}{\dashv}}& \flat & \stackrel{Repulsion}{} & \stackrel{Sein-fuer-Anderes}{} \\ && && &&&& \bot &\stackrel{Eichmass}{}& \bot \\ && && \stackrel{Endlichkeit}{} &&\stackrel{Dasein}{}&\stackrel{Diskretion}{}& \flat &\stackrel{Quantitaet}{\dashv}& \sharp & \stackrel{Kontinuitaet}{} }
    • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 4th 2015

    I don’t really understand in what sense a consistent mathematical theory can be said to be “inherently contradictory”.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2015

    The point is of course exactly that: what sounds (e.g. to Zermelo above) like a contradiction when phrased in everyday language, is actually an adjoint modality.

    With some imagination however one may still see in the adoint modality the “opposing forces” which in the coarser reasoning seemed to lead to a paradox or contradiction. Reversing this process next time that one runs into an apparent paradox may be helpful in formulating ones mathematical theory, whence the “credit Lawvere with the idea that such conceptual contradictions can be productive in mathematics.”

    • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 4th 2015

    I don’t even see why it sounds like a contradiction in the first place. If I have a bag of identical beads, they have no distinguishing features allowing me to tell one from the other, but they are still different beads.

    • CommentRowNumber31.
    • CommentAuthorThomas Holder
    • CommentTimeFeb 4th 2015
    • (edited Jul 4th 2015)

    The above remarks are meant to make sense of the somewhat mysterious passage of Cantor and help evaluating the claim that Cantor has anticipated an adjoint cylinder. Certainly, it wasn’t meant to imply that contemporary set theory is inconsistent. But Cantor’s ideas seem to me to ride on a quite a mixed bag of intuitions nevertheless his intellectual courage to treat ∞\infty as just another number is admirable as it defied almost all tradition.

    That a consistent definition of the concept of natural number was indeed tricky to define in the 19th century is illustrated by the pleasure that Frege took in critizing the attempts of the ancients and his contemporaries so this was really an intuitively simple concept with many strings attached at the time. The beginning of calculus illustrates in a more dramatic form a similar phenomenon were conflicting intuitions actually came to the surface as contradictory propositions.

    I think it offers a reasonable story on an enigmatic passage but one would have to take a closer look at Cantor’s other texts and especially at the exact criticism of Zermelo who was apparently the first to object to the lauter Einsen to back this up.

    [addendum]: Just for the record some corrections and expansions on my above remarks.

    Zermelo’s comment on the passage can be found on p.351 of Cantor’s ’Gesammelte Abhandlungen’ (available via gdz) and his criticism is not that the abstraction is inconsistent but merely ’unfortunate’ as he interprets the introduction of the units as a renaming {a,b,c}→{1,1′,1″}\{a,b,c\} \to \{1,1',1''\} after which one still faces the task to define a cardinality for the renamed set.

    Contrary to what I suggested above, Zermelo was also far from being the first one to object to Cantor’s abstraction to a set of units. Actually, Frege in the ’Grundlagen’ had already criticised such accounts and engaged Cantor-Dedekind in a debate on these issues in the 1880s. In particular, he insisted on the inconsistency of this abstraction process which he saw not as an ’objective contradiction’ but rather a result of deficient conceptualization though. Zermelo presumably wrote his comment 1932 under the influence of his reading Frege as suggested already by McLarty in his introduction to the TAC reprint of Lawvere’s ETCS paper.

    This controversy is discussed in a 1996 paper by William Tait ’Frege versus Cantor and Dedekind: on the concept of number’. Tait provides a lot of historical background on the issue (e.g. he points out that numbers are defined as ’multiplicities of units’ already in book VII, def.2 of Euclid) and touches on wider issues relevant to a structuralist account of mathematics.

    [addendum to the addendum]: This concerns my claim in #26 that the first abstraction is from ⟨a,b,c⟩\langle a,b,c\rangle to {a,b,c}\{a,b,c\} - in fact it is to ⟨1,1′,1″⟩\langle 1,1',1''\rangle (an ordinal) and then from there to the cardinal {1,1′,1″}\{1, 1', 1''\}.

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