FOM: Dedekind on Numbers

Vaughan Pratt pratt at cs.Stanford.EDU
Fri Mar 13 14:53:24 EST 1998

From: Walter Felscher <walter.felscher at>
> [citation and translation of]
>Dedekind's ... letter to Weber of January
>24, 1888, as reprinted in Gesammelte Mathematische Werke,
>Band 3 (ed. R.Fricke, E.Noether, ™.Ore), Braunschweig 1932,
>p. 488-490.

Your translation is greatly appreciated.  Dedekind is among my favorite
19th century contributors to foundations, along with C.S. Peirce, Cantor,
and Schroeder.  I only wish there were a counterpart for him of the
immensely useful "Collected Works" of Goedel edited by Sol Feferman et al.
Wooster Beman's translations of "Stetigkeit un irrationale Zahlen" and
"Was sind und was sollen die Zahlen" give us non-German-speakers some
insight into Dedekind's foundational contributions, which a century of
practice has taught us all to express with greater clarity than Dedekind
could muster but which with some reading between the lines can be seen
to contain intrinsically clear and very original conceptions.

>     while I prefer to create
>     something New (distinct from the cut) which corresponds
>     to the cut

I would think that the closest thing in modern practice (even in
Dedekind's day though perhaps less clearly seen then) to this "Something
new" notion is what takes place with the axiomatic method, in which one
postulates a universe whose elements are not specified but whose structure
is constrained by axioms.  Dedekind is certainly explicit about rejecting
the identification of a real number with the set of rationals below it.
What other methodology in wide use, then or today, could interpret
"something new" better than the axiomatic method?

>D1. Dedekind proposes to view numbers as creations of the
>    human mind.

This seems clear from the passage you cite.  Given that the axiomatic
method does not pretend to define its objects (I mean here the structures
themselves, not the individuals therein) up to better than isomorphism,
what Dedekind's mind would appear to have created here was less the
individual numbers themselves than their totality as a concrete object.
(I emphasize concrete, i.e. the reals as a structured set for which
extensionality is irrelevant, to avoid being misconstrued as happened
recently as trying to inject category theory into a discussion that
neither contained nor needed it.  On FOM one lives and learns.:)

Whether Dedekind would have quarrelled with this reading is, if not moot,
at least mootable, as you yourself observe.

>    That they are manifestations of its creative
>    power is the principal reason which Dedekind puts
>    forward against Weber's reductionist propositions.

As you point out yourself this is not much of a reason.  I am not
convinced that Dedekind is here advancing it as his principal reason.
I would prefer to say that Dedekind begins his argument by first
advocating axiomatics over reductionism, and then moving on to a more
technically sustainable argument.  Let me elaborate.

>D2. Only with a "auszerdem = besides" does Dedekind mention
>    his mathematical, internal reasons, which Mr. Tait so
>    aptly referred to as the prevention of ungrammatical
>    usages brought about by a reductionist terminology.

I'm not sure I can agree with the "only".  The innocent "auszerdem"
leads into a strong argument against identifying the reals with the
Dedekind cuts, of sufficient importance as to warrant my bringing it
up in my freshman seminar "Paradox: Bug or Feature" last quarter in
a lecture on Conway ("surreal") numbers vs. Dedekind cuts.  Such are
the arguments that carry weight with mathematicians, who may tolerate
polemics but depend on them only at the risk of their reputation.

The objection to Dedekind cuts that Dedekind himself develops in
this passage is best described as their lack of homogenity: all but
countably many of the cuts are open on both sides of the cut, but the
cuts at the rationals are open on only one side.  (That the choice of
side is an artifact of the definition might lessen the nonhomogeneity,
but it does not remove it.)  The essence of this objection is found also
in Conway numbers, where the finite binary rationals emerge on finite
"days" and the rest (including both 1/3 and pi) only on day omega.

When one visualizes the real line as a continuum along which a
parameterized point may smoothly glide, it goes counter to that image
to have these speed bumps disturbing the smooth ride.  One wants to
suppress at least that side effect of the identification of the reals
with either the Dedekind cuts or the Conway numbers of rank \omega+1,
those born on day < \omega + 1.

One can of course attack this argument.  My point here is only that it is
(my interpretation of) Dedekind's own argument against the identification
of reals with cuts.  This is a serious argument (which my interpretation
aims only to clarify, not strengthen) despite its chinks.

>It is obvious that D1 is not a mathematical argumention, and
>it may go against many a contemporary reader's grain to even
>listen to it. Yet for Dedekind it is a foundational, epistemic
>argumentation, and historical justice requires us to
>acknowledge this fact.

I hope I've made clear that I attach more significance to Dedekind's
"auszerdem", and that the technically substantive argument is in D2.
I would prefer to read D1 as a polemical lead-in to D2, as I feel that
that interpretation does more justice to Dedekind's mathematical skills
and intuitions than taking D1 as the primary argument.

I could however be persuaded of the primacy of D1 in Dedekind's mind
if you would allow it as an endorsement of the axiomatic method, whose
advantages Dedekind presumably at least intuits then if not sees as
clearly as he might have later.  What I have more difficulty with is
the unqualified idea, which I inferred from the second sentence of your
D1, that Dedekind would regard "manifestation of creative power" as a
substantive rationale.  At a minimum I would ask for some qualification
mentioning axiomatics or other technically substantive notion.

>Also, it is evident from Dedekind's own commentaries - e.g. in his letter
>to Lipschitz from June 10, 1876 , reprinted l.c., p.470 - that he does
>not consider his view in D1 as particularly novel or original; rather,
>he sees it as a description of the ways mathematicians commonly invent
>new numbers.

This would be lend support to the view of D1 as an endorsement of
axiomatics, whose intrinsic rationale is both clear and strong today.

>I hope to find the agreement of Messrs. Tait and Pratt that
>it is only an inessential rephrasing of their words when I
>formulate the insight, which they draw from Dedekind's work, as
>E   "what" the numbers "are" is explained by exhibiting the
>    structure of the system of numbers, and the numbers are
>    better understood in terms of their structure than by
>    the nature of their elements.

All of this is agreeable to me except for the last line, which I would
rather phrase as "representing them as sets".

>Of course, these are not the words of Dedekind, and so the
>question arises whether the insight E is either explicitly
>formulated in different words in Dedekind's work, or is at
>least implicit in his commentaries.

I'd certainly agree with that.  Dedekind calls for much reading between
the lines.

>It seems to be a matter of debate whether Dedekind's
>awareness here can be counted as witnessing implicitly his
>support of the insight E ; based on Dedekind's general
>methodological attitudes, I am inclined to do so.  Still, a
>reference to Dedekind for insight E about the reals cannot,
>it seems, be supported by an explicit quotation, but would
>require a closer report on the tangled web within which the
>'foundations of analysis' developed one hundred years ago.

Again my sentiments precisely!

>It is, of course, most uplifting a situation when we can
>quote the great men of the past to support the insights of
>the present. But history is not just a quarry from which we
>may choose isolated blocks to embellish our present designs.

Indeed.  To do so in the case of the history of continuity would destroy
that fascinating subject's own continuity!

Vaughan Pratt

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