Dedekind Numbers and Frege Numbers

Around the turn of the 20th century foundational work in mathematics was divided between approaches based on ordinality and approaches based on cardinality.

The ordinal notion of number was understood in connection with with Kant’s theory that arithmetical judgment is synthetic a priori, having its origin in the form of the intuition of time.  The ordinals are understood in connection with the relations “before”, “after” and “next”.   Dedekind thought about the ordinals in terms of a mental capacity to freely generate a new object of thought, and it is natural to think of such objects as ordered according to the temporal order of their generation.  I don’t know of anyplace where Dedekind says so in exactly these terms, but it also seems natural to suppose that although each mind has its own private sequence of thought-objects mathematical communication is nevertheless facilitated because our seperate sequences are one by isomorphism.  At this same time, significant cultural pressure toward axiomatization came with growing value placed on rigor in the mathematical community.  Dedekind was a careful mathematician who set a standard for rigor; it was Dedekind to whom Cantor turned for validation of results he himself “could scarcely believe”.  The work of Dedekind and others, including of course Peano, came to be axiomatized by the Peano axioms.

An opposing (or, perhaps, it’s better to say “complementary”) approach was pursued by Frege.  It was Frege’s conviction that while the epistemic status of geometry was properly given by Kant, arithmetic is in fact analytic.  This he sought to demonstrate by reducing arithmetic to logic.  It should be noted that the proposal should seem rather hopeful.  After all, the complex numbers are “reducible” to the reals, as are the reals to the rationals by Dedekind cuts (or Cauchy limits), as are the rationals to the integers, and the integers to the natural numbers.  By “reducible” I mean something like this: constructible given certain existence assumptions (e.g., the existence of limits or of ordered pairs from a given domain).  To accomplish his reduction to logic Frege adopted an approach based on the notion of cardinalty.  Cardinality answers the question “how many?”.  It is worth noting that this notion is implicitly disconnected from temporality.  After all, when we ask “how may?” we mean to ask how many right now.  The notion of cardinality has its philosophical pedigree with the ancients.  For instance in Metaphysics N, Aristotle defines One in opposition to Plurality.  To be one is to be an instance of a form and to fall under some sortal concept.  Frege numbers were defined as follows:

0 is defined: ε(ίX[X<=>ίx[x≠x])

You read that aloud as follows “0 is the extension of the concept of being a concept that is equinumerous with the concept of being non-self identical”.  Close interpreters may prefer “course of values” to “extension” but I haven’t been able to make out what difference it would make other than terminological. Hopefully someone can make this clear for me if there is clarity to be had.

There are a few notational points worth clarifying.  The capital letter X is used as a variable ranging over first-order concepts.  The lower case letter x is used as a variable ranging over objects.  Each variable is bound by an abstraction comprehension operator: “ίX” and “ίx”, respectively.  The abstraction comprehension operator may be thought of as a function mapping open formulas to concepts one order higher than the open place in the formula.  For instance the formula x≠x has open places in the subject position, so the concept ίx[x≠x] is first order.  In turn, the concept ίX[X<=>ίx[x≠x] is second order.

It helps me to keep things straight to think of “ί” as the “ascending” operator because it, in a manner of speaking, bumps you up a level.  By contrast, “ε” can be thought of as the “descending” operator.  It may be thought of as a function from a concept to an object.  On Frege’s view, concepts may be formed by abstraction comprehension from formulas and numbers may be identified with the extensions of such concepts. The object 0 having been defined as above, Frege then proceeds thusly:

1 is defined: ε(ίX[X<=>ίx[x=0])
2 is defined: ε(ίX[X<=>ίx[x≤1])
etc.

There would be much more to say if we were to completely cash in the logicist promissory note to give a reduction to logic.  In what sense do the relation signs (“<=>”, “≤”, and “=”) occuring in these expressions stand for “logical” relations, for instance?  Perhaps I’ll address this question in a future post; for now I’ve already digressed more than I had planned.  At this point it will be helpful to recall Hume’s Principle.  Hume’s Principle introcuces a decendent operator “#”, and it is stated thusly:

HP: #F=#G .<–>. F<=>G

Read aloud: “the number of Fs is equal to the number of Gs if and only if F is equinumerous with G”.  It is pretty straightforward to prove Hume’s Principle from Frege’s definition of numbers with the operator # defined thusly:

#F is defined:  ε(ίX[X<=>F)

It is also reasonably straightforward to prove that the numbers as Frege defines them when ordered by the relation “≤” satisfy the Dedekind-Peano Axioms.  Again, the logicist program requires cashing in the promise to define all of those relations logically, but that is really no problem if Frege’s conception of logic as second-order and including functions is granted.  The trouble comes with the validation of the ascendent and descendent operators by the principles of abstraction comprehension (i.e., comprehension of formulas) and extensionality abstraction (i.e., Basic Law V), which assert the existence of the concepts and objects indicated by the expressions formed by the operators.  These existential principles were considered by Frege to be properly logical axioms.  They were shown by Russell to be inconsistent.

As I’ve suggested by indicating it is perhaps best to see the approaches as complimentary, I think it would be folly to presume that the subsequent development of mathematics has vindicated either Dedekind’s approach or Frege’s.  Of course, both ordinality and cardinality are perfectly respectable mathematical notions.  Furthermore, there’s little important philosophical or mathematical point in disputing whether it’s the ordinals or the cardinals that are the natural numbers, especially if the answer to this question just depends on what the folk have in mind when they’re using numerals and even if it depends on which notion is prior cognitively or developmentally.  There is always more to say, but this seems a good place to leave off.  In a future post I will discuss developments in the methodology of mathematics during the 20th century in light of the approaches of Dedekind and Frege to arithmetic.  I will also return to the question of the analyticity or syntheticity of mathematics in later posts.