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I’m slightly confused by what Dedekind means by “characteristic for the concepts they designate” in the quote below:

"But [. . . ] these extensions of definitions no longer allow scope for arbitrariness; on the contrary, they follow with compelling necessity from the earlier restricted definitions, provided one applies the following principle: Laws which emerge from the initial definitions and which are characteristic for the concepts that they designate are to be considered as of general validity"

Here is his example for context:

"We already have a definite example in multiplication. This operation arose from the requirement that a multiply-repeated performance of an operation of the next lower rank [Ordnung] — namely the addition of a fixed positive or negative addend (the so-called multiplicand) — be collected together into a single act. The multiplier — that is, the number which states how often the addition of the multiplicand is to be thought of as repeated — is therefore at the outset necessarily a positive integer; a negative multiplier would, under this first definition of multiplication, make absolutely no sense.

A special definition is therefore needed in order to admit negative multipliers as well, and thereby to liberate the operation from the initial constraint; but such a definition involves a priori complete arbitrariness, and it would only later be decided whether then this arbitrarily chosen definition would bring any real use to arithmetic; and even if the definition succeeded, one could only call it a lucky guess, a happy coincidence — the sort of thing a scientific method ought to avoid.

So let us instead apply our general principle. We must investigate which laws govern the product if the multiplier undergoes in succession the same general alterations which led to the creation of the sequence of negative integers out of the sequence of positive integers. For this it suffices if we determine the alteration which the product undergoes if one makes the simplest numerical operation with the multiplier, namely, allowing it to go over into the next-following number. By successive repetition of this operation we obtain the familiar addition theorem for the multiplier: in order to multiply a number by a sum, one multiplies it by each summand and then adds these partial products together.

From this theorem a subtraction theorem immediately follows for the case where the minuend is greater than the subtrahend. If one now declares this law to be valid in general (that is, to hold also when the difference which the multiplier represents is negative) then one obtains the definition of multiplication with negative multipliers; and it is then of course no accident that the general law which multiplication obeys is exactly the same for both cases." [Dedekind, 1854, §8]

Also, was Dedekind influenced by group theory in his treatment of multiplication?

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  • $\begingroup$ He means constructing objects so as to extend algebraic laws to them. That the laws for "initial definitions" are made "characteristic" of the new objects means that his "extensions of definitions" of algebraic operations to them are designed to conform to the original laws. In his example, he verifies that multiplication laws for positive numbers extend to negative numbers by tracking how extended multiplication interacts with the construction of negative numbers. So "it is then of course no accident that the general law which multiplication obeys is exactly the same for both cases." $\endgroup$
    – Conifold
    Feb 25 at 7:23
  • $\begingroup$ @Conifold Doesn’t this fail the problem of induction $\endgroup$
    – Jerry
    Feb 25 at 8:54
  • $\begingroup$ He is motivating and explaining extensions of definitions, not inductively justifying empirical generalizations. The definitions once made stand on their own, there is no problem of induction. $\endgroup$
    – Conifold
    Feb 25 at 9:47
  • $\begingroup$ @Conifold right so it’s more of a principle that allows him to make “systematic” definitions instead of “arbitrary” ones, which is more scientific. $\endgroup$
    – Jerry
    Feb 25 at 11:09

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