In his Stetigkeit Dedekind raises an objection to the then prevalent Newtonian concept of number, that it depends on the confused and ill defined concept of magnitude. In a footnote he raises the further objection that, after all, it could not be extended to complex numbers, presumably because there are no "complex magnitudes". If there are no complex magnitudes then we cannot speak of the ratio of a complex magnitude to a unit magnitude.
This statement has always bothered me and I have tried to get around it. But first let us return to the main objection. There has been no attempt that I am aware of to provide a general definition of magnitude and certainly such a definition would be very difficult as can be seen from the following incomplete and very rough definition.
What is meant by "magnitude" is the incomplete abstraction of a quality of a geometric figure which is independent of the location of the figure. By incomplete is meant that when we speak of adding, subtracting, multiplying magnitudes we must always refer to the geometric object and construct the sum, difference or product. In the case of ratio, however, we do not refer to the geometric object. A ratio is not constructed because not a geometric object but a being of a different sort entirely.
Now given a line AB as unit and another line CD, according to Newton's definition, the ratio of CD to AB is a number. If a whole number, then AB multiplied, that is made multiple, that many times is the length CD. If a fraction, then we must first subdivide AB to create a unit which can measure CD. If neither a whole number nor a fraction, then we must appeal to more powerful techniques, which, we being Newton, we are confident we have available.
Theoretically, the same number can be determined using instead areas, volumes or any other sort of magnitude for which ratios are defined, but, since Descartes at least, usually lines have been used because the geometric constructions are simpler. These magnitudes then are lines, fixed in size, but abstracted from their location and movable at will. Since in the traditional concept a line has two distinct endpoints, the notion of a zero magnitude is impossible, so we have defined positive real numbers (non zero). In this view, multiplication has been extended to include the stretching or shrinking of the unit AB to match the size CD.
To introduce zero and negative numbers we introduce the concept of directed line segment or directed magnitude. Take a directed line OA as unit and, on the same line, any other directed line OB. The ratio OB to OA then defines a real number. If B is on the same side of O as A, the number will be positive, if on the opposite side, negative. We agree that A must be distinct from O but B can coincide with O in which case the number is zero. We have now to extend the concept of multiplying the unit OA by zero and by a negative number
To define complex numbers, we need only release the restriction that OA and OB be in the same line. Multiplication (of a directed line) by a complex number involves a stretching (or shrinking) s and a rotation r. If a unit line OB is erected perpendicular to the unit line OA then the stretching s = 1 and the rotation is pi/2 and we may call the ratio i and see immediately that i squared is -1.
So from this view a complex number is the ratio of a directed line segment to a directed line segment not in the same line which is taken as a unit. Since, with the substitution of directed line segment for magnitude, this definition is essentially the same as Newton's, it seems to me Dedekind's secondary objection fails, although his primary objection, i.e. that the concept of magnitude is very murky and ill defined, still stands. (However my objection that the abandonment of the Newtonian concept may have been premature and that the modern concept is quite as murky as that of magnitude, also still stands. I do not see that the concept of directed magnitude or directed line segment is any more murky than that of magnitude.)
I have tried to extend the above analysis to quaternions with very limited success.
Regards from Comer, Georgia where the weather has been painfully beautiful, Bob
Robert Eldon Taylor philologos at mindspring dot com