"When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power." (Benjamin Peirce, "On the Uses and Transformations of Linear Algebra", 1875)
"How are complex numbers to be given to us then […]? If we turn for assistance to intuition, we import something foreign into arithmetic; but if we only define the concept of such a number by giving its characteristics, if we simply require the number to have certain properties, then there is no guarantee that anything falls under the concept and answers to our requirements, and yet it is precisely on this that proofs must be based." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"[…] it is not immaterial to the cogency of our proof whether 'a + bi' has a sense or is nothing more than printer's ink. It will not get us anywhere simply to require that it have a sense, or to say that it is to have the sense of the sum of a and bi, when we have not previously defined what 'sum' means in this case and when we have given no justification for the use of the definite article." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"Nothing prevents us from using the concept 'square root of-1'; but we are not entitled to put the definite article in front of it without more ado and take the expression 'the square root of -' as having a sense." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
"What is commonly called the geometrical representation of complex numbers has at least this advantage […] that in it 1 and i do not appear as wholly unconnected and different in kind: the segment taken to represent i stands in a regular relation to the segment which represents 1. […] A complex number, on this interpretation, shows how the segment taken as its representation is reached, starting from a given segment (the unit segment), by means of operations of multiplication, division, and rotation." (Gottlob Frege, "Grundlagen der Arithmetik" ["Foundations of Arithmetic"], 1884)
“A satisfactory theory of the imaginary quantities of ordinary algebra, which is essentially a simple case of multiple algebra, with difficulty obtained recognition in the first third of this century. We must observe that this double algebra, as it has been called, was not sought for or invented; - it forced itself, unbidden, upon the attention of mathematicians, and with its rules already formed. [...] But the idea of double algebra, once received, although as it were unwillingly, must have suggested to many minds more or less distinctly the possibility of other multiple algebras, of higher orders, possessing interesting or useful properties.” (Josiah W Gibbs, “On multiple Algebra”, Proceedings of the American Association for the Advancement of Science Vol. 35, 1886)
"Judged by the only standards which are admissible in a pure doctrine of numbers i is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers)." (Henry B Fine, "The Number-System of Algebra", 1890)
“It is generally true, that wherever an imaginary expression occurs the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real.” (Augustus de Morgan, “On the Study and Difficulties of Mathematics”, 1898)
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