"It is common - especially in algebraic questions - to encounter essentially new difficulties when one breaks away from those cases which are customarily designated as general. As soon as one penetrates beneath the surface of the so-called singularities, the real difficulties of the investigation are usually first encountered but, at the same time, also the wealth of new viewpoints and phenomena contained in its depths." (Leopold Kronecker, 1874)
"What good is your beautiful investigation regarding π? Why study such problems, since irrational numbers do not exist?" (Leopold Kronecker [letter to Ferdinand von Lindemann] 1882)
"The dear God has made the whole numbers, all the rest is man's work." (Leopold Kronecker, [Speech] 1886)
"And I also believe that some day we will succeed in 'arithmetizing' the whole content of these mathematical disciplines [algebra, analysis], i.e., in basing them exclusively upon the notion of number, taken in the most restricted sense, and thus in eliminating again the modifications and extensions of this notion [note: I mean here especially the addition of irrational and continuous magnitudes], which have mostly been motivated by applications to geometry and mechanics." (Leopold Kronecker, 1887)
"Every scientific task aims at the determination of equivalences and at the discovery of their invariants and for it the following line of poetry holds: 'the wise one seeks a stable pole amid the flight of phenomena'." (Leopold Kronecker,"Zur Theorie der elliptischen Functionen", 1889-1890)
"With the happy expression 'Invariants' chosen by Mr. Sylvester, and quite appropriate to the meaning of the matter, one originally denotes only rational functions of the coefficients of forms that remain unchanged under certain linear transformations of the variables of the forms. But the same expression has since then also been extended to some other entities [Bildungen] that remain unchanged under transformation. This multiple applicability of the concept of invariants rests upon the fact that it belongs to a much more general and abstract realm of ideas. In fact, when the concept of invariants is separated from the direct formal relation to a process of transformation and it is tied rather to the general concept of equivalence, then the concept of invariants reaches the most general realm of thought. For, every abstraction, - say an abstraction from certain differences that are presented by a number of objects, - states an equivalence and the concept originating from the abstraction, for instance the concept of a species, represents the 'invariant of the equivalence'." (Leopold Kronecker,"Zur Theorie der elliptischen Functionen", 1889-1890)
"[...] for each arbitrary equivalence there are always invariants in the above mentioned sense. This is indeed a trivial and completely uninformative truth, for it is the broad concept of function that puts us in a position to give an answer. Let us consider again the already mentioned equivalence for a system of three whole numbers, then we can conceive of the elements of an arbitrarily selected system as characteristic invariants. The elements are invariants because I can reach this system from any other system and they are the characteristic invariants because they represent a system that belongs to one definite class but to none of the other classes. In this typical case of quadratic forms [which are identified as triples of numbers, PM], one chooses as the characteristic invariants of the class the elements of that system that have the smallest value and calls, since Lagrange’s time, the form represented by them the reduced form (forme réduite). From it one can of course generate the entire class of forms. I distinguish the invariants as arithmetical, algebraical, and analytical, according to the method through which they are derived from the elements of a system." (Leopold Kronecker, 1891)
"Number theorists are like lotus-eaters - having once tasted of this food they can never give it up." (Leopold Kronecker)
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