Georg Kreisel - Collected Quotes

"For foundations it is important to know what we are talking about; we make the subject as specific as possible. In this way we have a chance to make strong assertions. For practice, to make a proof intelligible, we want to eliminate all properties which are not relevant to the result proved, in other words, we make the subject matter less specific." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"Foundations provide an analysis of practice. To deserve this name, foundations must be |  expected to introduce notions which do not occur in practice. Thus in foundations of set theory, types of sets are treated explicitly while in practice they are generally absent; and in foundations of constructive mathematics, the analysis of the logical operations involves (intuitive) proofs while in practice there is no explicit mention of the latter." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"Foundations and organization are similar in that both provide some sort of more systematic exposition. But a step in this direction may be crucial for organization, yet foundationally trivial, for instance a new choice of language when (i) old theorems are simpler to state but (ii) the primitive notions of the new language are defined in terms of the old, that is if they are logically dependent on the latter. Quite often, (i) will be achieved by using new notions with more ‘structure’, that is less analyzed notions, which is a step in the opposite direction to a foundational analysis. In short, foundational and organizational aims are liable to be actually contradictory." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"[...] in foundations we try to find (a theoretical framework permitting the formulation of) good reasons for the basic principles accepted in mathematical practice, while the latter is only concerned with derivations from these principles. The methods used in a deeper analysis of mathematical practice often lead to an extension of our theoretical understanding. A particularly important example is the search for new axioms, which is nothing more than a continuation of the process which led to the discovery of the currently accepted principles." (Georg Kreisel & Jean-Louis Krivine, "Elements of Mathematical Logic: Model Theory", 1967)

"Before going further into the relation between mathematical practice and foundations, it is worth noting the obvious distinction between (i) foundational analysis (which is specifically concerned with validity) and (ii) general conceptual analysis (which, in the traditional sense of the word, is certainly a philosophical activity). As mentioned above, the working mathematician is rarely concerned with (i), but he does engage in (ii), for instance when establishing definitions of such concepts as length or area or, for that matter, natural transformation. For this activity to be called an analysis the principal issue must be whether the definitions are correct, not merely, for instance, whether they are useful technically for deriving results not involving the concepts (when their correctness is irrelevant). In short, it’s not (only) what you do it’s the way that you do it." (Georg Kreisel, "Observations on popular discussions of foundations", 1971)