Elements of Intuitionism Quotes

“From an intuitionistic standpoint, mathematics, when correctly carried on, would not need any justification from without, a buttress from the side or a foundation from below: it would wear its own justification on its face.”
― Michael Dummett, Elements of Intuitionism

“From an intuitionistic standpoint, the platonistic conception is the result of blatantly transferring, from the finite case to the infinite one, a picture appropriate only to the former. In making this transference, the platonist destroys the whole essence of infinity, which lies in the conception of a structure which is always in growth, precisely because the process of construction is never completed. The platonistic conception of an infinite structure as something which may be regarded both extensionally, that is, as the outcome of the process, and as a whole, that is, as if the process were completed, thus rests on a straightforward contradiction: an infinite process is spoken of as if it were merely a particularly long finite one. On an intuitionistic view, neither the truth-value of a statement nor any other mathematical entity can be given as the final result of an infinite process, since an infinite process is precisely one that does not have a final re- sult: that is why, when the domain of quantification is infinite, an existentially quantified statement cannot be regarded in advance as determinately either true or false, and a universally quantified one cannot be thought of as being true accidentally, that is independently of there being a proof of it, a proof which must depend intrinsically upon our grasp of the process whereby the domain is generated.”
― Michael Dummett, Elements of Intuitionism

“[T]he [intuitionistic] thesis that there is no completed infinity means, simply, that to grasp an infinite structure is to grasp the process which generates it, that to refer to such a structure is to refer to that process, and that to recognize the structure as being infinite is to recognize that the process will not terminate. In the case of a process that does terminate, we may legitimately distinguish between the process itself and its completed output: we may be presented with the structure that is generated, without knowing anything about the process of generation. But, in the case of an infinite structure, no such distinction is permissible: all that we can, at any given time, know of the output of the process of generation is some finite initial segment of the structure being generated. There is no sense in which we can have any conception of this structure as a whole save by knowing the process of generation.”
― Michael Dummett, Elements of Intuitionism