Resumo de The Nature And Role Of Intuition

Atthe heart of these debates lies the task of isolating precisely what it is thatour intuition provides us with, and deciding when we should be particularlycircumspect about applying it. Nevertheless, those who seek anepistemologically satisfying account of the role of intuition in mathematicsare often faced with an unappealing choice, between the smoky metaphysics ofBrouwer, and the mystical affidavit of Gödel and the Platonists that we canintuitively discern the realm of mathematical truth. In the proposedthesis I hope to supply, as an alternative, the lineaments of a more plausibleand naturalistic account of mathematical intuition. And the conjecturingof those types of metaphors which can safely and profitably be cashed, lies atthe heart of intuition's fundamental role in mathematics. Consequently,the intuitions of which mathematicians speak are exercised more in the solutionof research problems than in the knowledge of axioms, and are not, crucially,those which Platonism requires. The metaphor works, according to MaxBlack (4) by transferring the associated ideas and implications of thesecondary to the primary system, and by selecting, emphasising and suppressingfeatures of the primary in such a way that new slants on it are illuminated. Now,of course, the most violent objection to intuitiveness counting in theprocess of justification could be some type of internalist stipulation that ajustification must always take the form of a convincing series of reasonsavailable, or cognitively accessible, to the knower. For instance, theeventual ability of set-theoretic methods to generate the analytic theoryof the continuum, a consistent rendering of our confused intuitive beliefsabout the relation of the line to its smallest parts, is one of its greatestachievements, and one of its strongest post facto supports. The ParisSchool attacked the blind cashing of unwarranted finitary schemas in widerdomains, criticising the unfettered development of the theory of selections andrejecting the tacit use of non-effective procedures in topology, measuretheory, and functional analysis. Thecommon uncircumspect belief in the applicability of traditional logic to mathematicswas caused historically," says Brouwer, "by the fact that, firstly,classical logic was abstracted from the mathematics of subsets of a definitefinite set, that, secondly, an a priori existence independent ofmathematics was ascribed to this logic,and finally, on the basis of thissuppositious apriority, it was unjustifiably applied to the mathematics ofinfinite sets". And that might help to extricate us from a situation inwhich 'the dial is tied to the hands of the clock'. The 19th centurybelief that our geometrical prejudices should be isolated and withdrawn fromthe formal presentation of proofs in analysis, led to the idea that our basicintuitions were too weak to have any decisive role to play in the subsequentdevelopment of mathematics. The prime instance of this was the case ofthe continuous but nowhere-differentiable function whose surprise presentationin a paper read to the Berlin Academy of Sciences in 1872, challenged the looseway in which geometical and other intuitive ideas were used in proofs. Inother words,"the appearance ... Shortly before this, the great Gauss and Cauchywent astray by surrendering themselves to the guidance of intuition, andearlier still many mathematicians of the 18th century believed in theself-evidence of the 'law of continuity', (which states that what holds up tothe limit, also holds at the limit). But then the geometry of thesurface (i.e. what the angle-sums of triangles composed of 3 geodesicswill be, and so forth) will be determined by the curvature of the surface, sothat any two regions that are similar in curvature, will be similar ingeometry. In the meantime then, in discussing our primitive inklings ofplausibility and the epistemological status of their currently-sanctioned extensions,what is at stake is not a simple ecal question of truth or falsity, nor isthe issue one of analysing the semantics of ordinary usage: this is the kind ofcase where our problem is to decide which are the deeper of many conflictingtendencies, all present in our usage of the terms involved, and each of whichenjoys its own ephemeral rise and fall in the conceptual evolution of ourparticular culture. of positive curvature. Of course, this is moreeasily said than done, in that we are largely the inheritors of conceptualsystems peculiar to our scientific heritage, and even more constrained by theidioms peculiar to the present stage of its development. Even the symbolsdesigned for the expression and development of mathematics have variablemeanings, often representing different things in the 19th and 20th centuries,by virtue of the underlying evolution of mathematical thought.

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