Oct 13, 2012
Thompson’s Notions of the Fallacies of Mathematical Intuition
Yogyakarta State University
Thompson, P.,1993, elaborated that those who are eager to argue how futile it is to try and demarcate or even seek out an epistemologically safe subsystem of pure intuitive propositions which could be used as the basis for an unproblematic branch of mathematics, also tend to emphasize how often we fail to discriminate
reliable intuitions from processes known, post facto, to lead to false beliefs. Thompson 1 argued that in his attack on various popular accounts of intuition, insofar as they claim that intuition provides us with an incorrigible a priori knowledge of mathematics, Philip Kitcher cites several episodes from the history of mathematics when mathematicians have hailed something as intuitively self-evident to give it much the same status as we give to the Zermelo-Fraenkel axioms of set theory, subsequently turned out to be false. Thompson 2 exposed that when Frege toke any property to determine a set in which this is by no means the only case of its kind; and the great Gauss and Cauchy went astray by surrendering themselves to the guidance of intuition, and earlier still many mathematicians of the 18th century believed in the self-evidence of the law of continuity which states that what holds up to the limit, also holds at the limit; this also turns out to be a natural fallacy. Thompson 3 argued that this lead to disconcerting cases show that we cannot always apply Gödel's wedge and discriminate reliable or even a priori intuitions from processes known to lead to false beliefs.
Further, Thompson, P.,1993, insisted that in cases where experience suggests that the intuitive belief we have formed is misguided and this provides a stumbling-block for the thesis that our intuitions occupy the position of being a privileged warrant, by their very nature, for our beliefs, and somehow continue to justify them, whatever recalcitrant experience we come up against; similarly, the set-theoretical paradoxes threaten not so much the possibility of mathematical knowledge, as they now threaten either an a priori, or any other unduly perspicuous account of its nature. Thompson 4 concluded that these fallacies of intuition then, have gained a significant in the contemporary epistemology of mathematics, in which Georg Kreisel suggests that it has been somewhat overplayed; he claimed that this, no doubt, results from our memory bias which makes us, for the most part, recall surprises, memorable cases in which strong initial impressions were later disconfirmed, and ultimately it also leads to an overestimate of the dangers of intuitive thinking; he then stated:
Favourite examples of intuition going astray are often cases of over-simplifications, of applying schemas too generously where their domain of application has to be more finally demarcated. This happened when the Weierstrass M-test undermined the epistemic status of most proofs with casual interchanges of limits in double-limit or integral-summation processes, and, similarly, Zermelo's separation axiom was designed to allow limited comprehension on previously-constructed sets. Some intuitive beliefs have in fact been falsified by the progress of science - for example, the belief that at any given moment, a physical object is in a certain location and moving at a certain speed (pre-Heisenberg), or the pre-relativistic belief that time doesn't slow down when you travel at ten miles an hour. But the feeling is, that these examples only replace one form of intuitive justification with a finer one, so that in scientific contexts intuitive beliefs must be tested like any other hypothesis - they are equally defeasible, can be outweighed by theoretical evidence, and, like any other hypothesis, they can be overthrown. In the words of Imre Lakatos why not honestly admit mathematical fallibility, and try to defend the dignity of fallible knowledge from cynical scepticism, rather than delude ourselves that we can invisibly mend the latest tear in the fabric of our 'ultimate' intuitions? 5
Next, Thompson, P.,1993, in the sense of catching strong postulates in a broader intuitive net, insisted that there are several types of cut-off arguments which seem devastating against any ramifying plan such as that advocated by Gödel; by way of illustration, one of Gödel's original arguments in favor of the un-solvability of the generalized continuum problem, seemed to indicate intuitively that the continuum hypothesis will ultimately turn out to be wrong, while, on the other hand, we know that its disproof is demonstrably impossible, on the basis of the axioms being used today. Thompson 6 indicated that even our schematic means of definition in creating an apparently substantial hierarchy by recursion of our intuitive operations over the countable ordinals, guarantees that we have insidiously conferred an unwanted simplicity on what point-sets we are equipped with, to act as feedstock for ramifying our intuition. Thompson 7 also indicated that in-building the cognitive tendency, which hampers our attempts to ramify our intuition, we extend our mathematics into strongly-axiomatized domains, where new principles have a much freer rein than before, so that the potential domain of their application outstrips what we can readily specify using our old schemas, even suitably bolstered by using transfinite induction, or recursion, as ramifiers. Thompson 8 argued that, consequently, any familiarity we pretend to develop with these domains will be largely mediated by schemas developed on the subsystem, which we must therefore guard ourselves against cashing - as far as is consciously possible that is in the surrounding global domain. Thompson 9 summed up that inability to escape from intuiting formally simple subsystems of those domains into which we extend our mathematics, guarantees that the progress of ramifying our intuition will inevitably be skeletal.
Further, Thompson, P.,1993, argued that the progressive insinuation into the epistemologically-safer sub-domains of mathematics, can be partially held back by a revisionist struggle, such as that advocated by Hermann Weyl which consists of successively: updating, altering and refining our naive intuitions to diminish Frege's qualm; and subsequently decreasing the shortfall between our formal systems and the intuitions of the day, which they claim to represent i.e. reducing Brouwer's qualm. Thompson 10 claimed by this way, the conclusions will not be intuitively false, but simply not intuitively true, and the candidates for appraisal will behave like targets which are no longer just very far from the archer, but no longer even visible at all. In the sense of the analysts distance themselves from geometrical intuition and its role in extension problems, Thompson 11 insisted that the 19th century belief that our geometrical prejudices should be isolated and withdrawn from the formal presentation of proofs in analysis, led to the idea that our basic intuitions were too weak to have any decisive role to play in the subsequent development of mathematics; this, however, often meant that we had now begun to notice when inappropriate schemas were being used, or that we had become impatient on noticing that their unquestionable success at the conjectural stage.
1 Thompson, P.,1993, The Nature And Role Of Intuition In Mathematical Epistemology, University College, Oxford University, U.K