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Originally, fallibilism (from Medieval Latin: fallibilis, "liable to error") is the philosophical principle that Proposition can be accepted even though they cannot be conclusively proven or justified,Peirce, Charles S. (1896–1899) "The Scientific Attitude and Fallibilism". In Buchler, Justus (1940). Philosophical Writings of Peirce. Routledge. p. 59.Haack, Susan (1979). "Fallibilism and Necessity". Synthese, Vol. 41, No. 1, pp. 37–63. or that neither knowledge nor belief is Certainty.Hetherington, Stephen. "Fallibilism". Internet Encyclopedia of Philosophy. The term was coined in the late nineteenth century by the American philosopher Charles Sanders Peirce, as a response to foundationalism. Theorists, following Austrian-British philosopher Karl Popper, may also refer to fallibilism as the notion that knowledge might turn out to be false.Anastas, Jeane W. (1999). Research Design for Social Work and the Human Services. Columbia University Press. p. 19. Furthermore, fallibilism is said to imply corrigibilism, the principle that propositions are open to revision.Levi, Isaac (1984). Messianic vs Myopic Realism. The University of Chicago Press. Vol. 2. pp. 617–636. Fallibilism is often juxtaposed with infallibilism.
Somewhere along the seventeenth century, English philosopher Thomas Hobbes set forth the concept of "infinite progress". With this term, Hobbes had captured the human proclivity to strive for perfection.Hobbes, Thomas (1974). De homine. Cambridge University Press. Vol. 20. Philosophers like Gottfried Wilhelm Leibniz, Christian Wolff, and Immanuel Kant, would elaborate further on the concept. Kant even went on to speculate that Immortality species should hypothetically be able to develop their capacities to perfection.Rorty, Amélie; Schmidt, James (2009). Kant's Idea for a Universal History with a Cosmopolitan Aim. Cambridge University Press.
Already in 350 B.C.E, Greek philosopher Aristotle made a distinction between potential and actual infinities. Based on his discourse, it can be said that actual infinities do not exist, because they are paradoxical. Aristotle deemed it impossible for humans to keep on adding members to indefinitely. It eventually led him to refute some of Zeno's paradoxes.Aristotle (350 B.C.E.). Physics. Massachusetts Institute of Technology. Other relevant examples of potential infinities include Galileo's paradox and the paradox of Hilbert's hotel. The notion that infinite regress and infinite progress only manifest themselves potentially pertains to fallibilism. According to philosophy professor Elizabeth F. Cooke, fallibilism embraces uncertainty, and infinite regress and infinite progress are not unfortunate limitations on human cognition, but rather necessary antecedents for knowledge acquisition. They allow us to live functional and meaningful lives.Cooke, Elizabeth F. (2006). Peirce's Pragmatic Theory of Inquiry: Fallibilism and Indeterminacy. Continuum.
Furthermore, Popper defended his critical rationalism as a Normativity and methodological theory, that explains how objective, and thus mind-independent, knowledge ought to work.Thornton, Stephen (2022). Karl Popper. Stanford Encyclopedia of Philosophy. Hungarian philosopher Imre Lakatos built upon the theory by rephrasing the problem of demarcation as the problem of normative appraisal. Lakatos' and Popper's aims were alike, that is finding rules that could justify falsifications. However, Lakatos pointed out that critical rationalism only shows how theories can be falsified, but it omits how our belief in critical rationalism can itself be justified. The belief would require an inductively verified principle.Forrai, Gábor (2002). Lakatos, Reason and History. p. 6–7. In Kampis, George; Kvasz, Ladislav & Stöltzner, Michael (2002). Appraising Lakatos: Mathematics, Methodology, and the Man. Vienna Circle Institute Library. Vol. 1. When Lakatos urged Popper to admit that the falsification principle cannot be justified without embracing induction, Popper did not succumb.Zahar, E. G. (1983). The Popper-Lakatos Controversy in the Light of 'Die Beiden Grundprobleme Der Erkenntnistheorie'. The British Journal for the Philosophy of Science. p. 149–171. Lakatos' critical attitude towards rationalism has become emblematic for his so called critical fallibilism.Musgrave, Alan; Pigden, Charles (2021). "Imre Lakatos". Stanford Encyclopedia of Philosophy.Kiss, Ogla (2006). Heuristic, Methodology or Logic of Discovery? Lakatos on Patterns of Thinking. MIT Press Direct. p. 314. While critical fallibilism strictly opposes Dogma, critical rationalism is said to require a limited amount of dogmatism.Lakatos, Imre (1978). Mathematics, Science and Epistemology. Cambridge University Press. Vol. 2. p. 9–23.Popper, Karl (1995). The Myth of the Framework: In Defence of Science and Rationality. Routledge. p. 16. Though, even Lakatos himself had been a critical rationalist in the past, when he took it upon himself to argue against the inductivist illusion that Axiom can be justified by the truth of their consequences. In summary, despite Lakatos and Popper picking one stance over the other, both have oscillated between holding a critical attitude towards rationalism as well as fallibilism.Kockelmans, Joseph J. Reflections on Lakatos’ Methodology of Scientific Research Programs. Boston Studies in the Philosophy of Science. Vol. 59. pp. 187–203. In Radnitzky, Gerard & Andersson, Gunnar. The Structure and Development of Science. D. Reidel Publishing Company.
Fallibilism has also been employed by philosopher Willard V. O. Quine to attack, among other things, the distinction between analytic and synthetic statements.Quine, Willard. V. O. (1951). "Two Dogmas of Empiricism". The Philosophical Review Vol. 60, No. 1, pp. 20–43. British philosopher Susan Haack, following Quine, has argued that the nature of fallibilism is often misunderstood, because people tend to confuse fallible propositions with fallible agents. She claims that logic is revisable, which means that analyticity does not exist and necessity (or Apriority) does not extend to logical truths. She hereby opposes the conviction that propositions in logic are infallible, while agents can be fallible.Haack, Susan (1978). Philosophy of Logics. Cambridge University Press. pp. 234; Chapter 12. Critical rationalist Hans Albert argues that it is impossible to prove any truth with certainty, not only in logic, but also in mathematics.Niemann, Hans-Joachim (2000). Hans Albert - critical rationalist. Opensociety.
In the philosophy of mathematics, a central tenet of fallibilism is undecidability (which bears resemblance to the notion of isostheneia, or "equal veracity"). Two distinct types of the word "undecidable" are currently being applied. The first one relates, most notably, to the continuum hypothesis, which was proposed by mathematician Georg Cantor in 1873.Gödel, Kurt (1940). The Consistency of the Continuum-Hypothesis. Princeton University Press. Vol. 3.Enderton, Herbert. "Continuum hypothesis". Encyclopædia Britannica. The continuum hypothesis represents a tendency for infinite sets to allow for undecidable solutions — solutions which are true in one constructible universe and false in another. Both solutions are independent from the axioms in Zermelo–Fraenkel set theory combined with the axiom of choice (also called ZFC). This phenomenon has been labeled the independence of the continuum hypothesis.Cohen, Paul (1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. Vol. 50, No. 6. pp. 1143–1148. Both the hypothesis and its negation are thought to be consistent with the axioms of ZFC.Goodman, Nicolas D. (1979) "Mathematics as an Objective Science". The American Mathematical Monthly. Vol. 86, No. 7. pp. 540-551. Many noteworthy discoveries have preceded the establishment of the continuum hypothesis.
In 1877, Cantor introduced the diagonal argument to prove that the cardinality of two finite sets is equal, by putting them into a Bijection.Cantor, Georg (1877). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die reine und angewandte Mathematik. Vol. 84. pp. 242-258. Diagonalization reappeared in Cantors theorem, in 1891, to show that the power set of any countable set must have strictly higher cardinality.Hosch, William L. "Cantor's theorem". Encyclopædia Britannica. The existence of the power set was postulated in the axiom of power set; a vital part of Zermelo–Fraenkel set theory. Moreover, in 1899, Cantor's paradox was discovered. It postulates that there is no set of all cardinalities. Two years later, polymath Bertrand Russell would invalidate the existence of the universal set by pointing towards Russell's paradox, which implies that no set can contain itself as an element (or member). The universal set can be confuted by utilizing either the axiom schema of separation or the axiom of regularity.Forster, Thomas. E. (1992). Set Theory with a Universal Set: Exploring an Untyped Universe. Clarendon Press. In contrast to the universal set, a power set does not contain itself. It was only after 1940 that mathematician Kurt Gödel showed, by applying inter alia the diagonal lemma, that the continuum hypothesis cannot be refuted, and after 1963, that fellow mathematician Paul Cohen revealed, through the method of forcing, that the continuum hypothesis cannot be proved either.Cohen, Paul (1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America. Vol. 50, No. 6. pp. 1143–1148. In spite of the undecidability, both Gödel and Cohen suspected dependence of the continuum hypothesis to be false. This sense of suspicion, in conjunction with a firm belief in the consistency of ZFC, is in line with mathematical fallibilism.Goodman, Nicolas D. (1979) "Mathematics as an Objective Science". The American Mathematical Monthly. Vol. 86, No. 7. pp. 540-551. Mathematical fallibilists suppose that new axioms, for example the axiom of projective determinacy, might improve ZFC, but that these axioms will not allow for dependence of the continuum hypothesis.Marciszewskip, Witold (2015). "On accelerations in science driven by daring ideas: Good messages from fallibilistic rationalism". Studies in Logic, Grammar and Rhetoric 40, no. 1. p. 39.
The second type of undecidability is used in relation to computability theory (or recursion theory) and applies not solely to statements but specifically to Decision problem; mathematical questions of decidability. An undecidable problem is a type of computational problem in which there are Countable set sets of questions, each requiring an effective method to determine whether an output is either "yes or no" (or whether a statement is either "true or false"), but where there cannot be any computer program or Turing machine that will always provide the correct answer. Any program would occasionally give a wrong answer or run forever without giving any answer.Rogers, Hartley (1971). "Theory of Recursive Functions and Effective Computability". Journal of Symbolic Logic. Vol. 36. pp. 141–146. Famous examples of undecidable problems are the halting problem, the Entscheidungsproblem, and the unsolvability of the Diophantine equation. Conventionally, an undecidable problem is derived from a Computable set, formulated in undecidable language, and measured by the Turing degree.Post, Emil L. (1944). Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society.Kleene, Stephen C.; Post, Emil L. (1954). "The upper semi-lattice of degrees of recursive unsolvability". Annals of Mathematics. Vol. 59. No. 3. pp. 379–407. Undecidability, with respect to computer science and mathematical logic, is also called unsolvability or non-computability.
Undecidability and uncertainty are not one and the same phenomenon. Mathematical theorems which can be formally proved, will, according to mathematical fallibilists, nevertheless remain inconclusive.Dove, Ian J. (2004). Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism. Dissertation, Rice University. pp. 120-122. Take for example proof of the independence of the continuum hypothesis or, even more fundamentally, proof of the diagonal argument. In the end, both types of undecidability add further nuance to fallibilism, by providing these fundamental thought-experiments.Lakatos, Imre; Worrall, John & Zahar, Elie (1976). "Proofs and Refutations". Cambridge University Press. p. 9.
Fallibilism claims that legitimate epistemic justifications can lead to false beliefs, whereas academic skepticism claims that no legitimate epistemic justifications exist (acatalepsy). Fallibilism is also different to epoché, a suspension of judgement, often accredited to Pyrrhonism.
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