Fallibilism

According to philosopher Scott F. Aikin, fallibilism cannot properly function in the absence of infinite regress.Aikin, Scott F. (2014). [https://onlinelibrary.wiley.com/doi/10.1111/meta.12071 "Prospects for Moral Epistemic Infinitism"]. Metaphilosophy. Vol. 45. No. 2. pp. 172–181. The term, usually attributed to Pyrrhonist philosopher Agrippa, is argued to be the inevitable outcome of all human inquiry, since every proposition requires justification.Annas, Julia & Barnes, Jonathan (2000). [https://www.cambridge.org/gb/academic/subjects/philosophy/philosophy-texts/sextus-empiricus-outlines-scepticism-2nd-edition Sextus Empiricus: Outlines of Scepticism]. Cambridge University Press. Infinite regress, also represented within the regress argument, is closely related to the problem of the criterion and is a constituent of the Münchhausen trilemma. Illustrious examples regarding infinite regress are the cosmological argument, turtles all the way down, and the simulation hypothesis. Many philosophers struggle with the metaphysical implications that come along with infinite regress. For this reason, philosophers have gotten creative in their quest to circumvent it.

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). [https://www.cambridge.org/core/journals/medical-history/article/thomas-hobbes-de-homine-traite-de-lhomme-translation-and-commentary-by-paulmarie-maurin-paris-a-blanchard-1974-8vo-pp-205-illus-fr30/C2E101FE9E2DE13A69BA98D4D90143B0 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 immortal species should hypothetically be able to develop their capacities to perfection.Rorty, Amélie; Schmidt, James (2009). [https://www.cambridge.org/core/books/kants-idea-for-a-universal-history-with-a-cosmopolitan-aim/13AE2197C66DD7BD05F22F1C3E616CCE 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 finite sets indefinitely. It eventually led him to refute some of Zeno's paradoxes.Aristotle (350 B.C.E.). [http://classics.mit.edu/Aristotle/physics.html 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). [http://www.commens.org/bibliography/manuscript/peirce-charles-s-1895-c-logic-quantity-ms-r-13?page=11 Peirce's Pragmatic Theory of Inquiry: Fallibilism and Indeterminacy]. Continuum.

{{Main|Critical rationalism}}

File:Karl Popper2.jpg

In the mid-twentieth century, several important philosophers began to critique the foundations of logical positivism. In his work The Logic of Scientific Discovery (1934), Karl Popper, the founder of critical rationalism, argued that scientific knowledge grows from falsifying conjectures rather than any inductive principle and that falsifiability is the criterion of a scientific proposition. The claim that all assertions are provisional and thus open to revision in light of new evidence is widely taken for granted in the natural sciences.Kuhn, Thomas S. (1962). [https://philpapers.org/rec/KUHTSO-3 The Structure of Scientific Revolutions.] University of Chicago Press.

Furthermore, Popper defended his critical rationalism as a normative and methodological theory, that explains how objective, and thus mind-independent, knowledge ought to work.Thornton, Stephen (2022). [https://plato.stanford.edu/entries/popper/ 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). [https://link.springer.com/chapter/10.1007/978-94-017-0769-5_5 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). [https://www.jstor.org/stable/i227794 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). [https://plato.stanford.edu/entries/lakatos/ "Imre Lakatos"]. Stanford Encyclopedia of Philosophy.Kiss, Ogla (2006). [https://direct.mit.edu/posc/article-pdf/14/3/302/1789393/posc.2006.14.3.302.pdf Heuristic, Methodology or Logic of Discovery? Lakatos on Patterns of Thinking]. MIT Press Direct. p. 314. While critical fallibilism strictly opposes dogmatism, critical rationalism is said to require a limited amount of dogmatism.Lakatos, Imre (1978). [https://philpapers.org/rec/LAKMSA Mathematics, Science and Epistemology]. Cambridge University Press. Vol. 2. p. 9–23.Popper, Karl (1995). [https://www.routledge.com/The-Myth-of-the-Framework-In-Defence-of-Science-and-Rationality/Popper-Notturno/p/book/9780415135559 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 axioms 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. [https://link.springer.com/chapter/10.1007/978-94-009-9459-1_7 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). [http://www.ditext.com/quine/quine.html "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 a priority) 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). [https://www.cambridge.org/core/books/philosophy-of-logics/951150505337A200EB152237BDF55A3B 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). [http://www.opensociety.de/Web1/Albert/krirat_e.htm Hans Albert - critical rationalist]. Opensociety.

File:Professor Imre Lakatos, c1960s.jpg

In Proofs and Refutations: The Logic of Mathematical Discovery (1976), philosopher Imre Lakatos implemented mathematical proofs into what he called Popperian "critical fallibilism".{{cite journal |last=Lakatos |first=Imre |author-link=Imre Lakatos |date=1962 |title=Infinite Regress and Foundations of Mathematics (in Symposium: The Foundations of Mathematics) |journal=Proceedings of the Aristotelian Society, Supplementary Volumes |volume=36 |pages=145–184 (165) |jstor=4106691 |quote=Popperian critical fallibilism takes the infinite regress in proofs and definitions seriously, does not have illusions about 'stopping' them, accepts the sceptic criticism of any infallible truth-injection.}} However, Lakatos' interpretation of Popper was not equivalent to Popper's philosophy: {{cite book |last1=Ravn |first1=Ole |last2=Skovsmose |first2=Ole |date=2019 |chapter=Mathematics as Dialogue |title=Connecting Humans to Equations: A Reinterpretation of the Philosophy of Mathematics |series=History of Mathematics Education |location=Cham |publisher=Springer-Verlag |pages=107–119 (110) |isbn=9783030013363 |doi=10.1007/978-3-030-01337-0_8 |s2cid=127561458 |quote=Lakatos also refers to the scepticist programme as a 'Popperian critical fallibilism.' However, we find that this labelling could be a bit misleading as the programme includes a good deal of Lakatos' own philosophy.}} Lakatos's mathematical fallibilism is the general view that all mathematical theorems are falsifiable.Kadvany, John (2001). [https://johnkadvany.com/s/guises-of-reason.pdf Imre Lakatos and the Guises of Reason.] Duke University Press. pp. 45, 109, 155, 323. Mathematical fallibilism deviates from traditional views held by philosophers like Hegel, Peirce, and Popper. Although Peirce introduced fallibilism, he seems to preclude the possibility of our being mistaken in our mathematical beliefs. Mathematical fallibilism appears to uphold that even though a mathematical conjecture cannot be proven true, we may consider some to be good approximations or estimations of the truth. This so called verisimilitude may provide us with consistency amidst an inherent incompleteness in mathematics.Falguera, José L.; Rivas, Uxía; Sagüillo, José M. (1999). [https://www.worldcat.org/oclc/432840477 Analytic Philosophy at the Turn of the Millennium: Proceedings of the International Congress: Santiago de Compostela, 1–4 December, 1999]. Servicio de Publicacións da Universidade de Santiago de Compostela. pp. 259–261. Mathematical fallibilism differs from quasi-empiricism, to the extent that the latter does not incorporate inductivism, a feature considered to be of vital importance to the foundations of set theory.Dove, Ian J. (2004). [https://scholarship.rice.edu/handle/1911/18625 Certainty and error in mathematics: Deductivism and the claims of mathematical fallibilism]. Dissertation, Rice University. pp. 120-122.

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). [https://press.princeton.edu/books/paperback/9780691079271/consistency-of-the-continuum-hypothesis-am-3-volume-3 The Consistency of the Continuum-Hypothesis]. Princeton University Press. Vol. 3.Enderton, Herbert. [https://www.britannica.com/science/continuum-hypothesis "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). [https://www.jstor.org/stable/71858 "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) [http://www.jstor.org/stable/2320581 "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 one-to-one correspondence.Cantor, Georg (1877). [http://eudml.org/doc/148353 "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. [https://www.britannica.com/science/Cantors-theorem "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). [https://philpapers.org/rec/EFOSTW 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). [https://www.jstor.org/stable/71858 "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) [http://www.jstor.org/stable/2320581 "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). [https://sciendo.com/pdf/10.1515/slgr-2015-0002 "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 problems; mathematical questions of decidability. An undecidable problem is a type of computational problem in which there are countably infinite 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). [https://philpapers.org/rec/ROGTOR-4 "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 recursive set, formulated in undecidable language, and measured by the Turing degree.Post, Emil L. (1944). [https://www.ams.org/bull/1944-50-05/S0002-9904-1944-08111-1/S0002-9904-1944-08111-1.pdf Recursively enumerable sets of positive integers and their decision problems]. Bulletin of the American Mathematical Society.Kleene, Stephen C.; Post, Emil L. (1954). [https://www.jstor.org/stable/1969708 "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). [https://scholarship.rice.edu/handle/1911/18625 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). [https://books.google.com/books?id=1n6SFdXCOBQC "Proofs and Refutations"]. Cambridge University Press. p. 9.

{{Main|Philosophical skepticism}}

Fallibilism should not be confused with local or global skepticism, which is the view that some or all types of knowledge are unattainable.

{{blockquote|text=But the fallibility of our knowledge — or the thesis that all knowledge is guesswork, though some consists of guesses which have been most severely tested — must not be cited in support of scepticism or relativism. From the fact that we can err, and that a criterion of truth which might save us from error does not exist, it does not follow that the choice between theories is arbitrary, or non-rational: that we cannot learn, or get nearer to the truth: that our knowledge cannot grow.|author=Karl Popper}}

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 Pyrrhonian skepticism.

Nearly all philosophers today are fallibilists in some sense of the term. Few would claim that knowledge requires absolute certainty, or deny that scientific claims are revisable, though in the 21st century some philosophers have argued for some version of infallibilist knowledge.Moon, Andrew (2012). [https://www.jstor.org/stable/i40068688 "Warrant does entail truth"]. Synthese. Vol. 184. No. 3. pp. 287–297.Dutant, Julien (2016). [https://onlinelibrary.wiley.com/doi/10.1111/phis.12085 "How to be an infallibilist"]. Philosophical Issues. Vol 26. pp. 148–171.Benton, Matthew (2021). [https://link.springer.com/journal/11229/volumes-and-issues/198-7/supplement "Knowledge, hope, and fallibilism"]. Synthese. Vol. 198. pp. 1673–1689. Historically, many Western philosophers from Plato to Saint Augustine to René Descartes have argued that some human beliefs are infallibly known. John Calvin espoused a theological fallibilism towards others beliefs.{{cite book | last=Friedman | first=J. | title=Intolerance: Premodern Roots and Modern Manifestations | publisher=Taylor & Francis | year=2022 | isbn=978-1-000-80294-8 | url=https://books.google.com/books?id=dIOKEAAAQBAJ&pg=PT12 | access-date=2023-06-01 | page=12}}{{cite book | last=Helm | first=P. | title=Calvin at the Centre | publisher=OUP Oxford | year=2010 | isbn=978-0-19-953218-6 | url=https://books.google.com/books?id=GFYVDAAAQBAJ&pg=PA92 | access-date=2023-06-01 | page=92}} Plausible candidates for infallible beliefs include logical truths ("Either Jones is a Democrat or Jones is not a Democrat"), immediate appearances ("It seems that I see a patch of blue"), and incorrigible beliefs (i.e., beliefs that are true in virtue of being believed, such as Descartes' "I think, therefore I am"). Many others, however, have taken even these types of beliefs to be fallible.

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• Defeasible reasoning
• Logical holism
• Nihilism
• Multiverse (set theory)
• Pancritical rationalism
• Perspectivism
• {{slink|Pragmatism#Reconciliation of anti-skepticism and fallibilism}}
• Probabilism
• Projectivism

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{{reflist}}

• Charles S. Peirce: Selected Writings, by Philip P. Wiener (Dover, 1980)
• Charles S. Peirce and the Philosophy of Science, by Edward C. Moore (Alabama, 1993)
• Treatise on Critical Reason, by Hans Albert (Tübingen, 1968; English translation, Princeton, 1985)

• {{PhilPapers|category|epistemic-fallibilism|Epistemic Fallibilism}}
• [https://iep.utm.edu/fallibil/ "Fallibilism"] by Stephen Hetherington in the Internet Encyclopedia of Philosophy
• [https://www.rep.routledge.com/articles/thematic/fallibilism/v-1 "Fallibilism"]{{closed access}} by Nicholas Rescher in the Routledge Encyclopedia of Philosophy

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