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Contents

1 Conceptualism 11.1 Conceptualism in scholasticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Modern conceptualism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Conceptualism and perceptual experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Conventionalism 32.1 Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Legal Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Dialetheism 53.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.1.1 Dialetheism resolves certain paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.2 Dialetheism and human reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.3 Apparent dialetheism in other philosophical doctrines . . . . . . . . . . . . . . . . . . . . 63.1.4 Dialetheism may be a more accurate model of the physical world . . . . . . . . . . . . . . 6

3.2 Formal consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Examples of True Contradictions that Dialetheists Accept . . . . . . . . . . . . . . . . . . . . . . 73.6 Modern Dialetheists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Epilogism 94.1 Epilogism in popular culture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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4.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Fictionalism 105.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Illuminationism 116.1 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.2 Iranian school of Illuminationism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Logical holism 167.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Logicism 178.1 Origin of the name “logicism” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Intent, or goal, of Logicism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 Epistemology behind logicism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.4 The Logistic construction of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.4.2 The definition of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.4.3 Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8.5 The unit class, impredicativity, and the vicious circle principle . . . . . . . . . . . . . . . . . . . . 278.5.1 A solution to impredicativity: a hierarchy of types . . . . . . . . . . . . . . . . . . . . . . 298.5.2 Gödel’s criticism and suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.6 Neo-logicism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.9 Annotated bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

9 Panlogism 359.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10 Polylogism 3610.1 Types of polylogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.1.1 Proletarian logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3610.1.2 Racialist polylogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CONTENTS iii

10.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Preintuitionism 3811.1 The introduction of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.2 The principle of complete induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.3 Arguments over the excluded middle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4 Other Pre-Intuitionists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

12 Ultrafinitism 4012.1 Main ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.2 People associated with ultrafinitism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.3 Complexity theory based restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Chapter 1

Conceptualism

For the postmodern art movement, see conceptual art.

Conceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks sit-uated within the thinking mind.[1] Intermediate between nominalism and realism, the conceptualist view approachesthe metaphysical concept of universals from a perspective that denies their presence in particulars outside of themind’s perception of them.[2]

1.1 Conceptualism in scholasticism

The evolution of late scholastic terminology has led to the emergence of Conceptualism, which stemmed from doc-trines that were previously considered to be nominalistic. The terminological distinction was made in order to stressthe difference between the claim that universal mental acts correspond with universal intentional objects and theperspective that dismissed the existence of universals outside of the mind. The former perspective of rejection ofobjective universality was distinctly defined as Conceptualism.Peter Abélard was a medieval thinker whose work is currently classified as having the most potential in representingthe roots of conceptualism. Abélard’s view denied the existence of determinate universals within things, proposingthe claim that meaning is constructed solely by the virtue of conception.[3] William of Ockham was another famouslate medieval thinker who had a strictly conceptualist solution to the metaphysical problem of universals. He arguedthat abstract concepts have no fundamentum outside the mind, and that the purpose they serve is the construction ofmeaning in an otherwise meaningless world.[4]

In the 17th century conceptualism gained favour for some decades especially among the Jesuits: Hurtado deMendoza,Rodrigo de Arriaga and Francisco Oviedo are the main figures. Although the order soon returned to the more realistphilosophy of Francisco Suárez, the ideas of these Jesuits had a great impact on the contemporary early modernthinkers.

1.2 Modern conceptualism

Conceptualism was either explicitly or implicitly embraced by most of the early modern thinkers like René Descartes,John Locke or Gottfried Leibniz – often in a quite simplified form if compared with the elaborate Scholastic the-ories. Sometimes the term is applied even to the radically different philosophy of Kant, who holds that universalshave no connection with external things because they are exclusively produced by our a priori mental structures andfunctions.[5] However, this application of the term “conceptualism” is not very usual, since the problem of universalscan, strictly speaking, be meaningfully raised only within the framework of the traditional, pre-Kantian epistemology.

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2 CHAPTER 1. CONCEPTUALISM

1.3 Conceptualism and perceptual experience

Though separate from the historical debate regarding the status of universals, there has been significant debate re-garding the conceptual character of experience since the release of Mind and World by John McDowell in 1994.[6]McDowell’s touchstone is the famous refutation that Wilfrid Sellars provided for what he called the “Myth of theGiven”—the notion that all empirical knowledge is based on certain assumed or 'given' items, such as sense data.[7]Thus, in rejecting the Myth of the Given, McDowell argues that perceptual content is conceptual “from the groundup”, that is, all perceptual experience is a form of conceptual experience. Put differently, there are no “bare” or“naked” sense data that serve as a foundation for all empirical knowledge—McDowell is not a foundationalist aboutperceptual knowledge.A clear motivation of conceptualism, in this sense, is that the kind of perception that rational creatures like humansenjoy is unique in the fact that it has conceptual character. McDowell explains his position in a recent paper as:

I have urged that our perceptual relation to the world is conceptual all the way out to the world’simpacts on our receptive capacities. The idea of the conceptual that I mean to be invoking is to beunderstood in close connection with the idea of rationality, in the sense that is in play in the traditionalseparation ofmature human beings, as rational animals, from the rest of the animal kingdom. Conceptualcapacities are capacities that belong to their subject’s rationality. So another way of putting my claimis to say that our perceptual experience is permeated with rationality. I have also suggested, in passing,that something parallel should be said about our agency.[8]

McDowell’s conceptualism, though rather distinct (philosophically and historically) from conceptualism’s genesis,shares the view that universals are not “given” in perception from outside of the sphere of reason. Particular objectsare perceived, as it were, already infused with conceptuality stemming the spontaneity of the rational subject herself.

1.4 See also• Problem of universals

• Pierre Abélard

• Conceptual art

• Lyco art (Lyrical Conceptualism), term coined by artist Paul Hartal

• Philosophical realism

• Conceptual architecture

1.5 References[1] See articles in Strawson, P. F. and Arindam Chakrabarti (eds.), Universals, concepts and qualities: new essays on the

meaning of predicates. Ashgate Publishing, 2006.

[2] “Conceptualism.” The Oxford Dictionary of Philosophy. Simon Blackburn. Oxford University Press, 1996. Oxford Ref-erence Online. Oxford University Press. 8 April 2008.

[3] “Aune, Bruce. “Conceptualism.” Metaphysics: the elements. U of Minnesota Press, 1985. 54.

[4] “Turner, W. "William of Ockham." The Catholic Encyclopedia. Vol. 15. New York: Robert Appleton Company, 1911.27 Oct. 2011

[5] “De Wulf, Maurice. "Nominalism, Realism, Conceptualism." The Catholic Encyclopedia. Vol. 11. New York: RobertAppleton Company, 1911. 27 Oct. 2011

[6] McDowell, John (1994). Mind and World. Cambridge: Harvard University Press. ISBN 978-0-674-57610-0.

[7] “Wilfrid Sellars”. Retrieved 2013-05-24.

[8] McDowell, J. (2007). “What Myth?". Inquiry 50 (4): 338–351. doi:10.1080/00201740701489211.

Chapter 2

Conventionalism

Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on (explicitor implicit) agreements in society, rather than on external reality. Although this attitude is commonly held withrespect to the rules of grammar, its application to the propositions of ethics, law, science, mathematics, and logic ismore controversial.

2.1 Linguistics

The debate on linguistic conventionalism goes back to Plato's Cratylus and the Mīmāṃsā philosophy of KumārilaBhaṭṭa. It has been the standard position of modern linguistics since Ferdinand de Saussure's l'arbitraire du signe, butthere have always been dissenting positions of phonosemantics, recently defended byMargaret Magnus and VilayanurS. Ramachandran.

2.2 Geometry

The French mathematician Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use ofnon-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry should notbe regarded as a priori truth. He held that axioms in geometry should be chosen for the results they produce, not fortheir apparent coherence with human intuitions about the physical world.

2.3 Philosophy

Conventionalism was adopted by logical positivists, chiefly AJ Ayer and Carl Hempel, and extended to both mathe-matics and logic. To deny rationalism, Ayer sees two options for empiricism regarding the necessity of the truth offormal logic (and mathematics): 1) deny that they actually are necessary, and then account for why they only appearso, or 2) claim that the truths of logic and mathematics lack factual content - they are not “truths about the world” -and then explain how they are nevertheless true and informative.[1] John Stuart Mill adopted the former, which Ayercriticized, opting himself for the latter. Ayer’s argument relies primarily on the analytic/synthetic distinction.The French philosopher Pierre Duhem espoused a broader conventionalist view encompassing all of science. Duhemwas skeptical that human perceptions are sufficient to understand the “true,” metaphysical nature of reality and arguedthat scientific laws should be valued mainly for their predictive power and correspondence with observations.

2.4 Legal Philosophy

Conventionalism, as applied to legal philosophy, provides a justification for state coercion. It is one of the three rivalconceptions of law constructed by American legal philosopher Ronald Dworkin in his work Law’s Empire. The othertwo conceptions of law are legal pragmatism and law as integrity.

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4 CHAPTER 2. CONVENTIONALISM

According to conventionalism as defined by Dworkin, a community’s legal institutions should contain clear socialconventions relied upon which rules are promulgated. Such rules will serve as the sole source of information for allthe community members because they demarcate clearly all the circumstances in which state coercion will and willnot be exercised.Dworkin nonetheless has argued that this justification fails to fit with facts as there are many occasions wherein clearapplicable legal rules are absent. It follows that, as he maintained, conventionalism can provide no valid ground forstate coercion. Dworkin himself favored law as integrity as the best justification of state coercion.One famous criticism of Dworkin’s idea comes from Stanley Fish who opines that Dworkin, like the Critical LegalStudies movement, Marxists and adherents of feminist jurisprudence, was guilty of a false 'Theory Hope'. Fishclaims that such mistake stems from their mistaken belief that there exists a general or higher 'theory' that explainsor constrains all fields of activity like state coercion.Another criticism is based on Dworkin’s assertion that positivists’ claims amount to conventionalism. H. L. A. Hart,as a soft positivist, denies such claim as he had pointed out that citizens cannot always discover the law as plain matterof fact. It is however unclear as to whether Joseph Raz, an avowed hard positivist, can be classified as conventionalistas Raz has claimed that law is composed “exclusively” of social facts which could be complex, and thus difficult tobe discovered.In particular, Dworkin has characterized law as having the main function of restraining state’s coercion. Nigel Sim-monds has rejected Dworkin’s disapproval of conventionalism, claiming that his characterization of law is too narrow.

2.5 References[1] Ayer, Alfred Jules. Language, Truth and Logic, Dover Publications, Inc.: New York. 1952. p. 73.

• The Internet Encyclopedia of Philosophy entry on Henri Poincaré

• “Pierre Duhem”. Notes by David Huron

• Mary Jo Nye, “The Boutroux Circle and Poincare’s Conventionalism,” Journal of the History of Ideas, Vol. 40,No. 1. (Jan. - Mar., 1979), pp. 107–120.

2.6 See also• phonosemantics

• true name

• Kumarila Bhatta

Chapter 3

Dialetheism

Dialetheism is the view that some statements can be both true and false simultaneously. More precisely, it is thebelief that there can be a true statement whose negation is also true. Such statements are called “true contradictions",dialetheia, or nondualisms.Dialetheism is not a system of formal logic; instead, it is a thesis about truth that influences the construction of aformal logic, often based on pre-existing systems. Introducing dialetheism has various consequences, depending onthe theory into which it is introduced. For example, in traditional systems of logic (e.g., classical logic and intuitionisticlogic), every statement becomes true if a contradiction is true; this means that such systems become trivialist whendialetheism is included as an axiom. Other logical systems do not explode in this manner when contradictions areintroduced; such contradiction-tolerant systems are known as paraconsistent logics.Graham Priest defines dialetheism as the view that there are true contradictions.[1] JC Beall is another advocate; hisposition differs from Priest’s in advocating constructive (methodological) deflationism regarding the truth predicate.[2]

3.1 Motivations

3.1.1 Dialetheism resolves certain paradoxes

The Liar’s paradox and Russell’s paradox deal with self-contradictory statements in classical logic and naïve settheory, respectively. Contradictions are problematic in these theories because they cause the theories to explode—ifa contradiction is true, then every proposition is true. The classical way to solve this problem is to ban contradictorystatements, to revise the axioms of the logic so that self-contradictory statements do not appear. Dialetheists, on theother hand, respond to this problem by accepting the contradictions as true. Dialetheism allows for the unrestrictedaxiom of comprehension in set theory, claiming that any resulting contradiction is a theorem.[3]

3.1.2 Dialetheism and human reasoning

Ambiguous situations may cause humans to affirm both a proposition and its negation. For example, if John standsin the doorway to a room, it may seem reasonable both to affirm that John is in the room and to affirm that John is notin the room.Critics argue that this merely reflects an ambiguity in our language rather than a dialetheic quality in our thoughts; ifwe replace the given statement with one that is less ambiguous (such as “John is halfway in the room” or “John is inthe doorway”), the contradiction disappears. The statements appeared contradictory only because of a syntactic play;here, the actual meaning of “being in the room” is not the same in both instances, and thus each sentence is not theexact logical negation of the other: therefore, they are not necessarily contradictory.The inadequacy of dialetheism to model human thoughts is shown by the appearance of cognitive dissonances whenfacing real contradictions.

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3.1.3 Apparent dialetheism in other philosophical doctrines

The Jain philosophical doctrine of anekantavada— non-one-sidedness— states that[4] all statements are true in somesense and false in another. Some interpret this as saying that dialetheia not only exist but are ubiquitous. Technically,however, a logical contradiction is a proposition that is true and false in the same sense; a proposition which is true inone sense and false in another does not constitute a logical contradiction. (For example, although in one sense a mancannot both be a “father” and “celibate”, there is no contradiction for a man to be a spiritual father and also celibate;the sense of the word father is different here. In another example, although at the same time George W. Bush cannotboth be President and not be President, he was President from 2001-2009, but was not President before 2001 or after2009, so in different times he was both President and not President.)The Buddhist logic system namedCatuṣkoṭi similarly implies that a statement and its negationmay possibly co-exist.[5][6]

Graham Priest argues in Beyond the Limits of Thought that dialetheia arise at the borders of expressibility, in a numberof philosophical contexts other than formal semantics.

3.1.4 Dialetheism may be a more accurate model of the physical world

This is a new area of study, so ideas are only just coming to light, but dialetheism allows the possibility that naturalthings may have contradictory properties. Whether Wave–particle duality is one such case is not established, but it isa possibility Are there non-semantic dialethia

3.2 Formal consequences

In some logics, we can show that taking a contradiction p ∧ ¬p as a premise (that is, taking as a premise the truth ofboth p and ¬p ), we can prove any statement q . Indeed, since p is true, the statement p∨q is true (by generalization).Taking p ∨ q together with ¬p is a disjunctive syllogism from which we can conclude q . (This is often called theprinciple of explosion, since the truth of a contradiction makes the number of theorems in a system “explode”.)Because dialetheists accept true contradictions, they reject that logic alone can prove anything at all because anythingat all is possible. According to dialetheists, evidence is always needed, and we cannot conclude anything for certainoutside of our own immediate experiences, which cannot be described perfectly with words.

3.3 Advantages

The proponents of dialetheism mainly advocate its ability to avoid problems faced by other more orthodox resolutionsas a consequence of their appeals to hierarchies. Graham Priest once wrote “the whole point of the dialetheic solutionto the semantic paradoxes is to get rid of the distinction between object language and meta-language”.[1]

There are also dialetheic solutions to the sorites paradox.

3.4 Criticisms

One important criticism of dialetheism is that it fails to capture something crucial about negation and, consequently,disagreement. Imagine John’s utterance of P. Sally’s typical way of disagreeing with John is a consequent utterance of¬P. Yet, if we accept dialetheism, Sally’s so uttering does not prevent her from also accepting P; after all, P may be adialetheia and therefore it and its negation are both true. The absoluteness of disagreement is lost. The dialetheist canrespond by saying that disagreement can be displayed by uttering "¬P and, furthermore, P is not a dialetheia”. Again,though, the dialetheist’s own theory is his Achilles’ heel: the most obvious codification of "P is not a dialetheia” is¬(P & ¬P). But what if this itself is a dialetheia as well? One dialetheist response is to offer a distinction betweenassertion and rejection. This distinction might be hashed out in terms of the traditional distinction between logicalqualities, or as a distinction between two illocutionary speech acts: assertion and rejection. Another criticism is thatdialetheism cannot describe logical consequences because of its inability to describe hierarchies.[1]

3.5. EXAMPLES OF TRUE CONTRADICTIONS THAT DIALETHEISTS ACCEPT 7

3.5 Examples of True Contradictions that Dialetheists Accept

According to dialetheists, there are some truths that can only be expressed in contradiction. Some examples include:The only certain knowledge we have outside of our immediate experience is that there is no certain knowledge outsideof our immediate experience.“All statements are false” is a true statement.“There are no absolutes” is an absolute.According to dialetheists, these statements are not derived from logic (which they say is false), but are instead de-scriptions of experience.

3.6 Modern Dialetheists

Many modern Zen Buddhists are dialetheists. They use the term nondualism to refer to true contradictions.

3.7 See also• Problem of future contingents

• Leibniz's compossibility

• Liar paradox

• Doublethink

• Trivialism

3.8 References[1] Whittle, Bruno. “Dialetheism, logical consequence and hierarchy.” Analysis Vol. 64 Issue 4 (2004): 318–326.

[2] Jc Beall in The Law of Non-Contradiction: New Philosophical Essays (Oxford: Oxford University Press, 2004), pp. 197–219.

[3] Transfinite Numbers in Paraconsistent Set Theory (Review of Symbolic Logic 3(1), 2010), pp. 71-92..

[4] Matilal, Bimal Krishna. (1998), “The character of logic in India” (Albany, State University of New York press), 127-139

[5] http://www.iep.utm.edu/nagarjun/#H2

[6] ed : Ganeri, J. (2002), “The Collected Essays of Bimal Krishna Matilal: Mind, Language and World” (Oxford UniversityPress), 77-79

3.9 Sources• Frege, Gottlob. “Negation.” Logical Investigations. Trans. P. Geach and R. H Stoothoff. New Haven, Conn.:Yale University Press, 1977. 31–53.

• Parsons, Terence. “Assertion, Denial, and the Liar Paradox.” Journal of Philosophical Logic 13 (1984): 137–152.

• Parsons, Terence. “True Contradictions.” Canadian Journal of Philosophy 20 (1990): 335–354.

• Priest, Graham. In Contradiction. Dordrecht: Martinus Nijhoff (1987). (Second Edition, Oxford: OxfordUniversity Press, 2006.)

• Priest, Graham. “What Is So Bad About Contradictions?" Journal of Philosophy 95 (1998): 410–426.

8 CHAPTER 3. DIALETHEISM

3.10 External links• Francesco Berto and Graham Priest. Dialetheism. In the Stanford Encyclopedia of Philosophy.

• JC Beall UCONN Homepage

• (Blog & ~Blog)

• Dialethiesm Web Page

• Kabay on dialetheism and trivialism (includes both published and unpublished works)

Chapter 4

Epilogism

Epilogism is a style of Inference invented by the ancient Empiric school of medicine. It is a theory-free methodof looking at history by accumulating fact with minimal generalization and being conscious of the side effects ofmaking causal claims (See also Causal inference). Epilogism is an inference which moves entirely within the domainof visible and evident things, it tries not to invoke unobservables. It is tightly knit to the famous “tripos of medicine”.See also Doctrines of the Empiric school.

4.1 Epilogism in popular culture

Epilogism is discussed as a way of viewing history in The Black Swan (Taleb book) by Nassim Nicholas Taleb.

4.2 See also• Transduction (machine learning)

4.3 External links• http://bmcr.brynmawr.edu/2004/2004-12-20.html

• repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/24239/1/nishimura.pdf

9

Chapter 5

Fictionalism

Fictionalism is the view in philosophy according to which statements that appear to be descriptions of the worldshould not be construed as such, but should instead be understood as cases of “make believe”, of pretending totreat something as literally true (a “useful fiction”). Two important strands of fictionalism are modal fictionalismdeveloped by Gideon Rosen, which states that possible worlds, regardless of whether they exist or not, may be a partof a useful discourse, and mathematical fictionalism advocated by Hartry Field, which states that talk of numbers andother mathematical objects is nothing more than a convenience for doing science. Also in meta-ethics, there is anequivalent position called moral fictionalism (championed by Richard Joyce). Many modern versions of fictionalismare influenced by the work of Kendall Walton in aesthetics.Fictionalism consists in at least the following three theses:

1. Claims made within the domain of discourse are taken to be truth-apt; that is, true or false.

2. The domain of discourse is to be interpreted at face value—not reduced to meaning something else.

3. The aim of discourse in any given domain is not truth, but some other virtue(s) (e.g., simplicity, explanatoryscope).

5.1 See also• Nancy Cartwright

• Philosophy of color

5.2 Further reading• Balaguer, Mark (1998). Platonism and Anti-Platonism in Mathematics. Oxford: Oxford University Press.ISBN 978-0-19-514398-0.

• Kalderon, Mark (2005). Moral Fictionalism. Oxford: Clarendon Press. ISBN 978-0-19-927597-7.

5.3 External links• Fictionalism in the Philosophy of Mathematics article in the Internet Encyclopedia of Philosophy

• Modal Fictionalism entry by Daniel Nolan in the Stanford Encyclopedia of Philosophy, 2007-12-11

• Fictionalism entry by Matti Eklund in the Stanford Encyclopedia of Philosophy, 2007-03-30

• Mathematical fictionalism entry by Mark Balaguer in the Stanford Encyclopedia of Philosophy, 2008-04-22

10

Chapter 6

Illuminationism

Illuminationism is a doctrine according to which the process of human thought needs to be aided by divine grace. It isthe oldest and most influential alternative to naturalism in the theory of mind and epistemology.[1] It was an importantfeature of ancient Greek philosophy, Neoplatonism, medieval philosophy, and in particular, the Illuminationist schoolof Islamic philosophy.

6.1 Early history

Socrates says in The Apology that he had a divine or spiritual sign that began when he was a child. It was a voicethat turned him away from something he was about to do, although it never encouraged him to do anything. Apuleiuslater suggested the voice was of a friendly demon [2] and that Socrates deserved this help as he was the most perfectof human beings.The early Christian philosopher Augustine (354 – 430) also emphasised the role of divine illumination in our thought,saying that “The mind needs to be enlightened by light from outside itself, so that it can participate in truth, because itis not itself the nature of truth. You will light my lamp, Lord [3] and “You hear nothing true from me which you havenot first told me.[4] Augustine’s version of illuminationism is not that God gives us certain information, but rathergives us insight into the truth of the information we received for ourselves.

If we both see that what you say is true, and we both see that what I say is true, then where do we seethat? Not I in you, nor you in me, but both of us in that unalterable truth that is above our minds.[5]

Augustine’s theory was defended by Christian philosophers of the later Middle Ages, particularly Franciscans suchas Bonaventura and Matthew of Aquasparta. According to Bonaventura:

Things have existence in the mind, in their own nature (proprio genere), and in the eternal art. So thetruth of things as they are in themind or in their own nature – given that both are changeable – is sufficientfor the soul to have certain knowledge only if the soul somehow reaches things as they are in the eternalart.[6]

The doctrine was criticised by John Pecham and Roger Marston, and in particular by Thomas Aquinas, who deniedthat in this life we have divine ideas as an object of thought, and that divine illumination is sufficient on its own,without the senses. Aquinas also denied that there is a special continuing divine influence on human thought. Peo-ple have sufficient capacity for thought on their own, without needing “new illumination added onto their naturalillumination”.[7]

The theory was defended by Henry of Ghent. Henry argued against Aquinas that Aristotle’s theory of abstractionis not enough to explain how we can acquire infallible knowledge of the truth, and must be supplemented by divineillumination. A thing has two exemplars against which it can be compared. The first is a created exemplar whichexists in the soul. The second is an exemplar which exists outside the soul, and which is uncreated and eternal. But nocomparison to a created exemplar can give us infallible truth. Since the dignify of man requires that we can acquiresuch truth, it follows that we have access to the exemplar in the divine mind.[8]

11

12 CHAPTER 6. ILLUMINATIONISM

Socrates

Henry’s defence of illuminationism was strongly criticised by the Franciscan theologian Duns Scotus, who arguedthat Henry’s version of the theory led to scepticism.

6.2 Iranian school of Illuminationism

Influenced by Avicennism and Neoplatonism, the Persian[9][10][11][12] or Kurdish,[13][14][15][16] philosopher Shahabal-Din Suhrawardi (1155–1191), who left over 50 writings in Persian and Arabic, founded the school of Illumination.He developed a version of illuminationism (Persian اشراق حكمت hikmat-i ishrāq, Arabic: الإشراق حكمة ḥikmat al-ishrāq). The Persian and Islamic school draws on ancient Iranian philosophical disciplines,[17][18] Avicennism (IbnSina’s early Islamic philosophy), Neoplatonic thought (modified by Ibn Sina), and the original ideas of Suhrawardi.In his Philosophy of Illumination, Suhrawardi argued that light operates at all levels and hierarchies of reality (PI,97.7–98.11). Light produces immaterial and substantial lights, including immaterial intellects (angels), human andanimal souls, and even 'dusky substances’, such as bodies.[19]

6.2. IRANIAN SCHOOL OF ILLUMINATIONISM 13

Augustine

Suhrawardi’s metaphysics is based on two principles. The first is a form of the principle of sufficient reason. Thesecond principle is Aristotle’s principle that an actual infinity is impossible.[20]

None of Suhrawardi’s works were translated into Latin, and so he remained unknown in the Latin West, although hiswork continued to be studied in the Islamic East.[21]

14 CHAPTER 6. ILLUMINATIONISM

6.3 See also• Augustine• Bonaventure• Henry of Ghent• Duns Scotus• Iranian philosophy• Early Islamic philosophy• Light (theology)• Mulla Sadra• Enlightenment (spiritual)

6.4 Notes[1] Stanford Encyclopedia of Philosophy

[2] De deo Socratis, XVII–XIX)

[3] Confessions IV.xv.25

[4] Confessions X.ii.2

[5] Confessions XII.xxv.35

[6] De scientia Christi, q.4 resp

[7] Summa theologiae 1a2ae 109.1c

[8] A Companion to Philosophy in the Middle Ages, ed. Gracia and Noone

[9] John Walbridge, “The leaven of the ancients: Suhrawardī and the heritage of the Greeks”, State University of New YorkPress, 1999. Excerpt: “Suhrawardi, a 12th-century Persian philosopher, was a key figure in the transition of Islamic thoughtfrom the neo-Aristotelianism of Avicenna to the mystically oriented philosophy of later centuries.”

[10] Seyyed Hossein Nasr, “The need for a sacred science”, SUNY Press, 1993. Pg 158: “Persian philosopher Suhrawardi refersin fact to this land as na-kuja abad, which in Persian means literally utopia.”

[11] Matthew Kapstein, University of Chicago Press, 2004, “The presence of light: divine radiance and religious experience”,University of Chicago Press, 2004. pg 285:"..the light of lights in the system of the Persian philosopher Suhrawardi”

[12] Hossein Ziai. Illuminationsim or Illuminationist philosophy, first introduced in the 12th century as a complete, reconstructedsystem distinct both from the Peripatetic philosophy of Avicenna and from theological philosophy. in: Encyclopaedia Iranica.Volumes XII & XIII. 2004.

[13] R. Izady, Mehrdad (1991). The Kurds: a concise handbook.

[14] Kamāl, Muḥammad (2006). Mulla Sadra’s transcendent philosophy.

[15] =C. E. Butterworth, M. Mahdi, The Political Aspects of Islamic Philosophy, Harvard CMES Publishers, 406 pp., 1992,ISBN 0-932885-07-1 (see p.336)

[16] M. Kamal, Mulla Sadra’s Transcendent Philosophy, p.12, Ashgate Publishing Inc., 136 pp., 2006, ISBN 0-7546-5271-8(see p.12)

[17] Henry Corbin. The Voyage and the Messenger. Iran and Philosophy. Containing previous unpublished articles and lecturesfrom 1948 to 1976. North Atlantic Books. Berkeley, California. 1998. ISBN 1-55643-269-0.

[18] Henry Corbin. The Man of Light in Iranian Sufism. Omega Publications, New York. 1994. ISBN 0-930872-48-7.

[19] Philosophy of Illumination 77.1–78.9

[20] Philosophy of Illumination 87.1–89.8

[21] Marcotte, Roxanne, “Suhrawardi”, The Stanford Encyclopedia of Philosophy (Summer 2012 Edition), Edward N. Zalta(ed.), URL = <http://plato.stanford.edu/archives/sum2012/entries/suhrawardi/>.

6.5. FURTHER READING 15

6.5 Further reading• Suhrawardi and the School of Illumination by Mehdi Amin Razavi

• Islamic Intellectual Tradition in Persia by Seyyed Hossein Nasr

6.6 External links• Stanford Encyclopedia of Philosophy, Divine Illumination

• Encyclopedia Britannica, Epistemology (philosophy)

• Augustine Stanford Encyclopedia of Philosophy

• Duns Scotus Stanford Encyclopedia of Philosophy

• suhrawardi Stanford Encyclopedia of Philosophy

• Illuminationist philosophy

Chapter 7

Logical holism

Logical holism is the belief that the world operates in such a way that no part can be known without the whole beingknown first.

7.1 See also• The doctrine of internal relations

• Holography

1. In optics:holography

2. In metaphysics:holonomic brain theory, holographic paradigm and The Holographic Universe (MichaelTalbot’s book)

Proponents: Michael Talbot, David Bohm, Karl H. Pribram

• 1. In quantum mechanics:holographic principle (the conjecture that all of the information about the realities in a volumeof space is present on the surface of that volume)

Proponents: Gerard 't Hooft, Leonard Susskind, John A. Wheeler

16

Chapter 8

Logicism

Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematicsis an extension of logic and therefore some or all mathematics is reducible to logic.[1] Bertrand Russell and AlfredNorth Whitehead championed this theory, created by mathematicians Richard Dedekind and Gottlob Frege.Dedekind’s path to logicism had a turning point when he was able to reduce the theory of real numbers to the rationalnumber system by means of set theory. This and related ideas convinced him that arithmetic, algebra and analysiswere reducible to the natural numbers plus a “logic” of sets; furthermore by 1872 he had concluded that the naturalsthemselves were reducible to sets and mappings. It is likely that other logicists, most importantly Frege, were alsoguided by the new theories of the real numbers published in the year 1872. This started a period of expansionof logicism, with Dedekind and Frege as its main exponents, which however was brought to a deep crisis with thediscovery of the classical paradoxes of set theory (Cantor 1896, Zermelo and Russell 1900–1901). Frege gave upon the project after Russell recognized and communicated his paradox exposing an inconsistency in naive set theory.On the other hand, Russell wrote The Principles of Mathematics in 1903 using the paradox and developments ofGiuseppe Peano's school of geometry. Since he treated the subject of primitive notions in geometry and set theory,this text is a watershed in the development of logicism. Evidence of the assertion of logicism was collected by Russelland Whitehead in their Principia Mathematica.[2]

Today, the bulk of modern mathematics is believed to be reducible to a logical foundation using the axioms ofZermelo-Fraenkel set theory (or one of its extensions, such as ZFC), which has no known inconsistencies (althoughit remains possible that inconsistencies in it may still be discovered). Thus to some extent Dedekind’s project wasproved viable, but in the process the theory of sets and mappings came to be regarded as transcending pure logic.Kurt Gödel's incompleteness theorem undermines logicism because it shows that no particular axiomatization ofmathematics can decide all statements. [3] Some believe that the basic spirit of logicism remains valid because thattheorem is proved with logic just like other theorems. However, that conclusion fails to acknowledge any distinctionbetween theorems of mathematical logic and theorems of higher-order logic. The former can be proven using thefundamental theorem of arithmetic (see Gödel numbering), while the latter must rely on human-provided models.Tarski’s undefinability theorem shows that Gödel numbering can be used to prove syntactical constructs, but notsemantic assertions. Therefore, any claim that logicism remains a valid concept must strictly rely on the dubiousnotion that a system of proof based on man-made models is precisely as powerful and authoritative as one based onthe existence and properties of the natural numbers.Logicism was key in the development of analytic philosophy in the twentieth century.

8.1 Origin of the name “logicism”

Grattan-Guinness states that the French word 'Logistique' was “introduced by Couturat and others at the 1904 Inter-national of Congress of Philosophy', and was used by Russell and others from then on, in versions appropriate forvarious languages” (G-G 2000:4502).Apparently the first (and only) usage by Russell appeared in his 1919: “Russell referred several time [sic] to Frege,introducing him as one 'who first succeeded in “logicising” mathematics’ (p. 7). Apart from the mis-representation(which Russell partly rectified by explaining his own view of the role of arithmetic in mathematics), the passage is

17

18 CHAPTER 8. LOGICISM

notable for the word which he put in quotation marks, but their presence suggests nervousness, and he never used theword again, so that 'logicism' did not emerge until the later 1920s” (G-G 2002:434).[4]

About same time as Carnap (1929), but apparently independently, Fraenkel (1928) used the word: “Without commenthe used the name 'logicism' to characterise the Whitehead/Russell position (in the title of the section on p. 244,explanation on p. 263)" (G-G 2002:269). Carnap used a slightly different word 'Logistik'; Behmann complainedabout its use in Carnap’s manuscript so Carnap proposed the word 'Logizismus’, but he finally stuck to his word-choice 'Logistik' (G-G 2002:501). Ultimately “the spread was mainly due to Carnap, from 1930 onwards.” (G-G2000:502).

8.2 Intent, or goal, of Logicism

Symbolic logic: The overt intent of Logicism is to reduce all of philosophy to symbolic logic (Russell), and/orto reduce all of mathematics to symbolic logic (Frege, Dedekind, Peano, Russell). As contrasted with algebraiclogic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks(non-arithmetic symbols), a (very-)few “logical” axioms that embody the three “laws of thought,” and a couple ofconstruction rules that dictate how the marks are to be assembled and manipulated—substitution and modus ponens(inference of the true from the true). Logicism also adopts from Frege’s groundwork the reduction of natural languagestatements from “subject|predicate” into either propositional “atoms” or the “argument|function” of “generalization”—the notions “all,” “some,” “class” (collection, aggregate) and “relation.”As perhaps its core tenet, logicism forbids any “intuition” of number to sneak in either as an axiom or by accident.The goal is to derive all of mathematics, starting with the counting numbers and then the irrational numbers, from the“laws of thought” alone, without any tacit (hidden) assumptions of “before” and “after” or “less” and “more” or to thepoint: “successor” and “predecessor.” Gödel 1944 summarized Russell’s logicistic “constructions,” when comparedto “constructions” in the foundational systems of Intuitionism and Formalism (“the Hilbert School”) as follows: “Bothof these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principalaims of Russell’s constructivism” (Gödel 1944 in Collected Works 1990:119).History: Gödel 1944 summarized the historical background from Leibniz’s in Characteristica universalis, throughFrege and Peano to Russell: “Frege was chiefly interested in the analysis of thought and used his calculus in thefirst place for deriving arithmetic from pure logic”, whereas Peano “was more interested in its applications withinmathematics”. But “It was only [Russell’s] Principia Mathematica that full use was made of the new method foractually deriving large parts of mathematics from a very few logical concepts and axioms. In addition, the youngscience was enriched by a new instrument, the abstract theory of relations” (p. 120-121).Kleene 1952 states it this way: “Leibniz (1666) first conceived of logic as a science containing the ideas and principlesunderlying all other sciences. Dedekind (1888) and Frege (1884, 1893, 1903) were engaged in defining mathematicalnotions in terms of logical ones, and Peano (1889, 1894–1908) in expressing mathematical theorems in a logicalsymbolism” (p. 43); in the previous paragraph he includes Russell and Whitehead as exemplars of the “logicisticschool,” the other two “foundational” schools being the intuitionistic and the “formalistic or axiomatic school” (p.43).Dedekind 1887 describes his intent in the 1887 Preface to the First Edition of his The Nature and Meaning ofNumbers. He believed that in the “foundations of the simplest science; viz., that part of logic which deals with thetheory of numbers” had not been properly argued -- “nothing capable of proof ought to be accepted without proof":

In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions of intuitions of space and time, that I consider it an immediateresult from the laws of thought . . . numbers are free creations of the human mind . . . [and] onlythrough the purely logical process of building up the science of numbers . . . are we prepared accuratelyto investigate our notions of space and time by bringing them into relation with this number-domaincreated in our mind” (Dedekind 1887 Dover republication 1963 :31).

Peano 1889 states his intent in his Preface to his 1889 Principles of Arithmetic:

Questions that pertain to the foundations of mathematics, although treated by many in recent times, stilllack a satisfactory solution. The difficulty has its main source in the ambiguity of language. ¶ That iswhy it is of the utmost importance to examine attentively the very words we use. My goal has been toundertake this examination” (Peano 1889 in van Heijenoort 1967:85).

8.3. EPISTEMOLOGY BEHIND LOGICISM 19

Frege 1879 describes his intent in the Preface to his 1879Begriffsschrift: He startedwith a consideration of arithmetic:did it derive from “logic” or from “facts of experience"?

“I first had to ascertain how far one could proceed in arithmetic by means of inferences alone, with thesole support of those laws of thought that transcend all particulars. My initial step was to attempt toreduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from thereto the concept of number. To prevent anything intuitive from penetrating here unnoticed I had to bendevery effort to keep the chain of inferences free of gaps . . . I found the inadequacy of language to bean obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, asthe relations became more and more complex, to attain the precision that my purpose required. Thisdeficiency led me to the idea of the present ideography. Its first purpose, therefore, is to provide us withthe most reliable test of the validity of a chain of inferences and to point out every presupposition thattries to sneak in unnoticed” (Frege 1879 in van Heijenoort 1967:5).

Russell 1903 describes his intent in the Preface to his 1903 Principles of Mathematics:

“THE present work has two main objects. One of these, the proof that all pure mathematics dealsexclusively with concepts definable in terms of a very small number of fundamental logical concepts,and that all its propositions are deducible from a very small number of fundamental logical principles”(Preface 1903:vi).

“A few words as to the origin of the present work may serve to show the importance of the questionsdiscussed. About six years ago, I began an investigation into the philosophy of Dynamics. . . . [Fromtwo questions -- acceleration and absolute motion in a “relational theory of space"] I was led to a re-examination of the principles of Geometry, thence to the philosophy of continuity and infinity, and then,with a view to discovering the meaning of the word any, to Symbolic Logic” (Preface 1903:vi-vii).

8.3 Epistemology behind logicism

TBD: [Dedekind’s and Frege’s epistemology needs expansion]Dedekind and Frege: The epistemology of Dedekind and Frege is not as well-defined as that of the philosopher Rus-sell, but both seem accepting as a priori the customary “laws of thought” concerning simple propositional statements(usually of belief); these laws would be sufficient in themselves if augmented with theory of classes and relations (e.g.x R y) between individuals x and y linked by the generalization R.Dedekind’s “free formations of the human mind” rebels against the strictures of Kronecker: Dedekind’sargument begins with “1. In what follows I understand by thing every object of our thought"; we humans use symbolsto discuss these “things” of our minds; “A thing is completely determined by all that can be affirmed or thoughtconcerning it” (p. 44). In a subsequent paragraph Dedekind discusses what a “system S is: it is an aggregate, amanifold, a totality of associated elements (things) a, b, c"; he asserts that “such a system S . . . as an object of ourthought is likewise a thing (1); it is completely determined when with respect to every thing it is determined whetherit is an element of S or not.*" (p. 45, italics added). The * indicates a footnote where he states that:

“Kronecker not long ago (Crelle’s Journal, Vol. 99, pp. 334-336) has endeavored to impose certainlimitations upon the free formation of concepts in mathematics which I do not believe to be justified”(p. 45).

Indeed he awaits Kronecker’s “publishing his reasons for the necessity or merely the expediency of these limitations”(p. 45).Leopold Kronecker, famous for his assertion that “God made the integers, all else is the work of man”[5] had his foes,among them Hilbert. Hilbert called Kronecker a "dogmatist, to the extent that he accepts the integer with its essentialproperties as a dogma and does not look back”[6] and equated his extreme constructivist stance with that of Brouwer’sintuitionism, accusing both of “subjectivism": “It is part of the task of science to liberate us from arbitrariness,sentiment and habit and to protect us from the subjectivism that already made itself felt in Kronecker’s views and,it seems to me, finds its culmination in intuitionism”.[7] Hilbert then states that “mathematics is a presuppositionlessscience. To found it I do not need God, as does Kronecker . . ."(p. 479).

20 CHAPTER 8. LOGICISM

[TBD: There is more discussion to be found in Grattan-Guinness re Kronecker, Cantor, the Crelle journal edited byKronecker et. al., philosophies of Cantor and Kronecker.]Russell the realist: Russell’s Realism served him as an antidote to British Idealism,[8] with portions borrowed fromEuropean Rationalism and British empiricism.[9] To begin with, “Russell was a realist about two key issues: universalsand material objects” (Russell 1912:xi). For Russell, tables are real things that exist independent of Russell theobserver. Rationalism would contribute the notion of a priori knowledge,[10] while empiricism would contribute therole of experiential knowledge (induction from experience).[11] Russell would credit Kant with the idea of “a priori”knowledge, but he offers an objection to Kant he deems “fatal": “The facts [of the world] must always conform to logicand arithmetic. To say that logic and arithmetic are contributed by us does not account for this” (1912:87); Russellconcludes that the a priori knowledge that we possess is “about things, and not merely about thoughts” (1912:89).And in this Russell’s epistemology seems different from that of Dedekind’s belief that “numbers are free creations ofthe human mind” (Dedekind 1887:31)[12]

But his epistemology about the innate (he prefers the word a priori when applied to logical principles, cf 1912:74) isintricate. He would strongly, unambiguously express support for the Platonic “universals” (cf 1912:91-118) and hewould conclude that truth and falsity are “out there"; minds create beliefs and what makes a belief true is a fact, “andthis fact does not (except in exceptional cases) involve the mind of the person who has the belief” (1912:130).Where did Russell derive these epistemic notions? He tells us in the Preface to his 1903 Principles of Mathematics.Note that he asserts that the belief: “Emily is a rabbit” is non-existent, and yet the truth of this non-existent propositionis independent of any knowing mind; if Emily really is a rabbit, the fact of this truth exists whether or not Russell orany other mind is alive or dead, and the relation of Emily to rabbit-hood is “ultimate” :

“On fundamental questions of philosophy, my position, in all its chief features, is derived from Mr G.E. Moore. I have accepted from him the non-existential nature of propositions (except such as happento assert existence) and their independence of any knowing mind; also the pluralism which regardsthe world, both that of existents and that of entities, as composed of an infinite number of mutuallyindependent entities, with relations which are ultimate, and not reducible to adjectives of their termsor of the whole which these compose. . . . The doctrines just mentioned are, in my opinion, quiteindispensable to any even tolerably satisfactory philosophy of mathematics, as I hope the following pageswill show. . . . Formally, my premisses are simply assumed; but the fact that they allow mathematics tobe true, which most current philosophies do not, is surely a powerful argument in their favour.” (Preface1903:viii)

Russell and the paradox: In 1902 Russell discovered a “vicious circle” (the so-called Russell’s paradox) in Frege’sBegriffsschrift and he was determined not to repeat it in his 1903 Principles of Mathematics. In two Appendices thathe tacked on at the last minute he devotes 28 pages to a detailed analysis of, first Frege’s theory contrasted againsthis own, and secondly a fix for the paradox. Unfortunately he was not optimistic about the outcome:

“In the case of classes, I must confess, I have failed to perceive any concept fulfilling the conditionsrequisite for the notion of class. And the contradiction discussed in Chapter x. proves that something isamiss, but what this is I have hitherto failed to discover. (Preface to Russell 1903:vi)"

“Fictionalism” and Russell’s no-class theory: Gödel in his 1944 would disagree with the young Russell of 1903("[my premisses] allow mathematics to be true”) but would probably agree with Russell’s statement quoted above(“something is amiss”); Russell’s theory had failed to arrive at a satisfactory foundation of mathematics: the resultwas “essentially negative; i.e. the classes and concepts introduced this way do not have all the properties required forthe use of mathematics” (Gödel 1944:132).How did Russell arrive in this situation? Gödel observes that Russell is a surprising “realist” with a twist: he citesRussell’s 1919:169 “Logic is concerned with the real world just as truly as zoology” (Gödel 1944:120). But heobserves that “when he started on a concrete problem, the objects to be analyzed (e.g. the classes or propositions)soon for the most part turned into “logical fictions” . . . [meaning] only that we have no direct perception of them.”(Gödel 1944:120)In an observation pertinent to Russell’s brand of logicism, Perry remarks that Russell went through three phases ofrealism -- extreme, moderate and constructive (Perry 1997:xxv). In 1903 he was in his extreme phase; by 1905 hewould be in his moderate phase. In a few years he would “dispense with physical or material objects as basic bits ofthe furniture of the world. He would attempt to construct them out of sense-data” in his next book Our knowledge ofthe External World [1914]" (Perry 1997:xxvi).

8.4. THE LOGISTIC CONSTRUCTION OF THE NATURAL NUMBERS 21

These constructions in what Gödel 1944 would call “nominalistic constructivism . . . which might better be calledfictionalism” derived from Russell’s “more radical idea, the no-class theory” (p. 125):

“according to which classes or concepts never exist as real objects, and sentences containing these termsare meaningful only as they can be interpreted as . . . a manner of speaking about other things” (p.125).

See more in the Criticism sections, below.

8.4 The Logistic construction of the natural numbers

The attempt to construct the natural numbers is summarized succinctly by Bernays 1930–1931.[13] But rather than useBernays’ précis, which is incomplete in the details, the construction is best given as a simple finite example togetherwith the details to be found in Russell 1919.In general the logicism of Dedekind-Frege is similar to that of Russell, but with significant (and critical) differences inthe particulars (see Criticisms, below). Overall, though, the logicistic construction-process [Dedekind-Frege-Russell]is very different from that of contemporary set theory. Whereas in set theory the notion of “number” begins froman axiom—the axiom of pairing that leads to the definition of “ordered pair”—no overt number-axiom exists inlogicism. Rather, logicism begins its construction of the numbers from “primitive propositions” that include “class”,“propositional function”, and in particular, “relations” of “similarity” (“equinumerosity": placing the elements ofcollections in one-to-one correspondence) and “ordering” (using “the successor of” relation to order the collectionsof the equinumerous classes)".[14] The logicistic derivation equates the cardinal numbers constructed this way to thenatural numbers, and these numbers end up all of the same “type”—as equivalence classes of classes—whereas in settheory each number is of a higher class than its predecessor (thus each successor contains its predecessor as a subset).Kleene observes the following. (Kleene’s assumptions (1) and (2) state that 0 has property P and n+1 has property Pwhenever n has property P.)

“The viewpoint here is very different from that of [Kronecker’s supposition that 'God made the integers’plus Peano’s axioms of number and mathematical induction], where we presupposed an intuitive con-ception of the natural number sequence, and elicited from it the principle that, whenever a particularproperty P of natural numbers is given such that (1) and (2), then any given natural number must havethe property P.” (Kleene 1952:44).

The importance to logicism of the construction of the natural numbers derives from Russell’s contention that “Thatall traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it hadlong been suspected” (1919:4). The derivation of the real numbers (rationals, irrationals) derives from the theory ofDedekind cuts on the continuous “number line”. While an example of how this is done is useful, it relies first on thederivation of the natural numbers. So, if philosophic problems appear in the logistic attempt to derive the naturalnumbers, these problems will be sufficient to stop the program until these are fixed (see Criticisms, below).

8.4.1 Preliminaries

For Dedekind, Frege and Russell, collections (classes) are aggregates of “things” specified by proper names, that comeabout as the result of propositions (utterances about something that asserts a fact about that thing or things). Russelltore the general notion down in the following manner. He begins with “terms” in sentences that he decomposes asfollows:Terms: For Russell, “terms” are either “things” or “concepts": “Whatever may be an object of thought, or mayoccur in any true or false proposition, or can be counted as one, I call a term. This, then, is the widest word in thephilosophical vocabulary. I shall use as synonymous with it the words, unit, individual, and entity. The first twoemphasize the fact that every term is one, while the third is derived from the fact that every term has being, i.e. is insome sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, issure to be a term; and to deny that such and such a thing is a term must always be false” (Russell 1903:43)Things are indicated by proper names; concepts are indicated by adjectives or verbs: “Among terms, it ispossible to distinguish two kinds, which I shall call respectively things and concepts; the former are the terms indicated

22 CHAPTER 8. LOGICISM

by proper names, the latter those indicated by all other words . . . Among concepts, again, two kinds at least mustbe distinguished, namely those indicated by adjectives and those indicated by verbs” (1903:44).Concept-adjectives are “predicates"; concept-verbs are “relations”: “The former kind will often be called pred-icates or class-concepts; the latter are always or almost always relations.” (1903:44)The notion of a “variable” subject appearing in a proposition: “I shall speak of the terms of a proposition as thoseterms, however numerous, which occur in a proposition and may be regarded as subjects about which the propositionis. It is a characteristic of the terms of a proposition that anyone of them may be replaced by any other entity withoutour ceasing to have a proposition. Thus we shall say that “Socrates is human” is a proposition having only one term;of the remaining component of the proposition, one is the verb, the other is a predicate.. . . Predicates, then, areconcepts, other than verbs, which occur in propositions having only one term or subject.” (1903:45)In other words, a “term” can be place-holder that indicates (denotes) one or more things that can be put into theplaceholder. (1903:45).Truth and falsehood: Suppose Russell were to point to an object and utter: “This object in front of me named“Emily” is a woman.” This is a proposition, an assertion of Russell’s belief to be tested against the “facts” of the outerworld: “Minds do not create truth or falsehood. They create beliefs . . . what makes a belief true is a fact, and thisfact does not (except in exceptional cases) in any way involve the mind of the person who has the belief” (1912:130).If by investigation of the utterance and correspondence with “fact”, Russell discovers that Emily is a rabbit, thenhis utterance is considered “false"; if Emily is a female human (a female “featherless biped” as Russell likes to callhumans), then his utterance is considered “true”.If Russell were to utter a generalization about all Emilys then these object/s (entity/ies) must be examined, one afteranother in order to verify the truth of the generalization. Thus if Russell were to assert “All Emilys are women”, thenthe “All” is a tipoff that the utterance is about all entities “Emily” in correspondence with a concept labeled “woman”and a methodical examination of all creatures with human names would have to commence.Classes (aggregates, complexes): “The class, as opposed to the class-concept, is the sum or conjunction of all theterms which have the given predicate” (1903 p. 55). Classes can be specified by extension (listing their members)or by intension, i.e. by a “propositional function” such as “x is a u” or “x is v”. But “if we take extension pure, ourclass is defined by enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, withinfinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent thepoint of view of intension is essential.” (1909 p. 66)Propositional functions: “The characteristic of a class concept, as distinguished from terms in general, is that “x isa u” is a propositional function when, and only when, u is a class-concept.” (1903:56)Extensional versus intensional definition of a class: “71. Class may be defined either extensionally or intensionally.That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: this isthe precise meaning of the opposition of extension and intension in this connection. But although the general notioncan be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be definedintensionally, i.e. as the objects denoted by such and such concepts. . . logically; the extensional definition appears tobe equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudableendeavour before it had attained its goal.(1903:69)

8.4.2 The definition of the natural numbers

The natural numbers derive from ALL propositions (i.e. completely unrestricted) in this and all other possible worlds,that can be uttered about ANY collection of entities whatsoever. Russell makes this clear in the second (italicized)sentence:

“In the first place, numbers themselves form an infinite collection, and cannot therefore be defined byenumeration. In the second place, the collections having a given number of terms themselves presumablyform an infinite collection: it is to be presumed, for example, that there are an infinite collection of triosin the world, for if this were not the case the total number of things in the world would be finite, which,though possible, seems unlikely. In the third place, we wish to define “number” in such a way that infinitenumbers may be possible; thus we must be able to speak of the number of terms in an infinite collection,and such a collection must be defined by intension, i.e. by a property common to all its members andpeculiar to them.” (1919:13)

8.4. THE LOGISTIC CONSTRUCTION OF THE NATURAL NUMBERS 23

To begin, devise a finite example. Suppose there are 12 families on a street. Some have children, some do not. Todiscuss the names of the children in these households requires 12 propositions asserting " childname is the name of achild in family Fn” applied this collection of households on the particular street of families with names F1, F2, . . .F12. Each of the 12 propositions regards whether or not the “argument” childname applies to a child in a particularhousehold. The children’s names (childname) can be thought of as the x in a propositional function f(x), where thefunction is “name of a child in the family with name Fn”.[15]

To keep things simple all 26 letters of the alphabet are used up in this example, each letter representing the nameof a particular child (in real life there could be repeats). Notice that, in the Russellian view these collections are notsets, but rather “aggregates” or “collections” or “classes”—listings of names that satisfy the predicates F1, F2, . . ..As noted in Step 1, for Russell, these “classes” are “symbolic fictions” that exist only as their aggregate members, i.e.as the extensions of their propositional functions, and not as unit-things in themselves.Step 1: Assemble ALL the classes: Whereas the following example is finite over the very-finite propositionalfunction " childnames of the children in family Fn'" on the very-finite street of a finite number of (12) families,Russell intended the following to extend to ALL propositional functions extending over an infinity of this and allother possible worlds; this would allow him to create ALL the numbers (to infinity).Kleene observes that already Russell has set himself up with an impredicative definition that he will have to resolve,or otherwise he will be confronted with his Russell paradox. “Here instead we presuppose the totality of all propertiesof cardinal numbers, as existing in logic, prior to the definition of the natural number sequence” (Kleene 1952:44).The problem will appear, even in the finite example presented here, when Russell confronts the unit class (cf Russell1903:517).The matter of debate comes down to this: what exactly is a “class"? For Dedekind and Frege, a class is a distinctentity all of its own, a “unity” that can be identified with all those entities x that satisfy the propositional function F(). (This symbolism appears in Russell, attributing it to Frege: “The essence of a function is what is left when the x istaken away, i.e in the above instance, 2( )3 + ( ). The argument x does not belong to the function, but the two togethermake a whole (ib. p. 6 [i.e. Frege’s 1891 Function und Begriff]" (Russell 1903:505).) For example, a particular“unity” could be given a name; suppose a family Fα has the children with the names Annie, Barbie and Charles:

[ a, b, c ]Fα

This Dedekind-Frege construction could be symbolized by a bracketing process similar to, but to be distinguishedfrom, the symbolism of contemporary set theory { a, b, c }, i.e. [ ] with the elements that satisfy the propositionseparated by commas (an index to label each collection-as-a-unity will not be used, but could be):

[a, b, c], [d], [ ], [e, f, g], [h, i], [j, k], [l, m, n, o, p], [ ], [q, r], [s], [t, u], [v, w, x, y, z]

This notion of collection-or or class-as-object, when used without restriction, results in Russell’s paradox; see morebelow about impredicative definitions. Russell’s solution was to define the notion of a class to be only those elementsthat satisfy the proposition, his argument being that, indeed, the arguments x do not belong to the propositionalfunction aka “class” created by the function. The class itself is not to be regarded as a unitary object in its own right,it exists only as a kind of useful fiction: “We have avoided the decision as to whether a class of things has in anysense an existence as one object. A decision of this question in either way is indifferent to our logic” (First edition ofPrincipia Mathematica 1927:24).Russell does not waver from this opinion in his 1919; observe the words “symbolic fictions":

“When we have decided that classes cannot be things of the same sort as their members, that theycannot be just heaps or aggregates, and also that they cannot be identified with propositional functions,it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And ifwe can find any way of dealing with them as symbolic fictions, we increase the logical security of ourposition, since we avoid the need of assuming that there are classes without being compelled to makethe opposite assumption that there are no classes. We merely abstain from both assumptions. . . . Butwhen we refuse to assert that there are classes, we must not be supposed to be asserting dogmaticallythat there are none. We are merely agnostic as regards them . . ..” (1919:184)

And by the second edition of PM (1927) Russell would insist that “functions occur only through their values, . . . allfunctions of functions are extensional, . . . [and] consequently there is no reason to distinguish between functions

24 CHAPTER 8. LOGICISM

and classes . . . Thus classes, as distinct from functions, loose even that shadowy being which they retain in *20” (p.xxxix). In other words, classes as a separate notion have vanished altogether.Given Russell’s insistence that classes are not singular objects-in-themselves, but only collected aggregates, the onlycorrect way to symbolize the above listing is to eliminate the brackets. But this is visually confusing, especiallywith regards to the null class, so a dashed vertical line at each end of the collection will be used to symbolize thecollection-as-aggregate:

a, b, c , d , , e, f, g , h, i , j, k , l, m, n, o, p , , q, r , s , t, u , v, w, x, y, z

Step 2: Collect “similar” classes into bundles (equivalence classes): These above collections can be put into a“binary relation” (comparing for) similarity by “equinumerosity”, symbolized here by ≈, i.e. one-one correspondenceof the elements,[16] and thereby create Russellian classes of classes or what Russell called “bundles”. “We can supposeall couples in one bundle, all trios in another, and so on. In this way we obtain various bundles of collections, eachbundle consisting of all the collections that have a certain number of terms. Each bundle is a class whose membersare collections, i.e. classes; thus each is a class of classes” (Russell 1919:14).Take for example h,i . Its terms h, i cannot be put into one-one correspondencewith the terms of a,b,c , d , , e,f,g ,etc. But it can be put in correspondence with itself and with j,k , q,r , and t,u . These similar collections canbe assembled into a “bundle” (equivalence class) as shown below.

h,i , j,k , q,r , t,u

The bundles (equivalence classes) are shown below.

a, b, c , e, f, gd , s,

h, i , j, k , q, r , t, ul, m, n, o, p , v, w, x, y, z

Step 3: Define the null-class: Notice that the third class-of-classes, , , is special because its classescontain no elements, i.e. no elements satisfy the predicates that created this particular class/collection. Example: thepredicates are:

“For all childnames: "childname is the name of a child in family Fᵨ".“For all childnames: "childname is the name of a child in family Fσ".

Thes particular predicates cannot be satisfied because families Fᵨ and Fσ are childless. There are no terms (names)that satisfy these particular predicates. Remarkably, the class of things, signified by the fictitious , that satisfyeach of these this classes is not only empty, it does not exist at all (more or less, for Russell the agnostic-about-class-existence); for Dedekind-Frege it does exist.This peculiar non-existent entity is nicknamed the “null class” or the “empty class”. This is not the same as theclass of all null classes : the class of all null classes is destined to become “0"; see below. Russell symbolizedthe null/empty class with Λ. So what exactly is the Russellian null class? In PM Russell says that “A class is saidto exist when it has at least one member . . . the class which has no members is called the “null class” . . . "α isthe null-class” is equivalent to "α does not exist”. One is left uneasy: Does the null class itself “exist"? This problembedeviled Russell throughout his writing of 1903.[17] After he discovered the paradox in Frege’s Begriffsschrift headded Appendix A to his 1903 where through the analysis of the nature of the null and unit classes, he discovered theneed for a “doctrine of types"; see more about the unit class, the problem of impredicative definitions and Russell’s“vicious circle principle” below.[18]

Step 4: Assign a “numeral” to each bundle: For purposes of abbreviation and identification, to each bundle assigna unique symbol (aka a “numeral”). These symbols are arbitrary. (The symbol ≡ means “is an abbreviation for” or“is a definition of”):

8.4. THE LOGISTIC CONSTRUCTION OF THE NATURAL NUMBERS 25

a, b, c , e, f, g ≡ ✖d , s ≡ ■

≡ ♣h, i , j, k , q, r , t, u ≡ ❥l, m, n, o, p , v, w, x, y, z ≡ ♦

Step 5: Define “0”: In order to “order” the bundles into the familiar number-line a starting point traditionally called“zero”, is required. Russell picked the empty or null class of classes to fill this role. This null class-of-classes

has been labeled “0” ≡ ♣[19]

Step 6: Define the notion of “successor”: Russell defined a new characteristic “hereditary”, a property of certainclasses with the ability to “inherit” a characteristic from another class (or class-of-classes) i.e. “A property is saidto be “hereditary” in the natural-number series if, whenever it belongs to a number n, it also belongs to n+1, thesuccessor of n.” (1903:21). He asserts that “the natural numbers are the posterity -- the “children”, the inheritors ofthe “successor”—of 0with respect to the relation “the immediate predecessor of (which is the converse of “successor”)(1919:23).Note Russell has used a fewwords here without definition, in particular “number series”, “number n”, and “successor”.Hewill define these in due course. Observe in particular that Russell does not use the unit class-of-classes “1” to constructthe successor (in our example d , s ≡ ■ ) . The reason is that, in Russell’s detailed analysis,[20] if a unit class■ becomes a entity in its own right, then it too can be an element in its own proposition; this causes the proposition tobecome “impredicative” and result in a “vicious circle”. Rather, he states (confusingly): “We saw in Chapter II that acardinal [natural] number is to be defined as a class of classes, and in Chapter III that the number 1 is to be definedas the class of all unit classes, of all that have just one member, as we should say but for the vicious circle. Of course,when the number 1 is defined as the class of all unit classes, unit classes must be defined so as not to assume that weknow what is meant by one (1919:181).For his definition of successor, Russell will use for his “unit” a single entity or “term” as follows:

“It remains to define “successor.” Given any number n let α be a class which has n members, and letx be a term which is not a member of α. Then the class consisting of α with x added on will have +1members. Thus we have the following definition:the successor of the number of terms in the class α is the number of terms in the class consisting of αtogether with x where x is not any term belonging to the class.” (1919:23)

Russell’s definition requires a new “term” (name, thing) which is “added into” the collections inside the bundles. Tokeep the example abstract this will be abbreviated by the name “Smiley” ≡ ☺ (on the assumption that no one has everactually named their child “Smiley”).Step 7: Construct the successor of the null class: For example into the null class Λ stick the smiley face. Fromthe previous, it is not obvious how to do this. The predicate:

“For all childnames: "childname is the name of a child in family Fα".

has to be modified to creating a predicate that contains a term that is always true:

“For all childnames: "childname is the name of a child in family Fα *AND* Smiley";

In the case of the family with no children, “Smiley” is the only “term” that satisfies the predicate. Russell fretted overthe use of the word *AND* here, as in “Barbie AND Smiley”, and called this kind of AND (symbolized below with*&* ) a “numerical conjunction":[21]

*&* ☺ → ☺

By the relation of similarity ≈, this new class can be put into the equivalence class (the unit class) defined by ■:

☺ ≈ d , s → ☺ , d , s ≡ ■, i.e.0 *&* ☺ → ■,

26 CHAPTER 8. LOGICISM

Step 8: For every equivalence class, create its successor: Note that the smiley-face symbol must be insertedinto every collection/class in a particular equivalence-class bundle, then by the relation of similarity ≈ each newlygenerated class-of-classes must be put into the equivalence class that defines n+1:

❥ *&* ☺ ≡ h,i , j,k , q,r , t,u *&* ☺ → h, i, ☺ , j, k, ☺ , q, r, ☺ , t, u, ☺ , a, b,c , e, f, g ≡ ✖, i.e.❥ *&* ☺ → ✖

And in a similar manner, by use of the abbreviations set up above, for each numeral its successor is created:

00 *&* ☺ = ■■ *&* ☺ = ❥❥ *&* ☺ = ✖✖ *&* ☺ = ? [no symbol]? *&* ☺ = ♦♦ *&* ☺ = etc, etc

Step 9: Order the numbers: The process of creating a successor requires the relation " . . . is the successor of .. .”, call it “S”, between the various “numerals”, for example ■ S 0, ❥ S ■, and so forth. “We must now considerthe serial character of the natural numbers in the order 0, 1, 2, 3, . . . We ordinarily think of the numbers as in thisorder, and it is an essential part of the work of analysing our data to seek a definition of “order” or “series " in logicalterms. . . . The order lies, not in the class of terms, but in a relation among the members of the class, in respect ofwhich some appear as earlier and some as later.” (1919:31)Russell applies to the notion of “ordering relation” three criteria: First, he defines the notion of “asymmetry” i.e.given the relation such as S (" . . . is the successor of . . . ") between two terms x, and y: x S y ≠ y S x. Second, hedefines the notion of transitivity for three numerals x, y and z: if x S y and y S z then x S z. Third, he defines the notionof “connected": “Given any two terms of the class which is to be ordered, there must be one which precedes and theother which follows. . . . A relation is connected when, given any two different terms of its field [both domain andconverse domain of a relation e.g. husbands versus wives in the relation of married] the relation holds between thefirst and the second or between the second and the first (not excluding the possibility that both may happen, thoughboth cannot happen if the relation is asymmetrical).(1919:32)He concludes: ". . . [natural] number m is said to be less than another number n when n possesses every hereditaryproperty possessed by the successor of m. It is easy to see, and not difficult to prove, that the relation “less than,” sodefined, is asymmetrical, transitive, and connected, and has the [natural] numbers for its field [i.e. both domain andconverse domain are the numbers].” (1919:35)

8.4.3 Criticism

The problem of presuming the “extralogical” notion of “iteration”: Kleene points out that, “the logicistic thesiscan be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation. In theIntuitionistic view, an essential mathematical kernel is contained in the idea of iteration” (Kleene 1952:46)Bernays 1930–1931 observes that this notion “two things” already presupposes something, even without the claim ofexistence of two things, and also without reference to a predicate, which applies to the two things; it means, simply,“a thing and one more thing. . . . With respect to this simple definition, the Number concept turns out to be anelementary structural concept . . . the claim of the logicists that mathematics is purely logical knowledge turns outto be blurred and misleading upon closer observation of theoretical logic. . . . [one can extend the definition of“logical"] however, through this definition what is epistemologically essential is concealed, and what is peculiar tomathematics is overlooked” (in Mancosu 1998:243).Hilbert 1931:266-7, like Bernays, detects “something extra-logical” inmathematics: “Besides experience and thought,there is yet a third source of knowledge. Even if today we can no longer agree with Kant in the details, neverthelessthe most general and fundamental idea of the Kantian epistemology retains its significance: to ascertain the intuitivea priori mode of thought, and thereby to investigate the condition of the possibility of all knowledge. In my opinion

8.5. THE UNIT CLASS, IMPREDICATIVITY, AND THE VICIOUS CIRCLE PRINCIPLE 27

this is essentially what happens in my investigations of the principles of mathematics. The a priori is here nothingmore and nothing less than a fundamental mode of thought, which I also call the finite mode of thought: somethingis already given to us in advance in our faculty of representation: certain extra-logical concrete objects that existintuitively as an immediate experience before all thought. If logical inference is to be certain, then these objects mustbe completely surveyable in all their parts, and their presentation, their differences, their succeeding one another ortheir being arrayed next to one another is immediately and intuitively given to us, along with the objects, as somethingthat neither can be reduced to anything else, nor needs such a reduction.” (Hilbert 1931 in Mancosu 1998: 266, 267).In brief: the notion of “sequence” or “successor” is an a priori notion that lies outside symbolic logic.Hilbert dismissed logicism as a “false path": “Some tried to define the numbers purely logically; others simply tookthe usual number-theoretic modes of inference to be self-evident. On both paths they encountered obstacles thatproved to be insuperable.” (Hilbert 1931 in Mancoso 1998:267) .Mancosu states that Brouwer concluded that: “the classical laws or principles of logic are part of [the] perceivedregularity [in the symbolic representation]; they are derived from the post factum record ofmathematical constructions. . . Theoretical logic . . . [is] an empirical science and an application of mathematics” (Brouwer quoted by Mancosu1998:9).Gödel 1944: With respect to the technical aspects of Russellian logicism as it appears in Principia Mathematic (eitheredition), Gödel is flat-out disappointed:

“It is to be regretted that this first comprehensive and thorough-going presentation of a mathematicallogic and the derivation of mathematics from it [is?] so greatly lacking in formal precision in the foun-dations (contained in *1 - *21 of Principia) that it presents in this respect a considerable step backwardsas compared with Frege. What is missing, above all, is a precise statement of the syntax of the formal-ism"(cf footnote 1 in Gödel 1944 Collected Works 1990:120).

In particular he pointed out that “The matter is especially doubtful for the rule of substitution and of replacing definedsymbols by their definiens" (Russell 1944:120)With respect to the philosophy that formed these foundations, Gödel would home in on Russell’s “no-class theory”, orwhat Gödel would call his “nominalistic kind of constructivism, such as that embodied in Russell’s “no class theory”. . . which might better be called fictionalism” (cf footnote 1 in Gödel 1944:119). See more in “Gödel’s criticismand suggestions” below.Grattan-Guinness: [TBD] A complicated theory of relations continued to strangle Russell’s explanatory 1919 In-troduction to Mathematical Philosophy and his 1927 second edition of Principia. Set theory, meanwhile had movedon with its reduction of relation to the ordered pair of sets. Grattan-Guinness observes that in the second edition ofPrincipia Russell ignored this reduction that had been achieved by his own student Norbert Wiener (1914). Perhapsbecause of “residual annoyance, Russell did not react at all”.[22] By 1914 Hausdorff would provide another, equivalentdefinition, and Kuratowski in 1921 would provide the one in use today.[23]

8.5 The unit class, impredicativity, and the vicious circle principle

A benign impredicative definition: Suppose the local librarian wants to catalog (index) her collection into a singlebook (call it Ι for “index”). Her index must list ALL the books and their locations in the library. As it turns out,there are only three books, and these have titles Ά, β, and Γ. To form her index-book I, she goes out and buys abook of 200 blank pages and labels it “I”. Now she has four books: I, Ά, β, and Γ. Her task is not difficult. Whencompleted, the contents of her index I is 4 pages, each with a unique title and unique location (each entry abbreviatedas Title.LocationT):

I ← { I.LI, Ά.LΆ, β.Lᵦ, Γ.LΓ}.

This sort of definition of I was deemed by Poincaré to be “impredicative”. He opined that only predicative definitionscan be allowed in mathematics:

“a definition is 'predicative' and logically admissible only if it excludes all objects that are dependent uponthe notion defined, that is, that can in any way be determined by it”.[24]

28 CHAPTER 8. LOGICISM

By Poincaré's definition, the librarian’s index book is “impredicative” because the definition of I is dependent upon thedefinition of the totality I, Ά, β, and Γ. As noted below, some commentators insist that impredicativity in commonsenseversions is harmless, but as the examples show below there are versions which are not harmless. In the teeth of these,Russell would enunciate a strict prohibition—his “vicious circle principle":

“No totality can contain members definable only in terms of this totality, or members involving or pre-supposing this totality” (vicious circle principle)" (Gödel 1944 appearing in Collected Works Vol. II1990:125).[25]

A pernicious impredicativity: α = NOT-α: To create a pernicious paradox, apply input α to the simple functionbox F(x) with output ω = 1 - α. This is the algebraic-logic equivalent of the symbolic-logical ω = NOT-α for truthvalues 1 and 0 rather than “true” and “false”. In either case, when input α = 0, output ω = 1; when input α = 1, outputω = 0.To make the function “impredicative”, wrap around output ω to input α, i.e. identify (equate) the input with (to) theoutput (at either the output or input, it does not matter):

α = 1-α

Algebraically the equation is satisfied only when α = 0.5. But logically, when only “truth values” 0 and 1 are permitted,then the equality cannot be satisfied. To see what is happening, employ an illustrative crutch: assume (i) the startingvalue of α = α0 and (ii) observe the input-output propagation in discrete time-instants that proceed left to right insequence across the page:

α0 → F(x) → 1-α0 → F(x) → (1 -(1-αₒ)) → F(x) → (1-(1-(1-αₒ))) → F(x) → ad nauseam

Start with α0 = 0:

α0 = 0→ F(x) → 1→ F(x) → 0→ F(x) → 1→ F(x) → ad nauseam

Observe that output ω oscillates between 0 and 1. If the “discrete time-instant” crutch (ii) is dropped, the function-box’s output (and input) is both 1 and 0 simultaneously.Fatal impredicativity in the definition of the unit class: The problem that bedeviled the logicists (and set theoriststoo, but with a different resolution) derives from the α = NOT-α paradox[26] Russell discovered in Frege’s 1879Begriffsschrift[27] that Frege had allowed a function to derive its input “functional” (value of its variable) not onlyfrom an object (thing, term), but from the function’s own output as well.[28]

As described above, Both Frege’s and Russell’s construction of natural numbers begins with the formation of equinu-merous classes-of-classes (bundles), then with an assignment of a unique “numeral” to each bundle, and then placingthe bundles into an order via a relation S that is asymmetric: x S y ≠ y S x. But Frege, unlike Russell, allowed theclass of unit classes (in the example above [[d]], [[s]]) to be identified as a unit itself:

[[d]], [[s]] ≡ ■ ≡ 1

But, since the class ■ or 1 is a single object (unit) in its own right, it too must be included in the class-of-unit-classesas an additional class [■]. And this inclusion results in an “infinite regress” (as Godel called it) of increasing “type”and increasing content:

[[d]], [s], [[■]] ≡ ■[[d]], [s], [[d]], [s], [■]]]] ≡ ■[[d]], [s], [[d]], [s], [[[d]], [s], [[d]], [s], [■]]]]]]]] ≡ ■, ad nauseam

Russell would make this problem go away by declaring a class to be a “fiction” (more or less). By this he meant thatthe class would designate only the elements that satisfied the propositional function (e.g. d and s) and nothing else.As a “fiction” a class cannot be considered to be a thing: an entity, a “term”, a singularity, a “unit”. It is an assemblagee.g. d,s but it is not (in Russell’s view) worthy of thing-hood:

8.5. THE UNIT CLASS, IMPREDICATIVITY, AND THE VICIOUS CIRCLE PRINCIPLE 29

“The class as many . . . is unobjectionable, but is many and not one. We may, if we choose, representthis by a single symbol: thus x ε u will mean " x is one of the u 's.” This must not be taken as a relationof two terms, x and u, because u as the numerical conjunction is not a single term . . . Thus a class ofclasses will be many many’s; its constituents will each be only many, and cannot therefore in any sense,one might suppose, be single constituents.[etc]" (1903:516).

This supposes that “at the bottom” every single solitary “term” can be listed (specified by a “predicative” predicate)for any class, for any class of classes, for class of classes of classes, etc, but it introduces a new problem—a hierarchyof “types” of classes.

8.5.1 A solution to impredicativity: a hierarchy of types

Classes as non-objects, as useful fictions: Gödel 1944:131 observes that “Russell adduces two reasons against theextensional view of classes, namely the existence of (1) the null class, which cannot very well be a collection, and(2) the unit classes, which would have to be identical with their single elements.” He suggests that Russell shouldhave regarded these as fictitious, but not derive the further conclusion that all classes (such as the class-of-classes thatdefine the numbers 2, 3, etc) are fictions.But Russell did not do this. After a detailed analysis in Appendix A: The Logical and Arithmetical Doctrines of Fregein his 1903, Russell concludes:

“The logical doctrine which is thus forced upon us is this: The subject of a proposition may be not asingle term, but essentially many terms; this is the case with all propositions asserting numbers otherthan 0 and 1” (1903:516).

In the following notice the wording “the class as many”—a class is an aggregate of those terms (things) that satisfythe propositional function, but a class is not a thing-in-itself:

“Thus the final conclusion is, that the correct theory of classes is even more extensional than that ofChapter VI; that the class as many is the only object always defined by a propositional function, and thatthis is adequate for formal purposes” (1903:518).

It is as if Russell-as-rancher were to round up all his critters (sheep, cows and horses) into three fictitious corrals (onefor the sheep, one for the cows, and one for the horses) that are located in his fictitious ranch. What actually existsare the sheep, the cows and the horses (the extensions), but not the fictitious “concepts” corrals and ranch.Ramified theory of types: function-orders and argument-types, predicative functions: When Russell pro-claimed all classes are useful fictions he solved the problem of the “unit” class, but the overall problem did not goaway; rather, it arrived in a new form: “It will now be necessary to distinguish (1) terms, (2) classes, (3) classes ofclasses, and so on ad infinitum; we shall have to hold that no member of one set is a member of any other set, andthat x ε u requires that x should be of a set of a degree lower by one than the set to which u belongs. Thus x ε x willbecome a meaningless proposition; and in this way the contradiction is avoided” (1903:517).This is Russell’s “doctrine of types”. To guarantee that impredicative expressions such as x ε x can be treated inhis logic, Russell proposed, as a kind of working hypothesis, that all such impredicative definitions have predicativedefinitions. This supposition requires the notions of function-"orders” and argument-"types”. First, functions (andtheir classes-as-extensions, i.e. “matrices”) are to be classified by their “order”, where functions of individuals areof order 1, functions of functions (classes of classes) are of order 2, and so forth. Next, he defines the “type” ofa function’s arguments (the function’s “inputs”) to be their “range of significance”, i.e. what are those inputs α(individuals? classes? classes-of-classes? etc.) that, when plugged into f(x), yield a meaningful output ω. Note thatthis means that a “type” can be of mixed order, as the following example shows:

“Joe DiMaggio and the Yankees won the 1947 World Series”.

This sentence can be decomposed into two clauses: "x won the 1947World Series” + "y won the 1947World Series”.The first sentence takes for x an individual “Joe DiMaggio” as its input, the other takes for y an aggregate “Yankees”as its input. Thus the composite-sentence has a (mixed) type of 2, mixed as to order (1 and 2).

30 CHAPTER 8. LOGICISM

By “predicative”, Russell meant that the function must be of an order higher than the “type” of its variable(s). Thusa function (of order 2) that creates a class of classes can only entertain arguments for its variable(s) that are classes(type 1) and individuals (type 0), as these are lower types. Type 3 can only entertain types 2, 1 or 0, and so forth.But these types can be mixed (for example, for this sentence to be (sort of) true: " z won the 1947 World Series "could accept the individual (type 0) “Joe DiMaggio” and/or the names of his other teammates, and it could acceptthe class (type 1) of individual players “The Yankees”.The axiom of reducibility: The axiom of reducibility is the hypothesis that any function of any order can be reducedto (or replaced by) an equivalent predicative function of the appropriate order.[29] A careful reading of the first editionindicates that an nth order predicative function need not be expressed “all the way down” as a huge “matrix” oraggregate of individual atomic propositions. “For in practice only the relative types of variables are relevant; thus thelowest type occurring in a given context may be called that of individuals” (p. 161). But the axiom of reducibilityproposes that in theory a reduction “all the way down” is possible.Russell 1927 abandons the axiom of reducibility, and the edifice collapses: By the 2nd edition of PM of 1927,though, Russell had given up on the axiom of reducibility and concluded he would indeed force any order of function“all the way down” to its elementary propositions, linked together with logical operators:

“All propositions, of whatever order, are derived from a matrix composed of elementary propositionscombined by means of the stroke” (PM 1927 Appendix A, p. 385),

(The “stroke” is Sheffer’s inconvenient logical NAND that Russell adopted for the 2nd edition—a single logicalfunction that replaces logical OR and logical NOT).The net result, though, was a collapse of his theory. Russell arrived at this disheartening conclusion: that “the theoryof ordinals and cardinals survives . . . but irrationals, and real numbers generally, can no longer be adequately dealtwith. . . .Perhaps some further axiom, less objectionable than the axiom of reducibility, might give these results, butwe have not succeeded in finding such an axiom.” (PM 1927:xiv).Gödel 1944 agrees that Russell’s logicist project was stymied; he seems to disagree that even the integers survived:

"[In the second edition] The axiom of reducibility is dropped, and it is stated explicitly that all primitivepredicates belong to the lowest type and that the only purpose of variables (and evidently also of con-stants) of higher orders and types is to make it possible to assert more complicated truth-functions ofatomic propositions” (Gödel 1944 in Collected Works:134).

Gödel asserts, however, that this procedure seems to presuppose arithmetic in some form or other (p. 134). Hededuces that “one obtains integers of different orders” (p. 134-135); the proof in Russell 1927 PM Appendix Bthat “the integers of any order higher than 5 are the same as those of order 5” is “not conclusive” and “the questionwhether (or to what extent) the theory of integers can be obtained on the basis of the ramified hierarchy [classes plustypes] must be considered as unsolved at the present time”. Gödel concluded that it wouldn't matter anyway becausepropositional functions of order n (any n) must be described by finite combinations of symbols (cf all quotes andcontent derived from page 135).

8.5.2 Gödel’s criticism and suggestions

Gödel in his 1944 bores down to the exact place where Russell’s logicism fails and offers a few suggestions to rectifythe problems. He submits the “vicious circle principle” to reexamination, tearing it apart into three phrases “definableonly in terms of”, “involving” and “presupposing”. It is the first clause that “makes impredicative definitions impos-sible and thereby destroys the derivation of mathematics from logic, effected by Dedekind and Frege, and a gooddeal of mathematics itself”. Since, he argues, mathematics is doing quite well, thank you, with its various inherentimpredicativities (e.g. “real numbers defined by reference to all real numbers”), he concludes that what he has offeredis “a proof that the vicious circle principle is false [rather] than that classical mathematics is false” (all quotes Gödel1944:127).Russell’s no-class theory is the root of the problem: Gödel believes that impredicativity is not “absurd”, as itappears throughout mathematics. Where Russell’s problem derives from is the “constructivistic (or nominalistic[30])standpoint toward the objects of logic and mathematics, in particular toward propositions, classes, and notions . . . anotion being a symbol . . . so that a separate object denoted by the symbol appears as a mere fiction” (p. 128).Indeed, this “no class” theory of Russell, Gödel concludes:

8.6. NEO-LOGICISM 31

“is of great interest as one of the few examples, carried out in detail, of the tendency to eliminate as-sumptions about the existence of objects outside the “data” and to replace them by constructions onthe basis of these data33. [33 The “data” are to understand in a relative sense here; i.e. in our case aslogic without the assumption of the existence of classes and concepts]. The result has been in this caseessentially negative; i.e. the classes and concepts introduced in this way do not have all the propertiesrequired from their use in mathematics. . . . All this is only a verification of the view defended abovethat logic and mathematics (just as physics) are built up on axioms with a real content which cannot beexplained away” (p. 132)

He concludes his essay with the following suggestions and observations:

“One should take a more conservative course, such as would consist in trying to make the meaning ofterms “class” and “concept” clearer, and to set up a consistent theory of classes and concepts as objectivelyexisting entities. This is the course which the actual development of mathematical logic has been takingand which Russell himself has been forced to enter upon in the more constructive parts of his work.Major among the attempts in this direction . . . are the simple theory of types . . . and axiomaticset theory, both of which have been successful at least to this extent, that they permit the derivation ofmodern mathematics and at the same time avoid all known paradoxes . . . ¶ It seems reasonable tosuspect that it is this incomplete understanding of the foundations which is responsible for the fact thatmathematical logic has up to now remained so far behind the high expectations of Peano and others . ..."(p. 140)

8.6 Neo-logicism

Neo-logicism describes a range of views claiming to be the successor of the original logicist program.[31] Morenarrowly, it is defined as attempts to resurrect Frege’s programme through the use of second order logic as applied toHume’s Principle.[32] This kind of neo-logicism is often referred to as neo-Fregeanism. Two of the major proponentsof neo-logicism are Crispin Wright and Bob Hale.[33]

8.7 See also

• Mathematical fictionalism

• Nominalism

8.8 Notes[1] Logicism

[2] Principia Mathematica entry in the Stanford Encyclopedia of Philosophy.

[3] On the philosophical relevance of gödel’s incompleteness theorems

[4] The exact quote from Russell 1919 is the following: “It is time now to turn to the considerations which make it necessaryto advance beyond the standpoint of Peano, who represents the last perfection of the “arithmetisation” of mathematics, tothat of Frege, who first succeeded in “logicising” mathematics, i.e. in reducing to logic the arithmetical notions which hispredecessors had shown to be sufficient for mathematics.” (Russell 1919/2005:17).

[5] For example, von Neumann 1925 would cite Kronecker as follows: “The denumerable infinite . . . is nothing more thegeneral notion of the positive integer, on which mathematics rests and of which even Kronecker and Brouwer admit that itwas “created by God"" (von Neumann 1925 An axiomatization of set theory in van Heijenoort 1967:413).

[6] Hilbert 1904 On the foundations of logic and arithmetic in van Heijenoort 1967:130.

[7] Pages 474–5 in Hilbert 1927, The Foundations of Mathematics in: van Heijenoort 1967:475.

[8] Perry in his 1997 Introduction to Russell 1912:ix)

32 CHAPTER 8. LOGICISM

[9] Cf Russell 1912:74.

[10] “It must be admitted . . . that logical principles are known to us, and cannot be themselves proved by experience, since allproof presupposes them. In this, therefore . . . the rationalists were in the right” (Russell 1912:74).

[11] “Nothing can be known to exist except by the help of experience” (Russell 1912:74).

[12] He drives the point home (pages 67-68) where he defines four conditions that determine what we call “the numbers” (cf(71).Definition, page 67: the successor set N' is a part of the collection N, there is a starting-point “1ₒ" [base number of thenumber-series N], this “1” is not contained in any successor, for any n in the collection there exists a transformation φ(n)to a unique (distinguishable) n(cf (26). Definition)). He observes that by establishing these conditions “we entirely neglectthe special character of the elements; simply retaining their distinguishability and taking into account only the relation toone another . . . by the order-setting transformation φ. . . . With reference to this freeing the elements from every othercontent (abstraction) we are justified in calling numbers a free creation of the human mind.” (p. 68)

[13] cf The Philosophy of Mathematics and Hilbert’s Proof Theory 1930:1931 in Mancosu p. 242.

[14] In his 1903 and in PM Russell refers to such assumptions (there are others) as “primitive propositions” (“pp” as opposedto “axioms” (there are some of those, too). But the reader is never certain whether these pp are axioms/axiom-schemas orconstruction-devices (like substitution or modus ponens), or what, exactly. Gödel 1944:120 comments on this absence offormal syntax and the absence of a clearly specified substitution process.

[15] To be precise both childname = variable x and family name Fn are variables. Childname 's domain is “all childnames inthis and every other possible world”, and family name Fn has a domain over the 12 families on the street.

[16] “If the predicates are partitioned into classes with respect to equinumerosity in such a way that all predicates of a class areequinumerous to one another and predicates of different classes are not equinumerous, then each such class represents theNumber, which applies to the predicates that belong to it” (Bernays 1930-1 in Mancosu 1998:240.

[17] section 487ff (pages 513ff in the Appendix A).

[18] Cf sections 487ff (pages 513ff in the Appendix A).

[19] Whether or not the null class is reducible to is unclear; the conclusion is not important for this example.

[20] 1909 Appendix A

[21] 1903:133ff, Section 130: “Numerical Conjunction” and plurality”.

[22] Russell deemed Wiener “the infant phenomenon . . . more infant than phenomenon"; see Russell’s confrontation withWiener in Grattan-Guinness 2000:419ff.

[23] See van Heijenoort’s commentary and Norbert Wiener’s 1914 A simplification of the logic of relations in van Heijenoort1967:224ff.

[24] Zermelo 1908 in van Heijenoort 1967:190. See the discussion of this very quotation in Mancosu 1998:68.

[25] This same definition appears also in Kleene 1952:42.

[26] An excellent source for details is Fairouz Kamareddine, Twan Laan and Rob Nderpelt, 2004, A Modern Perspective onType Theory, From its Origins Until Today, Kluwer Academic Publishers, Dordrecht, The Netherlands, ISBN. They give ademonstration of how to create the paradox (pages 1–2), as follows: Define an aggregate/class/set y this way: ∃y∀x[ x ε y↔ Φ(x)]. (This says: There exists a class y such that for ANY input x, x is an element of set y if and only if x satisfies thegiven function Φ.) Note that (i) input x is unrestricted as to the “type” of thing that it can be (it can be a thing, or a class),and (ii) function Φ is unrestricted as well. Pick the following tricky function Φ(x) = ¬(x ε x). (This says: Φ(x) is satisfiedwhen x is NOT an element of x)). Because y (a class) is also “unrestricted” we can plug “y” in as input: ∃y[ y ε y ↔ ¬(yε y)]. This says that “there exists a class y that is an element of itself only if it is NOT and element of itself. That is theparadox.

[27] Russell’s letter to Frege announcing the “discovery”, and Frege’s letter back to Russell in sad response, together withcommentary, can be found in van Heijenoort 1967:124-128. Zermelo in his 1908 claimed priority to the discovery; cffootnote 9 on page 191 in van Heijenoort.

[28] van Heijenoort 1967:3 and pages 124-128

[29] “The axiom of reducibility is the assumption that, given any function φẑ, there is a formally equivalent, predicative function,i.e. there is a predicative function which is true when φz is true and false when φz is false. In symbols, the axiom is: ⊦:(∃ψ) : φz. ≡ .ψ!z.” (PM 1913/1962 edition:56, the original uses x with a circumflex). Here φẑ indicates the functionwith variable ẑ, i.e. φ(x) where x is argument “z"; φz indicates the value of the function given argument “z"; ≡ indicates“equivalence for all z"; ψ!z indicates a predicative function, i.e. one with no variables except individuals.

8.9. ANNOTATED BIBLIOGRAPHY 33

[30] Perry observes that Plato and Russell are “enthusiastic” about “universals”, then in the next sentence writes: " 'Nominalists’think that all that particulars really have in common are the words we apply to them"(Perry in his 1997 Introduction toRussell 1912:xi). Perry adds that while your sweatshirt and mine are different objects generalized by the word “sweatshirt”,you have a relation to yours and I have a relation to mine. And Russell “treated relations on par with other universals” (p.xii). But Gödel is saying that Russell’s “no-class” theory denies the numbers the status of “universals”.

[31] What is Neologicism?

[32] PHIL 30067: Logicism and Neo-Logicism

[33] http://www.st-andrews.ac.uk/~{}mr30/papers/EbertRossbergPurpose.pdf

8.9 Annotated bibliography

• Richard Dedekind, circa 1858, 1878, Essays on the Theory of Numbers, English translation published by OpenCourt Publishing Company 1901, Dover publication 1963, Mineola, NY, ISBN 0-486-21010-3. Contains twoessays—I. Continuity and Irrational Numbers with original Preface, II. The Nature and Meaning of Numberswith two Prefaces (1887,1893).

• Howard Eves, 1990, Foundations and Fundamental Concepts ofMathematics Third Edition, Dover Publications,Inc, Mineola, NY, ISBN 0-486-69609-X.

• I. Grattan-Guinness, 2000, The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and TheFoundations of Mathematics from Cantor Through Russell to Gödel, Princiton University Press, Princeton NJ,ISBN 0-691-05858-X.

• Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, 3rdprinting 1976, Harvard University Press, Cambridge, MA, ISBN 0-674-32449-8. Includes Frege’s 1879 Be-griffsschrift with commentary by van Heijenoort, Russell’s 1908 Mathematical logic as based on the theory oftypes with commentary by Willard V. Quine, Zermelo’s 1908 A new proof of the possibility of a well-orderingwith commentary by van Heijenoort, letters to Frege from Russell and from Russell to Frege, etc.

• Stephen C. Kleene, 1971, 1952, Introduction To Metamathematics 1991 10th impression,, North-Holland Pub-lishing Company, Amsterdam, NY, ISBN 0-7204-2103-9.

• Mario Livio August 2011 “WhyMathWorks: Is math invented or discovered? A leading astrophysicist suggeststhat the answer to the millennia-old question is both”, Scientific American (ISSN 0036-8733), Volume 305,Number 2, August 2011, Scientific American division of Nature America, Inc, New York, NY.

• Bertrand Russell, 1903, The Principles of Mathematics Vol. I, Cambridge: at the University Press, Cambridge,UK.

• Paolo Mancosu, 1998, From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s,Oxford University Press, New York, NY, ISBN 0-19-509632-0.

• Bertrand Russell, 1912, The Problems of Philosophy (with Introduction by John Perry 1997), Oxford UniversityPress, New York, NY, ISBN 0-19-511552-X.

• Bertrand Russell, 1919, Introduction to Mathematical Philosophy, Barnes & Noble, Inc, New York, NY, ISBN978-1-4114-2942-0. This is a non-mathematical companion to Principia Mathematica.

• Amit Hagar 2005 Introduction to Bertrand Russell, 1919, Introduction to Mathematical Philos-ophy, Barnes & Noble, Inc, New York, NY, ISBN 978-1-4114-2942-0.

• Albert North Whitehead and Bertrand Russell, 1927 2nd edition, (first edition 1910–1913), Principia Math-ematica to *56,1962 Edition, Cambridge at the University Press, Cambridge UK, no ISBN. Second edition,abridged to *56, with Introduction to the Second Edition pages Xiii-xlvi, and new Appendix A (*8 PropositionsContaining Apparent Variables) to replace *9 Theory of Apparent Variables, and Appendix C Truth-Functionsand Others.

34 CHAPTER 8. LOGICISM

8.10 External links• Logicism at Encyclopaedia of Mathematics

• Logicism

Chapter 9

Panlogism

In philosophy, panlogism is a Hegelian doctrine that holds that the universe is the act or realization of Logos.[1][2]According to the doctrine of panlogism, logic and ontology are the same study.[3]

9.1 References[1] “Dagobert D. Runes, Dictionary of Philosophy, 1942” Retrieved September 1, 2009

[2] “Panlogism” at the free dictionary Retrieved September 1, 2009

[3] “Panlogism” at About.com Retrieved September 1, 2009

35

Chapter 10

Polylogism

Polylogism is the belief that different groups of people reason in fundamentally different ways (coined from Greekpoly=many + logos=logic).[1] The term is attributed to Ludwig von Mises,[2] who claimed that it described Marxismand other social philosophies.[3] In the Misesian sense of the term, a polylogist ascribes different forms of “logic” todifferent groups, which may include groups based on race,[1][4] gender, class, or time period.

10.1 Types of polylogism

A polylogist would claim that different groups reason in fundamentally different ways: they use different “logics”for deductive inference. Normative polylogism is the claim that these different logics are equally valid. Descriptivepolylogism is an empirical claim about different groups, but a descriptive polylogism need not claim equal validity fordifferent “logics”.[5] That is, a descriptive polylogist may insist on a universally valid deductive logic while claimingas an empirical matter that some groups use other (incorrect) reasoning strategies.An adherent of polylogism in the Misesian sense would be a normative polylogist. A normative polylogist mightapproach an argument by demonstrating how it was correct within a particular logical construct, even if it wereincorrect within the logic of the analyst. AsMises noted “this never has been and never can be attempted by anybody.”

10.1.1 Proletarian logic

The term 'proletarian logic' is sometimes taken as evidence of polylogism.[6] This term is usually traced back to JosephDietzgen in his 11th letter on logic.[7][8] Dietzgen is the now obscure philosophical monist of the 19th century whocoined the term 'dialectical materialism' and was praised by communist figures such as Karl Marx and V. I. Lenin.[9]His work has received modern attention primarily from the philosopher Bertell Ollman. As a monist, Dietzgen insistson a unified treatment of mind and matter. As Simon Boxley puts it, for Dietzgen “thought is as material an eventas any other”. This means that logic too has “material” underpinnings. (But note that Dietzgen’s “materialism” wasexplicitly not a physicalism.)

10.1.2 Racialist polylogism

Racialist polylogism is often identified with the Nazi era. It has been proposed that the ferment around Einstein’stheory of relativity is an example of racialist polylogism. Some of the criticisms of relativity theory were mixedwith racialist resistance that characterized the physics as an embodiment of Jewish ideology. (For example, NobelPrize winner Philipp Lenard claimed scientific thought was conditioned by “blood and race”, and he accused WernerHeisenberg of teaching "Jewish physics".[10]) However this appears to be an argument ad hominem, not polylogism.Modern examples of supposed racialist polylogism are generally misleading. For example, US Supreme Court JusticeSotomayor has been accused of racialist polylogism for suggesting that a “wise Latina” might come to different legalconclusions than a white male. Although generally given the interpretation that life experience can influence one’sability to understand the practical implications of a legal argument, some commentators suggested that Sotomayorsupported the idea that Latinas have a unique “logic”.[11][12]

36

10.2. REFERENCES 37

10.2 References[1] Percy L. Greaves Jr. (1974). “Glossary, Panphysicalism - Pump-priming”. Mises Made Easier. Retrieved 2011-01-13.

[2] Perrin, Pierre (2005). “Hermeneutic economics: Between relativism and progressive polylogism”. Quarterly Journal ofAustrian Economics 8 (3). pp. 21–38. doi:10.1007/s12113-005-1032-3.

[3] Ludwig von Mises. “Chapter 3, Section 1”. Human Action (PDF) (1996 ed.). pp. 72–75.

[4] Alexander Moseley (2002). A Philosophy of War. Algora Publishing. p. 239. ISBN 978-1-892941-94-7.

[5] Roderick Long. “Anti-Psychologism in Economics: Wittgenstein and Mises” (PDF).

[6] " “Polylogism”.

[7] Emmett, Dorothy (1928). “Joseph Dietzgen: The Philosopher of Proletarian Logic”. Journal of Adult Education 3. pp.26–35.

[8] The Positive Outcome of Philosophy; Letters on Logic, Especially Democratic Proletarian Logic.

[9] A Dictionary of Marxist thought

[10] Joseph W. Bendersky (2000). A history of Nazi Germany: 1919-1945. p. 140.

[11] Rich Lowry. “How Sotomayor Misspoke”.

[12] Peter Wehner. “Judge Sotomayor, in Her Own Words”.

• Boxley, Simon, (2008), Red, Black andGreen: Dietzgen’s PhilosophyAcross theDivide. http://www.anarchist-studies-network.org.uk/documents/Conference%20Papers/Simon%20Boxley.doc

• Ollman, B. (1976) Alienation: Marx’s Conception of Man in Capitalist Society, Cambridge: Cambridge Uni-versity Press

• Ollman, B. (2003a) Dance of the Dialectic: Steps in Marx’s Method, Chicago: University of Illinois Press

• Ollman, B. (2003b) ‘Marx’s Dialectical Method is more than a Mode of Exposition: A Critique of SystematicDialectics’ in Albritton, R. & Siloulidis, J. (Eds.) NewDialectics and Political Economy, Basingstoke: PalgraveMacmillan

• Perrin, Pierre, “Hermeneutic economics: Between relativism and progressive polylogism”, Quarterly Journalof Austrian Economics, Volume 8, Number 3, 21-38, doi:10.1007/s12113-005-1032-3

10.3 External links• Theory and History by Ludwig von Mises, for an exposition.

Chapter 11

Preintuitionism

In themathematical philosophy, the pre-intuitionistswere a small but influential group who informally shared similarphilosophies on the nature of mathematics. The term itself was used by L. E. J. Brouwer, who in his 1951 lectures atCambridge described the differences between intuitionism and its predecessors:

Of a totally different orientation [from the “Old Formalist School” of Dedekind, Cantor, Peano,Zermelo, andCouturat, etc.] was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue.These thinkers seem to have maintained a modified observational standpoint for the introduction ofnatural numbers, for the principle of complete induction [...] For these, even for such theorems aswere deduced by means of classical logic, they postulated an existence and exactness independent oflanguage and logic and regarded its non-contradictority as certain, even without logical proof. For thecontinuum, however, they seem not to have sought an origin strictly extraneous to language and logic.

11.1 The introduction of natural numbers

The Pre-Intuitionists, as defined by Brouwer, differed from the Formalist standpoint in several ways, particularly inregard to the introduction of natural numbers, or how the natural numbers are defined/denoted. For Poincaré, thedefinition of a mathematical entity is the construction of the entity itself and not an expression of an underlyingessence or existence.This is to say that no mathematical object exists without human construction of it, both in mind and language.

11.2 The principle of complete induction

This sense of definition allowed Poincaré to argue with Bertrand Russell over Giuseppe Peano’s axiomatic theory ofnatural numbers.Peano’s fifth axiom states:

• Allow that; zero has a property P;

• And; if every natural number less than a number x has the property P then x also has the property P.

• Therefore; every natural number has the property P.

This is the principle of complete induction, which establishes the property of induction as necessary to the system.Since Peano’s axiom is as infinite as the natural numbers, it is difficult to prove that the property of P does belong toany x and also x+1. What one can do is say that, if after some number n of trials that show a property P conserved inx and x+1, then we may infer that it will still hold to be true after n+1 trials. But this is itself induction. And hencethe argument is a vicious circle.From this Poincaré argues that if we fail to establish the consistency of Peano’s axioms for natural numbers withoutfalling into circularity, then the principle of complete induction is not provable by general logic.

38

11.3. ARGUMENTS OVER THE EXCLUDED MIDDLE 39

Thus arithmetic and mathematics in general is not analytic but synthetic. Logicism thus rebuked and Intuition is heldup. What Poincaré and the Pre-Intuitionists shared was the perception of a difference between logic and mathematicswhich is not a matter of language alone, but of knowledge itself.

11.3 Arguments over the excluded middle

It was for this assertion, among others, that Poincaré was considered to be similar to the intuitionists. For Brouwerthough, the Pre-Intuitionists failed to go as far as necessary in divesting mathematics from metaphysics, for they stillused principium tertii exclusi (the "Law of excluded middle").The principle of the excluded middle does lead to some strange situations. For instance, statements about the futuresuch as “There will be a naval battle tomorrow” do not seem to be either true or false, yet. So there is some questionwhether statements must be either true or false in some situations. To an intuitionist this seems to rank the law ofexcluded middle as just as unrigorous as Peano’s vicious circle.Yet to the Pre-Intuitionists this is mixing apples and oranges. For them mathematics was one thing (a muddledinvention of the human mind (aka. synthetic)), and logic was another (analytic).

11.4 Other Pre-Intuitionists

The above examples only include the works of Poincaré, and yet Brouwer named other mathematicians as Pre-Intuitionists too; Borel and Lebesgue. Other mathematicians such as Hermann Weyl (who eventually became disen-chanted with intuitionism, feeling that it places excessive strictures on mathematical progress) and Leopold Kroneckeralso played a role - though they are not cited by Brouwer in his definitive speech.In fact Kronecker might be the most famous of the Pre-Intuitionists for his singular and oft quoted phrase, “Godmade the natural numbers; all else is the work of man.”Kronecker goes in almost the opposite direction from Poincaré, believing in the natural numbers but not the law ofthe excluded middle. He was the first mathematician to express doubt on non-constructive existence proofs. That is,proofs that show that something must exist because it can be shown that it is “impossible” for it not to.

11.5 External links• Logical Meanderings - a brief article by Jan Sraathof on Brouwer's various attacks on arguments of the Pre-Intuitionists about the Principle of the Excluded Third.

• Proof And Intuition - an article on themany varieties of knowledge as they relate to the Intuitionist and Logicist.

• Brouwer’s Cambridge Lectures on Intuitionism - Wherein Brouwer talks about the Pre-Intuitionist School andaddresses what he sees as its many shortcomings.

Chapter 12

Ultrafinitism

In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, andstrong-finitism is a form of finitism. There are various philosophies of mathematics that are called ultrafinitism.A major identifying property common among most of these philosophies is their objections to totality of numbertheoretic functions like exponentiation over natural numbers.

12.1 Main ideas

Like other strict finitists, ultrafinitists deny the existence of the infinite set N of natural numbers, on the grounds thatit can never be completed.In addition, some ultrafinitists are concerned with acceptance of objects in mathematics that no one can constructin practice because of physical restrictions in constructing large finite mathematical objects. Thus some ultrafinitistswill deny or refrain from accepting the existence of large numbers, for example, the floor of the first Skewes’ number,which is a huge number defined using the exponential function as exp(exp(exp(79))), or

eee79 .

The reason is that nobody has yet calculated what natural number is the floor of this real number, and it may noteven be physically possible to do so. Similarly, 2 ↑↑↑ 6 (in Knuth’s up-arrow notation) would be considered only aformal expression which does not correspond to a natural number. The brand of ultrafinitism concerned with physicalrealizability of mathematics is often called actualism.Edward Nelson criticized the classical conception of natural numbers because of the circularity of its definition. Inclassical mathematics the natural numbers are defined as 0 and numbers obtained by the iterative applications of thesuccessor function to 0. But the concept of natural number is already assumed for the iteration. In other words, toobtain a number like 2 ↑↑↑ 6 one needs to perform the successor function iteratively, in fact exactly 2 ↑↑↑ 6 times to0.Some versions of ultrafinitism are forms of constructivism, but most constructivists view the philosophy as unworkablyextreme. The logical foundation of ultrafinitism is unclear; in his comprehensive survey Constructivism inMathematics(1988), the constructive logician A. S. Troelstra dismissed it by saying “no satisfactory development exists at present.”This was not so much a philosophical objection as it was an admission that, in a rigorous work of mathematical logic,there was simply nothing precise enough to include.

12.2 People associated with ultrafinitism

Serious work on ultrafinitism has been led, since 1959, by Alexander Esenin-Volpin, who in 1961 sketched a programfor proving the consistency of ZFC in ultrafinite mathematics. Other mathematicians who have worked in the topicinclude Doron Zeilberger, Edward Nelson, and Rohit Jivanlal Parikh. The philosophy is also sometimes associatedwith the beliefs of Ludwig Wittgenstein, Robin Gandy, and J. Hjelmslev.

40

12.3. COMPLEXITY THEORY BASED RESTRICTIONS 41

Shaughan Lavine has developed a form of set-theoretical ultra-finitism that is consistent with classical mathematics.[1]Lavine has shown that the basic principles of arithmetic such as “there is no largest natural number” can be upheld,as Lavine allows for the inclusion of “indefinitely large” numbers. [2]

12.3 Complexity theory based restrictions

Other considerations of the possibility of avoiding unwieldy large numbers can be based on computational complexitytheory, as in Andras Kornai's work on explicit finitism (which does not deny the existence of large numbers[3]) andVladimir Sazonov's notion of feasible number.There has also been considerable formal development on versions of ultrafinitism that are based on complexity theory,like Samuel Buss's Bounded Arithmetic theories, which capture mathematics associated with various complexityclasses like P and PSPACE. Buss’s work can be considered the continuation of Edward Nelson's work on PredicativeArithmetic as bounded arithmetic theories like S12 are interpretable in Raphael Robinson's theory Q and thereforeare predicative in Nelson's sense. The power of these theories for developing mathematics is studied in BoundedReverse Mathematics as can be found in the works of Stephen A. Cook and Phuong The Nguyen. However theseresearches are not philosophies of mathematics but rather the study of restricted forms of reasoning similar to ReverseMathematics.

12.4 Notes[1] http://plato.stanford.edu/entries/philosophy-mathematics/

[2] http://plato.stanford.edu/entries/philosophy-mathematics/

[3] http://kornai.com/Drafts/fathom_3.html

12.5 References• Ésénine-Volpine, A. S. (1961), “Le programme ultra-intuitionniste des fondements des mathématiques”, In-

finitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959), Oxford: Pergamon, pp. 201–223, MR0147389 Reviewed by Kreisel, G.; Ehrenfeucht, A. (1967), “Review of Le Programme Ultra-Intuitionniste desFondements des Mathematiques by A. S. Ésénine-Volpine”, The Journal of Symbolic Logic (Association forSymbolic Logic) 32 (4): 517, doi:10.2307/2270182, JSTOR 2270182

• Lavine, S., 1994. Understanding the Infinite, Cambridge, MA: Harvard University Press.

12.6 External links• Explicit finitism by Andras Kornai

• On feasible numbers () by Vladimir Sazonov

• “Real” Analysis Is A Degenerate Case Of Discrete Analysis by Doron Zeilberger

• Discussion on formal foundations on MathOverflow

• History of constructivism in the 20th century by A. S. Troelstra

• Predicative Arithmetic by Edward Nelson

• Logical Foundations of Proof Complexity by Stephen A. Cook and Phuong The Nguyen

• Bounded Reverse Mathematics by Phuong The Nguyen

• Reading Brian Rotman’s “Ad Infinitum…” by Charles Petzold

42 CHAPTER 12. ULTRAFINITISM

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