Saturday, March 27, 2021

Appendix F: Bertrand Russell's Critique of Fregean Logico-Mathematical Objects

 

Bertrand Russell's Critique of Fregean Logico-Mathematical Objects


“…the arithmetic of cookies and pebbles.”—Gottlob Frege

 

Introduction

Last July 1, 2020 I began a reading regime of Ernst Cassirer’s three-volume work on his philosophy of symbolic form and thirteen books by other authors. These volumes were written between 1923 and 1929 with the titles of The Philosophy of Symbolic Forms: Vol. 1, Language (pdf.): Vol. 2, Mythic Thought (pdf.), and, Vol. 3, Phenomenology of Knowledge (pdf.). I included in my reading routine other works by Cassirer such as, An Essay on Man,”(1944)(pdf.) and Language and Myth (1925).

 I knew there would be a big pay-off reading Ernst Cassirer’s three volumes on the philosophy of symbolic forms, and in particular the third book on the phenomenology of knowledge in Chapter 4, “The Object of Mathematics.”  Numbers, sets, and classes are examples of mathematical objects. This fourth chapter is complex, and examines a famous contradiction discovered in May 1901 by Bertrand Russell within Gottlob Frege’s set theory logic that attempted to establish mathematics on logic as its foundation. This discovered antinomy is referred to in the history of philosophy as “Russell’s Paradox,” and is relevant to many of the issues I have written about in this collection of essays concerning classic philosophical problems that are still relevant today.

My thinking was already moving toward this direction of Russell’s paradox in the essay, The Struggle Against Solipsism,” but stopped short of exploring the structure of mathematical objects. I will further develop the critique of solipsism by further examining Russell’s paradox that resulted from Frege’s conception of number. Russell sums up the solipsistic worldview in the saying, “I alone exist.”

The timeline of persons, publications, and events are important for understanding the philosophical differences between Bertrand Russell and Wittgenstein’s later radically changed view of language. Gottlob Frege published “The Foundations of Arithmetic, in 1884 (pdf.). Russell also published “The Principles of Mathematics, 1903 (pdf.).”  Russell authored “Introduction to Mathematical Philosophy,"(1919)(pdf.)while imprisoned six months for his history of opposing the First World War.

Wittgenstein did not meet Russell in person until 1911. Wittgenstein completed the Tractatus (pdf.in August 1918 shortly before the Italians held him as an enemy war prisoner for nine months in 1919. In both cases, the two philosophers were not treated as ordinary prisoners. By 1929 Wittgenstein’s view of logic had so evolved away from the Tractatus that he returned to Cambridge and become a lecturer of Trinity College after submitting the Tractatus, finally published in 1922, as his doctoral thesis. It was actually Russell that did all the necessary footwork to get the Tractatus published, then Wittgenstein did not like Russell’s introductory preface to the book.

Many authors have written about Russell’s discovered contradiction in Frege’s logical system, but often fail to explain what this contradiction actually means. How important could an abstract logical contradiction possibly be?  Russell’s paradox also sheds light on Wittgenstein’s motivations for studying philosophy; why he changed his views about language later in his life; and his many cryptic aphorisms about language, logic, and philosophy. 

Russell’s discovered antinomy cannot be fully understood without first briefly describing the relevant schools of mathematics; views on the nature of logical systems; and theories of truth that were common among philosophers in the early 1900s. I will then attempt to generally classify key contributing scholars including Gottlob Frege (1848-1925), G.E. Moore (1873-1958 ), Bertrand Russell (1872-1970), Ludwig Wittgenstein (1889-1951), mathematicians David Hilbert (1862-1943), Heinrich Rickert (1863-1936), and L.E.J. Brouwer (1881-1966). These academicians will sometimes change schools of thought, or hold incompatible doctrines causing multiple variations so that classification of each thinker can be ambiguously on the edge of another group. Yet, this kind of school classification is in the academic literature, and is useful for getting a wide view of the historical philosophical landscape during this era of changing mathematical thinking.

To better understand Frege’s contradiction, Russell’s paradox will be expressed both in terms of ordinary language that does not presuppose knowledge of logical symbolism, and in symbolic notation for the interested reader. The logical sign, Cassirer observed, compels “thought to come forth from its inner workshop and manifest itself in its involvements and complexities.” Lastly, I want to comment on Ernst Cassirer (1874-1945) and Wittgenstein’s insightful critiques of logic resulting from the antinomy and what the implications mean for philosophy and the physical sciences.

While researching this essay I got perverse pleasure reading that until Wittgenstein was 20 years old, he was a bad speller.


Part I

Paradigms of Truth And Logic


"If I designate phenomenal insight as knowledge, then theoretical insight rests on faith--faith in the reality of one's own ego and that of others, or in the reality of the outside world or of God"—Mathematician, Hermann Weyl (1885 – 1955)


1.)    The Pragmatic Theory of Truth is often associated with the American philosopher William James (1842-1910). The Greek term “πρᾶγμα” (prag-ma) is translated as “deed,” “fact,” or “business” and in this context truth means the utility of achieving a goal or end. A statement is true based on its usefulness as a belief: “X is useful; therefore, X is true.” Pragmatism does not seem to be an appropriate theory of truth in mathematics; however, some mathematicians note that pragmatism is important in the creation and use of signs in symbolic logic notation and math. Pragmatism in science is the successful manipulation of objects for a desired purpose. However, ordinary language philosophers have the recurrent theme of pragmatism throughout their understanding of how language actually works and how systematic thinking emerges. 

 2.)    The Coherence Theory of Truth is a more common concept of truth embraced by modern mathematicians by defining logical truth in relation to consistency and not by some content in the world; but instead, by the relation of content to other content, or by belief in relation to another belief. In symbolic logic, a valid argument is merely non-contradictory propositions in relation with another proposition; for example, “A is not non-A” is a consistent proposition, or coherent, and merely tautologically true (A is A). 

3.)    The Correspondence Theory of Truth holds that truth is the correspondence of some propositional content within the world since mere consistency alone is an inadequate criterion for truth. Propositional content is understood in this theory as descriptive statements such as “A is B.” The argument “All men are cows, and John is a man; therefore, John is a cow,” is a perfectly valid (consistent) argument. We can rewrite the same propositions as “All M are C, and J is M; therefore J is C” which is a valid argument form known as a hypothetical syllogism. In addition to a valid (consistent) inference, a propositional argument must also have true premises before it can be called a “sound argument.” Sound arguments are what philosophers seek to present into the market place of ideas since, in a deductive argument, if the premise are true, and the inferences are valid; then, the conclusion must be true. 

A true proposition in this theory is one that corresponds to the state of something in the world, or a fact. A belief is true if it corresponds to an existing factual object. There are differences among logicians and philosopher on whether truth is an attribute of an object, or a proposition, or belief.

Wittgenstein influenced Russell into accepting the correspondence theory of truth although their ontologies are different (Dr. K. Banick video lecture on Russell and Wittgenstein’s dissimilar metaphysics). Interestingly, around 1904, Russell and Moore held the identity theory of truth in which a true proposition is the object of belief: “When a proposition is true, it is identical to a fact, and a belief in that proposition is correct.” In other words, truth was a property of a proposition itself and not something else external to it. The nature of correspondence and truth bearing propositions forced Russell and Moore to abandon the identity theory after they were unable to define the difference between true and false propositions within this particular theory of truth.


Theories of Logic and Mathematics

 

1.)    Logical Conventionalism is the view that logical systems are invented cultural norms, signs, and linguistic conventions. The denial of logical necessity is key to conventionalism. All logical analytic a priori necessity (A in not non-A) is a disguise of collective non-arbitrary agreement on procedural and definitional rules that work when applied to the world. There are no real necessary independent truths of meaning, logic, or mathematics, but instead only a set of axiomatic rules for the manipulation of intuitive empirical signs. This school is also called “mathematical terminism” and views signs as mere “intuitive figures without any real independent meaning (Cassirer, Vol. 3, p. 381).”  Mathematician David Hilbert wrote, "In the beginning was the sign (Ibid., p. 380).” Conventionalism is associated with pragmatism in its way of analyzing how ordinary language is actually used.

Henri Poincaré (1855-1912) held to the coherence theory of truth and was among the first conventionalist mathematicians who viewed all mathematical objects as independent human constructions that have no reality in existence. He believed that natural numbers were innate to human understanding and could not be reduced to symbolic logic set theory objects. The logical positivists also held a conventionalist viewpoint of logic and mathematics as pure form, and completely independent of the world.

The late Wittgenstein is an extreme conventionalist by understanding language as a public tool consisting of norms of linguistic usage created from a certain way of life, and not possessing a hidden necessary essence of some ideal language. Wittgenstein’s late lecture notes were published posthumously in 1953 as, Philosophical Investigations that treats language and logic like the intuitionists who believe that mathematics emerge from a “primordial intuition” of space and time, which we impose on language and the world. Many of his ideas about language toward the end of his life were antithetical to the early Tractatus that instead attempted to discover a hidden calculus concealed within ordinary language. Wittgenstein’s philosophical “investigation” was about why the formalist Tractatus conception of language is wrong.

Wittgenstein respected ordinary language; and even though it can be entangled in misunderstandings from misuse, “philosophy may in no way interfere with the actual use of language; it can in the end only describe it…it cannot give it any foundation either, It leaves everything as it is….(Philosophical Investigations, paragraph 124).”

We can also place early Wittgenstein (Tractatus) in the conventionalist group since he has described logical propositions (A is not non-A) as senseless (Sinnolos) tautologies. Furthermore, Wittgenstein posited the existence of “simple objects” (Tractatus; 2.02) that are the smallest possible entities that names denote so that there is a connection between existence and logic through language. He was unable to give a actual example of the posited a priori simple object. And yet Wittgenstein writes of the simple object as both physical and as a non-physical object so that he is straddling two different schools of thought (conventionalism and intuitionism) if we try to force him into a category. Wittgenstein is again being deliberately ambiguous. 

2.)    Logical Formalism is a modern form of Platonism that included such thinkers as Frege, Russell, Hilbert, and the early Wittgenstein. This simplest version of Platonism believes in a totality of natural numbers (counting numbers) that exist independently of mind and can be symbolically defined as quantifying propositions (All X is Y) bearing decidable truth-values. Formalism typically holds to the consistency, or coherence theory of truth, and to sign conventionalism. As the leading figure of formalism, Hilbert viewed mathematics as the “manipulation of symbols” according to an axiomatic decision procedure for proof of consistency in a logical system. 

However, Hilbert also held some theoretical positions that were consistent with logical intuitionalism so that he is called by some as a “passive logical intuitionist” rejecting Russell and Frege’s version of Platonic objective realism while at the same time he sought to establish mathematics on a Platonist mathematical foundation in opposition to intuitionalism. 

Hilbert thought he could achieve his goal by the notion of a pure theory of mathematical “signs.” His project is to require verification of consistency with his theory of proof so “the process of verification is shifted from the sphere of content to that of symbolic thinking. As precondition for the use of logical inferences… certain sensuous and intuitive characters must always be given to us (Cassirer, Vol. 8, p. 379)."  There is seemingly an inherent conflict between these two doctrines: intuitionalism understands the symbol, or sign as an essential expression of thinking, but the formalist sees the symbol as merely marks on paper (Ibid., p. 381 ff). With empirical signs, mathematical-logical proofs seeking to identify contradictory signs could be completed by machine. Hilbert held to the doctrine that “mathematical symbols themselves, and not any meaning that might be ascribed to them, that are the basic objects of mathematical thought (see, Formalism: Britannica).” Cassirer’s comment on Hilbert’s project is “… that mathematics can retrieve its threatened autonomy only by becoming a pure theory of signs. Among present day mathematicians it is Hilbert who has drawn this conclusion most decisively. In direct opposition to intuitionism he strives to rehabilitate the classical form of analysis and theory of sets (Ibid., p. 379)." 

However, terminism transforms mathematics into a “monstrous tautology.” Authors Russell and Alfred North Whitehead believed they were able to develop a coherent and complete minimalist set of logical symbols as presented in their famous three volume work,“Principia Mathematica,”(1910,1912, and 1913). 

Logicism is consistent with formalism and is the effort by mathematicians to show mathematics has its foundation in logic, or that some mathematics can be reduced to logic. Frege, the early Wittgenstein, and Russell as the leading member of this school, attempted to prove this relationship using the theory of sets, and the concept of classes. Wittgenstein’s Tractatus is a classic example of logicism. The assumption of the Tractatus, shared by young Wittgenstein and all the Logical Positivists, is that natural language has a concealed calculus that can be symbolized and expressed as logical rules. 

3.)  Intuitionism is a rejection of mathematical Platonism. L.E.J. Brouwer (1881-1966) represents the position that mathematical objects (numbers) are mentally constructed and that mathematics is inherent in human experience. Brouwer believes that mathematics is "far more an action than a theory (Ibid., p. 371)." This inherent ability of reasoning is enabled by the intuitions of space and time conceptual totalities. Whereas mathematical objects such as numbers are autonomous entities in Platonism, they are instead reified “constructed” symbolic objects for the intuitionist. Brouwer’s intuitionism required a defining procedure for constructing a mathematical object before asserting its existence (see, Russell’s Paradox: Stanford Encyclopedia of Philosophy). Brouwer is anti-formalistic and was opposed to set-theory ideas even before the discovery of Russell’s paradox (Encyclopedia of Philosophy,1967, Macmillan, Vol. 5, p 204). Wittgenstein started out as a formal logicist, but ended up a Brouwerian intuitionalist concluding that language--like a game--has no essence, but is a pragmatic constructivist activity. Philosopher William Barrett summed up the insight of intuitionism as: “Technique has no meaning apart from some informing vision (The Illusion of Technique, Barrett, William, Anchor/Doubleday, 1979, p. 88).”



Part II

 

The Contradiction

 

“Alas, arithmetic totters.”--Gottlieb Frege

 

A tautology is a proposition that is necessarily true such as with the proposition, “Either it is going to Rain, or not Rain.” Every possible combination of the truth-value of ‘R’ in a disjunctive proposition is true, “R v ~R”.

On the other hand, the denial of a tautology is necessarily false, “It is false that either it is going to rain, or not rain,” ~(R v ~R). 


Every possible combination of truth-values that compose a contradiction result in a proposition that is false necessarily as in the first truth-table column.

Typically, we think of a noun as a name denoting some ‘thing’ in the world—some corresponding referent that is the sense of a name. However, we also use names to describe something that has no referent because it does not exist, but still has sense like the name “unicorn” in a folk tale. On the other hand, self-referential propositions refer to themselves as propositions--“Everything I say is false”--creating endless paradoxes (see, Graham Priest on Paradoxes). 

A contradiction can have many disguises, and some might appear as the simple equivocation of words. I have discussed contradiction is the essay, “The Machine Paradigm of Nature And Disenchantment” and there are other well-known examples of stated contradictions such as the self-referential contradiction, or the “Liar’s Paradox in which a liar states that everything they say is a lie which leads to the contradiction that if it is the case the liar is lying, then the liar is telling the truth, but then it also means he, the liar, just lied. These contradictions run in a vicious circle. 

Another well-known contradiction is the “Barber’s Paradox” that even puzzled Russell. This example is a story telling of a barber on an island who shaves all those, and only those, who do not shave themselves. The question posed is “Does the barber shave himself?” If the barber shaves himself then he is a man on the island who shaves himself; hence he, the barber, does not shave himself. However, if the barber does not shave himself then he is a man on the island who does not shave himself; hence he, cannot be the barber. In other words:

·       If he does, he cannot be a barber, since a barber does not shave himself.

·       If he doesn't, he falls in the category of those who do not shave themselves, and so, cannot be a barber. 

We can express this paradox in symbolic notation as the following:

{(x)[Bx * (y)(Py ⊃ Sxy)] ≡ ~Syy}

“A Barber on an island (x) Shaves all those island Persons (y), and only those () who do Not shave himself."

When we replace variable “x” with any person “y” we get the contradiction:

Syy ≡ ~Syy

In my thinking the Barber’s Paradox might be more of an analogy to Anthony Flew’s “No true Scotsman,” informal fallacy. A factual statement such as “The circle is green,” is a “synthetic” statement that may be true or false. The statement “Circles are round” is true by definition, or “analytically,” and cannot be false for it is a tautology.  This fallacy is committed, for example, if a person declares as a fact, “No true Scotsman puts sugar in their beer.” However, when the speaker is faced with a counter-example of a Scotsman putting sugar in their beer, they immediately equivocate from the initial factual (synthetic) proposition by declaring by definition (analytically), “He is not a true Scotsman!” So the Barber’s Paradox, depending on how it is stated, could be interpreted as another informal fallacy with “barber” defined analytically and then equivocates to a synthetic description of the barber in an endless circle. 

The essential feature of the No True Scotsman fallacy can be described in this way: the faulty argument takes an empirical proposition and covertly shifts it into a postulate making it a methodological proposition which becomes a norm of description. The No True Scotsman fallacy is not addressed by Wittgenstein, but this interpretation of how the fallacy works is inspired by his last essay, “On Certainty,” (1950)(pdf.), lines: 318, and 321.

A Formal Symbolic Expression of Russell’s Paradox

Our immediate interest is in Russell’s paradox and so we are going to turn away from lairs, barbers, and Scotsmen to class attributes which are “predicable,” and “impredicable.” I am going to present a slightly expanded argument form of Russell’s paradox by borrowing some logical symbols provided by logician I.M. Copi in his textbook, “Symbolic Logic: Fifth edition,” p. 153 (pdf.). We must have some symbols to represent a formal expression of the contradiction in order to symbolize quantitative propositions like, “Socrates has some attribute.” 

Definitions: 

‘s’ = Socrates

~ = negation

≡ Equivalent truth-value of proposition.

F = Predicate variable for the concept of an “attribute”.

(∃F) = This is a special symbol to mean “At least one attribute.”

 (∃F)Fs means “some attribute” as in writing, “Socrates has some attribute.”

(x)(F) means “all attributes.” 


These symbols now enable us to symbolize the following statements such as: 

(x)(F)Fx

“Everything has an attribute.”

(x)( ∃F)Fx

“Everything has some attribute or other.”

(F)(∃x)Fx

“Every attribute belongs to some thing or other.” 

Now we need a symbol to represent “impredicable attributes” of classes. For example, the class of pebbles is not itself a pebble. The class of pebbles does not share any attribute of any of its members (pebbles) so the class is defined as “impredicable” represented by the symbol “I”. Impredicable class attributes are ordinary classes. 

On the other hand, there are some unordinary classes that share predicable attributes the same as its members such as the class of all abstract ideas which is itself abstract. Predicable class attributes are defined as “FF”. Predicable class attributes are attributes of attributes. 

 

1.) IF df ~FF

Impredicable class attributes are now defined (≡ df) as non-predicable class attributes:

 

2.) (F)(IF ≡ ~FF)

All impredicable class attributes (ordinary classes) are defined as non-predicable class attributes.

 

3.) Therefore: II ≡ ~II

Universal Instantiation, applied to premise 2.

Impredicable class attributes are not impredicable class attributes,” by replacing “F” with “I” in premise 2 resulting in a contradiction. 

The rule of universal instantiation applied to premise 2 is analogous to the argument:

Definition: Zeus is not mortal.


1.) All Greeks are mortal.

2.) Zeus is Greek. 

3.) Therefore, Zeus is mortal.

But Zeus is by definition not mortal! 

Likewise, there are no predicable class attributes either: “FF” and “~FF” turn out to be meaningless symbols—pseudo-universals.

 

Part III

 

The Depth of Reason

 

 

"There is a crack in everything. That's how the light gets in."--Leonard Cohen 

 

The closed system of schematized formalistic logicism had a flaw. Much has been written about the contradiction found in Fregean set-theoretic logic of classes and in the process of review turned attention toward a new realm of logic that moves way from mathematical objects to the “extra-logical discrete objects” of mathematical “signs” themselves (Cassirer, p. 378 ff). Hilbert points out mathematical problems have a completely different meaning in relation to signs as “concrete forms.” Signs are an independent power in themselves by ignoring content and ideas; yet, contradiction is still effectively detected by “the appearance of certain constellations of signs (Ibid., p. 381).” 

Wittgenstein defined three intentionally ambiguous logical categories: “sense,” “senseless,” and “nonsense.” He writes, “Tautology, and contradiction are without sense (Tractatus, 4.461).  However, they are not metaphysical nonsense, “Unsinning.” He further clarifies what he means: “Tautology and contradiction are, however, not senseless; they are part of the symbolism, in the same way that “0” is part of the symbolism of Arithmetic (Tractatus, 4.4611).” The original English editions of the Tractatus mistranslated both “Unsinning,” and “Sinnolos” as “senseless.” (see, Wittgenstein’s Conception of Philosophy, K.T. Fann, Univer. of Cal. Press, 1969, p. 25). 

 

Cassirer reminds us that Leibnitz called formulas the “logic of discovery,” the “logica inventionis (Cassirer, p. 440).” The sign represents not a real order, but a possible order according to some fixed principle: “atomic weight” as a unit of the periodic table is one example of a principled ordered system based on fixed principles. From similarly ordered totalities, mathematicians are able to create Imaginary numbers: negative numbers, irrational numbers, squares, and square root of a negative number. Now, the “heuristic maxim” of mathematicians is the unlimited use of form (Cassirer, p. 442). However, a contradiction is the sign of an impossible object. 

Fregean logicism is based on the concepts of sets, classes, and member objects; also known generally as the theory of groups. Russell writes, “But all finite collections of individuals form classes, so that what results is after all the number of a class (Principles of Mathematics, POM, p. 113).” The idea of numbers as properties of classes arose from the mathematics of Giuseppe Peano arguing that arithmetic and algebra could be derived from idea of class (A History of Philosophy: Modern Philosophy, Fichte to Hegel, Vol. 8, Part II, Frederick Copleston, Doubleday, 1967, p. 198).” As a Platonist Russell adopts this idea of number in 1900. Both Russell, and early Wittgenstein, viewed language as ultimately an “object-language.” Gradually, Russell became uncertain of the term “class” comparing it to a parenthesis, “The parenthesis is clear only if what falls inside the two parentheses is clear and definite (Barrett, p. 97). However, unconscious Platonism allowed logicism to reify the concept of class giving priority to the object by hypostatizing the object. 

Χωρισμός is the Greek term for “separation,” “abstraction,” and is used to describe the “secretion of sap.” Set theory applies to first level abstractions, but not abstractions of abstractions such as Russell’s example of the class of attributes of attributes. Every proposition has a range of significance: “The class of men is mortal” is meaningless since a class of men is not man, no more than “Socrates is human,” means “Socrates is humanity” for humanity is not a thing. Russell concluded that the entire concept of “a class is a member of itself” is nonsensical and a “incomplete symbol.”

 

Logicism attempted to reduce mathematics on the underlying stratum of the “reality of things,” not unlike empiricism, by attempting to turn mathematics into physics (Cassirer, p. 375 ff). For logicism the meaning of number rests on some empirical matter, or “the existence of things,” usurping the autonomy of logic. The Neo-Kantian Cassirer instead argues, "The world of mathematical forms is a world of ordinative forms, not of thing forms (Ibid., p.383).” There is also a purely non-material “functional” meaning of number based not on the existence of things, but on the ordinative form of constructed concepts. The logicist concept of number must be de-materialized and detached from “thing-hood.”

 

"To convince someone of the truth, it is not enough to state it, but rather one must find the path from error to truth."-- Wittgenstein

 

In Cassirer’s first volume of his philosophy of symbolic form addressing language, the evolutionary development of the concept of number is traced from an ancient manual concept such as grasping, pointing, and counting objects; then, to the concept of number having no attributes; and finally, to number as de-materialized pure form. 

The Fuji Islanders treated numbers as a quality of things so that there were different numbers for different classes of things in their language: “…different words are used to designate groups of two, ten, a hundred, a thousand coconuts, or a group of ten canoes, ten fish, etc. (Cassirer, Vol. 1, p. 233 ff).”  For the Brazilian Bakairi tribe numbers were so laden to content that a tribesman could not count on their fingers even a few grains of rice without physically touching them for number had to be translated into a specified body part. Cassirer observed that “…it does not suffice at this stage for counted objects to be referred to the parts of the body; in order to be counted, they must in a sense be immediately transposed into parts of the body or bodily sensations (Ibid., p. 230).” In American Indian languages different numbers represent the same quantity of things as opposed to humans, or to animate and inanimate things. Only later in the evolution of language does number gradually shed material content and become independent of things as pure form. Similarly for Russell, he must also de-objectify reason so that thing-constants and thing-unties give way to the universal unities of mathematical function.

 

Fregean logicism is “Arithmetic with frills.”-Wittgenstein.

 

The inventive mind of Ludwig Wittgenstein understood the problem with Frege’s project of establishing mathematics on logic. Logicism instead based logic on mathematics by smuggling in the concept of number within the concept of class. After the contradiction, logic now appears as an unessential decoration attached to mathematics. Other critics suggest quantitative predicate logic should be promptly return to the humanities department. The problem with sets is that the logician must one way or another individually count the number of members within a class. We say that a dozen is a class of twelve members, but we still must conceptually translate “dozen” into twelve members of a class using natural numbers. In addition to the circularity of using number to define number, mathematician Hermann Weyl argued that simply pairing sets is sorely inadequate for establishing the notion of number. Neither is the fundamental logical idea of identity and difference is enough to derive the concept of number. On this issue, Cassirer writes perceptively: “In their attempts to derive the concept of number from the concept of sets, the logicians have always argued most emphatically against any imputation of a petitio principii; they have pointed out that the sense in which logic speaks of ‘identity’ and ‘difference’ does not include the numerical one and the numerical many, and that it is consequently a decided advance in knowledge if we can reduce the numerical sense to a purely logical sense (Cassirer, Vol. 3, p. 377).” Class-theoretic quantification logic was at this stage of development simply the arithmetic of cookies and pebbles. 

"...numbers have neither substance, nor meaning, nor qualities. They are nothing but marks, and all that is in them we have put into them by the simple rule of straight succession." --Hermann Weyl


Concluding Part III:

 The Depth of Reason


Many of Russell’s beliefs concerning logic originated, or were influenced by the early Wittgenstein such as the theory of logical atomism; that atomic facts are logically independent of one another; picture-theory of propositions; and the tautologous circularity of logical-deductive propositions (Copleston, p. 199 ff). With the exception of the circularity of logic, Wittgenstein abandoned those theories by 1933. After hearing intuitionist Brouwer lecture in Vienna during 1928, Wittgenstein returned within one year to Cambridge as a lecturer with a newly evolving philosophy of language and logic (Barrett, p. 87).

Russell commented that once he realized that logic says no more than “a four legged animal is an animal,” he lost interest in logic. Wittgenstein argued in the Tractatus that there is no symbolism that is able to say anything about it own structure: “3.333 A function cannot be its own argument, because the functional sign already contains the prototype of its own argument and it cannot contain itself.” Logic, in the relatively mechanistic Tractatus, was a discovery disguised by language, but later he viewed logic more as an inventive activity as presented in Philosophical Investigations (1953) moving philosophically toward pragmatism, and conventionalism while being open to a variety of new mathematical systems (Barrett, p. 90-91).”  Wittgenstein liked Goethe’s famous saying, “In the beginning was the deed. Brouwer similarly concluded that mathematics is "far more an action than a theory (Cassirer, p. 371)." Wittgenstein still held to the view there are limits to language; “7 Whereof one cannot speak, thereof one must be silent.” (video lecture:, “Ludwig Wittgenstein, the Great War and the Unsayable,” by Ray Monk).

“The Tractatus was not all wrong: it was not like a bag of junk professing to be a clock, but like a clock that did not tell you the right time.”--Wittgenstein

Sentential propositional logic (“Socrates is mortal”) is a finite syllogistic system. It is perfectly internally consistent and decidable as true of false. However, quantification categorical propositional logic uses universal quantifiers such as, “Everything has some attribute or other,” whose meaning encompasses a possible infinite manifold of things. It is within quantification logic that the contradiction appears. In his “Principles of Mathematics,” Russell writes: “For publishing a work containing so many unsolved difficulties, my apology is, that investigation revealed no near prospect of adequately resolving the contradiction discussed in Chapter x, [The Contradiction] or of acquiring a better insight into the nature of classes (p. vii).” 

Russell’s first attempt to resolve the contradiction was to construct formal syntactical rules prohibiting the inferential steps that give rise to the contradiction. His newly formulated “Simple Theory of Types,” presented a lexicon of symbols to represent types of attributes on different levels in a hierarchy of abstract idealizations; individuals, then attributes of individuals, then attributes of attributes of individuals, and so on. As a rule, the attribute of one individual cannot be predicated of entities of a different abstract hierarchical type such that  “impredicable” class attributes cannot even be defined to produce a contradiction. "Tâtonnement," is the ugly way logical sausage is made, i.e.,  “groping one’s way moving forward by trail and error.” If one wants to avoid an inference leading to a contradiction, then don’t violate the “limitative theorems.”  Russell eventually realized that he was not dealing with different types of entities, but different types of linguistic functions (Copleston, p. 190).  Mathematics advanced with this insight that “…the validity of a mathematical object is not its construction but its ‘constructability’ (Cassirer, p. 372).” Later, other mathematicians created new functions for better versions of Russell’s first theory of types.

“...in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes."-- Hermann Weyl

In contemplation of Russell’s paradox, Cassirer caught sight of the important themes of freedom, possibility, and creativity. He believed, “No mathematical concept...can be gained through mere abstraction from the given; a mathematical concept always comprises a free act of combination, an act of synthesis (Cassirer, p. 361)." Mathematical symbolism is an instrument like a microscope, or telescope that enhance our vision (Ibid., p. 386). Mathematics is not a once completed project. The synthetic function is produced from the creative acts of intuition (experience) and the understanding. Cassirer has been called a Hegelian, in addition to a Kantian philosopher for his view of the essential character of the logos as going through a process of self-alienation, and then intellectual reunification through the act of conceptual synthesis (Ibid., p. 432).

Wittgenstein’s Loophole Metaphysics 

“Vorbei redden (speak past) Gödel.”—Wittgenstein


In 1930 Kurt Friedrich Gödel published his incompleteness theorems that state: “A.) If a logical system is consistent, it cannot be complete. B.) The consistency of axioms cannot be proved within their own system.” Wittgenstein was not alarmed by Russell’s paradox, but instead tossed the ladder of logicism away after he climbed it (see, Tractatus, 6.45). He viewed the inconsistency of logicism the same as the ancient paradoxes such as Zeno’s paradox of motion. If the axioms of a logical system cannot prove themselves, then a “hierarchy of languages,” with multiple logics could say what a logical-scientific structure—like the Tractatus—cannot say about itself. Russell correctly understood that Wittgenstein’s Tractatus has a loophole: “… Mr. Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the skeptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit. The whole subject of ethics, for example, is placed by Mr. Wittgenstein in the mystical, inexpressible region (Tractatus, p. 18 )." Philosophy is just one loophole avoiding solipsism with others being literature, music, art, ethics, and religion.

"Don't for heaven's sake, be afraid of talking nonsense! But you must pay attention to your nonsense."--Wittgenstein, 1947

In addition to scientific positivism, a constellation of other methodologies is required for studying human behavior, language, and cultural history. The objectivating attitude of natural-scientific reductionism blends into culture in a horrible transposition by which human beings are perceived as mere objects while things are seen as possessing human attributes and value (see, The Objectification of Human Beings And Animistic Commodities).

In another example of misplaced scientific reductionism, economist Steve Keen has critiqued Neo-classical and Neo-liberal economic theories for treating macroeconomics as applied microeconomics: these two methodologies are on completely different levels of abstraction. Dr. Keen argues that “…psychology is not applied biology, nor is biology applied chemistry…nor is macro-economics applied microeconomics (video lecture: 6:51 min., Debunking Economics: the Failure of Neo-classical Economics with Steve Keen). Keen quotes 1977 Nobel Prize winning theoretical physicist, Philip Anderson's saying of multi-levels of scientific abstraction: “Instead at each level of complexity entirely new properties appear, and the understanding of the new behavior requires research which I think is as fundamental in its nature as any other (video: 6:41 min.)”.

Mathematician Hermann Weyl studied under formalist David Hilbert, but later adopted the views of intuitionist Brouwer in 1919.  By 1928, Weyl changed his views again and moved toward the constructivism of Neo-Kantian Ernest Cassirer's philosophy of symbolic form. Another figure involved in the crisis of Fregean logicism is Neo-Kantian Heinrich Rickert who rejects any effort to establish mathematics in empirical reality, or “the thing-sphere of the countable.” Consequently, he believed it impossible to derive mathematics from logic for even the logical unity of “1 = 1” requires experience (intuition) which is ultimately ‘alogical’ (Ibid., p. 346.)”  Interestingly, Rickert acted as an advisor to a young theology student named Martin Heidegger at the University of Freiburg and approved his doctoral thesis in 1916. Rickert was a strong influence on Heidegger’s interpretation of Kant.

By critically applying a constellation of scientific paradigms to the complexities of human life, we might be able to live in truth, and then live in freedom.


“With mathematics we stand precisely at that intersection of bondage and freedom that is the essence of the human itself.”― Hermann Weyl


No comments:

Post a Comment