Is Carnap's Program Viciously Circular? Or Perhaps He's a

Is Carnap’s Program Viciously Circular?Or Perhaps He’s a Relativist?

Emerson P. Doyle

The University of Western OntarioDepartment of Philosophy

[email protected]

December 9, 2013

Preliminary Remarks Relativism? Larry 6= Carnap Explication

Two Questions

Rudolf Carnap(1891–1970)

Answer: No.

1. Carnap is forwarding the doctrinesof Mathematical Conventionalismand Empiricism (Viciously Circular)

2. Carnap suggests a radical shift awayfrom asserting philosophical theses,and toward a Tolerant Relativism(Descends into Just Relativism)


Two Questions


Answer: No.




Two Questions


Answer: No.




Two Questions


Answer: No.




Two Questions


Answer: No.




In This Talk I Will Show. . .

Carnap’s position is more careful than eitherof these two alternatives.

The Take Home Message:Carnap’s program aims to replacetraditional philosophical methodology withthe Explication of the concepts, methods,theories, and languages of the sciences,including mathematics.

Carnap forwards the doctrines of Empiricismand Mathematical Conventionalism as thebest explication of the methodologicalorientation of the sciences.












Example: Mathematical Conventionalism

Mathematical Conventionalism:Purely mathematical languages are free fromcontent, or in other words are empty calculi,formal auxiliaries of our language utilized forthe inferential manipulation of contentful,empirical sentences—thus there’s no questionof absolute correctness in the choice ofmathematical principles.

Carnap: This is the maximal account weneed to give to explain mathematics’ role inthe practice of science.

Anything further seems to be difficult tosupport, mere philosophical speculation.












What About A Tolerant Relativism?

Carnap thinks most philosophical assertionsseem to be contentful assertions about theworld, but are better interpreted as about thelanguage we use to describe the world.

Realist: Things are material objects.Idealist: Things are phenomena.

The Deflationary Reading:This becomes the most fundamental aspect ofCarnap’s program for Goldfarb and Ricketts,so Carnap is interpreted as advocating wegive up making any kind of philosophicalassertion, philosophy just a matter oflanguage construction and investigation.

Larry(Deflationary/Relativist

Stand-In)

The World

Science

Philosophy







Stand-In)

The World

Science

Philosophy







Stand-In)

The World

Science

Philosophy







Stand-In)

The World

Science

Philosophy







Stand-In)

The World

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Philosophy


Outlines(s)

The Methodological Roles of Toleranceand Conventionalism in the Philosophy of Mathematics:

Reconsidering Carnap’s Logic of Science

1. A Bittersweet Renaissance for Logical Empiricism2. Is Carnap’s Meta-Philosophy Viciously Circular?3. ‘Empirical Fact’, Tolerance, and Conventionalism: A Reply to

Goldfarb and Ricketts

4. The Boundless Ocean of Unlimited Possibilities: Tolerance inFoundations


Outlines(s)



1. A Bittersweet Renaissance for Logical Empiricism

2. Is Carnap’s Meta-Philosophy Viciously Circular?3. ‘Empirical Fact’, Tolerance, and Conventionalism: A Reply to




Outlines(s)



1. A Bittersweet Renaissance for Logical Empiricism2. Is Carnap’s Meta-Philosophy Viciously Circular?

3. ‘Empirical Fact’, Tolerance, and Conventionalism: A Reply toGoldfarb and Ricketts



Outlines(s)







Outlines(s)







Outlines(s)







Carnap’s Program

Logical Empiricism

Empiricism:Sense experience or observation is our only source of substantiveknowledge, and this is exclusively facts about the world.

Problem:What about logico-mathematical knowledge? Seems substantive,but not empirical—rather, a priori and necessary.

Solution:Logico-mathematical knowledge is not substantive knowledge, it’sConventional: the inferential residue of the tacit (or explicit)syntactical rules of our language.


Carnap’s Program

Logical Empiricism





Carnap’s Program

Logical Empiricism





Carnap’s Program

Linguistic Frameworks (or Formal Languages)

The various proposals for the structure of our language should beexplicitly laid out (including frameworks for the total language ofscience), and they can help to uncover conceptual confusions too.

Components of a Framework:Logico-Mathematicalrules/vocab.Physical rules/vocab.Synthetic Sentences

As a formal system, we can specifyany framework we’d like, but usuallyour frameworks are constructed forsome particular purpose.

Empirical FactsThe World

LinguisticFramework

Syntactic

RulesRational/JustificatoryStructure


Carnap’s Program



Components of a Framework:

Logico-Mathematicalrules/vocab.Physical rules/vocab.Synthetic Sentences



LinguisticFramework

Syntactic



Carnap’s Program



Components of a Framework:Logico-Mathematicalrules/vocab.

Physical rules/vocab.Synthetic Sentences



LinguisticFramework

Syntactic



Carnap’s Program



Components of a Framework:Logico-Mathematicalrules/vocab.Physical rules/vocab.

Synthetic Sentences



LinguisticFramework

Syntactic



Carnap’s Program






LinguisticFramework

Syntactic



Carnap’s Program






LinguisticFramework

Syntactic



Carnap’s Program






LinguisticFramework

Syntactic



Carnap’s Program

The Principle of Tolerance

It is not our business to set upprohibitions, but to arrive atconventions. [. . . ]

In logic, there are no morals.Everyone is at liberty to buildup his own logic, i.e., his ownform of language, as hewishes. All that is required ofhim is that, if he wishes todiscuss it, he must state hismethods clearly, and givesyntactical rules instead ofphilosophical arguments.

The World

Linguistic Frameworks

The choice between frameworks is amatter of practical considerations:fruitfulness, expedience, simplicity. . .


Carnap’s Program

The Principle of Tolerance

It is not our business to set upprohibitions, but to arrive atconventions. [. . . ]

In logic, there are no morals.Everyone is at liberty to buildup his own logic, i.e., his ownform of language, as hewishes. All that is required ofhim is that, if he wishes todiscuss it, he must state hismethods clearly, and givesyntactical rules instead ofphilosophical arguments.

The World

Linguistic Frameworks

The choice between frameworks is amatter of practical considerations:fruitfulness, expedience, simplicity. . .


Kurt Godel

Godel’s Observation

Kurt Godel(1906–1978)

Observation: Carnap, you’re advocating aMathematical Conventionalism, in otherwords, that mathematics is withoutcontent, and really just the formal syntaxof our language.

If this is the case, then mathematics reallyis just the Logico-Mathematical rules thatwe’ve selected as that component of ourframework.

Don’t you need to argue for this position?


Kurt Godel







Kurt Godel







Kurt Godel

Godel’s Circularity Objection

Question: Carnap, how can we be sure that a givensyntactical rule has no empirical content?

A rule needs to be demonstrably consistent in order toensure it does not imply any factual sentences, and sois properly logical.

My Second Incompleteness Theorem entails that aninteresting proposed syntactical rule cannot prove itsown consistency.

You can prove this from a meta-language, but thislanguage will have to presuppose the same“mathematical content” as the language you’re tryingto show consistent. (And that’s circular)

Therefore you cannot justify the thesis thatmathematics is syntax of language.


Kurt Godel








Kurt Godel








Kurt Godel








Kurt Godel








Kurt Godel








Larry’s Response

The Deflationary Response

Godel, in forwarding your objection, you’veforgotten that certain epistemic concepts arerelative to the framework in question.

You are presupposing that the world can beinvestigated prior to or outside the purview ofsome or other linguistic framework.

But having a Linguistic Framework in place isa precondition for inquiry and discourse, soconcepts like ‘Empirical Fact’ can only bedefined within the context of someframework; it makes no sense to consider alanguage-transcendent notion by which tocompare our framework’s syntactical rules.


Larry’s Response






Larry’s Response






Larry’s Response






Larry’s Response


The World

LinguisticFrameworks

Empiricism

Platonism

Conventionalism


Larry’s Response


The World


Empiricism

Platonism

Conventionalism


Larry’s Response


The World


Empiricism

Platonism

Conventionalism


Larry’s Response

Textual Evidence

Ricketts (1994)This notion of empirical fact imposes morals in logic on theconventionalist. Carnap, in adopting the Principle of Tolerance,rejects any such language-transcendent notions. (p. 180)

Goldfarb (1996)However, as the Principle of Tolerance indicates, it is central to themetaphysics of Logical Syntax that any such languagetranscendence be rejected. Rather, the notion of empirical fact isgiven by way of the distinction between what follows from the rulesof a particular language and what does not, so that differentlanguages establish different domains of fact. In this way, Carnapundercuts the very formulation of Godel’s argument. (p. 227)


Larry’s Response

Textual Evidence

Ricketts (1994)This notion of empirical fact imposes morals in logic on theconventionalist. Carnap, in adopting the Principle of Tolerance,rejects any such language-transcendent notions. (p. 180)

Goldfarb (1996)However, as the Principle of Tolerance indicates, it is central to themetaphysics of Logical Syntax that any such languagetranscendence be rejected. Rather, the notion of empirical fact isgiven by way of the distinction between what follows from the rulesof a particular language and what does not, so that differentlanguages establish different domains of fact. In this way, Carnapundercuts the very formulation of Godel’s argument. (p. 227)


Larry’s Response

Larry’s Program

Larry’s primary insight is that our concepts arelanguage-relative (Logocentrism).

This is what prompts him to adopt thePrinciple of Tolerance, which is an attitudethat we can have toward philosophicalpositions: Give them up and engage inlanguage construction instead.

This attitude extends to empiricism andmathematical conventionalism as well.

Tolerance repudiates the idea that argumentsare the correct sort of method in philosophy.


Larry’s Response

Larry’s Program






Larry’s Response

Larry’s Program






Larry’s Response

Larry’s Program






Larry’s Response

Textual Evidence

Goldfarb & Ricketts (1992)To think there is some foundational question that will concernrelative strengths of metalanguage and object language, and thathas the effect of imposing restrictions on the metalanguage, is toaccept some kind of epistemic relation that applies across language.But that is precisely what Carnap’s basic view disallows. (p. 69)

Again: Comparing frameworks requires a third that canencompass both, but if I’m using that framework for comparison,I’ve then already adopted it.

So we seem to lose an important element of objectivity for ourframeworks, and Larry comes quite close to the idea that allpositions deserve equal consideration.


Larry’s Response

Textual Evidence





Larry’s Response

Textual Evidence





One Thing Larry is Missing

Carnap: A Scientific Philosophy

The Task of the Logic of Science (1934)[I]n our own discipline, logic or the logic ofscience, is in the process of cutting itselfloose from philosophy and of becoming aproperly scientific field, where all work is doneaccording to strict scientific methods and notby means of “higher” or “deeper” insights.

Carnap is trying to replace traditionalphilosophical methods with the logicalanalysis of the concepts, methods, andlanguages of science.

This all involves taking the methods andresults of science seriously.














Empirical Facts

Carnap Seems to Disagree With Larry

Intellectual Autobiography (1963)—Reply to CohenTo Cohen’s criticism of conventionalism I should like to say that apure conventionalism [. . . ] was never maintained by any adherentof logical empiricism, nor by Mach or Poincare. [. . . ] Cohenbelieves that my so-called principle of tolerance in the logicalsyntax contains a “doctrine of conventionally-chosen basic-truths”.But this is not the case. The principle referred only to the freechoice of the structure of the language, and not to the content ofsynthetic sentences. I emphasized the non-conventional, objectivecomponent in the knowledge of facts [. . . ] I also pointed out thatthe first operation in the testing of synthetic statements is theconfrontation of the statement with observed facts. Thereby I tooka position clearly opposed to a pure conventionalism and to anycoherence theory of truth.


Empirical Facts

Carnap Seems to Disagree With Larry

Intellectual Autobiography (1963)—Reply to CohenTo Cohen’s criticism of conventionalism I should like to say that apure conventionalism [. . . ] was never maintained by any adherentof logical empiricism, nor by Mach or Poincare. [. . . ] Cohenbelieves that my so-called principle of tolerance in the logicalsyntax contains a “doctrine of conventionally-chosen basic-truths”.But this is not the case. The principle referred only to the freechoice of the structure of the language, and not to the content ofsynthetic sentences. I emphasized the non-conventional, objectivecomponent in the knowledge of facts [. . . ] I also pointed out thatthe first operation in the testing of synthetic statements is theconfrontation of the statement with observed facts. Thereby I tooka position clearly opposed to a pure conventionalism and to anycoherence theory of truth.


Empirical Facts

Protocol Sentences

“On Protocol Sentences” (1932) (same in Logical Syntax)

Protocols: Synthetic sentences which represent our observations.

Realist Idealist

LinguisticFramework

Deduction

Synthetic ProtocolSentence


Observation


Empirical Facts

Protocol Sentences



Realist Idealist

LinguisticFramework

Deduction



Observation


Empirical Facts

Protocol Sentences



Realist Idealist

LinguisticFramework

Deduction



Observation


Empirical Facts

Protocol Sentences



Realist Idealist

LinguisticFramework

Deduction



Observation


Empirical Facts

Protocol Sentences

Logical Syntax, §82Syntactical rules will have to be stated concerning the forms whichthe protocol-sentences, by means of which the results ofobservations are expressed, may take. [On the other hand, it is notthe task of syntax to determine which sentences of the establishedprotocol form are to be actually laid down as protocol-sentences[. . . ] the statement of protocol-sentences is the affair of thephysicist who is observing and making protocols.]

Constraint: The evidential bases of our scientific theories.

Constraint: The actual practice of science.


Empirical Facts

Protocol Sentences





Empirical Facts

Protocol Sentences





What About An Analytic Domain?

Example: Intuitionism and Classical MathematicsCarnap treats this debate in the foundations of mathematicsanalogously to the debate between Realism and Idealism—hebegins by reconstructing each theory:

Intuitionism → LI (More Safety)Conservative set of axioms/inference rulesOnly bounded quantificationOnly primitive-recursive predicates/functor symbols

Classical Mathematics → LII (More Expressive)Higher-Order Type Theory with ChoiceUnlimited quantificationCan recover ZF set-theory.

It’s not a question of which methods of proof are correct, but ofwhich language is most useful.















The Reconstruction of Classical Mathematics

Carnap proves that the Axiom of Choice and thePrinciple of Complete Induction are analytic in LII.

But this doesn’t seem very interesting, because it’s ameta-linguistic proof, and Carnap’s informalmeta-language already includes Choice and Induction.

So clearly the proofs have no justificatorypurpose—Carnap actually says as much.

So what’s their point?

LII ML








LII ML








LII ML








LII ML




Logical Syntax, §34hThe proofs [. . . ] must not be interpreted asthough by means of them it were proved that[Induction and Choice] were materially true.They only show that our definition of ‘analytic’effects on this point what it is intended to effect,namely, the characterization of a sentence asanalytic if, in material interpretation, it isregarded as logically valid.

In other words: The proofs act as evidence thathis formal definition of ‘Analytic’ for LII is anadequate explication of our informal concept ofclassical mathematical truth.




Logical Syntax, §34hThe proofs [. . . ] must not be interpreted asthough by means of them it were proved that[Induction and Choice] were materially true.They only show that our definition of ‘analytic’effects on this point what it is intended to effect,namely, the characterization of a sentence asanalytic if, in material interpretation, it isregarded as logically valid.

In other words: The proofs act as evidence thathis formal definition of ‘Analytic’ for LII is anadequate explication of our informal concept ofclassical mathematical truth.




So what Carnap’s doing here is assuming thatclassical mathematics is valid, so far as it goes,and trying to offer a rigorization of our notion ofmathematical truth.

This is what it means to say that Carnap takesthe results of the sciences seriously: Philosophershave no special authority to tell mathematicianswhich methods or principles are correct.

But we can still contribute via explication, tomake concepts clearer or help to understandtheir essential characteristics.















What Does Larry Think?

Goldfarb & Ricketts (1992)Clearly, if the metalanguage is a rich one, and ifour understanding of it cannot be exhaustivelyexplicated in terms of rules, deductive proceduresin axiomatic systems, or the like, then Carnap’s“presupposition” is an admission that much cannever be made explicit, but must simply be tacitlyrelied upon. This fits poorly with Carnap’sproclaimed standards of exactitude and rigor.

What is missed here is that Actual Carnap ishappy to presuppose an informal meta-language,he’s trying to provide insight into those concepts,formalization is a means to that end.



What Does Larry Think?

Goldfarb & Ricketts (1992)Clearly, if the metalanguage is a rich one, and ifour understanding of it cannot be exhaustivelyexplicated in terms of rules, deductive proceduresin axiomatic systems, or the like, then Carnap’s“presupposition” is an admission that much cannever be made explicit, but must simply be tacitlyrelied upon. This fits poorly with Carnap’sproclaimed standards of exactitude and rigor.

What is missed here is that Actual Carnap ishappy to presuppose an informal meta-language,he’s trying to provide insight into those concepts,formalization is a means to that end.


Explaining Explication

The Explication of the Concepts of Science

1961 Preface to the Aufbau (1928)By rational reconstruction is here meant thesearching out of new definitions for oldconcepts. The old concepts did not ordinarilyoriginate by way of deliberate formulation, butin more or less unreflected and spontaneousdevelopment. The new definitions should besuperior to the old in clarity and exactness, and,above all, should fit into a systematic structureof concepts. Such a clarification of concepts,nowadays frequently called “explication”, stillseems to me one of the most important tasks ofphilosophy, especially if it is concerned with themain categories of human thought.




Carnap most frequently sites:

The Frege-Russell definition of ‘Number’and the numeralsRussell’s analysis of definite descriptionsTarski’s definition of ‘Truth’ for formallanguages

What’s common to all of these examples is thatthe authors attempt to show their explicationsadequate by arguing that they recover essentialcharacteristics of the informal concepts.




Carnap most frequently sites:The Frege-Russell definition of ‘Number’and the numerals

Russell’s analysis of definite descriptionsTarski’s definition of ‘Truth’ for formallanguages





Carnap most frequently sites:The Frege-Russell definition of ‘Number’and the numeralsRussell’s analysis of definite descriptions

Tarski’s definition of ‘Truth’ for formallanguages





Carnap most frequently sites:The Frege-Russell definition of ‘Number’and the numeralsRussell’s analysis of definite descriptionsTarski’s definition of ‘Truth’ for formallanguages





Carnap most frequently sites:The Frege-Russell definition of ‘Number’and the numeralsRussell’s analysis of definite descriptionsTarski’s definition of ‘Truth’ for formallanguages




Obligatory Frege Quotation

Grundlagen der Arithmetik, §70 (1884)Definitions show their worth by proving fruitful.[. . . ] Let us try, therefore, whether we can derivefrom our definition of the Number which belongsto the concept F any of the well-known propertiesof numbers.

Carnap seems to be doing exactly the same thingwith LI and LII.

He proposes each language as a reconstruction ofa mathematical theory, and then argues that it isadequate in that regard.

Constraint: Recover the essential characteristicsof the explicandum.

Gottlob Frege(1848–1925)


























Carnap’s Program’s Not Circular

Responding to Godel’s Circularity Objection

Godel: Worried that we have to prove syntacticalrules are factually non-creative in order to justifythe thesis that mathematics is syntax of language.

Carnap: I’m not trying to justify this thesis, atleast, not in the foundational way that you suppose.

What I’m trying to do is show that myreconstruction of mathematics provides an adequateand fruitful account of the role of mathematics inthe sciences:

1. Tolerance is a typical mathematical method.2. Conventionalism is enough to account for the

use of mathematics in science.























1. Tolerance is a typical mathematical method.

2. Conventionalism is enough to account for theuse of mathematics in science.










A Methodological Argument for Conventionalism

Tolerance in Mathematics

Carnap argues that the Principle of Tolerance is a typicalmathematical attitude:

Logical Syntax, §17The tolerant attitude here suggested is, as far as specialmathematical calculi are concerned, the attitude which is tacitlyshared by the majority of mathematicians.

The idea is that most mathematicians are happy to explore theconsequences of various mathematical theories or the properties ofvarious mathematical structures, using whatever methods of proofso long as they state their assumptions clearly.















The Role of Mathematical Sentences

Carnap Asks: What should a logical foundationof mathematics achieve?

Foundations of Logic and MathematicsThe chief function of a logical calculus in itsapplication to science is not to furnish logicaltheorems, i.e., L-true sentences, but to guide thededuction of factual conclusions from factualpremises. (p. 35)

Carnap argues that a logical interpretation ofmathematical sentences achieves the requisitegenerality, and in the context of a languageincluding also descriptive symbols and syntheticsentences it accounts for this role.














Summing Up

Summing Up

Scientific Philosophy: The Explication of theconcepts, methods, theories, and languages of thesciences, including mathematics.

He is in a sense trying to develop a formal andsystematic context for performing explications.

Program isn’t circular—he provides methodologicalreasons for thinking that Conventionalism is anadequate account of mathematics.

Nor is it relativism, since the purpose is to takeaccount of the actual methods, practices, andconcepts of the sciences—explications can’t bestrictly wrong, but they can be inadequate.


Summing Up

Summing Up






Summing Up

Summing Up






Summing Up

Summing Up






Summing Up

Obligatory Russell Quotation

Congress of Scientific Philosophy (1936)Modern science arose from the marriage ofmathematics and empiricism; three centurieslater the same union is giving birth to asecond child, scientific philosophy, which isperhaps destined to as great a career. For italone can provide the intellectual temper inwhich it is possible to find a cure for thediseases of the modern world.

Bertrand Russell(1872–1970)


Thanks!

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Is Carnap's Program Viciously Circular? Or Perhaps He's a