commentary on `the philosophy of arithmetic and the ... · (4)natural numbers do not exist. barlew...

Commentary on ‘The Philosophy of Arithmetic and theSemantics of Number Words’ by Thomas Hofweber

Jefferson [email protected]

The Ohio State University

Workshop on the Semantics of Cardinals6-7 January 2014

Barlew (OSU) Commentary on Hofweber Cardinals Workshop 1 / 42

Initial data

Originally from Frege (1980):

(1) Jupiter has four moons.

(2) The number of moons of Jupiter is four.

Standard observations:

(1) is innocent in that it is ontologically committed only to existenceof Jupiter and moons.

(2) is loaded in that it appears to be ontologically committed to theexistence of natural numbers.

Initial data

Originally from Frege (1980):

(1) Jupiter has four moons.

Standard observations:

(1) is innocent in that it is ontologically committed only to existenceof Jupiter and moons.

(2) is loaded in that it appears to be ontologically committed to theexistence of natural numbers.

Standard program of research

Research question:

Do natural numbers exist?

Subquestions:

Do so-called “easy arguments” provide evidence that numbers exist?

Do other natural language data provide evidence?

Standard program of research

Research question:

Do natural numbers exist?

Subquestions:

Do so-called “easy arguments” provide evidence that numbers exist?

Do other natural language data provide evidence?

Previous research

Neo-Fregean easy arguments:

Example (2) constitutes evidence that natural numbers exist.

Hofweber’s ground breaking work (Hofweber 2005, 2007, to appear):

Introduce sophisticated linguistic analyses to the discussion of easyarguments.

(3) Example (2), under a proper linguistic analysis, does not constituteevidence that natural numbers exist.

(4) Natural numbers do not exist.

Previous research

Claim (3): Easy arguments do not work

1 Number expressions are quantificational determiners in their basicuses in natural language, exemplified in (1).

2 In loaded sentences such as (2), number expressions are determinersbut are in a non-canonical syntactic position for pragmatic reasons.They merely appear to be referring expressions.

3 Therefore, loaded sentences are not evidence for the existence ofnumbers.

Claim (4): Natural numbers do not exist

1 Number terms without arguments are bare determiners:quantificational determiners with a contextually supplied, possiblygeneric argument.

2 Quantificational determiners are not referring expressions.

3 In some cases where number expressions appear to be referential (e.g.two and two is four), number terms are syntactically (but notsemantically) type shifted due to type coercion in the language ofthought.

4 By [1]-[3], number expressions are not referring expressions in anycontext.

5 By [4], there are no entities such that those entities are referred to bynumber expressions.

6 Line [5] is evidence that natural numbers do not exist.

This commentary

1 Introduction

2 Can loaded sentences really be unloaded?“The number of moons of Jupiter...”“...is four”

3 Worries about a determiner analysisCognitive type coercionPotential problems for cognitive type coercion

Cognitive type coercion and the syntax semantics interfaceAn argument from Arapesh

Why not linguistic type shifting?

4 Final worry: Arguing from null results

5 Conclusion

Hofweber’s analysis of loaded sentences

The number of moons of Jupiter is not a definitedescription/referring expression.

Four is not a referring expression.

Evidence: (2) is not an identity.

Hofweber on the number of moons of Jupiter

From Hofweber (to appear 46)

(5) The composer of Tannhauser is Wagner.

entails

(6) A composer of Tannhauser is Wagner.

entails

(7)??A number of moons of Jupiter is four.

Note: The judgment is mine, though it corresponds to Hofweber’s (to appear: 46)

description of (7) as “quite awkward and seem[ing] to involve an agreement violation

with respect to singular or plural.”

Hofweber on the number of moons of Jupiter

entails

(6) A composer of Tannhauser is Wagner.

entails

(7)??A number of moons of Jupiter is four.

Note: The judgment is mine, though it corresponds to Hofweber’s (to appear: 46)

description of (7) as “quite awkward and seem[ing] to involve an agreement violation

with respect to singular or plural.”

Potential example-specific pitfalls

Pragmatic and encyclopedic facts about planets and moons make (7)implausible.

The idiomatic determiner a number of, meaning ‘several,’ blocks thereading in which a quantifies over numbers (see also Balcerak Jackson2013).

Solution

Manipulate context to control these factors.

Solution

Loaded sentence with indefinite entailment

(8) Context: Alice, and Bob are applying for a mechanic job, and they have beengiven a test. The test involves one machine that does not work and seven bolts.A and B must use the bolts to fix the machine. There are 3 possible solutions,each of which involves a different number of bolts. A finishes first.A: I’m done! Six bolts works.B: Ah, so the number of bolts that works is six.Test administrator (TA): No, a number of bolts that works is six.

Additional quantification over numbers

(9) Context: As B continues to work, the test administrator begins giving hints.

a. Some number (of bolts) that works is higher than 5.

b. Every number (of bolts) that works is higher than 2.

c. No number (of bolts) that works is lower than 2.

(10) Context: For a task with a larger number of bolts and a larger number of correctsolutions:Most numbers (of bolts) that work are higher than 7.

Overloaded version

(11) Context: The same scenario as above, but in this case, on a task for which thereis only one correct answer. Neither A nor B has finished, and B becomesfrustrated. He says, ‘This can’t be done. No matter how many bolts I use, Ican’t make it work. Two bolts doesn’t work; seven bolts doesn’t work. I give up.’

Test Administrator (TA): You’re right, two and seven don’t work. However,there is a number which is the number of bolts that works, namely five.

Interpreting the data

The number of moons of Jupiter is a definite description in loadedsentences.

The is a determiner with its usual meanings. On a quantificationalanalysis of determiners, it quantifies over numbers.

Interpreting the data

The number of moons of Jupiter is a definite description in loadedsentences.

The is a determiner with its usual meanings. On a quantificationalanalysis of determiners, it quantifies over numbers.

Hofweber on ...is four

In loaded sentences, four is an extracted determiner with the syntacticproperties of an NP (but with determiner semantics).Evidence:

1 If a loaded sentence is an identity sentence, then four is a referringexpression.

2 Identity sentences do not have structural focus properties.

3 Loaded sentences do have structural focus properties.

4 Therefore, loaded sentences are not identity sentences.

5 Therefore, in a loaded sentence, four is not a referring expression.

Hofweber’s argument for focus

The question-answer congruence properties of loaded sentences mirrorthose of innocent sentences with prosodic stress on the number expression.

(12) A: How many moons does Jupiter have?

a. B: Jupiter has FOUR moons.

(2) B: The number of moons of Jupiter is four.

Potential pitfalls

Other factors besides structural focus (e.g. order of constituents,nature of the question under discussion) can affect question answercongruence.

Solution: Compare question-answer congruence properties of a loadedsentence and a syntactically equivalent identity sentence ininformation-structurally equivalent contexts.

Preview of results:

Loaded sentences and identity sentences pattern exactly the same withrespect to question-answer congruence.

Potential pitfalls

Preview of results:

Potential pitfalls

Preview of results:

Which identity sentence?

From Hofweber (to appear: 45):

Adapted to be syntactically equivalent to (2):

(14) The first chair of clarinets for Tannhauser is Harold.

Which identity sentence?

From Hofweber (to appear: 45):

Adapted to be syntactically equivalent to (2):

(14) The first chair of clarinets for Tannhauser is Harold.

Subject question: moons

(15) Which planet has four moons?

(1) ?Jupiter has four moons.

a. JUPITER has four moons.

b.??Jupiter has FOUR moons.

(2)??The number of moons of Jupiter is four.

c. ?The number of moons of JUPITER is four.

d. It is Jupiter that has four moons.

e.??It is Jupiter that has FOUR moons.

f. Four is the number of moons of Jupiter.

g.#FOUR is the number of moons of Jupiter.

Subject question: clarinetists

(16) Which production features Harold?

a. ?Tannhauser features Harold.

b. TANNHAUSER features Harold.

c.??Tannhauser features HAROLD.

d.??The first chair of clarinets for Tannhauser is Harold.

e. ?The first chair of clarinets for TANNHAUSER is Harold.

f. It is Tannhauser that features Harold.

g.??It is Tannhauser that features HAROLD.

h. Harold is the first chair of clarinets for Tannhauser.

i.#HAROLD is the first chair of clarinets for Tannhauser.

How many question: moons

(17) How many moons does Jupiter have?

(1) ?Jupiter has four moons.

a.#JUPITER has four moons.

b. Jupiter has FOUR moons.

c.??The number of moons of JUPITER is four.

d.#It is Jupiter that has four moons.

e.#It is Jupiter that has FOUR moons.

f. ?Four is the number of moons of Jupiter.

g. FOUR is the number of moons of Jupiter.

How many question: clarinetists

(18) Which clarinetist does Tannhauser feature?

a. ?Tannhauser features Harold.

b.#TANNHAUSER involves Harold.

c. Tannhauser features HAROLD.

d. The first chair of clarinets for Tannhauser is Harold.

e.??The first chair of clarinets for TANNHAUSER is Harold.

f.#It is Tannhauser that features Harold.

g.#It is Tannhauser that features HAROLD.

h. ?Harold is the first chair of clarinets for Tannhauser.

i. HAROLD is the first chair of clarinets for Tannhauser.

Corrective statement: moons

(19) A: Jupiter has four moons.B: Really? I had no idea Jupiter has forty moons.

a. A: No, Jupiter has FOUR moons.

(2) ?A’: No, the number of moons of Jupiter is four.

b. A: No, the number of moons of Jupiter is FOUR.

c. ?A: No, four is the number of moons of Jupiter.

d. A: No, FOUR is the number of moons of Jupiter.

e. A: No, it is four moons that Jupiter has.

Corrective statement: clarinetists

(20) Context: Both the speaker and the addressee know that Harold and Gerold areprominent clarinetists.A: Tannhauser features Harold.B: Really? I had no idea Tannhauser features Gerold.

a. A: No, Tannhauser features HAROLD.

b. ?A: No, the first chair of clarinets for Tannhauser is Harold.

c. A: No, the first chair of clarinets for Tannhauser is HAROLD.

d. ?A: No, Harold is the first chair of clarinets for Tannhauser.

e. A: No, HAROLD is the first chair of clarinets for Tannhauser.

f. A: No, it is Harold that Tannhauser features.

“Loaded” question and answer order: moons andclarinetists

(21) What is the number of moons of Jupiter?

a. ?Four is the number of moons of Jupiter.

b. FOUR is the number of moons of Jupiter.

(22) Who is the first chair of clarinets for Tannhauser?

a. ?Harold is the first chair of clarinets for Tannhauser.

b. HAROLD is the first chair of clarinets for Tannhauser.

c. The first chair of clarinets for Tannhauser is Harold.

Interpretation of the data

Whatever focus properties loaded sentences have are shared bysyntactically equivalent identity sentences.

The argument from focus does not show that four is not a referringexpression.

Interpretation of the data

Whatever focus properties loaded sentences have are shared bysyntactically equivalent identity sentences.

The argument from focus does not show that four is not a referringexpression.

Hofweber’s determiner analysis

(23) Four boys ate a pizza.

1 Number expressions are quantificational determiners of type〈〈e, t〉, 〈〈e, t〉, t〉〉. This accounts for examples such as (23).

2 Where there is no overt first argument (boys in (23)) as in (24),number expressions are bare determiners, with a covert, contextuallysupplied argument. Here, four has the meaning ‘four Fs’ for some Fand the type 〈〈e, t〉, t〉.

(24) Two and two are four.

Hofweber’s determiner analysis

(23) Four boys ate a pizza.

1 Number expressions are quantificational determiners of type〈〈e, t〉, 〈〈e, t〉, t〉〉. This accounts for examples such as (23).

2 Where there is no overt first argument (boys in (23)) as in (24),number expressions are bare determiners, with a covert, contextuallysupplied argument. Here, four has the meaning ‘four Fs’ for some Fand the type 〈〈e, t〉, t〉.

(24) Two and two are four.

Naive question for a determiner analysis

But there sure are times where it looks like a number expression isreferring to an abstract object...

Cognitive type coercion

1 In the syntax, number expressions remain bare determiners.

2 In the semantics of natural language, number expressions remaingeneralized quantifiers (〈〈e, t〉, t〉).

3 In mental representation, the syntax of the language of thought,number expressions become individuals of type e to facilitatecalculation.

4 In the syntax of English, the syntactic category of number expressionsmakes an analogous change.

Evidence for cognitive type coercion from acquisition

1 After learning to count, children learn number expressions asquantifiers over physical objects.

2 Initially children learn to calculate using collections of physical objects.

3 Later, when doing more complex calculations, children use cognitivetype coercion to make numbers more manipulable in the language ofthought.

4 Therefore, in natural language, number expressions are determinersand type coercion happens cognitively.

Aside: A different, equally plausible interpretation of theacquisition data

1 Number expressions are referential and refer to abstract objects,namely numbers.

2 Reference to abstract objects is difficult for kids.

3 Therefore, in the LOT, kids type coerce number expressions up todeterminers, and use them to quantify over physical objects.

4 Once they have learned certain facts about numbers in this way, theydispense with the type coercion.

Cognitive type coercion and the syntax semantics interface

Potentially puzzling data from Hofweber (to appear: 175)

(25) a. Two and two are/make four.

b. Two and two is/makes four.

What is the nature of the syntactic difference between (25a) and(25b)?

Not a syntactic category change: NP subject.A feature change? [+plural] to [-plural]?

Not a syntactic category change: NP subject.

A feature change? [+plural] to [-plural]?

Not a syntactic category change: NP subject.A feature change? [+plural] to [-plural]?

Questions continued

Why does a change of syntactic type (〈〈e, t〉, t〉 to 〈e〉) in thelanguage of thought correspond to a change of syntactic feature([+plural] to [-plural]) in English?

Given plural individuals of type 〈e〉 (Link, 1983), why should changingthe type change the number?

Is there independent motivation for syntactic type changes in thelanguage of thought conditioning syntactic feature changes in Englishor other natural languages?

Questions continued

Arapesh (Torricelli, Papua New Guinea)

Arapesh numbers as described by Fortune (1942)

Base 3: applies to one set of individuals, call it A.

5 big yams = “three two” big yams

Base 4: applies to a different set of individuals, call it B.

5 coconuts = “four one” coconuts

Arapesh thought experiment

Questions:

(26) Given a pile of 5 big yams and a pile of 5 coconuts, would anArapesh speaker say...The number of big yams is identical to the number of coconuts.

(27) When doing arithmetic, would an Arapesh speaker say...“Three two” {is/is identical to/equals} “four one.”

Let’s assume the answer is yes.

Questions:

Implications of (27)

Arapesh under a determiner analysis:

(28) “three two” = three two FAs

(29) “four one” = four one GBs

(30) “Three two” {is/is identical to/equals} “four one.”

1 There are no individuals satisfying property F such that “three two”Fs {is/is identical to/equals} “four one” Fs.

2 Use cognitive type coercion to avoid quantification over individualswith property F.

3 Problem: Only the syntactic category shifts, not the semantic type.4 So what equivalence is being predicated?5 Is this the Hesperous is Phosphorous of number expressions?

Note: If it turns out that Arapesh speakers cannot affirm (27), then theArapesh data may provide support for Hofweber’s analysis.

3 Problem: Only the syntactic category shifts, not the semantic type.

4 So what equivalence is being predicated?5 Is this the Hesperous is Phosphorous of number expressions?

3 Problem: Only the syntactic category shifts, not the semantic type.4 So what equivalence is being predicated?

5 Is this the Hesperous is Phosphorous of number expressions?

Linguistic type shifting as an alternative

See Partee (1987), Eric yesterday, and Chris today.

Data from Hofweber (to appear: 186) used as evidence againstlinguistic type shifting:

(31) I want two or three beers.

No reading under which two is interpreted as ‘the number two.’

But this is an example-specific problem. Controlling for world knowledgeand plausibility is the solution.

Linguistic type shifting as an alternative

See Partee (1987), Eric yesterday, and Chris today.

Data from Hofweber (to appear: 186) used as evidence againstlinguistic type shifting:

(31) I want two or three beers.

No reading under which two is interpreted as ‘the number two.’

But this is an example-specific problem. Controlling for world knowledgeand plausibility is the solution.

Acceptable examples with a ‘number two’ reading

(32) I’d like to study the Riemann Hypothesis or seven.

(33) Context: The speaker is a mathematician with a drinking problem talking to hisnew girlfriend.I’d be glad to get a call from you any time. When I’m up late, you can bet mymind is being ovewhelmed by either {two/the lowest prime number} or fortyounces of Malt liquor. Either way, an interruption would be awesome.

Acceptable examples with a ‘number two’ reading

(32) I’d like to study the Riemann Hypothesis or seven.

(33) Context: The speaker is a mathematician with a drinking problem talking to hisnew girlfriend.I’d be glad to get a call from you any time. When I’m up late, you can bet mymind is being ovewhelmed by either {two/the lowest prime number} or fortyounces of Malt liquor. Either way, an interruption would be awesome.

Final worry: Arguing from null results

The final two points of Hofweber’s argumentation:

(34) There are no entities such that those entities are referred to bynatural numbers.

(35) (34) constitutes evidence that natural numbers do not exist.

Analog in experimental work

1 Hypothesis: Experimental result y is evidence for factor x .

2 Data: Result y does not obtain.

3 Acceptable conclusion: Our experiment does not provide evidence forthe influence of factor x .

4 Unacceptable conclusion: Factor x does not obtain.

Null results and Hofweber’s argument

1 Hypothesis: English expressions referring to natural numbers areevidence for the existence of natural numbers.

2 Data (on Hofweber’s analysis): English expressions referring tonatural numbers do not exist.

3 Acceptable conclusion: English does not provide evidence for theexistence of natural numbers.

4 Unacceptable conclusion: Natural numbers not exist.

Conclusion: 3 concerns

1 Loaded sentences cannot be unloaded.

2 Cognitive type coercion has undesirable consequences for analyses ofArapesh and the syntax-semantic interface.

3 The argument against natural numbers requires reasoning from nullresults.

Balcerak Jackson, B. (2013). Defusing easy arguments for numbers.36:447–461.

Fortune, R. F. (1942). Arapesh. J.J. Augustin.

Frege, G. (1980). The Foundations of Arithmetic. Harper and Brothers.

Hofweber, T. (2005). Number determiners, numbers, and arithmetic.

Hofweber, T. (2007). Innocent statements and their metaphysically loadedcounterparts.

Link, G. (1983). The logical analysis of plurals and mass terms: A latticetheoretical approach. In Bauerle, R., Schwarze, C., and von Stechow,A., editors, Meaning, use, and the interpretation of language. deGruyter, Berlin.

Partee, B. H. (1987). Noun phrase interpretation and type-shiftingprinciples. In Studies in discourse representation theory and the theoryof generalized quantifiers, volume 8, pages 115–143.

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