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  1. Existence by Definition Tom Donaldson

  2. Section 1. Introduction

  3. Existence by Definition: Some Examples • (Some versions of) the ontological argument • Analytic phenomenalism • Thomasson on ‘ordinary objects’. Wherever there are some particles arranged table-wise, there is a table.

  4. Existence by Definition: Some Examples • (Some versions of) the ontological argument • Analytic phenomenalism • Thomasson on ‘ordinary objects’ • Schiffer on properties and propositions If Palo Alto is a town, then Palo Alto has the property of being a town.

  5. Existence by Definition: Some Examples • (Some versions of) the ontological argument • Analytic phenomenalism • Thomasson on ‘ordinary objects’ • Schiffer on properties and propositions • Neofregeanism • Postulationism

  6. Existence by Definition: Some Examples • (Some versions of) the ontological argument • Analytic phenomenalism • Thomasson on ‘ordinary objects’ • Schiffer on properties and propositions • Neofregeanism • Postulationism

  7. Reactionaries David Lewis mocks those who deny the existence of classes: I’m moved to laughter at the thought of how presumptuous it would be to reject mathematics for philosophical reasons. How would you like the job of telling the mathematicians that they must change their ways … now that philosophy has discovered that there are no classes? ...

  8. Reactionaries …If they challenge your credentials, will you boast of philosophy’s other great discoveries: that motion is impossible, ... that it is unthinkable that anything exists outside the mind, that time is unreal, ... ad nauseam? Not me! David Lewis

  9. Reactionaries Reactionaries are (typically) platonists, in the sense that (1) they believe that abstract mathematical objects exist, and (2) they believe that these objects are (setting aside the odd mistake) accurately described by current mathematical theories.

  10. Section 2. Postulationism

  11. Reichenbach’sPostulationism The axiomatic method has most frequently been applied in mathematics. It has become customary to reduce a controversy about the logical status of mathematics to a controversy about the logical status of the axioms. Nowadays one can hardly speak of a controversy any longer. The problem of the logical status of the axioms of mathematics was solved by the discovery that they are definitions, that is, arbitrary stipulations … Reichenbach (1924)

  12. Postulationism: The Basic Idea One can learn mathematical truths by deducing them from stipulative definitions: both explicit definitions, and implicit definitions (which are more usually called ‘axioms’). For example, one might learn number theory by stipulating that the Peano axioms are implicit definitions of ‘number’, ‘plus’, ‘times’, ‘successor’ and ‘zero’, and then deducing theorems from these axioms, introducing additional terms by explicit definition as necessary. This is the ‘definitions and deductions method’.

  13. Some Points of Clarification • Most postulationists think that mathematical knowledge is a priori.

  14. Some Points of Clarification • Most postulationists think that mathematical knowledge is a priori.

  15. Some Points of Clarification • Most postulationists think that mathematical knowledge is a priori. • The postulationists’ view is that the definitions and deductions method can be used to acquire mathematical knowledge ab initio. • Postulationism is an answer to the ‘How possibly?’ question, not the ‘How actually?’ question. • I’ll focus on the definitions of words rather than concepts.

  16. Objections to postulationism • The historical objection

  17. Objections to postulationism • The historical objection[Postponed] • The infinity objection • The coherence objection

  18. Goals for the talk I want to… • …solve the infinity problem, on behalf of the postulationist. • …convince you that the coherence objection is fatal to traditional postulationism. • …suggest a new answer to the ‘How possibly?’ question, which incorporates postulationist insights while escaping the coherence problem.

  19. Section 3. The Infinity Problem

  20. A simple example • Clare sets out to learn some number theory… • …having no previous mathematical knowledge. • Her axioms might be the Peano axioms, or something similar. • She uses a ‘many-sorted’ language. • She works in isolation. • Her language (initially) is free from ambiguity and vagueness/open-texture.

  21. Success conditions A definition. D: Goliath is a horse at least 10cm taller than any other horse. Its success condition. There exists a horse at least 10cm taller than any other horse.

  22. Clare, again The function of a definition is to assign referents to the newly introduced words. A definition will succeed just in case suitable referents exist. Clare’s terms include: 0, 0′, 0′′, 0′′′, 0′′′′, … Plausible conclusion: the success condition of Clare’s definition is at least that there exist infinitely many things.

  23. How could Clare show that there exist infinitely many things? • Empirical arguments? • Appeal to existing mathematical theories? • A priori, but non-mathematical arguments?

  24. The infinity objection (P1) In order to learn a priori that her definitions are true, Clare must establish independently and a priori that their success condition is met. (P2) The success condition of Clare’s definition is at least that there exist infinitely many things. (P3) Clare cannot establish independently and a priori that there exist infinitely many things. (C) Clare cannot learn a priori that her definitions are true.

  25. Section 4. Solving The Infinity Problem

  26. Another way of thinking about success conditions The function of a definition is to assign referents to the newly introduced words. A definition will succeed just in case suitable referents exist.

  27. Another way of thinking about success conditions The function of a definition is to assign referents to the newly introduced words. A definition will succeed just in case suitable referents exist. The function of a definition is to assign truth conditions to sentences in the agent’s newly extended language. A definition will succeed just in case a suitable assignment of truth conditions exists.

  28. Some notation • L is the agent’s ‘old’ language. • L+is the agent’s newly extended language. • The definitions are δ1, …, δn. • J is a function which assigns truth conditions to the sentences in L. • We are looking for conditions on the suitability of a function J+ which assigns truth conditions to the sentences in the agent’s newly extended language.

  29. Conditions on suitability (C1) For each definition δi, J+(δi) is the set of all possible worlds. (C2) For any ‘old’ sentence φ, J(φ)=J+(φ). (C3) J+ should ‘respect logical necessitation’ in the following sense. If Γ is a set of sentences, and possible world wJ+(γ) for each γΓ, and if Γ logically necessitates φ, then wJ+(φ).

  30. Conditions on suitability (C1) For each definition δi, J+(δi) is the set of all possible worlds. (C2) For any ‘old’ sentence φ, J(φ)=J+(φ). (C3) J+ should ‘respect logical necessitation’ in the following sense. If Γ is a set of sentences, and possible world wJ+(γ) for each γΓ, and if Γ logically necessitatesφ, then wJ+(φ).

  31. The application to Clare’s definitions An assignment function that meets conditions (C1)-(C3) will exist provided that the following condition is met: The Modal Conservativeness Condition For any possible world w, {δ1, … , δn}∪{α∊L:w∊J(α)} is coherent.

  32. The application to Clare’s definitions An assignment function that meets conditions (C1)-(C3) will exist provided that the following condition is met: The Modal Conservativeness Condition For any possible world w, {δ1, … , δn}∪{α∊L:w∊J(α)} is coherent.

  33. The application to Clare’s definitions [I]f the arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined by the axioms exist. This is for me the criterion of truth and existence. Hilbert

  34. The infinity objection (P1) In order to learn a priori that her definitions are true, Clare must establish independently and a priori that their success condition is met. (P2) The success condition of Clare’s definition is at least that there exist infinitely many things. (P3) Clare cannot establish independently and a priori that there exist infinitely many things. (C) Clare cannot learn a priori that her definitions are true.

  35. Aside for metaphysicians: Do numbers exist in the perfectly natural sense? “There exists a hole in every bagel.”

  36. Goals for the talk I want to… • …solve the infinity problem, on behalf of the postulationist. • …convince you that the coherence problem is fatal to traditional postulationism. • …suggest a new answer to the ‘How possibly?’ question, which incorporates postulationist insights while escaping the coherence problem.

  37. Section 5. The coherence problem

  38. The infinity objection (P1) In order to learn a priori that her definitions are true, Clare must establish independently and a priori that their success condition is met. (P2) The success condition of Clare’s definition is at least that there exist infinitely many things. (P3) Clare cannot establish independently and a priori that there exist infinitely many things. (C) Clare cannot learn a priori that her definitions are true.

  39. Externalism Goliath is a horse at least 10cm taller than any other horse. (Strong Externalism) If an agent uses a definition , and if the success condition of  is met, then the agent is in a position to know a priori that  is true, in the absence of a defeater.

  40. Externalism (Weak Externalism) If an agent uses a definition , and if the success condition of  is met, and if the definition meets constraint C, then the agent is in a position to know a priori that  is true, in the absence of a defeater. Case (i)Clare uses as her definitions the Peano axioms. Case (ii)Clare uses as her definitions the Peano axioms, together with some difficult number-theoretic result, such as Fermat’s Last Theorem.

  41. What is coherence? First thesis: It is possible for Clare, who has no pre-existing mathematical knowledge, to establish a priori that (say) the Peano axioms are coherent. Second thesis: Whenever Clare’s definitions are coherent, there will exist a ‘suitable’ assignment of truth conditions to the sentences in her newly extended language.

  42. What is coherence? • Coherence as model-theoretic satisfiability First thesis: It is possible for Clare, who has no pre-existing mathematical knowledge, to establish a priori that (say) the Peano axioms are coherent.

  43. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency Second thesis: Whenever Clare’s definitions are coherent, there will exist a ‘suitable’ assignment of truth conditions to the sentences in her newly extended language.

  44. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency α1, …, αn⊢Tβ

  45. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency (i) PA2⊢Tφ(n) (for each numeral n). (ii)PA2⊬T ∀x φ(x).

  46. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency (i) PA2⊢Tφ(n) (for each numeral n). (ii)PA2⊬T ∀x φ(x). PA* = PA2∪{⌜¬∀xφ(x)⌝}

  47. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency J+(⌜¬∀xφ(x)⌝) = the set of possible worlds

  48. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency J+(⌜∀xφ(x)⌝) = ∅ J+(φ(n)) = the set of possible worlds The Generalization Constraint: J+(⌜∀x α(x)⌝) = J+( α(o)) ∩J+( α(1)) ∩J+(α(2)) ∩ …

  49. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency • Coherence as informal deductive consistency First thesis: It is possible for Clare, who has no pre-existing mathematical knowledge, to establish a priori that (say) the Peano axioms are coherent.

  50. What is coherence? • Coherence as model-theoretic satisfiability • Coherence as proof-theoretic consistency • Coherence as informal deductive consistency Second thesis: Whenever Clare’s definitions are coherent, there will exist a ‘suitable’ assignment of truth conditions to the sentences in her newly extended language.