Existence by DefinitionTom Donaldson

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Section 1. Introduction

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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.

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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.

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

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

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ReactionariesDavid 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? ...

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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

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ReactionariesReactionaries 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.

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Section 2. Postulationism

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Reichenbach’s PostulationismThe 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)

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Postulationism: The Basic IdeaOne 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’.

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

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

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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.

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Objections to postulationism• The historical objection

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

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Goals for the talkI 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.

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Section 3. The Infinity Problem

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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.

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Success conditionsA 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.

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Clare, againThe 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.

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

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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.

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Section 4. Solving The Infinity Problem

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Another way of thinking aboutsuccess conditionsThe function of a definition is to assign referents to the newly introduced words. A definition will succeed just in case suitable referents exist.

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Another way of thinking aboutsuccess conditionsThe 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.

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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.

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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 wJ+(γ) for each γΓ, and if Γ logically necessitates φ, then wJ+(φ).

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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 wJ+(γ) for each γΓ, and if Γ logically necessitates φ, then wJ+(φ).

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

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

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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

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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.

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

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Goals for the talkI 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.

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Section 5. The coherence problem

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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.

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ExternalismGoliath 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.

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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.

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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.

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What is coherence?1. Coherence as model-theoretic satisfiabilityFirst thesis: It is possible for Clare, who has no pre-existing mathematical knowledge, to establish a priori that (say) the Peano axioms are coherent.

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistencySecond thesis: Whenever Clare’s definitions are coherent, there will exist a ‘suitable’ assignment of truth conditions to the sentences in her newly extended language.

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency

α1, …, αn ⊢T β

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency(i) PA2 ⊢T φ(n) (for each numeral n).(ii) PA2 ⊬T ∀x φ(x).

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency(i) PA2 ⊢T φ(n) (for each numeral n).(ii) PA2 ⊬T ∀x φ(x).

PA* = PA2∪{⌜¬∀xφ(x)⌝}

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistencyJ+(⌜¬∀xφ(x)⌝) = the set of possible worlds

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistencyJ+(⌜∀xφ(x)⌝) = ∅J+(φ(n)) = the set of possible worldsThe Generalization Constraint:J+(⌜∀x α(x)⌝) = J+( α(o)) ∩ J+( α(1)) ∩ J+(α(2)) ∩ …

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency3. Coherence as informal deductive consistencyFirst thesis: It is possible for Clare, who has no pre-existing mathematical knowledge, to establish a priori that (say) the Peano axioms are coherent.

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency3. Coherence as informal deductive consistencySecond thesis: Whenever Clare’s definitions are coherent, there will exist a ‘suitable’ assignment of truth conditions to the sentences in her newly extended language.

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency3. Coherence as informal deductive consistency4. Coherence as informal satisfiability

∀x Rxx ∀x∀y∀z[(RxyRyz) Rxz]

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What is coherence?1. Coherence as model-theoretic satisfiability2. Coherence as proof-theoretic consistency3. Coherence as informal deductive consistency4. Coherence as informal satisfiability5. Field’s primitivism

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The vicious circleMathematicalKnowledge

Knowledge ofCoherence

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Goals for the talkI 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.

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Section 6. A solution (?)

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The vicious circle: A Solution?Method 1: Suppose that some set of mathematical terms are entirely defined by definitions δ1,…, δn, and that no other non-logical terms occur in δ1,…, δn. Then if you can show that δ1,…, δn are coherent, you may infer that they are all true.Method 2: Suppose that S is a set of sentences, and that T is a mathematical theory you know to be true. Then if you can interpret S in T, you may infer that S is coherent.

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The vicious circle: A Solution?Method 3: Suppose that S is a set of sentences, and T is an empirically well-confirmed scientific theory. Then if you can interpret S in T, you have good reason to believe that S is coherent.Method 4: If a mathematical theory T plays an indispensable role in an empirically well-confirmed scientific theory, one has good reason for believing that T is true. (See: Quine)

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The vicious circle: A Solution?Method 4: Suppose that S is a set of sentences. Then if you can deduce ⊥ from S, you may conclude that S is incoherent.Method 5: Changes to one’s overall system of logico-mathematical beliefs can be justified on grounds of simplicity, elegance etc.. (See: Penelope Maddy)

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Section 7. Postscript:Is there a ‘reliability problem’ for logic?

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Field’s Challenge• Platonists believe that ‘we’ are ‘reliable’ in what we believe about mathematics. • How can this ‘reliability’ be explained?A parallel (?) challenge: How can we explain the reliability of our beliefs about which collections of sentences are coherent?

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Field’s Argument(1) Were the mathematical facts different, the our mathematical beliefs would be exactly the same. (Premise)(2) It could only be a huge coincidence if our mathematical beliefs are true. (From 1)(3) “Our belief in a theory should be undermined if the theory requires that it would be a huge coincidence if what we believed about its subject matter were correct.” (Premise)(4) Our belief in current mathematical theories is undermined. (From 2, 3)

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Field’s Argument(1) Were the mathematical facts different, our mathematical beliefs would be exactly the same. (Premise)(1*) Were the mathematical logical facts different, our mathematical logical beliefs would be exactly the same. (Premise)

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Response to Field(1) Were the mathematical facts different, our mathematical beliefs would be exactly the same. (Premise)(2) It could only be a huge coincidence if our mathematical beliefs are right. (From 1)

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A lottery caseThe people in Dave’s department organize a lottery syndicate. They buy 100 tickets from the national lottery. Dave understands that each ticket has only a tiny chance of winning the jackpot, and he is a pessimist. So he forms the belief that the first ticket will not win, and the belief that the second ticket will not win, and the belief that the third ticket will not win, … and so on. As it happens, Dave is right: none of the tickets will win. There is thus a perfect correlation between his beliefs of the form [ticket n will not win] and the facts.

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Response to Field(1) Were the mathematical facts different, our mathematical beliefs would be exactly the same. (Premise)(2) It could only be a huge coincidence if our mathematical beliefs are right. (From 1)

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Section 8. Closing thoughts

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Conclusion: My take on postulationism• The postulationist is right that mathematical knowledge can be achieved using the definitions and deductions method. • However, this is of limited value, in the absence of some account of how one might establish the coherence of systems of mathematical definitions.• In order to obtain a satisfactory answer to the ‘How possibly?” question, we must add further methods of obtaining logico-mathematical knowledge.

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Some other possible applications• Compound objects• Social objects• Propositions(etc.)

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The End!