Sunday, 29 July 2012

Example of Analytic Metaphysics: Russell's Principles of Mathematics

A comment on the "List of achievements" asked me if I am serious. Well, yes: deadly serious. Here's a list of the contents of one of the founding pieces of research in analytic metaphysics by one of the founders of what is now called "analytic philosophy", The Principle of Mathematics (1903), by Bertrand Russell. I take it from this site. This is analytic metaphysics. (One major topic missing from Russell's topics is modality. Whether Russell's work has been superceded in some aspects is entirely beside the point. This is a methododological issue.)

The Principles of Mathematics (1903)
Bertrand Russell

Table of Contents
Preface
Part I. The Indefinables of Mathematics

Chapter I. Definition of Pure Mathematics
§ 1. Definition of pure mathematics
§ 2. The principles of mathematics are no longer controversial
§ 3. Pure mathematics uses only a few notions, and these are logical constants
§ 4. All pure mathematics follows formally from twenty premisses
§ 5. Asserts formal implications
§ 6. And employs variables
§ 7. Which may have any value without exception
§ 8. Mathematics deals with types of relations
§ 9. Applied mathematics is defined by the occurrence of constants which are not logical.
§ 10. Relation of mathematics to logic.

Chapter II. Symbolic Logic
§ 11. Definition and scope of symbolic logic
§ 12. The indefinables of symbolic logic
§ 13. Symbolic logic consists of three parts
The Propositional Calculus
§ 14. Definition
§ 15. Distinction between implication and formal implication.
§ 16. Implication indefinable
§ 17. Two indefinables and ten primitive propositions in this calculus
§ 18. The ten primitive propositions
§ 19. Disjunction and negation defined
The Calculus of Classes
§ 20. Three new indefinables
§ 21. The relation of an individual to its class
§ 22. Propositional functions
§ 23. The notion of such that
§ 24. Two new primitive propositions
§ 25. Relation to propositional calculus
§ 26. Identity
The Calculus of Relations
§ 27. The logic of relations essential to mathematics
§ 28. New primitive propositions
§ 29. Relative products
§ 30. Relations with assigned domains
Peano's Symbolic Logic
§ 31. Mathematical and philosophical definitions
§ 32. Peano’s indefinables
§ 33. Elementary definitions
§ 34. Peano’s primitive propositions
§ 35. Negation and disjunction
§ 36. Existence and the null-class

Chapter III. Implication and Formal Implication
§ 37. Meaning of implication
§ 38. Asserted and unasserted propositions
§ 39. Inference does not require two premisses
§ 40. Formal implication is to be interpreted extensionally
§ 41. The variable in formal implication has an unrestricted field
§ 42. A formal implication is a single propositional function, not a relation of two
§ 43. Assertions
§ 44. Conditions that a term in an implication may be varied
§ 45. Formal implication involved in rules of inference

Chapter IV. Proper Names, Adjectives and Verbs
§ 46. Proper names, adjectives and verbs distinguished
§ 47. Terms
§ 48. Things and concepts
§ 49. Concepts as such and as terms
§ 50. Conceptual diversity
§ 51. Meaning and the subject-predicate logic
§ 52. Verbs and truth
§ 53. All verbs, except perhaps is, express relations
§ 54. Relations per se and relating relations
§ 55. Relations are not particularized by their terms

Chapter V. Denoting
§ 56. Definition of denoting
§ 57. Connection with subject-predicate propositions
§ 58. Denoting concepts obtained from predicates
§ 59. Extensional account of all, every, any, a and some
§ 60. Intensional account of the same
§ 61. Illustrations
§ 62. The difference between all, every, etc. lies in the objects denoted, not in the way of denoting them.
§ 63. The notion of the and definition
§ 64. The notion of the and identity
§ 65. Summary

Chapter VI. Classes
§ 66. Combination of intensional and extensional standpoints required
§ 67. Meaning of class
§ 68. Intensional and extensional genesis of classes
§ 69. Distinctions overlooked by Peano
§ 70. The class as one and as many
§ 71. The notion of and
§ 72. All men is not analyzable into all and men
§ 73. There are null class-concepts, but there is no null class
§ 74. The class as one, except when it has one term, is distinct from the class as many
§ 75. Every, any, a and some each denote one object, but an ambiguous one
§ 76. The relation of a term to its class
§ 77. The relation of inclusion between classes
§ 78. The contradiction
§ 79. Summary

Chapter VII. Propositional Functions.
§ 80. Indefinability of such that
§ 81. Where a fixed relation to a fixed term is asserted, a propositional function can be analysed into a variable subject and a constant assertion
§ 82. But this analysis is impossible in other cases
§ 83. Variation of the concept in a proposition
§ 84. Relation of propositional functions to classes
§ 85. A propositional function is in general not analysable into a constant and a variable element

Chapter VIII. The Variable.
§ 86. Nature of the variable
§ 87. Relation of the variable to any
§ 88. Formal and restricted variables
§ 89. Formal implication presupposes any
§ 90. Duality of any and some
§ 91. The class-concept propositional function is indefinable
§ 92. Other classes can be defined by means of such that
§ 93. Analysis of the variable

Chapter IX. Relations
§ 94. Characteristics of relations
§ 95. Relations of terms to themselves
§ 96. The domain and the converse domain of a relation
§ 97. Logical sum, logical product and relative product of relations
§ 98. A relation is not a class of couples
§ 99. Relations of a relation to its terms

Chapter X. The Contradiction
§ 100. Consequences of the contradiction
§ 101. Various statements of the contradiction
§ 102. An analogous generalized argument
§ 103. Various statements of the contradiction
§ 104. The contradiction arises from treating as one a class which is only many
§ 105. Other primâ facie possible solutions appear inadequate
§ 106. Summary of Part I

Part II. Number

Chapter XI. Definition of Cardinal Numbers
§ 107. Plan of Part II
§ 108. Mathematical meaning of definition
§ 109. Definitions of numbers by abstraction
§ 110. Objections to this definition
§ 111. Nominal definition of numbers

Chapter XII. Addition and Multiplication
§ 112. Only integers to be considered at present
§ 113. Definition of arithmetical addition
§ 114. Dependence upon the logical addition of classes
§ 115. Definition of multiplication
§ 116. Connection of addition, multiplication, and exponentiation

Chapter XIII. Finite and Infinite
§ 117. Definition of finite and infinite
§ 118. Definition of a0
§ 119. Definition of finite numbers by mathematical induction

Chapter XIV. Theory of Finite Numbers
§ 120. Peano's indefinables and primitive propositions
§ 121. Mutual independence of the latter
§ 122. Peano really defines progressions, not finite numbers
§ 123. Proof of Peano's primitive propositions

Chapter XV. Addition of Terms and Addition of Classes
§ 124. Philosophy and mathematics distinguished
§ 125. Is there a more fundamental sense of number than that defined above?
§ 126. Numbers must be classes
§ 127. Numbers apply to classes as many
§ 128. One is to be asserted, not of terms, but of unit classes
§ 129. Counting not fundamental in arithmetic
§ 130. Numerical conjunction and plurality
§ 131. Addition of terms generates classes primarily, not numbers
§ 132. A term is indefinable, but not the number 1

Chapter XVI. Whole and Part
§ 133. Single terms may be either simple or complex
§ 134. Whole and part cannot be defined by logical priority
§ 135. Three kinds of relation of whole and part distinguished
§ 136. Two kinds of wholes distinguished
§ 137. A whole is distinct from the numerical conjunctions of its parts
§ 138. How far analysis is falsification
§ 139. A class as one is an aggregate

Chapter XVII. Infinite Wholes
§ 140. Infinite aggregates must be admitted
§ 141. Infinite unities, if there are any, are unknown to us
§ 142. Are all infinite wholes aggregates of terms?
§ 143. Grounds in favour of this view

Chapter XVIII. Ratios and Fractions
§ 144. Definition of ratio
§ 145. Ratios are one-one relations
§ 146. Fractions are concerned with relations of whole and part
§ 147. Fractions depend, not upon number, but upon magnitude of divisibility
§ 148. Summary of Part II

Part III. Quantity

Chapter XIX. The Meaning of Magnitude
§ 149. Previous views on the relation of number and quantity
§ 150. Quantity not fundamental in mathematics
§ 151. Meaning of magnitude and quantity
§ 152. Three possible theories of equality to be examined
§ 153. Equality is not identity of number of parts
§ 154. Equality is not an unanalyzable relation of quantities
§ 155. Equality is sameness of magnitude
§ 156. Every particular magnitude is simple
§ 157. The principle of abstraction
§ 158. Summary
Note to Chapter XIX.

Chapter XX. The Range of Quantity
§ 159. Divisibility does not belong to all quantities
§ 160. Distance
§ 161. Differential coefficients
§ 162. A magnitude is never divisible, but may be a magnitude of divisibility
§ 163. Every magnitude is unanalyzable

Chapter XXI. Numbers as Expressing Magnitudes: Measurement
§ 164. Definition of measurement
§ 165. Possible grounds for holding all magnitudes to be measurable
§ 166. Intrinsic measurability
§ 167. Of divisibilities
§ 168. And of distances
§ 169. Measure of distance and measure of stretch
§ 170. Distance-theories and stretch-theories of geometry
§ 171. Extensive and intensive magnitudes

Chapter XXII. Zero
§ 172. Difficulties as to zero
§ 173. Meinong's theory
§ 174. Zero as minimum
§ 175. Zero distance as identity
§ 176. Zero as a null segment
§ 177. Zero and negation
§ 178. Every kind of zero magnitude is in a sense indefinable

Chapter XXIII. Infinity, the Infinitesimal, and Continuity
§ 179. Problems of infinity not specially quantitative
§ 180. Statement of the problem in regard to quantity
§ 181. Three antinomies
§ 182. Of which the antitheses depend upon an axiom of finitude
§ 183. And the use of mathematical induction
§ 184. Which are both to be rejected
§ 185. Provisional sense of continuity
§ 186. Summary of Part III

Part IV. Order

Chapter XXIV. The Genesis of Series
§ 187. Importance of order
§ 188. Between and separation of couples
§ 189. Generation of order by one-one relations
§ 190. By transitive asymmetrical relations
§ 191. By distances
§ 192. By triangular relations
§ 193. By relations between asymmetrical relations
§ 194. And by separation of couples

Chapter XXV. The Meaning of Order
§ 195. What is order?
§ 196. Three theories of between
§ 197. First theory
§ 198. A relation is not between its terms
§ 199. Second theory of between
§ 200. There appear to be ultimate triangular relations
§ 201. Reasons for rejecting the second theory
§ 202. Third theory of between to be rejected
§ 203. Meaning of separation of couples
§ 204. Reduction to transitive asymmetrical relations
§ 205. This reduction is formal
§ 206. But is the reason why separation leads to order
§ 207. The second way of generating series is alone fundamental, and gives the meaning of order

Chapter XXVI. Asymmetrical Relations
§ 208. Classification of relations as regards symmetry and transitiveness
§ 209. Symmetrical transitive relations
§ 210. Reflexiveness and the principle of abstraction
§ 211. Relative position
§ 212. Are relations reducible to predications?
§ 213. Monadistic theory of relations
§ 214. Reasons for rejecting the theory
§ 215. Monistic theory and the reasons for rejecting it
§ 216. Order requires that relations should be ultimate

Chapter XXVII. Difference of Sense and Difference of Sign
§ 217. Kant on difference of sense
§ 218. Meaning of difference of sense
§ 219. Difference of sign
§ 220. In the cases of finite numbers
§ 221. And of magnitudes
§ 222. Right and left
§ 223. Difference of sign arises from difference of sense among transitive asymmetrical relations

Chapter XXVIII. On the Difference Between Open and Closed Series
§ 224. What is the difference between open and closed series?
§ 225. Finite closed series
§ 226. Series generated by triangular relations
§ 227. Four-term relations
§ 228. Closed series are such as have an arbitrary first term

Chapter XXIX. Progressions and Ordinal Numbers
§ 229. Definition of progressions
§ 230. All finite arithmetic applies to every progression
§ 231. Definition of ordinal numbers
§ 232. Definition of “nth”
§ 233. Positive and negative ordinals

Chapter XXX. Dedekind's Theory of Number
§ 234. Dedekind's principal ideas
§ 235. Representation of a system
§ 236. The notion of a chain
§ 237. The chain of an element
§ 238. Generalized form of mathematical induction
§ 239. Definition of a singly infinite system
§ 240. Definition of cardinals
§ 241. Dedekind's proof of mathematical induction
§ 242. Objections to his definition of ordinals
§ 243. And of cardinals

Chapter XXXI. Distance
§ 244. Distance not essential to order
§ 245. Definition of distance
§ 246. Measurement of distances
§ 247. In most series, the existence of distances is doubtful
§ 248. Summary of Part IV

Part V. Infinity and Continuity

Chapter XXXII. The Correlation of Series
§ 249. The infinitesimal and space are no longer required in a statement of principles
§ 250. The supposed contradictions of infinity have been resolved
§ 251. Correlation of series
§ 252. Independent series and series by correlation
§ 253. Likeness of relations
§ 254. Functions
§ 255. Functions of a variable whose values form a series
§ 256. Functions which are defined by formulae
§ 257. Complete series

Chapter XXXIII. Real Numbers
§ 258. Real numbers are not limits of series of rationals
§ 259. Segments of rationals
§ 260. Properties of segments
§ 261. Coherent classes in a series

Chapter XXXIV. Limits and Irrational Numbers
§ 262. Definition of a limit
§ 263. Elementary properties of limits
§ 264. An arithmetical theory of irrationals is indispensable
§ 265. Dedekind's theory of irrationals
§ 266. Defects in Dedekind's axiom of continuity
§ 267. Objections to his theory of irrationals
§ 268. Weierstrass's theory
§ 269. Cantor's theory
§ 270. Real numbers are segments of rationals

Chapter XXXV. Cantor's First Definition of Continuity
§ 271. The arithmetical theory of continuity is due to Cantor
§ 272. Cohesion
§ 273. Perfection
§ 274. Defect in Cantor's definition of perfection
§ 275. The existence of limits must not be assumed without special grounds

Chapter XXXVI. Ordinal Continuity
§ 276. Continuity is a purely ordinal notion
§ 277. Cantor's ordinal definition of continuity
§ 278. Only ordinal notions occur in this definition
§ 279. Infinite classes of integers can be arranged in a continuous series
§ 280. Segments of general compact series
§ 281. Segments defined by fundamental series
§ 282. Two compact series may be combined to form a series which is not compact

Chapter XXXVII. Transfinite Cardinals
§ 283. Transfinite cardinals differ widely from transfinite ordinals
§ 284. Definition of cardinals
§ 285. Properties of cardinals
§ 286. Addition, multiplication, and exponentiation
§ 287. The smallest transfinite cardinal a0
§ 288. Other transfinite cardinals
§ 289. Finite and transfinite cardinals form a single series by relation to greater and less

Chapter XXXVIII. Transfinite Ordinals
§ 290. Ordinals are classes of serial relations
§ 291. Cantor's definition of the second class of ordinals
§ 292. Definition of ω
§ 293. An infinite class can be arranged in many types of series
§ 294. Addition and subtraction of ordinals
§ 295. Multiplication and division
§ 296. Well-ordered series
§ 297. Series which are not well-ordered
§ 298. Ordinal numbers are types of well-ordered series
§ 299. Relation-arithmetic
§ 300. Proofs of existence-theorems
§ 301. There is no maximum ordinal number
§ 302. Successive derivatives of a series

Chapter XXXIX. The Infinitesimal Calculus
§ 303. The infinitesimal has been usually supposed essential to the calculus
§ 304. Definition of a continuous function
§ 305. Definition of the derivative of a function
§ 306. The infinitesimal is not implied in this definition
§ 307. Definition of the definite integral
§ 308. Neither the infinite nor the infinitesimal is involved in this definition

Chapter XL. The Infinitesimal and the Improper Infinite
§ 309. A precise definition of the infinitesimal is seldom given
§ 310. Definition of the infinitesimal and the improper infinite
§ 311. Instances of the infinitesimal
§ 312. No infinitesimal segments in compact series
§ 313. Orders of infinity and infinitesimality
§ 314. Summary

Chapter XLI. Philosophical Arguments Concerning the Infinitesimal
§ 315. Current philosophical opinions illustrated by Cohen
§ 316. Who bases the calculus upon infinitesimals
§ 317. Space and motion are here irrelevant
§ 318. Cohen regards the doctrine of limits as insufficient for the calculus
§ 319. And supposes limits to be essentially quantitative
§ 320. To involve infinitesimal differences
§ 321. And to introduce a new meaning of equality
§ 322. He identifies the inextensive with the intensive
§ 323. Consecutive numbers are supposed to be required for continuous change
§ 324. Cohen's views are to be rejected

Chapter XLII. The Philosophy of the Continuum
§ 325. Philosophical sense of continuity not here in question
§ 326. The continuum is composed of mutually external units
§ 327. Zeno and Weierstrass
§ 328. The argument of dichotomy
§ 329. The objectionable and the innocent kind of endless regress
§ 330. Extensional and intensional definition of a whole
§ 331. Achilles and the tortoise
§ 332. The arrow
§ 333. Change does not involve a state of change
§ 334. The argument of the measure
§ 335. Summary of Cantor's doctrine of continuity
§ 336. The continuum consists of elements

Chapter XLIII. The Philosophy of the Infinite
§ 337. Historical retrospect
§ 338. Positive doctrine of the infinite
§ 339. Proof that there are infinite classes
§ 340. The paradox of Tristram Shandy
§ 341. A whole and a part may be similar
§ 342. Whole and part and formal implication
§ 343. No immediate predecessor of ω or a0
§ 344. Difficulty as regards the number of all terms, objects, or propositions
§ 345. Cantor's first proof that there is no greatest number
§ 346. His second proof
§ 347. Every class has more sub-classes than terms
§ 348. But this is impossible in certain cases
§ 349. Resulting contradictions
§ 350. Summary of Part V

Part VI. Space

Chapter XLIV. Dimensions and Complex Numbers
§ 351. Retrospect
§ 352. Geometry is the science of series of two or more dimensions
§ 353. Non-Euclidean geometry
§ 354. Definition of dimensions
§ 355. Remarks on the definition
§ 356. The definition of dimensions is purely logical
§ 357. Complex numbers and universal algebra
§ 358. Algebraical generalization of number
§ 359. Definition of complex numbers
§ 360. Remarks on the definition

Chapter XLV. Projective Geometry
§ 361. Recent threefold scrutiny of geometrical principles
§ 362. Projective, descriptive, and metrical geometry
§ 363. Projective points and straight lines
§ 364. Definition of the plane
§ 365. Harmonic ranges
§ 366. Involutions
§ 367. Projective generation of order
§ 368. Möbius nets
§ 369. Projective order presupposed in assigning irrational coordinates
§ 370. Anharmonic ratio
§ 371. Assignment of coordinates to any point in space
§ 372. Comparison of projective and Euclidean geometry
§ 373. The principle of duality

Chapter XLVI. Descriptive Geometry
§ 374. Distinction between projective and descriptive geometry
§ 375. Method of Pasch and Peano
§ 376. Method employing serial relations
§ 377. Mutual independence of axioms
§ 378. Logical definition of the class of descriptive spaces
§ 379. Parts of straight lines
§ 380. Definition of the plane
§ 381. Solid geometry
§ 382. Descriptive geometry applies to Euclidean and hyperbolic, but not elliptic space
§ 383. Ideal elements
§ 384. Ideal points
§ 385. Ideal lines
§ 386. Ideal planes
§ 387. The removal of a suitable selection of points renders a projective space descriptive

Chapter XLVII. Metrical Geometry
§ 388. Metrical geometry presupposes projective or descriptive geometry
§ 389. Errors in Euclid
§ 390. Superposition is not a valid method
§ 391. Errors in Euclid (continued)
§ 392. Axioms of distance
§ 393. Stretches
§ 394. Order as resulting from distance alone
§ 395. Geometries which derive the straight line from distance
§ 396. In most spaces, magnitude of divisibility can be used instead of distance
§ 397. Meaning of magnitude of divisibility
§ 398. Difficulty of making distance independent of stretch
§ 399. Theoretical meaning of measurement
§ 400. Definition of angle
§ 401. Axioms concerning angles
§ 402. An angle is a stretch of rays, not a class of points
§ 403. Areas and volumes
§ 404. Right and left

Chapter XLVIII. Relation of Metrical to Projective and Descriptive Geometry
§ 405. Non-quantitative geometry has no metrical presuppositions
§ 406. Historical development of non-quantitative geometry
§ 407. Non-quantitative theory of distance
§ 408. In descriptive geometry
§ 409. And in projective geometry
§ 410. Geometrical theory of imaginary point-pairs
§ 411. New projective theory of distance

Chapter XLIX. Definitions of Various Spaces
§ 412. All kinds of spaces are definable in purely logical terms
§ 413. Definition of projective spaces of three dimensions
§ 414. Definition of Euclidean spaces of three dimensions
§ 415. Definition of Clifford's spaces of two dimensions

Chapter L. The Continuity of Space
§ 416. The continuity of a projective space
§ 417. The continuity of metrical space
§ 418. An axiom of continuity enables us to dispense with the postulate of the circle
§ 419. Is space prior to points?
§ 420. Empirical premisses and induction
§ 421. There is no reason to desire our premisses to be self-evident
§ 422. Space is an aggregate of points, not a unity

Chapter LI. Logical Arguments Against Points
§ 423. Absolute and relative position
§ 424. Lotze's arguments against absolute position
§ 425. Lotze's theory of relations
§ 426. The subject-predicate theory of propositions
§ 427. Lotze's three kinds of Being
§ 428. Argument from the identity of indiscernibles
§ 429. Points are not active
§ 430. Argument from the necessary truths of geometry
§ 431. Points do not imply one another

Chapter LII. Kant's Theory of Space
§ 432. The present work is diametrically opposed to Kant
§ 433. Summary of Kant's theory
§ 434. Mathematical reasoning requires no extra-logical element
§ 435. Kant's mathematical antinomies
§ 436. Summary of Part VI

Part VII. Matter and Motion

Chapter LIII. Matter
§ 437. Dynamics is here considered as a branch of pure mathematics.
§ 438. Matter is not implied by space
§ 439. Matter as substance
§ 440. Relations of matter to space and time
§ 441. Definition of matter in terms of logical constants

Chapter LIV. Motion
§ 442. Definition of change
§ 443. There is no such thing as a state of change
§ 444. Change involves existence
§ 445. Occupation of a place at a time
§ 446. Definition of motion
§ 447. There is no state of motion

Chapter LV. Causality
§ 448. The descriptive theory of dynamics
§ 449. Causation of particulars by particulars
§ 450. Cause and effect are not temporally contiguous
§ 451. Is there any causation of particulars by particulars?
§ 452. Generalized form of causality

Chapter LVI. Definition of a Dynamic World
§ 453. Kinematical motions
§ 454. Kinetic motions

Chapter LVII. Newton's Laws of Motion
§ 455. Force and acceleration are fictions
§ 456. The law of inertia
§ 457. The second law of motion
§ 458. The third law
§ 459. Summary of Newtonian principles
§ 460. Causality in dynamics
§ 461. Accelerations as caused by particulars
§ 462. No part of the laws of motion is an à priori truth

Chapter LVIII. Absolute and Relative Motion
§ 463. Newton and his critics
§ 464. Grounds for absolute motion
§ 465. Neumann's theory
§ 466. Streintz's theory
§ 467. Mr Macaulay's theory
§ 468. Absolute rotation is still a change of relation
§ 469. Mach's reply to Newton

Chapter LIX. Hertz's Dynamics
§ 470. Summary of Hertz's system
§ 471. Hertz's innovations are not fundamental from the point of view of pure mathematics
§ 472. Principles common to Hertz and Newton
§ 473. Principle of the equality of cause and effect
§ 474. Summary of the work

Part Appendices.

Appendix A. The Logical and Arithmetical Doctrines of Frege
§ 475. Principal points in Frege's doctrines
§ 476. Meaning and indication
§ 477. Truth-values and judgment
§ 478. Criticism
§ 479. Are assumptions proper names for the true or the false?
§ 480. Functions
§ 481. Begriff and Gegenstand
§ 482. Recapitulation of theory of propositional functions
§ 483. Can concepts be made logical subjects?
§ 484. Ranges
§ 485. Definition of ∈ and of relation
§ 486. Reasons for an extensional view of classes
§ 487. A class which has only one member is distinct from its only member
§ 488. Possible theories to account for this fact
§ 489. Recapitulation of theories already discussed
§ 490. The subject of a proposition may be plural
§ 491. Classes having only one member
§ 492. Theory of types
§ 493. Implication and symbolic logic
§ 494. Definition of cardinal numbers
§ 495. Frege's theory of series
§ 496. Kerry's criticisms of Frege

Appendix B. The Doctrine of Types
§ 497. Statement of the doctrine
§ 498. Numbers and propositions as types
§ 499. Are propositional concepts individuals?
§ 500. Contradiction arising from the question whether there are more classes of propositions than propositions

Friday, 27 July 2012

2 Post-doc jobs in logic and phil of sci, Ghent


The Centre for Logic and Philosophy of Science at Ghent University (http://logica.ugent.be/centrum/) is looking for 2 post-doctoral researchers for a research project entitled /Contextual and formal-logical approach to scientific problem solving processes /supervised by //Prof. Dr. Erik Weber and Prof. Dr. Joke Meheus.
We offer 2-year post-doctoral positions (October/November 2012 tillSeptember/October 2014). The research areas are the following:
(1) Position 1: Methodological and epistemological analysis of scientific reasoning processes.
(2) Position 2: Logical analysis of scientific reasoning processes.
Researchers with a PhD in philosophy or a related discipline (e.g. science studies or philosophy of a specific scientific discipline) who are interested in these positions can apply by sending an e-mail to Prof. Dr. Erik Weber (Erik.Weber@ugent.be ) before *12 August 2012.*Applications must contain a CV with list of publications and a 1 page research proposal that fits into one of the areas outlined above.
We particularly welcome proposals which relate to one of the topics thatare the currently investigated at the CLSP, such as: abduction, analogical reasoning, causation, belief revision, conceptual change, explanation, induction, scientific discovery, adaptive logics, erotetic logics, paraconsistent logics, ...

Individuation for Structured Sets and Leibniz Abstraction

The notion of a structured set appears throughout mathematics. A simple example might be a linear ordering $(X, \leq)$ of some kind. In this case, $X$ is called the domain, or carrier set. And then $\leq$ is simply a binary relation on $X$.

Note that we normally represent this structured set as $(X, \leq)$. That is, as the ordered pair of the carrier set and the distinguished relation. More generally, a structured set is represented as an ordered tuple,
$(X, R_1, \dots, R_n)$
So far as I can see, nothing hinges on the fact that there are finitely many distinguished $R_i$s. One might have an infinite sequence $(R_{\alpha} \mid \alpha \in I)$ of distinguished relations, but it is simpler to consider the case of a "finite signature", as the lingo puts it. So, in the simplest case of a domain $X$ and a single distinguished relation $R$, we have a structured set usually represented by,
$(X, R)$
One might wonder why we take the ordered pair of $X$ and $R$ in that order. Could we not take
$(R, X)$
as a representation of the structured set with domain $X$ and relation $R$?
The answer seems to be: yes. In fact, all that seems to be required is that given the "components" -- the domain $X$ and the distinguished relation $R$ - the corresponding structured set is uniquely determined.
So, instead of representing the structured set with domain $X$ and distinguished relation $R$ as the pair $(X,R)$, let us write $\sigma_{X,R}$ to mean "the structured set whose domain is $X$ and whose distinguished relation is $R$. Then the sole individuation condition is:
Individuation Principle for Structured Sets
$\sigma_{X,R} = \sigma_{X^{\prime}, R^{\prime}}$ iff $X = X^{\prime}$ and $R = R^{\prime}$.
Taking $\sigma_{X,R}$ to be the ordered pair $(X,R)$ verifies this, and also taking it to be the ordered pair $(R,X)$ would verify it too.
So: structured sets are the same structured set exactly if they have the same domain and their distinguished relations are identical.

But this clearly allows that structured sets can be distinct but isomorphic. In other words, there will be lots of cases where,
$\sigma_{X,R} \neq \sigma_{X^{\prime}, R^{\prime}}$ and $\sigma_{X,R} \cong \sigma_{X^{\prime}, R^{\prime}}$
Example. Consider $(\mathbb{N}, \leq)$ and let $\pi : \mathbb{N} \rightarrow \mathbb{N}$ be the transposition that swaps $0$, and $1$, leaving all other $n \in \mathbb{N}$ alone. Then define:
$n \leq^{\prime} k$ iff $\pi(n) \leq \pi(k)$.

Clearly, the new ordered set $(\mathbb{N}, \leq^{\prime})$ is distinct from $(\mathbb{N}, \leq)$, since $\leq$ is extensionally distinct from $\leq^{\prime}$. However, $(\mathbb{N}, \leq^{\prime}) \cong (\mathbb{N}, \leq)$, since the transposition $\pi$ is an isomorphism.

That is, structured sets violate the following indiscernibility principle for abstract structures:
If $A_{X,R} \cong A_{X^{\prime}, R^{\prime}}$ then $A_{X,R} = A_{X^{\prime}, R^{\prime}}$
So, how do we find entities that play this role?
It looks like they can't be structured sets, for structured sets can be distinct but isomorphic. (On the other hand, one might argue that the abstract structure $A_{X,R}$ might not be required to be the same structured set as one started with! So which structured set is it? What is its domain? One idea is that one has a background universe of "nodes" (or maybe collections $D_{\kappa}$ of nodes, for any given cardinality $\kappa$), from which to build abstract structured sets. This approach might work. I'm not sure.)
To generalize somewhat, suppose that $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are structured sets. We'd like to identify $A_{\mathcal{M}}$ and $A_{\mathcal{M}^{\prime}}$ as the corresponding abstract structures, but we don't know what such a thing is. All we know is that we want to have an individuation condition, a kind of abstraction principle,
Leibniz Abstraction
$A_{\mathcal{M}} = A_{\mathcal{M}^{\prime}}$ iff $\mathcal{M} \cong \mathcal{M}^{\prime}$.
That is, given structured sets, the abstract structures are identical exactly if the structured sets are isomorphic. However it is extremely unclear how to define such entities $A_{\mathcal{M}}$ satisfying the required condition.

Thursday, 26 July 2012

Ramsey sentence theorem (two-sorted)

From time to time I'd like to post on the topic of Ramsey sentence structuralism. This post describes a result concerning the truth conditions of Ramsey sentences. It is one way of trying to make precise the Newman objection to Ramsey sentence structuralism about scientific theories (originally going back to Newman 1928 and Demopoulos & Friedman 1985) and summarizes the main idea, and the main result (Theorem 6) from a 2004 BJPS paper "Empirical adequacy and ramsification".

1. Syntax
Suppose $\mathcal{L}$ is a two-sorted first-order language, with variables partitioned into what one might call primary and secondary variables (following the terminology of Burgess & Rosen 1997).
The primary sublanguage, obtained by deleting secondary variables and any secondary and mixed predicates is called $\mathcal{L}^{\circ}$.
Let $\mathcal{L}_2$ be the result of adding primary, mixed and secondary second-order variables or all arities (and corresponding atomic formulas of the right kind) to $\mathcal{L}$.
Let $(\mathcal{L}_2)^{\circ}$ be the primary restriction of this language (obtained by eliminating secondary variables).
Finally, let $(\mathcal{L}_2)^{c}$ be the sublanguage of $\mathcal{L}_2$ obtained by eliminating all non-logical mixed and secondary predicates.

The language $(\mathcal{L}_2)^{c}$ is a two-sorted rendition of the mature Carnapian "observational language": it allows observational predicates, and first-order observational variables; in addition, it has first-order variables ranging over unobservable objects; and it has primary, mixed and secondary second-order variables, giving what amounts to a general theory of sets of, and relations amongst, the first-order entities (either observable or unobservable). In principle, one could add third-order, fourth-order, etc., variables, giving type hierarchy. It makes no difference to the result below.

2. Semantics
If $\mathcal{M}$ is an two-sorted $\mathcal{L}$-structure, the primary domain is called $\mathsf{dom}^{\circ}(\mathcal{M})$ and the secondary domain is called $\mathsf{dom}^{\dagger}(\mathcal{M})$.
Furthermore, the reduct of $\mathcal{M}$ to the primary part (i.e., just the primary domain and the distinguished relations on the primary domain) is called $\mathcal{M}^{\circ}$.
Let $\mathcal{I}$ be any full $\mathcal{L}_2$-structure. So, $(\mathcal{L}_2, \mathcal{I})$ is an interpreted language, and $((\mathcal{L}_2)^{\circ}, \mathcal{I}^{\circ})$ is the interpreted primary language.
Any full $\mathcal{L}_2$-structure $\mathcal{M}$ can be regarded as an $(\mathcal{L}_2)^{c}$-structure $\mathcal{M}^c$, by just forgetting the secondary and mixed relations, but not the secondary domain. So, $((\mathcal{L}_2)^{c}, \mathcal{I}^{c})$ is the interpreted "Carnapian" language.

3. Ramsey sentence
Suppose $\Theta(M_1, \dots, M_k, P_1, \dots, P_n)$ is a finitely axiomatized theory in $\mathcal{L}_2$ containing precisely the mixed predicates $M_1, \dots, M_k$ and the secondary predicates $P_1, \dots, P_n$. Then the Ramsey sentence of $\Theta$, written $\Re(\Theta)$, is:
$\exists X_1 \dots X_k \exists Y_1 \dots Y_n \Theta(M_1/X_1, \dots, M_k/X_k, P_1/Y_1, ..., P_n/Y_n)$
where the mixed predicates $M_i$ are replaced by second-order variables $X_i$ (of the right arities) and the secondary predicates $P_i$ are replaced by second-order variables $Y_i$ (of the right arities): we say that the mixed and secondary predicates have been "ramsified".

Note that $\Re(\Theta)$ is a sentence of the language $(\mathcal{L}_2)^{c}$, the Carnapian "observational" language, which has first-order variables ranging over observable and unobservable objects, and it has second-order variables ranging over all sets and relations amongst these.

4. Ramsey sentence theorem
Let $\mathcal{I}$ be a full $\mathcal{L}_2$-structure. Thus, $((\mathcal{L}_2)^{c}, \mathcal{I}^{c})$ is the corresponding interpreted "Carnapian" language. The Ramsey sentence $\Re(\Theta)$ is a sentence in this language.
$\Re(\Theta)$ is true in $((\mathcal{L}_2)^{c}, \mathcal{I}^{c})$
iff
there is a full $\mathcal{L}_2$-structure $\mathcal{M}$ such that
i. $\mathcal{M} \models \Theta$;
ii. $|\mathsf{dom}^{\dagger}(\mathcal{M})| = |\mathsf{dom}^{\dagger}(\mathcal{I})|$;
iii. $\mathcal{M}^{\circ} \cong \mathcal{I}^{\circ}$.

Metaphysics as Über-theory and Metaphysics as Meta-theory

1. Understanding how things hang together ...

There is a view of metaphysics -- a very standard view of metaphysics from the pre-Socratics to the post-Quinians -- which takes metaphysics to be trying to say very broad things about reality, at the most abstract and general level. Quine says things along these lines, but not having my books with me means I can't find a nice quote to that effect. But there is a similar formulation due to Sellars, which I take from the SEP article on Wilfrid Sellars (by Willem deVries),
"The aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term” (Sellars, 1962, "Philosophy and the Scientific Image of Man").
As good as this sounds, I think there's an interesting distinction to be drawn here. Consider the phrase "things in the broadest sense of the term". Does this mean,
"How does everything there is hang together"?
Or does it mean,
"How do our theories and representations of everything there is hang together"?
Sellars is not focussing here on metaphysics, but rather on philosophy as a whole. Still, let's say that the aim to understand how every thing there is hangs together is über-theory; while the aim to understand how our theories, representations and concepts of everything there is hang together I shall call meta-theory.

(I'm not sure if the slightly jokey name "über-theory" has been used much before. I take it from a 2003 Scientific American article, "A Unified Physics by 2050?", by the theoretical physicist, Steven Weinberg.)

Über-theorizing is doctrinal in the sense that it usually concerns competing doctrines as to what there is (in the broadest sense of the term). For example, "are there abstract objects?", "are there possible worlds?", etc. Meta-theorizing is conceptual in the sense that it usually concerns trying to analyse or explicate broad notions - such as existence, identity, indiscernibility, causation, possibility, truth, reference, and so on. (It also counts if one does not explicitly analyse/explicate, but merely relates the concepts.)

This is not to suggest that über-theorizing and meta-theorizing need be disjoint! They may overlap considerably. The metaphysician may be both über-theorist and meta-theorist in what they aim to do and what their work achieves. So, there may be significant overlap. For example:
(i) What is a relation? (über-theory)
(i)* What does "relation" mean? (meta-theory)
(ii) What is truth? (über-theory)
(ii)* What does "true" mean? (meta-theory)
In such cases, it may be quite hard to see a sharp difference between the two. The reason is that there is an intimate link between (knowing) what "F" means and (knowing) what Fs are. If one's semantic theory is suitably disquotational, then x is true iff "true" is true of x, and x is a relation iff "relation" is true of x, and so on.

2. Metaphysics as über-theory

Considering again Sellars's phrase, "to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term", the qualifier "in the broadest sense of the term" is what make such understanding philosophical, or metaphysical, rather than merely scientific.

Again, it may be both. If Quine, say, suggests that a scientific theory has less and more philosophical aspects, this doesn't imply that the more philosophical aspects are non-scientific. It might imply that they are less directly empirically checkable. To infer the conclusion that the philosophical or metaphysical parts are non-scientific, one would need further empiricist assumptions.

So, the über-theorist might, and in fact normally does, judge the more mundane parts of physics (e.g., solid state or astrophysics) as lacking much metaphysical import, while focusing on the more fundamental and foundational parts (e.g., getting conservation laws from symmetries of a lagrangian; general relativity; quantum theory; theoretical parts of statistical mechanics) as providing a guide to über-theoretic issues about space, time, identity, objects, causation, etc. In such cases, there is overlap between scientific inquiry and metaphysics as über-theory. Indeed, the projects of interpreting GR and QM (and this includes such programmes as ontic structuralism) are all examples of über-theory. Questions about the nature of time, for example, are über-theoretic, but are linked with foundational parts of general relativity. This is why I would classify work on Gödel spacetimes and hypercomputation/supertasks as belonging to both theoretical physics and metaphysics. For similar reasons, I would classify theories of continuous quantities, infinite sets/structures, and computable functions as belonging to both mathematics and metaphysics. (Examples of these are listed in the list of achievements.)

But there is, in addition, metaphysical über-theory which has no obvious connection with science. Examples include possible worlds, theology, moral facts, and perhaps the metaphysics of abstract objects. Clearly, there is sometimes an antagonism to this, sometimes for allegedly naturalistic reasons. I disagree quite strongly with the suggestion that such work "should be discontinued". It is anti-intellectualism. In fact, I think the naturalistic reasons given are not even good naturalistic reasons, but that would be a topic for another post. (But as a quick example, the best argument we have against nominalism is the indispensability of mathematics in science.)

3. Metaphysics as meta-theory

One might be suspicious of metaphysics as über-theory - perhaps because one thinks that "understanding how things hang together" is the job of science. Full stop. How could one come to know "how things hang together" in an a priori way, without proceeding through empirically-condtioned scientific inquiry? Or perhaps because one has read The Phenomenology of Spirit and drawn the conclusion that Hegel is bonkers. Or perhaps because one accepts some kind of verification principle (identifying meaningfulness with empirical content).

For whatever reason is dominant here, whatever good metaphysics might be in that case, it couldn't be just more scientific theory. For it would just be more speculative scientific theory. For example, Hegel (and German Idealism more generally) is metaphysics, but mathematically, logically and scientifically ignorant metaphysics. Then, what would good metaphysics be?

Thanks to Frege and Russell, followed by Wittgenstein and then Carnap, we do have another way of thinking about all this. Metaphysics, done properly, is meta-theory. I.e., meta-theoretical logical analysis. One finds views like this expressed sometimes by Russell and Carnap. For example, Russell's "Philosophy of Logical Analysis" (from his History of Western Philosophy (1945)) and Carnap's Logische Aufbau.

This sort of view is linked to deflationism. It isn't that metaphysics should be eliminated; rather it is that metaphysical problems should be reconceptualized in a deflationary manner which doesn't have the pretensions of being a kind of über-science: discovering how things are merely from the armchair.

How to characterize deflationary metaphysics is something I thought about for a long time during my PhD between 1994 and 1998, and I focused on truth (though there are analogous approaches relating to abstract objects and possible worlds). Roughly, truth can be deflated by focusing on the principles governing the truth predicate (e.g., the T-scheme; or perhaps compositional principles), and showing that they are conservative with respect to non-truth-theoretic matters. Stewart Shapiro suggested this in a 1998 J. Philosophy article "Proof and Truth - Through Thick and Thin"; I suggested it in a 1999 Mind article "Deflationism and Tarski's Paradise". Shapiro and I both argued in this context (it is well-known, and not a major discovery) that a compositional truth theory (like Tarski's, or modern variants like Kripke-Feferman or Friedman-Sheard) is usually non-conservative, allowing one to prove things like reflection principles and consistency sentence (which the underlying theory cannot prove). Such results are consequences of Gödel's incompleteness theorems.

Arguing that deflationary truth should be conservative is analogous to deflating talk of infinite sets by arguing that infinitary set theory is conservative with respect to finitary matters (i.e., computations on finite objects). Such a result would sanction the inferential use of infinite sets, but while showing this to be merely instrumental -- i.e., merely increasing the usefulness of the theory. The instrumentalist programme in this case was Hilbert's Programme. But this came unstuck when it was recognized, by Kurt Gödel, that finitary arithmetic can encode, but not prove its own consistency, while infinitary set theory can prove the consistency of finitary arithmetic; and so, non-conservatively extends finitary arithmetic.

Tuesday, 24 July 2012

What is Analytic Philosophy? What is Analytic Metaphysics?

This definition of "analytic philosophy", by Brian Leiter, seems to me to be ok:
"Analytic" philosophy today names a style of doing philosophy, not a philosophical program or a set of substantive views. Analytic philosophers, crudely speaking, aim for argumentative clarity and precision; draw freely on the tools of logic; and often identify, professionally and intellectually, more closely with the sciences and mathematics, than with the humanities. (It is fair to say that "clarity" is, regrettably, becoming less and less a distinguishing feature of "analytic" philosophy.) The foundational figures of this tradition are philosophers like Gottlob Frege, Bertrand Russell, the young Ludwig Wittgenstein and G.E. Moore; other canonical figures include Carnap, Quine, Davidson, Kripke, Rawls, Dummett, and Strawson.
The definition (preceding the examples) is methodological, rather than historical, and identifies three characteristic features:
1. Analytic philosophy draws on the tools of logic.
2. Analytic philosophy identifies with the sciences and mathematics.
3. Analytic philosophy aims for argumentative clarity and precision.
Admittedly these criteria will be rather vague in application to many individual cases. Still, it seems adequate to a good approximation. Also, no specific big metaphysical or epistemological doctrine follows (realism vs anti-realism; rationalism vs empiricism).
Now define "analytic metaphysics" by replacing "philosophy" by "metaphysics":
1. Analytic metaphysics draws on the tools of logic.
2. Analytic metaphysics identifies with the sciences and mathematics.
3. Analytic metaphysics aims for argumentative clarity and precision.
(It does not define what a "metaphysical claim" is, though.)

Monday, 23 July 2012

The Principle of Naturalistic Closure and Naughty Metaphysics

A couple of people asked me what I meant by "polemics", in the previous post. I was referring to a recent book, Ladyman & Ross et al. 2007, Every Thing Must Go: Metaphysics Naturalized, whose Preface begins:
This is a polemical book. One of its main contentions is that contemporary analytic metaphysics, a professional activity engaged in by some extremely intelligent and morally serious people, fails to qualify as part of the enlightened pursuit of objective truth, and should be discontinued.
As a kind of criterion for identifying the allegedly naughty metaphysics which "should be discontinued", Ladyman & Ross et al. introduce to The Principle of Naturalistic Closure, denoted PNC:
The Principle of Naturalistic Closure
Any new metaphysical claim that is to be taken seriously at time t should be motivated by, and only by, the service it would perform, if true, in showing how two or more specific scientific hypotheses, at least one of which is drawn from fundamental physics, jointly explain more than the sum of what is explained by the two hypotheses taken separately, where this is interpreted by reference to the following terminological stipulations:

Stipulation: A ‘scientific hypothesis’ is understood as an hypothesis that is taken seriously by institutionally bona fide science at t.

Stipulation: A ‘specific scientific hypothesis’ is one that has been directly investigated and confirmed by institutionally bona fide scientific activity prior to t or is one that might be investigated at or after t, in the absence of constraints resulting from engineering, physiological, or economic restrictions or their combination, as the primary object of attempted verification, falsification, or quantitative refinement, where this activity is part of an objective research project fundable by a bona fide scientific research funding body.

Stipulation: An ‘objective research project’ has the primary purpose of establishing objective facts about nature that would, if accepted on the basis of the project, be expected to continue to be accepted by inquirers aiming to maximize their stock of true beliefs, notwithstanding shifts in the inquirers’ practical, commercial, or ideological preferences. (Ladyman & Ross (et al) 2007, Every Thing Must Go, pp. 37-8.)
It seems to me that there are problems with PNC, at least as stated, and the crucial phrase is
"metaphysical claim"
A definition of this is repudiated. But, for example, are claims like,
i) spacetime contains closed timelike curves
ii) spacetime permits hypercomputation
to be counted as metaphysical? The question is not whether they are scientific or not (they are scientific, in some reasonable sense). The question is whether they are metaphysical or not.

One might think, with some logical empiricists, that no scientific claim is a metaphysical claim. But then that just rules out naturalized metaphysics.

If, on the other hand (as seems reasonable to me), there is non-trivial overlap between metaphysical claims and scientific claims, then it would be useful to obtain some guidance as to how to identify the claims in the overlap, even if the guidance is a bit vague.

There presumably are also metaphysical claims which are somehow non-scientific (perhaps the existence of possible worlds, or transfinite cardinals, the existence of angels dancing on pins, etc.), but this raises demarcation questions about what a scientific claim is. If one adopts a sufficiently holistic epistemology, then - at least in principle, through pragmatic, coherence and simplicity considerations, but still conditioned by sensory experience - such questions become amenable to rational acceptance/rejection. (Quine's view.)

Tuesday, 17 July 2012

A List of Achievements of Analytic Metaphysics

Somewhat tongue in cheek. Motivated by some of the recent polemics against metaphysics.

A List of Achievements of Analytic Metaphysics

1. Leibniz’s Principle of Identity of Indiscernibles.
2. Theory of continuous quantities, from Leibniz to Robinson.
3. Frege’s analysis of cardinality.
4. Abstraction and abstraction principles (Frege, Dedekind).
5. Invention of quantification theory; predication; what variables are (Frege).
6. Existence not a predicate, rather a quantifer (Frege).
7. Concepts as functions (Frege).
8. Theory of infinity (Bolzano, Cantor).
9. Mereology (Lesniewski, et al.).
10. Theory of relations (Russell).
11. Non-classical logics.
12. Incompleteness of formal systems (Gödel).
13. Concept of a computable function (Gödel, Turing, Church, et al).
14. Rotating solutions of Einstein's equations with CTCs (Gödel).
15. Tarski’s theory of truth; object language/metalanguage; undefinability theorem.
16. Kripke models; possible worlds anaylsis (Kripke, Lewis, et al.).
17. Kripke’s fixed-point theory of truth; grounding.
18. Formal semantics & pragmatics.
20. Supervenience (Kim, et al.)
21. Representation theorems; applicability of analysis.
22. Field’s theory of applicability of mathematics; conservation theorems.
23. Properties of identity and indiscernibility.
24. Dependence?

Of course, many of the contributions to metaphysics listed here were made by individuals working in the intersection of mathematics, logic and philosophy.

Abstracta - the way of modal invariance

Debates about how to analyse the distinction between abstract and concrete entities have normally focused on four "ways", introduced by David Lewis (1986, The Plurality of Worlds). In his very nice SEP article on "Abstract Objects", Gideon Rosen labels and describes these as follows:
(i) The Way of Negation ("abstract objects are defined as those which lack certain features possessed by paradigmatic concrete objects")
(ii) The Way of Example ("it suffices to list paradigm cases of abstract and concrete entities in the hope that the sense of the distinction will somehow emerge")
(iii) The Way of Conflation ("the abstract/concrete distinction is to be identified with one or another metaphysical distinction already familiar under another name: as it might be, the distinction between sets and individuals, or the distinction between universals and particulars")
(iv) The Way of Abstraction ("The simplest version of this strategy would be to say that an object is abstract if it is (or might be) the referent of an abstract idea, i.e., an idea formed by abstraction"; then goes on to discuss Fregean abstraction principles)
I'd suggest another approach: that concreteness/abstractness is connected to modal variation. In particular, abstracta are modally invariant.

Canonical examples of abstracta are entities like numbers (e.g., $3$, or $\sqrt 2$, or $\pi$ or $i$ or $\aleph_{57}$) or pure sets (e.g., $\varnothing$ or $\omega$ or $V_{\omega + \omega}$). Or the assorted entities of modern algebra (e.g., groups) and geometry (e.g., manifolds). Additionally, we have the abstracta of traditional philosophy - forms, universals, properties, relations, essences.

Notice that these entities do not "change", either temporally or modally.

On the other hand, concrete - ordinary physical objects for example - do "change", both temporally and modally. Concrete entities undergo temporal and modal change.

So, it seems to me that the abstract/concrete distinction can perhaps be explained modally along these lines.

Monday, 16 July 2012

Eliminating Relata, II

The underlying problem for the ontic structuralism programme is to make mathematical sense of the notion of an "abstract structure", a structure lacking some special domain or carrier set upon which the distinguished relations live. This domain would contain the relata which one wants to eliminate. But the usual notion of a structure, or model, is $\mathcal{A} = (A, R_1, ...)$, where $A$ is the domain or carrier set. The distinguished relations $R_i$ are then subsets of Cartesian products of $A$. In other words, such a model is a "structured set". E.g., the ordering $(\mathbb{N}, <)$ or the field $(\mathbb{R}, 0, 1, +, \times)$. (We think of $\mathbb{N}$ as a set; though it doesn't matter which one it is: usually, it's the finite ordinals $\omega$.) Consequently, if you eliminate the carrier set, then everything else goes with it.

The approach described in the previous post starts with a model (structured set) $\mathcal{A}$, and then identifies the abstract structure with a kind of ramsified proposition that categorically axiomatizes $\mathcal{A}$.

There is another approach. I'd read about this several years ago, but I'd forgotten about it. It's category-theoretic and it was described to me by one of our MCMP graduate students, Hans-Christoph Kotzsch, a couple of weeks ago. On this view, abstract structures are objects in categories. The objects in a category $C$ needn't be regarded as built-up from a carrier set. And one can talk about something akin to "elements" of an object $X$ in a category $C$ by identifying such elements with morphisms $1 \rightarrow X$, where $1$ is a terminal object of $C$.

The idea is developed in a 2011 Synthese article, "Category-Theoretic Structure and Radical Ontic Structural Realism" by Jonathan Bain (and in these slides too).

Eliminating Relata

There's a view in philosophy of science called "ontic structuralism" and it is sometimes expressed by saying it wishes to eliminate relata, or to take "structure" to be primary, or something along those lines.

But the main problem, as I've always seen it, is that a structure (or model) $\mathcal{A}$ in the usual mathematical sense is a mathematical object of the form $(A, R_1, ...)$ with a domain $A$. The domain is some set of objects, and the $R_i$ are relations on $A$, and are called the "distinguished relations" in $\mathcal{A}$. But this isn't structuralism in the required sense because one has a domain. (One kind of structuralism does retain a domain of "nodes": this is Shapiro's ante rem structuralism - or at least I think Shapiro's ante rem structuralism retains a domain.)

So one wants somehow to achieve two goals:
(i) identify something as the "abstract structure" of a given model, or system, etc.;
(ii) while also getting rid of the domain.
The only way I know of that might, in some sense, "eliminate" the domain is to consider a certain (usually very long, possibly infinitely long) sentence which defines $\mathcal{A}$ up to isomorphism. First, consider some first-order language $\mathcal{L}_{\mathcal{A}}$, with identity, whose signature fits that of $\mathcal{A}$ and which also has a unique constant $\underline{c}$ for each element $c \in A$. Suppose that the predicate symbol for $R_i$ is $P_i$, and suppose $P_0$ is $=$. So, $(P_i)^{\mathcal{A}} = R_i$ and $\underline{c}^{\mathcal{A}} = c$. Next take the diagram $D(\mathcal{A})$ of $\mathcal{A}$ in the language $\mathcal{L}_{\mathcal{A}}$. That is the set of all literals true in $\mathcal{A}$. Let $\delta_{\mathcal{A}}$ be (possibly infinitary) conjunction of all elements of $D(\mathcal{A})$ and let $\theta_{\mathcal{A}}$ be the (possibly infinitary) formula:
$\forall x \bigvee_{c \in A} (x = \underline{c})$
Let $\Sigma_{\mathcal{A}}$ be the formula:
$\delta_{\mathcal{A}} \wedge \theta_{\mathcal{A}}$
Then
$\mathcal{B} \models \Sigma_{\mathcal{A}}$ iff $\mathcal{B} \cong \mathcal{A}$.
So, $\Sigma_{\mathcal{A}}$ categorically axiomatizes $\mathcal{A}$.

Example: let $A = \{0,1\}$, let $R = \{(0,0),(0,1)\}$ and let $\mathcal{A} = (A,R)$. Then $\Sigma_{\mathcal{A}}$ is the formula
$\underline{0} \neq \underline{1} \wedge P\underline{0}\underline{0} \wedge P\underline{0}\underline{1} \wedge \neg P\underline{1}\underline{0} \wedge \neg P\underline{1}\underline{1} \wedge \forall x(x = \underline{0} \vee x = \underline{1}).$
As one can intuitively see, this "tells you everything you need to know" about the structure $\mathcal{A}$. All identity facts are in there, and all the atomic truths about $R$ are in there too (with the negated ones).

Next, we want to find something to be the abstract structure of $\mathcal{A}$, as well as "eliminating the relata". To do this, one can just quantify all these individual constants away, by a kind of ramsification. Suppose $(\underline{c}_{\alpha})_{\alpha \in I}$ is a non-repeating enumeration of the constants, and $(y_{\alpha})_{\alpha \in I}$ is an enumeration of distinct new variables with the same index set $I$. Then the new ramsified formula, $\Re(\mathcal{A})$, is:
$\exists y_0 \dots \exists y_{\alpha} \dots \Sigma_{\mathcal{A}}(\underline{c}_0/y_0, \dots \underline{c}_{\alpha}/y_{\alpha}, \dots)$.
(The initial string of quantifiers may be infinitary.) This procedure makes no difference to the result above, and we can now forget the constants.
Example again: $\Re(\Sigma_{\mathcal{A}})$ is the formula:
$\exists y_0 \exists y_1(y_0 \neq y_1 \wedge Py_0 y_0 \wedge P y_0 y_1 \wedge \neg Py_1 y_0 \wedge \neg Py_1 y_1 \wedge \forall x(x = y_0 \vee x = y_1)).$
What one obtains here has the right categoricity property: it determines $\mathcal{A}$ up to isomorphism. Notice that in order for this work, one must keep identity as a primitive (a point made, in this context, in a 2006 paper). Even so, $\Re(\Sigma_{\mathcal{A}})$ is a syntactic entity---a string of symbols---and so will not be invariant under even trivial changes (e.g., relabellings of variables, or switching logical conjuncts). So, one has not found a unique entity to be the abstract structure.

However, one can "quotient this out" by considering the Fregean proposition (the abstract content) expressed by this syntactic string. One can then say that the abstract structure of $\mathcal{A}$ is this proposition: the Fregean proposition expressed by $\Re(\Sigma_{\mathcal{A}})$. This in some important sense has no special individuals associated with it, for the constants used to denote individuals have been quantified away. However, it does in some sense retain all the primitive concepts/relations that one started with.

Example again: let $A$ and $R$ be as before and let $\mathcal{A} = (A,R)$. Then the abstract structure for $\mathcal{A}$ is the proposition that there are exactly two things such that one of them bears $R$ to itself and to the other, but the other does not bear $R$ to itself or the other.

(If one identifies possible worlds with such things, one gets rid of purely haecceistic differences; it's one way of responding to the hole argument and the "Leibniz equivalence" of isomorphic spacetimes.)

Wednesday, 4 July 2012

Announcing the MCMP Round Table on Coherence (20 July)

The Munich Center for Mathematical Philosophy has laid the cloth once again: You are invited to join us for the Round Table #2 on Coherence with Richard Pettigrew (Bristol) and Branden Fitelson (Rutgers). This public MCMP event will take place on July 20th, 2012, and center about recent arguments for probabilistic norms that have attempted to justify coherence requirements for (sets of) degrees of confidence solely by appeal to considerations involving their accuracy. Pettigrew and Fitelson (and their collaborators) have worked extensively on various arguments of this sort. This public MCMP event is to bring their differing approaches and perspectives to one table.
Download more info about venue and programme here (PDF).