Philosophy of Mathematics Ways Forward from Logicism & Intuitionism Know Your Number System! A Version of the ZF Axioms Extensionality: Two sets that have exactly the same elements are the same set. Regularity: All sets are well-founded. Replacement: This is an axiom schema that allows us to build up sets from other sets by saying things like X is the set of all x such that without falling into Russells Paradox. Youll sometimes see an Axiom of Separation as an alternative. Null set: there exists a set of which nothing is an element Unordered pairs: given sets x and y, there exists a set {x, y} Union: given sets x and y, x union y exists. Power set: given x, its power set exists. Infinity: A specific infinite set exists. Russells Paradox The Principle of Comprehension licenses us to declare: S is the set of all sets that dont contain themselves Is R a member of itself? If it is, then it must satisfy the condition of not being a member of itself and so it is not. If it is not, then it must not satisfy the condition of not being a member of itself, and so it must be a member of itself. Since by classical logic one case or the other must hold either R is a member of itself or it is not it follows that the theory implies a contradiction. S is a contradictory object; any theory that allows it is absurd: ex falso quodlibet. PART 5 LOGICISM IN RUSSELL & WHITEHEADS PRINCIPIA MATHEMATICA Logicism Reduction of mathematics to a founding discipline, logic & set theory Requirement: founding discipline is consistent Arithmetic would be branch of logic Logic - picks out generality no special domain of knowledge - but all domains objects of any kind can be numbered all objects belong to a class Numbers themselves need not be objects Theory of Types Principia Mathematica Russell & Whitehead ( ) Functions and statements can be arranged into hierarchy Sets, or classes, at different level (Type 1) from individuals (Type 0) Sets of sets (Type 2 ) at different levels from sets (Type 1) etc. Avoids the Paradox No propositional function can be defined prior to specifying the function's scope of application Domain of objects, individuals, must first be specified before function is defined Avoids problem of illegitimate totalities Whatever involves all of a collection must not be one of the collection. Numbers become classes of classes rather than objects The Axiom of Infinity This axiom guarantees the existence of a set that can be interpreted as the set of natural numbers. There are various ways to build sets that can be interpreted as natural numbers the details arent important. This axiom plays a similar role to induction in Peano Arithmetic. Without it we have finite set theory, which would appear to be a theory of the potential infinite only. It can say this even number is the successor of an odd number for any even number you choose. It can say in this set of even numbers, every one is the successor of an odd number for any set of numbers you like, however large. But it cant say every even number is the successor of an odd number. The point to note is that we cant get our first infinite set just by juggling with finite sets. We must import or assert it. Failure of Logicism? Stephen Shapiro: "The contrast with Frege is stark. Frege proved that each natural number exists, but his proof is impredicative, violating the type restrictions. Russell had to assume the existence of enough individuals for each natural number to exist. This puts a damper on logicism." Discuss: Logicism failed because it couldnt guarantee its objects existed But... Gdels incompleteness theorems also struck a blow 1.Always at least one sentence that is true but unprovable 2.No formal system can prove its own consistency Gdel: Principia Mathematica a considerable step backwards as compared with Frege (1944) Dummett on PM [Russell & Whiteheads] attempt ran against the difficulty that would have supplied the only valid ground for Frege's insistence that numbers are genuine objects, the impotence of logic (at least as they understood it) to guarantee that there are sufficiently many surrogate objects for the purposes of mathematics, forcing them to make assumptions far from being logically true, and probably not true at all: to secure the infinity of the natural-number sequence, they had to assume their axiom of infinity,....' Ways forward Most philosophy of mathematics is post-Fregean, largely accepting the semantics but concerned over the status of mathematical objects (e.g. fictionalism) Dummett: 'we may conclude that Frege proved all arithmetical propositions that do not require the existence of infinitely many natural numbers are analytic (Frege returned to Kantian ideas) Other, more radical options: 1.Gdel - special arithmetical intuition 2.Empiricism or Naturalism dispense with the special status of arithmetic & mathematical objects Quine maths is subordinate to physics 3.Intuitionism classical logic & mathematics is a problem Intuitionism LEJ Brouwer ( ) Dutch mathematician early 20 th Century Involved in controversies over foundational programme with David Hilbert Mathematics is based on the mental construction of mathematical objects Subjective idealism? Brouwer: Kant is 'old form of intuitionism' Maths is not founded on logic The fundamental faculty for arithmetic is intuition Succession in time & counting But broader faculty of grasping first principles not just spatiotemporal Intuitionism & Constructivism Constructivism Mathematical objects are constructed by (human) activity Entities are equivalent to their process of construction (steps) Intuitionism Series of mathematical positions regarding real numbers & nature of continuum Alternative Logic based on semantic consequences of proof, not truth Not every mathematical statement has determinate truth-value Vs double negation elimination & law of excluded middle Reformulation of classical mathematics sought New Interpretation of real numbers & continuum Infinity as potential infinity to grasp an infinite structure is to grasp the process that generates it NOT A COMPLETED TOTALITY Dummett on intuitionism 1 Classical logic is fine for empirical statements number of stars or comets is not generated by counting "A realistic conception of the external world assures us that, once we are satisfied that the concept is sharp in these respects, we need do no more to guarantee determinate truth-values for quantified statements involving it, statements to the effect that there is a comet satisfying some condition, or that all comets satisfy some other condition. In general, the determination of the truth-values of our sentences is effected jointly by our attaching particular senses to them and by the way things are. We do not need to specify what comets there are, once we have rendered our concept of a comet sharp: reality does that job for us, and reality therefore determines the truth or falsity of our quantified statements. So, at least, realism assures us. Dummett on intuitionism 2 Classical logic is inadequate for mathematical statements, where numbers are generated: "Hardly anyone is realist enough about mathematics to think in the same way about quantified mathematical statements. A fundamental mathematical concept, say real number, which determines the domain of quantification of a mathematical theory, must indeed have a criterion of application and a criterion of identity. Given a mathematical object, specified in some legitimate way, we must know what has to hold good of it for it to be a real number; and, given two such specifications, we must know the condition for them to pick out the same real number. Few suppose, however, that, once these two criteria have been fixed, statements involving quantification over real numbers have thereby all been rendered true or false; to achieve that, it would be generally agreed that further specifications on our part were required, in some fashion circumscribing the totality of real numbers and laying down what real numbers there are to be taken to be." [197] Motivation for intuitionism Disjunction same form of expression but I has a determinate truth- value while II does not Classical II is true or false, we just dont know the case yet Intuitionistic II is not well-defined because we cannot construct both sides of the disjunction "There is an infinite number of prime pairs." (prime pair ) Either true or false? Neither true nor false? "If a Well formed formula is not constructively provable using the rules of inference, then it is not meaningful, for we cannot know what its truth consists in, since we have no justification." Quantification differences Classical says A andmeans A is true Intuitionist says A but means we have a proof for A Classical: Not all xs are F therefore there is at least one x that is not F VALID Intuitionistic: We do not have a proof that all xs are F therefore we have a proof that there is at least one x that is not F INVALID Double negation double negation introduction is allowed A A If we know that A, then it follows that we know that it is not the case that we do not know that A double negation elimination is not allowed A A If we do not know that we do not know A - we cannot conclude A Classical vs Intuitionist Real numbers Infinite decimal expansion of real numbers (or Cauchy sequences of rationals or continued fractions) Cannot assume these numbers exist such that nth digit is fixed Early Brouwer: Only a real number constructed according to a rule or law is legitimate Late Brouwer: Free choice sequences by creative subject Dummett: "As soon as we consider a domain which cannot be effectively enumerated, the quantifiers need to be explained... by reference to a proof that a given object belongs to the domain." Must intuitionism be wrong if it demands revisions to mathematical practice? Continued Fractions Hilbert wrote these words in 1922, responding to the intuitionist programme of Weyl, Brouwer and others. His aim, then, is to preserve the achievements of the C19 against skepticism and paradox. Hilberts Programme 1.Express statements about infinite objects as finite strings of symbols in a precisely-defined formal language. 2.Express rules of inference as precisely-defined operations on symbol strings. 3.Derive theorems from the axiom strings by finitely many applications of rules of inference. 4.Prove that the rules of inference produce no contradictory sentence, by purely finite reasoning about strings of symbols. APPENDIX ON LOGIC Syntax vs Semantics What weve just described is syntax: mere symbol-shuffling. We like to think theres another layer: the meaning of the symbols. This is their semantics. Without this, a formal language is just a kind of abstract game. To matter to us, it must be about something. Still, we can use it to state (purely abstract) axioms and proofs of theorems. Given a set of formally-encoded axioms, the set of all the theorems we can prove from them is called their theory. This is captured more formally by the idea of an interpretation of the pure symbolic syntax. We find a model something outside the theory that its suppposed to be about. Then we map the symbols of the language, one by one, onto features of the model. Contradiction A theory is consistent if it doesnt prove any contradictions. If you can produce a proof of P and of not P, the theory is inconsistent. Assumption: if a theory is inconsistent, it has no model. That is, there are no real contradictions. It follows that if a theory demonstrably has a model, its consistent. Ex falso quodlibet Its a feature of classical logic that if you can prove a contradiction, you can use that contradiction to prove any other statement you like, even if the two have nothing to do with each other. Suppose we, who are working in a formalized language, have a theory that we interpret as being about sea creatures. Suppose this theory can prove Whales are fish and also, by a different line of argument, Whales are not fish. We may now prove Humans are fish as follows: Whales are fish is true, so Either whales are fish or humans are fish is also true. This is because either X or Y is true if X is true, regardless of whether Y is true. But Whales are not fish Well, Either whales are fish or humans are fish and Whales are not fish. So it must be that humans are fish! This is because when either X or Y is true and X is false, Y must be true. Nobody thinks this is a good proof that humans are fish. But its not so easy to point to one step that causes the trouble. Propositional Logic This formal language uses letters to represent propositions, such as Whales are fish. The built-in symbols, besides letters, are (not), | (or) and & (and), along with parentheses ( and ). The rules for forming expressions are: Every letter on its own is an expression If X is an expression, so is (X) If X and Y are expressions, so is (X | Y) If X and Y are expressions, so is (X & Y) Nothing else is an expression We then need a set of rules for combining expressions into proofs. These are usually pretty short but in the interests of space we wont set this out here. Here is the proof we just did on the previous slide: P (P | Q) (P) ((P) & (P | Q)) Q Predicate Logic