Philosophy of Mathematics Ways Forward from Logicism & Intuitionism

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 couldn’t guarantee its objects existed But ... Gödel’s incompleteness theorems also struck a blow Always at least one sentence that is true but unprovable No formal system can prove its own consistency Gödel: Principia Mathematica ‘a considerable step backwards as compared with Frege’ (1944) Failure of Logicism?

Dummett on Principia Mathematica ‘[Russell & Whitehead’s] 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, ... .' Dummett on Principia Mathematica

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: Gödel - special arithmetical intuition Empiricism or Naturalism – dispense with the special status of arithmetic & mathematical objects Quine – maths is subordinate to physics Intuitionism – classical logic & mathematics is a problem Ways forward

Intuitionism

LEJ Brouwer (1881-1966) Dutch mathematician – early 20th Century Involved in controversies over foundational programme with David Hilbert “What is this Frog and Mouse Battle?” (Einstein c. 1928) 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 Cf Descartes vs Leibniz on proof: Does a proof have to be “grasped”? Or is it the stipulation of a correct algorithm? LEJ Brouwer (1881-1966)

Intuitionism & Constructivism Mathematical objects are constructed by (human) activity Entities are equivalent to their process of construction (steps) Intuitionism as form of constructivism If one sticks to finitary arithmetic, there is not much difference with classical (cf .Russell on Axiom of Infinity) No intutionist ‘truths’ that aren’t also true in classical maths It drops assumption that there must be determinate truth value when quantifying ( “there exists …” or “for all …” ) over any purportedly infinite totality: Natural numbers Decimal places of real numbers Alternative Logic based on semantic consequences of proof, not truth Not every mathematical statement has determinate truth-value “A or ¬A” is not a tautology – we need to be able to assert one or other of disjunction Reformulation of classical mathematics sought New Interpretation of real numbers & continuum Infinity as potential infinity (Aristotle) to grasp an infinite structure is to grasp the process that generates it NOT A COMPLETED TOTALITY Intuitionism & Constructivism

Dummett on intuitionism 1

Intuitionist: we can in principle carry this out this construction Finist: we have to be able to carry it out in practice Cf “Letter” in Heyting’s “Disputation” Phenomenologist à la Husserl in Origins of Geometry: Mathematics escapes problem of subjective, mental activity by writing Accumulation & Sedimentation of mathematics as intergenerational practice “Classical” platonist: The objects or the facts exist whether we know them or not. To Summarise

Intuitionist Logic

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 don’t know the case yet Assumes object exists and we are trying to find out what is the case about them 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 <11,13>) 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." Motivation for intuitionism

Motivation for intuitionism Heyting: “Classical mathematics neglects altogether the obvious difference in character between these definitions. k can actually be calculated (k=3), whereas we possess no method for calculating l, as it is not known whether the sequence of pairs of twin primes p, p+ 2 is finite or not. Therefore intuitionists reject II as a definition of an integer; they consider an integer to be well defined only if a method for calculating it is given. Now this line of thought leads to the rejection of the principle of the excluded middle, for if the sequence of twin primes were either finite or not finite, II would define an integer.” Motivation for intuitionism

Semantic differences For any proposition A: Classical – asserts ‘A’ and means ‘A is true’ Bivalence – there are only two truth values (true or false), one or other must be the case Cannot be both, cannot be neither “A or not-A” is then a tautology (necessarily true) A can only be true or false Law of Excluded Middle – there is no third option or middle way! Intuitionist – asserts ‘A’ but means: ‘we have a proof for A’; asserts ‘All x are F’ and means ‘we have a proof that all x are F’ “A or not-A” is not a tautology It means: “We have a proof that A is the case or a proof that A is not the case” We may have neither Semantic differences

Negation differences What is the negation of “All x are F”? Classical: ‘All x are F’ is false therefore ‘Not all x’s are F’ therefore ‘there is at least one x that is not F’ – VALID inference Intuitionistic: ‘We do not have a proof that all x’s are F’ therefore ‘we have a proof that there is at least one x that is not F’ INVALID inference Negation differences

Double negation in Classical Logic Classical: Law of Excluded Middle does apply – the only two options are A or not-A [¬A] double negation introduction & elimination is allowed A Ⱶ ¬¬A If A is true, then not-A is false [ ¬(¬A)]. ¬¬A Ⱶ A If it’s not not-A [ie not-A is false], then A is true. Double negation in Classical Logic

Double negation in Intuitionism Intuitionism: Law of Excluded Middle does not apply double negation introduction is allowed A Ⱶ ¬¬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 Ⱶ A If we do not know that we do not know A, we cannot conclude A Double negation in Intuitionism

Classical vs Intuitionist “A or ¬A” is true for Intuitionist if we have a proof of A or a proof of ¬A If we have neither then this sentence cannot be asserted, whereas for classical logic it is a tautology! Classical vs Intuitionist

Dummett on intuitionism 2 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

Dummett on intuitionism 3 Classical logic is inadequate for mathematical statements, where numbers are generated by us (i.e. not about external world – no checks!): "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] Dummett on intuitionism 3

Gödel (1944) is such a realist! “… the vicious circle … applies only if the entities are constructed by ourselves. In this case, there must exist a definition … which does not refer to a totality to which the object defined belongs. … If, however, it is a question of objects that exist independently of our constructions, there is nothing in the least absurd in the existence of totalities containing members, which can be described only by reference to this totality. “Classes and concepts may be conceived as real objects existing independently of us and our definitions and constructions. It seems to me that the assumption of such objects is quite legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.” Gödel (1944) is such a realist!

Intuitionist Mathematics

Real numbers Infinite decimal expansion of real numbers Cannot assume these numbers exist such that nth digit is fixed – cf. Discussion of Pi’s digits Early Brouwer: Only a real number constructed according to a rule or law is legitimate That is minor subset of the reals! Intuitionists have fewer numbers & fewer digits! Late Brouwer: Allows “free choice sequences” by creative subject so we can continue sequence as far as we like (but sequence is not there already as totality) Dummett: "As soon as we consider a domain which cannot be effectively enumerated [an infinite domain], the quantifiers need to be explained ... by reference to a proof that a given object belongs to the domain." Real numbers

Continued Fractions Continued Fractions! Proofs of the irrationality of particular numbers are used by intuitionists to construct those numbers & their decimal expansions Continued Fractions! Continued Fractions

Intuitionists are so opposed to the appeal to infinite totalities in classical mathematics that they construct a new logic & a new mathematics to avoid the philosophical and foundational problems. They have fewer numbers & can prove fewer things – largely restricted to “finite” mathematics Central place for the activity of the mathematician who constructs proofs and numbers Must intuitionism be wrong if it demands revisions to mathematical practice? Summary

Appendix on logic

What we’ve just described is syntax: mere symbol-shuffling. We like to think there’s 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 it’s suppposed to be “about”. Then we map the symbols of the language, one by one, onto features of the model. Syntax vs Semantics

A theory is consistent if it doesn’t 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, it’s consistent. Ex falso quodlibet It’s 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. Contradiction

We may now prove “Humans are fish” as follows: 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 it’s not so easy to point to one step that causes the trouble.

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 won’t set this out here. Here is the proof we just did on the previous slide: P (P | Q) ¬(P) (¬(P) & (P | Q)) Q Propositional Logic

Predicate logic allows us to “break apart” propositions a bit Predicate logic allows us to “break apart” propositions a bit. It’s a bit more complicated as a result. To express “Whales are fish”, we might write ∀(𝑥)(𝑤 𝑥 →𝑓 𝑥 ) Which we would read aloud as “for every creature, x, if x is a whale (“w(x)”) then x is a fish (“f(x)”). Predicate logic gives us a way to express mathematical statements. For example, let’s change the model: Let the variable, x, range over the natural numbers instead of animals. Let w(x) mean “x is an even number greater than 2” Let f(x) mean “x is the sum of two prime numbers” Now our symbols express Goldbach’s Conjecture! Predicate Logic