RI Math Lecture 2 2023

Mathematics Lecture 2: Logicism, Intuitionism, Formalism and the 3 crises

OverviewuLogicismuIntuitionismuFormalismu3 crises

RecapuExamined how Math is constructed and its nature (deductive, a priori)uTwo of the usual candidates on the nature of Math –Platonism and Empiricism

LogicismuAnother Realist, Rationalist view of Math, closely linked to PlatonismuMain proponents: Frege, Bertrand Russell, Peano and WhiteheaduPurpose: give math a firm foundation by showing that classical math is part of logic uWhy? So that mathematics can be shown to be free of contradictions and therefore a body of knowledge that is certain and absolute.uFoundationalist tendencies are present here.uIndeed, it seems weird to think of Math as anything but foundational. uIt’s how Euclid started out in his Elements of Mathematics uThinking of Math as coherent or reliabilist is just plain weird.uAlso, if successful, questions like “why is classical math free of contradictions?” can be reduced to “why is logic free of contradictions?” uThe latter Qn is something that philosophers can at least have a thorough handle on.

LogicismuRussell and Whitehead showed in their Principia Mathematica that all classical math, known in their time, can be derived from set theory and hence from the axioms of the Principia (or other formal set theories like that developed by Zermelo and Fraenkel)u(Using the ZF theory) Since the ZF theory has only 9 axioms, the logicist program is thus to show that all 9 ZF axioms belong to logic, i.e. can be reduced to logical propositions.uLogical Proposition: one that has complete generality and is true in virtue of its form rather than its content. uE.g. Law of the excluded middle: if p is a proposition, then either p or its negation ~p is trueuThis law does not hold because of any special content of the proposition p; it doesn’t matter whether p is a proposition of math or physics etcuRather, this law holds with “complete generality”, i.e. for any proposition p whatsoever. uWhy then does this hold? Logicists: because of its form!

CounterargumentuDid the logicists succeed in reducing all 9 ZF axioms to logical propositions? uNo; at least 2 axioms (infinity and choice) cannot be considered as logical propositionsuE.g. the axiom of infinity states that there exist infinite sets.uWhy do we accept it? Because we are familiar with quite a few infinite sets, say, the set of natural no. or the set of points in Euclidean 3-space.uBut then this shows that we accept this axiom by virtue of its content and not by virtue of its form!uIn general, when an axiom claims the existence of objects with which we are familiar on grounds of our common everyday experience, it is pretty certain that this axiom is not a logical proposition in the sense of Logicism.

Problemu1st crisis in math: at least 2 out of the 9 ZF axioms are not logical propositions in the sense of Logicism. uHence, this school failed by about 20% in its effort to give math a firm foundation!uWithout this firm foundation, math is then open to sceptical attacks like the infinite regress.

IntuitionismuAnti-Realist, A PrioriuMain proponent: BrouweruRadically different from the logicists who thought that there was never anything wrong with classical mathuIntuitionists thought that there was plenty wrong with classical mathuE.g. Cantor’s set theory had several paradoxes/contradictions – the logicists thought of them as common errors caused by erring mathematicians and not by a faulty mathematics; the intuitionists thought otherwise.uHence, they thought that math had to be rebuilt from the bottom on up.

IntuitionismuThis bottom, this beginning of math, is their explanation of what the natural no. 1,2,3… are (doesn’t include 0).uFor the intuitionists, all human beings have an innate primordial intuition for the natural numbers (not unlike Kant) (c.f. Article C, 243-4) ui.e. that we have an immediate certainty as to what is meant by the number 1 uand that the mental process which goes into the formation of the number 1 can be repeated to get 2, and on and on and on…uThis process is both inductive and effectiveuInductive: if one wants to construct 3, then one must go through constructing 1 and then 2.uEffective: once the construction of a natural no. is finished, that natural no. is entirely constructed and a complete finished mental construct, ready for our study of it.

IntuitionismuMath for the Intuitionists is a mental activity (of inductive and effective constructions) and not a set of theorems (which is what Logicism held)ui.e. Math is the mental activity which consists in carrying out constructs one after the other and is thus a priori (and therefore necessary).uUpshot? All intuitionistic math, like proofs, theorems and definitions, is constructed.uFurthermore, any mathematical theorem, proof etc that is not composed of constructs are seen as meaningless combinations of words.uThis also includes some logical rules such as the law of excluded middleuIn this way, the intuitionists have come up with their own intuitionistic arithmetic, algebra, analysis, set theory etc