A STRICTLY FINITISTIC SYSTEM FOR APPLIED MATHEMATICS FENG

A STRICTLY FINITISTIC SYSTEM FOR APPLIED MATHEMATICS FENG YE Dept. of Philosophy, Peking University fengye 63@gmail. com http: //sites. google. com/site/fengye 63/ 11/5/2008 1

1. Introduction 11/5/2008 2

Background l General 1. Introduction background: ¡ Realism vs. anti-realism (nominalism) debates in philosophy of mathematics. l. A research project: ¡ Explore a truly naturalistic philosophy of mathematics. ¡ Support anti-realism (nominalism). ¡ http: //sites. google. com/site/fengye 63 11/5/2008 3

Background l. A 1. Introduction topic in the project: ¡ Explaining the applicability of mathematics. l The strategy for explaining applicability requires ¡ Developing Strict Finitism, essentially a fragment of the quantifier-free PRA, with recognized functions limited to elementary recursive functions. ¡ Prove that applied mathematics can be developed in Strict Finitism. 11/5/2008 4

This Talk 1. Introduction l Present a formal system SF, the basis for strict finitism l Explain how to do mathematics in strict finitism l Illustrate the mathematics developed within strict finitism so far l Briefly discuss the potential philosophical implications and/or uses of this technical work 11/5/2008 5

Warning 1. Introduction I’m not suggesting that strict finitism is the only meaningful mathematics, or the true foundation of mathematics, or even a better mathematics. l My own philosophical position is naturalism or physicalism, while ‘finitism’ as a philosophical position can be ambiguous. l Strict finitism is used only as an assistant analytical tool for explaining the applicability of classical mathematics to this finite physical world. l 11/5/2008 6

2. The System SF 11/5/2008 7

2. The System SF The Language of SF l A summary: ¡ The language of typed λ-calculus, plus constants for base elementary recursive functions, and operators for bounded primitive recursion, finite product, and finite sum. ¡ No quantifiers; equalities between numerical terms only, i. e. no equalities between functionals. 11/5/2008 8

2. The System SF The Language of SF l Details: 11/5/2008 9

2. The System SF The Language of SF l Details: 11/5/2008 10

2. The System SF The Axioms of SF l Common axioms: ¡ Tautologies, axioms for equalities ¡ Axioms of the typed λ-calculus for functional application and λ-abstraction, ¡ Arithmetic axioms characterizing S, ¡ Recursive definitions of base elementary recursive functions and recursive characterizations of finite product and finite sum. 11/5/2008 11

2. The System SF The Axioms of SF l Selection axioms: indicates an occurrence of a subterm t in the term s, ¡ s{t} 11/5/2008 12

2. The System SF The Axioms of SF l Recursion ¡ i, 11/5/2008 axioms: j do not occur free in r, b, s, 13

2. The System SF Rules ¡ The 11/5/2008 induction is a quantifier-free induction 14

2. The System SF SF Is Strictly Finitistic normal form theorem for typed calculus holds for SF. l A numerical term t [m 1, …, mn] with numerical free variables only (no free variables of higher types) represents an elementary recursive function. l SF is strictly finitistic in this sense. l The 11/5/2008 15

3. Mathematics in Strict Finitism 11/5/2008 16

3. Mathematics in Strict Finitism A Summary Doing mathematics in strict finitism means constructing terms and proving that they satisfy some conditions expressed as quantifier-free formulas in SF. l Similar to a programmer’s work: designing programs and proving that the programs meet some specifications. l 11/5/2008 17

3. Mathematics in Strict Finitism Mathematical Claims in Strict Finitism A claim in strict finitism is denoted as: (Fin. C) where is a formula of SF, meaning that we have constructed a sequence of terms t (not depending on y) of appropriate types and prove that l l Note: and are not classical or intuitionistic quantifiers; they are used in this context only and cannot be nested otherwise; they are used only for suppressing mentioning the constructed terms. 11/5/2008 18

3. Mathematics in Strict Finitism Mathematical Claims in Strict Finitism l A proof of the claim consists of the required terms t (called witnesses for the claim) and a proof of the condition in SF: 11/5/2008 19

3. Mathematics in Strict Finitism Defined Logical Constants defined logical constants *, *, *, and * in our semi-formal language, to allow expressing claims in strict finitism in simplified and more readable formats. l Introduce ¡ Caution: They are not equivalent to the corresponding classical or intuitionistic logical constants. ¡ After defined logical constants are eliminated, claims are reduced to the format 11/5/2008 20

3. Mathematics in Strict Finitism Defined Logical Constants l Let 11/5/2008 21

3. Mathematics in Strict Finitism Defined Logical Constants They are Gödel’s Dialectica interpretation. l The definition of * is Bishop’s numerical implication. l Starred logical constants are consistent with non-starred logical constants in SF and in claims in the format (Fin. C) when both are meaningful in a context. So, we can omit the stars. l 11/5/2008 22

3. Mathematics in Strict Finitism Defined Logical Constants defined logical constants *, *, *, and * follow the intuitionistic logical laws, including the axiom of choice. l All ¡ for the logical axiom , this means that when the defined constant is eliminated, we get a claim in the format (Fin. C), and the required witnesses for the claim can be automatically constructed. 11/5/2008 23

3. Mathematics in Strict Finitism Proving Claims l We can use intuitionistic logic for deriving a claim of strict finitism stated using defined logical constants; witnesses for the derived claim can be automatically extracted from the proof. 11/5/2008 24

3. Mathematics in Strict Finitism Compare with Bishop’s Constructive Mathematics l Differences: Strict finitism allows ¡ ¡ 11/5/2008 bounded primitive recursions on numerical terms only, and no recursive constructions on higher-type terms (i. e. functionals) quantifier-free inductions on equalities between numerical terms only, and no inductions on quantified statements, or on equalities between higher-type terms no quantifications over sets 25

3. Mathematics in Strict Finitism Compare with Bishop’s Constructive Mathematics l Part of our work in developing mathematics within strict finitism consists in ¡ ¡ 11/5/2008 unraveling the recursive constructions and inductions in Bishop’s constructive mathematics and reducing them to the those available in strict finitism, eliminating quantifications over sets 26

4. Sets and Functions in Strict Finitism 11/5/2008 27

4. Sets and Functions in Strict Finitism A Summary Sets are formulas viewed as conditions for classifying terms, and functions are terms that apply to terms satisfying some conditions and produce other terms satisfying some other conditions. l The ideas are from Bishop’s constructive mathematics, with some modifications. l 11/5/2008 28

4. Sets and Functions in Strict Finitism Defining Sets l Let A [a], [a, b] be a pair of claims, a, b be of the same type . Write [a] as a A and write [a, b] as a=Ab. ‘A is a set of the type ’ as a claim is the conjunction of where l If a A is the claim denote 11/5/2008 , we use a x. A to , “x is the witness for a A”. 29

4. Sets and Functions in Strict Finitism Example: Sets of Real Numbers 11/5/2008 30

4. Sets and Functions in Strict Finitism Defining Functions l Suppose A and B are sets, ‘f is a function from A to B’ is the claim l Functions operate on witnesses, if any: ¡ The function witness for 11/5/2008 must operate on the 31

4. Sets and Functions in Strict Finitism What Are Sets and Functions Used for? Sets and functions allow stating complex conditional constructions in familiar and simplified formats. l The following becomes a very complex claim when spelled out: l 11/5/2008 32

5. Applied Mathematics in Strict Finitism 11/5/2008 33

5. Applied Mathematics in Strict Finitism Calculus Mostly follows Bishop l A sequence (an) converges to y, if l l A function is continuous, if l g = f’ on I if there exists , such that (n)>0 for all n>0, and for all x, y I, |x-y|< (n). 11/5/2008 34

5. Applied Mathematics in Strict Finitism Calculus Riemann integrations for continuous functions are constructed as limits of partial sums. l Basic theorems of calculus then follow, including an approximate form of the Intermediate Value Theorem, an approximate form of Roll’s Theorem, Taylor series theorem, and the fundamental theorem of calculus and so on. l 11/5/2008 35

5. Applied Mathematics in Strict Finitism Metric Space A set is a pair of claims, or a pair of formulas in the extended language. We cannot quantify over sets. l The theory of metric spaces is presented as schematic claims containing an arbitrary set satisfying some conditions. l 11/5/2008 36

5. Applied Mathematics in Strict Finitism Metric Space l Suppose that X is a set. is a metric on X, or (X, ) is a metric space, if and for all x, y X, 11/5/2008 37

5. Applied Mathematics in Strict Finitism Metric Space Familiar notions such as boundedness, completeness, total boundedness, and compactness can be defined as in Bishop’s constructive mathematics. l Completion of a metric space can be constructed and other theorems in Bishop’s constructive mathematics can be carried over, including the Stone-Weierstrass theorem. l 11/5/2008 38

5. Applied Mathematics in Strict Finitism Lebesgue Integration Lebesgue integrable functions on are partial functions on represented by sequences of continuous functions that converge in some way. (The idea is adapted with some revisions from Bishop and Bridges. ) l They are natural extensions of continuous functions and they include familiar partial functions such as characteristic of intervals, step functions, and so on. l 11/5/2008 39

5. Applied Mathematics in Strict Finitism Lebesgue Integration l We can prove the completeness of Lebesgue integration; we can define L 1, L 2 and prove their separability and completeness; we can define measurable functions, various notions of convergence and prove convergence theorems. 11/5/2008 40

5. Applied Mathematics in Strict Finitism Hilbert Space Theory l The definitions for linear space, Banach space and Hilbert space are also schematic definitions involving an arbitrary set. l We can define familiar notions including self-adjointness and prove the spectral theorem and Stone’s theorem for unbounded self-adjoint operators. 11/5/2008 41

6. Philosophical Implications or Uses of Strict Finitism 11/5/2008 42

6. Philosophical Implications or Uses Conjecture of Finitism l Strict finitism is in principle sufficient formulating theories and representing proofs and calculations in the ordinary sciences. 11/5/2008 43

6. Philosophical Implications or Uses Reasons Supporting The Conjecture Some significant applied mathematics has been developed and it appears that the method can advance further. l Current sciences deal with discrete things from the Planck scale to the cosmological scale; functions not essentially bounded by the power function never appear in natural scientific contexts. l Infinity, continuity, and so on are glosses over details; they ought not to be logically strictly indispensable l 11/5/2008 44

6. Philosophical Implications or Uses Counter Examples? l Beliefs about concrete things derived from consistency beliefs about ZFC or its extensions are not derivable from strict finitism: ¡ Example: A computer proving theorems in ZFC will not output a contradiction. l But ¡ they do not belong to the ordinary sciences， ¡ they do not rely on the axioms of ZFC; they rely on consistency beliefs only, which are about concrete things and are inductive in nature 11/5/2008 45

6. Philosophical Implications or Uses Possible Philosophical Implications l Cast doubts on the indispensability argument for mathematical realism. l Hilbert’s instrumentalist interpretation of classical mathematics may still be true: ¡ Hilbert’s proof theory program sets its goal too high: classical mathematics is not conservative over finitism. ¡ The part of classical mathematical that is actually applied in the sciences may still be conservative over finitism. 11/5/2008 46

6. Philosophical Implications or Uses My Own Use l Strict Finitism is designed as an assistant tool for explaining the applicability of classical mathematics to strictly discrete and finite things in this universe: ¡ The applications of classical mathematics can in principle be reduced to the applications of strict finitism. ¡ The applications of strict finitism can be interpreted as valid logical deductions from true premises about strictly finite concrete things, to true conclusions about them. 11/5/2008 47

Comparisons with Other Approaches There have been several nominalization programs, predicativism and so on. l This approach differs from them mainly in that its basis is strictly finitistic: l Committing to the reality of infinity (including potential infinity) in any format means committing to things not in this universe; ¡ It is not nominalism anymore and it must face its epistemological difficulty. ¡ Only a strictly finitistic system can be used as an assistant tool for explaining the applicability to strictly finite things in this universe. ¡ 11/5/2008 48