A STRICTLY FINITISTIC SYSTEM FOR APPLIED MATHEMATICS

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A STRICTLY FINITISTIC SYSTEM FOR APPLIED
MATHEMATICS

• FENG YE
• Dept. of Philosophy, Peking University
• fengye63_at_gmail.com
• http//sites.google.com/site/fengye63/

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1. Introduction
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1. Introduction
Background

• General background
• Realism vs. anti-realism (nominalism) debates in
  philosophy of mathematics.
• A research project
• Explore a truly naturalistic philosophy of
  mathematics.
• Support anti-realism (nominalism).
• http//sites.google.com/site/fengye63

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1. Introduction
Background

• A topic in the project
• Explaining the applicability of mathematics.
• 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.

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1. Introduction
This Talk

• Present a formal system SF, the basis for strict
  finitism
• Explain how to do mathematics in strict finitism
• Illustrate the mathematics developed within
  strict finitism so far
• Briefly discuss the potential philosophical
  implications and/or uses of this technical work

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1. Introduction
Warning

• Im not suggesting that strict finitism is the
  only meaningful mathematics, or the true
  foundation of mathematics, or even a better
  mathematics.
• My own philosophical position is naturalism or
  physicalism, while finitism as a philosophical
  position can be ambiguous.
• Strict finitism is used only as an assistant
  analytical tool for explaining the applicability
  of classical mathematics to this finite physical
  world.

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2. The System SF
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2. The System SF
The Language of SF

• A summary
• The language of typed ?-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.

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2. The System SF
The Language of SF

• Details

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2. The System SF
The Language of SF

• Details

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2. The System SF
The Axioms of SF

• Common axioms
• Tautologies, axioms for equalities
• Axioms of the typed ?-calculus for functional
  application and ?-abstraction,
• Arithmetic axioms characterizing S,
• Recursive definitions of base elementary
  recursive functions and
  recursive characterizations of finite product and
  finite sum.

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2. The System SF
The Axioms of SF

• Selection axioms
• st indicates an occurrence of a subterm t in
  the term s,

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2. The System SF
The Axioms of SF

• Recursion axioms
• i, j do not occur free in r, b, s,

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2. The System SF
Rules

• The induction is a quantifier-free induction

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2. The System SF
SF Is Strictly Finitistic

• The normal form theorem for typed ?-calculus
  holds for SF.
• A numerical term t m1, , mn with numerical
  free variables only (no free variables of higher
  types) represents an elementary recursive
  function.
• SF is strictly finitistic in this sense.

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3. Mathematics in Strict Finitism
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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.
• Similar to a programmers work designing
  programs and proving that the programs meet some
  specifications.

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3. Mathematics in Strict Finitism
Mathematical Claims in Strict Finitism

• A claim in strict finitism is denoted as
• (FinC)
• 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
• 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.

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3. Mathematics in Strict Finitism
Mathematical Claims in Strict Finitism

• A proof of the claim
• consists of the required terms t (called
  witnesses for the claim) and a proof of the
  condition in SF

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3. Mathematics in Strict Finitism
Defined Logical Constants

• Introduce defined logical constants ?, ?, ?,
  ?, ?, and ? in our semi-formal language, to
  allow expressing claims in strict finitism in
  simplified and more readable formats.
• 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

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3. Mathematics in Strict Finitism
Defined Logical Constants

• Let

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3. Mathematics in Strict Finitism
Defined Logical Constants

• They are Gödels Dialectica interpretation.
• The definition of ? is Bishops numerical
  implication.
• Starred logical constants are consistent with
  non-starred logical constants in SF and ? and ?
  in claims in the format (FinC) when both are
  meaningful in a context. So, we can omit the
  stars.

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3. Mathematics in Strict Finitism
Defined Logical Constants

• All defined logical constants ?, ?, ?, ?, ?,
  and ? follow the intuitionistic logical laws,
  including the axiom of choice.
• for the logical axiom , this
  means that when the defined constant is
  eliminated, we get a claim in the format (FinC),
  and the required witnesses for the claim can be
  automatically constructed.

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3. Mathematics in Strict Finitism
Proving Claims

• 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.

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3. Mathematics in Strict Finitism
Compare with Bishops Constructive Mathematics

• Differences Strict finitism allows
• 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 equalities between higher-type terms
• no quantifications over sets

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3. Mathematics in Strict Finitism
Compare with Bishops Constructive Mathematics

• Part of our work in developing mathematics within
  strict finitism consists in
• unraveling the recursive constructions and
  inductions in Bishops constructive mathematics
  and reducing them to the those available in
  strict finitism,
• eliminating quantifications over sets

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4. Sets and Functions in Strict Finitism
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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.
• The ideas are from Bishops constructive
  mathematics, with some modifications.

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4. Sets and Functions in Strict Finitism
Defining Sets

• 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 aAb. A is a set of the type ?
  as a claim is the conjunction of
• where
• If a?A is the claim , we use
  a?xA to denote , x is the
  witness for a?A.

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4. Sets and Functions in Strict Finitism
Example Sets of Real Numbers
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4. Sets and Functions in Strict Finitism
Defining Functions

• Suppose A and B are sets, f is a function from A
  to B is the claim
• Functions operate on witnesses, if any
• The function
  must operate on the witness for

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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.
• The following becomes a very complex claim when
  spelled out

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5. Applied Mathematics in Strict Finitism
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5. Applied Mathematics in Strict Finitism
Calculus

• Mostly follows Bishop
• A sequence (an) converges to y, if
• A function is continuous, if
• g f on I if there exists ?, such that ?(n)gt0
  for all ngt0, and
• for all x,y?I, x-ylt?(n).

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5. Applied Mathematics in Strict Finitism
Calculus

• Riemann integrations for continuous functions are
  constructed as limits of partial sums.
• Basic theorems of calculus then follow, including
  an approximate form of the Intermediate Value
  Theorem, an approximate form of Rolls Theorem,
  Taylor series theorem, and the fundamental
  theorem of calculus and so on.

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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.
• The theory of metric spaces is presented as
  schematic claims containing an arbitrary set
  satisfying some conditions.

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5. Applied Mathematics in Strict Finitism
Metric Space

• Suppose that X is a set. ? is a metric on X, or
  (X,?) is a metric space, if
• and for all x,y?X,

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5. Applied Mathematics in Strict Finitism
Metric Space

• Familiar notions such as boundedness,
  completeness, total boundedness, and compactness
  can be defined as in Bishops constructive
  mathematics.
• Completion of a metric space can be constructed
  and other theorems in Bishops constructive
  mathematics can be carried over, including the
  Stone-Weierstrass theorem.

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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.)
• They are natural extensions of continuous
  functions and they include familiar partial
  functions such as characteristic of intervals,
  step functions, and so on.

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5. Applied Mathematics in Strict Finitism
Lebesgue Integration

• We can prove the completeness of Lebesgue
  integration we can define L1, L2 and prove their
  separability and completeness we can define
  measurable functions, various notions of
  convergence and prove convergence theorems.

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5. Applied Mathematics in Strict Finitism
Hilbert Space Theory

• The definitions for linear space, Banach space
  and Hilbert space are also schematic definitions
  involving an arbitrary set.
• We can define familiar notions including
  self-adjointness and prove the spectral theorem
  and Stones theorem for unbounded self-adjoint
  operators.

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6. Philosophical Implications or Uses of Strict
Finitism
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6. Philosophical Implications or Uses
Conjecture of Finitism

• Strict finitism is in principle sufficient for
  formulating theories and representing proofs and
  calculations in the ordinary sciences.

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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.
• 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.
• Infinity, continuity, and so on are glosses over
  details they ought not to be logically strictly
  indispensable

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6. Philosophical Implications or Uses
Counter Examples?

• 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.
• 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

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6. Philosophical Implications or Uses
Possible Philosophical Implications

• Cast doubts on the indispensability argument for
  mathematical realism.
• Hilberts instrumentalist interpretation of
  classical mathematics may still be true
• Hilberts 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.

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6. Philosophical Implications or Uses
My Own Use

• 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.

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Comparisons with Other Approaches

• There have been several nominalization programs,
  predicativism and so on.
• This approach differs from them mainly in that
  its basis is strictly finitistic
• 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.