“KWARC was!”

Notes on Cristian S. Calude et al. (2007) Proving and Programming (see also previous post, see slides). Please note that the authors do not necessarily share my personal opinions and summaries on this page.

There is a strong analogy between proving theorems in mathematics and writing programs in computer science. Programming gives mathematics a new form of understanding. The computer is the driving force behind these changes.

• “Theorems (in mathematics) correspond to algorithms and not programs (in computer science)”
• “Programs (in computer science) correspond to mathematical models.”
• Proving the correctness of algorithms is seldom a focus in software projects (but: even testing/ documenting is sometimes neglected)
• Do some programming language facilitate/ required to put more focus on the correctness of the algorithms (e.g. functional language such as scala or ML; where programs are closer to mathematical models than in imperative languages)?
• Can bugs be avoided? Can the use of rigorous mathematical proofs guarantee that software and hardware perform as expected?
• Many projects and products dedicated to automated testing of program correctness, for example, VIPER or TestEra (automated testing of Java programs).
• What do proof obligations (Verpflichtung) have to do with the correctness of programs?: e.g. Barthe et al. discuss the “interaction between compilation and verification condition generators (VC generators), which are used in many interactive verification environments to guarantee the correctness of source programs, and by several proof carrying code (PCC) architectures to check the correctness of compiled programs. Such VC generators operate on annotated programs that carry loop invariants and procedure specifications expressed as preconditions and postconditions, and yield a set of proof obligations that must be discharged in order to establish the correctness of the program.” (loop and procedure level; see also specification languages such as CASL for the representation of precondition, invariants, and postconditions, i.e. proofs obligation)
• But correctness proof for a program, adds very little to one’s confidence in the program: “Beware of bugs in the above code: I have only proved it correct, not tried it. (Knuth)” (p.310)
• Checking new types of proofs: probabilistic, experimental, hybrid proofs (computation plus theoretical arguments) (p.311 ff.)

Further readings

• Hayashi (1998): Testing Proofs by Examples

Notes on (1) Cristian S. Calude et al. (2003) Passages of Proof and (2) Cristian S. Calude et al. (2004): Mathematical Proofs at a Crossroad .

Please note that the authors do not necessarily share my personal opinions and summaries on this page.

Stages of mathematics

1. pre-Greek mathematics dominated by observation, intuition, experience
2. Greek deductive mathematics based on theorems; Euclid’s geometry; see also Pythagoras,Thales, Aristotle, Euclid, Archimedis. (Euclid became the reference for axiomatic-deductive thinking, see more recent works in mathematics, physics, computer science, biology, linguistics, which follow this tradition)
3. Mathematical language triggered by need for precision and rigour. Previously ordinary language was used “dominated by imprecision resulting from its predominantly spontaneous use, where emotional factors and lack of care have an impact”. Galilei, Descartes, Newton, Leibniz, … contribute to a shift from ordinary to a mixed language, i.e. ordinary language supplemented by an artificial component of symbols, formulas, and equations
4. The epsilon rigour; 19th century, Cauchy, Weierstrass (coping with processes with infinitely many steps such as limit, continuity, differentiability, and integrability)
5. The challenge of the principle of non–contradiction and the logical crisis (Russell–Whitehead, Hilbert, Brouwer); 19th/20th century; optimistic towards the possibility to arrange the whole mathematics as a formal system and to decide for any possible statement whether it is true or false.
6. Gödel’s incompleteness theorem (1931): “every formal system which is finitely specified, rich enough to include the arithmetic, and consistent (free of contradiction), is incomplete”, distinction between truth and provability (1:p.173 ff.); Chaitin (1975) suggests that complexity is a source of incompleteness
7. Reconciliation of empirical–experimental mathematics with deductive mathematics (today): Four-colour Problem (4CT; 1976; by Kenneth Appel & Wolfgang Haken; proof of 4CT by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas ; or the Kepler Conjecture by Thomas Hales) realized by the use of computer programs as pieces of mathematical proof
8. Quantization: Proofs are no longer exclusively based on logic and deduction, but also empirical and experimental.

What is mathematical proof?

• “A proof is a series of logical steps based on some axioms and deduction rules which reaches a desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound.” (1:p. 171)
• But who checks the proof (as human and Computer agents make mistakes)? Often proofs are falsified after having been published and then corrected (e.g. Appel/ Haken take on Kempe’s ideas). Some proofs cannot be checked by single humans (being to long and/or accomplished by computer-assistance) such as 4CT. (1:p. 171)
• “Mathematics cannot be conceived without proofs: [..] proofs and theorems go together; the object of a proof is to reach a theorem, while theorems are validated by proofs” (1:p.172).
• But: Mathematics is more than proofs. “What the mathematical community seems to value most are ideas. The most respected mathematicians are those with strong intuitions. (Harris)”. (1:p.172)
• Three dimensions of proofs:
  • syntactically: the formal proof, but “proof is only one step in the direction of confidence” (1:p.174, see Edmund Landau)
  • semantically: (truth value) “correctness by itself does not validate a proof, it is also necessary to understand it” (1:p.175; René Thom, Daniel Cohen, William Thurston) “the mission of mathematics is understanding”, consequently, computer-assisted proofs are harder to accept (see 4CT), but also deductive proofs are sometimes only understood by few mathematicians (see 1:p.176), “a theorem is validated if it has been accepted by a general agreement of the mathematical community” (Thom, 1:p.178), “a theorem is a statement that could be checked individually by a mathematician and confirmed also individually by at least two or three mathematicians, each of them working independently” (1:p.178) thus proposing the notion of an “agnogram … a theorem-like statement that we have verified as best we could, but whose truth is not know with the kind of assurance we attach to theorems and about which we must thus remain, to some extend, agnostic” (Swart, 1:p.178), “we believe the experts and we cannot tell for ourselves” (1:p.179), “mathematics occupies a special place [among other disciplines], where we believe anyone who claims to have proved a theorem on the say – so of just a few people – that is, until we hear otherwise” (1:p.179), “… in mathematics you can really argue that this is as close to absolute truth as you can get (Joe Spencer)” (1:p.179)
  • pragmatical: (relevance & use) “truth is not where you find it, but where you put it” (Perlis). “no matter how precise the rules are, we need human consciousness to apply the rules and to understand them and their consequences” (1:p.183)
• Aspects of mathematical proofs: deduction/ syllogistic reasoning (most visible aspect), but also: observation, intuition, experiment, visual representations, induction, analogy, and examples; some belonging to the preliminary steps, whose presence is not made explicit (when finally presenting the proof), but without which proofs cannot be conceived (2-p.17; see previous post); proving is a very heterogeneous process

In real life proofs may be different …

• Proof by obviousness: “The proof is so clear that it need not be mentioned.”
• Proof by general agreement: “All in favour?”
• Proof by calculus: “This proof requires calculus, so we’ll skip it.”
• Proof by lost reference: “I know I saw it somewhere”
• Proof by necessity: “It had better be true, or the entire structure of mathematics would crumble to the ground.”
• Proof by plausibility: “It sounds good, so it must be true.”
• Proof by intimidation: “Don’t be stupid; of course it’s true.”
• Proof by terror: When intimidation fails
• Proof by lack of sufficient time: “Because of the time constraint, I’ll leave the proof to you.”
• Proof by tessellation: “This proof is the same as the last.”
• Proof by majority rule: Only to be used if general agreement is impossible
• Proof by authority: “Well, Don’t Knuth says it’s true, so it must be!”
• Proof by intuition: “I just have this gut feeling”

Towards Artificial Mathematics and quasi-empirical proofs

• equivalence between the logical and computational proofs
• logical/ conventional proofs: traditional; reasoning of humans (see Euclid, …); the logical process, i.e. finding a finite sequence of sentences strictly obeying some axioms and inference rules
• computational/ unconventional: computational process (machines are constructed based on sequences of sentences by humans) producing these sequences; but: proofs can contain steps that can never be verified by humans (based on the equivalence: development of artificial mathematicians, i.e. theorem provers such as Mathematica, Maple, MathLab)
• classical but unconventional proofs also comprise classical proof of excessive length and complexity (e.g. the classification of finite simple groups; 1:p.176/183)
• Artificial mathematicians are far less ingenious and subtle than human mathematicians, but they surpass their human counterparts by being infinitely more patient and diligent.
• Towards quantum computational proofs: conversion from computation into a sequence of sentences may not longer be possible, quantum automation are able to check a proof, but fail to reveal a “trace” of the proof (we don’t why it true), (quasi-empirical) quantum proofs might influence how we learn/ understand mathematics; leading to new ways to understand (and practice) mathematics, although we might not fully accept/trust unconventional proofs, the computational result is “a mathematical activity because it advances our knowledge of mathematics” (1:p.184)
• Overall: “There is little difference between traditional and unconventional types of proofs as [..] i) mathematical truth cannot always be certified by proof, ii) correctness is not absolute, but almost certain, as mathematics advance by making mistakes and correcting and re-correcting them (see Lakatos), iii) non-deterministic and probabilistic proofs do not allow mistakes in the applications of rules, the are just indirect forms of checking, iv) the explanatory component, the understanding emerging from proofs [..] is subjective and has no bearing on formal correctness.” (1:p.184)
• “Experimental Mathematics – as systematic mathematical experimentation ranging from hypotheses building to assisted proofs and automated proof-checking-will play an increasingly important role and will become part of the mainstream of mathematics. There are many reasons for this trend: They range from logical (the absolute truth simply doesn’t exist), sociological (correctness is not absolute [..]), economic (powerful computers will be accessible to more and more people), and psychological (results and success inspire emulation)” (2-p.26)

Knowledge is acquired through reason and by experiment: Should proofs belong exclusively to logic? Or should we also accept empirical-experimental arguments? Towards blending logical and empirical-experimental arguments. There is hope for integration! (see also next post)