Explanation and Argument in Mathematical Practice

Transcription

1 Explanation and Argument in Mathematical Practice Andrew Aberdein Humanities and Communication, Florida Institute of Technology, 50 West University Blvd, Melbourne, Florida , U.S.A. my.fit.edu/ aberdein Association for the Philosophy of Mathematical 4th Congress of Logic, Methodology & Philosophy of Science Université de Nancy 2, July 20

2 Outline. What is explanation in mathematics? 2. Kitcher s account and its problems. 3. How to remedy this? A detour via argumentation theory and mathematical practice. 5. An example from Euler. 6. Conclusions.

3 Outline. What is explanation in mathematics? 2. Kitcher s account and its problems. 3. How to remedy this? A detour via argumentation theory and mathematical practice. 5. An example from Euler. 6. Conclusions.

4 Outline. What is explanation in mathematics? 2. Kitcher s account and its problems. 3. How to remedy this? A detour via argumentation theory and mathematical practice. 5. An example from Euler. 6. Conclusions.

5 Outline. What is explanation in mathematics? 2. Kitcher s account and its problems. 3. How to remedy this? A detour via argumentation theory and mathematical practice. 5. An example from Euler. 6. Conclusions.

6 Outline. What is explanation in mathematics? 2. Kitcher s account and its problems. 3. How to remedy this? A detour via argumentation theory and mathematical practice. 5. An example from Euler. 6. Conclusions.

7 Outline. What is explanation in mathematics? 2. Kitcher s account and its problems. 3. How to remedy this? A detour via argumentation theory and mathematical practice. 5. An example from Euler. 6. Conclusions.

8 Kitcher on Explanation as Unification Science advances our understanding of nature by showing us how to derive descriptions of many phenomena, using the same patterns of derivation again and again, and, in demonstrating this, it teaches us how to reduce the number of types of facts that we have to accept as ultimate (or brute). So the criterion of unification I shall try to articulate will be based on the idea that E(K) [the most explanatory systematization of K] is a set of derivations that makes the best tradeoff between minimizing the number of patterns of derivation employed and maximizing the number of conclusions generated. P. Kitcher, 989, Explanatory Unification and the Causal Structure of the World.

9 Kitcher s Argument Pattern A schematic sentence is an expression obtained by replacing some or all of the non-logical expressions in a sentence by schematic letters. A set of filling instructions tells us how the dummy letters in a schematic sentence are to be replaced. A schematic argument is a sequence of schematic sentences. A classification for a schematic argument is a set of sentences which tell us exactly what role each sentence in a schematic argument is playing, e.g. whether it is a premise, which sentences are inferred from which and according to what rules, etc. A general argument pattern s, f, c is a triple consisting of a schematic argument s, a set of f of sets of filling instructions, and a classification c for s. J. Hafner & P. Mancosu, 2008, Beyond unification.

10 Kitcher s Argument Pattern A schematic sentence is an expression obtained by replacing some or all of the non-logical expressions in a sentence by schematic letters. A set of filling instructions tells us how the dummy letters in a schematic sentence are to be replaced. A schematic argument is a sequence of schematic sentences. A classification for a schematic argument is a set of sentences which tell us exactly what role each sentence in a schematic argument is playing, e.g. whether it is a premise, which sentences are inferred from which and according to what rules, etc. A general argument pattern s, f, c is a triple consisting of a schematic argument s, a set of f of sets of filling instructions, and a classification c for s. J. Hafner & P. Mancosu, 2008, Beyond unification.

11 Kitcher s Argument Pattern A schematic sentence is an expression obtained by replacing some or all of the non-logical expressions in a sentence by schematic letters. A set of filling instructions tells us how the dummy letters in a schematic sentence are to be replaced. A schematic argument is a sequence of schematic sentences. A classification for a schematic argument is a set of sentences which tell us exactly what role each sentence in a schematic argument is playing, e.g. whether it is a premise, which sentences are inferred from which and according to what rules, etc. A general argument pattern s, f, c is a triple consisting of a schematic argument s, a set of f of sets of filling instructions, and a classification c for s. J. Hafner & P. Mancosu, 2008, Beyond unification.

12 Kitcher s Argument Pattern A schematic sentence is an expression obtained by replacing some or all of the non-logical expressions in a sentence by schematic letters. A set of filling instructions tells us how the dummy letters in a schematic sentence are to be replaced. A schematic argument is a sequence of schematic sentences. A classification for a schematic argument is a set of sentences which tell us exactly what role each sentence in a schematic argument is playing, e.g. whether it is a premise, which sentences are inferred from which and according to what rules, etc. A general argument pattern s, f, c is a triple consisting of a schematic argument s, a set of f of sets of filling instructions, and a classification c for s. J. Hafner & P. Mancosu, 2008, Beyond unification.

13 Kitcher s Argument Pattern A schematic sentence is an expression obtained by replacing some or all of the non-logical expressions in a sentence by schematic letters. A set of filling instructions tells us how the dummy letters in a schematic sentence are to be replaced. A schematic argument is a sequence of schematic sentences. A classification for a schematic argument is a set of sentences which tell us exactly what role each sentence in a schematic argument is playing, e.g. whether it is a premise, which sentences are inferred from which and according to what rules, etc. A general argument pattern s, f, c is a triple consisting of a schematic argument s, a set of f of sets of filling instructions, and a classification c for s. J. Hafner & P. Mancosu, 2008, Beyond unification.

14 Hafner & Mancosu on Kitcher One of the reasons for Kitcher s failure may lie in the fact that his account, although much more sophisticated than Friedman s model, still shares the latter s basic intuition, namely that unifying and explanatory power can be accounted for on the basis of quantitative comparisons alone. However, in the controversy over the use of transcendental methods in real algebraic geometry the point at issue concerns qualitative differences in the proof methods. J. Hafner & P. Mancosu, 2008, Beyond unification.

15 Kitcher on Unrigorous Reasonings For my purposes, the most interesting subset of the accepted reasonings of a practice will not be those which are accepted as proofs nor those which are clearly unable to discharge the functions of proofs, but those which occupy an intermediate status. I shall call these the unrigorous reasonings of the practice. They are analogous to proofs in promising to figure in the system of proofs, but disanalogous in that they cannot be integrated within that system. They suppose that a proof can be given along the lines they set down. Yet it is currently impossible to reconstruct them in a way which will accord with the system of proofs. P. Kitcher, 983, The Nature of Mathematical Knowledge.

16 The Textbook Account of Fallacies A fallacy is a defective argument. It is an error in reasoning. An argument commits the fallacy of appeal to illegitimate authority if it appeals to someone or something as an authority on a particular subject who is not an authority on that subject. I. M. Copi, C. Cohen & D. Flage, 2007, Essentials of Logic.

17 The Textbook Account of Fallacies A fallacy is a defective argument. It is an error in reasoning. An argument commits the fallacy of appeal to illegitimate authority if it appeals to someone or something as an authority on a particular subject who is not an authority on that subject. I. M. Copi, C. Cohen & D. Flage, 2007, Essentials of Logic.

18 Argumentation Schemes Argumentation schemes are stereotypical patterns of defeasible reasoning that typically occur in common, everyday arguments. A standard account of argumentation schemes describes them as representing different types of plausible argument which, when successfully deployed, create presumptions in favor of their conclusions and thereby shift the burden of proof to an objector. Associated with each argumentation scheme is a set of critical questions to be used in the evaluation of arguments of the corresponding type. The posing of a critical question has the effect of defeating the initial presumption and shifting the burden of proof back on to the initial proponent. D. Godden & D. Walton, 2007, Advances in the Theory of Argumentation Schemes and Critical Questions

19 Appeal to Expert Opinion Argument Scheme for Appeal to Expert Opinion Major Premise Source E is an expert in subject domain S containing proposition A. Minor Premise E asserts that proposition A is true (false). Conclusion A is true (false). Critical Questions:. Expertise Question: How credible is E as an expert source? 2. Field Question: Is E an expert in the field that A is in? 3. Opinion Question: What did E assert that implies A? 4. Trustworthiness Question: Is E personally reliable as a source? 5. Consistency Question: Is A consistent with what other experts assert? 6. Backup Evidence Question: Is E s assertion based on evidence? D. Walton, C. Reed, & F. Macagno, 2008, Argumentation Schemes.

20 Comparisons Argument Patterns & Argumentation Schemes Argument Patterns Arbitrary Deductive Apodeictic Argumentation Schemes Stereotypical Deductive or Defeasible Dialectical Mathematical Reasoning Fallacy : Argumentation Scheme :: Unrigorous Reasoning : Mathematical Reasoning

21 Comparisons Argument Patterns & Argumentation Schemes Argument Patterns Arbitrary Deductive Apodeictic Argumentation Schemes Stereotypical Deductive or Defeasible Dialectical Mathematical Reasoning Fallacy : Argumentation Scheme :: Unrigorous Reasoning : Mathematical Reasoning

22 The Dance of Mathematical Practice Human mathematics consists in fact in talking about formal proofs, and not actually performing them. One argues quite convincingly that certain formal texts exist, and it would in fact not be impossible to write them down. But it is not done: it would be hard work, and useless because the human brain is not good at checking that a formal text is error-free. Human mathematics is a sort of dance around an unwritten formal text, which if written would be unreadable. This may not seem very promising, but human mathematics has in fact been prodigiously successful. David Ruelle, 2000, Conversations on mathematics with a visitor from outer space.

23 Hardy on Two Senses of Proof If we were to push it to its extreme we should be led to a rather paradoxical conclusion; that we can, in the last analysis, do nothing but point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils.... On the other hand it is not disputed that mathematics is full of proofs, of undeniable interest and importance, whose purpose is not in the least to secure conviction. Our interest in these proofs depends on their formal and aesthetic properties. Our object is both to exhibit the pattern and to obtain assent. G. H. Hardy, 928, Mathematical proof.

24 Epstein s Picture of Mathematical Proof A Mathematical Proof Assumptions about how to reason and communicate. A Mathematical Inference Premises argument necessity Conclusion The mathematical inference is valid. R. L. Epstein, 2008, Logic as the Art of Reasoning Well.

25 The Parallel Structure of Mathematical Proof Argumentational Structure: Mathematical Proof, P n Endoxa: Data accepted by mathematical community argument Inferential Structure: Mathematical Inference, I n Premisses: Axioms or statements formally derived from axioms derivation Claim: I n is sound Conclusion: An additional formally expressed statement

26 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

27 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

28 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

29 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

30 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

31 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

32 A Famous Argument of Euler s that If we expand sin x in a power series, we have n 2 = π2 6 sin x = x x 3 3! + x 5 5! x 7 7! +... Hence sin x = x 2 x 3! + x 4 5! x 6 7! +... Consider this as an infinite polynomial whose roots are ±π, ±2π, ±3π,... This enables us to represent sin x/x as an infinite product ( x 2 π )( x 2 2 4π )( x 2 2 9π )... 2 When this product is expanded, the coefficient of x 2 will be ( π 2 + 4π 2 + 9π ) Identifying this with the coefficient of x 2 given by the power series, we have π 2 ( ) = 3! whence we obtain = π2 6 Quoted in P. Kitcher, 983, The Nature of Mathematical Knowledge.

33 The Parallel Structure of Euler s Argument P : Prior work I P 2 : Elementary operations I 2 P 3 : Analogy I 3? P 4 : Elementary operations I 4 P 5 : Elementary operations I 5 P 6 : Elementary operations I 6 P 7 : Elementary operations I 7

34 Argument from Analogy Argumentation Scheme for Argument from Analogy Similarity Premise Generally, case C is similar to case C 2. Base Premise A is true (false) in case C. Conclusion A is true (false) in case C 2. Critical Questions:. Are there differences between C and C 2 that would tend to undermine the force of the similarity cited? 2. Is A true (false) in C? 3. Is there some other case C 3 that is also similar to C, but in which A is false (true)? D. Walton, C. Reed, & F. Macagno, 2008, Argumentation Schemes.

35 Conclusions Kitcher is right about explanation, but is undercut by a limited appreciation of mathematical practice; Mathematical proof has a parallel structure: argumentational and inferential; Explanation is a property of the argumentational structure, not the inferential structure; Substituting argumentation schemes for argument patterns shifts focus to argumentational structure, but also leads to qualitative comparison, since schemes are stereotypical.

36 Conclusions Kitcher is right about explanation, but is undercut by a limited appreciation of mathematical practice; Mathematical proof has a parallel structure: argumentational and inferential; Explanation is a property of the argumentational structure, not the inferential structure; Substituting argumentation schemes for argument patterns shifts focus to argumentational structure, but also leads to qualitative comparison, since schemes are stereotypical.

37 Conclusions Kitcher is right about explanation, but is undercut by a limited appreciation of mathematical practice; Mathematical proof has a parallel structure: argumentational and inferential; Explanation is a property of the argumentational structure, not the inferential structure; Substituting argumentation schemes for argument patterns shifts focus to argumentational structure, but also leads to qualitative comparison, since schemes are stereotypical.

38 Conclusions Kitcher is right about explanation, but is undercut by a limited appreciation of mathematical practice; Mathematical proof has a parallel structure: argumentational and inferential; Explanation is a property of the argumentational structure, not the inferential structure; Substituting argumentation schemes for argument patterns shifts focus to argumentational structure, but also leads to qualitative comparison, since schemes are stereotypical.

39 Hafner & Mancosu s Test Case: Proof I Theorem. A polynomial f (x,..., x n ) assumes a maximum value on any bounded closed semi-algebraic set S R n. Proof I. Within RCF the result cannot be proved as stated for we cannot quantify over polynomials and semi-algebraic sets. The best we could do is to prove its instances. For instance, RCF proves that x 3 has a maximum on [0, 2]. We could, in principle arrive at this proof by running the Tarski-Seidenberg decision procedure and getting as output. From there it would be a mechanical task to provide the explicit proof in RCF. And every single instance of the theorem could be obtained in this way. In other words, if we let P RCF be the set containing all the instances of the theorem in RCF, then we could thus arrive at a systematization of the whole set P on the basis of the Tarski-Seidenberg decision procedure. J. Hafner & P. Mancosu, 2008, Beyond unification.

40 Hafner & Mancosu s Test Case: Proof II Theorem. A polynomial f (x,..., x n) assumes a maximum value on any bounded closed semi-algebraic set S R n. Proof II. The following proof strategy draws on transcendental methods. This approach is based not just on the Tarski-Seidenberg decision procedure but rather on one of its consequences, namely that RCF is a complete theory. This gives rise to the following transfer principle. If a sentence ϕ in the language of RCF can be shown to hold in one particular real closed field, say, the real numbers R, then it must be true for any real closed field because of the completeness of RCF. Now, the theorem can of course be established for R relying on, among other things, the Bolzano-Weierstrass theorem (every bounded sequence has a convergent subsequence) and the least upper bound principle. These basic properties of R don t hold in general for real closed fields. For instance, they both fail for R alg. However, once the theorem has been proved for R, by whatever means, we can conclude by appeal to the transfer principle that all its instances, i.e. all sentences in P, hold for real closed fields in general. J. Hafner & P. Mancosu, 2008, Beyond unification.

41 Hafner & Mancosu s Test Case: Proof III Theorem. A polynomial f (x,..., x n ) assumes a maximum value on any bounded closed semi-algebraic set S R n. Proof III. Another way of establishing the theorem relies on purely algebraic means exploiting the fact that if A R n is a closed and bounded semi-algebraic set and g : A R p a continuous semi-algebraic mapping, then g(a) is a closed and bounded semi-algebraic set. Since a polynomial f (x,..., x n ) is a continuous semi-algebraic mapping and we assume a a closed and bounded semi-algebraic set S R n to be given, it follows that f (S) is a closed and bounded semi-algebraic set. But since f (S) R, it is a finite union of points and of closed and bounded intervals. And so f (S) has a maximum. J. Hafner & P. Mancosu, 2008, Beyond unification.