Diversity in Proof Appraisal
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1 Diversity in Proof Appraisal Matthew Inglis and Andrew Aberdein Mathematics Education Centre Loughborough University homepages.lboro.ac.uk/ mamji School of Arts & Communication Florida Institute of Technology my.fit.edu/ aberdein Buffalo Annual Experimental Philosophy Conference, September 19, 2014
2 Outline Good Mathematics Human Personalities Original Study Short Scale Against the Exemplar Philosophers Conclusions
3 Most frequent adjectives for proofs on MathOverflow Cluster Raw Freq % Freq elementary proof simple proof original proof short proof direct proof standard proof formal proof algebraic proof complete proof nice proof usual proof rigorous proof new proof easy proof first proof constructive proof combinatorial proof simpler proof quick proof geometric proof theoretic proof Cluster Raw Freq % Freq bijective proof full proof general proof alternative proof detailed proof slick proof analytic proof mathematical proof elegant proof classical proof inductive proof conceptual proof correct proof consistency proof shortest proof topological proof beautiful proof similar proof probabilistic proof published proof valid proof
4 Most frequent adjectives for proofs on MathOverflow Cluster Raw Freq % Freq elementary proof simple proof original proof short proof direct proof standard proof formal proof algebraic proof complete proof nice proof usual proof rigorous proof new proof easy proof first proof constructive proof combinatorial proof simpler proof quick proof geometric proof theoretic proof Cluster Raw Freq % Freq bijective proof full proof general proof alternative proof detailed proof slick proof analytic proof mathematical proof elegant proof classical proof inductive proof conceptual proof correct proof consistency proof shortest proof topological proof beautiful proof similar proof probabilistic proof published proof valid proof
5 What is Good Mathematics? the concept of mathematical quality is a high-dimensional one Terence Tao, 2007, What is good mathematics? Bulletin of the American Mathematical Society, 44(4).
6 What is Good Mathematics? the concept of mathematical quality is a high-dimensional one Terence Tao, 2007, What is good mathematics? Bulletin of the American Mathematical Society, 44(4). How many dimensions?
7 An Analogy: Human Personalities Very many adjectives are used to describe human characteristics.
8 An Analogy: Human Personalities Very many adjectives are used to describe human characteristics. For example, you might hear someone say David is loud, talkative, excitable, outgoing, shameless...
9 An Analogy: Human Personalities Very many adjectives are used to describe human characteristics. For example, you might hear someone say David is loud, talkative, excitable, outgoing, shameless... How many basic ways of characterizing a person are there?
10 An Analogy: Human Personalities Very many adjectives are used to describe human characteristics. For example, you might hear someone say David is loud, talkative, excitable, outgoing, shameless... How many basic ways of characterizing a person are there? This sounds like it s an impossibly complicated question to answer.
11 An Analogy: Human Personalities Very many adjectives are used to describe human characteristics. For example, you might hear someone say David is loud, talkative, excitable, outgoing, shameless... How many basic ways of characterizing a person are there? This sounds like it s an impossibly complicated question to answer. Actually it s not. The answer s five.
12 The Big Five Personality Traits Probably the greatest achievement of 20th century psychology was the discovery that there are only five broad dimensions on which a person s personality varies.
13 The Big Five Personality Traits Probably the greatest achievement of 20th century psychology was the discovery that there are only five broad dimensions on which a person s personality varies. This was discovered by asking people to think of a person, then rate how well a long long list of adjectives described them, then looking at the correlations between these ratings (performing a PCA).
14 The Big Five Personality Traits Probably the greatest achievement of 20th century psychology was the discovery that there are only five broad dimensions on which a person s personality varies. This was discovered by asking people to think of a person, then rate how well a long long list of adjectives described them, then looking at the correlations between these ratings (performing a PCA). Those characteristics which almost always go together are in some sense the same characteristic.
15 The Big Five Openness to Experience: inventive/curious vs consistent/cautious
16 The Big Five Openness to Experience: inventive/curious vs consistent/cautious Conscientiousness: efficient/organised vs easy-going/careless
17 The Big Five Openness to Experience: inventive/curious vs consistent/cautious Conscientiousness: efficient/organised vs easy-going/careless Extraversion: outgoing/energetic vs solitary/reserved
18 The Big Five Openness to Experience: inventive/curious vs consistent/cautious Conscientiousness: efficient/organised vs easy-going/careless Extraversion: outgoing/energetic vs solitary/reserved Agreeableness: compassionate/friendly vs cold/unkind
19 The Big Five Openness to Experience: inventive/curious vs consistent/cautious Conscientiousness: efficient/organised vs easy-going/careless Extraversion: outgoing/energetic vs solitary/reserved Agreeableness: compassionate/friendly vs cold/unkind Neuroticism: sensitive/nervous vs secure/confident
20 Original Study We created a list of eighty adjectives which have often been used to describe mathematical proofs.
21 Original Study We created a list of eighty adjectives which have often been used to describe mathematical proofs. Each had more than 250 hits on Google for adjective proof and mathematics.
22 Original Study We created a list of eighty adjectives which have often been used to describe mathematical proofs. Each had more than 250 hits on Google for adjective proof and mathematics. For example: 21,000 webpages contained the phrase conceptual proof and mathematics ; 1290 contained the phrase obscure proof and mathematics.
23 Eighty Adjectives definitive clear simple rigorous strong striking general non-trivial elegant obvious practical trivial intuitive natural conceptual abstract efficient careful effective incomplete precise useful beautiful minimal unambiguous accurate tedious ambitious elaborate weak ingenious clever applicable robust sharp intricate loose pleasing sketchy dull innovative cute worthless explanatory plausible illustrative creative insightful deep profound awful ugly speculative confusing dense expository lucid obscure delicate meticulous subtle clumsy flimsy informative crude appealing careless enlightening inspired bold polished charming unpleasant sublime awkward exploratory inefficient shallow fruitful disgusting
24 Empirical Work Participants were 255 research mathematicians based in US universities (follow-up studies with British, Irish and Australian participants give similar results).
25 Empirical Work Participants were 255 research mathematicians based in US universities (follow-up studies with British, Irish and Australian participants give similar results). Asked to participate by via their department secretaries.
26 Empirical Work Participants were 255 research mathematicians based in US universities (follow-up studies with British, Irish and Australian participants give similar results). Asked to participate by via their department secretaries. Participants asked to pick a proof they d recently read or refereed and to state how accurately each of our 80 adjectives described it.
27 Empirical Work Empirical Work We then asked 255 mathematicians to pick a proof that they d recently read or refereed and to rate how well each adjective described it (5 point scale from very inaccurate to very accurate ). Participants were US-based research mathematicians contacted by through their departments.
28 Analysis Component 2 seemed to be those words with low ratings.
29 Analysis Analysis We correlated each word s loadings on Component 2 with its mean We rating correlated on the five-point each scale. word s loadings on Suggests that Component Component 2 2 was just a measure of with its mean non-use. rating on the In other words, five-point mathematicians scale. tend not to think that mathematical proofs are crude, careless, shallow or flimsy. Loading on Component r = Mean Rating (1-5)
30 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong
31 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics
32 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics dense difficult intricate unpleasant confusing tedious not simple
33 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics dense difficult intricate unpleasant confusing tedious not simple Intricacy
34 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics dense difficult intricate unpleasant confusing tedious not simple Intricacy precise careful meticulous rigorous accurate lucid clear
35 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics dense difficult intricate unpleasant confusing tedious not simple Intricacy precise careful meticulous rigorous accurate lucid clear Precision
36 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics dense difficult intricate unpleasant confusing tedious not simple Intricacy precise careful meticulous rigorous accurate lucid clear Precision practical efficient applicable informative useful
37 Four Factors striking ingenious inspired profound creative deep sublime innovative beautiful elegant charming clever bold appealing pleasing enlightening ambitious delicate insightful strong Aesthetics dense difficult intricate unpleasant confusing tedious not simple Intricacy precise careful meticulous rigorous accurate lucid clear Precision practical efficient applicable informative useful Utility
38 Short Scale ingenious inspired profound striking careless crude flimsy shallow dense difficult intricate unpleasant careful meticulous precise rigorous applicable efficient informative practical Aesthetics Non-Use Intricacy Precision Utility
39 Short Scale ingenious inspired profound striking careless crude flimsy shallow dense difficult intricate not simple careful meticulous precise rigorous applicable useful informative practical Aesthetics Non-Use Intricacy Precision Utility
40 Empirical Semantics and the Oslo Group I think that even superficial questioning of nonphilosophers makes it hard for anyone to believe that the philosopher has got his knowledge about peasants and others use of the word true or about the views of nonphilosophers on the notion of truth by asking any other person than himself. Arne Naess, 1938, Common sense and truth, Theoria, 4.
41 Empirical Semantics and the Oslo Group I think that even superficial questioning of nonphilosophers makes it hard for anyone to believe that the philosopher has got his knowledge about peasants and others use of the word true or about the views of nonphilosophers on the notion of truth by asking any other person than himself. Arne Naess, 1938, Common sense and truth, Theoria, 4. The analyst or investigator makes a single subject, namely himself, object of an investigation and records the ideas immediately. The analysis might also include a criticism (unfavourable) of the accessible or potential, but frequently less successful attempts of other authors. Or the analyst may back up his hypotheses of usage by quotations which may be interpreted in such away that they directly or indirectly are supporting his ideas. Herman Tönnessen, 1951, The fight against revelation in semantical studies, Synthese, 8.
42 Exemplar Philosophers Step 1: Offer an example of a proof or a mathematical object
43 Exemplar Philosophers Step 1: Offer an example of a proof or a mathematical object Step 2: Assert that the proof or object has a given property
44 Exemplar Philosophers Step 1: Offer an example of a proof or a mathematical object Step 2: Assert that the proof or object has a given property Step 3: Appeal to the readers intuitions for agreement
45 Exemplar Philosophers Step 1: Offer an example of a proof or a mathematical object Step 2: Assert that the proof or object has a given property Step 3: Appeal to the readers intuitions for agreement
46 Exemplar Philosophers Step 1: Offer an example of a proof or a mathematical object Step 2: Assert that the proof or object has a given property Step 3: Appeal to the readers intuitions for agreement A final example of an explanatory proof Our examples of explanation in mathematics are all analyzable this way. Steiner, 1978, Mathematical explanation, Philosophical Studies, 34. we wish to propose a proof that meets Steiner s criterion but doesn t explain and one which ought to explain if any proof does but fails to meet Steiner s criterion. Resnik & Kushner, 1987, Explanation, independence and realism in mathematics, British Journal of the Philosophy of Science, 38.
47 A Proof from The Book Theorem. In any configuration of n points in the plane, not all on a line, there is a line which contains exactly two of the points. Proof. Let P be the given set of points and consider the set L of all lines which pass through at least two points of P. Among all pairs (P, l) with P not on l, choose a pair (P 0, l 0 ) such that P 0 has the smallest distance to l 0, with Q being the point on l 0 closest to P 0 (that is, on the line through P 0 vertical to l 0 ). l 1 P 0 Claim: This line l 0 does it! l 0 If not, then l 0 contains at least three points of P, and thus two of them, say P 1 and P 2, lie on the same side of Q. Let us assume that P 1 lies between Q and P 2, where P 1 possibly coincides with Q. The figure below shows the configuration. It follows that the distance of P 1 to the line l 1 determined by P 0 and P 2 is smaller than the distance of P 0 to l 0, and this contradicts our choice for l 0 and P 0. Q P 1 P 2
48 How participants rated the proof on the four dimensions Frequency Aesthetics Intricacy Precision Utility Score (4 to 20)
49 Clusters Mean Score (4 to 20) Aesthetics Intricacy Precision Utility 4 Cluster 1 Cluster 2 Cluster 3 Cluster The mean ratings on each dimension of the three clusters. Error bars show ±1 SE of the mean.
50 Conclusions Mathematical proofs have personalities.
51 Conclusions Mathematical proofs have personalities. They can be characterized, roughly speaking, with four dimensions.
52 Conclusions Mathematical proofs have personalities. They can be characterized, roughly speaking, with four dimensions. There was no between-mathematician agreement on any dimension for the proof we investigated.
53 Conclusions Mathematical proofs have personalities. They can be characterized, roughly speaking, with four dimensions. There was no between-mathematician agreement on any dimension for the proof we investigated. Exemplar philosophers rely upon their own intuitions or those of individual mathematicians about the qualities of proofs.
54 Conclusions Mathematical proofs have personalities. They can be characterized, roughly speaking, with four dimensions. There was no between-mathematician agreement on any dimension for the proof we investigated. Exemplar philosophers rely upon their own intuitions or those of individual mathematicians about the qualities of proofs. There is no empirical support that there is a consensus behind these intuitions
55 Future Work Comparison of two proofs.
56 Future Work Comparison of two proofs. Non-mathematical aesthetic appraisal.
57 Future Work Comparison of two proofs. Non-mathematical aesthetic appraisal. Other languages?
58 Diversity in Proof Appraisal Matthew Inglis and Andrew Aberdein Mathematics Education Centre Loughborough University homepages.lboro.ac.uk/ mamji School of Arts & Communication Florida Institute of Technology my.fit.edu/ aberdein Buffalo Annual Experimental Philosophy Conference, September 19, 2014
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