Diversity in Proof Appraisal

Matthew Inglis and Andrew Aberdein

Mathematics Education CentreLoughborough University

m.j.inglis@lboro.ac.uk

homepages.lboro.ac.uk/∼mamji

School of Arts & CommunicationFlorida Institute of Technologyaberdein@fit.edu

my.fit.edu/∼aberdein

Buffalo Annual Experimental Philosophy Conference,September 19, 2014

Outline

Good Mathematics

Human Personalities

Original Study

Short Scale

Against the Exemplar Philosophers

Conclusions

Most frequent adjectives for proofs on MathOverflow

Cluster Raw Freq % Freq

elementary proof 269 1.27simple proof 223 1.05

original proof 164 0.77short proof 156 0.74

direct proof 147 0.69standard proof 117 0.55

formal proof 107 0.50algebraic proof 104 0.49complete proof 95 0.45

nice proof 92 0.43usual proof 91 0.43

rigorous proof 84 0.40new proof 83 0.39easy proof 82 0.39first proof 80 0.38

constructive proof 78 0.37combinatorial proof 77 0.36

simpler proof 61 0.29quick proof 59 0.28

geometric proof 55 0.26theoretic proof 54 0.25

Cluster Raw Freq % Freq

bijective proof 47 0.22full proof 42 0.20

general proof 42 0.20alternative proof 41 0.19

detailed proof 41 0.19slick proof 38 0.18

analytic proof 37 0.17mathematical proof 37 0.17

elegant proof 36 0.17classical proof 35 0.17

inductive proof 32 0.15conceptual proof 31 0.15

correct proof 29 0.14consistency proof 28 0.13

shortest proof 28 0.13topological proof 28 0.13

beautiful proof 23 0.11similar proof 23 0.11

probabilistic proof 21 0.10published proof 21 0.10

valid proof 20 0.09

Most frequent adjectives for proofs on MathOverflow

Cluster Raw Freq % Freq

elementary proof 269 1.27simple proof 223 1.05

original proof 164 0.77short proof 156 0.74

direct proof 147 0.69standard proof 117 0.55

formal proof 107 0.50algebraic proof 104 0.49complete proof 95 0.45

nice proof 92 0.43usual proof 91 0.43

rigorous proof 84 0.40new proof 83 0.39easy proof 82 0.39first proof 80 0.38

constructive proof 78 0.37combinatorial proof 77 0.36

simpler proof 61 0.29quick proof 59 0.28

geometric proof 55 0.26theoretic proof 54 0.25

Cluster Raw Freq % Freq

bijective proof 47 0.22full proof 42 0.20

general proof 42 0.20alternative proof 41 0.19

detailed proof 41 0.19slick proof 38 0.18

analytic proof 37 0.17mathematical proof 37 0.17

elegant proof 36 0.17classical proof 35 0.17

inductive proof 32 0.15conceptual proof 31 0.15

correct proof 29 0.14consistency proof 28 0.13

shortest proof 28 0.13topological proof 28 0.13

beautiful proof 23 0.11similar proof 23 0.11

probabilistic proof 21 0.10published proof 21 0.10

valid proof 20 0.09

What is Good Mathematics?

‘the concept of mathematical quality is ahigh-dimensional one’

Terence Tao, 2007, What is good mathematics?Bulletin of the American Mathematical Society, 44(4).

How many dimensions?

What is Good Mathematics?

‘the concept of mathematical quality is ahigh-dimensional one’

Terence Tao, 2007, What is good mathematics?Bulletin of the American Mathematical Society, 44(4).

How many dimensions?

An Analogy: Human Personalities

I Very many adjectives are used to describehuman characteristics.

I For example, you might hear someone say‘David is loud, talkative, excitable, outgoing,shameless...’

I How many basic ways of characterizing a personare there?

I This sounds like it’s an impossibly complicatedquestion to answer.

I Actually it’s not. The answer’s five.

An Analogy: Human Personalities

I Very many adjectives are used to describehuman characteristics.

I For example, you might hear someone say‘David is loud, talkative, excitable, outgoing,shameless...’

I How many basic ways of characterizing a personare there?

I This sounds like it’s an impossibly complicatedquestion to answer.

I Actually it’s not. The answer’s five.

An Analogy: Human Personalities

I Very many adjectives are used to describehuman characteristics.

I For example, you might hear someone say‘David is loud, talkative, excitable, outgoing,shameless...’

I How many basic ways of characterizing a personare there?

I This sounds like it’s an impossibly complicatedquestion to answer.

I Actually it’s not. The answer’s five.

An Analogy: Human Personalities

I Very many adjectives are used to describehuman characteristics.

I For example, you might hear someone say‘David is loud, talkative, excitable, outgoing,shameless...’

I How many basic ways of characterizing a personare there?

I This sounds like it’s an impossibly complicatedquestion to answer.

I Actually it’s not. The answer’s five.

An Analogy: Human Personalities

I Very many adjectives are used to describehuman characteristics.

I For example, you might hear someone say‘David is loud, talkative, excitable, outgoing,shameless...’

I How many basic ways of characterizing a personare there?

I This sounds like it’s an impossibly complicatedquestion to answer.

I Actually it’s not. The answer’s five.

The Big Five Personality Traits

I Probably the greatest achievement of 20thcentury psychology was the discovery that thereare only five broad dimensions on which aperson’s personality varies.

I This was discovered by asking people to think ofa person, then rate how well a long long list ofadjectives described them, then looking at thecorrelations between these ratings (performing aPCA).

I Those characteristics which almost always gotogether are in some sense the samecharacteristic.

The Big Five Personality Traits

I Probably the greatest achievement of 20thcentury psychology was the discovery that thereare only five broad dimensions on which aperson’s personality varies.

I This was discovered by asking people to think ofa person, then rate how well a long long list ofadjectives described them, then looking at thecorrelations between these ratings (performing aPCA).

I Those characteristics which almost always gotogether are in some sense the samecharacteristic.

The Big Five Personality Traits

I Probably the greatest achievement of 20thcentury psychology was the discovery that thereare only five broad dimensions on which aperson’s personality varies.

I This was discovered by asking people to think ofa person, then rate how well a long long list ofadjectives described them, then looking at thecorrelations between these ratings (performing aPCA).

I Those characteristics which almost always gotogether are in some sense the samecharacteristic.

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

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

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

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

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

Original Study

I We created a list of eighty adjectives which haveoften been used to describe mathematicalproofs.

I Each had more than 250 hits on Google for“〈adjective〉 proof” and “mathematics”.

I For example: 21,000 webpages contained thephrase “conceptual proof” and “mathematics”;1290 contained the phrase “obscure proof” and“mathematics”.

Original Study

I We created a list of eighty adjectives which haveoften been used to describe mathematicalproofs.

I Each had more than 250 hits on Google for“〈adjective〉 proof” and “mathematics”.

I For example: 21,000 webpages contained thephrase “conceptual proof” and “mathematics”;1290 contained the phrase “obscure proof” and“mathematics”.

Original Study

I We created a list of eighty adjectives which haveoften been used to describe mathematicalproofs.

I Each had more than 250 hits on Google for“〈adjective〉 proof” and “mathematics”.

I For example: 21,000 webpages contained thephrase “conceptual proof” and “mathematics”;1290 contained the phrase “obscure proof” and“mathematics”.

Eighty Adjectivesdefinitiveclearsimplerigorousstrongstrikinggeneralnon-trivialelegantobviouspracticaltrivialintuitivenaturalconceptualabstractefficientcarefuleffectiveincomplete

preciseusefulbeautifulminimalunambiguousaccuratetediousambitiouselaborateweakingeniouscleverapplicablerobustsharpintricateloosepleasingsketchydull

innovativecuteworthlessexplanatoryplausibleillustrativecreativeinsightfuldeepprofoundawfuluglyspeculativeconfusingdenseexpositorylucidobscuredelicatemeticulous

subtleclumsyflimsyinformativecrudeappealingcarelessenlighteninginspiredboldpolishedcharmingunpleasantsublimeawkwardexploratoryinefficientshallowfruitfuldisgusting

Empirical Work

I Participants were 255 research mathematiciansbased in US universities (follow-up studies withBritish, Irish and Australian participants givesimilar results).

I Asked to participate by email via theirdepartment secretaries.

I Participants asked to pick a proof they’drecently read or refereed and to state howaccurately each of our 80 adjectives described it.

Empirical Work

I Participants were 255 research mathematiciansbased in US universities (follow-up studies withBritish, Irish and Australian participants givesimilar results).

I Asked to participate by email via theirdepartment secretaries.

I Participants asked to pick a proof they’drecently read or refereed and to state howaccurately each of our 80 adjectives described it.

Empirical Work

I Participants were 255 research mathematiciansbased in US universities (follow-up studies withBritish, Irish and Australian participants givesimilar results).

I Asked to participate by email via theirdepartment secretaries.

I Participants asked to pick a proof they’drecently read or refereed and to state howaccurately each of our 80 adjectives described it.

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 email through their departments.

Analysis

Component 2 seemed to be those words with low ratings.

Analysis

I We correlated each word’sloadings on Component 2with its mean rating onthe five-point scale.

I Suggests that Component2 was just a measure ofnon-use.

I In other words,mathematicians tend notto think thatmathematical proofs are‘crude’, ‘careless’,‘shallow’ or ‘flimsy’.

Analysis

We correlated each word’s loadings on Component 2 with its mean rating on the five-point scale.

r = -.94

Load

ing

on C

ompo

nent

2!0.5

0

0.5

Mean Rating (1-5)2 3 4

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Four Factors

strikingingeniousinspiredprofoundcreativedeepsublimeinnovativebeautifulelegantcharmingcleverboldappealingpleasingenlighteningambitiousdelicateinsightfulstrong

densedifficultintricateunpleasantconfusingtediousnot simple

Intricacy

precisecarefulmeticulousrigorousaccuratelucidclear

Precision

practicalefficientapplicableinformativeuseful

Utility

Aesthetics

Short Scale

ingeniousinspiredprofoundstriking

Aesthetics

carelesscrudeflimsyshallow

Non-Use

densedifficultintricateunpleasant

Intricacy

carefulmeticulouspreciserigorous

Precision

applicableefficientinformativepractical

Utility

Short Scale

ingeniousinspiredprofoundstriking

Aesthetics

carelesscrudeflimsyshallow

Non-Use

densedifficultintricatenot simple

Intricacy

carefulmeticulouspreciserigorous

Precision

applicableusefulinformativepractical

Utility

Empirical Semantics and the Oslo Group

I think that even superficial questioning of nonphilosophers makesit hard for anyone to believe that the philosopher has got his‘knowledge’ about peasants’ and others’ use of the word true—orabout the views of nonphilosophers on the notion of truth—byasking 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. Theanalysis might also include a criticism (unfavourable) of theaccessible or potential, but frequently less successful attempts ofother authors. Or the analyst may back up his hypotheses of usageby quotations which may be interpreted in such away that theydirectly or indirectly are supporting his ideas.

Herman Tonnessen, 1951, The fight against revelation insemantical studies, Synthese, 8.

Empirical Semantics and the Oslo Group

I think that even superficial questioning of nonphilosophers makesit hard for anyone to believe that the philosopher has got his‘knowledge’ about peasants’ and others’ use of the word true—orabout the views of nonphilosophers on the notion of truth—byasking 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. Theanalysis might also include a criticism (unfavourable) of theaccessible or potential, but frequently less successful attempts ofother authors. Or the analyst may back up his hypotheses of usageby quotations which may be interpreted in such away that theydirectly or indirectly are supporting his ideas.

Herman Tonnessen, 1951, The fight against revelation insemantical studies, Synthese, 8.

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 analyzablethis way.’

Steiner, 1978, Mathematical explanation, Philosophical Studies, 34.

‘we wish to propose a proof that meets Steiner’s criterion butdoesn’t explain and one which ought to explain if any proof doesbut fails to meet Steiner’s criterion.’

Resnik & Kushner, 1987, Explanation, independence and realism in mathematics,British Journal of the Philosophy of Science, 38.

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 analyzablethis way.’

Steiner, 1978, Mathematical explanation, Philosophical Studies, 34.

‘we wish to propose a proof that meets Steiner’s criterion butdoesn’t explain and one which ought to explain if any proof doesbut fails to meet Steiner’s criterion.’

Resnik & Kushner, 1987, Explanation, independence and realism in mathematics,British Journal of the Philosophy of Science, 38.

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 analyzablethis way.’

Steiner, 1978, Mathematical explanation, Philosophical Studies, 34.

‘we wish to propose a proof that meets Steiner’s criterion butdoesn’t explain and one which ought to explain if any proof doesbut fails to meet Steiner’s criterion.’

Resnik & Kushner, 1987, Explanation, independence and realism in mathematics,British Journal of the Philosophy of Science, 38.

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 analyzablethis way.’

Steiner, 1978, Mathematical explanation, Philosophical Studies, 34.

‘we wish to propose a proof that meets Steiner’s criterion butdoesn’t explain and one which ought to explain if any proof doesbut fails to meet Steiner’s criterion.’

Resnik & Kushner, 1987, Explanation, independence and realism in mathematics,British Journal of the Philosophy of Science, 38.

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 analyzablethis way.’

Steiner, 1978, Mathematical explanation, Philosophical Studies, 34.

‘we wish to propose a proof that meets Steiner’s criterion butdoesn’t explain and one which ought to explain if any proof doesbut fails to meet Steiner’s criterion.’

Resnik & Kushner, 1987, Explanation, independence and realism in mathematics,British Journal of the Philosophy of Science, 38.

A Proof from The BookTheorem. In any configuration of n points inthe plane, not all on a line, there is a line whichcontains exactly two of the points.

Proof. Let P be the given set of points andconsider the set L of all lines which passthrough at least two points of P. Among allpairs (P, `) with P not on `, choose a pair(P0, `0) such that P0 has the smallest distanceto `0, with Q being the point on `0 closest toP0 (that is, on the line through P0 vertical to`0).

Claim: This line `0 does it!

If not, then `0 contains at least three points ofP, and thus two of them, say P1 and P2, lie onthe same side of Q. Let us assume that P1 liesbetween Q and P2, where P1 possibly coincideswith Q. The figure below shows theconfiguration. It follows that the distance of P1

to the line `1 determined by P0 and P2 issmaller than the distance of P0 to `0, and thiscontradicts our choice for `0 and P0.

`0

`1

P2

P0

P1Q

How participants rated the proof on the four dimensions

Aesthetics

Intricacy

Precision

Utility

Freq

uenc

y

05

101520

05

101520

05

101520

05

101520

Score (4 to 20)4 6 8 10 12 14 16 18 20

Clusters

AestheticsIntricacyPrecisionUtility

Mea

n Sc

ore (

4 to

20)

4

8

12

16

20

ClusterCluster 1 Cluster 2 Cluster 3

The mean ratings on each dimension of the three clusters. Error bars

show ±1 SE of the mean.

Conclusions

I Mathematical proofs have {personalities|.

I They can be characterized, roughly speaking,with four dimensions.

I There was no between-mathematicianagreement on any dimension for the proof weinvestigated.

I ‘Exemplar philosophers’ rely upon their ownintuitions or those of individual mathematiciansabout the qualities of proofs.

I There is no empirical support that there is aconsensus behind these intuitions

Conclusions

I Mathematical proofs have {personalities|.

I They can be characterized, roughly speaking,with four dimensions.

I There was no between-mathematicianagreement on any dimension for the proof weinvestigated.

I ‘Exemplar philosophers’ rely upon their ownintuitions or those of individual mathematiciansabout the qualities of proofs.

I There is no empirical support that there is aconsensus behind these intuitions

Conclusions

I Mathematical proofs have {personalities|.

I They can be characterized, roughly speaking,with four dimensions.

I There was no between-mathematicianagreement on any dimension for the proof weinvestigated.

I ‘Exemplar philosophers’ rely upon their ownintuitions or those of individual mathematiciansabout the qualities of proofs.

I There is no empirical support that there is aconsensus behind these intuitions

Conclusions

I Mathematical proofs have {personalities|.

I They can be characterized, roughly speaking,with four dimensions.

I There was no between-mathematicianagreement on any dimension for the proof weinvestigated.

I ‘Exemplar philosophers’ rely upon their ownintuitions or those of individual mathematiciansabout the qualities of proofs.

I There is no empirical support that there is aconsensus behind these intuitions

Conclusions

I Mathematical proofs have {personalities|.

I They can be characterized, roughly speaking,with four dimensions.

I There was no between-mathematicianagreement on any dimension for the proof weinvestigated.

I ‘Exemplar philosophers’ rely upon their ownintuitions or those of individual mathematiciansabout the qualities of proofs.

I There is no empirical support that there is aconsensus behind these intuitions

Future Work

I Comparison of two proofs.

I Non-mathematical aesthetic appraisal.

I Other languages?

Future Work

I Comparison of two proofs.

I Non-mathematical aesthetic appraisal.

I Other languages?

Future Work

I Comparison of two proofs.

I Non-mathematical aesthetic appraisal.

I Other languages?

Diversity in Proof Appraisal

Matthew Inglis and Andrew Aberdein

Mathematics Education CentreLoughborough University

m.j.inglis@lboro.ac.uk

homepages.lboro.ac.uk/∼mamji

School of Arts & CommunicationFlorida Institute of Technologyaberdein@fit.edu

my.fit.edu/∼aberdein

Buffalo Annual Experimental Philosophy Conference,September 19, 2014