Diversity in Proof Appraisal
Matthew Inglis and Andrew Aberdein
Mathematics Education Centre
Loughborough University
[email protected]
homepages.lboro.ac.uk/∼mamji
School of Arts & Communication
Florida Institute of Technology
[email protected]
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
elementary
simple
original
short
direct
standard
formal
algebraic
complete
nice
usual
rigorous
new
easy
first
constructive
combinatorial
simpler
quick
geometric
theoretic
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
Raw Freq
% Freq
269
223
164
156
147
117
107
104
95
92
91
84
83
82
80
78
77
61
59
55
54
1.27
1.05
0.77
0.74
0.69
0.55
0.50
0.49
0.45
0.43
0.43
0.40
0.39
0.39
0.38
0.37
0.36
0.29
0.28
0.26
0.25
Cluster
bijective
full
general
alternative
detailed
slick
analytic
mathematical
elegant
classical
inductive
conceptual
correct
consistency
shortest
topological
beautiful
similar
probabilistic
published
valid
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
Raw Freq
% Freq
47
42
42
41
41
38
37
37
36
35
32
31
29
28
28
28
23
23
21
21
20
0.22
0.20
0.20
0.19
0.19
0.18
0.17
0.17
0.17
0.17
0.15
0.15
0.14
0.13
0.13
0.13
0.11
0.11
0.10
0.10
0.09
Most frequent adjectives for proofs on MathOverflow
Cluster
elementary
simple
original
short
direct
standard
formal
algebraic
complete
nice
usual
rigorous
new
easy
first
constructive
combinatorial
simpler
quick
geometric
theoretic
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
Raw Freq
% Freq
269
223
164
156
147
117
107
104
95
92
91
84
83
82
80
78
77
61
59
55
54
1.27
1.05
0.77
0.74
0.69
0.55
0.50
0.49
0.45
0.43
0.43
0.40
0.39
0.39
0.38
0.37
0.36
0.29
0.28
0.26
0.25
Cluster
bijective
full
general
alternative
detailed
slick
analytic
mathematical
elegant
classical
inductive
conceptual
correct
consistency
shortest
topological
beautiful
similar
probabilistic
published
valid
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
proof
Raw Freq
% Freq
47
42
42
41
41
38
37
37
36
35
32
31
29
28
28
28
23
23
21
21
20
0.22
0.20
0.20
0.19
0.19
0.18
0.17
0.17
0.17
0.17
0.15
0.15
0.14
0.13
0.13
0.13
0.11
0.11
0.10
0.10
0.09
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).
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?
An Analogy: Human Personalities
I
Very many adjectives are used to describe
human characteristics.
An Analogy: Human Personalities
I
I
Very many adjectives are used to describe
human characteristics.
For example, you might hear someone say
‘David is loud, talkative, excitable, outgoing,
shameless...’
An Analogy: Human Personalities
I
I
I
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?
An Analogy: Human Personalities
I
I
I
I
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.
An Analogy: Human Personalities
I
I
I
I
I
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.
The Big Five Personality Traits
I
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.
The Big Five Personality Traits
I
I
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).
The Big Five Personality Traits
I
I
I
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.
The Big Five
Openness to Experience:
inventive/curious vs consistent/cautious
The Big Five
Openness to Experience:
inventive/curious vs consistent/cautious
Conscientiousness:
efficient/organised vs easy-going/careless
The Big Five
Openness to Experience:
inventive/curious vs consistent/cautious
Conscientiousness:
efficient/organised vs easy-going/careless
Extraversion:
outgoing/energetic vs solitary/reserved
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
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 have
often been used to describe mathematical
proofs.
Original Study
I
I
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
“hadjectivei proof” and “mathematics”.
Original Study
I
I
I
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
“hadjectivei proof” and “mathematics”.
For example: 21,000 webpages contained the
phrase “conceptual proof” and “mathematics”;
1290 contained the phrase “obscure proof” and
“mathematics”.
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
Empirical Work
I
Participants were 255 research mathematicians
based in US universities (follow-up studies with
British, Irish and Australian participants give
similar results).
Empirical Work
I
I
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 email via their
department secretaries.
Empirical Work
I
I
I
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 email 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.
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.
I
I
I
We correlated each word’s
loadings on Component 2
Werating
correlated
with its mean
on
each
word’s
the five-point scale.
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
tend not
scale.
to think that
mathematical proofs are
‘crude’, ‘careless’,
‘shallow’ or ‘flimsy’.
Analysis
r = -.94
Loading on Component 2
Analysis
0.5
0
−0.5
2
3
Mean Rating (1-5)
4
Four Factors
striking
ingenious
inspired
profound
creative
deep
sublime
innovative
beautiful
elegant
charming
clever
bold
appealing
pleasing
enlightening
ambitious
delicate
insightful
strong
Four Factors
striking
ingenious
inspired
profound
creative
deep
sublime
innovative
beautiful
elegant
charming
clever
bold
appealing
pleasing
enlightening
ambitious
delicate
insightful
strong
Aesthetics
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
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
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
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
precise
careful
meticulous
rigorous
accurate
lucid
clear
Intricacy
Precision
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
precise
careful
meticulous
rigorous
accurate
lucid
clear
Intricacy
Precision
practical
efficient
applicable
informative
useful
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
precise
careful
meticulous
rigorous
accurate
lucid
clear
Intricacy
Precision
practical
efficient
applicable
informative
useful
Utility
Short Scale
ingenious
inspired
profound
striking
careless
crude
flimsy
shallow
Aesthetics Non-Use
dense
difficult
intricate
unpleasant
careful
meticulous
precise
rigorous
Intricacy
Precision
applicable
efficient
informative
practical
Utility
Short Scale
ingenious
inspired
profound
striking
careless
crude
flimsy
shallow
Aesthetics Non-Use
dense
difficult
intricate
not simple
careful
meticulous
precise
rigorous
Intricacy
Precision
applicable
useful
informative
practical
Utility
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.
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.
Exemplar Philosophers
Step 1: Offer an example of a proof or a mathematical object
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
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
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
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.
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, `) with P not on `, choose a pair
(P0 , `0 ) such that P0 has the smallest distance
to `0 , with Q being the point on `0 closest to
P0 (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 of
P, and thus two of them, say P1 and P2 , lie on
the same side of Q. Let us assume that P1 lies
between Q and P2 , where P1 possibly coincides
with Q. The figure below shows the
configuration. It follows that the distance of P1
to the line `1 determined by P0 and P2 is
smaller than the distance of P0 to `0 , and this
contradicts our choice for `0 and P0 .
`1
P0
`0
Q P
1
P2
Frequency
How participants rated the proof on the four dimensions
20
15
10
5
0
Aesthetics
20
15
10
5
0
Intricacy
20
15
10
5
0
Precision
20
15
10
5
0
Utility
4
6
8
10
12
14
Score (4 to 20)
16
18
20
Clusters
Mean Score (4 to 20)
20
Aesthetics
Intricacy
Precision
Utility
16
12
8
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.
Conclusions
I
Mathematical proofs have {personalities|.
Conclusions
I
I
Mathematical proofs have {personalities|.
They can be characterized, roughly speaking,
with four dimensions.
Conclusions
I
I
I
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.
Conclusions
I
I
I
I
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.
Conclusions
I
I
I
I
I
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
Future Work
I
Comparison of two proofs.
Future Work
I
I
Comparison of two proofs.
Non-mathematical aesthetic appraisal.
Future Work
I
I
I
Comparison of two proofs.
Non-mathematical aesthetic appraisal.
Other languages?
Diversity in Proof Appraisal
Matthew Inglis and Andrew Aberdein
Mathematics Education Centre
Loughborough University
[email protected]
homepages.lboro.ac.uk/∼mamji
School of Arts & Communication
Florida Institute of Technology
[email protected]
my.fit.edu/∼aberdein
Buffalo Annual Experimental Philosophy Conference,
September 19, 2014