Explanation and Explicationin Mathematical Practice

Andrew Aberdein∗ and Matthew Inglis†

∗School of Arts & CommunicationFlorida Institute of Technology

aberdein@fit.edu

my.fit.edu/∼aberdein

†Mathematics Education CentreLoughborough Universitym.j.inglis@lboro.ac.uk

homepages.lboro.ac.uk/∼mamji

APMP2, University of Illinois at Urbana-Champaign,October 3, 2013

Outline

Explanatoriness

Good Mathematics

Human Personalities

Empirical Work

Analysis

Explanation and Explication

Conclusion

When is a Mathematical Proof Explanatory?

1. Explanatory proofs turn on a characterizing property (MarkSteiner).

I A characterizing property is ‘unique to a given entity orstructure within a family or domain of such entities orstructures’.

I But characterizing properties are often very difficult to find:What’s the characterizing property in Euclid’s proof ofPythagoras’s Theorem?

2. Explanatory proofs unify—they derive a lot of material from alittle (Philip Kitcher, Michael Friedman).

I Unification entails minimizing the number of ‘argumentpatterns’ or distinctive forms of argument.

I But ‘nuclear flyswatter’ proofs end up rated as moreexplanatory than elegant combinations of distinct but simpletechniques, contrary to mathematical practice.

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.

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

Empirical Work

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 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

Aesthetics

When is a Mathematical Proof Said to be Explanatory?

I Explanatoriness is a multi-dimensional concept.

I ‘Explanatory’ loaded negatively onto intricacy, positively ontoprecision and positively (albeit weakly) onto utility:

aesthetics intricacy precision utility

explanatory 0.096 -0.314 0.472 0.228

I Suggests that proofs which are (i) not intricate, (ii) preciseand (iii) maybe useful, have a chance of being called‘explanatory’ by mathematicians.

I Doesn’t support either the Steiner ‘characterizing property’account, or the Kitcher/Friedman ‘unification’ account.

I What, if anything, does it support?

Explication

1. The explicatum [the thing which explicates] is to be similar tothe explicandum [the thing requiring explication] in such a waythat, in most cases in which the explicandum has so far beenused, the explicatum can be used; however, close similarity isnot required, and considerable differences are permitted.

2. The characterization of the explicatum, that is, the rules of itsuse (for instance, in the form of a definition), is to be given inan exact form, so as to introduce the explicatum into awell-connected system of scientific concepts.

3. The explicatum is to be a fruitful concept, that is, useful forthe formulation of many universal statements (empirical lawsin the case of a nonlogical concept, logical theorems in thecase of a logical concept).

4. The explicatum should be as simple as possible; this means assimple as the more important requirements (1), (2), (3)permit.

Rudolf Carnap, 1950, Logical Foundations of Probability. Routledge & Kegan Paul.

Proof as Explication?

I We have empirical evidence that proofs described asexplanatory are also seen to be simple (not intricate), exact(precise) and fruitful (useful).

I So, explanatory proofs are explications?I But in describing a proof as explicatory we are ascribing these

properties not to the proof as a whole, but to a concept usedin it. Doesn’t applying a property of the whole to a properpart commit the fallacy of division?

I No: Some properties of the whole do distribute over (essential)parts, apparently including simple, exact and fruitful.

I If a proof is simple, exact and fruitful, then concepts that areemployed essentially in the proof should also be simple, exactand fruitful.

I That is, explanatory proofs make essential use of explicata, inCarnap’s sense.

Proof as Explication?

I But explication is a process of replacement of explicanda byexplicata. Must proofs involve such replacement?

I Some do: the proof of V − E + F = 2, as discussed byLakatos, is a paradigm case of explicatory proof in this sense.

I Many don’t: What explicanda are replaced by explicata inEuclid’s proof of Pythagoras’s Theorem?

I Explanatory proofs contain explicata, but the explicationmight have gone on elsewhere.

I In other words, explanatory proofs are explicit.

Mathematics as Explication

‘I would like to claim that the basic similarity between philosophyand mathematics is the focus on the explication of informalconcepts’

Theo Kuipers, 2005, Cognitive Structures in Scientific Inquiry. Rodopi.

‘getting a clearly articulated grasp of the concepts is not merelyprerequisite for mathematical knowledge: it is the whole story’

Antony Eagle, 2008, Mathematics and conceptual analysis. Synthese, 161(1).

‘conceptual analysis, as exemplified by the famous Gettierprogramme in the analysis of knowledge, has the heuristic form ofproofs and refutations that Lakatos identifies’

Alexander Harper, 2012, An oblique epistemic defence of conceptual analysis.Metaphilosophy, 43(3).

Conclusion

I Mathematical proofs have {personalities|.

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

I Explanatoriness is a multi-dimensional concept.

I Those dimensions resemble the dimensions ofexplication, in Carnap’s sense.

I Explanatory proofs make essential use ofexplicata.