how do we explain mathematical explanation of scienti c facts? · 2011. 3. 18. · why one should...

Explanation Mathematical Explanations of Scientific Facts Steiner Baker Batterman Conclusions

How do we explain mathematicalexplanation of scientific facts?

Daniele MolininiRESHEIS, Paris

Mexico, March-April 08

Daniele Molinini RESHEIS, Paris How do we explain mathematical explanation of scientific facts?

A naıf definition

An explanation must teach something new, that was notknown before

I to teach: to give an information to someone in order to makethem think in a new or different way.→ How? Via numbers? Music? Empirical facts?

I something new : Behavior of a system? A fact about theworld? Fire is dangerous and I dont have to put my hand onit? Love also could be dangerous? What kind of information?

Only to the extent that we are able to explain empiricalfacts we can attain the major objective of scientificresearch, namely not merely to record the phenomena ofour experience but to learn from them 1

1Hempel and Oppenheim, Studies in the Logic of Explanation , 1948

A naıf definition

3 features of Explanation

3 general features of Explanation

I Gap between Knowledge and Understanding.

I Knowledge That: a descriptive knowledgeI Knowledge Why (Understanding): an explanatory knowledge 2

Knowledge of the latter type is explanatory. It is explanatoryknowledge that provides scientific understanding of our world 3

I Why regress: the possibility of having explanations that are notthemselves been explained

I Self-evidencing explanations: The possibility of explaining aphenomenon where the phenomenon itself provides an essential partof the reason for believing that the explanation is correct. Hexplains E while E justifies H.

2The explanation-seeking why-question Why did X occur? is different from the evidence-seeking why-question

Why one should believe that X occurred? Ex: Why Lev Trotsky died? 1) Ramon Mercader smashed the pick of anice axe into his skull 2) was reported in the press

3Kitcher Salmon 1989

I Gap between Knowledge and Understanding.I Knowledge That: a descriptive knowledgeI Knowledge Why (Understanding): an explanatory knowledge 2

Mathematical Explanation

Senses of Mathematical Explanation

Two different senses of Mathematical Explanation:

I Mathematical explanation of mathematical facts→ Informal and formal proofs within mathematics.

I Mathematical explanation of scientific facts→ Explanation in natural science as carried out by essentialappeal to mathematical facts.

Mathematical Explanation of Scientific Facts

An explanation must teach something new ...→ Explanation must teach something new about a physical factknown or not known before.→ How? Via Mathematics.The Concise Oxford Dictionary: to explain 4 means

1. Make clear or intellegible with detailed information

2. Account for

3. Minimize the significance of (a difficulty or a mistake) byexplanation

(a better definition): An explanation must make clear or intellegible aphenomenon with detailed information.

4The verb is borrowed from the classical latin verb explanare, which means “tomake plain, to flatten” ( ex -out-, planus -plain-)

Mathematical Explanation of Scientific Facts

An explanation must teach something new ...→ Explanation must teach something new about a physical factknown or not known before.→ How? Via Mathematics.The Concise Oxford Dictionary: to explain 4 means

1. Make clear or intellegible with detailed information

2. Account for

3. Minimize the significance of (a difficulty or a mistake) byexplanation

(a better definition): An explanation must make clear or intellegible aphenomenon with detailed information.

4The verb is borrowed from the classical latin verb explanare, which means “tomake plain, to flatten” ( ex -out-, planus -plain-)

Into the problem I: Why?

Mathematical Explanations of Scientific Facts ♠

I Renewed interactions between mathematics and physicsduring the XXth century. 5

I Impossibility to apply theories of Scientific Explanation for ♠.

In particular: ♠ seem to be counterexamples to the causal theoryof explanation.

The existence of mathematical explanations of naturalphenomena is widely recognized in the literature. However,until recently very little attention has been devoted to them ...In short, articulating how mathematics works in sciencerequires an account of how mathematics hooks on to reality,e.g. an account of the applicability of mathematics to reality 6

5Alasdair Urquhart, in Mancosu Paolo (Ed), The Philosophy of Mathematical Practice, forthcoming for OUP

6Mancosu Paolo, The Philosophy of Mathematical Practice

Into the problem II: How?

The arena: philosophy of mathematics and physics, methaphysics, history and

philosophy of science in general

The topic of explanation is deeply connected to the different topicsof application of mathematics to natural sciences, mathematizationand ontological questions which are well-expressed by the famousindispensability argument (IA).→ Scientific explanatory IA: If apparent reference to some entity(or class of entity) ξ is indispensable to our best scientific theories,then we ought to believe in the existence of ξ.7

Quine - Putnam IA: replace ξ with “mathematical entities”.−→ Debate: Nominalism vs Platonism.

7Colyvan, The Indispensability of Mathematics, p. 7

Into the problem III: What?

Application of Mathematics and Mathematization

I Application of Mathematics: a natural phenomenon that ascience is studying is described or explained usingmathematics, or when a problem the science aims to solve isanswered by means of mathematical techniques.

I Mathematization: the object of the science becomes amathematical object, i. e. mathematics provides a model or ascheme of a natural or social phenomenon and this model orscheme becomes the real object of studying. 8

8Panza, M., “Mathematisation of the Science of Motion and the Birth ofAnalytical Mechanics: A Historiographical Note”, 2002

Into the problem IV: Who?

I A first tentative in the analytic literature: Steiner’s account[Steiner, 1978b]9.

I a different account: Baker’s [Baker, 2005]10.

9“Mathematics, explanation and scientific knowledge”, Nous, 12, 17-2810“Are there genuine mathematical explanations of physical phenomena?”,

Mind, 114. 223-238Daniele Molinini RESHEIS, Paris How do we explain mathematical explanation of scientific facts?

Steiner’s questions

1. Do physical phenomena have mathematical explanations?

2. If so, what existential conclusions follow? Do suchexplanations make reasonable the existence of mathematicalentities?

He discusses a single example from the kinematics of rigid bodymotion: the motion of a rigid body about a fixed point.

Steiner’s example I: a case of application of mathematics

The motion of a rigid body about a fixed point

Figure: Unprimed axes represent an external reference set of axes; theprimed axes are fixed in the rigid body

Euler’s Theorem: The general displacement of a rigid bodywith one point fixed is a rotation about some axis 11.

11Goldstein, H., Classical Mechanics, 3rd edition, Addison Wesley

Steiner’s example II

At any instant the orientation of the body can be specified by anortoghonal tranformation. The matrix of this transformation isA.

What states the theorem?

The real orthogonal matrix specifying the physical motion of arigid body with one point fixed always has the eigenvalue +1

The matrix A must correspond to a proper rotation(det [A] = +1). There is a real vector x such that Ax = x ; andthis vector x is an axis of rotation.Steiner claims this is a mathematical explanation of a physical fact,though physical assumptions enter: that physical space isthree-dimensional. His point is that we have a mathematicalexplanation of a physical fact when we remove the physics and weare left with a mathematical explanation of a mathematical facts.

At any instant the orientation of the body can be specified by anortoghonal tranformation. The matrix of this transformation isA.What states the theorem?

The matrix A must correspond to a proper rotation(det [A] = +1). There is a real vector x such that Ax = x ; andthis vector x is an axis of rotation.

Steiner claims this is a mathematical explanation of a physical fact,though physical assumptions enter: that physical space isthree-dimensional. His point is that we have a mathematicalexplanation of a physical fact when we remove the physics and weare left with a mathematical explanation of a mathematical facts.

Steiner’s explanation ♠

Steiner:

I shall not reproduce my analysis of mathematical explanationhere, but assume that mathematical explanation ofmathematical truth exists.The difference betweenmathematical and physical explanations of physical phenomenais now amenable to analysis. In the former, as in the latter,physical and mathematical truths operate. But only inmathematical explanation is this the case: when we removethe physics, we remain with a mathematical explanation of amathematical truth! In our example, the “bridge” betweenphysics and mathematics is the assumptions that space isthreedimensional Euclidean, and that the rotation of a rigidbody around a point generates an orthogonal, real, propertransformation (to use the lingo). Deleting these assumptions,we obtain an explanatory proof of a theorem concerningtransformations and eigenvectors. In standard scientificexplanations, after deleting the physics nothing remains

Steiner’s answers

I “Do physical phenomena have mathematical explanations?”

→ Yes.

I “Do such explanations make reasonable the existence ofmathematical entities?”→ No. Mathematical explanation of physical phenomenacould not be used to infer the existence of mathematicalentities, for the existence of mathematical entities ispresupposed in the description of the phenomena to beexplained (explananda)12.

12this is very different, as Steiner points out, from the presupposition of lightquanta in the photoelectric effect.

Steiner’s answers

I “Do physical phenomena have mathematical explanations?”→ Yes.

Steiner’s answers

I “Do such explanations make reasonable the existence ofmathematical entities?”

→ No. Mathematical explanation of physical phenomenacould not be used to infer the existence of mathematicalentities, for the existence of mathematical entities ispresupposed in the description of the phenomena to beexplained (explananda)12.

Steiner’s answers

Remarques on Steiner’s account.

Testing Steiner’s account on the 3 features of explanation

I Gap Knowledge - Understanding? Why there is a fixed axis ofrotation? Why should I believe there is a fix axis of rot?Answer of Salmon for Sc. Expl: causal account doesn’t work.No causal connection here. Problematic.

I Benign Why regress? Yes. If we stop to eigenvalues and wedon’t touch ontological questions (Steiner’s question 2 goes inthis direction... ).

I self-evidencing explanation H (matrix of rotations) explains E(rotations of body) while E justifies H: OK. A sort of feedbackin explanation.

An observation

In Physics, II Aristotle distinguish between a substance as described in anexplanatory way (Polyclitus qua sculptor) and as not so described(Polyclitus qua pale man). Aristotle calls Polyclitus qua sculptor the perse cause of the statue, Polyclitus qua pale man the incidental cause ofthe statue. For Aristotle, per se causes are explanatory causes, whileincidental causes are non-explanatory.The thing that explains is the thing described in the terms under which itis desiderable to the agent.Polyclitus ! a mathematical object.Polyclitus qua sculptor ! orthogonal matrix qua orthogonaltransformations in R3.Polyclitus qua pale man ! orthogonal matrix qua just orthogonalmatrix.

Baker’s account I: a case of application of mathematics

Baker’s account: starting point

I Melya-Colyvan debate: theoretical role which mathematicsplays in science. Melya: mathematics must be indispensablein the right way.

I Focus on external applications of mathematics (in order toavoid circularity). When does the postulation of mathematicalobjects yield explanatory power?

I Are there genuine mathematical explanations of physicalphenomena?

Baker’s account II

Baker’s test case I

A case-study from evolutionary biology: explaining the life-cycle ofthe so called periodical cicada13.Three species of cicada of the genus Magicicada share the sameunusual life-cycle: in each specie the nymphal stage remains in thesoil for a lenghty period, then the adult cicada emerges after 13 or17 years depending on the geographical area.

13large fly-life insects distributed more in the northern states of the easternUnited States

Baker’s account III

Baker’s test case II

Five features of their life-cycle:

1. The great duration of the cicada life-cycle �

2. The presence of 2 separate life-cycle durations (within eachcicada species) in different regions �

3. The periodic emergence of adult cicadas ♦

4. The synchronized emergence of adult cicadas ♦

5. The prime-numbered-year cicada life-cycle lenghts. F

�: explicable in terms of ecological constraints.♦: explicable in terms of biological lawsF Why prime? No explanation

Baker IV

Magicicada...

Figure: A 17-year Periodical cicada

Baker’s account V

Baker’s test case

Two accounts for F (advantage of prime cycle periods), based on:

I avoiding predators 14

I ibridation with similar subspecies 15

The are two biological theories which share number theoreticalbases. The number theoretic theorem “prime periods minimizeintersection (compared to non-prime periods)” is essential to thestructure of the general explanation (which makes also use ofspecific ecological facts and general biological laws) and answers tothe particular question: ”Why the prime periods are evolutionarilyadvantageous?”

14Goles, E., Schulz, O. and M. Markus (2001). “Prime number selection of cycles in a predator-prey model”,

Complexity 6(4): pp. 33-38.15

Cox, Randel C E Carlton (1998), “A commentary on Prime Numbers and Life Cycle of Periodical Cicadas”,American Naturalist 152, pp. 162-4. Yoshimura, Jin (1997): “The Evolutionary Origins of Periodical Cicadasduring Ice Ages”, American Naturalist 152, pp. 112-124.

Baker’s account V

Baker’s test case

Two accounts for F (advantage of prime cycle periods), based on:

I avoiding predators 14

I ibridation with similar subspecies 15

The are two biological theories which share number theoreticalbases. The number theoretic theorem “prime periods minimizeintersection (compared to non-prime periods)” is essential to thestructure of the general explanation (which makes also use ofspecific ecological facts and general biological laws) and answers tothe particular question: ”Why the prime periods are evolutionarilyadvantageous?”

14Goles, E., Schulz, O. and M. Markus (2001). “Prime number selection of cycles in a predator-prey model”,

Complexity 6(4): pp. 33-38.15

Cox, Randel C E Carlton (1998), “A commentary on Prime Numbers and Life Cycle of Periodical Cicadas”,American Naturalist 152, pp. 162-4. Yoshimura, Jin (1997): “The Evolutionary Origins of Periodical Cicadasduring Ice Ages”, American Naturalist 152, pp. 112-124.

Is the cicada example a genuinely mathematicalexplanation?

Is the explanation a genuine explanation? The case-study of cicadais usefull to the platonist only if:

1. the application is external to mathematics

2. the phenomenon must be in need of explanation

3. the phenomenon must have been identified independently ofthe putative explanation

Baker: Ok. It is a genuine explanation, and it involves reference tomathematical objects, but: Is it a genuine mathematicalexplanation?

Baker’s test case

We’ve to check: Is the mathematical component of the explanationexplanatory in its own right? We adopt an account of explanation.

I A casual-account (explaining a phenomenon involves giving adescription of his causes): No. Problematic.

I we adopt the deductive-nomological model (D-N model)(constructing an inference of the phenomenon from premisseswhich include statements of general laws of nature): Ok.

I a pragmatic account (providing an answer to a why-questionwhich shows how the phenomenon is more likely than itsalternatives): Ok.

=⇒ Baker: we can assert that the mathematical component isexplanatory in its own right.

Baker’s test case

Baker’s conclusions

Baker’s conclusion

... there are genuine mathematical explanations ofphysical phenomena, and that the explanation of theprime cycle lenghts of periodical cicadas using numbertheory is one example of such. If this is right, thenapplying inference to the best explanation in the cicadaexample yields the conclusion that number exist.

Batterman’s account: new perpectives on explanation

Batterman’s general idea 16

Asymptotic reasoning: ignoring ot throwing away various details(explanatory noise...).Idea: Science procedes eliminating details and, in some sense,precision (asymptotic methods); the details of a system are, for themost part, unnecessary to the comprehension of the his behavior.In this sense, asymptotic methods play an explanatory andinterpretative role, expecially when we take into account propertieswhich emerge in the asymptotic domains between two theories (asort of no man’s land between two theories). They might play animportant role in the philosophical investigation of intra-theoreticaldomains and contexts of reduction.

16Batterman, R., The devil in the details, Oxford University Press, 2001

an example of Batterman’s Devil: Where it is?

Asymptotic Expanation: example: universality 17 of critical phenomenaprovided by renormalization group (RG) arguments in physics

I Different systems (magnets and fluids) with distinct microstructuresexibit the same behavior by the same critical exponents18

I An account of this universal behaviour is provided by RG agumentsin modern statistical physics.

I The RG transformations eliminates degree of freedom (microscopicdetails) that are inessential or irrelevant for characterizing thesystem’s behavior at criticality.

I The RG type analysis demonstrates that many of the details thatdistinguish the physical sistems from one another are irrelevant fortheir universal behavior. At the same time, it allows for thedetermination of those physical features that are relevant for thatbehavior.

17identical behavior in different physical systems

18numbers which characterize the behavior of a system near his critical point.

Asymptotic Expanation

The RG is a method for extracting structures that are, with respectto the behaviour of interest, detail independent. RG method allowsfor a construction of a kind of limiting macroscopicphenomenology -an upper level generalization- that is not tied toany unique microstructural account.

Remarque: importance of the mathematical component here (RGtransformations).Asymptotic Expanation: a strategy in which when any time onewants to explain some “upper level” generalization, one is trying toexplain a universal pattern of behaviour.

Asymptotic Expanation

The RG is a method for extracting structures that are, with respectto the behaviour of interest, detail independent. RG method allowsfor a construction of a kind of limiting macroscopicphenomenology -an upper level generalization- that is not tied toany unique microstructural account.Remarque: importance of the mathematical component here (RGtransformations).Asymptotic Expanation: a strategy in which when any time onewants to explain some “upper level” generalization, one is trying toexplain a universal pattern of behaviour.

a second example: the rainbow

I Certain features of the rainbows “emerge” in the asymptoticdomain as the wave theory approaches the ray theory in thelimit as the wavelenght of light approaches zero.

I The phenomenon inhabiting this borderland are notexplainable in purely wave theoretic or ray theoretic terms(failure of reducibility -irriducibility- is a necessary conditionfor emergence but does not necessarily entail a failure ofexplanability -because emergent properties are universal).

I A third explanatory theory (Batterman) is required for thisasymptotic domain.

I The new theory incorporates features of both the wave andray theories.

Batterman’s conclusion

Asymptotic reasoning offers better results in the comprehension of:

I Explanation

I Reduction

I Emergency

Conclusions and Perspectives

How do we explain Math. explanations of Scientific Facts?

I ∃ different types of mathematical explanation of scientificfacts.

I Familiar → Unfamiliar. Familiar → Familiar.I How acasual entities enter the explanation our scientific

theories?I role of analogies and models 19

I distinction of senses of explanation in mathematization andapplication of mathematics.

I the explanatory power of such entities depends to the statusof the entity in the description under consideration.

I Asymmetry in explanation (in physical laws asserting anumerical equivalence): if p explain q, does it follow that qexplain p? 20

19see Giorgio Israel on models: Israel, G., La mathematisation du reel. Essai sur la modelisation mathematique,

Paris, Editions du Seuil.20

An example of this is the case of the law of simple pendulum: from period T to lenght L ok, but could theprediction of L from T be considered an explanation of pendulum lenght?

I Familiar → Unfamiliar. Familiar → Familiar.

I How acasual entities enter the explanation our scientifictheories?

I role of analogies and models 19

New directions

I benefit from the interplay between history and philosophy ofmathematics

I the role of mathematical practice and the development of theconcept of explanation through the history of application ofmathematics and of mathematization

I Improving the ontological debate

I New case-studies (in which ♠ are know and unknown -as inBatterman example of rainbow-)

I Comparison between different conceptions of ScientificExplanation and expanation as ♠.

New directions

Acknoledgements

Carlos Alvarez and UNAM, ECOS, Equipe RESHEIS Paris, MarcoPanza, Davide Crippa.

how do we explain mathematical explanation of scienti c facts? · 2011. 3. 18. · why one should...

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