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Readings on Explanation

PHIL 484

Table of contents
  1. Scientific Explanation (SEP)
  2. Mathematical Explanation (SEP)

Scientific Explanation (SEP)


Background and Introduction

  • What is ‘scientific explanation’?
    • What is ‘scientific’ about an explanation?
    • What makes science ‘explanatory’?
  • It is possible for claims to be true and unexplanatory.
  • Description vs explanation.
  • Are all scientific explanations causal?
  • Causal vs non-causal explanations

The DN Model

  • Deductive-Nomological Model: a scientific explanation has an explanandum and an explanans
    • Explanandum: describing the phenomenon to be explained
    • Expanans: the class of sentences adduced to account for the phenomenon
  • The explanation is a sound deductive argument which follows from the explanans (deductive)
  • The explanans must have at least one law of nature (essential premise) (nomological)
  • What is a law of nature? SOme generalizations are laws and others are accidentally true.
  • Generalizations which are laws can explain why… maybe?
  • Are laws exceptionless generalizations?
  • DN captures explanation via deduction. What about statistical laws?
  • Deductive-statistical explanation: deduction of narrower statistical uniformity.
  • Inductive-Statistical explanation: explanandum and explanans relationship is inductive
  • Which should all or some explanations have DN or IS structure?
  • We need to show that the phenomenon resulted from particular circumstances.

“the essence of scientific explanation can be described as nomic expectability—that is expectability on the basis of lawful connections.”

  • Role of causal claims in scientific explanation: are all explanations causal?
  • All causal claims imply some sort of regularity – law
  • Explanations should satisfy the DN/IS model because of the requirement of causal claims.
  • DN as necessary vs sufficient conditions for explanations
  • Is explanation asymmetric? You can only go in one direction of initial conditions towards an outcome
  • A derivation can satisfy DN and still contain irrelivances
  • DN does not even state necessary conditions for successful explanation – maybe.

The SR Model

  • Wesley Salmon’s statistical relevance model.
  • Class/population \(A\); attribute \(C\) is statistically relevant to another attribute \(B\) iff \(P(B \mid A \wedge C) \neq P(B \mid A)\)
  • Statistically relevnat properties are explanatory.
  • Homogenous partition: a set of subclasses which are mutually exclusive adn exhausted. Allows us to characterize SR mdoels more precisely.
  • An explanation is not an argument (vs. DN/IS). It is an assembly of statistically relevant information.
  • Irrelevancies are harmless in arguments but fatal in explanations – Salmon
  • Once an SR model has been constructed, no other factors can distinguish outcomes.
  • Generally – what do statistical theories explain?
  • Does it make sense to use a single model to capture all examples?
  • A more radical question: does it even make sense to think statistically to begin with?
  • Explanations should cite causal relationships

The Casual Mechanical Model

  • Causal process theories of causation – overcome facts of statistical relevance
  • Causal process: a physical process which transmits a mark in a continuous way.
  • A mark is a local modification on the structure of a process.
  • Marks persist at other spatio-temporal locations even without more interaction
  • Pseudo-processes cannot transmit marks.
  • Genuine causal processes must structurally leave marks which retain over time.
  • A process can be causal even if it does not transmit any mark.
  • Causal interaction. A causal interaction involves a spatio-temporal intersection between causal processes which changes the structure of both
  • CM leaves out causal relationships which exist between properties and magnitude rather than strictly processes and interactions
  • Mark transmission may not help us understanding which features of the causal process are causally relevant.
  • Generalization about the distribution of a large quantity of interactions – ‘radical abstraction’
  • Causal explanation which avoids appeals to counterfactuals
  • Causal process: a process which transmits non-zero conserved quantity at each moment in history.

A Unificationist Account of Explanation

  • Scientific explanation is about providing a unified account of different phenomena.
  • Does unification allow us to better understand what explanations are good?
  • Schematic sentence: nonlogical vocabulary has been substituted for predicates. Filling instructions sepcify how to replace predicates in dummy sentences. Schematic arguments – sequences of schematic sentences.
  • Kitcher: an explanation is about deriving descriptions for as many phenomena as possible with as few argument patterns as possible. How to reduce certain facts as ultimate? How to reach the explanatory store?
  • What is the origin of the theory and its development? Origin & development pattern of explanation
  • Unificationism does not require all explanations to be deductive.
  • ‘because’ in causation is always derivative of ‘because’ in explanation. Causal judgements simply reflect explanatory relationships
  • Epistemic inaccessibility of causal claims

Pragmatic Theories of Explanation

  • Pragmatic theories of explanation:
    • Irreducible reference to facts about interests, beliefs, features of psychology
    • Irreducible reference to the context in which the explanation occurs
  • What is the role of pragmatic elements in theories of explanation?
  • There is thought to be a non-pragmatic core to the notion of explanation, without reference to pscyhology.
  • How much does explanation ahve a pragmatic dimension?
  • Stronger claim of pragmatic approaches: constructing a model of explanation which is abstractly universe is unsuccessfuly because of the ineliminability of pragmatic and contextual factors. A diagnosis of the failure of accounts.
  • Pragmatic is generally associated with the consideration of pyschology and other attempts to localize models of explanation.
  • Pragmatism may not be in conflict with the traditional goals of the theory of explanation
  • Bas van Fraassen: the aim of science is the construction of empirically adequate theories, not to tell true stories about unobservables. Explanations are answers to questions. What is a why question? – queries about why a particular explanandum rather than any other member of a contrast class.
  • Explanation is not a relation (between theory and fact), but between theory, fact, and context. Being an explanation is relative, because an explanation is an answer, to a question.
  • Where is the source of asymmetry in the flagpole vs shadow case? In nature or in the psychology of the viewer.
  • Perhaps the relevance relation here is too broad. A relevance relation can be held for any two true propositions
  • However van Fraassen’s account still leaves us with an unexciting and uninteresting theory of explanation.

Conclusions, Open Issues

  • The role of causation – how interested are we in causal forms of scientific explanation?
  • Why takl about explanation at all?
  • What sorts of distinctions matter?W hat is relevant?
  • Are there forms of why-explanations which are non-causal?
  • Is there a single general model of causation? Or is it precluded

Questions and Notes

  • Unificationism seems to be related to the impulse to identify purpose or reason against coincidence.
  • General question, more of a meta-philosophical one pointing towards a pragmatist view: what is the point of defining explanation and what are properties of a good model of explanation? And is this neccessarily an explanation of explanation?
  • Rephrased: Perhaps explanation cannot be determined because from a meta-philosophical perspective to determine what explanation is we must produce an explanation, that is of what explanation is. And so it is fundamentally recursive and circular.
    • Are descriptions explanations? Possibly. Explanations apply too in reasoning and argumentation.
    • Are models of explanation self-consistent in their justification? Maybe.
    • Are all models of explanation inductively formed and validated?

Mathematical Explanation (SEP)


Mathematical explanations in the natural sciences

  • How to understand the appearance of mathematical phenomena in science?
  • How to distinguish from the mathematical and physical components of an explanation?
  • Dominant accounts of scientici explanation are causal
  • What about non-causal explanations?

The explanatory role of mathematics in science: some historical remarks

  • How do we obtain scientific knowledge? Through demonstration.
  • Aristotle: of the fact vs of the reasoned fact.
  • There are mathematical explanations of physical phenomena
  • Can mathematics give explanations of natural phenomena?

Philosophical relevance of mathematical explanations in science

  • Unification and explanation often pull in different directions.
  • Mathematics, modeling, and idealization
    • Moving away from explanations of mechanical causes, rather, emphasis on mathematical forms.
    • Mathematical formalism helps unify but doesn’t really explain the how and why.
    • Asumptotic reasoning – idealisms. Batterman: ‘‘asymptotic analyses so often provide physical insight is that they illuminate structurally stable aspects of the phenomenon and its governing equations’’.
    • Details are ignored, but one still arrives at correct explanations
    • How does math ‘hook onto’ reality?
    • Optimality models: explain without asserting causal structure
    • Topological explanations
    • What is distinctively mathematical about mathematical explanations of scientific facts?
  • Indispensability arguments
    • Descriptive vs explanatory role of mathematics in scientific explanations
    • Math is indispensable for science
    • There are genuinely mathematical explanations of empirical phenomena. We ought to be committed to the theoretical posits postulated by such explanations; thus, We ought to be committed to the entities postulated by the mathematics in question.
    • Maybe what needs explanation cannot even be descreibed without mathematical language.
    • Are numbers part of our symbolic universe?
    • Maybe mathematics only indexes physical facts.
    • Is mathematics purely representational.
    • Perhaps mathematics has an ontological commimtnet to concrete entities.
    • Optimality fallacy – solutions to biological problems
    • Lyon: alternative causal explanations present in mathematical explanations. Process explanation: the actual causes. Program explanation: cites a property which ensures causality.
    • Existence of alternative mathematical explanations for the same phenomenon – negates indispensability of either entity

Mathematical explanations within mathematics

  • Proofs: decision procedures, transfer principles
  • Explanations do not just come in the form of proofs. (really?)

Mathematical explanations: some historical remarks

  • Aristotle: of the fact vs. of the reasoned fact (provides explanations)
  • Proof by contradiction: not a demonstration of reasoned fact.
  • Mathematical demonstrations are not causal? Or is causal?
  • Bolzano and Cournot: central problem of philosophy of math being explanatory vs non-explanatory demonstrations.

Two classical models for mathematical explanation: Steiner and Kitcher

  • Explanitoriness is a local property of proofs vs a global property of the whole theory.
  • A local model of explanation: Steiner
    • To explain the behavior of a thing, one deduces behavior from the entity’s essence
    • Characterizing property: a property unique to a given structure within a domain of structures
    • An explanatory proof has to make reference to the characterizing property such that the result is dependent on that property.
  • A holistic model of explanation: Kitcher
    • Unification as the model for explanation in science and mathematics
    • Understanding phenomena: not just reduction to axioms, but seeing connections and patterns. From premise-conclusion to derivation.
    • Derivation and redundancy
    • Explanatory store: the best systemization over \(K\).
    • Systemization: arguments deriving some elements from \(K\) from other elements from \(K\).

Recent developments

  • ‘Bottom up’ vs ‘top down’ approaches
  • Marc Lange: mathematical induction fails to be explanatory.
  • Mathematical explanations do not follow a general pattern
  • Coincidence vs non-coincidence
  • Context-dependency of mathematical explanations: what are the salient features?

Connections to other debates

  • Mathematical beauty? Explanatory proofs are not necessarily beautiful proofs.
  • Are aesthetic properties of proofs dependent on the epistemic property of explanitoriness.
  • Pure vs impure
  • Mathematical depth


  • Better understnading of how mathematicals applies in the physical world
  • Mathematical explanations of scientific fasts being tested in theories of scientific explanation
  • Metaphyiscal arena


  • Why cannot mathematical facts be explanations?
  • If proofs are citing evidence, then why cannot the axioms of our system (theorems) and properties be cited?
  • Is there a difference between being explained by [x] and being explained? Can a theory be universally explained?
  • Are explanations unique, scientific or mathematic? – unique, all explanations can be trivially transformed into another.
    • Euclid: sum of internal angles of a triangle is equal to two right angles. vs. 180 degrees. vs. pi radians. Is there a difference?
  • Maybe mathematics as representation is functionally explanatory?
  • How much of explanation requires the subject’s view?
    • Is an explanation for how to construct an explanation an explanation? (induction)
    • Banach-Tarski theorem?
  • Proofs as a process? Center the subject? How did an individual come to get the result?
  • Random question: Are humans restricted to proving computable proofs?
  • Grad-Cam and causality in AI explanation approaches.
  • Is AI an interesting subject of explanation thinking?
  • Math as reducing results to axioms, fundamental results?
  • Is causality ultra-saturated in math in that everything necessary becomes part of the causal process?
  • What is a coincidence?
  • Might the question beg the answer in certain scenarios, esp using mathematical facts for scientific phenomena?
  • Explanations in AI ‘meta over-fitting’?
  • Does explanation ‘exist’ always?
  • Do arguments on causality by necessity/removal also threaten the conditions which make the removed necessity necessary?