PHI2391: Logical Empiricism I 8.0

1 2 3 4 5 PHI2391: Logical Empiricism I 8.0 Hume and Kant! Remember Hume s question:! Are we rationally justified in inferring causes from experimental observations?! Kant s answer: we can give a transcendental deduction of why we must do so! Of course both agree that we do think causally, the question is! By what right do we think causally? Kant s Answer(s)! Kant s answer:! Psychological interpretation:! Without the categories (causality, substance, etc.), I wouldn t have a consciousness! The world would just be a mass of sensations, with no order! There would be no subject, because mind would be fragmented! Compare: Anthropic Principle in contemporary physics Kant s Answer(s)! Science-theoretical:! A world that was not interpreted by means of the categories and pure mathematics would not be unified! If there is to be a unified world, then! Since all objects appear in space and time, and,! Time and space have distinct parts! They can only be unified if! The parts are filled with substances that causally interact Kant s Answer(s)! Science-theoretical:! Time and space can only be unified if their parts are filled with substances that causally interact! Example (Einsteinian simultaneity):! When can I tell something on my video screen is live?! Only when I cause what I see to change. How can I know that events in the Rideau Centre are 1&

6 7 8 9! Only when I cause what I see to change.! How can I know that events in the Rideau Centre are happening now?! Under what circumstances is a body orbiting Pluto part of my world? Science-theoretical answer! There can be no guarantee that the world will be comprehensible as a unified mathematical system! But, if there is to be a body of knowledge corresponding to the whole of nature! Then it must have this mathematical, nomological (lawgoverned) structure Space-time and mathematics! So far have talked about the categories that synthetise intuitions! What gets synthesised - the intuitions - are sensations in space and time! They are in space and time, meaning! In space their shape, magnitude, and relation to one another is determined [B37, p. 174]! simultaneity or succession would not come into perception if time did not ground them a priori. Only under its presupposition can one represent that several things exist at one and the same time (simultaneously) or in different times (successively). [B46, p. 178]! Compare: Einstein on EPR Mathematics is synthetic a priori! Kant claims that all mathematical propositions are synthetic a priori! Arithmetic [Introduction B15, p. 142; Axioms of Intuition B205, p. 288]]:! 7+5=12 may seem analytic! Then it should be the case that the concepts 7, 5, and + parts of the the concept 12. But (says Kant) they are not! So 7+5=12 is synthetic! But we know it a priori! So it is synthetic a priori Mathematics is synthetic a priori Geometry [B16, p.145]: 2&

10 11 12 13 Mathematics is synthetic a priori! Geometry [B16, p.145]:! The concepts shortest and straightest do not contain one another (one concerns distance, the other direction)! But we know a priori (says Kant) that a straight line is the shortest distance between two points! So the proposition is synthetic a priori How is it possible that mathematics be synthetic a priori?! Kant sees a problem that his predecessors can t solve! They believe that space exists as a property of things in themselves! One group (the mathematical investigators ) believes that space is a thing that exists on its own (so it s a substance)! Another group (the metaphysicians ) believe that it is a property of things that exist on their own (substances, Leibnizian monads) How can mathematics be synthetic a priori?! Mathematician-physicists like Newton must assume two eternal and infinite self-subsisting non-entities [TA B56, p. 184]! They can explain why mathematics holds of physical objects - they are in space and time! But (so Kant) they become confused when attempting to explain how there could be anything beyond space and time (e.g. God and angels)! More importantly, they must explain in what sense these things exist, since they seem to be a pure emptiness (thus non-entities ) How can mathematics be synthetic a priori?! Metaphysicians like Leibniz think that space and time are human constructions! We have confused knowledge of things (monads), and we organise this confused knowledge with space and time! But, in the things themselves, space and time are not present How can mathematics be synthetic a priori?! Leibniz (and presumably Hume) must dispute the validity of a priori mathematical doctrines [B57]! Why? Because space and time are confused creatures of the imagination [B57], i.e. we make them up 3&

14 15 16! Why? Because space and time are confused creatures of the imagination [B57], i.e. we make them up! How could we have precise knowledge of something confused?! If S&T were merely inductive generalizations of observations, how can we explain the Law of Inertia?! Don t bodies have to obey (Euler) the straight lines and follow inertial paths? How can mathematics be synthetic a priori?! Kant claims his theory solves all these problems:! Mathematics is true of things insofar as they are objects of possible experience! So: insofar as they are represented in pure space and time, the forms of experience! We know it a priori, because we know the structure of our own minds! NB Kant also admits that mathematics doesn t hold of things in themselves! But he can explain why it holds of all objective experience Importance of Kantian Doctrines! 19th c. philosophy, and thus philosophy of science profoundly affected by Kant! But also science itself:! Kantian theories (apparently) lay to rest questions about absolute space and mathematics! Important assumptions: there are synthetic a priori truths! There are intuitions (raw data) at the basis of mathematics that are - in being spatial - essentially the same as our everyday intuitions Importance of Kantian Doctrines! Kant gives a non-metaphysical explanation of the task of science! We are built to explain events in space-time in terms of causal relations between events! Our reason imagines a total system in which all events that occur could be predicted as instances of general mathematical laws! This completed system is an ideal (Kant s term) towards which we strive 4&

17 18 19 20 21! This completed system is an ideal (Kant s term) towards which we strive! The ideal system has an a priori core which Kant calls [Metaphysical Foundations, p.4ff.] the pure part of science Importance of Kantian Doctrines! Kant s system lays down model for future philosophy of science! Pure logic and mathematics form the a priori core of science! They lay down basic patterns of reasoning! These patterns allow us to think about the data of our experience - to draw conclusions! And to predict Emergence of logical empiricism! Question: Is Kant right to say that mathematics is synthetic a priori?! Two fundamental 19th c. developments:! Emergence of new logics (Boole, Peano, Frege)! Emergence of new geometries (Gauss, Bolyai, Lobachevsky, Helmholtz, Lie) Emergence of logical empiricism! Emergence of new logics led to new possibility:! Perhaps Kant though that arithmetic was analytic because his logic was too weak! Improved logic of relations, with quantification, opens new possibilities Logicism! Leads to foundation of new school of thought: logicism (Frege, Russell)! Logicists explicitly opposed to Kant:! Mathematics is not synthetic a priori! The essential concepts of arithmetic can be defined in pure logic Logicism: definition of number! Definition of number 2! 2 = the class of all classes A s.t.! ( x,y,z)((xεa & yεa & zεa & x y) z=x v z=y)! Means: For all x,y,z, such that x and y are in A, and are not identical (=the same), 5&

22 23 24 25 26 identical (=the same),! then any other z that is in A is identical to either x or y! Now I can define equations as statement asserting the identity of classes! The concept 7+5 may not contain 12 Logicism: definition of number! But from the concepts of 7, +, 5, and 12 we can logically deduce the equation! NB Only concepts, not intuitions are required! Arithmetic expresses conceptually necessary truths! This is just what Kant denied (assuming conceptually necessary = analytic ) Logicism & Empiricism! Why does this matter?! How many ingredients go into science?! Logic, Mathematics, Theory of induction, general empirical laws, experimental data! Some are logical, some empirical! But for Kant there had been mixed knowledge: pure, but intuitional Logicism & Empiricism! British proponent of logicism: Bertrand Russell! Prominent student: Ludwig Wittgenstein! Difficult to assess influence on: Vienna Circle [Schlick, Carnap, Feigl, Frank, Gödel, Hahn, Menger, Neurath, Waismann, et alia]! We will due a fusion of Carnap and Wittgenstein Logicism & Empiricism! This fusion is similar to the hypothetico-deductive model in PS 2.2! It is the first truly modern theory of science! Advocated in general outlines by Wittgenstein in his Tractatus Logico-philosophicus! Developed into fully articulated theory by Carnap in his Logical Structure of the World! Characterised by a clean separation between logical and empirical elements of science Carnap-Wittgenstein Theory 6&

27 28 29 30 Carnap-Wittgenstein Theory! Fundamental division: language and world! World consists of facts! Language of propositions! Correspondence theory: the propositions are true or false when the corresponding facts exist or not! But propositions are logically articulated Kinds of propositions! Propositions are divided in 2 kinds:! Elementary propositions: consist of concatenations of names! Molecular propositions: consist of logical connections between (truth-functions of) elementary propositions! Truth-functions: and, or, if-then, not, etc. Elementary propositions! Elementary propositions express observations or empirical information! In Wittgenstein s version: elementary propositions are concatenations of primitive names! The names stand for primitive objects, which are indestructible! When a group of objects is configured as its corresponding set of names in the elementary propositions says it is! Then the proposition is true Elementary propositions! In later versions of this theory, e.g. Carnap and Neurath,! Elementary propositions are loosened! First, the doctrine of simple and unchanging objects is rejected! Finally, the doctrine of correspondence itself is weakened Molecular propositions! Most propositions - scientific or otherwise - are not elementary! They are molecular : they are composed of elementary propositions! Composed in a truth-functional sense! So if P and Q are elementary propositions, P & Q is molecular! A molecular proposition asserts that some elementary 7&

31 32! A molecular proposition asserts that some elementary propositions depend on one another logically Molecular propositions! E.g. If Pa then Qa says:! Whenever Pa is true, Qa is true! I.e. If Pa then Qa is true so long as we do not have the following case:! Pa is true and Qa is false Molecular propositions! Often expressed as a truth-table : 33 34 35 Molecular propositions! The truth-table shows the logical dependence of the molecular proposition on its components! Note that in the truth-table, we do not say whether the components are elementary or not! We can engage in recursive construction of molecular propositions, e.g. P (Q R)! We get a nesting of truth-functions, thus of truth-tables! But: the truth or falsity of the molecular proposition depends entirely on the truth or falsity of the elementary propositions! The values of all the elementary propositions = The World Meaning! Since the molecular proposition depends on the elementary ones! We can say:! The meaning of the elementary propositions is the elementary fact they refer to! The meaning of the molecular propositions is the truthdependency they assert among the elementary ones! Important consequence:! Molecular propositions don t mean anything more than can be said with elementary ones Meaning and logic! The definition of molecular prop s as truth-functional 8&

36 37 38 39! The definition of molecular prop s as truth-functional combinations of elementary ones leads to 2 special classes of m. propositions:! Tautologies and Contradictions ( Is Metaphysics Meaningless? pp. 170-1)! Tautologies are always true, whatever the values of their components! Contradictions are always false, whatever etc.! NB that the negation of every tautology is a contradiction and vice versa Tautologies! The class of tautologies is of particular interest! It includes propositions such as! P (Q P)! P v ~P! (P & (P Q)) Q! These are proposition that Russell and Frege called the logical propositions! For Russell and Frege, these propositions were theorems following from basic logical axioms Tautologies! Wittgenstein s definition of a tautology is a semantic as opposed to a syntactic definition of logical truth! Semantics = having to do with meaning! Tautologies are propositions that say (Gr. legein, logos) the same (tautos) thing! In the Carnap-Wittgenstein theory, this has a definite significance Tautologies! The tautologies (and contradictions) are the propositions that do not depend on the elementary propositions! So if the elementary propositions are observation-statements! The tautologies are statements that:! (1) Are true! (2) Make no empirical claims! They are therefore strictly speaking meaningless Tautologies & mathematics! Theory gets special importance from the logicist background Frege and Russell have (almost) reduced mathematics to 9&

40 41 42 43! Theory gets special importance from the logicist background! Frege and Russell have (almost) reduced mathematics to logic! Wittgenstein defines logic as a special class of propositions, that are in fact meaningless! So mathematics becomes meaningless! Strange result! Mathematics & meaninglessness! Isn t it absurd to say that mathematics is meaningless?! Not necessarily: meaning is being used in a particular sense! What is meaningful is what corresponds to a world-element that may or may not exist! Meanings describe the possibility of the existence of an empirical datum! In Kantian terms, meanings refer to possible intuitions Mathematics & meaninglessness! When we say mathematics is meaningless, we mean that! It lacks any empirical content: it makes no claims about what is actually the case! As in Kant, mathematics and logic are formal characteristics of thought, or language! But unlike in Kant, mathematics has no synthetic content! It is a purely general, and in this sense empirically vacuous body of knowledge Empirical content! The logico-mathematical propositions are opposed to the empirical ones! These do make statements about observations - what is actually going on! So the other pole of the theory concerns the status of these observation-statements, or elementary propositions! Wittgenstein declined to explain what they were! Some evidence that they were primitive observation statements à la Kant Observation statements! Vienna Circle wants to apply the theory (Witt s conclusions purely negative)! First stage: abandon the metaphysical components of Wittgenstein s theory 10&

44! First stage: abandon the metaphysical components of Wittgenstein s theory! Wittgenstein himself opposed to metaphysics! Here means: stop talking about elementary facts and objects! Stop talking about the world-elements! That s all metaphysics is (talk of world-elements) Observation statements! Carnap and Neurath: We don t need the elementary facts! All that matters for the theory is the elementary propositions! They can be singled out without supplementary hypotheses about the world! Call these elementary statements protocol-sentences ( On the Logical Positivists pp. 181-182) 11&