Equally surprisingly, Descartes’s paranoid belief was shared by several contemporary mathematicians, among them Isaac Barrow, John Wallis and Edmund Halley. (Huxley 1959, pp. 354-355.) In the light of our fuller knowledge of history it is easy to smile at Descartes. It has even been argued by Netz that analysis was in fact for ancient Greek geometers a method of presenting their results (see Netz 2000). But in a deeper sense Descartes perceived something interesting in the historical record. We are looking (...) in vain in the writings of Greek mathematicians for a full explanation of what this famous method was. And I will argue for an answer to the question why this lacuna is there: Not because Greek geometers wanted to hide this method, but because they did not fully understand it. It is instructive to note the ambivalent attitude of the most rigorous mathematician of the period, Isaac Newton, to the method of analysis. He used it himself in his own mathematical work and in the expositions of that work. Yet when the mathematical push came to physical and cosmological shove, he formulated his Principia entirely in.. (shrink)
Jaakko Hintikka 1. How to Study Set Theory The continuum hypothesis (CH) is crucial in the core area of set theory, viz. in the theory of the hierarchies of infinite cardinal and infinite ordinal numbers. It is crucial in that it would, if true, help to relate the two hierarchies to each other. It says that the second infinite cardinal number, which is known to be the cardinality of the first uncountable ordinal, equals the cardinality 2 o of the continuum. (...) (Here o is the smallest infinite cardinal.). (shrink)
particular alternative logic could be relevant to another one? The most important part of a response to this question is to remind the reader of the fact that independence friendly (IF) logic is not an alternative or “nonclassical” logic. (See here especially Hintikka, “There is only one logic”, forthcoming.) It is not calculated to capture some particular kind of reasoning that cannot be handled in the “classical” logic that should rather be called the received or conventional logic. No particular epithet (...) should be applied to it. IF logic is not an alternative to our generally used basic logic, the received first-order logic, aka quantification theory or predicate calculus. It replaces this basic logic in that it is identical with this “classical” first-order logic except that certain important flaws of the received first-order logic have been corrected. But what are those flaws and how can they be corrected? To answer these questions is to explain the basic ideas of IF logic. Since this logic is not as well known as it should be, such explanation is needed in any case. I will provide three different but not unrelated motivations for IF logic. (shrink)
What one can say about the past, present and future of set theory depends on what one expects or at least hopes set theory will accomplish. In order to gauge the early expectations, I begin with a quote from the inaugural lecture in 1903 of my mathematical grandfather, the internationally known Finnish mathematician Ernst Lindelöf. The subject of his lecture was – guess what – Cantor’s set theory. In his conclusion, Lindelöf says of Cantor’s results: For mathematics they have lent (...) new tools and opened up new fields of research, they have thrown entirely new light on the foundations of analysis and brought clarity and order where there was only disorder and contradictions. Thus they have greatly contributed to the harmony that is the essence of mathematics, a harmony a grasp of which is the reward of mathematical research. (Quoted in Olli Lehto, Tieteen aatelia, Otava, Helsinki, 2008, p. 263) We can all agree with the compliments Lindelöf pays to set theory as an impressive specimen of mathematical research, including the theory of infinite cardinals and ordinals. But as far as the foundational role of set theory is concerned, in the perspective of the subsequent century his words read as an example of supreme historical irony. Far from bringing harmony into the foundations of mathematics, problems arising from set theory led to a schism between different schools of thought. Few mathematicians think of set theory as a tool for reaching new results outside set theory.. On the contrary, an interesting rich tradition called reverse mathematics takes significant mathematical results and asks what set-theoretical assumptions are needed to prove them. Set-theoretical paradoxes have greatly increased mathematicians’ concerns about contradictions instead of assuaging them. Many foundationalists would blandly deny that we have even now, more than a hundred years later, reached “clarity and order” about the foundations of analysis. What Lindelöf took to be the results of set theory thus were in reality so many hopes that set theory was expected to fulfill.. (shrink)
The world of philosophy can perhaps be seen as a microcosm of the world at large. In the course of the last few decades, the world has seen the collapse of the communist system of Russia, a major crisis of the free market economy in the USA, Europe and Japan, and massive economic changes in China. One perspective on contemporary philosophical research is reached by asking what crises the major philosophical traditions, if not literally “systems”, are likewise undergoing and what (...) can be done to find a road ahead. What might a “stimulus package” for philosophy be like (except for inevitably being controversial)? (shrink)
By speaking of the bald man, I am of course referring to the most clear-cut of the paradoxes of vagueness, the sorites paradox. Or, strictly speaking, I am referring to one of the dramatizations of this paradox. This case is nevertheless fully representative of the general issues involved. (For the sorites paradox in general, see e.g. Keefe and Smith 1987 or Sainsbury 1995, ch.2.) The allegedly paradoxical argument is well known. It might be formulated as follows.
In a definition (∀ x )(( x є r )↔D[ x ]) of the set r, the definiens D[ x ] must not depend on the definiendum r . This implies that all quantifiers in D[ x ] are independent of r and of (∀ x ). This cannot be implemented in the traditional first-order logic, but can be expressed in IF logic. Violations of such independence requirements are what created the typical paradoxes of set theory. Poincaré’s Vicious Circle Principle (...) was intended to bar such violations. Russell nevertheless misunderstood the principle; for him a set a can depend on another set b only if ( b є a ) or ( b ⊆ a ). Likewise, the truth of an ordinary first-order sentence with the Gödel number of r is undefinable in Tarki’s sense because the quantifiers of the definiens depend unavoidably on r. (shrink)
The main tool of the arithmetization and logization of analysis in the history of nineteenth century mathematics was an informal logic of quantifiers in the guise of the “epsilon–delta” technique. Mathematicians slowly worked out the problems encountered in using it, but logicians from Frege on did not understand it let alone formalize it, and instead used an unnecessarily poor logic of quantifiers, viz. the traditional, first-order logic. This logic does not e.g. allow the definition and study of mathematicians’ uniformity concepts (...) important in analysis. Mathematicians’ stronger logic was rediscovered around 1990 as the form of independence-friendly logic which hence is not a new logic nor a further development of ordinary first-order logic but a richer version of it. (shrink)
The ancient Greek method of analysis has a rational reconstruction in the form of the tableau method of logical proof. This reconstruction shows that the format of analysis was largely determined by the requirement that proofs could be formulated by reference to geometrical figures. In problematic analysis, it has to be assumed not only that the theorem to be proved is true, but also that it is known. This means using epistemic logic, where instantiations of variables are typically allowed only (...) with respect to known objects. This requirement explains the preoccupation of Greek geometers with questions as to which geometrical objects are ?given?, that is, known or ?data?, as in the title of Euclid's eponymous book. In problematic analysis, constructions had to rely on objects that are known only hypothetically. This seems strange unless one relies on a robust idea of ?unknown? objects in the same sense as the unknowns of algebra. The Greeks did not have such a concept, which made their grasp of the analytic method shaky. (shrink)
The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation (...) of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation. (shrink)
Most current work in epistemology deals with the evaluation and justification of information already acquired. In this book, Jaakko Hintikka instead discusses the more important problem of how knowledge is acquired in the first place. His model of information-seeking is the old Socratic method of questioning, which has been generalized and brought up-to-date through the logical theory of questions and answers that he has developed.
One of the characteristic features of contemporary logic is that it incorporates the Frege-Russell thesis according to which verbs for being are multiply ambiguous. This thesis was not accepted before the nineteenth century. In Aristotle existence could not serve alone as a predicate term. However, it could be a part of the force of the predicate term, depending on the context. For Kant existence could not even be a part of the force of the predicate term. Hence, after Kant, existence (...) was left homeless. It found a home in the algebra of logic in which the operators corresponding to universal and particular judgments were treated as duals, and universal judgments were taken to be relative to some universe of discourse. Because of the duality, existential quantifier expressions came to express existence. The orphaned notion of existence thus found a new home in the existential quantifier. (shrink)
Omitting experimental data is often considered a violation of scientific integrity. If we consider experimental inquiry as a questioning process, omitting data is seen to be merely an example of tentatively rejecting (‘bracketing’) some of nature’s answers. Such bracketing is not only occasionally permissible; sometimes it is mandated by optimal interrogative strategies. When to omit data is therefore a strategic rather than ethical question. These points are illustrated by reference to Millikan’s oil drop experiment.
In order to be able to express all possible patterns of dependence and independence between variables, we have to replace the traditional first-order logic by independence-friendly (IF) logic. Our natural concept of truth for a quantificational sentence S says that all the Skolem functions for S exist. This conception of truth for a sufficiently rich IF first-order language can be expressed in the same language. In a first-order axiomatic set theory, one can apparently express this same concept in set-theoretical terms, (...) since the existence of functions can be expressed there. Because of Tarski's theorem, this is impossible. Hence there must exist set-theoretical statements, even provable ones, which are said to be true in first-order models of axiomatic set theory but whose Skolem functions do not all exist. Hence there are provable sentences in axiomatic set theory that are false in accordance with our ordinary conceptions of set-theoretical truth. Such counter-intuitive propositions have been known to exist, but they have been blamed on the peculiarities of very large sets. It is argued here that this explanation is not correct and that there are intuitively false theorems not involving very large sets. Hence the provability or unprovability of a set-theoretical statement, e.g. of the continuum hypothesis (CH) in axiomatic set theory is not necessarily relevant to the truth of CH. (shrink)
Finland is internationally known as one of the leading centers of twentieth century analytic philosophy. This volume offers for the first time an overall survey of the Finnish analytic school. The rise of this trend is illustrated by original articles of Edward Westermarck, Eino Kaila, Georg Henrik von Wright, and Jaakko Hintikka. Contributions of Finnish philosophers are then systematically discussed in the fields of logic, philosophy of language, philosophy of science, history of philosophy, ethics and social philosophy. Metaphilosophical reflections on (...) the nature of philosophy are highlighted by the Finnish dialogue between analytic philosophy, phenomenology, pragmatism, and critical theory. (shrink)
My propositions are elucidatory in this way: he who understands them eventually recognizes them as senseless [unsinnig], when he has climbed out through them, on them, over them… He must surmount these propositions; then he sees the world rightly. (Tractatus 6.54) These statements must be taken seriously and therefore must be interpreted as literally possible. They have nevertheless been experienced by some philosophers as posing a major interpretational problem. For if Wittgenstein’s words are taken literally, we seem to have a (...) major problem in our hands. If what Wittgenstein had said before proposition 6.54 is literally nonsense, we apparently cannot understand his book at its face value. And, as was pointed out, this face value is that of a treatise in logical semantics. Hence.. (shrink)
The following analysis shows how developments in epistemic logic can play a nontrivial role in cognitive neuroscience. We argue that the striking correspondence between two modes of identification, as distinguished in the epistemic context, and two cognitive systems distinguished by neuroscientific investigation of the visual system (the "where" and "what" systems) is not coincidental, and that it can play a clarificatory role at the most fundamental levels of neuroscientific theory.
In game-theoretical semantics, perfectlyclassical rules yield a strong negation thatviolates tertium non datur when informationalindependence is allowed. Contradictorynegation can be introduced only by a metalogicalstipulation, not by game rules. Accordingly, it mayoccur (without further stipulations) onlysentence-initially. The resulting logic (extendedindependence-friendly logic) explains several regularitiesin natural languages, e.g., why contradictory negation is abarrier to anaphase. In natural language, contradictory negationsometimes occurs nevertheless witin the scope of aquantifier. Such sentences require a secondary interpretationresembling the so-called substitutionalinterpretation of quantifiers.This interpretation is sometimes impossible,and (...) it means a step beyond thenormal first-order semantics, not an alternative to it. (shrink)
The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann "quantum logic" (...) can be interpreted by taking their "disjunction" to be ¬(∼A & ∼ B). Their logic can thus be mapped into a Boolean structure to which an additional operator ∼ has been added. (shrink)
In the present day and age, it seems that every constructivist philosopher of mathematics and her brother wants to be known as an intuitionist. In this paper, It will be shown that such a self-identification is in most cases mistaken. For one thing, not any old (or new) constructivism is intuitionism because not any old relevant construction is carried out mentally in intuition, as Brouwer envisaged. (edited).
We introduce several senses of the principle ofcompositionality. We illustrate the difference between them with thehelp of some recent results obtained by Cameron and Hodges oncompositional semantics for languages of imperfect information.
How does language represent ("mirror") the world it can be used to talk about? Or does it? A negative answer is maintained by one of the main traditions in language theory that includes Frege, Wittgenstein, Heidegger, Quine and Rorty. A test case is offered by the question whether the critical ''mirroring'' relations, especially the notion of truth, are themselves expressible in language. Tarski's negative thesis seemed to close the issue, but dramatic recent developments have decided the issue in favour of (...) the expressibility of truth. At the same time, the "mirroring" relations are not natural ones, but constituted by rule-governed human activities a la Wittgenstein's language games. These relations are nevertheless objective, because they depend only on the rules of these "games", not on the idiosyncrasies of the players. It also turns out that the "truth games" for a language are the same as the language games that give it its meaning in the first place. Thus truth and meaning are intrinsically intertwined. (shrink)
Pretheoretically, truth is a correspondence between a sentence and facts. Other so-called theories of truth have typically been resorted to because such a correspondence is thought of as being inexpressible or as being incapable of yielding a definition of truth which expresses what we actually mean. It can be shown that truth is indefinable in the paradigm case of ordinary first-order languages only because they cannot express informational independence. As soon as this is corrected, as in independence-friendly first-order logic, truth (...) predicates are readily definable, Tarski notwithstanding. Hence, there is no reason to think that truth cannot also be defined for our actual working language—Tarski’s “colloquial language.”. (shrink)