Fonksiyon, Kavram, Anlam

by Gottlob Frege, M. Sami Türk (translator)

Modern matematiksel mantığın kurucusu olan Gottlob Frege, Aristoteles’ten sonraki zamanların en büyük mantıkçısı olarak kabul edilmektedir. Eserleri kendi döneminde çok fazla üne kavuşmamış, fikirleri daha çok kendisinden etkilenen düşünürler aracılığı ile yayılmıştır. Birbiri ardına devrim niteliğinde eserler neşreden Frege, modern mantığın gelişimini mümkün kılan temel fikirleri kendi başına keşfetmiştir ve bütün disiplini adeta tek başına kurmuştur. Matematik felsefesi, matematiksel mantık, dil felsefesi gibi alanları derinden etkileyen Frege’nin fikirleri, Ludwig Wittgenstein ve Edmund Husserl gibi isimlere de tesir etmiştir.
Bu eser, Frege’nin günümüzün felsefi tartışmalarında en etkili teorilerini içeren, matematikle alakalı meselelere çok fazla aşina olmayan okurun da kolaylıkla anlayabileceği makalelerden oluşmaktadır.

Genres: Philosophy, Mathematics

96 pages, Paperback

First published November 30, 2011

About the author

Friedrich Ludwig Gottlob Frege (German: [ˈɡɔtloːp ˈfreːɡə]) was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics. While he was mainly ignored by the intellectual world when he published his writings, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of logicians and philosophers.

Reviews

[William Bies · Rating 4 out of 5]

It proves to be a common occurrence in intellectual history that one can make a fresh start on a subject by disregarding all the subtleties that may have accumulated and reanalyzing the simple terms. Thus, Galileo breaks with scholastic physics in the Aristotelian mode (at least in intention, if not in fact) and founds classical mechanics by zeroing in on the problem of free fall of a projectile; around the same time, Descartes, by treating animals as machines and by ignoring teleology, launches a new chapter in biology which was to lead to fruitful discoveries all through the subsequent centuries. Along these lines, the great late nineteenth-century logician Gottlob Frege prepares the ground for twentieth-century analytic philosophy and its linguistic turn by means of a radical simplification of philosophical terminology.

Frege’s concern is always to ground our knowledge of things and, to this purpose, he takes mathematical knowledge as paradigmatic. This decision accounts for both his strengths and his weaknesses. For in view of its exactness, mathematical reasoning is ideally suited to an exhaustive analysis of the kind our author wishes to carry out. But, on the other hand, this same property renders mathematics remote from everyday affairs, where approximation and heuristics rule – not in itself a problem, but it remains by no means clear that in investigating the world of man’s experience one may continue to employ a method worked out only for the ideal case of mathematics.

Let us enter a little more deeply into this point, since it is crucial and indicates that Frege’s preeminence these days among Anglo-American analytic philosophers may be somewhat misplaced. Bertrand Russell, for one, wants his logical researches to have a wider relevance than just to the foundations of mathematics (in which, to be sure, he reigns supreme) – for instance, he certainly supposes that his Fregean-inspired approach to logic gives insight into physics, such as what to make of the age-old problem of causality in light of modern discoveries?

What does Frege himself omit, as far as the real world goes apart from pure mathematics? In the world of pure mathematics, everything is susceptible of exact definition. There can be no doubt as to the meaning of the terms and, in any particular application, the objects about which one is reasoning are sharply defined so that one will be in a position to determine what is true or false about them. As Frege understands very well, his characterization of the logical connectives departs to some degree from how speakers of a natural language understand and use them. He views this fact as a condition that has to be accepted, if one is to arrive at something absolutely unambiguous to submit to rigorous analysis. Thus, the dichotomy expressed by the law of the excluded middle – perfectly fine within the demarcated domain of mathematics itself, just unrealistic if pressed too far when brought to bear on real-world matters. In a similar vein, Frege goes on to reduce natural language, with all its suppleness and expressiveness, to the formulaic language of the predicate calculus [Formelsprache]. Not only this, Frege knows very well that he is suppressing many of the features traditionally ascribed to grammar in the interest of having materials of the greatest possible logical simplicity with which to work. Thus, he eliminates predication and the difference between hypothetical, apodictic and assertorial judgments. For him, the conceptual content alone expressed by a proposition is what matters and he does not want to be diverted into seemingly metaphysical disquisitions on the difference between a subject and a predicate, say. Thus, also, he is willing to consider statements in the indicative only. Hence, he suppresses several further qualifications that figure prominently in everyday speech and would conventionally be regarded as belonging to grammar: tense, mood, aspect, voice, pragmatics. We are left with a timeless intelligible world of ideal mathematical objects about which one can make only descriptive statements.

Very well – we agreed at the outset that such a radical mode of proceeding can have its justification. What does Frege go on to accomplish with it? The relatively short papers reprinted in this volume may serve as an ideal introduction into Frege’s project.

In the first, Funktion und Begriff (1891), he lays out his ontology of completed object [Gegenstand] versus incomplete function [Funktion]: why refer to the latter as incomplete? Because a function depends on a variable, hence it doesn’t become a statement about anything until a variable is supplied. Therefore, Frege represents a continuation of the older view of a function as a rule in contrast to the modern view of a function as a mapping which, for practical purposes may be identified with its graph. Note, the graph of a function would be an actual completed totality entertained by the mind. Thus, it would surely repay further reflection to ponder the difference between Frege’s view and that of the most up-to-date modern mathematicians.

Über Sinn und Bedeutung (1892), the second paper in the series, portrays what is perhaps Frege’s most celebrated distinction (among linguists), between sense [Sinn] and reference [Bedeutung]. We won’t spoil the reader’s chance to encounter this distinction in the original, for a gentle lead into it, if wanted, see the relevant paragraphs in Textor’s introduction to the present edition. Let us merely advert to what Frege’s motivations must have been for conceiving it in the first place: his Erkenntniswert-Argument, turning on the meaning of identity respectively of proper names versus general terms designating concepts. Thus, for him, the distinction is intended as a tool with which to investigate the epistemological issues of concern to him.

Third essay, Über Begriff und Gegenstand (1892). These two are central terms about which Frege has distinctive ideas. He sees every concept [Begriff] as requiring completion, hence the meaning of a concept-term [Begriffswort] is a special kind of function. The object [Gegenstand], on the other, may be viewed as the intention of a thought [Gedanke]. Thus, Frege’s strict dictum that no concept is an object. This reviewer, once again, has nothing in particular in mind to comment upon here other than to recommend the exploration of Frege’s ideas in the original. It may be remarked, though, that perhaps Frege is naïve in the sense mooted above in this review since he never does any serious work on anything apart from the logical foundation of arithmetic. This must be borne in mind when he discusses issues such as identity, scope and range. Typical of the concerns fielded here would be the following question: what is the difference between specifying a number via several of its properties that single it out uniquely from among all other numbers, and its definition or concept? See p. 59, we can view it as a logical equivalence, say, 4 is nothing other than the result of adding 3 and 1. In the closing paragraphs to this essay, Frege vents a few pointed remarks to illustrate his position. Thus, for instance, he disallows relation [Beziehung] without some kind of connection [Bindemittel] when one says 2 is a prime number although primeness is not part of its definition. All around, Frege is very clear and his position certainly deserves consideration.

Was ist eine Funktion? (1904) – Here, Frege begins by commenting on the meaning of function in analysis: as a Rechnungsausdruck versus as a Veränderliche (close to, but not the same as the set-theoretic concept of a mapping). But our author sees a difficulty in the latter in that it would require a notion of time that is alien to mathematics! Then the question inevitably arises, what is it that changes? For it would be all right for infinitesimal analysis to conceive of a variable quantity but in arithmetic, integers cannot change continuously. Thus, there are no variable numbers. Another possible view: Veränderlich als unbestimmte Zahl – in Frege’s opinion, this also doesn’t work:

Hinsichtlich des Veränderlichen hat sich uns folgendes ergeben. Veränderliche Größen können zwar anerkannt werden, gehören aber nicht der reinen Analysis zu. Veränderliche Zahlen gibt es nicht. Das Wort »Veränderliche« hat mithin in der reinen Analysis keine Berechtigung. [p. 65]

Why won’t Frege ascribe any credit to Dedekind’s cuts in pure analysis? Russell obviously allows them. Perhaps Frege is everywhere too disinclined to contemplate infinitary processes. In his Grundlagen der Arithmetik of 1884 (see our review here), for instance, he is willing to admit the first countably infinite ordinal as a number but stops short of Cantor’s transfinitum. In this reviewer’s opinion, this hesitation represents a disabling shortcoming in Frege. Remarks in this connection dismissive of Czuber’s idea of Zuordnung illustrate Frege’s extreme literal-mindedness, for Czuber’s is in fact the accepted modern way to view the matter. It would seem that the whole field of the philosophy of mathematics is ripe for a reappraisal, in that there subsists a latent tension between the fact, on the one hand, that Frege’s logical formalism is foundational to modern mathematics (especially as taken up by Russell) and, on the other, that since his time set theorists by and large have thrust aside Frege’s wariness and forged boldly ahead in their investigation of the higher infinite, with many deep results to show for it!

Conclusion: Frege still thinks of a function as a rule not as a mapping from domain to codomain. He fails moreover to distinguish between the function and its value at a given point in its domain. Fortunately, modern mathematicians haven’t followed his usage; on Frege’s terms, it would be nearly impossible to conceive of a functional analysis in which functions themselves are treated as but individual points in an infinite-dimensional space of functions: thus, the greater part of the advances of the twentieth century would thereby be blocked, if one were to adopt a Fregean view uncritically.

The final essay, Über die wissenschaftliche Berechtigung einer Begriffschrift (1882), proffers Frege’s reflections on the nature of language in general leading to a spirited answer as to why his program of defining an ideal Formelsprache has justification.

One should read Frege’s original text first, only then the extended Einleitung [pp. ix-xliii] by Mark Textor for its reception and his Erläuterungen [pp. 79-91] for useful clarifications while consulting the originals. To close, let us spell out the place of the essays reprinted here in Frege’s overall work relative to the Begriffsschrift and the Grundgesetze? They should be seen as pedagogical introductions, suited to alert one to what he is up to but for the real thing one has to go to his Hauptwerke.

[Liedzeit Liedzeit · Rating 3 out of 5]

Dieser kleine Band enthält fünf logische Schriften, davon die bekanntesten und einflussreichsten sicher Über Sinn und Bedeutung und Über Begriff und Gegenstand.

Frege meint, dass ein Eigenname sowohl Sinn als auch Bedeutung habe. Wobei die Bedeutung der gemeinte Gegenstand ist. Und Eigenname auch Bezeichnungen beinhaltet, die einen Eigennamen vertreten. Morgenstern und Abendstern, so sein berühmtes Beispiel, haben unterschiedlichen Sinn, aber dieselbe Bedeutung, nämlich den Planeten Venus.

Anders verhält es sich mit der Bedeutung von Sätzen. Der Gedanke ist der Sinn eines Satzes, die Bedeutung jedoch sein Wahrheitswert. Also gibt es für Sätze nur zwei Bedeutungen, das Wahre und das Falsche. „Das Streben nach Wahrheit also ist es, was uns überall vom Sinn zur Bedeutung vorzudringen treibt.”

Das Wahre und Falsche sind einfache Begriffe, die nicht definiert werden können. Das Urteil kann, sagt er, als Fortschreiten von einem Gedanken zu seinem Wahrheitswert gefaßt werden. (Das will er aber nicht als Definition verstanden wissen.)

Was passiert, wenn ein Name keine Bedeutung hat? Der Satz „Odysseus wurde tief schlafend in Ithaka an Land gesetzt” hat offenbar Sinn, obwohl es zweifelhaft ist, dass Odysseus ein Bedeutung hat. In diesem Fall kann der Satz Vorstellungen und Gefühle erwecken, hat aber keine Bedeutung.

Über Begriff und Gegenstand ist eine Erwiderung auf Einwände, die ein Bruno Kerry gegen Frege erhoben hat. Dieser Aufsatz ist berühmt und berüchtigt für die Aussage „der Begriff Pferd ist kein Begriff.” (Während zum Beispiel die Stadt Berlin eine Stadt ist.)

Frege nennt seine Behauptung immerhin eine „sprachliche Härte”. Zwar ist seine Argumentation nachvollziehbar, absonderlich bleibt seine Behauptung dennoch. Für Frege ist der Begriff Pferd ein Eigenname und als solcher bezeichnet er einen Gegenstand.

Die sprachliche Härte entsteht, weil Frege meint, dass die Sprache mangelhaft ist und uns beim Denken zu Fehlern führt. Zum Beispiel wird Pferd sowohl für ein Einzelwesen als auf für die Art verwendet. Mit seiner Begriffsschrift versuchte Frege das zu schaffen, was man später by Idealsprache nannte, eine Sprache, die unter anderem dadurch ausgezeichnet ist, dass sie keine Mehrdeutigkeiten zulässt.

[A YOGAM · Rating 4 out of 5]

Mein Exemplar dieser von Günther Patzig herausgegebenen Sammlung, „Funktion, Begriff, Bedeutung“, habe ich am 12.6.1990 für fast schon lächerlich günstige 9,80 DM erworben. Diese Sammlung war bei ihrem Erscheinen 1962 eine editorische Pionierleistung, die den damals fast vergessenen Gottlob Frege wieder als Schlüsselfigur der Gegenwartsphilosophie ins Bewusstsein rückte. Die fünf hier versammelten Aufsätze – allen voran das weltberühmte „Über Sinn und Bedeutungsowie Funktion und Begriff“ – sind heute unsterbliche Klassiker, die mit vorbildlicher Klarheit das Fundament der modernen Logik und Sprachphilosophie legten. Patzigs einordnendes Vorwort, das Freges Bedeutung damals erst sichtbar machte, liest sich heute als ideale Einführung in Freges kühne Thesen.

[Kaizhi Liu · Rating 5 out of 5]

厉害，而且也还能读懂

[Diogo (Amaral) · Rating 4 out of 5]

Bom livro sobre as bases da matemática, seja definindo conceito, como expressão e função, tem alguns debates que pode ser postos entre Frege a crítica de Deleuze à metafísica platónica, será interessante ver como os dois convergiam e divergiam