Abstract
Frege’s conception of science includes three features: (1) a science is applicable to other sciences, or even to itself, (2) a science consists of a more or less rigid system of judgements and (3) a science presupposes elucidations, illustrative examples and a “catch on” among scientists. Together, I label these three features “The scientific Picture”. Both logic and mathematics are included among the sciences and are covered by the scientific picture. As I understand Frege, this picture guides his logical and philosophical reflections. Here it is invoked in a treatment of two well-known and controversial Fregean topics: His claim, often repeated, that the axioms of Begriffsschrift and Grundgesetze are obvious and stand in no need of justification, and his use of a Kantian terminology in classifying judgements as analytic or synthetic, a priori or a posteriori. The most significant consequence of my reading is that it underscores the epistemological nature of Frege’s thinking and, at the same time, downplays a current, and in my mind unfortunate, trend of ascribing to Frege a rather “thick” metaphysics. Towards the end, I discuss different aspects of the notion of a judgment at play in Frege’s discussions: judgement as movement from thought to truth-value and judgement as represented by the judgement-stroke. These aspects point back to the distinction, so nicely illustrated by Frege’s own writings, between a scientist, engaged in scientific research, and a philosopher, explicating the scientific activity and its general presuppositions, respectively.
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Notes
- 1.
- 2.
That is to say, the claim is that his understanding of logic was transformed when he reached the second stage of the programme outlined in the preface to Begriffsschrift : “I sought first to reduce the concept of ordering-in-a-sequence to the notion of logical ordering, in order to advance from here to the concept of number ” (Frege 1879a/1972, 104).
- 3.
By this, I do not mean that to Frege, epistemology covers all of philosophy , but only that I am focusing on the part of philosophy that concerns my present purpose.
- 4.
- 5.
This is made evident throughout the mentioned series of papers.
- 6.
- 7.
Cf. also “I now turn to the second of the two views that may be called formal theories […]. This view has it that the signs of the numbers 1/2, 1/3, of the number π, etc. are empty signs . This cannot very well be extended to cover whole numbers , since in arithmetic , we cannot do without the content of the signs 1, 2, etc., and because otherwise no equation would have a sense which we could state—in which case we should have neither truths , nor a science, of arithmetic . It is curious, that it is precisely the lack of its consequential application, that has made the continuous existence of this opinion possible ” (Frege 1885/1984, 114).
Michael Dummett illuminates the two facets nicely: “It is when he is criticising empiricism that Frege insists on the gulf between the senses of mathematical propositions and their applications; it is when he criticises formalism that he stresses that applicability is essential to mathematics ” (Dummett 1991a, 60) .
- 8.
For a corresponding passage , cf. Frege (1881/1979, 13). Øystein Linnebo takes the first passage as evidence for the claim that Frege at the time subscribed to a Kantian formal understanding of logic (Linnebo 2003, 2013). In my view, he does not take into consideration the wider context, namely, the attempt on Frege’s part to uncover the difference in ambition between his Begriffsschrift or formula-language and that of Boole .
- 9.
The expression contains three content -strokes ; one connected to the judgement-stroke , and one preceding each of the arithmetical equations.
- 10.
- 11.
Cf. Frege (1880?/1979, 7, 1897b/1979, 139, 1906b/1979, 185, 1906c/1979, 198, 1919/1979, 253, 1925/1979, 267). To Frege , after introducing the sense-meaning distinction, a sentence is a name that expresses a thought and means an object, either the True or the False. I shall refrain from raising the complex issue as to whether Frege’s position is adequately captured by the label “realism”. For two competing accounts, cf. Carl (1994, 137–160) and Dummett (1981, 428–473) .
The reson I refer to “Logic” by “1880?” is that the editors of Frege, Posthumous Writings are uncertain about when it was drafted; they suggest sometimes between 1879 and 1891.
- 12.
In the normative phrasing of Gilead Bar-Elli , Frege , in common with Quine , subscribes to the “homogeneity of truth” principle: “Whatever your metaphysical view of truth is, it should not recognize radically different kinds or notions of truth but should be homogenous across the board” (Bar-Elli 2010, 180) . Thomas Ricketts uses the phrase “monolithic view of truth” (Ricketts 1996, 126–127) .
- 13.
Jeshion ascribes to Frege “a foundational hierarchy” of propositions (Joshin 2001, 939), the view that there are simples that “constitute the essence of a discipline” and an idea about “the natural order of truth” (Jeshion 2001, 947) . Steven Shapiro , in an account deeply influenced by Jeshion , label their common reading a “metaphysical-cum-epistemic account” (Shapiro 2009, 184), and he ascribes to Frege “a large dose of pre-established harmony” (Shapiro 2009, 185). To ascribe to Frege such a “thick” rationalistic and pre-Kantian metaphysics —unprovable and speculative as it is—goes against the view on science and justification formulated here.
- 14.
Cf. Weiner (1990, 230) and Ricketts (2010, 191–192) . As Frege says in “On the Foundations of Geometry : Second Series”: “Just as the concept point belongs to geometry , so logic, too, has its own concepts and relations; and it is only in virtue of this that it can have a content […] to logic for instance, there belongs the following: negation , identity , subsumption, subordination of concepts” (Frege 1906a/1984, 338). John MacFarlane (2002, 33) invokes this passage in his criticism of Ricketts ’s ascription to Frege of the view that, “in contrast to the laws of special sciences like geometry or physics, the laws of logic do not mention this or that thing. Nor do they mention properties whose investigation pertains to a particular discipline” (Ricketts 1985, 4–5) . MacFarlane , however, misses the point, as no serious scholar, certainly not Ricketts, would ever dream of denying that there are logical constants; see for instance Ricketts (1996, 123) .
- 15.
In their respective articles in the present volume, Gisela S. Bengtsson (Chap. 7) and Gottfried Gabriel (Chap. 2) prefer other renderings of “verständnisvollen Entgegenkommen” than “meeting of minds”. Bengtsson prefers “willingness to understand”, and Gabriel , “accommodating understanding ”. I have changed the translation of “Erläuterungen” to “elucidate” and “elucidations ” rather than keeping “illustrative examples ” and “new meanings ”.
- 16.
Frege is fully aware that there are always alternative axiomatic systems , so that a judgement could be an axiom in one system and a theorem in another (Frege 1879a/1972, § 7; 1914/1979, 205). I cannot see that this poses a problem for the present reading, as Frege does not accept a holistic or pragmatic strategy for justifying axioms ; cf. Shapiro (2009).
- 17.
For the term “primitive truth”, cf. Frege (1914/1979, 204).
- 18.
The full passage is cited in Ricketts (1996, 126) ; Gisela Bengtsson drew it to my attention. The very same allusion (although concerning arithmetic and not logic) is made also in another context, namely in the criticism of formalism :
A clear-cut separation of the domains of the sciences may be a good thing, provided no domain remains for which no one is responsible. We know that the same quantitative ratio (the same number ) may arise with lengths, time interval, masses, moments of inertia, etc.; and for this reason it is likely that the problem of the usefulness of arithmetic is to be solved—in part, at least—independently of those sciences to which it is to be applied. Therefore, it is reasonable to ask the arithmetician to undertake the task, so far as he can accomplish it without encroaching on the domains of the other special sciences. To this end it is necessary, above all things, that the arithmetician attach a sense to his formulas; and this will then be so general that, with the aid of geometrical axioms and physical and astronomical observations and hypotheses, manifold applications can be made to the sciences . (Frege 1903/1952, § 92)
- 19.
The alternation of capital and small letters between the explication and the axiom is because in the former case a relation is introduced, while in the latter case we have an assertion .
- 20.
- 21.
On the confusions involved in the latter idea , cf. Frege (1880?/1979, 2–3).
- 22.
This passage stands in debt to comments by Joan Weiner .
- 23.
Here, as in the passage from Grundgesetze looked at above, the German “Verwirrung” is translated as “confusion”.
- 24.
M. Dummett and R. Heck relate the issue about the justification of the logical law to the familiar Cartesian Circle, according to which an attempt at justifying a logical axiom must itself presuppose logic. Both try to escape, or at least minimise the damage of the circle by way of semantic considerations (Dummett 1991b, 200–215; Heck 2010, 342–358) . In opposition to this, Frege thinks that his basic laws cannot, and need not be justified, but might be elucidated. Accordingly, we should not ascribe to Frege the task that occupies these semanticists.
- 25.
There is an illuminating discussion about the non-definitional, non-stipulative status of Axiom V in Frege (1903a/1952, § 146).
- 26.
- 27.
In his Begriffsschrift account of identity , Frege maintained that “Identity of content differs from conditionality and negation by relating to names , not to contents ”, and that “with the introduction of a symbol for identity of content , a bifurcation is necessarily introduced into the meaning of every symbol ” (Frege 1879a/1972, § 8). By the new account, this bifurcation of meaning is avoided.
- 28.
I have modified Max Black ’s translation by translating “Erkenntniswert” as “epistemic value ” and not “cognitive value ”—this translation connects to the numerous places where Frege uses “Erkenntnis”, “Erkenntnistheorie” and constructions such as “kennst man”—and by rendering “Sätze” as “sentences” rather than “statements”.
- 29.
Cf. also Frege (1903a, § 138) .
- 30.
I here sidestep the rather complex issue as to whether Frege’s reconstruction of his former account of identity is adequate.
- 31.
The German original of, “In spite of this, the sense of ‘b’ may differ from that of ‘a’, and thereby the thought expressed in ‘a = b’ differs from that of ‘a = a’” is “Trotzdem kann der Sinn von “b” von dem Sinn von “a” verschieden sein, und mithin auch der in “a = b” ausgedrückte Gedanke verschieden von dem [in] “a = a” ausgedrückten sein”. Cf. also the first sentence of the concluded passage of “On Sense and Meaning ”, cited above.
- 32.
Within the context of the philosophy of propositional attitudes and other (purported) intensional contexts , this crucial presupposition for epistemic value is often overlooked. Nathan Salmon , for instance, translates Erkenntniswert by the highly loaded, semantic, and misleading “cognitive information content ” (Salmon 1986, 47).
- 33.
Cf. Frege (1903b/1984, 274), where Frege talks about one epistemic value as “no greater” than another; see also the quote below.
- 34.
For a simple and clear presentation of the different account of axioms by Frege and Hilbert , cf. Eder (2015) . (I do not agree to all of the author’s claims about Frege and independence proofs, but that is more or less independent of the exposition of the basic disagreement at stake.)
- 35.
Cf. the aspect about kinds of stipulations towards the end of Part II.
- 36.
Cf. Floyd (1998) .
- 37.
Towards the end of this part, I present my understanding of the role of the Kantian distinctions in Frege’s philosophy .
- 38.
- 39.
- 40.
Cf. the discussion about Begriffsschrift formula 133, below.
- 41.
Cf. Frege (1893/1967, § § 3, 9 and 11). A major consequence of these new complex requirements of the Zerlegungen is that they no longer carry the strong intuitive appeal of those of Begriffsschrift .
- 42.
Cf. note 12 and the discussion leading up to it.
- 43.
In Grundlagen, Frege paraphrases the judgement thus: “If the relation of every member of a series to its successor is one-or many-one, and if m and y follow in that series after x, then either y comes in that series before m, or it coincides with m, or it follows after m” (Frege 1884/1968, § 91).
- 44.
- 45.
As said in one of the reviews of Begriffsschrift :
I cannot completely agree with the comments which Frege makes about the relations which the fundamental concepts of mathematics have to each other. I cannot agree that the concept of ordering-in-a-sequence can be reduced to that of logical ordering, let alone the concept of number can be advanced by investigations into ordering-in-a-sequence. On the contrary, the concept of ordering-in-a-sequence is a secondary one, dependent upon the concept of time; while the concept of number is a primary mathematical one—indeed the simplest, most general concept of all (Michaëlis 1880/1972, 217) .
C. Th. Michaëlis not only takes formula 133 to be synthetic and based in intuition , he further warns Frege against pursuing the project announced in the introduction to Begriffsschrift (cited in note 2 above).
- 46.
I use the German terms since Frege’s play on words is lost in the translation.
- 47.
- 48.
On this very page, the plural “Erkenntnisquellen” is translated as “springs of knowledge ”, and the singular “Erkenntnisquelle” as “source of cognition”. This slide between “knowledge ” and “cognition” is characteristic of the standard translations of Frege’s writings .
- 49.
Cf. also Weiner (2004, 120 (n6))
- 50.
My reasoning about Frege’s use of the Kantian notions in his classification of judgements should have made it clear that Carl ’s worry (in the cited passage from Carl (1994, 147) ), that an alternative rendering of epistemic value would mean that Frege changed his view after Grundlagen, is unwarranted.
- 51.
Cf. also the discussion about the judgement-stroke in Martin Gustafsson’s article (Chap. 6) in this volume.
- 52.
As argued by Juliet Floyd , since definitions proper belong to the system, the Begriffsschrift has a potentiality for an ever-growing vocabulary (Floyd 1998) .
- 53.
Grundlagen, Part V. Conclusion, begins thus: “I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgements and consequently a priori. Arithmetic does become simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one ” (Frege 1884/1968, § 87). One reason why logicism is probable, but not certain, is that this guarantee can be obtained only by way of an axiomatic system .
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Acknowledgments
This article is a modified version of the talk “Logic as Substantial Science”, held at the Conference Frege zwischen Dichtung und Wissenschaft, University of Bergen, December 5, 2014. I am grateful to the audience and the organisers of the conference for putting together a stimulating group of scholars and for ensuring an excellent social atmosphere. Although I am thankful to all participants for comments and reflections, I would like to mention three of them in particular. Juliet Floyd gave a most useful response to my talk, and Gisela Bengtsson and Joan Weiner, in addition to verbal responses, wrote comprehensive and quite instructive comments. This improved my line of thought and made possible the transformation of my talk into an article. The editors Gisela Bengtsson and Simo Säätelä suggested several useful improvements during the writing of the final versions.
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Alnes, J.H. (2018). Frege’s Unquestioned Starting Point: Logic as Science. In: Bengtsson, G., Säätelä, S., Pichler, A. (eds) New Essays on Frege. Nordic Wittgenstein Studies, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-71186-7_3
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