The Centrality of Simplicity in Frege's Philosophy

Abstract

It is widely recognized that Frege's systematic conception of science has a major impact on his work. I argue that central to this conception and its impact is Frege's Simplicity Requirement that a scientific system must have as few primitive truths as possible. Frege states this requirement often, justifies it in several ways, and appeals to it to motivate important aspects of his broader views. Acknowledging its central role illuminates several aspects of his work in new ways, including his treatment of truth values, logical concepts, and the dependence relations that induce a ‘natural order’ of truths.

Keywords:

• Frege
• science
• philosophy of science
• Simplicity
• cognitive value

Acknowledgments

I've been working on some form of this paper for a long time. I learned a lot about it from many members of the UC Berkeley philosophical community, especially John Campbell, Klaus Corcilius, Tim Crockett, Richard Lawrence, John MacFarlane, Paolo Mancosu, Hans Sluga, Justin Vlasits, and Daniel Warren. More recently, the paper benefitted from discussions with Kirsten Pickering, Gurpreet Rattan, Yuan Wu, and participants at the Boston meeting of the Society for the Study of Analytic Philosophy, especially Thomas Ricketts and Joan Weiner.

Notes

1 Frege 1914, 261. When no translator is mentioned in the References, any translations are my own.

2 Frege 1914, 220–221, reviews the basics. E.g.: ‘once we successfully discovered these primitive truths […] then [the science] will show itself as a system of truths that are connected with each other by logical inferences.’ Once each other truth is connected to the primitive truths by chains of such inferences, ‘All of these inference-chains constitute the proof of our theorem.’

3 See de Jong and Betti 2010 on the influence of the systematic conception—which ‘count[ed] among its pronounced followers Newton, Pascal, Spinoza, Descartes, Leibniz, Wolff and Kant, and still later Bolzano, Husserl, Frege and Leśniewski’—and discussion of some variations.

4 Views about especially valuable cognitive states today still have much in common with traditional views of scientific systems: compare, for example, the state of understanding described in chapter 8 of Kvanvig 2003.

5 Recall that to guide his inquiry into what numbers are, Frege 1884, § 5, cites a purported ‘need of reason’ concerning primitive truths in scientific systems—where its being reason in particular that strives for scientific systems is a theme of the Appendix to the Transcendental Dialectic in Kant 1781/1787. Less obviously, Frege's view on the seemingly peripheral question whether ‘the primitive truths of a systematic natural [i.e. empirical] science […] include supporting data for its laws’ or whether they are all themselves laws proves relevant to the central question why he tries to demonstrate logicism about arithmetic. See Weiner 2004, 123; Jeshion 2001; and Jeshion 2004. For examples involving other thinkers, see, e.g. Sorell et al. 2010.

6 Frege 1884, § 2.

7 Frege 1880–1881, 40–42. A mistake in the standard translation obscures the connection between simplicity and science in this paper. Frege is comparing his Begriffsschrift, which is meant to express a scientific system, to Boole's ‘logical calculus’, which is meant only as a problem-solving tool. Since minimizing the number of primitive truths is a requirement of science, Frege observes that since Boole is ‘only concerned to solve his problems in a brief and practical way’ rather than to express a scientific system, it is no surprise that ‘in his case, the striving to manage everything with as few primitive laws as possible is not noticeable.’ (‘Bei ihm ist ein solches Streben, mit möglichst wenigen Urgesetzen alles zu leisten, nicht bemerkbar.’) But in their translation for Frege 1979, 37, Long and White have him say: ‘In his case there is nothing remarkable in the attempt to manage everything with the fewest possible primitive laws,’ making it sound like what is no surprise is that Boole does attempt to minimize their number.

8 Frege 1893, vi. Note that here and in the Early quote, since Frege is discussing sciences all of whose primitive truths are laws, he puts the requirement in terms of those things.

9 Frege 1914, 221.

10 Sober 2015, 2, 4; Sober 2002, 14

11 Of course, in addition to accepting the Simplicity Requirement, he might also think we must minimize something else; and minimizing the number of primitive truths might even require minimizing something else. We will soon see what Frege thinks about some other things that one might minimize.

12 Frege 1914, 261; Frege 1880–1881 44, 40.

13 Frege 1914, 221.

14 Other references to the primitive truths as a ‘seed’ are made at, e.g. Frege 1879, § 19 and Frege 1885, 96—although strictly speaking, the latter claims the content of a science to be contained in certain properties as in a seed, and goes on to claim that these properties are ‘expressed’ in the primitive truths. A similar metaphor appears when Frege 1884, § 88, claims that consequences can be contained in definitions either as plants are in seeds, or as beams are in houses. Despite the common metaphor, though, it is hard to see any connection between this point about definitions and the minimization of primitive truths. Thanks to an anonymous reviewer for noting the possible relevance of the definition passage.

15 Swinburne 1997, 1.

16 Weiner 1990 emphasizes this goal.

17 Schlimm 2017, § 3.3.1, thinks Frege's Simplicity Requirement is justified by the requirement that proofs be gap-free, since more primitive truths ‘would make it more difficult to keep track […] easier to overlook some […] putting at risk the overall goal of gap-free derivations.’ But I do not see Frege give this justification in the paragraph on which Schlimm is commenting, nor anywhere else. Nor, I think, could he: again, as long as the number is manageable, there is no real danger of losing ‘track’.

18 Friedman 1974. As with Frege, this reduction is a matter of ‘deriving’ one thing from another, and Friedman sometimes talks (like Frege) of reducing ‘the total number of […] sentences’ rather than of ‘phenomena.’ Friedman appears to have neglected Frege when he claims that ‘the only writer that I am aware of who has suggested that this [simplification/unification] is the essence of explanation…is William Kneale,’ 15.

19 Readers who wish to evaluate this sort of view in a non-historical way can consult discussions surrounding the unification theory, and those already familiar with some of its challenges will recognize Frege addressing forms of them below. For example, §§ 3.1–3.2 discusses challenges posed by the possibility of trivially reducing primitive truths by conjoining them or adding inference rules. Addressing the former, which had been pointed out by Hempel and Oppenheim 1948, footnote 28, was a major goal of Friedman 1974. Recent discussions of the unification theory largely focus on the variation provided by Kitcher 1989 which, as pp 431–432 makes clear, aims to preserve Friedman's core idea while avoiding technical problems that arose from his attempt to address such challenges.

20 E.g. Frege 1914, 221. ‘Science demands that we prove whatever is susceptible of proof.’

21 Frege 1884, § 2.

22 Frege 1893, vi: ‘Es muss danach gestrebt werden, die Anzahl dieser Urgesetze möglichst zu verringern, indem man Alles beweist, was beweisbar ist.’ (My italics.)

23 Frege 1880–1881, 40, 52. In discussing Frege's effort to minimize the number of primitive signs, Bellucci et al. 2017 fail to note that he does so in order to minimize the number of primitive truths, which leads their comparison with Peirce to miss a potentially illuminating connection to Peirce's notion of analysis, on which ‘the fewer the primitive symbols, the fewer the axioms; and the fewer the symbols and axioms the more analytic the system.’ Schlimm 2017, by contrast, notices this justification, and partly for this reason, his brief discussion of these issues, §§ 3.3.1–3.3.2, is the best I know—though see footnotes 17 above and 40 below.

24 Frege 1880–1881, 42.

25 Frege 1893, vi, § 14.

26 Frege 1893, x. Dummett 1973, 183, thinks Frege's talk of simplicity here instead concerns a ‘simplification in […] ontology,’ while Burge 1986, 113–115, thinks it is about making available ‘analogies that are quite natural within a formal context,’ and allowing ‘the simplest construal of the Composition Principle.’ But these readings are not very plausible. For unlike minimizing the number of primitive truths, Frege has not mentioned the importance of such ‘simplifications’ in this passage—nor indeed does he ever, to my knowledge. Moreover, since there is no requirement to pursue them—at least, Frege certainly never endorses any—Dummett is probably right that it would be a ‘blunder’ to achieve these sorts of ‘simplification […] at the price of a highly implausible analysis of language.’ Frege surely has not made this blunder; surely he had his Simplicity Requirement in mind instead.

27 The discussion of the conditional is at Frege 1880–1881, 40–44; of negation, at Frege 1918–1919, 154. Interestingly, his justification for including the concept of generality may also appeal implicitly to the Simplicity Requirement. Frege 1923/1925, 278, claims that the ‘value’ that justifies the choice of this concept is found in the way it enables us to make a claim that ‘contains many—indeed infinitely many—particular facts as special cases.’ He does not here say why this is valuable, but he has elsewhere: doing so ‘controls a large—possibly unsurveyable—manifold through one or a few sentences.’

28 Previous footnotes have pointed out how commentators have missed the role of the Simplicity Requirement in justifying particular decisions. General discussions of Frege's version of the systematic conception of science exhibit a similar neglect. de Jong 1996, for example, introduces Frege's view of systematic science with a large block quote, in the course of which Frege says ‘we must try to diminish the number of these primitive laws as far as possible.’ Soon after, de Jong apparently glosses this as the claim that ‘every proposition should be proved from a limited number of principles,’ which becomes, later, ‘there are […] a (finite) number of fundamental propositions.’ To be ‘limited’ and ‘finite’ is not to be minimized; so that is not the Simplicity Requirement. That requirement is never formulated clearly by de Jong; despite being quoted, it has been, at best, passed over without comment. If mentioned at all, it is either misinterpreted or breezed over by Detlefsen 1988, Jeshion 2001, Shapiro 2009 and Macbeth 2016.

29 Schröder 1880 charged that Frege's Begriffsschrift was not very interesting because it ‘does not differ essentially from Boole's formula language.’ The Early passage is from Frege's response. See Sluga 1987 for discussion.

30 Frege 1884, § 89. For Kant's endorsement of the systematic conception and of the minimization of the number of ‘principles,’ see Kant 1781/1787, A649/B677-A650/B678. On the dominance of the Neo-Kantians, see, e.g. the General Introduction to Beiser 2014. Such influential Neo-Kantian figures as Liebmann 1876 had affirmed (my italics) that ‘No science can be counted as perfect and completed […] until it […] forms a logical whole, in which […] an absolutely minimized number of […] primitive sentences […] [a] narrow tip of primitive thoughts […] flows into the broad—indeed infinite—group of […] details’ (8). I think it worth noting the parallel between the construction of Liebmann's ‘broad—indeed infinite’ and two of Frege's in making the same point, noted in footnote 27 above: ‘many—indeed infinitely many’ and ‘large—possibly unsurveyable.’

31 Thanks to Robert May for suggesting the relevance of this point.

32 To get the idea, compare the truth that if P, then $\mathrm{\neg }\mathrm{\neg }$P with an inference rule allowing one to always conclude $\mathrm{\neg }\mathrm{\neg }$P from P.

33 Many contemporary logicians prefer to operate exclusively with inference rules rather than with Hilbert-and-Frege-style axiomatic systems. Dummett 1973, 433-434, credits Gentzen as ‘the first to correct [Frege's] distorted perspective.’

34 Frege 1879, § 13; Frege's term is ‘Abbild.’

35 Frege 1914, 219; Frege 1880–1881, 42.

36 Frege 1879, § 13.

37 Depending on how literally Frege takes his claim that inference rules and primitive truths are the ‘same thing’, it may be that in the end, he really accepts not the Simplicity Requirement as he states it, but a requirement to minimize the number of primitive truths and inference rules.

38 Or at least, unless all true sentences refer to one object, and all false sentences to another.

39 Recall that Begriffsschrift has no primitive conjunction function, and that Frege recognizes distinct but logically equivalent truths when he identifies truths with thoughts and takes logical equivalence not to imply sameness of thought. See Frege 1902 and Frege 1914, 253.

40 Wittgenstein 1921, 6.1271. See, e.g. Frege 1879–1891, 3, and Frege 1884, § 5, on this requirement, and Weiner 2020, 142 and Weiner 2007, 681 for discussion. Schlimm 2017, § 3.3.2, seems to treat the constraint on minimizing the number of primitive truths provided by this other requirement as a reason to downplay the Simplicity Requirement: to describe Frege, for example, as merely ‘hinting’ at a ‘quantitative assessment’ of a system's contribution to cognitive value. But it no more does so than does the constraint placed by the requirement that primitive truths be true, without which many new possibilities for reduction would appear.

41 E.g. Frege 1884, § 3; Frege1924/1925a, 286–288.

42 Frege 1884, § 5; Frege 1924/1925, 288–292. Though what Frege means by ‘self-evidence’ is a difficult interpretive question, on no reasonable construal—e.g. Burge 1998, Jeshion 2001, Hutchinson 2021—does the fact that two truths are self-evident guarantee that a truth conjoining them is, or the fact that a truth is self-evident guarantee that more complex but logically equivalent truths are too.

43 This is Basic Law V; see the Appendix to Frege 1903.

44 They may also do so in other ways. For example: where Frege 1879, §§ 20–21, had relied on two primitive truths involving identity, both are proved from the single Basic Law III in § 50 of Frege 1893—and the truth-values are implicated in these proofs by allowing the negation and horizontal functions as substitutions for a second-level variable. Thanks to an anonymous reviewer for suggesting this.

45 Frege 1893, vi and § 14.

46 Frege 1893, vi.

47 Frege 1884, § 2.

48 Frege 1879, § 13, contrasts the goal to ‘make [truths] more certain’ with that of letting ‘the relations of the [truths] to one another emerge,’ while Frege 1914, 220, contrasts the goal to ‘convince us of the truth of what is proved’ with that to ‘bring out logical connections among truths.’ Though Frege talks only of ‘relations’ and ‘logical connections’ in these passages, the goal of proof is presumably not to reveal any old relations among truths, nor even any logical ones. He is presumably talking about the same relations that he elsewhere describes in terms of dependence.

49 Frege 1884, § 17. Detlefsen 1988, 98, Jeshion 2001, 945, and Shapiro 2009, 183 read it this way.

50 Leibniz 1765 Book IV, Chapter VII, § 9: ‘It is here not about the history of our discoveries, which is different in different men, but the connection and natural order of truths, which is always the same.’ The view that proofs should follow the order of discovery was attributed by Leibniz to his opponent, Locke. See Wilson 1967.

51 Leibniz 1680–1684, 267.

52 Leibniz 1765, Book IV, Chapter XII, § 6. The speaker is Theophilus, who represents Leibniz's view.

53 Frege 1884, § 15. It is worth mentioning that Leibniz 1765 thinks—Book IV, Chapter VII, § 7—that humans cannot do most proofs from identities: I cannot, for example, prove that I exist from the claim that I am me, because ‘only God sees how these two terms me and existence are connected.’ Leibniz thus contrasts a natural order which only God can follow with a second natural order which he recommends for us. Frege cannot think of dependence relations as inducing Leibniz's second order either, because in it, we do not prove anything that is already maximally certain unless we can do so from identities. This is why the Cartesian Principle ‘I am, is an axiom, and […] a primitive truth […] in the natural order of our knowledge.’

54 Bolzano 1837, § 525.

55 Thanks to Joan Weiner for pointing this out to me. Frege does talk of ‘on what, in the deepest grounds, the justification for holding something to be true rests,’ ‘grounds of proof,’ ‘grounds of judgement,’ and ‘grounds of justification,’ but this talk is not distinctively Bolzanian, and he never talks of grounding relations among truths. At one point, he mentions in passing the ‘relation of ground and consequent,’ but never elaborates. See Frege 1884 §§ 3, 17; Frege 1879–1891, 3; and Frege 1880–1881, 42. Some commentators on Frege talk interchangeably of ‘dependence’ and ‘grounding,’ but they mislead themselves by doing so. Jeshion 2001, for example, argues that Frege sees an important difference between a priori sciences and empirical ones partly because he ‘never says that primitive truths of empirical sciences ‘ground’ other non-basic truths.’ I think that is no difference: I think he never says that any truths ‘ground’ any others.

56 It receives serious attention from, e.g. Detlefsen 1988, Jeshion 2001, and Shapiro 2009.

57 Shapiro 2009, 184, claims that ‘like Bolzano's […] ground-consequence relation, Frege's dependency relation is asymmetric: if proposition A depends on proposition B, then B does not depend on A.’ Detlefsen 1988, endnote 7, agrees that ‘the […] relation is asymmetrical.’

58 Frege 1914, 222. He makes the same point at Frege 1879, § 13 and Frege 1923–1926, 49.

59 Perhaps in an effort to resolve the conflict between this passage and the asymmetry assumption, Jeshion 2001, 951, claims that cases like these are ones in which neither A nor B depends on the other: whichever truth we prove ‘still does not admit of proof in the sense that its truth is not grounded on any other propositions.’ But she gives no evidence that Frege thinks this, and indeed he cannot: the whole reason he brings up dependence relations is in discussing why we should prove precisely truths like these, which are acceptable as axioms, and hence not in doubt.

60 To take one more example: Jeshion 2001, 945, observes that Frege 1885 claims that primitive truths involve simple concepts, and extrapolates that the order induced by the dependence relation ‘is a structuring of propositions […] according to their relative simplicity and complexity.’ But even if Frege thinks this order starts from truths involving only simple concepts, this does not imply that the rest of the order continues in order of increased complexity. And he probably does not think it does. After all, a chain of proof in mathematics often starts from primitive truths involving only simple concepts, detours through truths involving complex ones, and ends with truths involving only simple ones again.

61 Suppose that a domain comprises just three truths: T1, T2, and T3, all of which are knowable without proof and otherwise suitable to be primitive. T1 is logically implied by T2 and T3 together, while T1 alone logically implies T2 and T3 both. The Simplicity Requirement requires us to prove T2 and T3 from T1. Recognizing that T1 is logically implied by T2 and T3, then, does not help to minimize the number of primitive truths.

62 Shapiro 2009, 186, points to another such coincidence: that for Frege's Dependence Requirement to be compatible with his requirement that we know primitive truths without proving them, some set of truths must both stand at the head of chains of dependence leading to every other truth and be among those of which we can know without proof. Shapiro sees a ‘large dose of preestablished harmony’ in the idea that the truths are ‘structured in such a pleasing way, a way designed to facilitate proper […] knowledge […] by beings just like us.’

63 Bolzano 1837, § 221.

64 I myself expect that Frege sees accounting for coincidences like these, which will often require reflection on the relative priority of ontological and cognitive notions, as a job for a metaphysician, and hence as inappropriate in ‘logical’ works like his. See Hutchinson 2022.

65 For example, recognizing the role of simplicity in the choice of primitive logical concepts helps Hutchinson 2021 explain why Frege gives arguments for axioms, serving to connect what Frege says with the Neo-Kantian ‘critical method’ for justifying axioms. For another example, see Hutchinson 2020, § 6.2.

66 Dummett 1973, 183–184; see footnote 26 above.

67 See Weiner 1996.

68 See, for example, the first section of Jeshion 2001.