Numbers in Elementary Propositions : Some remarks on writings before and after Some Remarks on Logical Form

It is often held that Wittgenstein had to introduce numbers in elementary propositions due to problems related to the so-called colour-exclusion problem. I argue in this paper that he had other reasons for introducing them, reasons that arise from an investigation of the continuity of visual space and what Wittgenstein refers to as ‘intensional infinity’. In addition, I argue that the introduction of numbers by this route was prior to introducing them via the colourexclusion problem. To conclude, I discuss two problems that Wittgenstein faced in the writings before Some Remarks on Logical Form (1929), problems that are independent of the colour-exclusion problem but dependent on the introduction of numbers in elementary propositions. 1. Numbers in Some Remarks on Logical Form In Some Remarks on Logical Form (RLF) Wittgenstein wrote: I wish to make my first definite remark on the logical analysis of actual phenomena: it is this, that for their representation numbers (rational and irrational) must enter into the structure of the atomic propositions themselves. (RLF: 165) In the same article, Wittgenstein justifies the need for introducing numbers in propositions consisting of an assignment of a degree of Anderson Luis Nakano CC-BY 86 a property that admits gradation (e.g. colour) to a certain object by showing (or rather arguing very briefly) the impossibility of an analysis in terms of a logical product and a “completing supplementary statement”. Given the fact that each degree of a quality excludes every other, Wittgenstein was led to abandon one of the cornerstones of the Tractatus, the thesis of the logical independence of elementary propositions. While some commentators have found Wittgenstein’s arguments for the unanalysability of statements of degree cogent and even obvious, others have raised doubts concerning the force of the argument vis-à-vis the Tractarian background. I shall not revisit this issue here but would point out that the introduction of numbers in elementary propositions does not occur only within the context of ascriptions of properties that admit gradation. Numbers were already used by Wittgenstein in passages pre-RLF and even in RLF itself to demarcate a place in a space (e.g. the visual space). In this connection, I shall argue that there is strong evidence in pre(and post-) RLF passages (from MS 105-106 as presented in the Wiener Ausgabe) for taking Wittgenstein to have had other reasons for introducing numbers in elementary propositions, reasons independent of the colour-exclusion problem. I begin by taking a closer look at the role played by numbers in RLF. After stating that numbers must enter into the structure of atomic 5 propositions, Wittgenstein asks the reader to imagine a 1 See Hacker 1986: 108-9; see also Marion 1998: 123. 2 See Ricketts 2014; see also Lugg 2015. 3 Wittgenstein’s manuscripts were transcribed and put into chronological order in the Wiener Ausgabe edition, from which I shall quote the relevant passages. In the manuscripts, Wittgenstein used first the recto and then the verso pages and for this reason the order of the pages is not the order in which they were written. The first volume of the Wiener Ausgabe edition (hereafter Wi1) contains the remarks made in MS 105 and 106 (for the most part undated). The material from the manuscripts that covers the text of RLF (1929) is found at MS 106 pp. 71–111 (Wi1, pp. 55–63), cf. Wittgenstein Source . By preand post-RLF passages I mean the passages written before and after these pages. For information about the chronological order of MS 105-6, see Engelmann 2013. 4 I am using the expression “colour-exclusion problem” to refer to the general issue (not limited to colours) of ascriptions of properties that admit gradation. 5 The term “atomic proposition” is used by Russell and is equivalent to the tractarian “elementary proposition”. Nordic Wittgenstein Review 6 (1) 2017 | pp. 85-103 | DOI 10.15845/nwr.v6i1.3438 87 system of rectangular axes drawn in the visual field together with an arbitrarily fixed scale (in short, a coordinate system). He continues: It is clear that we then can describe the shape and position of every patch of colour in our visual field by means of statements of numbers which have their significance relative to the system of co-ordinates and the unit chosen. Again, it is clear that this description will have the right logical multiplicity, and that a description which has a smaller multiplicity will not do. (RLF: 165) He then gives the example of the use of such a coordinate system to describe a patch and attribute to it the colour red. He takes the proposition to be symbolized as “[6–9, 3–8] R” and argues that the unanalysed term “R” must contain numbers when properly analyzed inasmuch as a specific degree of red is being assigned to the patch. It is tempting to think that an examination of this proposition suffices to show that numbers are already present in elementary propositions. After all, “[6–9, 3–8]” includes numbers, these numbers being used to designate the “object” of which red is predicated. Since “R” is the only “unanalyzed term”, an analysis of this proposition, whatever it may be, will have to include numbers in elementary propositions. It is hard to see, then, why Wittgenstein needs to show that “R” too includes numbers. This temptation should be resisted, however. For it may be the case that the representation of a place in visual space by means of numbers is a merely feature of a particular symbolism. That it is not, i.e. that it is, as Wittgenstein puts it, “an essential and, consequently, unavoidable feature of the representation” (RLF: 166) has to be justified. I take the argument that Wittgenstein presents in the fourth paragraph of RLF as a justification to introduce numbers in statements expressing the degree of a quality. But it is important to note that, although the quality that is assigned to the patch “[6–9, 3–8]” admits gradation, the patch itself is not the degree of any quality. The reason for this is that the description of the spatial characteristics of a patch in visual space (its size and position) is not a (true/false) proposition at all, the patch being identifiable only by its size and position. That is, the size and position of a Anderson Luis Nakano CC-BY 88 patch in visual space are not (external) properties of a thing that can be independently identified inasmuch as the size and position of a patch constitute the patch. The ascription of a colour to a patch in visual space, by contrast, is a proposition, because the object to which the colour is predicated is identified independently of its colour, namely, by its size and position. Therefore, the numbers appearing in the symbol “[6–9, 3–8]” do not refer to the degree of any quality that would be ascribed to a thing. In this case, the problem of how to analyze propositions expressing the degree of a quality simply does not exist. Because of this asymmetry, I do not take the argument that Wittgenstein presents in the fourth paragraph of RLF as providing 6 There is a passage in WWK: 75 which seems to contradict what I am saying. In this passage Wittgenstein apparently treats the description of a rectangle by giving its coordinates and the description of its colour as having the same status. The most problematic sentence from this passage is the following: “Jedes Rechteck kann ich beschreiben durch vier Zahlenangaben, nämlich durch die Koordinaten des linken oberen Eckpunktes, durch seine Länge und durch seine Breite, also durch (x, y; u, v). Die Angabe dieser vier Koordinaten ist mit jeder anderen Angabe unverträglich. Ebenso ann ich die Farbe des Rechtecks beschreiben, indem ich gleichsam die Farbenskala anlege”. The idea would be that two spatial specifications (x1, y1; u1, v1) and (x2, y2; u2, v2) are incompatible because they are specifications of the same rectangle (they predicate incompatible properties of the same substrate). I cannot see, however, how the identity of a rectangle, in the context of a complete description of visual space, could be given independently of its size and position. For the size and position of a thing can only be predicated of it if they are not criteria for identifying it, if they are external properties of it (i.e., if it is thinkable that the thing does not possess this property). In cases like “red is a colour”, where it seems that an internal property is being predicated of a thing, what we actually have is not a proposition, but the specification of a variable's value (see PB: I-3b). So it seems that in the above passage Wittgenstein is simplifying the matter to make a point (in the broader context of the conversation) that is independent of the distinction I am drawing. This distinction, however, is present in some passages where Wittgenstein considers the matter more thoroughly. For instance, in PB: IX-96b, he says that “red” and “circle” are not “properties” that are on the same level, for “it is easy to imagine what is red but difficult to imagine what is circular”. He goes on to say that “the position is part of the form”. In other words, position is a formal (internal) property, not a material (external) one, and this is the point I am stressing. It is worth also taking a look at WWK: 54, where Wittgenstein says that “von diesen zwei bestimmten Strecken ist es freilich nicht denkbar, daß die eine länger oder kürzer ist als die andere”. The idea here is that since the length of a line segment is an internal property of it, it does not makes sense to say it is longer or shorter than another given line segment. This would show itself in the symbolism for representing these line segments. I would like to thank an anonymous referee for drawing my attention to the passage of WWK: 75. Nordic Wittgenstein Review 6 (1) 2017 | pp. 85-103 | DOI 10.15845/nwr.v6i1.3438 89 a reason for thinking that numbers should occur in symbols f