Wittgenstein and Frege on Arithmetical Objects and Concepts
I. Introduction and Preliminary Remarks
This post aims at presenting two essential notions in Wittgenstein’s philosophical account of mathematics duting the middle period of his philosophical carreer: mathematical objects and concepts.
Let me start by making two preliminary remarks: First, to specify to which Frege and which Wittgenstein’s I refer to, and, second, to clarify the scope of Wittgenstein’s philosophy of mathematics during this period.
First of all, it must be recognized that both Wittgenstein and Frege were thinkers whose philosophical views, specially on mathematics –a central topic for both of them–, radically mutated throughout their philosophical careers. In this paper, I focus on two particular periods of each of these philosopher’s careers. For Frege, I focus on his late XIXth century work, specially on The Foundations of Arithmetic (1884) and contemporary articles. For Wittgenstein, I consider only the work of his middle period, the early thirties, as it is expressed in his Philosophical Remarks (PR), Philosophical Grammar (PG) and Big Typescript (BT).
Second, it must be remarked that, for Wittgenstein, ‘mathematics’ just means calculation. This is obviously a radical narrowing of scope for a philosophy of mathematics, since it seems obvious that mathematical practice covers a much wider spectrum of activities than mere calculation. It could even be said that the philosophically interesting aspects of mathematics are precisely those involving much more than mere calculation. Furthermore, it also stands in direct opposition to Frege, for whom theorem proving was the paradigmatic mathematical process. While, for Wittgenstein, mathematical calculations were basic, for Frege, it was mathematical proofs that were basic. It would seem therefore, that the opposition between Frege’s and Wittgenstein’s views could be easily dismissed by arguing that, since their theories covered different aspects of mathematics, they could not come in contention with each other. However, in his Foundations of Arithmetic, Frege aimed not only at giving a logical analysis (and foundation) to the mathematical practice of theorem proving, but also to offer a semantic account of arithmetical statements. It as at this point that Wittgenstein’s and Frege’s thought finally reach a point of conflict, since each one of them attempted at giving an interpretation of such statements based on calculations and proofs, respectively.
Still, Wittgenstein’s philosophical account of mathematical calculi aimed at covering the whole palette of existing mathematics. Nevertheless, in this period, Wittgenstein develops his ideas on mathematics largely through examples from elementary arithmetic. Following his lead, this article explores Wittgenstein’s ideas through his analysis of numerical expressions.
II. Wittgenstein’s Criticism of Frege’s Concept-Object Distinction
A. Numerical Statements
Wittgenstein devotes Section XI of the Philosophical Remarks to numerical statements, which he calls Zahlangaben. A numerical statement is any expression or diagram equivalent to a statement of the form ‘there are n Xs (that are) Y’, where n is a number, X is a common noun and Y a adjective phrase. This category covers both what Wittgenstein calls ‘genuine’ statements like ‘There is a man on this island’ and mathematical ones like ‘There are 6 permutations of 3 elements’. Furthermore, it also includes arithmetical equations. For example, the equation ‘3 + 3 = 6’ is a numerical statement, because it is equivalent to ‘There are six units in 3 + 3’.
One can also say that the proposition ‘there are 6 permutations of 3 elements’ is related to the proposition ‘There are 6 people in this room’ in precisely the same way as is ‘3 + 3 =6’, which you could also express in the form ‘There are 6 units in 3 + 3’. [PR §117 p. 139]
Wittgenstein makes it fairly obvious that not all numerical statements are arithmetical and, furthermore, that they are not all mathematical. Wittgenstein offers a definite way to grammatically distinguish mathematical numerical statements like ‘The interval AB is divided into two (3, 4 etc.) equal parts’, from non-mathematical ones like ‘There are four men in this room’. For Wittgenstein, the distinction between mathematical and non-mathematical numerical statements is grammatical. This means that, at least one important context, they are not exchangeable without grammatical loss. The relevant context is the direct object of the German verb zu berechen [to calculate]. Let p be a mathematical numerical statement and q a non-mathematical one. The sentence ‘I calculate whether p (is the case/true)’ makes sense, but ‘I calculate whether q (is the case/true)’ is nonsense. It makes sense to talk of calculating p, but not q. For example, it makes sense to say ‘I calculate whether there are 5 prime numbers less than 11’ or ‘I calculate whether 3 + 3 = 6’, but not ‘I calculate whether there are four men in this room’ or ‘I calculate whether there are a dozen bowls in my cupboard’. Another way of characterizing this grammatical difference is to let ‘There are n Xs such that Y’ be a mathematical numerical statement, and ‘There are m Zs such that W’ be a non-mathematical one. It makes sense to say ‘I calculate how many X are Y’, but not ‘I calculate how many Z are W’. For example, it makes sense to say ‘I calculate how many permutations are of AB’ but not ‘I calculate how many bowls are in my cupboard’.
Using this grammatical criterion, Wittgenstein divides numerical statements into two major groups. The first group includes statements like ‘There are 4 men in this room’, ‘There are two circles in this square’ and ‘I have as many spoons as can be put in 1-1 correspondence with a dozen bowls’. The second group contains statements like ‘There are 6 permutations of 3 elements’, ‘3+3=6’, ‘There are 6 units in 3+3’, ‘there are 4 prime numbers between 10 and 20’ and ‘A quadratic equation has two roots’. Only the first group expresses genuine [eigentlich] propositions. Genuine propositions are asserted and negated, true or false. Statements in the second group, so-called ‘statements of grammar’, do not describe any possible states of affairs. For Wittgenstein, this means that they do not express genuine propositions. In this sense, mathematical numerical propositions are pseudo-propositions.
This distinction is a special case of the more general distinction between genuine and pseudo-propositions in Wittgenstein. At this point, I agree with Stuart G. Shanker’s Wittgenstein and the Turning-Point in the Philosophy of Mathematics on the view that Wittgenstein’s claim that mathematical propositions are grammatical must be understood in the context of Wittgenstein’s comparison of mathematical and genuine propositions (what Shanker calls “empirical propositions”). In particular, it must be perceived in the context of Wittgenstein’s comparison between “the meaning of a mathematical proposition and its method of proof to the meaning of an empirical proposition and its method of verification.” Wittgenstein purports to state the similarities and differences between these two sorts of propositions, their meanings and verification methods. He aims at answering two questions: ‘Why are they both propositions?’ and ‘Why are mathematical propositions not empirical?’ Wittgenstein claim that mathematical propositions are grammatical answers both questions.
B. Frege’s Grammatical Enquiry into the Concept of Number
Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician. Gottlob Frege
Wittgenstein starts his analysis of numerical statements from Frege’s own investigation on The Foundations of Arithmetic (1884). In this seminal work, Frege defined the notion ‘cardinal number’ through the primitive notion of a concept’s extension or ‘value-range’. The insight behind Frege’s definition was that a cardinal number statement such as ‘There are n F-things’ predicates the number n as a higher-order concept of F Namely, it says that n things fall under F Frege defines the cardinal number of concept F (i.e., the number of Fs) as the concept ‘being a concept equinumerous to F’s extension’. This definition identifies the number of planets as the extension of the concept being a concept ‘equinumerous to the concept of being a planet’. The number of planets is an extension containing all and only those concepts which nine objects exemplify, like the concept ‘being a planet’. Frege writes,
§46. It may throw some light on the matter to consider number in the context of a judgement which brings out the way in which it is in origin applied. While looking at one and the same external phenomenon, I can say with equal truth both “It is a copse” and “It is five trees” or both “Here are four companies” and “Here are 500 men.” Now what changes here from one judgement to the other is neither any individual object, nor the whole, the agglomeration of them, but only my terminology. But that is of itself only a sign that one concept has been substituted for another. This suggests as the answer to the first of the questions left open in our last paragraph [when we make a statement of number, what is that of which we assert something?], that a statement of number contains an assertion about a concept. This is perhaps clearer with the number 0. If I say “Venus has 0 moons”, there simply does not exist any moon or agglomeration of moons for anything to be asserted of; but what happens is that a property is assigned to the concept ”Moon of Venus”, namely that of including nothing under it. If I say “The King’s carriage is drawn by four horses,” then I assign the number four to the concept “horse that draws the King’s carriage.” [p. 59e]
In 1892, Frege expanded his treatment of the ‘concept’ notion in an article for the Vierteljahrsschrift für Wissenschaftliche Philosophie. He used this paper to counteract certain criticisms of the Foundations, especially those from Benno Kerry. Frege found that these objections stemmed from a misunderstanding of his ‘concept’ notion. Previously, in the article “Function and Concept” from 1891, he had defined concepts as functions mapping objects to truth values. Later, Frege needed to clarify his distinction between concepts and objects. In his 1892 presentation, he based his new formulation on the grammatical distinction between proper names and predicates. For Frege, both distinctions are parallel because names refer to objects, while predicates refer to concepts.
The concept (as I understand the word) is predicative [footnote: It is, in fact, the referent of a grammatical predicate]. On the other hand, a name of an object, a proper name, is quite incapable of being used as a grammatical predicate.
From this we can conclude that, at the end of the XIXth Century, Frege saw a close link between the grammatical distinction between subject and predicate and the logical distinction between concept and object.
C. The Grammatical Method
Wittgenstein and Frege shared a strong belief in grammar’s philosophical significance. Both believed that logical distinctions were ultimately grammatical. For Wittgenstein as well as Frege, inventing distinctions not existing in natural language grammar is idle philosophical speculation. For every philosophical category L, and every expression x, the replacement of ‘a’ in at least one grammatically acceptable statement by x makes sense if and only if (the meaning of) x belongs to L. In this sense, every significant philosophical distinction and category corresponds to a grammatical one. The grammatical analysis that Wittgenstein endorses already appears in many of Frege’s arguments. For Frege, two objects or concepts are ontologically different only if they are the meanings of expressions with substantially different grammar. For example, in §29 of the Foundations of Arithmetic, Frege argues that the number word ‘one’ does not stand for a property of objects, because it is not a grammaticaicate.
If it were correct to take “one man” in the same way as “wise man”, we should expect to be able to use “one” also as a grammatical predicate, and to be able to say “Solon was one” just as much as “Solon was wise” . . . This is even clearer if we take the plural. Whereas we can combine “Solon was wise” and “Thales was wise” into “Solon and Thales were wise”, we cannot say “Solon and Thales were one.” But it is hard to see why this should be impossible, if “one” was a property both of Solon and of Thales in the same way that “wise” is. [Pp. 40e, 41e]
In this example, the logical category ‘predicate of objects’ corresponds to the context x (‘Solon was x’). Only expressions that the word ‘wise’ substitute for in the statement ‘Solon was wise’ stand for object predicates. ‘One’ is not among them. In consequence, ‘one’ does not stand for a property of objects. In a similar fashion, in §38, Frege argues that one is a unique object, because ‘one’ functions as a proper name. It makes sense to say ‘the number one’, but not ‘a number one’. Furthermore, it does not have a plural form any more than ‘Frederick the Great’, ‘the chemical element gold’ or any other proper name. ‘One’ is a proper noun. In consequence, one is a unique object. For Frege, the category ‘unique objects’ corresponds to the grammatical category ‘proper names’.
However, it is important not to mistakenly attribute to Frege a view according to which grammatical categories are philosophically prior to ontological ones, as Matthias Schirn has done. For Frege, logical distinctions are read off grammatical ones, indeed. However, there is important philosophical work to do in order to do this reading off. Remember that for Frege, grammar, which has a significance for language analogous to that which logic has for judgement, is a mixture of the logical and the psychological. The work of the logician is precisely to separate the logical from the psychological in grammar.
D. Wittgenstein’s Criticism of Frege’s Grammatical Analysis of numerical statements
Wittgenstein criticizes the grammatical distinction behind the concept and object characterization in Frege’s analysis of numerical statements. The core of this criticism appears in the aptly titled Appendix 2 “Concept and Object. Property and Substrate” of the first part of the Philosophical Grammar. Wittgenstein questions the philosophical value of Frege’s grammatical analysis of statements in the subject-predicate form.
When Frege and Russell speak of concept and object they really mean property and thing; and here I’m thinking in particular of a spatial body and its colour. Or one can say: concept and object are the same as predicate and subject. The subject-predicate form is one of the forms of expression that occur in human languages. It is the form “x is y” (“x = y”): “My brother is tall”, “The storm is nearby”, “This circle is red”, “Augustus is strong”, “2 is a number”, “This thing is a piece of coal.” [PG Pt. I Appendix 2. p. 202],
Wittgenstein complains that the distinction between subject and predicate at the base of Frege’s analysis is undeveloped. “The subject-predicate form serves as a projection of countless different logical forms.” Frege’s analysis is not mistaken, it is limited in scope. It ignores important differences within the ‘object’ and ‘concept’ categories. In particular, it fails to distinguish between genuine objects and mathematical ones.
This notation is built up after the analogy of subject-predicate propositions in ordinary language, such as those describing physical objects. . . And propositions having different grammars, both mathematical and nonmathematical propositions, are dealt with in the same way, e. g., “All men are mortal,” “All men in this room have hats,” “All rational numbers are comparable in respect to magnitude.” [WL Philosophy for Mathematicians 1932-33 §1 p.205]
For Wittgenstein, mathematical and non-mathematical numerical statements involve different concepts. Both sorts of numerical statements can take the form ‘There are n Xs (such that) Y’. However, the concepts in X and Y are different in each. Concepts such as ‘persons’, ‘spoons’, ‘this room’, ‘this square’, ‘my mother’s cupboard’ usually occur in non-mathematical numerical statements. Mathematical numerical statements, on the contrary, contain terms like ‘pure colors’, ‘units’, ‘permutations’ and numerical expressions like ‘3 + 3’, ‘two’ and ‘as many as can be out in 1-1 correspondence with a dozen bowls’. For a numerical statement to be mathematical, Y must be a calculation concept in the arithmetic of X.
The terms occurring in non-mathematical numerical statements are genuine names of objects or concepts. The terms in mathematical numerical statements, on the contrary, do not refer to any genuine concepts. The objects that fall under them are improper [uneigentlich].
We do indeed talk about a circle, its diameter, etc. etc. As if we were describing a concept in complete abstraction from the objects falling under it. – But in that case ‘circle’ is not a predicate in the original sense. [PG Pt. I. Appendix 2. p. 207]
Wittgenstein’s distinction between ‘genuine’ and ‘improper’ concepts and objects springs from his criticism of Frege’s account of numerical statements. The heart of Wittgenstein’s objection is that Frege’s distinction between object and concept is an insufficient analysis of mathematical numerical statements. It misses important conceptual differences between mathematical propositions and genuine ones. Frege was blind to the fact that genuine and mathematical concepts’ cardinalities are radically different. Wittgenstein’s criticism clarifies this difference.
E. Mathematical Objects and Concepts
Wittgenstein’s criticism of Frege’s analysis of numerical statement starts from his account of what Wittgenstein calls ‘descriptions’. Wittgenstein calls ‘description’ any statement of the subject-predicate form. According to Wittgenstein, Frege reads every description as saying that an object, the referent of the subject, falls under a concept, the referent of the predicate. Wittgenstein considers two different kinds of description: internal and external. Internal descriptions ascribe to objects the properties essential for their existence, while external descriptions ascribe accidental properties to them. For Wittgenstein, a property is essential for the existence of an object if its absence “would reduce the existence of the object itself to nothing.” According to Wittgenstein, Frege’s analysis holds for external descriptions, but it fails when applied to internal descriptions. In the aforementioned appendix, Wittgenstein offers the following example:
What is necessary to a description that – say – a book is in certain position? The internal description of the book, i.e. of the concept, and the description of its place which it would be possible to give by giving the co-ordinates of three points. The proposition “Such book is here” would mean that it had these three coordinates. For the specification of the “here” must not prejudge what is here. [PG Pt. I, Appendix 2 pp. 206, 207]
Consider the case in which, pointing at the same object, one makes the following two statements: ‘This book has pages’ and ‘This book is here’. Since having pages is an essential property of any book, the first statement is an internal description. In contrast, the second statement is an example of an external description. It says something about the book. It gives its spatial location. This property is independent of being a book. In the other case, on the contrary, the property of ‘being a book’ already includes ‘having pages’. The internal description does not actually say anything about the object, but about the concept of book under which it falls. The internal description does not describe the book as an object, but the concept ‘book’.
In this sense, the distinction between internal and external descriptions ultimately depends on the concept under which the described object falls. Instead of an object and a concept, every description involves two concepts. The internal or external nature of the description depends on the relation between these two concepts. If the concept in the subject includes or implies the concept in the predicate, the description is internal. Otherwise, the description is external. The spatial location of a book externally describes it, because the concept of book does not include its location. ‘Having pages’ describes it internally, because the concept ‘book’ includes ‘having pages’. This later case describes the concept ‘book’, not any particular book. In consequence, Frege’s analysis mistakenly says that every description describes some object. For Wittgenstein, internal descriptions do not describe objects. They state conceptual relations.
As in Wittgenstein’s example of the book, the same object under the same concept can be subject of internal and external description. For Wittgenstein, this means that books are genuine objects. Describing genuine objects both externally and internally is possible. However, other objects may only have internal descriptions. Under some concepts no proper objects may fall. These concepts occur only in internal descriptions. In the aforementioned appendix, Wittgenstein gives shapes and colors as examples of these concepts. In section XI of the Philosophical Remarks, he adds mathematical concepts to this list.
You might of course write it like this: there are 3 circles with the property of being red. But at this point the difference emerges between improper objects – colour patches in a visual field, etc. etc. – and the elements of knowledge, the genuine objects. [PR §115 p. 136]
For Wittgenstein, mathematical objects are not genuine objects. They have no external properties. All their properties are essential, and essential properties do not describe objects. They describe concepts.
All mathematical terms correspond to mathematical concepts. Even in statements of the subject-predicate form, the subject does not refer. For example, the statement ‘4 is a number’ does not ascribe the property of being a number to the object 4. The mathematical statement does not express a proposition of the form F(a), because subject term a and predicate term F are inseparable. For any mathematical concept, it makes sense to ask whether or not something satisfies it. However, it does not make sense to ask whether or not something that satisfies it exists. It makes sense to ask whether or not there are prime numbers between any two given natural numbers, or filters for a determined algebraic structure. However, ‘there are circles’ does not mean that circles exist besides performing their mathematical role. ‘There are numbers’, ‘circles’, or ‘sets’ only means that those concepts are not extensionally empty. However, the extension of a mathematical concept does not have an existence external to the concept and its intension. In mathematics, ‘there is’ does not equal ‘exists’. Mathematics has no proper existential propositions. For Wittgenstein, mathematical propositions of the form ‘(x) · Fx’ do not mean that an object x (such that Fx) exists. Existential propositions only make sense with genuine concepts. Books exist, but numbers do not. Saying that there are numbers only means that the concept ‘number’ is not empty.
However, this sense of empty is more like consistent, except that consistency meant, literally, without contradiction and Wittgenstein does not think that contradiction plays any privileged role in determining whether a concept is empty in my sense. Instead, all it means is that the rules which determine its calculus can be followed, in a purely pragmatic sense.
In summary, Wittgenstein thought that Frege’s main flaw was not considering his own context principle with sufficient seriousness. If he had, he would have noticed that, just like words, propositions have sense only inside a larger context. He would have noticed that different numerical statements play different roles in language. Some of them are external descriptions – genuine propositions–, while others are internal ones – grammatical rules–. Mathematical numerical statements are the latter kind. The objects and concepts in mathematical propositions are completely different from those in genuine propositions. Mathematical concepts are actually grammatical categories. Mathematical objects are not genuine objects, but roles within a calculus.
III. On Mathematical Equations
A. Mathematical Equations as Identity Statements
‘Every symbol is what it is and not another symbol’. Wittgenstein PR §163 p. 196
After abandoning the semantic outlook of the Conceptual Notation, Frege failed to distinguish between mathematical and non-mathematical numerical statements. For Frege, at this period, all numerical statements are arithmetical equations. On §57 of The Foundations of Arithmetic, he writes:
For example, the proposition “Jupiter has four moons” can be converted into “the number of Jupiter’s moons is four”. Here the word “is” should not be taken as a mere copula, as in the proposition “the sky is blue”. This is shown by the fact that we can say: “the number of Jupiter’s moons is the number four, or 4”. Here “is” has the sense of “is identical with” or “is the same as”. So what we have is an identity, stating that the expression “the number of Jupiter’s moons” signifies the same object as the word “four”. And identities are, of all forms of propositions, the most typical of arithmetic. [P. 69e]
Frege viewed numerical equations – and, in consequence, all numerical statements – as identity statements. In ‘On Sense and Reference’ [Über Sinn und Bedeutung], Frege interpreted statements of the form ‘a = b’ as identity statements. At the very beginning of ‘On Sense and Reference’, he writes in a footnote,
I use this word [Equality] in the identity sense and I understand ‘a = b’ in the sense of ‘a is the same as b’ or ‘a and b agree’.
Under this point of view, arithmetical equations are similar to identity statements like ‘the morning star is the evening star’. The arithmetical equation ‘3 + 4 = 7’, for example, expresses the referential identity between the numerical terms ‘3 + 4’ and ‘7’. In other words, seven is the sum of three and four means that the expressions ‘the sum of three and four’ and ‘seven’ refer to the same number: seven.
Wittgenstein finds Frege’s account problematic, because by treating numerical expressions as names, it separates the sameness of the number from the sameness of the sign. Wittgenstein, in contrast, does not differentiate between numerical identity and syntactic identity. For him, asking when different signs –– represent the same number makes no sense. It is like asking when do different tokens belong to the same sign-type. The relation between numeral schemata and types is like that between tokens and types? Representing the same number is being the same sign.
The question of numerical identity is a question for the identity of sign types. Numerical identity is a grammatical matter. I mean: numbers are what I represent in my language by number schemata.That is to say, I take (so to speak) the number schemata of the language as what I know, and say that numbers are what these represent [(Later marginal note): Instead of a question of the definition of number, it’s only a question of the grammar of numerals].This is what I once meant when I said, it is with the calculus [system of calculation] that numbers enter into logic. [PR §107 p. 129]
Accordingly, an answer in terms of mere perception is unsuitable. Seeing directly the number a symbol represents is impossible. Just as it is impossible to see that the signs ‘a’ and ‘a’ are the same letter, to see that ‘|||||||’ and ‘7’ are the same number is impossible too.
How am I to know that |||||||| and |||||||| are the same sign? It isn’t enough that they look alike. For having approximate similarity in Gestalt can’t be what is to constitute the identity of the signs, but just being the same in number. [PR §103 p. 125. Cf. PG Pt. II §18 p. 331]
Wittgenstein found that different numerical signs are the same number if and only if they obey the same rules. Determining the numerical identity of different numerical signs requires a grammatical investigation into the rules of the calculus. It must be the result of a ‘comparison of the structures’ [Vergleiching der Strukturen PR §104 p. 117 (p. 126)]. Expressing numerical identity in a mathematical proposition is impossible, because it is not the result of calculation. The problem of numerical identity involves the totality of the calculus. It is not the result of a calculation, but a condition for it. The knowledge that ‘|||||||’ and ‘7’ are the same number does not result from arithmetical calculation. A criterion for numerical identity is necessary for doing arithmetic. However, ‘3 + 4’ = 7’ is the result of an arithmetical calculation and does not express the referential identity of the signs ‘3 + 4’ and ‘7’. Despite both being numerical expressions, they belong to different grammatical categories.
Wittgenstein emphatically rejects the view of mathematical equations as identity statements. Frege bases his interpretation on the view that arithmetical expressions like ‘3 + 4’, or ‘the sum of three and four’ and ‘7’ are names of numbers and that numerals and other, complex numerical expressions are not philosophically different. For Wittgenstein, mathematical terms are not names. Since they do not refer, it does not make sense to talk about them having the same referent.
Second, arithmetical expressions like ‘3+4’, or ‘the sum of three and four’ and numerals like ‘7’ belong to different grammatical categories. According to Wittgenstein’s method, grammatical distinctions are prior to logical ones. In this case, calculation expressions like ‘the product of 3 by 4’ and result expressions like ‘7’ are not exchangeable in the context of the verb to calculate. In other words, it makes sense to say ‘I calculate the product of 3 by 4’, but not ‘I calculate 7’. Mathematical expressions which can substitute for ‘the product of 3 by 4’ in the aforementioned context stand for calculations. They are both numerical expressions, only because they belong to the same grammatical system operating with numbers, i. e., arithmetic.
In German grammar, the verb ‘berechnen’ [to calculate] is an active or transitive verb. It requires a direct object. This means that using ‘berechen’ without a direct object is grammatically incorrect. Determining the calculation object is essential. In a well constructed sentence of German, the direct object usually follows the verb ‘berechen’ in a sentence. Not any sort of expression can serve this function. Answering the question ‘What is to be calculated?’ with a verb, adjective or adverb is grammatically incorrect. The answer is always an expression like ‘the successor of four’, ‘the supremum of set E’ or ‘the product of three by four’. These expressions behave like complex nouns. They are either singular or plural. It makes sense to talk about calculating the smallest element in a well ordered set (singular), or about calculating the square roots of a positive number among the reals (plural). And they play the sort of grammatical roles in sentences nouns usually play. Besides the role of direct object, they also play the role of subject in certain sentences. For example, it makes sense to say that the supremum of E is less than or equal to any upper bound of E. However, not every mathematical term can play this role. Not any sentence with a numerical expression following the verb ‘to calculate’ makes sense. It makes sense to talk about calculating the product of three by four, but not calculating twelve, for example. It makes sense to talk about calculating the square roots of four, but not about calculating two and minus two, even though ‘twelve’, ‘two’, ‘minus two’, just like ‘the square roots of four’ and ‘the product of three by four’ all behave like nouns. In this particular grammatical case, apparently synonymous terms like ‘twelve’ and ‘the product of three by four’ cannot substitute for each other without loss of grammatical correctness.
Mathematical expressions referring to the final results of calculations cannot be direct objects of the verb ‘to calculate’. This latter sort of expressions – sometimes called canonical numerals – behave just like proper names (or lists of them). Talking about the number two, instead of a number two or some number two is grammatically correct. These kinds of mathematical terms are irreducible. After arriving at one of them, taking the calculation further is impossible. Instead, the sort of expressions playing the role of direct object of ‘to calculate’ behave more like adjective phrases in natural language. In general, most mathematical terms are either calculation terms or final result terms. Most philosophers of mathematics conceiving both kinds of expressions as names referring to abstract objects overlook this distinction. However, it is essential for the analysis of numerical statements.
B. Equations and Calculations
The distinction between calculation expressions and final result expressions sheds some new light on the distinction between obviously tautological equations like ‘7 = 7’ or ‘3 + 4 = 3 + 4’ and genuine equations like ‘3 + 4 = 7’, which have puzzled so many philosophers like Frege. Despite the superficial similarity, in the second sort of equations, the terms on both sides of the ‘=’ are grammatically different. Call them ‘calculation’ and ‘final result terms’ respectively. One expresses a calculation, and the other its final result. In consequence, the sign ‘=’ in these equations does not work as a copula. It connects the calculation with its final result. ‘3 + 4 = 7’, for example, is a mathematical proposition saying that seven is the final result of adding three plus four. In it ‘3 + 4’ expresses the calculation of adding three plus four, while ‘7’ is the final result of such calculation.
By contrast, ‘7 = 7’ is not a genuine mathematical proposition, because it says nothing about any calculation. Seven is not a calculation. Seven is not the result of calculating seven. Such nonsense results from treating identity statements like ‘7 = 7’ as equations on a par with ‘3 + 4 = 7’.
If we ask: But what then does ‘5 + 7 = 12’ mean – what kind of significance or point is left for this expression – the answer is, this equation is a rule for signs which specifies which sign is the result of applying a particular operation (addition) to two other particular assigns. The content of 5 + 7 = 12 (supposing someone didn’t know it) is precisely what children find difficult when they are learning this proposition in arithmetic lessons. [PR §103 p. 126]
‘The equations yields a’ means: If I transform the equation in accordance with certain rules, I get a, just as the equation 25 x 25 = 620 says that I get 620 if I apply the rules of multiplication to 25 x 25. [PR §150 p. 175]
Mathematical equations are statements of the form a = b where one of the terms ‘a’ or ‘b’ is a calculation term and the other is a final result term. In consequence, equations of the form ‘7 = 7’ or ‘3 + 4 = 3 + 4’ or the false ‘3 = 12’, where the expressions on both sides of the ‘=’ sign belong to the same grammatical category, are not calculation statements. Every mathematical equation of the form a = b or ‘b is a’ expresses that b is the final result of calculating a (or vice versa). This holds not only of equations expressed with the help of the ‘=’ sign, but also of equations expressed in prose like ‘two is the positive square root of four’ or ‘seven is the least common denominator of twenty one and fifty six’. The particle ‘is’ does not work as a copula here either. It connects a calculation with its final result. Two is the positive square of four means that two is the correct final result of calculating the positive square of four. In general, calculation statements of the form ‘a is b’ say that a is the correct result of calculating b. Despite their surface grammar, these are not propositions of the form F(x), where x is an object, and F is a property. The number seven is not an object, and being the least common denominator of twenty-one and fifty-six is not one of its properties. Wittgenstein addresses this in §102 of the Philosophical Remarks, where he says that using ‘=’ can make numerical assertions appear to refer to genuine concepts, when they do not.
IV. An Extension of Frege’s Context Principle
A. The Context Principle
The Context Principle is a fundamental principle [Grundsätze] of Frege’s philosophical enquiry. In the introduction to his Foundations of Arithmetic, he formulates this principle as “never to ask for the meaning of a word in isolation, but only in the context of a proposition” [nach der Bedeutung der Wörter muss in Satzzussammenhänge, nicht in ihrer Vereinzelung gefragt werden].
But we ought always to keep before our eyes a complete proposition. Only in a proposition have the words really a meaning. It may be that mental pictures float above us all the while, but these need not correspond to the logical elements in the judgement. It is enough if the proposition taken as a whole has a sense; it is this that confers on its parts also their content.
Wittgenstein thought that the main flaw of Frege’s analysis was not considering his own Context Principle seriously enough. Wittgenstein agrees with Frege that words have meaning only in the context of a proposition. However, Wittgenstein also recommended asking for the meaning of a proposition not in isolation but in its larger context of use. For mathematical propositions, this context is their use in calculation. Wittgenstein’s extension of Frege’s Context Principle commands asking for the meaning of mathematical propositions only in the context of their calculi.
B. Specification and Calculation Propositions and Concepts
According to Wittgenstein’s reading of Frege, in the later’s analysis, both mathematical numerical statements like ‘4 is a number’ and equations like ‘3 + 4 = 7’ are propositions of the subject-predicate form F(a), where ‘F’ is a predicate – referring to a concept – and ‘a’ is a name – whose referent is an object. Furthermore, since all numerical statements are arithmetical equations, the concept involved in a numerical statement is always of the form ‘ . . . = b’ where ‘b’ is also the name of a number. For Wittgenstein, Frege’s distinction overlooks the real difference between these two sorts of mathematical propositions. This difference stems from the different roles they play in their calculi. Both sorts of propositions are rules of mathematical calculi. But they are rules of different sorts. Every calculus has two different sorts of propositions. Propositions of one sort connect calculations with their results. Call these ‘calculation propositions’. Propositions of the second sort specify the calculus’ elements. Call propositions of this sort ‘specification propositions’. In elementary arithmetic, the calculation propositions are the equations. Mathematical equations connect calculations with their final results. For example, an arithmetical equation like ‘3 + 4 = 7’ says that adding three plus four results in eight. A mathematical proposition like ‘7 is a number’ does not. It says that ‘7’ belongs to the category of number. ‘Being a number’ is not a calculation, but a calculus category. A category’s elements share the same role in the calculus. Saying that 7 is a number specifies the role of ‘7’ in the calculus. Frege overlooks this difference as well, while Wittgenstein assigns it a central role in his philosophy of mathematics.
A previous section showed how most arithmetical terms in a calculus are either calculation or result terms. However, not all mathematical terms fit into these categories. Consider the term ‘number’. Even though basic in arithmetic, ‘number’ is neither a result nor a calculation term. It does not occur in either side of the ‘=’ sign in arithmetical equations. Other examples of arithmetical terms of this sort are ‘addition’, ‘unit’, ‘equation’, etc. They also describe calculation and result terms internally, but they are not calculation or result terms themselves. They occur in internal descriptions not corresponding to any calculation whatsoever. Call these descriptions ‘specification propositions’.
The distinction between calculation propositions and specification propositions corresponds to a distinction at the level of mathematical concepts. The concepts that occur in specification propositions differ from those in calculation propositions. On a superficial level, statements of the subject + predicate form express propositions of both sorts. However, the concepts that occur as predicates in the specification propositions do not occur as subjects or predicates in calculation statements.
In the case of elementary arithmetic, ‘number’, ‘addition’, ‘equation’, etc. are specification concepts. In consequence, ‘4 is a number’ and ‘3 + 4 is an addition’ are specification propositions. They differ from calculation propositions in that they do not describe the result of any calculation in the calculus they specify. They can play the role of calculation propositions in other calculi, but not in the calculus they specify. For example, constructing a calculus where propositions like ‘4 is a number’ or ‘3 + 4 = 7 is an equation’ are calculation propositions is possible. Wittgenstein found that much of the logicists’ work on the Foundations of Arithmetic is of this sort. What Frege achieved by giving a formal definition of number was a new calculus in which propositions like ‘4 is a number’ are calculation propositions. He has constructed a calculus where the concept of number is not a specification one, but a calculation one. Hence, it makes sense to say in Frege’s framework ‘I calculate whether 4 is a number’.
This we can conclude that mathematical propositions may be either calculation or specification propositions. Calculation propositions connect calculations with their results. Specification propositions distribute the the elements of the calculus into categories.
Wittgenstein developed his ideas of arithmetics in response to Frege’s seminal work on the concept of number. Wittgenstein shared two substantial, methodological principles with Frege: the Grammatical Principle and the Context Principle. For Frege, as well as Wittgenstein, grammatical distinctions have strong philosophical significance. Both believe that every significant philosophical distinction is already present in the grammar of language. For them, the meaning of two expressions is logically different if and only if their replacement affects the grammar of the expression in which they occur.
Frege and Wittgenstein also share a strong faith in the philosophical importance of context. For Frege, words have meaning only in the context of propositions. Understanding the meaning of a word requires analyzing its role in complete sentences. Wittgenstein extends Frege’s principle to cover sentences as well as words. For him, understanding the meaning of a sentence requires an analysis of its role in a larger system of propositions (or in other sentences), too. In particular, understanding the meaning of numerical statements [numerical statements] requires an analysis of their roles in the contexts of their use. In the case of mathematical numerical statements, it requires understanding their roles inside systems of calculation. In the case of non-mathematical ones, it requires an analysis of their role in the application of mathematics. Both analyses are essential for the full understanding of numbers in mathematics. If mathematical propositions are grammatical, this would manifest itself in their roles in both arithmetical calculation and application.