Does Arithmetic Consist Only of Analytic Statements: A Study of the Logical System of Gottlob Frege

Gottlob Frege begins Foundations of Arithmetic[1] by posing the very important question, “what is number?” The following exposition found in Foundations is one of the fundamental texts upon which a large portion of both analytic philosophy and the entire study of the mathematics of arithmetic are based. Frege openly acknowledges that, for many people, the question “what is the number one?” will appear trivial and, to some extent, meaningless, yet, at the same time, states that most mathematicians don’t understand the basic concepts of number with which they work. Indeed, a “mere moral conviction, supported by a mass of successful applications is not enough” (FA, sec. 1) to confirm mathematical rigor, and we necessarily need proof and certainty. In fact, Frege claims that such proof of the concept of the cardinal numbers is “demanded.” (FA, sec. 1 and 2) Prior to the work of Frege, the entire structure of the most basic mathematic science, arithmetic, was completely unknown. Essentially, mathematicians had assumed the truths of arithmetic without ever completely seeking to gain the insights necessary to prove certainty. Frege, through his project, wished to establish certainty in the mathematical sciences by proving the truths of arithmetic to be analytically derived through the laws of logic.

Prior to Frege, two views on the source of all human knowledge existed. Empiricists, like John Stuart Mill and David Hume, postulated that sensory experience is the source of all knowledge. While Frege certainly considers the empiricist argument that it appears that the most basic arithmetical truths, such as 2 + 2 = 4, are proved in the physical realm of sensory experience,[2] he argues that the truths of arithmetic cannot be learned empirically, only shown to be true. Because the investigation into the source of knowledge has, according to Frege, nothing to do with how we come to believe truths, such an empirical method is invalid in considering the nature of arithmetic. The source of our knowledge of a truth, says Frege, is determined not by how we come to believe it, but rather by what it is we actually use to justify or establish the truth. (Weiner, 8) Even though our knowledge of the physical sciences is established by appealing to sensory experiences, i.e. a posteriori, our knowledge of the truths of mathematics cannot be established by the empirical method. This must be the case, as “we can figure out a proof without writing it down” (Weiner, 10) because a proof can be convincing without a sensory experience.[3]

The other view on the source of all knowledge was the Kantian view, established in the works of Immanuel Kant.[4] For Kant, truths were either analytic, what we can establish through analysis of concepts, or synthetic, which we cannot establish through analysis of concepts, but only through an appeal to something beyond the concept involved in the statement. Kant further separates analytic statements as statements which contained the predicate of the sentence in the subject of the sentence, whereas synthetic statements do not contain the predicate in the subject of the sentence. Prior to Kant, synthetic statements were considered to only derive their truth or falsity from an appeal to empirical evidence, or, a posteriori. Kant, however, claimed that the truths of geometry, though synthetic (because, for Kant, space is a synthetic concept and geometry is founded on spatial properties), were also a priori truths. Kant claimed that the basic axioms of geometry existed in the mind outside of and independent from sensory experiences. The truths of geometry, according to Kant, must be justified by a formal proof from an axiom set, and, because the axiom set is “self-evident” and a priori, the source of the justification must come from “pure intuition.” Pure intuition, to Kant, is a faculty which underlies our perceptions of spatial objects. Because all geometric truths are general truths which apply to all spatial objects, the axioms of geometry must be self-evident because we do not need sensory evidence to understand them.

Frege recognized the power of this notion of the synthetic a priori truth. Frege maintained that such synthetic a priori truths govern “all that is spatially intuitable,” (FA, sec. 14) and do, in fact, hold over a more general domain than the a posteriori laws of the physical sciences. Though Frege maintained the Kantian view of geometry[5], he disagreed with Kant about the absolute extension of the synthetic a priori truths over the entire domain of mathematics. Frege assumes the apriority of both geometry and arithmetic, but “that they differ in that geometry rests on intuition and is synthetic.” (Burge, 360)

While Kant maintained that all of arithmetic must also be founded upon synthetic a priori truths which hold truth in every domain, Frege recognized that synthetic a priori laws do not hold everywhere. In fact, the laws of Euclidean geometry do not hold everywhere, like Kant believed.[6] Yet, asserted Frege, there must be laws which will hold in “the widest domain of all … everything thinkable.” (FA, sec. 14) These laws are the laws of logic, and are necessarily analytic a priori laws.

For Frege, the laws of logic must be necessarily analytic. Because any truth whose justification is pure logic must, by definition, be an a priori truth, a law of logic cannot be a posteriori. Furthermore, synthetic truths hold only what is spatially intuitable and logical justification is not spatially intuitable. Thus, logical truths cannot be synthetic, and must be analytic and a priori. This is Frege’s major insight, and major contribution to the Kantian notion of analyticity.
Frege’s notion of analyticity is also different from the Kantian view, though Frege maintains that Kant truly meant what Frege proposed.[7] Kant merely stated that an analytic truth was a truth where the predicate was contained in the subject of the proposition. Yet, this leaves out obvious analytic truths, says Frege. Consider any statement in the conventional form of (P v ~P), such as, “it is cloudy or it is not cloudy.” Surely this statement is analytic, as it is always true. As such statements must be included in the definition of “analytic,” Frege expands the definition of the criterion of analyticity. The Fregean definition of analytic truth is “a truth that can be established by a derivation that relies only on definitions and general logical laws.” (Weiner, 15)[8] Through this adaptation and separation of the Kantian notions of analyticity and the foundations of arithmetic, Frege was able to set a distinct goal for his project: to provide the logical laws necessary to prove that arithmetic was analytic, and necessarily based upon logical, a priori laws. Tyler Burge writes, “he [Frege] seeks to isolate basic concepts and basic principles in trying to demonstrate … that the mathematics of number is reducible to logic.” (Burge, 7)

At the time of Frege’s conception of his own project, the only means for evaluating the validity of logical arguments was Aristotelian logic, which consisted only of 256 possible syllogisms. Frege noticed that such a logical system, founded only upon a limited number of arguments, cannot possibly provide any new knowledge, because the analytic premises of Aristotelian logic only proved analytic conclusions, as each sentence was strictly regimented into distinctions of subject and predicate.[9] This was one of the major reasons why Kant was believed that all of mathematics must be synthetic, not analytic. Kant believed that math did in fact lead to knowledge, thus, mathematical truths must exist outside of the analytic Aristotelian logic. Prior to Frege, all logic could do was prove analytic truths, which, under the Aristotelian system, did not equate to any gain of knowledge for the logician undertaking the project. As Frege’s project sought to find the source of all knowledge of arithmetical truths, the Aristotelian system would not suffice. Frege’s “primary interest lies in the nature of human knowledge of mathematics,” (Burge, 8) and, as such, he sought to explain the source of arithmetical knowledge.

Frege argues that all arithmetical truths are analytic, opposing the Kantian position. As Frege’s view of arithmetic applied “to a realm wider than the spacial,” (Weiner, 20) he needed a logical system which applied to the most general realm: everything. For Frege to truly argue his view of the nature of arithmetic, he had to develop an entirely new logical system.

This was Frege’s first project. Upon completion, in 1879, and entitled Begriffsschrift, the project was a massive logical undertaking and is now heralded as one of the most important logical projects in the history of the science. For brevity, we are not concerned about the details of the logical system laid out in the Begriffsschrift, as it is merely the tool which Frege uses to attempt to prove the analyticity of all arithmetical statements. It is nevertheless important to note that Frege’s work in the Begriffsschrift enabled Frege to replace the principle of mathematical induction with a principle based solely upon logical laws already contained in his system.[10] This principle is the hereditary sequence principle, also called the immediate successor principle, and, with Frege’s exposition of it, could now be expressed solely in terms of logic, no longer relying upon induction as a method. Missing from the Begriffsschrift, however, is a formal proof of the concept which would allow for the immediate and immaculate use of Frege’s immediate successor principle—the concept of number. For this reason, as well as a further proof of the analytic nature of arithmetic, Frege writes Foundations.

Foundations is comprised solely of the exposition of a proof of the concept of number in order to prove the analyticity of arithmetic as shown through logical laws. Yet, if Frege’s proofs of these logical laws are to demonstrate that the truths of arithmetic are analytic, then he must be able to properly define the concept of the numbers within his very own system. Frege writes, “it is above all Number which has to be either defined or recognized as indefinable … On the outcome of this task will depend the decision as to the nature of the laws of arithmetic.” (FA, sec. 4)[11] Because concept is important, Frege must be able to define the numbers in such a way as their content is included within the definitions. Thus, Frege devotes Foundations to defining the numbers and satisfying his own requirements the concepts of the numbers necessarily entails in his system. Throughout the project of Foundations, Frege adheres to three cardinal principles, which he himself outlines in the preface to the work. They are:

“always to separate sharply the psychological from the logical, the subjective from the objective; never to ask for the meaning of a word in isolation, but only in the context of a proposition; never to lose sight of the distinction between concept and object.” (FA, x)

Throughout the piece, to adhere to the first principle, Frege uses the word “idea” solely in the psychological sense to distinguish it from the notions of concept and object. This distinction is important to consider because Frege wishes to separate his project from that of psychological thought. Frege rightly believes that psychological thought stems from looking at epistemology through the lens of the historical method. Such a method, claims Frege, “is certainly perfectly legitimate, but it also has limitations.” (FA, iii) If the method is limited, as Frege showed, such a method could never be comprehensive. The historical approach cannot establish the truths which Frege wishes to prove through the project of Foundations, because the method “makes everything subjective … and does away with truth.” (FA, vii) If the truths of mathematics, and in particular, arithmetic, were psychological, they would necessarily be subjective, and a demonstration of the objective truths of arithmetical principles could never be achieved. Yet, because the psychological and historical approach is limited and non-comprehensive, there is still room to believe that arithmetic can be analytically derived and objective truth can exist.
Frege, in his fundamental principles, has stated the importance of treating the meaning of words in the context in which they are used, and never to solely define a word in isolation. This remains important because, as Frege says, if the second proposition is broken, “one is almost forced to take as the meanings of words mental pictures or acts of the individual mind, and so offend the first principle as well.” (FA, x) Frege’s third principle only affirms that there is a distinction between “concept” and “object,” and any perversion of this distinction would alter the meaning of either concept or object. As Frege puts it, “it is mere illusion to suppose that a concept can be made an object without altering it.” (FA, x)

After setting the guidelines of the project, Frege can then undertake his exposition to prove and define the concept of number in order to demonstrate that all arithmetical statements can be derived through logic and must therefore be analytic and a priori. According to Michael Dummett, a Frege scholar, “the fundamental laws of logic should be included among the analytic truths.” (Dummett, 132) Immediately, Frege delves into the synthetic/analytic distinction and the swirling discussion which surrounds it. The distinctions, writes Frege, “between a priori and a posteriori, synthetic and analytic, concern, as I see it, not the content of the judgement but the justification for making the judgement.”[12] (FA, sec. 3) Thus, when Frege refers to a proposition using any of these terms, he cautions, the proposition isn’t about the way men have psychologically come to believe it to be true (thus adhering to his first principle), it is rather “a judgement about the ultimate ground upon which rests the justification for holding it to be true.” (FA, sec. 3) Frege has therefore reduced the debate about the concept of number solely to the mathematical sphere—the problem becomes “that of finding the proof of the proposition.” (FA, sec. 3) According to Frege, if we are successful in following the truth of the proposition back to the truth of the premises and only use general logical laws, then we have proven that the truth is an analytic one. Nevertheless, Frege does recognize that if the proof necessitates an appeal to a “special science,” i.e. an empirical, physical science, the truth would have to be synthetic. Similarly, if a proof cannot be derived without an appeal to sensory experience or facts, then it must be a posteriori, but if we can derive a proof only using the general laws of logic, “which themselves neither need nor admit of proof,” (FA, sec. 3) then the truth can be called a priori.

From here, Frege can begin to discuss the concept of number in order to find the basic grounding upon which all of arithmetic is based. From the definition of the “Numbers,” Frege could then formulate the basic and fundamental logical laws upon which arithmetic is based. Upon completion of this project, since all of arithmetic could be based on logical principles, arithmetical propositions would necessarily be analytic. This is Frege’s ultimate goal. In order to begin this process, Frege asserts that the definition of the concept of number can only be given if the principles of arithmetic are analytic, (and thus, a priori). If the principles of arithmetic happen to be a posteriori, then no definition of the concept of number can be given. Though this appears to be quite a strong statement and position, Frege does accept the fact that synthetic a priori truths exist,[13] so Frege does not leave out the possibility that arithmetic, too, consists of synthetic a priori truths. He just does not believe that the truths of arithmetic aren’t analytic.
Frege next addresses the positions of the philosophers and mathematicians who preceded him, borrowing both from Kantian and from Leibnitzian appeals about the truths of arithmetic.[14] Frege accepts the Leibnitzian rationalism inherent in Leibnitz’s exposition and goes on to refute J.S. Mill’s empirical approach to logical proof. Whereas Mill would say (according to Frege), “we ought not to form the definitions [2=1+1, 3=2+1, etc] unless and until the facts he [Mill] refers to have been observed,” Frege argues that such an approach would itself be meaningless. Obviously, the problem with the empirical approach is that observation must occur before accepting a proposition of mathematics as true; otherwise, propositions made about mathematics have no meaning. If Mill is correct in this notion, he must also be asserting that mathematics consists of synthetic and a posteriori notions, claims Frege. Furthermore, says Frege, we wouldn’t be able to make serious statements about mathematics. Consider the notion of “0.” How would we ever be able to make a serious claim about it—we can’t have any sensory information of the concept of zero, yet we can make serious claims involving the concept of zero. Mill is wrong because he is using an empirical approach which is derived from a psychological view rather than a logical position, and, as such, he gives only psychological justification with no logical formulation.

Mill, claims Frege, is also wrong about the nature of mathematical induction. In fact, “in order to be able to call arithmetical truths laws of nature, Mill attributes to them a sense which they do not bear.” (FA, sec. 9) Mathematical truths, claims Frege, cannot be established through induction, like Mill believes, for Mill “always confuses the applications that can be made of arithmetical propositions.” (FA, sec. 9) Mill wishes to do something with arithmetic that cannot be done—establish truth through induction. As discussed previously, Frege seeks to replace mathematical induction with logical certainty, thus proving that arithmetic does not require any induction and can be proven deductively from within the system.

Frege again asserts that all arithmetical propositions are analytic in nature, and thus, are necessarily a priori, but now wishes to address the Kantian notion of all of mathematics consisting of synthetic a priori truths. This is important, because, until this point in his exposition, Frege has only presented the Kantian position, and has not actually addressed Kantian thought. Frege realizes that he has been able to provide a sound argument against arithmetical truths being a posteriori in any sense, and thus, out of the four possible combinations able to be made from the synthetic/analytic distinction and the a priori/a posteriori distinction, only two remain: arithmetical truths must be either analytic and a priori or they must be synthetic a priori. Kant asserts that all mathematical truths, including the truths of arithmetic, are the latter, and, as such, there is no alternative than to “invoke a pure intuition as the ultimate ground [source] of knowledge of such judgements.” (FA, sec. 12)
Frege disagrees, writing, “we are all too ready to invoke inner intuition, whenever we cannot produce any other ground [source] of knowledge. But we have no business, in doing so, to lose sight altogether of the sense of the word ‘intuition’.” (FA, sec. 12) Kant, according to Frege, makes a distinction about the use of the word ‘intuition’. For Kant, “an intuition is an individual idea” (FA, sec. 12) This is why Frege must necessarily disagree with Kant’s assertion that it is intuition upon which all mathematical truths are based.[15] Frege strongly states, “an intuition in this sense cannot serve as the ground [source] of our knowledge of the laws of arithmetic.” (FA, sec. 12) This must be the case, as the truths of arithmetic can only be found by general concepts, not by individual ideas. Because intuition discounts general concepts in favor of individual ideas, and the notions of the synthetic truths of mathematics depends upon intuition, the truths of arithmetic can not be synthetic a priori, like Kant claims. While Frege is fine in accepting that geometry, which is spatially bound by individual and synthetic concepts, he will not accept that arithmetic is similarly based or bound. Frege claims that, because we can always assume an axiom of geometry without involving ourselves in contradictions despite the conflict between intuition and assumption, the axioms are synthetic. For Frege, “synthetic” means “not derivable from logic.” (Burge, 375) It remains, though, that “the basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even that of geometry.” (FA, sec 14) This base is analytic a priori truths.

Frege further clarifies his own position by arguing against Leibnitz, who similarly asserted that all arithmetical truths must be analytic. Leibnitz, according to Frege, based algebra’s benefits solely on logical truths, yet “this view, too, has its difficulties.” (FA, sec. 16) Leibnitzian doctrine, according to Frege, only has in the axiomatic set a series of identities. Since an entire logical system cannot be based solely upon identities, the Leibnitzian system must be incomplete. In fact, this is true, as Leibnitz depends upon mathematical induction to provide the arithmetic truths unattainable through identity arguments. Again, Frege seeks to defeat the mathematical induction claim, by stating, “it cannot be denied that the laws established by induction [those of the identities] are not enough.” (FA, sec. 16) Nevertheless, if we can first adopt the content in the form of a conditional, substitution would allow the reduction of arithmetic to a distinct form (made dependent upon certain conditions by the general laws of logic), then truth can be established by thought alone. As such, these truths would be derived completely external from any sensory experience, i.e. completely rationally, i.e. analytically. Furthermore, the judgments resulting from this experiment must also be analytic, as the truth of the judgments would also result from thought alone. Only then could observation confirm whether or not the conditions included in the axiom set of the laws of logic and arithmetic were actually fulfilled. This would once and for all “link the chain” of direct deductions of logically-deduced arithmetical truths to already proven observable “matters of fact.” (FA, sec. 17) This procedure, claims Frege, ought to be the preferred procedure of establishing arithmetical proofs because such a method can establish the deductions of the entire proof structure as a whole rather than one by one through, say, Aristotelian logic. “If this be so,” writes Frege, “the prodigious development of arithmetical studies … will suffice to put an end to the widespread contempt for analytic judgements.” (FA, sec. 17)

Without a doubt, “the philosophical purpose of Foundations … is to establish beyond any possible doubt that arithmetical truths are a priori” (Sluga, 101) and “also that they are analytic, logical truths.” (Sluga, 101) Does Frege actually succeed in this project? The answer is, unfortunately, no. Frege himself realizes that Foundations does not, by itself, establish deductive certainty for his belief that arithmetic truths are analytic and a priori, writing in his conclusion, “from all the preceding it thus emerged as a very probable conclusion that the truths of arithmetic are analytic and a priori.” (FA, sec. 109, my italics) Notice that Frege does not write that it is deductively certain, only that it is very probable.

In this sense, Frege’s project can be called “pragmatic,”[16] as it is concerned only with the justification for believing in a truth of a statement, and the statement “all arithmetical truths are analytic” is a statement, which, in fact, fits into Frege’s own pragmatic notion of justification of the source of knowledge. Nevertheless, Frege’s project in Foundations, and in Begriffsschrift, signified a heroic undertaking in the realm of mathematics and philosophy. Frege created a logical system, which, since its creation, has been found to be inconsistent. Nevertheless, his system paved the way for modern logic and for the entire analytic philosophy movement.

While Frege continually maintained that “the laws of arithmetic are analytic judgements and consequently a priori,” (FA, sec. 87) he was unable to provide a consistent deductive proof in his lifetime, and, ultimately failed in providing a proof of the analyticity of arithmetic.[17] Though it may not be the case, as Frege believed, that arithmetic is just a development of logic, and every proposition of arithmetic is bound by the general analytic laws of logic, he did still provide a valuable contribution to the field of mathematics, philosophy and logic. Frege’s definitions of the concept of number were revolutionary, as he claimed that the laws of number “are not really applicable to external things; they are not laws of nature. They are, however, applicable to judgements holding good of things in the external world: they are the laws of the laws of nature.” Throughout Foundations, Frege proposes a system and an explanation for number, and without his contributions to mathematical logic and philosophy, the development of these sciences would have taken many, many years to develop.

Sources Cited and References Used

1. Burge, Tyler. Truth, Thought, Reason: Essays on Frege. (New York: Oxford University Press, 2005).

2. Dummett, Michael. Frege and Other Philosophers. (New York: Oxford University Press, 1991).

3. Dummet, Michael. The Interpretation of Frege’s Philosophy. (Cambridge: Harvard University Press,
1981). Note: book used as reference. In-text citations come from Frege and Other Philosophers.

4. Frege, Gottlob. Foundations of Arithmetic, translated into English by J.L. Austin. (Oxford: Basil Blackwell & Mott, Limited, 1953).

5. Frege, Gottlob. The Basic Laws of Arithmetic, translated into English by M. Furth. (Berkeley: University of California Press, 1964).

6. Sluga, Hans. Gottlob Frege. (London: Routledge & Kegan Paul, 1980).

7. Weiner, Joan. Frege. (New York: Oxford University Press, 1999).

8. Van Heijenoort, Jean, ed. Frege and Gödel: Two Fundamental Texts in Mathematical Logic. (Cambridge: Harvard University Press, 1970).





[1] Hereafter referred to as “Foundations.” In text footnotes will be noted as “FA, sec. #” where # stands for the section in which the quotation or source documentation can be found.
[2] For example, this is often shown by teaching a child that two apples and two more apples are the same as three apples and one more apple and that this can be called “four.”
[3] Both Frege and Weiner use the example of the following property of arithmetic algebra: 5a+5b=5*(a+b).
[4] See Foundations, sections 4, 5, 12, 88, 89, and 93 for further information than is contained within the following paragraph.
[5] For example, see Foundations, section 89. Frege writes, “In calling the truths of geometry synthetic and a priori, he [Kant] revealed their true nature.” Hans Sluga responds to this statement by affirming, “Frege never abandoned the conception of geometry as synthetic a priori. It remained one of the stable elements in his thought.” (Sluga, 45)
[6] The obvious example is that of Non-Euclidean geometry. For further reading, I suggest a study of mathematicians Beltrami, Klein or Poincaré.
[7] See Foundations, section 3, footnote 1. Or, view footnote 12 of this paper for clarification.
[8] I have chosen to quote Weiner, rather than Frege himself. I believe that Weiner has presented this information more clearly. Further discussion of this concept will occur later in this paper. For Frege’s definition, see Foundations, section 3.
[9] The Fregean system allowed a more general learning to occur, i.e. because in Frege’s system, the premises do not have to be true for the conclusion to be valid, as we are only concerned with the form of the argument, thus the acquisition of further knowledge can occur through logical deduction of the Fregean system whereas the Aristotelian system was unable to provide a means for further knowledge gain.
[10] See Begriffsschrift, section 24. This section also touches on the notion of the analyticity of logical arithmetic and discusses the hereditary property.
[11] Frege, by “Number” of course means cardinal and “natural” numbers, though his main concern throughout the book is providing a “general means of defining individual natural numbers.” (Dummett, 19-20) Frege hopes to show that these definitions are analytic ones, found within the laws of logic.
[12] Frege also inserts a footnote in his paper, writing, “By this I do not, of course, mean to assign a new sense to these terms, but only to state accurately what earlier writers, Kant in particular, have meant by them.”
[13] Michael Dummett writes that it is “quite likely to be true that Frege always regarded the truths of geometry as synthetic a priori.” (Dummett, 128)
[14] Gottfried Leibnitz (1646-1716), famous modern rationalist, presented an “incomplete” proof that “numerical formulae are actually provable” (FA, sec. 6) Frege writes that his proof is incomplete because he neglects to insert parenthesis as proper operators within his exposition of the proof of mathematical formulas. See Foundations, section 6, for Frege’s own discussion of Leibnitz.
[15] Tyler Burge has written a very informative and helpful essay “Frege on Apriority” which could provide further clarification of the relationship between Frege, Kant, arithmetic, geometry and analyticity. Read Burge, 370-372.
[16] Pragmatism, developed by Thomas Peirce and William James in the late 19th and early 20th centuries, holds that a belief or opinion can be considered true if the opinion brings “cash value” to the life of the individual belief holder. It is in this sense which I use the term “pragmatic,” as I see Frege’s entire project in Foundations as a quest to provide justification for his own belief that arithmetic is analytic.
[17] The Basic Laws of Arithmetic, published in 1893, contained Basic Law V, which led to the discovery of “Russell’s Paradox.” Ultimately, Frege was unable to resolve the paradox, and, as such, his Basic Laws were proven to be inconsistent, as a contradiction could be derived from Basic Law V.