Does the Caesar Problem Prevent the Reduction of Arithmetic to Logic?

Introduction

Reducing arithmetic to logic is the aim of the logicist programme, or as the position is held today, the neologicist programme. Frege’s derivation of arithmetic from logic in his logicist programme has many successes, but has one major objection: the Caesar problem. The Caesar problem is the largest obstacle in the path of the logicist, and neologicist. It shows that, from Frege’s formulation of arithmetic, any object is allowed to be a number, even Julius Caesar. If it were to remain unsolved then numbers would not be fully defined, rendering the neologicist project incomplete. Frege’s own attempt at resolving this objection resulted in his greatest failure, emphasising the gravity of the problem it poses for logicism. Frege’s abandonment of the logicist programme did not extinguish the field, however – instead, new attempts have continued to develop, making up the neologicist programme. A compelling solution to the Caesar problem is the structuralist neologicism approach constructed by Boccuni and Woods (2020). This position combines elements of structuralism and of neologicism to result in a theory in which arithmetic is still reduced to logic, without inviting the Caesar problem – a monumental success for neologicism. I therefore conclude that the Caesar problem is not an obstacle to reducing arithmetic to logic.

Frege’s Theorem

Frege ingeniously derived arithmetic from logic, resulting in a consistent framework which is thus a great explanation for where arithmetic comes from. Frege’s logicist programme was an attempt to demonstrate that arithmetic is analytic, as well as a priori. For Frege, analyticity is the idea that something has purely logical justifications, whether that be from logical proofs or logical axioms (Shapiro, 2000). This is in contrast with a synthetic statement - one which does not have a purely logical justification. An a priori truth is one which either is or comes from a fundamental conceptual truth, independent from any real-world experiences. For example, the statement ‘all circles are round’ is a priori, because it is true in virtue of its definition. So, to show that arithmetic is analytic a priori, Frege had to derive arithmetic, with all its rules, from fundamental logical concepts and using only logic, thus demonstrating that arithmetic can be reduced to logic.

Frege successfully derived natural numbers (hereafter occasionally referred to as numbers) and their relations from an abstraction principle he termed Hume’s Principle (HP), through a process that we now call ‘Frege’s Theorem’. An abstraction principle is a conceptual truth which provides identity conditions for and therefore introduces abstract objects, like numbers. HP is the idea that for any two concepts, their number is identical if they are equinumerous (Frege, 1980). That is, if they can be put in a one-to-one correspondence, then they are identical in number. Here when mentioning the number of a certain concept, this is to be taken as a proper name that denotes an object, so that the definition is not circular. From HP, Frege then defined a concept, call it Z, which means ‘not identical to itself’, under which no object falls since every object is identical to itself. Because no object falls under it, Frege defined 0 as the number belonging to the concept Z. Next, Frege defined the successor relation such that when we have a number of objects, n, and take one object away so that there are m objects, then n is a successor to m (Frege, 1980; Shapiro, 2000).

Using the successor relation and the defined number 0, Frege could then define the number 1 and show it to be the successor to 0. The number 1 is defined as the number belonging to the concept of ‘identical with 0’. Since there is only one thing identical with 0 – 0 itself – this is the number 1. When we take away one object from the number 1 we are left with 0, so 1 is the successor to 0. This process can be repeated for all of the natural numbers. Further, a natural number can be defined using the number 0 and the successor relation as any number that can be obtained from applying the successor relation to the number 0. It is also possible to show that there are infinite natural numbers: with a series of natural numbers whose final number is n, there is a number n+1 that is a successor to n. From these definitions, Frege then proceeded to derive the Peano axioms, a set of logical statements that define all of arithmetic (Shapiro, 2000). All of this can be obtained simply from HP.

The derivation of arithmetic from this process is seen as a major success in the philosophy of mathematics, and as a truly brilliant derivation (Salmón, 2018). HP is consistent, and so is a reliable starting point for the derivation of arithmetic as it doesn’t lead to any inconsistencies later on. This is important because a mathematical system must be consistent, otherwise it is not useful nor reliable as a framework upon which to build other systems and disciplines. HP gives all the necessary properties of the natural numbers for use in arithmetic, and they are defined in relation to each other. Through Frege’s theorem then, we seemingly can consistently reduce arithmetic to logic. However, despite these strengths, there remains a major problem with logicism: the Caesar problem.

The Caesar Problem

HP, while providing identity conditions for numbers, fails to specify their intrinsic nature - their content isn’t defined - meaning that absolutely anything, even Julius Caesar, could be a number (Frege, 1980). This is called the Caesar problem, and it poses a significant problem for logicism because it questions the completeness of their theory.

Logicists are realists about the ontology of natural numbers (Shapiro, 2000). This means that they take natural numbers to be independent objects that objectively exist. Logicists also take numbers to be uniquely identifiable, an addition to the realist position assumed to be necessary by the typical logicist and which Boccuni and Woods term “perfect individuation” (2020, p.301). Logicists are also realists about the truth-value of number (Shapiro, 2000), so that statements about the numbers must have objective truth-values, including mixed-identity statements such as ‘Caesar is the number 2’. Since the numbers are defined as new objects by HP (MacBride, 2003), and the logicist takes these objects to be real and unique, HP must provide this full account of what the numbers are to be a complete definition. Otherwise, HP and Frege’s Theorem are incomplete and we do not have a full theory of what numbers are or of how to reduce arithmetic to logic; logicism is incomplete, and therefore philosophically inviable. According to the logicist, to properly define number, HP must fulfil two criteria: the criterion of application – telling us which objects can and cannot be numbers, and the criterion of identity – telling us which numbers are different from each other or the same (MacBride, 2003).

While HP can tell us whether two numbers are the same or not, as if they are equinumerous they are identical, it does not tell us which objects can or cannot be a number. Not only does it not tell us, but it cannot tell us. If HP could determine that Caesar is not a number, it would be asserting something else about the state of the world beyond pure logical truth, and so would no longer be a purely logical, mathematical truth (Salmón, 2018; Boccuni and Woods, 2020). This undermines the foundational premise of logicism as an analytic a priori framework independent of contingent empirical claims, and such a conclusion would render logicism defunct. We cannot also rely on using HP along with some extra facts about the world, because then it would not be HP alone defining number (Boccuni and Woods, 2020). Therefore, numbers are introduced not only as objects, but objects of the same kind as basic objects like tables and chairs (Heck, 1997). This failure of HP to fulfil the criterion of application is why the Caesar objection arises; if numbers are treated as objects of the same kind as basic objects, nothing can prevent a Roman Emperor from being a number (Heck, 1997; Hale and Wright, 2001).

Basic Law V and Neologicism

Frege attempted to resolve the Caesar problem using an axiom called Basic Law V (BLV). Unfortunately, the cost of resolving the Caesar problem in this way was creating an inconsistent theory, ending in Frege’s greatest failure. Subsequently, neologicism avoids BLV, instead focusing solely on HP.

Frege could not solve the Caesar problem using the current theories he had built up, and so he attempted to find a way to go even further and derive HP from a more fundamental axiom that could define number entirely; BLV (Shapiro, 2000). Frege defined an extension of a concept as the class of objects that satisfies that concept; for example the extension of ‘the number 2’ would be all pairs of objects. From here he defined BLV to state that for any concepts, their extensions are the same if the concepts are all the case for the same objects. Defining numbers as extensions of concepts leaves no doubt about which objects are numbers – they are extensions, a specific kind of object, and so the Caesar problem should not arise (MacBride, 2003).

However, BLV was shown by Russell to be inconsistent due to Russell’s Paradox (RP) (Frege, 1995; Shapiro, 2000). Though the full explanation of RP is too complex to deconstruct here, in brief RP asks whether the class of all classes which do not contain themselves will contain itself. If the class of all classes which do not contain themselves fails to include itself, it therefore must include itself – and vice versa, leading to contradiction. Because Frege’s extensions are classes of objects, they are subject to RP, and since BLV is the source of this contradiction, it is inconsistent. RP meant that this avenue for resolving the Caesar problem was no longer viable, and so unable to find another solution, Frege abandoned the logicist programme: “I do not see… how numbers can be apprehended as logical objects… unless we are permitted… to pass from a concept to its extension.” (Frege, 1995, p.214). This demonstrates the significance and difficulty of the Caesar problem.

With Frege’s failure, our contemporary adaptation, neologicism, seeks to reduce arithmetic to logic using only HP due to its own strength – as described earlier, natural numbers and Peano’s arithmetic were consistently derived from HP alone. The neologicist does away with the idea of deriving the abstraction principles, but holds them as a starting point. Further, neologicists rely on something called the ‘syntactic priority thesis’ to explain how the abstraction principles introduce objects. The syntactic priority thesis is an idea used to try and join language with reality in such a way that the structure of reality reflects our language, and which states that what our language describes (when speaking truly) actually exists (MacBride, 2003). Specifically, it states that when we talk syntactically of a singular number, we mean this semantically as well, and that the singular number exists. An abstraction principle like HP works by introducing numbers into our language, thereby asserting their ontological presence. This means that when we mention, for example, the number 2, we must actually be referring to something that is or plays the role of the number 2. The syntactic priority thesis is essential for maintaining that arithmetic can be reduced to logic since it is necessary for explaining how HP works. However, it also leads the typical neologicist into believing that since the numbers exist individually in reality, they must also be perfectly individuated. As I will explain in the next section, this is not necessarily the case, and insisting upon it is what results in the Caesar problem.

Because HP does not provide truth-values for mixed-identity statements, it follows for the typical neologicist that either HP is insufficient in providing a full account of number, or that the neologicist’s formulation of number is not viable (Hale and Wright, 2001). As evidenced by Frege’s attempt at reconstructing his theory in terms of extensions, he opted to retain his ontology and instead to conclude that HP is incomplete. This strategy did not prove fruitful, and so I will be covering an attempt at the opposite; to maintain that HP is complete, and to instead change the ontology regarding number.

Solving the Caesar Problem - Structuralist Neologicism

As mentioned above, HP only needs to fulfil the criterion of application because the typical neologicist assumes natural numbers are perfectly individuated. Perfect individuation is not enforced by HP, but is instead an ontology that one can choose to adopt. Alternative ontologies are viable, and a strong approach which utilises such an alternative – structuralist neologicism (SNL) – avoids the Caesar problem entirely, successfully reducing arithmetic to logic.

Since SNL utilises aspects of structuralism, I will first briefly cover what this entails. Structuralism states that “mathematics is the science of structure” (Shapiro, 2000, p.257); all that mathematics is comprised of is various structures, such as arithmetic being the structure of a successor relation and a starting point. Structuralism can be nicely complemented by logicism, as the mathematical structures that structuralism uses to build up its picture of mathematics can also be reduced to and described in terms of logic; they are logical structures. Because structuralism only cares about the structures themselves, it does not matter what plays the role of number. This is the general idea behind SNL – using arbitrary objects to represent number. This means that SNL does not perfectly individuate numbers, instead holding that it does not matter which object is taken to be a certain number as long as it has the minimum required properties, much like what structuralism posits. This is the key point to resolving the Caesar problem. Since HP is now treated as being complete, then as long as something has the properties that are derivable directly from HP, it can count as a number.

Boccuni and Woods describe “canonical reference” (2020, p.301) such that canonically referring to an object means to refer to a perfectly individuated object. Rather than canonical reference, then, SNL opts for arbitrary reference (Boccuni and Woods, 2020). When saying, for example, ‘the number 2’, the SNL is not canonically referring to an object that is the number 2, but instead referring arbitrarily to some object that has the necessary properties of the number 2 – the properties derivable from HP. Numerical identity is grounded purely in these properties, and truth-value realism is retained for them. However, as the SNL is no longer referring canonically, there are no truth-values attached to mixed-identity statements like ‘Caesar is the number 2’, because such statements have no meaning. They are missing the point, because an arbitrary reference is not referring uniquely to something which is the number 2 and which therefore must have a truth-value for such statements. Therefore, there is no danger of Caesar really being a number in any meaningful way.

Arbitrary reference still functions in practise like canonical reference, in the sense that when one talks about ‘the number 2’, one is still talking syntactically and semantically about one thing. The difference with using arbitrary reference is that the one thing is an arbitrary object. As the SNL is semantically talking about one thing, they retain the neologicist’s syntactic priority thesis, since when they say ‘the number 2’ our language can still reflect a real object (Boccuni and Woods, 2020). The SNL is still allowed to be an ontological realist. This means that they are still justified in treating HP as a special kind of statement which can introduce the concept of number into reality. However, unlike the typical neologicist’s use of HP, the SNL can use HP without incurring the Caesar problem. The criteria laid out by MacBride (2003) that a typical neologicist holds HP to no longer have to apply. The criterion of identity still applies, but the criterion of application is left redundant.

This is a novel and compelling approach which retains all the necessary elements of neologicism as well as some of the optional ontological commitments; the ontological and truth-value realism. With the arbitrary reference formulation of number, it is consistent to reduce arithmetic to logic without compromising the integrity of HP, thus giving a stronger foundation than the formulation favoured by the typical neologicist. Further, SNL works well with how mathematics is actually used; it emphasises the practical and relational aspects of number without getting weighed down by ontological commitments.

Conclusion

In this essay I have outlined Frege’s Theorem – the original derivation of arithmetic from logic and a monumental success for the philosophy of mathematics, showing how formulating arithmetic in terms of logic is both consistent and elegant. I covered the nature and cause of the Caesar problem, the primary objection to reducing arithmetic to logic, as well as Frege’s failed attempt to resolve it, illuminating how for the typical neologicis t, this problem is insurmountable. Through utilising arbitrary reference as in the SNL account of number formulated by Boccuni and Woods (2020), I have explained how the Caesar objection is overcome, so that it no longer stands in the way of deriving arithmetic from logic. SNL is a tremendous success, as through combining aspects of structuralism into this neologicist approach, the mathematical structures are fundamentally based upon the logic of HP, meaning that the mathematics which is represented using this philosophy is still reduced to logic. Due to its relatively recent development, direct objections to this approach have not been widely explored, though this could be a promising avenue for future research.

By rejecting the perfect individuation of number and embracing a more nuanced perspective, we not only salvage neologicism but also reinforce its philosophical robustness. Due to the success of Frege’s Theorem, combined with the adaptability of structuralist neologicism, I therefore argue that arithmetic is fundamentally and successfully reducible to logic.

References

Boccuni, F. and Woods, J. 2020. Structuralist Neologicism. Philosophia Mathematica. [Online]. 28(3), pp.296–316. [Accessed 17 April 2025]. Available from: https://doi.org/10.1093/philmat/nky017

Frege, G. 1980. The foundations of arithmetic: a logico-mathematical enquiry into the concept of number. 2nd rev. ed. Translated by J.L. Austin. Evanston, Ill: Northwestern University Press.

Frege, G. 1995. Frege on Russell’s Paradox. In: Geach, P. and Black, M. eds. Translations from the philosophical writings of Gottlob Frege. 3rd ed., reprint. Translated by P.T. Geach. Oxford: Blackwell.

Hale, B. and Wright, C. 2001. The Reason’s Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics. [Online]. Oxford University Press. [Accessed 23 April 2025]. Available from: https://doi.org/10.1093/0198236395.001.0001

Heck, R.G. 1997. The Julius Caesar Objection. In: R.G. Heck, ed. Language, Thought, and Logic: Essays in Honour of Michael Dummett. [Online]. Oxford University Press, pp.273–308. [Accessed 17 April 2025]. Available from: https://doi.org/10.1093/oso/9780198239208.003.0011

MacBride, F. 2003. Speaking with Shadows: A Study of Neo‐Logicism. The British Journal for the Philosophy of Science. [Online]. 54(1), pp.103–163. [Accessed 15 April 2025]. Available from: https://doi.org/10.1093/bjps/54.1.103

Salmón, N. 2018. Julius Caesar and the numbers. Philosophical Studies. [Online]. 175(7), pp.1631–1660. [Accessed 18 April 2025]. Available from: https://doi.org/10.1007/s11098-017-0927-0 Shapiro, S. 2000. Thinking about mathematics: the philosophy of mathematics. New York: Oxford University Press.