Did a Number Conquer Gaul

Did a Number Conquer Gaul?

At first, the question posed by the title of this paper might seem so pointless as to be absurd. Of course Julius Caesar isn’t a number. Just look at the numbers themselves. One isn’t Caesar, two isn’t Caesar, seventy-three million isn’t Caesar. Unless a person says “Julius Caesar” at some point in the normal process of counting, which seems unlikely, the person to which the name “Julius Caesar” refers cannot be a number. However, this is presupposing that we already know how to count in the first place, and that we know, based on whatever rules we use while counting, that we will never reach a point when those rules instruct us to name a Roman emperor. Furthermore, how sure are we that the number two, for instance, is indeed something other than Julius Caesar? In The Foundations of Arithmetic, Gottlob Frege first formulates an imprecise definition of number, from which he admits it is impossible to determine whether or not Caesar is himself a number. He later defines numbers as being the extensions of certain concepts. This latter definition solves some of the problems of the former, but whether it takes care of the Julius Caesar problem is not clear. There is no explanation of how we know Caesar is not the extension of any of the concepts under consideration. Is it possible to show that Caesar is not an additional member of some basic set of numbers, and thus that his being in that set would not present new difficulties? From there, can we show that he is not an additional member of any more complex set of numbers, or perhaps that he is not a number at all? If this cannot be shown, what problems might remain for mathematics, if any?

Before any of these questions can be adequately addressed, it would be helpful to look at how Frege himself thought of the Julius Caesar problem. In The Foundations of Arithmetic, he begins his own approach to the concept of Number by saying individual numbers can “belong to” concepts. Zero could then be defined as the number belonging to a concept for which it is universally true that a does not fall under that concept, whatever a may be. The number one belongs to a concept for which it is not universally true that a does not fall under that concept, but for which it is universally true that, given some a that does fall under the concept, anything else that falls under the same concept must be the same as a. The number n+1 belongs to a concept F when, for some a falling under that concept, the number n falls under the concept “falling under F, but not the same as a”.[1] But nowhere in this definition is there any statement of what it means for a number to “belong to” a concept, nor is there any explanation of what exactly these things are, that seem so fond of belonging to concepts. Once can, by repeatedly applying the definition for the step from a given number to the next, say what is meant by “the number 1+1+…+1 belongs to the concept F”, but this doesn’t explain what thing, in particular, the number 1+1+…+1 really is. If we cannot tell what kind of objects numbers are, then there is no justification for saying whether any specific object is a number or not. “We can never—to take a crude example—decide by means of our definitions whether any concept has the number Julius Caesar belonging to it, or whether that same familiar conqueror of Gaul is a number or is not.”[2]

Eventually, Frege works out a far more precise and useful set of definitions, including one that explains what sorts of things numbers themselves are: “The Number which belongs to the concept F is the extension of the concept ‘equinumerous to the concept F’”.[3] Two concepts are equinumerous if and only if there is some one-to-one relationship between all the objects falling under one concept and all the objects falling under the other. There is another problem with the earlier set of definitions that is perhaps more mathematically concerning whether or not Caesar is a number. It does not follow from those definitions that, if the number m belongs to a concept and the number n belongs to that same concept, m and n are necessarily the same. Fortunately, this does follow from the new set of definitions. The extension of a concept is a definite, individual thing, so if the concept “equinumerous to the concept F” has n as its extension, and also has m as its extension, then m and n must be the same. Furthermore, by definition, n is a number because it is the extension of the right kind of concept.

Defining numbers as the extensions of concepts clearly offers some advantages over the earlier definition, but does this solve the Julius Caesar problem? At best, it might do so in a very non-obvious way. Frege gives no explanation of what sorts of things extensions of concepts can be, and in fact seems to at one point go back on his assertion that concepts and objects must be kept separate (assuming that he sees extensions as objects, which is reasonable given the time he spends arguing that the numbers themselves are objects). Rather than fully defining what numbers are, Frege seems merely to have defined them in terms of something else whose nature may not be all that much clearer. What, if anything, prevents us from legitimately claiming that Julius Caesar himself might be the extension of a concept of equinumerosity? If there is nothing that prevents us from doing so, then the question remains as to whether Julius Caesar is a number, and it appears that no progress was made by adopting the new set of definitions, at least when it comes to deciding which objects cannot be numbers.

An approach that might get around the mathematical side of the problem would be to show that at least Caesar is not some additional element in a simple set of numbers. It is the natural numbers, including 0, that Frege spends most of his time on, so it is them we will deal with here. If it can be shown that Julius Caesar is not a new natural number, apart from all the more familiar integers, then perhaps the mathematical side of the problem will be taken care of. This would not constitute a proof that Julius Caesar is not any kind of number, or even that he is not one of the natural numbers. It would merely imply that, if it turned out that he is the extension of a concept of equinumerosity, he must be equal to one of the more familiar numbers, which are themselves extensions of concepts of equinumerosity.[4] Thus any calculation that involved the number Julius Caesar could be done just as well after substituting the more familiar integer in place of Julius Caesar, and then proceeding according to the normal rules of arithmetic.

The number (referred to by the symbol) 0 is the extension of the concept “equinumerous to the concept of not being self-identical”. The number n is the successor of the number m if n is the extension of the concept “equinumerous to the concept ‘falling under the concept F’” and m is the extension of the concept “equinumerous to the concept ‘falling under the concept F but not identical to a’”, for some object a which falls under the concept F and may itself be a number. Additionally, it is possible to define the successor of 0 as the extension of the concept “equinumerous to the concept of being identical with 0”.[5] We can define the set of natural numbers as the set N of objects that are elements of every set X that includes the number 0 and the successor of every number in X. Since 0, the successor of 0, and the successor of any number m are all extensions of concepts of equinumerosity, the only objects that would fall in every set X must themselves be numbers. Any object apart from 0 and those numbers that can be reached by repeatedly moving from one number to its successor, starting with 0, could not be in every set X, and so must not be in the set N of natural numbers.

We may not be able to say precisely what objects extensions of concepts are, but we can develop a system of symbols to refer to the extensions that are natural numbers. Take 0 as defined above, and take 1 to be (the symbol for) the successor of 0. Then, given the string of 1’s and 0’s referring to a number n, produce the string of 1’s and 0’s referring to the successor of n as follows: If the rightmost symbol is 0, replace it with 1 and stop. If the rightmost symbol is 1, replace it with 0 and shift attention to the symbol immediately to the left. If that symbol is 0, replace it with 1 and stop. If that symbol is 1, replace it with 0 and shift attention to the left again. Repeat this process, replacing 0 with 1 and stopping, or shifting attention immediately to the left each time 1 is replaced with 0. If there is no symbol to the left, append a 1 to the leftmost end of the string of symbols and stop.[6]

Every element of N is either 0 or can be reached by moving from 0 to its successor to its successor’s successor and so on, and every move from a number to its successor corresponds to some change in the string of 1’s and 0’s referring to that number. Therefore, every natural number can be represented by some string of 1’s and 0’s, including Julius Caesar if he turns out to be one of the nonnegative integers himself. If some poor soul was unfortunate enough to be faced with an arithmetical calculation involving the natural number Julius Caesar, he or she could rest assured that the calculation would go perfectly well if Caesar was represented in the more familiar binary notation. Of course, it is generally preferable to know what numbers one is dealing with when doing arithmetic, so this assurance would not be particularly helpful unless we also had some way of knowing where Julius Caesar falls in the sequence of natural numbers.

We are now at a point where we may look at whether late emperors of Rome are equally “unwelcome” as additional members in any other set of numbers. A rigorous and detailed discussion of this is outside the scope of the present paper, so the following brief outline will have to suffice. First, suppose we already have suitable definitions for the addition and multiplication of natural numbers. There are different formulations of these two operations, but they all amount to essentially the same thing. The additive inverse of a natural number n is then a number that, when combined with n through the process of addition, results in 0. The additive inverse of 0 is, of course, 0 itself. While it is difficult to see how Frege’s definition of the natural numbers using equinumerosity could be extended to the negative integers, we can infer from the process of addition itself that each natural number will have exactly one additive inverse. If the numbers b and c are not the same, then the number resulting from the operation a + b is not the same as that resulting from a + c. So if a + b has the same result as a + c, and both result in 0, then b is the same as c and is, by definition, the additive inverse of a. The set of natural numbers, when combined with the set of additive inverses of each natural number other than 0, gives the set of integers. Each additive inverse corresponds to one and only one natural number, so because we represent the additive inverse of a natural number n by appending the symbol `-` to the leftmost end of the string of digits (1’s and 0’s) used to represent n, every possible member of the set of additive inverses must correspond to a number whose binary representation we are already familiar with. Therefore, if Caesar is any integer at all, he must be representable by a string of 1’s and 0’s, possibly with a `-` sign on the left end of the string.

The multiplicative inverse of an integer n is that number which, when combined with n through the process of multiplication, results in 1. The multiplicative inverse of 1 is 1, the multiplicative inverse of -1 is -1, and 0 has no multiplicative inverse, because it has the property of resulting in 0 when multiplied by any number whatsoever. The result of multiplying any integer n by the additive inverse of a natural number m is the additive inverse of the number that results from the operation n * m. (Note that the additive inverse of the additive inverse of n is n itself.) By an argument analogous to the one above, each integer will have exactly one multiplicative inverse, representable by appending ‘1/’ to the left of the string of 1’s and 0’s (and possible `-` sign) that represents the integer itself. The entire set of rational numbers can be obtained by combining integers and multiplicative inverses of integers through the process of multiplication. Symbolically, this is done by replacing the 1 to the left of the / in the representation for a multiplicative inverse with the string of 1’s and 0’s used to represent the integer. Combining 0 with any other number through the process of multiplication yields 0. Because the multiplicative inverse of 1 is 1, 1/1 = 1, and so multiplying 1/1 by any integer n results in the integer n, meaning that every integer is contained in the set of rational numbers. Any possible rational number is representable by placing / between the representations for any two integers, so if Caesar is in the set of rational numbers, he must again be representable by some string of symbols with which we are already familiar.

The set of real numbers as much more to it than any of the sets described above, so it will have to be covered in even less detail here. We will have to take as given that every real number is expressible as the sum of an infinite sequence of rational numbers, without taking time to define sequences or infinite sums. Further, we will have to take as given that every real number is either expressible as the sum of a natural number and 1 or 0 multiplied by the multiplicative inverses of 10, 100, 1000, 10000, and so on, or it is the additive inverse of a number expressible in this manner. By the conventions of binary notation, each real number can then be represented by an possible `-` sign, followed by the string of symbols for the natural number component, followed by a `.` sign, followed by 1’s and 0’s corresponding to which is multiplied by the successive multiplicative inverses of 10, 100, 1000, and so on. Thus “1.1101…” represents the number resulting from 1+1/10+1/100+0/1000+1/10000 and so on. Even if Julius Caesar is some real number apart from the integers or rational numbers, he is still the same number as one for which we already have a more familiar (and practical) symbolic representation.

As far as calculations are concerned, the preceding discussion gets around most problems caused by the possibility of Julius Caesar being a number. Someone wanting to use complex numbers, vectors, or any other type of number might want to carry out a further proof that Caesar is not an additional element in any of those sets. Once such a proof is completed, and I am quite confident that this can be done without difficulty, a mathematician can forget entirely about whether one of the numbers he or she is using might have been assassinated some two millennia ago. (Of course, if one believes in ghosts, it might be advisable not to be too hasty in multiplying Caesar by 0 and thus annihilating him from an equation.) However, most people probably agree that Julius Caesar is not a number, and would thus appreciate a proof to that effect. Such a proof may be possible, but this appears not to be the case if we are working from Frege’s definition alone. While intuition may tell us that numbers are not material things, there is no such intuition about extensions of concepts in general. For instance, the extension of the concept, “Roman emperor killed in 44 BC”, is none other than Julius Caesar. Thus there is no immediately apparent reason to conclude that the extensions of some particular type of concept (i.e. concepts of equinumerosity) are never material objects.

Certainly one is still entitled to believe that Caesar is not a number, and indeed that no concept of equinumerosity has a material object as its extension, but if this belief somehow turned out to be false, what sorts of problems would result? If it did turn out to be true that Caesar is a number, there would be serious implications for the philosophy of math, not to mention the study of history. Classical historians the world over would have to seriously reconsider what they thought they knew about the identity of the first Caesar. Some might even be driven to abandon their careers altogether and become professors or some such thing. Philosophically, the problems presented are largely of the metaphysical variety. Perhaps the most obvious problem would be for the view that mathematical objects are subjective or at least immaterial and abstract, and who think other things, such as emperors, are objective. This view would have to be changed to account for both the objectivity and the materiality of at least one of the numbers. Platonists and nominalists alike would have to reevaluate their philosophical views. For someone who is a subjectivist about everything, a less obvious but possibly more serious problem would exist. If it is objectively and literally true that Caesar is a number, then not everything can be subjective, and the entire basis of a radical subjectivist’s philosophy would be undermined. However, one might dismiss evidence or proof of the existence of a number Julius Caesar as no more literally and objectively true than any other mathematical “proof”. Instead, the evidence or proof might simply be an imagined perception leading up to a change in subjective belief about the nature of Julius Caesar and the other numbers. If everything is in the mind anyway, there shouldn’t be a problem with deciding to believe that Caesar is a number.

There are of course countless other philosophical issues at hand, but perhaps more important are issues within mathematics itself. As stated above, mathematical calculations seem to be completely unaffected by the possibility that certain material things are numbers. (The preceding arguments about Julius Caesar are, of course, equally valid for every other material object.) There is no obvious logical contradiction that arises from the assumption that any particular number is in fact a material object. Of course, it would be necessary to somehow work around the problem of Caesar’s no longer being a material object. Otherwise it does seem that some difficulty would arise when trying to work with a number that had been gone for over two thousand years. Assuming such a problem can be dealt with, perhaps by understanding the entire history of the universe as some completed, timeless entity, there seems to be no further problem on the mathematical side of the question. It seems that no truth within arithmetic, or any higher math, will be in any jeopardy if we never solve the Julius Caesar “problem”.

[1] Frege, Gottlob. The Foundations of Arithmetic. Translated by J. L. Austin. §55.

[4] It should be pointed out that here we are not dealing with just any concepts of equinumerosity, but only those concepts whose extensions are the natural numbers, the set of which is to be defined below.

[6] I should probably discuss “1” and “0” rather than 1and 0 here, because we are manipulating symbols referring to objects, rather than the objects themselves. But doing so would have been both cumbersome and visually confusing.