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Gauss’ quadratic reciprocity theorem and mathematical fruitfulness
Studies in History and Philosophy of Science 42 (2011) 410–415
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Gauss’ quadratic reciprocity theorem and mathematical fruitfulness Audrey Yap Department of Philosophy, University of Victoria, P.O. Box 3045, Victoria, Canada BC V8W 3P4
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Article history: Received 3 May 2010 Received in revised form 28 September 2010 Available online 9 April 2011 Keywords: Number theory Philosophy of mathematics Carl Friedrich Gauss Mathematical fruitfulness Congruence notation
a b s t r a c t This paper presents an account of the fruitfulness of new mathematical calculi in terms of their relationship to existing mathematical methods which is suggested by Carl Friedrich Gauss. This is done by considering some remarks that Gauss made explaining the fruitfulness of new calculi. These can be clarified in the context of his own (very fruitful) theory of congruences, which is considered as a case study for this alternative account. Such an account has the benefit of not being dependent on a particular metaphysical view in the philosophy of mathematics. 2011 Elsevier Ltd. All rights reserved.
When citing this paper, please use the full journal title Studies in History and Philosophy of Science
1. What are fruitful mathematical calculi? Georg Cantor justified the free exploration of conceptual possibilities (such as set theory) in mathematics by claiming that Every mathematical concept carries within itself the necessary corrective: if it is fruitless or unsuited to its purposes, then that appears very soon through its uselessness, and it will be abandoned for lack of success. (Cantor, 1883) But how are we to account for the fact that some mathematical concepts or methods seem to be more fruitful than others? One option is to take a realist line of explanation. Tappenden (2008), for instance, describes the fruitfulness of the Legendre symbol in terms of its carving mathematical reality at the joints. Such an account has its appeal, but in this paper, I would like to suggest another plausible line of explanation that does not appeal to mathematical realism. In this paper, I will consider some remarks that C. F. Gauss made on the fruitfulness of new mathematical methods; these remarks can be clarified in the context of his own (very fruitful) theory of congruences. I will then consider the extent to which this case study can help us account for mathematical fruitfulness as being relative to background mathematical methods. Considering
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mathematical fruitfulness as relative in this fashion provides an alternative to a realist explanation, though it does not preclude mathematical realism. 1.1. Characterizing fruitful methods In a letter, dated May 15th, 1843, addressed to Heinrich Christian Schumacher, one of his first students at Göttingen, Gauss wrote the following: In general the position as regards all such new calculi is this— that one cannot attain by them anything that could not be done without them: the advantage, however, is, that if such a calculus corresponds to the innermost nature of frequent wants, every one who assimilates it thoroughly is able—without the unconscious inspiration of genius which no one can command—to solve the respective problems, yes, even to solve them mechanically in complicated cases where genius itself becomes impotent. So it is with the invention of algebra generally, so with the differential calculus, so also—though in more restricted regions—with Lagrange’s calculus of variations, with my calculus of congruences, and with Möbius’s calculus. Through such
A. Yap / Studies in History and Philosophy of Science 42 (2011) 410–415
conceptions countless problems which otherwise would remain isolated and require every time (larger or smaller) efforts of inventive genius, are, as it were, united into an organic whole. (Gauss, 1900, p. 298) (quoted and translated in (Merz, 1912, p. 724)) In this letter, his claim seems to be the following: there are cases in which new mathematical calculi have been developed that were conservative extensions of the old theory, which nevertheless allowed mathematicians to solve problems previously thought to be intractable. Furthermore, these calculi could give solutions which we can call mechanical, even when the original problem was a complicated one. Restricting our attention here to conservative extensions then means that fruitfulness is not to be understood in terms of adding deductive power to the theory. Rather, we are to understand fruitfulness in terms of these calculi ‘‘being assimilated thoroughly’’ and corresponding to ‘‘the innermost nature of frequent wants.’’ Then we will be able to solve problems even ‘‘mechanically in complicated cases when genius itself becomes impotent’’. After some initial remarks about these phrases, I will outline the specifics of Gauss’ theory of congruences, and the complicated case whose solution it enabled: the proof of the quadratic reciprocity theorem. The first complete proof of this number-theoretic theorem appeared in Gauss’ Disquisitiones Arithmeticae, the work in which he also introduced the theory of congruences for the first time. First, the idea of ‘‘the innermost nature of frequent wants’’ immediately brings up the idea of there being certain ‘‘natural’’ mathematical methods or operations. This idea certainly admits of a psychological interpretation, but I want to consider how it could be understood in more mathematical terms. Perhaps our ‘‘frequent wants’’ arise when we find ourselves rederiving certain results again and again in proofs, without a general theorem to cover all of the individual cases. Second, we have the question of how one might assimilate a calculus thoroughly. This is likely more than a superficial recognition of its usefulness. And even though there could be a psychological interpretation of this phrase as well, a more mathematical one is also available. Given that a new calculus provides us with new mathematical tools, we could explain its assimilation as the integration of these tools with existing methods. Then they can subsequently be used in proofs alongside methods of the background theory. Finally, how can it be possible to give a mechanical proof of a theorem? One straightforward answer would be that there is a kind of effective procedure allowing the truth of the theorem to be ‘‘calculated’’, so to speak. But this is surely too strong a condition. It would not be plausible to claim that new calculi generally yield computationally effective procedures by which theorems can be decided. An alternative explanation is that our new calculi prove to be such useful analytical tools for the problems we apply them to, that they allow us to break them down in systematic ways, and tackle them piece by piece. Although in order for this to be useful, the number of cases yielded ought to remain tractable. The mechanical nature, in the sense of yielding a useful analytical tool, of Gauss’ first proof of quadratic reciprocity is what makes it unappealing to follow, yet methodologically interesting. 2. Congruences and quadratic reciprocity The theory of congruences was a new calculus in that it added a new relation symbol to number theory. And it is also a case in which, strictly speaking, one cannot ‘‘attain by it anything that could not be done without it’’. For congruence notation is not only 1 2 3
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a conservative extension, but merely a definitional extension, in which the new symbol added can be seen as an abbreviation for a longer statement in the language of the original theory. Gauss defines congruence by writing that If a number a divides the difference of the numbers b and c, b and c are said to be congruent relative to a; if not, b and c are noncongruent (Gauss, 1801/1965) Symbolically, this relationship is represented by the statement b c (mod a). So whenever we say that b and c are congruent relative to a modulus a, we mean that a divides the difference of b and c. Further, all of the defining axioms of the congruence relation that Gauss subsequently mentions correspond to already provable statements about divisibility relations. So congruence notation adds no prooftheoretic strength whatsoever. In fact, as they are provable propositions, its axioms are not axioms in the strictest sense of the word, although they do resemble axioms in the structural sense,1 in that they serve a descriptive purpose. Congruence notation can first be explained logically, though somewhat anachronistically. Suppose our initial language is just the language of number theory hZ, +, , 6i. Then, let us extend it by a new three-place relation symbol , where
a b ðmod pÞ iff9xða ¼ b þ ðx pÞÞ: Below are the properties of congruences that Gauss outlines at the start of Disquisitiones Arithmeticae, formulated in symbolic language for the sake of clarity. 1. a a (mod m) (Reflexivity) 2. a b (mod m) M b a (mod m) (Symmetry)2 V 3. a b (mod m) c b (mod m) ? a c (mod m) (Transitivity) V 4. "x $ y (0 6 y < m x y (mod m) (Existence of Least Residues) V 5. a A (mod m) b B (mod m) ? a + b A + B (mod m) (Preservation Under Addition) V 6. a A (mod m) b B (mod m) ? a b A B (mod m) (Preservation Under Subtraction) 7. a A (mod m) ? ka kA (Preservation Under Scalar Multiplication) V 8. a A (mod m) b B (mod m) ? ab AB (mod m) (Preservation Under Multiplication) All of these properties are provable, fairly straightforwardly, from the definitions and basic algebra. For instance, we can show preservation under (a) addition, (b) scalar, and (c) regular multiplication as follows: Suppose a A (mod m) and b B (mod m). Then there are x and y such that a A = xm and b B = ym. (a) (a A) + (b B) = xm + ym. Regrouping, we obtain a + b (A + B) = (x + y)m, which is equivalent to a + b A + B (mod m). (b) k(a A) = kxm, which implies ka kA = (kx)m, which is equivalent to ka kA (mod m). (c) By preservation under scalar multiplication, we know that AB Ab ab (mod m). So each result follows quite quickly, but proving any one requires at least one extra variable. This is because the formula which congruence notation abbreviates, contains an extra variable which it masks. The introduction of the new notation provides several shortcuts which eliminate the need for some variables being used in the proof, and hence some added complexity.3 Then
I refer to Feferman’s distinction between ideal and structural axioms in (Feferman, 1999). Gauss does not actually prove reflexivity or symmetry, but they are obvious properties of the relation, which he assumes. Avigad (2006) makes essentially the same points about the advantages of congruence notation.
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‘‘assimilating the methods of a new calculus’’ could just involve adding these defining axioms to the regular stock of results to which our proofs appeal. We can also connect this notation to the idea of a quadratic residue. Given a modulus m, we have a complete system of residues for that modulus: the numbers 0, 1, 2, . . . (m 1). However, we can also consider a system of residues prime to m, which is a complete system from which every residue has been omitted which has any common divisor with the modulus. In the case of a prime modulus p, these two notions coincide. The numbers 1, 2, . . . , (p 1) divide themselves evenly into quadratic residues and quadratic nonresidues. We call q a quadratic residue of p whenever there is an x such that p divides x2 q. Stated using congruences, we call q a quadratic residue of p whenever there is a solution to the congruence x2 q (mod p), and a nonresidue whenever that congruence is irresolvable. An example of a result which Gauss proves quite quickly using his ‘‘lemmas’’ about congruence notation is that the product of two quadratic residues is also a quadratic residue: Suppose a and b are both quadratic residues of p. Then there are x and y such that x2 a (mod p) and y2 b (mod p). By preservation under multiplication, we know that ab x2y2 (xy)2 (mod p). So the product is also a quadratic residue. A proof of this without using congruence notation is the following: Suppose a and b are both quadratic residues of p. Then there are x and y such that x2 a and y2 b are divisible by p. This implies that there are u, v such that a = x2 pu and b = y2 pv. Multiplying, we obtain ab = x2y2 x2pv y2pu + p2uv. But then we see that ab = x2y2 p(x2v y2u + puv). So ab is also a quadratic residue, since x2y2 ab is divisible by p. This proof is not only longer, but also messier than the previous one, since it requires the introduction of four new variables, where the former only required two. Also, the two ‘‘extra’’ variables are inessential, since we do not care about the multiplicity by which a x2 (for instance) is divisible by p. All the matters is the fact that it is divisible, which is straightforwardly represented by the congruence statement. It even uses the same property of equality as the proof above did for congruences, which is preservation under multiplication. Then the ‘‘innermost nature of frequent wants’’ can be explained in terms of the applicability of the new lemmas, once they have been assimilated. 2.1. The quadratic reciprocity theorem The quadratic reciprocity theorem is a general law in number theory having to do with quadratic residues. Gauss called it ‘the gem of the higher arithmetic,’ and Henry Smith, in his writing about the history of number theory, calls it ‘‘the most important general truth in the science of integral numbers which has been discovered since the time of Fermat’’ (Smith, 1859, p. 56). The theorem itself is the following: For odd primes p and q, they are either both residues of one another, or both non-residues of one another, unless they are both of the form 4k + 3, in which case exactly one is a residue of the other. And at first glance, it does seem quite non-obvious and surprising. Why would we connect the existence of a solution to x2 q (mod p) to the existence of a solution to x2 p (mod q)? (Or connect the existence of an x such that p divides x2 q to the existence of a y such that q divides y2 p?) A version of the theorem was first stated by Euler in an article entitled ‘‘Observationes circa divisionem quadratorum per
numeros primos’’ (Euler, 1783), but only has the status of a conjecture, which he believed on inductive evidence, confirmed by testing many cases. The first time the theorem was given in its modern form—the form in which it was presented here—was in a Memoir by Adrien-Marie Legendre, for the ‘Histoire de L’Academie des Sciences’ in 1785. However, the accompanying demonstration was incomplete, due to an undercharged assumption which it makes. The first complete proof was Gauss’, from his Disquisitiones Arithmeticae, which appeared in 1801. However, the first proof was also extremely inelegant, and Smith refers to it as being ‘‘presented by Gauss in a form very repulsive to any but the most laborious students (Smith, 1859, p. 59).’’ So we will next turn to this proof, in order to see in what sense it might count as a mechanical solution to the problem. 2.2. Gauss’ proof The first proof that appears in Disquisitiones Arithmeticae is an inductive one. The overall strategy is fairly simple, though the execution requires an exhaustive treatment of cases. Gauss’ statement of the theorem differs only slightly from the modern version: If p is a prime number of the form 4n + 1, +p will be a residue or nonresidue of any prime number which taken positively is a residue or nonresidue of p. If p is of the form 4n + 3, p will have the same property. (Art. 130) To see that his formulation is in fact equivalent to our original statement of the theorem, note that for primes of the form 4n + 1, if r is a residue, r will also be a residue, and for primes of the form 4n + 3, if r is a residue, r will be a non-residue. In carrying out the proof, Gauss first deduces several consequences of the quadratic reciprocity theorem, and notes that, if the theorem holds up to some number n, so do these consequences. The base case of the induction is established by verifying the theorem for 3 and 5. We then suppose that the quadratic reciprocity theorem holds up to some number T, and consider T + 1. If it were to fail at T + 1, this would mean that there are two prime numbers that contradict the theorem when compared, and the larger of these prime numbers is T + 1. But then we can still assume that all of our consequences of the quadratic reciprocity theorem hold for pairs of numbers both of which are less than or equal to T. Now we can distinguish eight ways in which the theorem can fail, two for each possible combination of primes. Let a and A be primes of the form 4n + 1 and b and B be primes of the form 4n + 3. We will write aRb to abbreviate that a is a residue of b and aNb to abbreviate that a is a nonresidue. The following table outlines the eight ways in which the theorem could fail.
I II III IV V VI VII VIII
If we have
Then
aRA aNA bRB bNB bRa bNa aRb aNb
ANa ARa BRb BNb aNb aRb bNa bRa
Gauss deals with each of the eight cases individually, showing that each one is impossible. We will only take up the first of these cases in detail here. So let us consider a = T + 1 and A, both of the
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form 4n + 1. We will suppose that A is a residue of a, and we will show that we cannot have a being a nonresidue of A. Once we assume ARa, meaning that there is an x2 such that A x2 (mod a), Gauss’ first step is to note that we can pick x such that x is positive, even and
3. Congruences, equivalence classes, and induction Congruence notation easily lends itself to the mathematical technique of induction. Every modulus yields a partition of the integers into equivalence classes made up of congruent integers with respect to that modulus. That this is possible follows directly from the axioms of least residues and transitivity. So even though Gauss does not appeal directly to the notion of an equivalence class, it is a very natural extension of the way in which he already works with congruences.
4 5
The only exception to this is where a appears as an exponent. For instance, Rosen (2000).
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We can see the way in which this plays out by invoking the preservation axioms. Congruent numbers (relative to some modulus) can be substituted for each other in congruence statements relative to that same modulus. This is a direct consequence of preservation under addition and multiplication. So, more generally, if we have a congruence statement modulo m in which a appears, we can substitute any member of a’s equivalence class for a, and preserve the truth of the statement.4 In fact, Gauss uses this technique quite frequently in his proof, substituting congruent numbers for one another, depending on the properties he wants them to have. Working with equivalences is much like working with identities, since an equivalence relation shares key properties with the identity relation. Substituting equals for equals, and carrying out fairly simple operations on identity statements, are straightforward and common algebraic tasks. When we have a mathematical equality between two terms, we can substitute those terms for each other in any mathematical proposition, while preserving the truth of that proposition. And just as we can substitute equals for equals, numbers congruent relative to a fixed modulus m can be substituted for each other in congruences (mod m). Thus one advantage of congruences is the similarity between working with congruence relations and working with identity statements. Indeed, when Gauss introduces the congruence symbol for the first time, he explicitly notes its resemblance to the identity symbol. Since we are already accustomed to working with identities, it is easy to ‘‘assimilate thoroughly’’ the methods congruence notation provides, since they are very similar to methods we already employ for dealing with identity statements. This latter fact about congruences is even cited in modern number theory textbooks when congruence notation is introduced.5 Given their relationship to equivalence classes and the manner in which congruence relative to m partitions the integers, the usefulness of congruences in induction becomes apparent. After all an inductive proof will make use of its inductive hypothesis, but can only do so to numbers which are below the threshold of induction. In Gauss’ proof, this allows us to exploit the fact that useful consequences of the quadratic reciprocity theorem hold for pairs of numbers smaller than a. But since a is also our modulus, we can work almost exclusively with positive numbers smaller than a; every integer that is not between 0 and a has an element of its equivalence class (mod a) which is. The existence of a least residue underlies much of the inductive step in Gauss’ proof. In particular, the proof that there was an x we could pick which was both even and
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of p. Its own defining axiom is: ðpaÞðbpÞ ¼ ðab Þ. This symbol is also one p which seems very natural—perhaps even more natural when it comes to tackling the theorem in question. This leads us to ask a (perhaps unfair) historical question. Why did Gauss manage to prove the theorem with the help of his new calculus, while Legendre never filled the last gap in his own proof? Even if the question is unfair, it can still provide us with some interesting insights, as long as we recognize its limitations. One obvious reason why it is unfair is the fact that the two calculi are perfectly compatible, and many modern treatments of quadratic reciprocity use both. Also, it is not simply the case that if Legendre had used congruence notation, the proof would have been successful. Gauss’ proof does not rely on any results which can only be proved using congruences, since congruences do not add prooftheoretic strength.6 The matter of Gauss’ mathematical talent should not be neglected either; we can almost certainly attribute some of Gauss’ success in giving a full proof, where Legendre did not, to Gauss’ greater skill as a mathematician. But perhaps part of being a more skilled mathematician is the development of better mathematical tools, such as the theory of congruences, which can then aid in proofs. So it is still worth looking at the reason why congruence notation might have been a better tool for proving this particular theorem. In looking at Legendre’s attempted proof, however, from Legendre (1808), we find many ways in which it resembles Gauss’ own. Both break up the problem into different cases, dealing with each one in turn, and their treatment of individual cases is very similar. The clear difference is in the overall proof strategy: Gauss’ proof uses induction, while Legendre carefully arranges the order in which he proves his cases. So while he has no inductive hypothesis to draw on, in order to use consequences of the reciprocity theorem, Legendre can still rely on results proved earlier on. But as we saw in the discussion of induction in Gauss’ proof, the fact that Gauss could appeal to least residues allowed him to choose a smaller member of the equivalence class, which allowed the inductive hypothesis to be applied. And as it turned out in Legendre’s proof, a key choice of number remained unjustified, which was the principal gap remaining. A difference between congruence notation and the Legendre symbol is their respective generality. Expressively, being a quadratic residue is a more ‘‘specific’’ relation than being congruent, in the sense that the former is expressible using the latter, but not vice versa: the Legendre symbol expresses that one number is a quadratic residue of another, which simply states that a congruence statement holds, and a congruence statement in turn expresses a divisibility relation. The ability of congruence notation to mask an extra variable in divisibility statements was an advantage, since it simplifies proofs. So with respect to the divisibility relation, its greater specificity is an advantage. However, we also saw that the Legendre symbol masks yet another variable, but that this was a disadvantage, since the x, such that a x2 (mod p) was a number which could be used in the proof. We can consider three ways logically to state that a is a quadratic residue of p: In the original language: $x $ y (x2 = a + py). Using congruences: $x (x2 a (mod p)). Using the Legendre symbol: ðpaÞ ¼ 1. In contrast with the Legendre symbol, the greater generality of congruence notation is an advantage. The notation allows us to present enough information in the proof that we can appeal to other results about divisibility, but manages to disguise some
6
irrelevant information as well, such as the multiplicity by which certain numbers divide each other. Perhaps congruence notation strikes a better balance between generality and specificity. This difference in generality might also have to do with the applicability of the defining axioms of each symbol. In terms of Gauss’ earlier remarks, perhaps congruences are better assimilated into existing methods of proof, because there are more situations in which the defining axioms for congruences can be applied. In fact, when we look at Legendre’s proof, and the section that corresponds to the part of Gauss’ proof outlined in this paper, we find only one place in which the defining axiom of the Legendre symbol is used. This is in the claim that ðab Þ ¼ ðAaÞðAbÞ. A But even though it is not an axiom, we can prove an equivalent theorem using congruences, and in fact, Gauss invokes that very theorem in his own proof. In contrast, though, the defining axioms for congruences were used in very many places throughout Gauss’ proof. The existence of least residues was invoked very often, and the substitutability of congruent numbers for each other, which relies on the preservation axioms, was also used quite often. None of this, however, should be taken as a criticism of the Legendre symbol as a piece of notation. Indeed, Gauss’ proof of the theorem can be simplified from eight cases to two cases through a generalization of the Legendre symbol (Tappenden, 2008, p. 263). So while the Legendre symbol is a useful definition in its own right, there are still reasons to think that, for the particular purpose of proving quadratic reciprocity, it was not as useful as congruence notation. And perhaps part of what makes the Legendre symbol seem so natural is that the quadratic reciprocity theorem is in fact true. So while Tappenden (2008) may be right that this symbol does carve nature at its joints, this alone does not make it a fruitful method—nor a fruitful method for all purposes. Given that such a relationship between quadratic residues does hold, a calculus which represents it, and embodies many of the more important theorems about it, is a very natural one. But if this is right, that could explain why it might not have been as useful as congruence notation in this initial proof of the reciprocity theorem. The usefulness of the Legendre symbol seems to be more specific, and tied to the truth of the reciprocity theorem itself, whereas the usefulness of congruences, being more general, is wider. Uses of the Legendre symbol in applications of quadratic reciprocity have vindicated its usefulness in many other contexts, but this is perfectly compatible with the claim that it, on its own, was not useful in proving the reciprocity theorem in the first place. Then, although we have spent most of our time discussing the idea of the ‘‘innermost nature of frequent wants’’, and what it means to ‘‘assimilate a calculus’’, these interpretive remarks can be brought to bear on the idea of enabling a mechanical proof. A new calculus can help us carry out a straightforward analysis of a problem, through its use of ‘‘natural’’ methods, and through the general applicability of its defining axioms. In the case of the quadratic reciprocity theorem, these calculi did not enable a solution that was really mechanical; we could not just calculate that the theorem holds. However, it is a result which divides up naturally into cases, and Gauss’ systematic analysis of each case using his new lemmas did allow him to prove each case in turn. Gauss went on to construct several different proofs of the theorem, most of which are seen as more intuitive and natural than the one given here, which was his first. The term ‘‘mechanical proof’’ here can perhaps be taken to mean a proof using caseby-case analysis.
Not to mention the fact that there are other proofs of the theorem, many of which are given by Gauss himself, which do not rely on congruences.
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3.2. Characterizing fruitfulness Congruences were a very natural tool in proving the quadratic reciprocity theorem, since they capture a relationship of divisibility, closely connected to the concept of a quadratic residue, but still general enough to allow for the application of other facts about divisibility. Whether a calculus corresponds to the innermost nature of frequent wants is a question of whether it provides useful lemmas for the proofs in which we want to use it. And whether it can be assimilated thoroughly has to do with its connection to our existing methods and how well it meshes with them. No definitive answer is being given here about exactly what makes some mathematical methods more fruitful than others in general. But Gauss’ remarks at least provide us with an outline of a non-realist (though not anti-realist) account of mathematical fruitfulness. While a realist account is certainly not ruled out by this discussion, my purpose here was to show that such an account is by no means the only way of explaining fruitfulness. Instead, the success of a new method can be explained as a matter of its relationship to the goals to which it is applied (the theorems we want to use it to prove) as well as to the existing methods of the field. Some methods may prove to be less useful in some contexts and
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more useful in others. What matters is the way in which the various fields and methods of mathematics fit together. References Avigad, J. (2006). Mathematical method and proof. Synthese, 153, 105–159. Cantor, G. (1883). Foundations of a general theory of manifolds: A mathematicophilosophical investigation into the theory of the infinite. In Ewald, W. (Ed.), (1996). From Kant to Hilbert: A sourcebook in the foundations of mathematics, volume II. Oxford: Clarendon Press. Euler, L. (1783). Opera Omnia, volume iii. Feferman, S. (1999). Does mathematics need new axioms? The American Mathematical Monthly, 106(2), 99–111. Gauss, C. F. (1801/1965). Disquisitiones arithmeticae. New Haven: Yale University Press. Gauss, C. F. (1900). Zu Möbius’ Barycentrischem Calcul. In Carl Friedrich Gauss Werke, VIII Band (pp. 301–304). Göttingen. Legendre, A. M. (1808). Essai sur la théorie des nombres. Paris: Courcier. Merz, J. T. (1912). A history of European thought in the nineteenth century. Edinburgh: Blackwood Press. Rosen, K. H. (2000). Elementary number theory and its applications. Reading: Addison-Wesley. Smith, H. J. (1859). Report on the theory of numbers. New York: Chelsea Publishing Company. Tappenden, J. (2008). Mathematical concepts: Fruitfulness and naturalness. In P. Mancosu (Ed.), The philosophy of mathematical practice. Oxford: Oxford University Press.
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Studies in History and Philosophy of Science journal homepage: www.elsevier.com/locate/shpsa
Gauss’ quadratic reciprocity theorem and mathematical fruitfulness Audrey Yap Department of Philosophy, University of Victoria, P.O. Box 3045, Victoria, Canada BC V8W 3P4
a r t i c l e
i n f o
Article history: Received 3 May 2010 Received in revised form 28 September 2010 Available online 9 April 2011 Keywords: Number theory Philosophy of mathematics Carl Friedrich Gauss Mathematical fruitfulness Congruence notation
a b s t r a c t This paper presents an account of the fruitfulness of new mathematical calculi in terms of their relationship to existing mathematical methods which is suggested by Carl Friedrich Gauss. This is done by considering some remarks that Gauss made explaining the fruitfulness of new calculi. These can be clarified in the context of his own (very fruitful) theory of congruences, which is considered as a case study for this alternative account. Such an account has the benefit of not being dependent on a particular metaphysical view in the philosophy of mathematics. 2011 Elsevier Ltd. All rights reserved.
When citing this paper, please use the full journal title Studies in History and Philosophy of Science
1. What are fruitful mathematical calculi? Georg Cantor justified the free exploration of conceptual possibilities (such as set theory) in mathematics by claiming that Every mathematical concept carries within itself the necessary corrective: if it is fruitless or unsuited to its purposes, then that appears very soon through its uselessness, and it will be abandoned for lack of success. (Cantor, 1883) But how are we to account for the fact that some mathematical concepts or methods seem to be more fruitful than others? One option is to take a realist line of explanation. Tappenden (2008), for instance, describes the fruitfulness of the Legendre symbol in terms of its carving mathematical reality at the joints. Such an account has its appeal, but in this paper, I would like to suggest another plausible line of explanation that does not appeal to mathematical realism. In this paper, I will consider some remarks that C. F. Gauss made on the fruitfulness of new mathematical methods; these remarks can be clarified in the context of his own (very fruitful) theory of congruences. I will then consider the extent to which this case study can help us account for mathematical fruitfulness as being relative to background mathematical methods. Considering
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mathematical fruitfulness as relative in this fashion provides an alternative to a realist explanation, though it does not preclude mathematical realism. 1.1. Characterizing fruitful methods In a letter, dated May 15th, 1843, addressed to Heinrich Christian Schumacher, one of his first students at Göttingen, Gauss wrote the following: In general the position as regards all such new calculi is this— that one cannot attain by them anything that could not be done without them: the advantage, however, is, that if such a calculus corresponds to the innermost nature of frequent wants, every one who assimilates it thoroughly is able—without the unconscious inspiration of genius which no one can command—to solve the respective problems, yes, even to solve them mechanically in complicated cases where genius itself becomes impotent. So it is with the invention of algebra generally, so with the differential calculus, so also—though in more restricted regions—with Lagrange’s calculus of variations, with my calculus of congruences, and with Möbius’s calculus. Through such
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conceptions countless problems which otherwise would remain isolated and require every time (larger or smaller) efforts of inventive genius, are, as it were, united into an organic whole. (Gauss, 1900, p. 298) (quoted and translated in (Merz, 1912, p. 724)) In this letter, his claim seems to be the following: there are cases in which new mathematical calculi have been developed that were conservative extensions of the old theory, which nevertheless allowed mathematicians to solve problems previously thought to be intractable. Furthermore, these calculi could give solutions which we can call mechanical, even when the original problem was a complicated one. Restricting our attention here to conservative extensions then means that fruitfulness is not to be understood in terms of adding deductive power to the theory. Rather, we are to understand fruitfulness in terms of these calculi ‘‘being assimilated thoroughly’’ and corresponding to ‘‘the innermost nature of frequent wants.’’ Then we will be able to solve problems even ‘‘mechanically in complicated cases when genius itself becomes impotent’’. After some initial remarks about these phrases, I will outline the specifics of Gauss’ theory of congruences, and the complicated case whose solution it enabled: the proof of the quadratic reciprocity theorem. The first complete proof of this number-theoretic theorem appeared in Gauss’ Disquisitiones Arithmeticae, the work in which he also introduced the theory of congruences for the first time. First, the idea of ‘‘the innermost nature of frequent wants’’ immediately brings up the idea of there being certain ‘‘natural’’ mathematical methods or operations. This idea certainly admits of a psychological interpretation, but I want to consider how it could be understood in more mathematical terms. Perhaps our ‘‘frequent wants’’ arise when we find ourselves rederiving certain results again and again in proofs, without a general theorem to cover all of the individual cases. Second, we have the question of how one might assimilate a calculus thoroughly. This is likely more than a superficial recognition of its usefulness. And even though there could be a psychological interpretation of this phrase as well, a more mathematical one is also available. Given that a new calculus provides us with new mathematical tools, we could explain its assimilation as the integration of these tools with existing methods. Then they can subsequently be used in proofs alongside methods of the background theory. Finally, how can it be possible to give a mechanical proof of a theorem? One straightforward answer would be that there is a kind of effective procedure allowing the truth of the theorem to be ‘‘calculated’’, so to speak. But this is surely too strong a condition. It would not be plausible to claim that new calculi generally yield computationally effective procedures by which theorems can be decided. An alternative explanation is that our new calculi prove to be such useful analytical tools for the problems we apply them to, that they allow us to break them down in systematic ways, and tackle them piece by piece. Although in order for this to be useful, the number of cases yielded ought to remain tractable. The mechanical nature, in the sense of yielding a useful analytical tool, of Gauss’ first proof of quadratic reciprocity is what makes it unappealing to follow, yet methodologically interesting. 2. Congruences and quadratic reciprocity The theory of congruences was a new calculus in that it added a new relation symbol to number theory. And it is also a case in which, strictly speaking, one cannot ‘‘attain by it anything that could not be done without it’’. For congruence notation is not only 1 2 3
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a conservative extension, but merely a definitional extension, in which the new symbol added can be seen as an abbreviation for a longer statement in the language of the original theory. Gauss defines congruence by writing that If a number a divides the difference of the numbers b and c, b and c are said to be congruent relative to a; if not, b and c are noncongruent (Gauss, 1801/1965) Symbolically, this relationship is represented by the statement b c (mod a). So whenever we say that b and c are congruent relative to a modulus a, we mean that a divides the difference of b and c. Further, all of the defining axioms of the congruence relation that Gauss subsequently mentions correspond to already provable statements about divisibility relations. So congruence notation adds no prooftheoretic strength whatsoever. In fact, as they are provable propositions, its axioms are not axioms in the strictest sense of the word, although they do resemble axioms in the structural sense,1 in that they serve a descriptive purpose. Congruence notation can first be explained logically, though somewhat anachronistically. Suppose our initial language is just the language of number theory hZ, +, , 6i. Then, let us extend it by a new three-place relation symbol , where
a b ðmod pÞ iff9xða ¼ b þ ðx pÞÞ: Below are the properties of congruences that Gauss outlines at the start of Disquisitiones Arithmeticae, formulated in symbolic language for the sake of clarity. 1. a a (mod m) (Reflexivity) 2. a b (mod m) M b a (mod m) (Symmetry)2 V 3. a b (mod m) c b (mod m) ? a c (mod m) (Transitivity) V 4. "x $ y (0 6 y < m x y (mod m) (Existence of Least Residues) V 5. a A (mod m) b B (mod m) ? a + b A + B (mod m) (Preservation Under Addition) V 6. a A (mod m) b B (mod m) ? a b A B (mod m) (Preservation Under Subtraction) 7. a A (mod m) ? ka kA (Preservation Under Scalar Multiplication) V 8. a A (mod m) b B (mod m) ? ab AB (mod m) (Preservation Under Multiplication) All of these properties are provable, fairly straightforwardly, from the definitions and basic algebra. For instance, we can show preservation under (a) addition, (b) scalar, and (c) regular multiplication as follows: Suppose a A (mod m) and b B (mod m). Then there are x and y such that a A = xm and b B = ym. (a) (a A) + (b B) = xm + ym. Regrouping, we obtain a + b (A + B) = (x + y)m, which is equivalent to a + b A + B (mod m). (b) k(a A) = kxm, which implies ka kA = (kx)m, which is equivalent to ka kA (mod m). (c) By preservation under scalar multiplication, we know that AB Ab ab (mod m). So each result follows quite quickly, but proving any one requires at least one extra variable. This is because the formula which congruence notation abbreviates, contains an extra variable which it masks. The introduction of the new notation provides several shortcuts which eliminate the need for some variables being used in the proof, and hence some added complexity.3 Then
I refer to Feferman’s distinction between ideal and structural axioms in (Feferman, 1999). Gauss does not actually prove reflexivity or symmetry, but they are obvious properties of the relation, which he assumes. Avigad (2006) makes essentially the same points about the advantages of congruence notation.
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‘‘assimilating the methods of a new calculus’’ could just involve adding these defining axioms to the regular stock of results to which our proofs appeal. We can also connect this notation to the idea of a quadratic residue. Given a modulus m, we have a complete system of residues for that modulus: the numbers 0, 1, 2, . . . (m 1). However, we can also consider a system of residues prime to m, which is a complete system from which every residue has been omitted which has any common divisor with the modulus. In the case of a prime modulus p, these two notions coincide. The numbers 1, 2, . . . , (p 1) divide themselves evenly into quadratic residues and quadratic nonresidues. We call q a quadratic residue of p whenever there is an x such that p divides x2 q. Stated using congruences, we call q a quadratic residue of p whenever there is a solution to the congruence x2 q (mod p), and a nonresidue whenever that congruence is irresolvable. An example of a result which Gauss proves quite quickly using his ‘‘lemmas’’ about congruence notation is that the product of two quadratic residues is also a quadratic residue: Suppose a and b are both quadratic residues of p. Then there are x and y such that x2 a (mod p) and y2 b (mod p). By preservation under multiplication, we know that ab x2y2 (xy)2 (mod p). So the product is also a quadratic residue. A proof of this without using congruence notation is the following: Suppose a and b are both quadratic residues of p. Then there are x and y such that x2 a and y2 b are divisible by p. This implies that there are u, v such that a = x2 pu and b = y2 pv. Multiplying, we obtain ab = x2y2 x2pv y2pu + p2uv. But then we see that ab = x2y2 p(x2v y2u + puv). So ab is also a quadratic residue, since x2y2 ab is divisible by p. This proof is not only longer, but also messier than the previous one, since it requires the introduction of four new variables, where the former only required two. Also, the two ‘‘extra’’ variables are inessential, since we do not care about the multiplicity by which a x2 (for instance) is divisible by p. All the matters is the fact that it is divisible, which is straightforwardly represented by the congruence statement. It even uses the same property of equality as the proof above did for congruences, which is preservation under multiplication. Then the ‘‘innermost nature of frequent wants’’ can be explained in terms of the applicability of the new lemmas, once they have been assimilated. 2.1. The quadratic reciprocity theorem The quadratic reciprocity theorem is a general law in number theory having to do with quadratic residues. Gauss called it ‘the gem of the higher arithmetic,’ and Henry Smith, in his writing about the history of number theory, calls it ‘‘the most important general truth in the science of integral numbers which has been discovered since the time of Fermat’’ (Smith, 1859, p. 56). The theorem itself is the following: For odd primes p and q, they are either both residues of one another, or both non-residues of one another, unless they are both of the form 4k + 3, in which case exactly one is a residue of the other. And at first glance, it does seem quite non-obvious and surprising. Why would we connect the existence of a solution to x2 q (mod p) to the existence of a solution to x2 p (mod q)? (Or connect the existence of an x such that p divides x2 q to the existence of a y such that q divides y2 p?) A version of the theorem was first stated by Euler in an article entitled ‘‘Observationes circa divisionem quadratorum per
numeros primos’’ (Euler, 1783), but only has the status of a conjecture, which he believed on inductive evidence, confirmed by testing many cases. The first time the theorem was given in its modern form—the form in which it was presented here—was in a Memoir by Adrien-Marie Legendre, for the ‘Histoire de L’Academie des Sciences’ in 1785. However, the accompanying demonstration was incomplete, due to an undercharged assumption which it makes. The first complete proof was Gauss’, from his Disquisitiones Arithmeticae, which appeared in 1801. However, the first proof was also extremely inelegant, and Smith refers to it as being ‘‘presented by Gauss in a form very repulsive to any but the most laborious students (Smith, 1859, p. 59).’’ So we will next turn to this proof, in order to see in what sense it might count as a mechanical solution to the problem. 2.2. Gauss’ proof The first proof that appears in Disquisitiones Arithmeticae is an inductive one. The overall strategy is fairly simple, though the execution requires an exhaustive treatment of cases. Gauss’ statement of the theorem differs only slightly from the modern version: If p is a prime number of the form 4n + 1, +p will be a residue or nonresidue of any prime number which taken positively is a residue or nonresidue of p. If p is of the form 4n + 3, p will have the same property. (Art. 130) To see that his formulation is in fact equivalent to our original statement of the theorem, note that for primes of the form 4n + 1, if r is a residue, r will also be a residue, and for primes of the form 4n + 3, if r is a residue, r will be a non-residue. In carrying out the proof, Gauss first deduces several consequences of the quadratic reciprocity theorem, and notes that, if the theorem holds up to some number n, so do these consequences. The base case of the induction is established by verifying the theorem for 3 and 5. We then suppose that the quadratic reciprocity theorem holds up to some number T, and consider T + 1. If it were to fail at T + 1, this would mean that there are two prime numbers that contradict the theorem when compared, and the larger of these prime numbers is T + 1. But then we can still assume that all of our consequences of the quadratic reciprocity theorem hold for pairs of numbers both of which are less than or equal to T. Now we can distinguish eight ways in which the theorem can fail, two for each possible combination of primes. Let a and A be primes of the form 4n + 1 and b and B be primes of the form 4n + 3. We will write aRb to abbreviate that a is a residue of b and aNb to abbreviate that a is a nonresidue. The following table outlines the eight ways in which the theorem could fail.
I II III IV V VI VII VIII
If we have
Then
aRA aNA bRB bNB bRa bNa aRb aNb
ANa ARa BRb BNb aNb aRb bNa bRa
Gauss deals with each of the eight cases individually, showing that each one is impossible. We will only take up the first of these cases in detail here. So let us consider a = T + 1 and A, both of the
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form 4n + 1. We will suppose that A is a residue of a, and we will show that we cannot have a being a nonresidue of A. Once we assume ARa, meaning that there is an x2 such that A x2 (mod a), Gauss’ first step is to note that we can pick x such that x is positive, even and
3. Congruences, equivalence classes, and induction Congruence notation easily lends itself to the mathematical technique of induction. Every modulus yields a partition of the integers into equivalence classes made up of congruent integers with respect to that modulus. That this is possible follows directly from the axioms of least residues and transitivity. So even though Gauss does not appeal directly to the notion of an equivalence class, it is a very natural extension of the way in which he already works with congruences.
4 5
The only exception to this is where a appears as an exponent. For instance, Rosen (2000).
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We can see the way in which this plays out by invoking the preservation axioms. Congruent numbers (relative to some modulus) can be substituted for each other in congruence statements relative to that same modulus. This is a direct consequence of preservation under addition and multiplication. So, more generally, if we have a congruence statement modulo m in which a appears, we can substitute any member of a’s equivalence class for a, and preserve the truth of the statement.4 In fact, Gauss uses this technique quite frequently in his proof, substituting congruent numbers for one another, depending on the properties he wants them to have. Working with equivalences is much like working with identities, since an equivalence relation shares key properties with the identity relation. Substituting equals for equals, and carrying out fairly simple operations on identity statements, are straightforward and common algebraic tasks. When we have a mathematical equality between two terms, we can substitute those terms for each other in any mathematical proposition, while preserving the truth of that proposition. And just as we can substitute equals for equals, numbers congruent relative to a fixed modulus m can be substituted for each other in congruences (mod m). Thus one advantage of congruences is the similarity between working with congruence relations and working with identity statements. Indeed, when Gauss introduces the congruence symbol for the first time, he explicitly notes its resemblance to the identity symbol. Since we are already accustomed to working with identities, it is easy to ‘‘assimilate thoroughly’’ the methods congruence notation provides, since they are very similar to methods we already employ for dealing with identity statements. This latter fact about congruences is even cited in modern number theory textbooks when congruence notation is introduced.5 Given their relationship to equivalence classes and the manner in which congruence relative to m partitions the integers, the usefulness of congruences in induction becomes apparent. After all an inductive proof will make use of its inductive hypothesis, but can only do so to numbers which are below the threshold of induction. In Gauss’ proof, this allows us to exploit the fact that useful consequences of the quadratic reciprocity theorem hold for pairs of numbers smaller than a. But since a is also our modulus, we can work almost exclusively with positive numbers smaller than a; every integer that is not between 0 and a has an element of its equivalence class (mod a) which is. The existence of a least residue underlies much of the inductive step in Gauss’ proof. In particular, the proof that there was an x we could pick which was both even and
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of p. Its own defining axiom is: ðpaÞðbpÞ ¼ ðab Þ. This symbol is also one p which seems very natural—perhaps even more natural when it comes to tackling the theorem in question. This leads us to ask a (perhaps unfair) historical question. Why did Gauss manage to prove the theorem with the help of his new calculus, while Legendre never filled the last gap in his own proof? Even if the question is unfair, it can still provide us with some interesting insights, as long as we recognize its limitations. One obvious reason why it is unfair is the fact that the two calculi are perfectly compatible, and many modern treatments of quadratic reciprocity use both. Also, it is not simply the case that if Legendre had used congruence notation, the proof would have been successful. Gauss’ proof does not rely on any results which can only be proved using congruences, since congruences do not add prooftheoretic strength.6 The matter of Gauss’ mathematical talent should not be neglected either; we can almost certainly attribute some of Gauss’ success in giving a full proof, where Legendre did not, to Gauss’ greater skill as a mathematician. But perhaps part of being a more skilled mathematician is the development of better mathematical tools, such as the theory of congruences, which can then aid in proofs. So it is still worth looking at the reason why congruence notation might have been a better tool for proving this particular theorem. In looking at Legendre’s attempted proof, however, from Legendre (1808), we find many ways in which it resembles Gauss’ own. Both break up the problem into different cases, dealing with each one in turn, and their treatment of individual cases is very similar. The clear difference is in the overall proof strategy: Gauss’ proof uses induction, while Legendre carefully arranges the order in which he proves his cases. So while he has no inductive hypothesis to draw on, in order to use consequences of the reciprocity theorem, Legendre can still rely on results proved earlier on. But as we saw in the discussion of induction in Gauss’ proof, the fact that Gauss could appeal to least residues allowed him to choose a smaller member of the equivalence class, which allowed the inductive hypothesis to be applied. And as it turned out in Legendre’s proof, a key choice of number remained unjustified, which was the principal gap remaining. A difference between congruence notation and the Legendre symbol is their respective generality. Expressively, being a quadratic residue is a more ‘‘specific’’ relation than being congruent, in the sense that the former is expressible using the latter, but not vice versa: the Legendre symbol expresses that one number is a quadratic residue of another, which simply states that a congruence statement holds, and a congruence statement in turn expresses a divisibility relation. The ability of congruence notation to mask an extra variable in divisibility statements was an advantage, since it simplifies proofs. So with respect to the divisibility relation, its greater specificity is an advantage. However, we also saw that the Legendre symbol masks yet another variable, but that this was a disadvantage, since the x, such that a x2 (mod p) was a number which could be used in the proof. We can consider three ways logically to state that a is a quadratic residue of p: In the original language: $x $ y (x2 = a + py). Using congruences: $x (x2 a (mod p)). Using the Legendre symbol: ðpaÞ ¼ 1. In contrast with the Legendre symbol, the greater generality of congruence notation is an advantage. The notation allows us to present enough information in the proof that we can appeal to other results about divisibility, but manages to disguise some
6
irrelevant information as well, such as the multiplicity by which certain numbers divide each other. Perhaps congruence notation strikes a better balance between generality and specificity. This difference in generality might also have to do with the applicability of the defining axioms of each symbol. In terms of Gauss’ earlier remarks, perhaps congruences are better assimilated into existing methods of proof, because there are more situations in which the defining axioms for congruences can be applied. In fact, when we look at Legendre’s proof, and the section that corresponds to the part of Gauss’ proof outlined in this paper, we find only one place in which the defining axiom of the Legendre symbol is used. This is in the claim that ðab Þ ¼ ðAaÞðAbÞ. A But even though it is not an axiom, we can prove an equivalent theorem using congruences, and in fact, Gauss invokes that very theorem in his own proof. In contrast, though, the defining axioms for congruences were used in very many places throughout Gauss’ proof. The existence of least residues was invoked very often, and the substitutability of congruent numbers for each other, which relies on the preservation axioms, was also used quite often. None of this, however, should be taken as a criticism of the Legendre symbol as a piece of notation. Indeed, Gauss’ proof of the theorem can be simplified from eight cases to two cases through a generalization of the Legendre symbol (Tappenden, 2008, p. 263). So while the Legendre symbol is a useful definition in its own right, there are still reasons to think that, for the particular purpose of proving quadratic reciprocity, it was not as useful as congruence notation. And perhaps part of what makes the Legendre symbol seem so natural is that the quadratic reciprocity theorem is in fact true. So while Tappenden (2008) may be right that this symbol does carve nature at its joints, this alone does not make it a fruitful method—nor a fruitful method for all purposes. Given that such a relationship between quadratic residues does hold, a calculus which represents it, and embodies many of the more important theorems about it, is a very natural one. But if this is right, that could explain why it might not have been as useful as congruence notation in this initial proof of the reciprocity theorem. The usefulness of the Legendre symbol seems to be more specific, and tied to the truth of the reciprocity theorem itself, whereas the usefulness of congruences, being more general, is wider. Uses of the Legendre symbol in applications of quadratic reciprocity have vindicated its usefulness in many other contexts, but this is perfectly compatible with the claim that it, on its own, was not useful in proving the reciprocity theorem in the first place. Then, although we have spent most of our time discussing the idea of the ‘‘innermost nature of frequent wants’’, and what it means to ‘‘assimilate a calculus’’, these interpretive remarks can be brought to bear on the idea of enabling a mechanical proof. A new calculus can help us carry out a straightforward analysis of a problem, through its use of ‘‘natural’’ methods, and through the general applicability of its defining axioms. In the case of the quadratic reciprocity theorem, these calculi did not enable a solution that was really mechanical; we could not just calculate that the theorem holds. However, it is a result which divides up naturally into cases, and Gauss’ systematic analysis of each case using his new lemmas did allow him to prove each case in turn. Gauss went on to construct several different proofs of the theorem, most of which are seen as more intuitive and natural than the one given here, which was his first. The term ‘‘mechanical proof’’ here can perhaps be taken to mean a proof using caseby-case analysis.
Not to mention the fact that there are other proofs of the theorem, many of which are given by Gauss himself, which do not rely on congruences.
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3.2. Characterizing fruitfulness Congruences were a very natural tool in proving the quadratic reciprocity theorem, since they capture a relationship of divisibility, closely connected to the concept of a quadratic residue, but still general enough to allow for the application of other facts about divisibility. Whether a calculus corresponds to the innermost nature of frequent wants is a question of whether it provides useful lemmas for the proofs in which we want to use it. And whether it can be assimilated thoroughly has to do with its connection to our existing methods and how well it meshes with them. No definitive answer is being given here about exactly what makes some mathematical methods more fruitful than others in general. But Gauss’ remarks at least provide us with an outline of a non-realist (though not anti-realist) account of mathematical fruitfulness. While a realist account is certainly not ruled out by this discussion, my purpose here was to show that such an account is by no means the only way of explaining fruitfulness. Instead, the success of a new method can be explained as a matter of its relationship to the goals to which it is applied (the theorems we want to use it to prove) as well as to the existing methods of the field. Some methods may prove to be less useful in some contexts and
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more useful in others. What matters is the way in which the various fields and methods of mathematics fit together. References Avigad, J. (2006). Mathematical method and proof. Synthese, 153, 105–159. Cantor, G. (1883). Foundations of a general theory of manifolds: A mathematicophilosophical investigation into the theory of the infinite. In Ewald, W. (Ed.), (1996). From Kant to Hilbert: A sourcebook in the foundations of mathematics, volume II. Oxford: Clarendon Press. Euler, L. (1783). Opera Omnia, volume iii. Feferman, S. (1999). Does mathematics need new axioms? The American Mathematical Monthly, 106(2), 99–111. Gauss, C. F. (1801/1965). Disquisitiones arithmeticae. New Haven: Yale University Press. Gauss, C. F. (1900). Zu Möbius’ Barycentrischem Calcul. In Carl Friedrich Gauss Werke, VIII Band (pp. 301–304). Göttingen. Legendre, A. M. (1808). Essai sur la théorie des nombres. Paris: Courcier. Merz, J. T. (1912). A history of European thought in the nineteenth century. Edinburgh: Blackwood Press. Rosen, K. H. (2000). Elementary number theory and its applications. Reading: Addison-Wesley. Smith, H. J. (1859). Report on the theory of numbers. New York: Chelsea Publishing Company. Tappenden, J. (2008). Mathematical concepts: Fruitfulness and naturalness. In P. Mancosu (Ed.), The philosophy of mathematical practice. Oxford: Oxford University Press.