Studies in History and Philosophy of Science 42 (2011) 410–415

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Studies in History and Philosophy of Science

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Gauss’ quadratic reciprocity theorem and mathematical fruitfulness

Audrey YapDepartment 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 2010Received in revised form 28 September2010Available online 9 April 2011

Keywords:Number theoryPhilosophy of mathematicsCarl Friedrich GaussMathematical fruitfulnessCongruence notation

0039-3681/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.shpsa.2010.09.002

E-mail address: ayap@uvic.ca

a b s t r a c t

This paper presents an account of the fruitfulness of new mathematical calculi in terms of their relation-ship to existing mathematical methods which is suggested by Carl Friedrich Gauss. This is done by con-sidering some remarks that Gauss made explaining the fruitfulness of new calculi. These can be clarifiedin the context of his own (very fruitful) theory of congruences, which is considered as a case study for thisalternative account. Such an account has the benefit of not being dependent on a particular metaphysicalview in the philosophy of mathematics.

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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 possi-bilities (such as set theory) in mathematics by claiming that

Every mathematical concept carries within itself the necessarycorrective: if it is fruitless or unsuited to its purposes, then thatappears very soon through its uselessness, and it will be aban-doned for lack of success. (Cantor, 1883)

But how are we to account for the fact that some mathematical con-cepts or methods seem to be more fruitful than others? One optionis to take a realist line of explanation. Tappenden (2008), for in-stance, describes the fruitfulness of the Legendre symbol in termsof its carving mathematical reality at the joints. Such an accounthas its appeal, but in this paper, I would like to suggest anotherplausible line of explanation that does not appeal to mathematicalrealism. In this paper, I will consider some remarks that C. F. Gaussmade on the fruitfulness of new mathematical methods; theseremarks can be clarified in the context of his own (very fruitful)theory of congruences. I will then consider the extent to which thiscase study can help us account for mathematical fruitfulness asbeing relative to background mathematical methods. Considering

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mathematical fruitfulness as relative in this fashion provides analternative to a realist explanation, though it does not precludemathematical realism.

1.1. Characterizing fruitful methods

In a letter, dated May 15th, 1843, addressed to Heinrich ChristianSchumacher, one of his first students at Göttingen, Gauss wrote thefollowing:

In general the position as regards all such new calculi is this—that one cannot attain by them anything that could not be donewithout them: the advantage, however, is, that if such a calcu-lus corresponds to the innermost nature of frequent wants,every one who assimilates it thoroughly is able—without theunconscious inspiration of genius which no one can com-mand—to solve the respective problems, yes, even to solve themmechanically in complicated cases where genius itself becomesimpotent. So it is with the invention of algebra generally, sowith the differential calculus, so also—though in more restrictedregions—with Lagrange’s calculus of variations, with my calcu-lus of congruences, and with Möbius’s calculus. Through such

A. Yap / Studies in History and Philosophy of Science 42 (2011) 410–415 411

conceptions countless problems which otherwise would remainisolated and require every time (larger or smaller) efforts ofinventive 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 casesin which new mathematical calculi have been developed that wereconservative extensions of the old theory, which nevertheless al-lowed mathematicians to solve problems previously thought to beintractable. Furthermore, these calculi could give solutions whichwe can call mechanical, even when the original problem was a com-plicated one. Restricting our attention here to conservative exten-sions then means that fruitfulness is not to be understood in termsof adding deductive power to the theory. Rather, we are to under-stand fruitfulness in terms of these calculi ‘‘being assimilated thor-oughly’’ and corresponding to ‘‘the innermost nature of frequentwants.’’ Then we will be able to solve problems even ‘‘mechanicallyin complicated cases when genius itself becomes impotent’’. Aftersome initial remarks about these phrases, I will outline the specificsof Gauss’ theory of congruences, and the complicated case whosesolution it enabled: the proof of the quadratic reciprocity theorem.The first complete proof of this number-theoretic theorem appearedin Gauss’ Disquisitiones Arithmeticae, the work in which he also intro-duced 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 admitsof a psychological interpretation, but I want to consider how itcould be understood in more mathematical terms. Perhaps our‘‘frequent wants’’ arise when we find ourselves rederiving certainresults again and again in proofs, without a general theorem tocover all of the individual cases.

Second, we have the question of how one might assimilate a cal-culus thoroughly. This is likely more than a superficial recognition ofits usefulness. And even though there could be a psychological inter-pretation of this phrase as well, a more mathematical one is alsoavailable. Given that a new calculus provides us with new mathe-matical tools, we could explain its assimilation as the integrationof these tools with existing methods. Then they can subsequentlybe used in proofs alongside methods of the background theory.

Finally, how can it be possible to give a mechanical proof of atheorem? One straightforward answer would be that there is akind of effective procedure allowing the truth of the theorem tobe ‘‘calculated’’, so to speak. But this is surely too strong a condi-tion. It would not be plausible to claim that new calculi generallyyield computationally effective procedures by which theoremscan be decided. An alternative explanation is that our new calculiprove to be such useful analytical tools for the problems we applythem to, that they allow us to break them down in systematicways, and tackle them piece by piece. Although in order for thisto be useful, the number of cases yielded ought to remain tractable.The mechanical nature, in the sense of yielding a useful analyticaltool, of Gauss’ first proof of quadratic reciprocity is what makes itunappealing to follow, yet methodologically interesting.

2. Congruences and quadratic reciprocity

The theory of congruences was a new calculus in that it added anew relation symbol � to number theory. And it is also a case inwhich, strictly speaking, one cannot ‘‘attain by it anything thatcould not be done without it’’. For congruence notation is not only

1 I refer to Feferman’s distinction between ideal and structural axioms in (Feferman, 192 Gauss does not actually prove reflexivity or symmetry, but they are obvious propertie3 Avigad (2006) makes essentially the same points about the advantages of congruence

a conservative extension, but merely a definitional extension, inwhich the new symbol added can be seen as an abbreviation fora longer statement in the language of the original theory. Gauss de-fines congruence by writing that

If a number a divides the difference of the numbers b and c, band c are said to be congruent relative to a; if not, b and c arenoncongruent (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 amodulus a, we mean that a divides the difference of b and c. Further,all of the defining axioms of the congruence relation that Gauss sub-sequently mentions correspond to already provable statementsabout divisibility relations. So congruence notation adds no proof-theoretic strength whatsoever. In fact, as they are provable proposi-tions, its axioms are not axioms in the strictest sense of the word,although they do resemble axioms in the structural sense,1 in thatthey serve a descriptive purpose.

Congruence notation can first be explained logically, thoughsomewhat anachronistically. Suppose our initial language is justthe language of number theory hZ, +, �, 6i. Then, let us extend itby a new three-place relation symbol �, where

a � b ðmod pÞ iff9xða ¼ bþ ðx� pÞÞ:

Below are the properties of congruences that Gauss outlines at thestart of Disquisitiones Arithmeticae, formulated in symbolic languagefor the sake of clarity.1. a � a (mod m) (Reflexivity)2. a � b (mod m) M b � a (mod m) (Symmetry)2

3. a � b (mod m)V

c � b (mod m) ? a � c (mod m) (Transitivity)4. "x $ y (0 6 y < m

Vx � y (mod m) (Existence of Least Residues)

5. a � A (mod m)V

b � B (mod m) ? a + b � A + B (mod m) (Pres-ervation Under Addition)

6. a � A (mod m)V

b � B (mod m) ? a � b � A � B (mod m) (Pres-ervation Under Subtraction)

7. a � A (mod m) ? ka � kA (Preservation Under ScalarMultiplication)

8. a � A (mod m)V

b � B (mod m) ? ab � AB (mod m) (Preserva-tion Under Multiplication)

All of these properties are provable, fairly straightforwardly, fromthe definitions and basic algebra. For instance, we can show pres-ervation under (a) addition, (b) scalar, and (c) regular multiplica-tion as follows:

Suppose a � A (mod m) and b � B (mod m). Then there are xand y such that a � A = xm and b � B = ym.

(a) (a � A) + (b � B) = xm + ym. Regrouping, we obtaina + 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 isequivalent to ka � kA (mod m).

(c) By preservation under scalar multiplication, we know thatAB � Ab � ab (mod m).

So each result follows quite quickly, but proving any one re-quires at least one extra variable. This is because the formulawhich congruence notation abbreviates, contains an extra variablewhich it masks. The introduction of the new notation provides sev-eral shortcuts which eliminate the need for some variables beingused in the proof, and hence some added complexity.3 Then

99).s of the relation, which he assumes.notation.

412 A. Yap / Studies in History and Philosophy of Science 42 (2011) 410–415

‘‘assimilating the methods of a new calculus’’ could just involve add-ing these defining axioms to the regular stock of results to which ourproofs appeal.

We can also connect this notation to the idea of a quadratic res-idue. Given a modulus m, we have a complete system of residuesfor that modulus: the numbers 0, 1, 2, . . . (m � 1). However, wecan also consider a system of residues prime to m, which is a com-plete system from which every residue has been omitted whichhas any common divisor with the modulus. In the case of a primemodulus p, these two notions coincide. The numbers 1, 2, . . . ,(p � 1) divide themselves evenly into quadratic residues and qua-dratic nonresidues. We call q a quadratic residue of p wheneverthere 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 tothe congruence x2 � q (mod p), and a nonresidue whenever thatcongruence is irresolvable.

An example of a result which Gauss proves quite quickly usinghis ‘‘lemmas’’ about congruence notation is that the product of twoquadratic residues is also a quadratic residue:

Suppose a and b are both quadratic residues of p. Then there arex and y such that x2 � a (mod p) and y2 � b (mod p). By preser-vation 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 arex and y such that x2 � a and y2 � b are divisible by p. Thisimplies that there are u, v such that a = x2 � pu and b = y2 � pv.Multiplying, we obtain ab = x2y2 � x2pv � y2pu + p2uv. But thenwe see that ab = x2y2 � p(�x2v � y2u + puv). So ab is also a qua-dratic 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 theformer only required two. Also, the two ‘‘extra’’ variables are ines-sential, since we do not care about the multiplicity by whicha � x2 (for instance) is divisible by p. All the matters is the fact thatit is divisible, which is straightforwardly represented by the congru-ence statement. It even uses the same property of equality as theproof above did for congruences, which is preservation under mul-tiplication. Then the ‘‘innermost nature of frequent wants’’ can beexplained in terms of the applicability of the new lemmas, oncethey have been assimilated.

2.1. The quadratic reciprocity theorem

The quadratic reciprocity theorem is a general law in numbertheory having to do with quadratic residues. Gauss called it ‘thegem of the higher arithmetic,’ and Henry Smith, in his writingabout the history of number theory, calls it ‘‘the most importantgeneral truth in the science of integral numbers which has beendiscovered since the time of Fermat’’ (Smith, 1859, p. 56). The the-orem itself is the following:

For odd primes p and q, they are either both residues of oneanother, or both non-residues of one another, unless they areboth of the form 4k + 3, in which case exactly one is a residueof the other.

And at first glance, it does seem quite non-obvious and surpris-ing. 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 con-nect the existence of an x such that p divides x2 � q to the exis-tence of a y such that q divides y2 � p?)

A version of the theorem was first stated by Euler in an articleentitled ‘‘Observationes circa divisionem quadratorum per

numeros primos’’ (Euler, 1783), but only has the status of a conjec-ture, which he believed on inductive evidence, confirmed bytesting many cases. The first time the theorem was given in itsmodern form—the form in which it was presented here—was in aMemoir by Adrien-Marie Legendre, for the ‘Histoire de L’Academiedes Sciences’ in 1785.

However, the accompanying demonstration was incomplete,due to an undercharged assumption which it makes. The firstcomplete proof was Gauss’, from his Disquisitiones Arithmeticae,which appeared in 1801. However, the first proof was also extre-mely inelegant, and Smith refers to it as being ‘‘presented byGauss in a form very repulsive to any but the most laborious stu-dents (Smith, 1859, p. 59).’’ So we will next turn to this proof, inorder to see in what sense it might count as a mechanical solu-tion to the problem.

2.2. Gauss’ proof

The first proof that appears in Disquisitiones Arithmeticae is aninductive one. The overall strategy is fairly simple, though the exe-cution requires an exhaustive treatment of cases. Gauss’ statementof 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 ornonresidue of any prime number which taken positively is aresidue or nonresidue of p. If p is of the form 4n + 3, �p willhave the same property. (Art. 130)

To see that his formulation is in fact equivalent to our ori-ginal statement of the theorem, note that for primes of theform 4n + 1, if r is a residue, �r will also be a residue, andfor primes of the form 4n + 3, if r is a residue, �r will bea non-residue.

In carrying out the proof, Gauss first deduces several conse-quences of the quadratic reciprocity theorem, and notes that, ifthe theorem holds up to some number n, so do these conse-quences. The base case of the induction is established by verifyingthe theorem for 3 and 5. We then suppose that the quadratic rec-iprocity theorem holds up to some number T, and consider T + 1. Ifit were to fail at T + 1, this would mean that there are two primenumbers that contradict the theorem when compared, and thelarger of these prime numbers is T + 1. But then we can stillassume that all of our consequences of the quadratic reciprocitytheorem hold for pairs of numbers both of which are less than orequal to T.

Now we can distinguish eight ways in which the theorem canfail, two for each possible combination of primes. Let a and A beprimes of the form 4n + 1 and b and B be primes of the form4n + 3. We will write aRb to abbreviate that a is a residue of band aNb to abbreviate that a is a nonresidue. The following tableoutlines the eight ways in which the theorem could fail.

If we have

Then

I

aRA ANa II aNA ARa III bRB BRb IV bNB BNb V bRa aNb VI bNa aRb VII aRb bNa VIII aNb bRa

Gauss deals with each of the eight cases individually, showingthat each one is impossible. We will only take up the first of thesecases in detail here. So let us consider a = T + 1 and A, both of the

A. Yap / Studies in History and Philosophy of Science 42 (2011) 410–415 413

form 4n + 1. We will suppose that A is a residue of a, and we willshow that we cannot have a being a nonresidue of A.

Once we assume ARa, meaning that there is an x2 such thatA � x2 (mod a), Gauss’ first step is to note that we can pick x suchthat x is positive, even and <a. He does not justify this claim, but itis easily verified: first, note that we can pick x such that 0 < x < a,due to the existence of least residues. Then ±x will be our tworoots. If x is odd, then we note that there is a least residue for�x, which we can call y. Since x � x � 0 (mod a), we knowx + y � 0 (mod a). Then since both x and y are positive and <a,we conclude x + y = a. Since a is an odd prime, one of x and y mustbe even, so y is even and can be our choice.

Gauss then distinguishes two further cases, of which we willconsider only the first, namely the case in which x is not divisibleby A. Let x2 = A + af. Gauss claims that f will be (i) positive, (ii) ofthe form 4n + 3, (iii) <a, (iv) and not divisible by A. Again, each ofthese claims can be justified:

(i) If f were negative, then af would be negative, and sinceA � x2 = af, this would imply A < x2. But by assumption, x2

is the least residue of A (mod a), so that would be impossible.(ii) Since x is even, x2 � 0 (mod 4), and by hypothesis, both a and

A are � 1 (mod 4). Then, since x2 � A + af (mod 4), we cansubstitute congruent numbers for each other to obtain0 � 1 + 1f (mod 4). So it must be that f � 3 (mod 4).

(iii) We already know x < a, so x2 < a2. Then, if it were true thata 6 f, then x2 < a2

6 af. But since x2 = A + af, A would haveto be negative, which is contrary to hypothesis. Thus, f < a.

(iv) Since x is not divisible by A, f cannot be either, or elseA + af = x2 would be divisible by A.

Having verified these claims, since x2 = A + af, we know x2 � A(mod f), and conclude ARf. Since both A and f are <a, we can invokethe inductive hypothesis, allowing us to appeal to the reciprocitytheorem, which tells us that fRA. Similar reasoning tells us thatafRA. But we cannot multiply a residue by a nonresidue to obtaina residue, so it must be that aRA (Gauss, 1801/1965).

Even in this small portion of the proof, the usefulness of congru-ence notation is clear. In particular, the existence of least residuesis invoked at the early stages in order to apply the inductivehypothesis later. Also, congruence notation gives us an alternaterepresentation for the form of a number, which eliminates theneed for extra variables. Instead of writing that a number is ofthe form 4k + 1, we can simply say that it is congruent to 1, modulo4. For instance, the reasoning step showing f � 3 (mod 4) wasstraightforward using the fact that congruent numbers can besubstituted for each other, but would have involved a messier alge-braic equation otherwise. Now, having laid out a part of Gauss’proof, in the last section of the paper I will discuss the role congru-ence notation played in that proof.

3. Congruences, equivalence classes, and induction

Congruence notation easily lends itself to the mathematicaltechnique of induction. Every modulus yields a partition of theintegers into equivalence classes made up of congruent integerswith respect to that modulus. That this is possible follows directlyfrom the axioms of least residues and transitivity. So even thoughGauss does not appeal directly to the notion of an equivalenceclass, it is a very natural extension of the way in which he alreadyworks with congruences.

4 The only exception to this is where a appears as an exponent.5 For instance, Rosen (2000).

We can see the way in which this plays out by invoking thepreservation axioms. Congruent numbers (relative to some modu-lus) can be substituted for each other in congruence statementsrelative to that same modulus. This is a direct consequence of pres-ervation under addition and multiplication. So, more generally, ifwe have a congruence statement modulo m in which a appears,we can substitute any member of a’s equivalence class for a, andpreserve the truth of the statement.4 In fact, Gauss uses this tech-nique quite frequently in his proof, substituting congruent numbersfor one another, depending on the properties he wants them to have.

Working with equivalences is much like working with identi-ties, since an equivalence relation shares key properties with theidentity relation. Substituting equals for equals, and carrying outfairly simple operations on identity statements, are straightfor-ward and common algebraic tasks. When we have a mathematicalequality between two terms, we can substitute those terms foreach other in any mathematical proposition, while preserving thetruth of that proposition. And just as we can substitute equals forequals, numbers congruent relative to a fixed modulus m can besubstituted for each other in congruences (mod m).

Thus one advantage of congruences is the similarity betweenworking with congruence relations and working with identitystatements. Indeed, when Gauss introduces the congruence symbolfor the first time, he explicitly notes its resemblance to the identitysymbol. Since we are already accustomed to working with identi-ties, it is easy to ‘‘assimilate thoroughly’’ the methods congruencenotation provides, since they are very similar to methods we al-ready employ for dealing with identity statements. This latter factabout congruences is even cited in modern number theory text-books when congruence notation is introduced.5

Given their relationship to equivalence classes and the mannerin which congruence relative to m partitions the integers, the use-fulness of congruences in induction becomes apparent. After all aninductive proof will make use of its inductive hypothesis, but canonly do so to numbers which are below the threshold of induction.In Gauss’ proof, this allows us to exploit the fact that useful conse-quences of the quadratic reciprocity theorem hold for pairs ofnumbers smaller than a. But since a is also our modulus, we canwork almost exclusively with positive numbers smaller than a;every integer that is not between 0 and a has an element of itsequivalence class (mod a) which is.

The existence of a least residue underlies much of the inductivestep in Gauss’ proof. In particular, the proof that there was an x wecould pick which was both even and <a depended on this fact. Todiscover that the sum of the two roots would be divisible by a,we chose a member of the equivalence class which was negative,since it is obvious that x + (�x) � 0 (mod m). But then we tookthe least residue of �x which is positive and <a. And it was this factthat gave us the two roots adding up to a. Thus with congruencenotation, it is very natural to interchange different members ofan equivalence class in different parts of the proof, depending onwhat properties we need them to have. In an inductive proof, ifwe can fix the modulus as the threshold of induction, the least res-idues axiom allows for the application of the inductive hypothesis.

3.1. Gauss and Legendre

Another advantage of congruence notation can be understood incontrast with another new piece of notation introduced inattempting to prove the quadratic residue theorem, namely, theLegendre symbol. This symbol is defined such that ðapÞ ¼ 1 iff a isa quadratic residue of p, and ðapÞ ¼ �1 iff a is a quadratic nonresidue

414 A. Yap / Studies in History and Philosophy of Science 42 (2011) 410–415

of p. Its own defining axiom is: ðapÞðbpÞ ¼ ðabp Þ. This symbol is also one

which seems very natural—perhaps even more natural when itcomes to tackling the theorem in question. This leads us to ask a(perhaps unfair) historical question. Why did Gauss manage toprove the theorem with the help of his new calculus, while Legen-dre never filled the last gap in his own proof?

Even if the question is unfair, it can still provide us with someinteresting insights, as long as we recognize its limitations. Oneobvious reason why it is unfair is the fact that the two calculi areperfectly compatible, and many modern treatments of quadraticreciprocity use both. Also, it is not simply the case that if Legendrehad used congruence notation, the proof would have been success-ful. Gauss’ proof does not rely on any results which can only beproved using congruences, since congruences do not add proof-theoretic strength.6 The matter of Gauss’ mathematical talentshould not be neglected either; we can almost certainly attributesome of Gauss’ success in giving a full proof, where Legendre didnot, to Gauss’ greater skill as a mathematician. But perhaps part ofbeing a more skilled mathematician is the development of bettermathematical tools, such as the theory of congruences, which canthen aid in proofs. So it is still worth looking at the reason why con-gruence notation might have been a better tool for proving this par-ticular theorem.

In looking at Legendre’s attempted proof, however, from Legen-dre (1808), we find many ways in which it resembles Gauss’ own.Both break up the problem into different cases, dealing with eachone in turn, and their treatment of individual cases is very similar.The clear difference is in the overall proof strategy: Gauss’ proofuses induction, while Legendre carefully arranges the order inwhich he proves his cases. So while he has no inductive hypothesisto draw on, in order to use consequences of the reciprocity theo-rem, Legendre can still rely on results proved earlier on. But aswe saw in the discussion of induction in Gauss’ proof, the fact thatGauss could appeal to least residues allowed him to choose a smal-ler member of the equivalence class, which allowed the inductivehypothesis to be applied. And as it turned out in Legendre’s proof,a key choice of number remained unjustified, which was the prin-cipal gap remaining.

A difference between congruence notation and the Legendresymbol is their respective generality. Expressively, being a qua-dratic residue is a more ‘‘specific’’ relation than being congruent,in the sense that the former is expressible using the latter, butnot vice versa: the Legendre symbol expresses that one numberis a quadratic residue of another, which simply states that a con-gruence statement holds, and a congruence statement in turn ex-presses a divisibility relation. The ability of congruence notationto mask an extra variable in divisibility statements was an advan-tage, since it simplifies proofs. So with respect to the divisibilityrelation, its greater specificity is an advantage. However, we alsosaw that the Legendre symbol masks yet another variable, but thatthis was a disadvantage, since the x, such that a � x2 (mod p) was anumber which could be used in the proof. We can consider threeways 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: ðapÞ ¼ 1.

In contrast with the Legendre symbol, the greater generalityof congruence notation is an advantage. The notation allows usto present enough information in the proof that we can appealto other results about divisibility, but manages to disguise some

6 Not to mention the fact that there are other proofs of the theorem, many of which ar

irrelevant information as well, such as the multiplicity by whichcertain numbers divide each other. Perhaps congruence notationstrikes a better balance between generality and specificity.

This difference in generality might also have to do with theapplicability of the defining axioms of each symbol. In termsof Gauss’ earlier remarks, perhaps congruences are better assim-ilated into existing methods of proof, because there are more sit-uations in which the defining axioms for congruences can beapplied. In fact, when we look at Legendre’s proof, and the sec-tion that corresponds to the part of Gauss’ proof outlined in thispaper, we find only one place in which the defining axiom of theLegendre symbol is used. This is in the claim that ðab

A Þ ¼ ðaAÞðbAÞ.But even though it is not an axiom, we can prove an equivalenttheorem using congruences, and in fact, Gauss invokes that verytheorem in his own proof. In contrast, though, the defining axi-oms for congruences were used in very many places throughoutGauss’ proof. The existence of least residues was invoked veryoften, and the substitutability of congruent numbers for eachother, which relies on the preservation axioms, was also usedquite often.

None of this, however, should be taken as a criticism of theLegendre symbol as a piece of notation. Indeed, Gauss’ proof ofthe theorem can be simplified from eight cases to two casesthrough a generalization of the Legendre symbol (Tappenden,2008, p. 263). So while the Legendre symbol is a useful definitionin its own right, there are still reasons to think that, for the par-ticular purpose of proving quadratic reciprocity, it was not as usefulas congruence notation. And perhaps part of what makes theLegendre symbol seem so natural is that the quadratic reciprocitytheorem is in fact true. So while Tappenden (2008) may be rightthat this symbol does carve nature at its joints, this alone doesnot make it a fruitful method—nor a fruitful method for all pur-poses. Given that such a relationship between quadratic residuesdoes hold, a calculus which represents it, and embodies many ofthe more important theorems about it, is a very natural one. Butif this is right, that could explain why it might not have been asuseful as congruence notation in this initial proof of the reciproc-ity theorem. The usefulness of the Legendre symbol seems to bemore specific, and tied to the truth of the reciprocity theorem it-self, whereas the usefulness of congruences, being more general,is wider. Uses of the Legendre symbol in applications of quadraticreciprocity 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 firstplace.

Then, although we have spent most of our time discussing theidea of the ‘‘innermost nature of frequent wants’’, and what itmeans to ‘‘assimilate a calculus’’, these interpretive remarks canbe brought to bear on the idea of enabling a mechanical proof.A new calculus can help us carry out a straightforward analysisof a problem, through its use of ‘‘natural’’ methods, and throughthe general applicability of its defining axioms. In the case ofthe quadratic reciprocity theorem, these calculi did not enable asolution that was really mechanical; we could not just calculatethat the theorem holds. However, it is a result which divides upnaturally into cases, and Gauss’ systematic analysis of each caseusing his new lemmas did allow him to prove each case in turn.Gauss went on to construct several different proofs of the theo-rem, most of which are seen as more intuitive and natural thanthe one given here, which was his first. The term ‘‘mechanicalproof’’ here can perhaps be taken to mean a proof using case-by-case analysis.

e given by Gauss himself, which do not rely on congruences.

A. Yap / Studies in History and Philosophy of Science 42 (2011) 410–415 415

3.2. Characterizing fruitfulness

Congruences were a very natural tool in proving the quadraticreciprocity theorem, since they capture a relationship of divisibil-ity, closely connected to the concept of a quadratic residue, but stillgeneral enough to allow for the application of other facts aboutdivisibility. Whether a calculus corresponds to the innermost nat-ure of frequent wants is a question of whether it provides usefullemmas for the proofs in which we want to use it. And whetherit can be assimilated thoroughly has to do with its connection toour existing methods and how well it meshes with them.

No definitive answer is being given here about exactly whatmakes some mathematical methods more fruitful than others ingeneral. But Gauss’ remarks at least provide us with an outline ofa non-realist (though not anti-realist) account of mathematicalfruitfulness. While a realist account is certainly not ruled out bythis discussion, my purpose here was to show that such an accountis by no means the only way of explaining fruitfulness. Instead, thesuccess of a new method can be explained as a matter of its rela-tionship to the goals to which it is applied (the theorems we wantto 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

more useful in others. What matters is the way in which the vari-ous fields and methods of mathematics fit together.

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