Gauss vs. Empiricism

by Jonathan Tennenbaum

Lyn has emphasized, how Carl Friedrich Gauss’ 1799 dissertation on the so-called “Fundamental Theorem of Algebra”, constituted a devastating refutation of the leading scientific authorities of his day, including Jean-Louis Lagrange and Leonard Euler.

Gauss first points out fundamental flaws in purported proofs of the “Fundamental Theorem”, put forward by D’Alembert, Euler and Lagrange in succession, showing that they were based on arbitrary assumptions and fell far short of actually establishing the proposition, they claimed to demonstrate. Then Gauss presents his own, rigorous proof, based on different principles.

So what’s the big deal? Certainly, Gauss’s ruthless exposure of gaping “holes” in D’Alembert’s, Lagrange’s, and Euler’s proofs was a scandal in and of itself, suggesting — as Gauss himself clearly intimates — a shocking degree of conceptual sloppiness on the part of men who were considered to be standards of scientific rigor. Also, Gauss’ concise proof, going to the heart of the matter in a few pages, contrasted with the voluminous and prolix treatises of Lagrange and Euler, that Gauss tore apart in the first half of his dissertation.

But the real issue is not one mathematics per se, but of physical principle. A PRELIMINARY access to that issue opens up, when we turn from the particularities of the Fundamental Theorem itself (although they are important and indispensible) and first ask the question: WHY did D’Alembert, Lagrange and Euler FAIL? What was wrong with their THINKING?

Someone might say: well, the 20-year-old Gauss was a towering genius. But in what did that genius really consist? Was Gauss’ proof itself so fantastically clever, complicated and ingenious? No, not at all! It is quite simple, natural and direct, once one masters the basic principles involved. From a purely {formal-technical} standpoint, there is nothing in Gauss’ proof, which was not well within the range of what Lagrange, Euler and many others could easily have done.

So, what prevented them from doing so? Ah! Here we come to the exact same “mechanism”, which causes the collapse of seemingly all-powerful empires — empires possessing vast, apparently overwhelming material and intellectual resources, but which collapsed nonetheless. Why is it, that the ruling elites of such empires — and their army of sometimes highly talented, skillful and experienced advisors, analysts and other “lackeys”, selected from the “cream of the cream” of the population — why did they invariably FAIL, at crucial junctures of history, to take actions that might have prevented, or at least greatly delayed, the collapse of their systems? Why is it, that we invariably discover, as the crucial element that finally dooms such empires to destruction, an obsessive insistence on expending every last ounce of resources, skill and cunning, in the attempt to make an intrinsically FAILED system “work” — even against the laws of the Universe –, rather than to accept a CHANGE in its the underlying, flawed axioms of that system.

So, we have Lagrange and Euler, both highly skillful, knowledgeable and experienced mathematicians, much more so than most leading academic authorities today, but who FAILED most decisively, where Gauss SUCCEEDED. The case of Euler is particularly instructive, because he was, to all accounts, extremely industrious and in some ways quite shrewd and perceptive, as well as a virtuouso master of algebraic methods. Euler also made some not-insignificant experimental discoveries, in number theory and other fields, as Gauss himself pointed out. But, at the same time, Euler was a crude phillistine in philosophical terms, and above all a fanatical EMPIRICIST, in the precise sense that Lyndon LaRouche has identified, with great precision, in a recent paper. Lyn writes, in part:

“Empiricism has the form of a synthesis of three, ostensibly mutually exclusive, categorical elements, as follows:

“1. First, the empiricist assumes that no experimentally verifiable knowledge exists outside the bounds of simple sense-certainty.

“2. Secondly, therefore, every cause-effect relationship which can not be located explicitly in an sense-observed agency, is related to a domain of such forms of attributed bias in statistical behavior of observable events, or to some anonymous agency to which neither sense-certainty nor cognitive reason provides access.

“3. Thirdly, the second element leaves available a niche for the creating the illusion of the existence of purely magical spiritual powers, operating entirely outside the reach of access by sense-certainty, but able to make arbitrary interventions, even capriciously, into the domain of sense-certainty.”

Let us compare this characterization by Lyn, with the clinical evidence Leonard Euler supplies for his own case. Take, for this purpose, the same paper, that is the immediate target of criticism in Gauss’ dissertation of 1799. This is Euler’s “Recherches sur les racines imaginaires des equations”, published in the “Memoires de l’Academie des sciences de Berlin” in the year 1749.

Euler begins by writing down the general form of an algebraic equation involving a single unknown, “X”. By “algebraic equation” Euler meant any formula, built up from the “unknown” X and specific integers, fractions or so-called irrational numbers, by means of the algebraic operations of addition, subtraction, multiplication and division, and then set equal to zero. So, for example: 2XX – 5X + 10 = 0, or XXX + 3 XX – 5X + 21 = 0 and so forth. (XX means X times X or “X squared”, XXX means “X cubed”, and so forth). (Apparently more complicated cases can occur, as for example (XXXX – 4)/X = 0 or others, that involve divisions. It turns out, that divisions can be eliminated by manipulations of the algebraic equation. But these technical aspects are not important for the point we want to make here. )

The problem (as Euler understands it) is to find a specific number or magnitude, which, when put in place of “X” in the formula, yields the value zero when the additions, subtractions, multiplications and divisions are all carried out. Such values became known as “solutions” or “roots of the equation”. Thus, the equation XX – 4 = 0 has two roots, namely 2 and -2. Referring implicitly to the work of Cardan and others in the 16th century, Euler notes (my emphasis):

“It happens (in general) that not all the roots are REAL quantities, and that some of them, or perhaps all, are IMAGINARY quantities.”

Thus, for example, the equation XX + 1 = 0, appears to have no solutions or roots among the magnitudes, that Euler regarded as “real”, namely positive or negative quantities corresponding to positions to the right or left of zero on the “number line” of standard textbook mathematics. For, whether X is positive or negative, XX (X squared) is always positive, so XX + 1 will always be at least 1, or larger, for any X on the “number line”. Hence, a solution or root of XX + 1 = 0, if one could speak of such a thing at all, could only be an “imaginary” entity, as when a formal algebraicist, merely playing with symbols, might reason:

XX + 1 = 0 implies

XX = -1, which implies

X = “the square root of -1”.

But such a value of “X” could have no real existence, because it corresponds to no point on the “number line”. Euler writes:

“One calls a magnitude IMAGINARY, when it is neither greater than zero, nor less than zero, nor equal to zero. This would therefore be something IMPOSSIBLE, as for example sqrt(-1), or in general a + b sqrt(-1), because such a quantity is neither positive, nor negative, nor zero.”

Thus, for Euler a magnitude such as sqrt(-1), which is neither more, nor less than, nor equal to zero, lies outside the domain of sense-certainty, and is therefore “impossible” or “imaginary”. On the other hand, a few paragraphs further down in his paper, Euler insists, that mathematicians must study and utilize these very same “impossible” quantities! Euler writes:

“Although it seems that the knowledge of the imaginary roots of an equation would be devoid of any use, since they furnish no (real) solutions to any problem, nevertheless it is very important in analysis to become familiar with the imaginary quantities. Because we thereby not only obtain a more perfect knowledge of the nature of equations; but the analysis of the infinite can enjoy considerable benefits.”

Euler goes on to remark, that various methods for the calculation of integrals and other mathematical problems, require the use of “imaginary quantities”, even though he himself has denounced those same quantities as “impossible”! Here Euler displays the second and third characteristics of empiricism, detailed by Lyn above, and which are curiously inconsistent with the first point: A mathematician must learn to communicate with GHOSTS, “imaginary quantities”, which are unreal and yet lend the mathematician MAGICAL POWERS to manipulate the visible universe!

Here Euler reveals exactly his empiricist problem. A true physical principle, which can only be generated as an IDEA in a single, sovereign human mind, cannot be known, he thinks. Instead, there are only sense perceptions, on the one hand, and “other-worldly” entities with magical powers, on the other. Above all, formalism itself — like a cult ritual — is supposed to convey magical powers, as many modern physicists for example, ascribe awesome powers to the so-called “quantum mechanical formalism” today. A revealing example is Euler’s contorted attempt to formally justify the “rules” for multiplication with simple NEGATIVE NUMBERS, in his algebra text from 1770:

“It remains still to solve the case where – is multiplied by – or, for example – a by – b. It is obvious initially that as for the letters, the product will be ab; but it is dubious still if it is the sign + or well the sign – that it is necessary to put in front of the product; all that one knows, it is that it will be one or the other of these signs. However I say that it cannot be the sign -; because – a by + b gives – ab and – a by – b cannot produce the same result that – a by + b; but it must result the opposite from it, i.e. + ab; consequently we have this rule: + multiplied by + made +, just as – multiplied by -.”

This, explicitly cultish, gobble-dee-gook formalism, became a paradigm for the teaching of mathematics, leading to generation after generation of crippled minds.

Only a bit less openly occult is Jean-Louis Lagrange, whose famous 1788 “Méchanique analytique” became a prototype for modern “systems analysis”. In his preface Lagrange writes:

“You will find no diagrams in this work. The methods I present, require neither constructions, nor geometrical or mechanical reasoning, but only algebraic operations, proceeding in regular and uniform manner. Those who love analysis will note with pleasure, how Mechanics thereby becomes one of its new branches, and will be grateful to me for having extended the domain of analysis in this way. “

Lagrange thus pretended, to MAKE PHYSICS INTO A BRANCH OF FORMAL MATHEMATICS — exactly the opposite of what Leibniz stood for, and what Nicolaus of Cusa and Plato before him had stood for. Common to Lagrange and Euler, is the demand, that no physical principles should be permitted to intrude upon the domain of mathematics! Yet, it is easy to demonstrate, in the typical contemporary physicist and physics student, a fanatical quality of belief in the supposed magical powers of the so-called “Lagrangian”.

As Gauss points out, that while pretending to “correct” the defects of Euler’s purported “proof” of the fundamental theorem, Lagrange maintains Euler’s implicit, though baseless assumption, that all roots of an algebraic equation, whether “real” or “imaginary”, must be capable of a formal algebraic representation, in terms of addition, subtraction, multiplication, division and the so-called extraction of roots (square roots, cube roots, fourth and fifth roots and so forth). But exactly the impossibility of such a universal formula for the roots of an equation, was key to Gauss’ s understanding of the significance of the complex domain.

Gauss himself repeatedly refers to the fundamental difference in method, between his approach and the stubborn empiricism of Euler, in his early writings in particular.

So, he writes in the introduction to his Disquisitiones arithmeticae, “When I, in the beginning of the year 1795 first took up this sort of (number theoretic) investigations, I knew nothing about the work of the moderns (Euler, Legendre et al) in this field … While I was engaged in other work, I chanced upon a remarkable arithmetic truth — when I am not mistaken, it is the theorem in D.A. Article 108 (**) –; and since I found it not only very beautiful in and of itself, but suspected that it must be connected with other remarkable properties, I devoted my entire energies to comprehending the PRINCIPLE upon which those properties are based, and obtaining a rigorous proof. When at last I succeeded in my wishes, the beauty of these investigations had taken such a hold over me, that I could not tear myself away from them; so it came about, that, as each thing led to another in turn, I had accomplished most of what is contained in the first four sections of this book, before I had seen anything of the similar work of other Geometers.”

In a letter to his former teacher at the Carolineum, E.A.W. Zimmermann in October 1795, soon after he had entered Göttingen University, Gauss wrote:

“I have seen the library and hope to derive from it a considerable contribution to a happy existence in Göttingen. I already have several volumes of the Commentaries to the Petersburg Academy (by Euler) at home, and I have looked through many more. I cannot deny, that it is very unpleasant for me, to see that the largest part of my discoveries in indeterminate Analysis were only discovered for the second time. What comforts me is this: All the discoveries of Euler, that I have found so far, I had also made by myself, plus some more, too. I have found a more general, and, I believe, more natural standpoint, and an immeasurable field (for further discoveries) lies in front of me; Euler made his discoveries over a period of many years, and only after many successive tentaminibus (attempts). “

Euler’s attitude toward the so-called “imaginary” or “impossible” numbers, reflected exactly his own crippling intellectual problem: for him, IDEAS — physical principles grasped by the mind — couldn’t really exist, as objects of concious deliberation. So, he was reduced to sniffing around, with his nose to the ground, for some sort of magic formulae by which he might manipulate the world. By concentrating on issues of PRINCIPLE, Gauss had overtaken a lifetime of trial-and-error-style number-theoretic investigations by Euler, within less than a year.


(**) It is worth giving here, at least briefly, some idea about the subject of Article 108 of Disquisitions Arithmeticae, as this is closely connected with the genesis of Gauss’ Fundamental Theorem of Algebra.

Gauss calls a given whole number A a “quadratic residue” relative to a prime number p, if there exists a “square number” — i.e. the square of a whole number: 1, 4, 9, 16, 25, 36, 49 etc — such that p divides the difference between that square and N. In Gauss’ language of congruences, the latter condition is expressed by saying, that “N is congruent to some square modulo p”. Of course, the square numbers themselves always fullfill that condition, but the more interesting case is when N is not the square of a whole number. For example, we can easily see that 2 is a quadratic residue relative to the prime number 7 — 7 divides the difference between 9 (a square!) and 2. Also 2 is a quadratic residue of 17 (the square 36 is congruent to 2 modulo 17), and of a whole series of other prime numbers. On the other hand, it turns out that 2 is NOT a quadratic residue relative to 5, nor is 2 a quadratic residue relative to 11, 13 and a whole series of other primes. Thus, the prime numbers fall into two series or species: those for which 2 is congruent to some square, and those for which 2 is not congruent to any square. Taking as N instead of 2 any other non-square number, we get a different division of the primes into species. (The harmonic interrelations of those species are the subject of an extraordinary discovery, which is the centerpiece of the Disquisitiones Arithmeticae — namely the so-called “Law of Quadratic Reciprocity”).

Another way to look at this, implied by Gauss, is to consider the realm of congruences relative to a given prime number, as defining a geometrical domain of a special type. In that domain, congruent numbers are considered to have the same “shape” and to be otherwise equivalent. So, for example, instead of saying, for example, that “2 is congruent to a square modulo 7”, we would actually regard 2 itself as a “square number” in the congruence domain defined by 7. Within that domain, for example, 2 and 9 ( = 3 squared) would be considered equivalent and indistinguishable. Thus, 2 will be “square-shaped” or “of the second power” in some prime domains, but not in others. The prime number p (the “modulus”) are thus not simple numbers, but topological characteristics.

Now, what about the number -1? This special case — which turns out to be of crucial importance in all of Gauss’ “higher arithmetic” — is the subject of the cited Article 108. As we noted above, the criterion for -1 to be a quadratic residue relative to a given prime p, is that p divides the difference between some square number and -1. But subtracting -1 from a number, means the same thing as adding 1 to it; so the condition is equivalent to saying, that there is a square number, such that when 1 is added to it, the result is divisible by p. Examples are easy to find. For example, 4 + 1 is divisible by 5, so -1 is a quadratic residue of 5. Similarly, -1 is a quadratic residue modulo 17. Also, 25 + 1 = 26 is divisible by 13, so -1 is a quadratic residue modulo 13, too, and so on for a certain series of prime numbers. But it turns out, for example, that -1 is NOT a quadratic residue relative to 7, nor relative to 23, nor 29, nor for a whole other series of prime numbers.

As a bit of reflection shows, one can characterize the two species of prime numbers also as follows: Take the series of squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 etc. and add one to each of them: 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122 etc. The primes of the first species — those for which -1 is a quadratic residue — are the ones which divide at least one of the numbers in the latter series. Those primes which do not divide any of the numbers 2, 5, 17, 26, 37, 50 etc, form the second species. (How many terms of the series do you need to check, to tell whether a given prime divides one of them?).

For the first species of prime numbers, “-1” is a square number in the corresponding congruence domain, i.e. sqrt(-1) corresponds to a specific value in the domain (for example, 5 is equivalent to sqrt (-1) in the domain of congruences modulo 13), while for the second species, the introduction of sqrt(-1) requires going OUTSIDE the domain.

Two years before his discovery of his first proof of the “Fundamental Theorem of Algebra” , Gauss uncovered the harmonic law governing the distribution of the primes into these two species: the first, as it turns out, are the primes which leave a remainder of one when divided by 4 (such as 5, 13, 17, 29…) and the second species are the primes leaving a remainder of 3 when divided by 4 (such as 3, 7, 1, 19, 23 …).

Most significant is the circumstance, that BOUNDING PRINCIPLE involved is physical in nature, and has nothing to do with any formula or formal proceedure. So. for example, in the case of primes of the first class, the existence of a value for sqrt(-1), is “forced”, as a singularity, by the overall geometry of the domain; while the specific value of sqrt(-1) assumes in that domain, must be discovered, a posteriori, by direct observation.