Somewhere I found this Quote but I didn't understand why?. Can anybody explain me?


I think that is a severe mis-interpretation of what Gauss actually wrote. The phrase "shadow of shadows" came up in Gauss' doctoral dissertation as the latin phrase "vera umbrae umbra" during an extended critique of Euler's proof of the fundamental theorem of algebra.

In the mid-to-late eighteen century, the fundamental theorem algebra is not stated as it is in modern times with the complex/imaginary roots. The statement is in fact that "polynomials can be factored into factors consisting entirely of linear and quadratic terms with real coefficients". See the discussion in Dunham's article in the CMJ. The statement of the theorem, in fact, refers not to any imaginary quantities at all.

Euler's proof of the theorem was not however, really a proof by modern standards. In fact, Gauss would argue that it is not even a proof by his, later eighteenth century standards. The main logical fallacy that he strives to points out is that Euler's deduction essentially boils down to:

  1. Assume the polynomial has (the right number of) roots, with the roots not necessarily real numbers.
  2. Prove that the roots must take the form $a + b \sqrt{-1}$.
  3. Since the numbers of that form are either all real, or satisfy quadratic equations (of the form $(x - a -b\sqrt{-1})(x - a + b \sqrt{-1}) = 0$) with real coefficients, the theorem is demonstrated.

What Euler has not provided a proof of, is that the polynomial does in fact contain the roots. At this point let me quote Gauss (translation from Latin by Ernest Fandreyer)

E[uler] tacitly assumes that the equation $X = 0$ [n.b. $X$ here denotes a polynomial of degree $2m$, with the coefficient of $x^{2m-1}$ being zero] has $2m$ roots and that their sum is 0 because $X$ has no second term. What I think of that license (which all authors use for this argument), I have made plain in art. 3 above. The assumption that the sum of all the roots of any equation is equal to the first coefficient, with opposite sign, does not seem applicable to other equations, but only to those that have roots. But as by this proof it must be shown that the equation $X = 0$ indeed has roots, it does not at all seem permissible to assume their existence. Without doubt, those who have not yet penetrated the fallaciousness of this paralogism will answer that here it is not to be proved that the equation $X = 0$ can be satisfied (for that is the meaning of the expression “it has roots”) but merely that the equation may be fulfilled by values of $x$ of the form $a + b \sqrt{-1}$. The former statement they assume indeed like an axiom. However, other forms for quantities beyond the real, and imaginary numbers $a + b \sqrt{−1}$ can not be conceived. And thus it does not appear clear in which way that which is to be proved may differ from that which is assumed like an axiom. If it were possible to devise yet other forms of quantities, say of the form $F, F', F''$ etc., then we should still not be obliged without a proof to admit that any equation whatever would be satisfied by a value of $x$ which is real or of the form $a + b \sqrt{−1}$ or of the form $F$, or $F'$ etc. For which reason that axiom cannot have any other meaning than this: An equation may be satisfied by a real value of the unknown, or by an imaginary value of the form $a + b \sqrt{−1}$, of perhaps by a value of a hitherto unknown form, or by a value which is not contained in any form whatsoever. But it can certainly not be understood with that clarity which must always be insisted upon in mathematics how quantities of such a nature, of which you cannot have any idea, may be added or multiplied. They are merely a shadow of a shadow.

It may be a bit hard to understand Gauss' point, which in more modern presentation may look like this:

  1. Euler (and other previous mathematicians) proved that $X = 0$ has roots of the form $a + b \sqrt{-1}$ assuming that the polynomial has the right number of roots, which are not necessarily real numbers.
  2. There are two cases: first, either the only allowed "non real" numbers are all of the form $a + b\sqrt{-1}$, or second, that there are other types of "non real numbers".
  3. In the former case the theorem is empty. So we can assume they try to prove it for the latter. But at no point are the arithmetic properties of these other types of "non real numbers" discussed. In particular: how do we know these "non real numbers" are closed under addition or multiplication? So how is a proof which is based on arithmetic properties of "non real numbers" supposed to be rigorous, when their arithmetic properties are not examined or defined?

(You should see in this discussion the roots of field theory taking form. In the comments below, Bill Dubuque provides a very simple example illustrating Gauss' concern.)

Returning to the original question:

Gauss' use of the phrase veritable shadow of shadows does not refer to the imaginary numbers. He dislikes the use of them in proofs (at that time in his life; this is what he wrote in a footnote to section 3 of his dissertation) because they are "imaginary" objects, and a proof of a "real" theorem (remember the statement of the fundamental theorem of algebra as conceived at that time period is purely a statement about polynomials of real coefficients and its factoring into simpler similar polynomials) should not rely on "imaginary" objects. But he accepts the use of the imaginary objects to describe, for example, the "roots" of an irreducible quadratic.

No, the use of shadow of shadows refers in fact to the "non real numbers" which are also not of the form $a + b\sqrt{-1}$. Gauss used this epithet to capture the fact that they are even less conceivable than just imaginary numbers: numbers of the form $a + b \sqrt{-1}$ at least can be used in computations with firm arithmetic rules regarding their addition and multiplication. Those other "numbers" do not even have an arithmetic rule associated to them.

  • 1
    $\begingroup$ To give an example of the sort of problems that Gauss is hinting at, the polynomial $\,x^2 -1\,$ has roots $\, x\equiv 1,3\,$ in $\,\Bbb Z/8 = $ integers mod $8,\,$ but the sum of the roots is $\,4\not\equiv 0\pmod 8.\,$ Thus the coefficient of the 2nd highest degree term need not always be the negative root sum for certain types of numbers. In fact this quadratic has $4$ roots $\,x\equiv\pm1,\pm 3,\,$ corresponding to the fact that odd squares are $\equiv 1\pmod 8.\,$ So one needs to make precise the algebraic properties of the extended number system to correctly employ such arguments. $\endgroup$ – Bill Dubuque Dec 9 '16 at 16:31

Here is some context for you here.

Basically he was saying that it had very little "physical meaning" because at the time there was not even a consensus on the meaning of a negative real number.


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