Give an example or a proof (from whatever branch of mathematics) where a desired thing is found by assuming tentatively its existence and thereby obtaining clue to its construction.
This appears a completely strange question to me and I wonder exactly what branch of mathematics could contain its proof. I got this question from a textbook of topology and how topology is related to the solution of this, I can't understand. Any help will be appreciated.
One example is the assumption of the existence of square roots of negative numbers. These were already considered in the beginning of 16th century, in relation to the general solution of the cubic equation, but it was not until the 19th century when they were better understood and generally accepted.
Section 1.3 The Solution of the Cubic Equation, of the text Complex Analysis by Joseph Bak and Donald J. Newman gives a short account of the inception of complex numbers. Here are some excerpts from it (lightly edited).
Numbers of the form $a+b\sqrt{−1}$, where $a$ and $b$ are real numbers — what we call complex numbers [today] — appeared as early as the 16th century. Cardan (1501–1576) worked with complex numbers in solving quadratic and cubic equations.
As more manipulations involving complex numbers were tried, it became apparent that many problems in the theory of real-valued functions could be most easily solved using complex numbers and functions. For all their utility, however, complex numbers enjoyed a poor reputation and were not generally considered legitimate numbers until the middle of the 19th century. ... The wider acceptance of complex numbers is due largely to the geometric representation of complex numbers.
... complex numbers were applied to the solution of quadratic and cubic equations as far back as the 16th century. ... Cubic equations presented a much more tantalizing situation. For one thing, every cubic equation with real coefficients has a real solution. The fact that such a real solution could be found through the use of complex numbers showed that the complex numbers were at least useful, even if somewhat illegitimate.
The first recorded solution for cubic equations involved a method for finding the real solution of the above “reduced” or “depressed” cubic in the form: $$x^3+px=q\ \ \ \ \ \ \ (2)$$
To find that solution, del Ferro (1465–1526) suggested setting $x=u+v$, so that $(2)$ could be rewritten as:
$$u^3+v^3+(3uv+p)(u+v)=q\ \ \ \ \ \ \ (3)$$
The solution to $(3)$ can be found, then, by solving the pair of equations: $3uv+p=0$ and $u^3+v^3=q$. Using the first equation to express $v$ in terms of $u$, and substituting into the second equation leads to:
$$u^6−u^3q−\frac {p^3}{27}=0$$
which is a quadratic equation for $u^3$ and has the solutions
$$u^3=\frac{q\pm\sqrt{q^2+4p^3/27}}2$$
The identical formula can be obtained for $v^3$, and since $u^3+v^3=q$,
$$x=u+v=\sqrt[3]{\frac{q+\sqrt{q^2+4p^3/27}}2}+\sqrt[3]{\frac{q-\sqrt{q^2+4p^3/27}}2}\ \ \ \ \ \ \ (4)$$
or, as del Ferro would have written it to avoid the cube root of a negative number,
$$x=u+v=\sqrt[3]{\frac{q+\sqrt{q^2+4p^3/27}}2}-\sqrt[3]{\frac{\sqrt{q^2+4p^3/27}-q}2}$$
For example, if $p=6$ and $q=20$ [that is $x^3+6x=20$], we find $x=\sqrt[3]{6\sqrt{3}+10}−\sqrt[3]{6\sqrt{3}-10}$ or (check this!) $x=2$.
[Edit. This was intended as a hint. All radicals involved are defined (i.e., exist, as reals) and positive, and obviously $x>0$. I thought for a moment that it would be easy to check that $x^3=8$, but I don't seem to know how to do this. It could be verified numerically (with a calculator) or trust WolframAlpha, or post this as a separate question, how to show that $\sqrt[3]{6\sqrt{3}+10}−\sqrt[3]{6\sqrt{3}-10}=2$. Clearly if $f(x)=x^3+6x$ then $f'(x)=3x^2+6>0$ so $f$ is strictly increasing, so $f(x)=x^3+6x=20$ has a unique real root $r$. One verifies directly that $2$ is a root, $2^3+6\cdot2=8+12=20$, so $\sqrt[3]{6\sqrt{3}+10}−\sqrt[3]{6\sqrt{3}-10}=r=2$, but I wonder how one proves $\sqrt[3]{6\sqrt{3}+10}−\sqrt[3]{6\sqrt{3}-10}=2$ in a more elementary way. Well, I found links on MSE, so here are a couple of elementary proofs, also related this and this questions.]
Although $(4)$ was originally intended to be applied with $p,q>0$, it can obviously be applied equally well for any values of $p$ and $q$. ... Changing $p$ into a negative number, however, can introduce complex values. To be precise, if $q^2+4p^3/27<0$; i.e., if $4p^3<−27q^2$, equation $(4)$ gives the solution as the sum of the cube roots of two complex conjugates. For example, if we apply $(4)$ to the equation $x^3−6x=4$, we obtain $$x=\sqrt[3]{2+2i}+\sqrt[3]{2-2i}\ \ \ \ \ \ \ (5)$$ Since we saw (in the last section) that we can calculate the three cube roots of any complex number using its polar form, and since the cube roots of a conjugate of any complex number are the conjugates of its cube roots, we realize that $(5)$ actually does give the three real roots of $x^3−6x=4$. To Cardan, however, who published formula $(4)$ in his Ars Magna (1545), the case: $4p^3<−27q^2$ presented a dilemma. We leave it as an exercise to verify that equation $(2)$ has three real roots if and only if $4p^3<−27q^2$. Ironically, then, precisely in the case when all three solutions are real, if formula $(4)$ is applicable at all, it gives the solutions in terms of cube roots of complex numbers! Moreover, Cardan was willing to try a direct approach to finding the cube roots of a complex number ..., but solving the equation $(x+iy)^3=a+bi$ by equating real and imaginary parts led to an equation no less complicated than the original cubic. Cardan, therefore, labeled this situation the “irreducible” case of the depressed cubic equation. Fortunately, however, the idea of applying $(4)$ even in the “irreducible” case, was never laid to rest. Bombelli’s Algebra(1574) included the equation $x^3=15x+4$, which led to the mysterious solution $$\sqrt[3]{2+11i}+\sqrt[3]{2-11i}\ \ \ \ \ \ \ (6)$$ By a direct approach, combined with the assumption that the cube roots in $(6)$ would involve integral real and imaginary parts, Bombelli was able to show that formula $(6)$ did “contain” the solution $x=4$ in the form of $(2+i)+(2−i)$. He did not suggest that $(6)$ might also contain the other two real roots nor did he generalize the method. In fact, over a hundred years later, the issue was still not resolved. Thus Leibniz (1646–1716) continued to question how “a quantity could be real when imaginary or impossible numbers were used to express it”. But he too could not let the matter go. Among unpublished papers found after his death, there were several identities similar to $\sqrt[3]{36+\sqrt{−2000}}+\sqrt[3]{36-\sqrt{−2000}}=−6$ which he found by applying $(4)$ to: $x^3−48x−72=0$.
So complex numbers maintained their presence, albeit as second-class citizens, in the world of numbers until the early 19th century when the spread of their geometric interpretation began the process of their acceptance as first-class citizens.
The above were excerpts from Newman and Bak text cited earlier. One may also browse Wikipedia, in particular Scipione del Ferro (6 February 1465 – 5 November 1526), The solution of the cubic equation, as well as Rafael Bombelli (baptised on 20 January 1526; died 1572), Bombelli's Algebra. In particular (from Wikipedia): Bombelli called the imaginary number $i$ “plus of minus” and used “minus of minus” for $-i$. Bombelli had the foresight to see that imaginary numbers were crucial and necessary to solving quartic and cubic equations. At the time, people cared about complex numbers only as tools to solve practical equations. As such, Bombelli was able to get solutions using Scipione del Ferro's rule, even in the irreducible case, where other mathematicians such as Cardano had given up. In his book, Bombelli explains complex arithmetic as follows:
"Plus by plus of minus, makes plus of minus.
Minus by plus of minus, makes minus of minus.
Plus by minus of minus, makes minus of minus.
Minus by minus of minus, makes plus of minus.
Plus of minus by plus of minus, makes minus.
Plus of minus by minus of minus, makes plus.
Minus of minus by plus of minus, makes plus.
Minus of minus by minus of minus makes minus."
Regarding something more closely related to topology, then take perhaps the construction of the completion of a metric space, in particular Cantor's construction of the real numbers as the completion of the rational numbers, assuming that the limit of every Cauchy sequence must exist. Then construct equivalence classes of Cauchy sequences, and use these as the elements (points) of the completion. Or, assume real numbers exist (based on our understanding of rational numbers and some real numbers like $\sqrt{2}$, $\pi$ and $e$, and experience working with them), and develop the real numbers axiomatically (as an ordered field), see e.g. e.g. Wikipedia, Construction of the real numbers, Synthetic_approach. Developing (or deducing) the properties of real numbers axiomatically is not exactly the same as constructing (models of) the real numbers and "observing" or "verifying" their properties, see also Wikipedia, Construction of the real numbers, Explicit constructions of models and BE Mathematical Extended Essay, Lothar Sebastian Krapp.
It seems to me that both in the case of real numbers, and in the case of complex numbers, historically people started working with them as if they existed, and expecting that they ought to satisfy properties that seemed natural, from what was already know of numbers (though certainly complex numbers seem to have been more challenging, to describe and develop their properties consistently), and only later some models were discovered. Real numbers were used for a long time (and calculus was already developed) before Cantor and Dedekind came up with models that behaved just like real numbers were expected to behave anyway. For complex numbers, their interpretation (model) as Cartesian complex plane or as Polar complex plane (discovered and developed by Hamilton, Wallis, Argand, Gauss), came after complex numbers were already thought of, used to solve equations (finding real solutions first), and discovering rules how to work with them (as discussed earlier).
For that matter, something similar happened with non-Euclidean geometries. Giovanni Girolamo Saccheri assumed that the parallel postulate was false and attempted to derive a contradiction. He was unable to derive a logical contradiction and instead derived many non-intuitive results; for example that triangles have a maximum finite area and that there is an absolute unit of length. In effect he developed many of the properties of hyperbolic geometry. (From Wikipedia: He finally concluded that: "the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines". Today, his results are theorems of hyperbolic geometry. There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.) Lobachevsky, Bolyai, Gauss discovered and developed non-Euclidean geometries by considering that it was possible that Euclid's Fifth postulate fails. It was only later that models of non-Euclidean geometries were discovered. Such models, as the pseudo-sphere proposed as a model by Eugenio Beltrami, or the Beltrami-Klein model, the Poincare disk model, or the Poincare half-plane model made it easier to accept non-Euclidean geometries (more or less on equal basis as the usual Euclidean geometry, though the latter remains most widely applicable on everyday basis), but these models came only after a long struggle to unsuccessfully derive Euclid's Fifth postulate from the remaining postulates, and after the question: What if we try to develop a geometry using alternatives to the Fifth postulate? The more I keep writing, the more examples come to mind ... let me post this.
Searching the web I found this historical claim copied verbatim in several places,
... so Fourier's contribution was mainly the bold claim
that an arbitrary function could be represented by a Fourier series.
concerning his work in 1807.
His work was reviewed by both Lagrange and Laplace in 1808; they objected to this central thesis. A few years later he was awarded a prize for these contributions, but the report contained this criticism
... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
See this biography on Jean Baptiste Joseph Fourier, where it states, in regard to Fourier's writings
All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views.
Indeed, Fourier was delineating his groundbreaking theory using the mathematical style/logic of that time.
This bring us to the paper
How did Cantor Discover Set Theory and Topology?
S M Srivastava
The opening paragraph:
In order to solve a precise problem on trigonometric series, “Can a function have more than one representation by a trigonometric series?”, the great German mathematician Georg Cantorc created set theory and laid the foundations of the theory of real numbers. This had a profound impact on mathematics. In this article, we narrate this fascinating story
Cantor produced his landmark results on Fourier series in 1870 -1871.
Fourier's discovery forced a change in mathematical logic and abstraction so that his theories could be made rigorous and precise. The amazing thing is how general set theory became a fundamental cornerstone of mathematics.
See also the video
HISTORY OF FOURIER SERIES AND FOURIER TRANSFORM| Signals and Systems
