Abstract
In our society serious questions regarding the future of science reflect a widespread fear that in certain important areas scientific progress is going down a blind alley.
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- 1.
For instance, we should remember that in computer programming of numerical calculations with two variables \(x\) and \(y\), representing “real numbers”, it is not allowed to use decisional statements of type “if (\(x\).equal.\(y\))...then ”, because the electronic processor is not in a position to decide on the equality of two real numbers. A serious limitation, for this indecision might be significant.
- 2.
The risk of losing a common sense of reality through scientific analysis was already clearly perceived in antiquity. Plato in his “Laws” recommended “not pushing too far the search for causes because this was not allowed (\(o\)ὐ \(\delta ' \)ὄ\(\sigma \iota o\nu \)) by divine law”.
- 3.
We note here as examples for the reader who has some familiarity with these problems the cases of the calculation of Dirac’s matrices in the most complex Feynman’s diagrams or in verifying Ward-Takahashi correlation-functions identities in problems of quantum gravity.
- 4.
The thesis of the theorem says that “Given two different rational numbers \(p < q\), there always exists a third number, \( r\), such that \(\vert q-p\vert > \vert r-p\vert \) ”.
- 5.
In set theory, a parameter, called cardinality, is defined and indicated with aleph (\(\aleph )\). The cardinality value \(\aleph _{\mathrm{0}}\) (aleph-zero) is that of sets of infinite, countable objects. Since a real number is defined by two successions (sub-sets) of rational numbers the cardinality of continuum is \(\aleph _{\mathrm{1}} =2^{\mathrm{\aleph } \mathrm{0}}\) (aleph-one). This subject is taken up in the next section .
- 6.
Considerations on relativistic quantum physics indicate that for lengths smaller than \(10^{-\mathrm{3}\mathrm{5}}\) m (called Planck length) and times shorter than \(10^{-\mathrm{4}\mathrm{4}}\) s (Planck time) use of continuum properties involves a certain theoretical incongruence. However, these figures are obtained from a dimensional analysis of universal constants and we don’t known what they really mean.
- 7.
The equation \(n ^{2} + m^{2} = p^{2}\) can be formulated in this manner: given a square of side \( n \), construct with all elements of a square of side \( m \) the gnomons which extend the square of side \( n \) to that of side \( p \). One can easily see that with this method general formulae are found which make it possible to calculate the Pythagorean triplets.
- 8.
During his work Cantor often kept in touch with philosophers and theologians in order to link his theory, and, especially its axiomatic aspects, with metaphysical speculation. Among his correspondents were Aloys von Schmid, Joseph Hontheim and Constantin Gutberlet, who were involved in the discussion concerning thermodynamics and cosmology. Tillmann Pesch [98], one of the moreimportant representatives of neo-scholasticism, contributed greatly to the diffusion of Cantor’s ideas of transfinite numbers in philosophic circles.
We should, however, mention that these interests provoked repeated violent attacks on Cantor from the aggressive positivistic milieu, attacks that hindered acceptance of his theory—now a pillar of modern mathematics—and, finally, ruined his health. Pesch was even arrested several times in Germany under the charge of being a Jesuit. A telling comment on the climate of European culture at the end of the 19th century.
- 9.
Note that this definition is different from that of infinite (\( \infty \)) as used in infinitesimal analysis, where it pertains to any kind of number. Cardinality is meant to define different “sizes” of different, infinite sets, but in a distinct way. Thus, for instance, sets of natural numbers and sets of prime numbers are both infinite and the latter type is a subset of the former; nonetheless the two sets have the same cardinality.
- 10.
A Dedekind cut is a partition of rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A does not contain any greatest element. If B has a smallest element, the cut corresponds to that rational number. Otherwise, the cut defines a unique irrational number, which is in neither set, and “fills” the gap between A and B. The set of irrational numbers is thus defined as the set of the set pairs of all rational numbers and hence has the cardinality \(\aleph _{\mathrm{1}}\).
- 11.
The war of intuitionism was indeed a strange one since, curiously enough, viewd from a distance of some decades, the sides and underlying interests were reversed. The first violent clash took place between Kronecker and Cantor, the creator of modern set theory and of transfinite numbers. Cantor represented the standpoint of the majority of mathematicians against intuitionism, but he was also working on pioneering theories which were being opposed to by the academic world. On the other hand, Kronecker was an influential member of the Berlin Academy and was personally against Cantor’s new ideas. The battle with intuititionists was very hard-fought and lasted more than ten years, finishing with the victory of Cantor, in spite of the incomprehension and the attacks on the part of some mathematicians of his time, a situation which ruined him mentally. The second battle broke out half a century later, between Brouwer and Hilbert. The latter, who represented the compact front of the academic establishment of his time, accused Brouwer of casting away all the precious fruits collected during three centuries in order to provide safer passage for, in his questionable opinion, the boat of mathematics. This time it was Brouwer, the great perfectionist of intuitionism, who lost his mental health in the battlefield.
- 12.
The most famous case is that of the equation \(x^{\mathrm{3}} + px + q = 0\), whose algebraic form is said to be “irreducible”. When its coefficients are such that \(27 \,q^{\mathrm{2}} + 4\,p^{\mathrm{3}} = 0\), the resolutive formulae yield three real solutions, while if this expression is greater than zero one of the solutions cannot be expressed with radicals of real numbers, even though the equation has in fact three real solutions (for example, when \(p = -2\) and \(q=1\)), but require complex numbers to express them in radicals. All this appeared odd, to put it mildly, since the results were in contrast with the geometric image of the problem.
- 13.
We note that a trigonometric form is the simplest derivation of analytic geometry which, however, involves an important property: while in vectorial form a complex number is defined by two single real numbers, \(x\) and \(y\), owing to the periodic character of functions sinus and cosinus, in a trigonometric form the same number is defined by its module, \(r\), and infinity of values of the argument \(\theta + 2n\pi \), with \(n = 0, 1, 2 ... \infty \). We shall see that this property is of primary importance in certain operations.
- 14.
The physical significance of a wave function, z, is given by its square only if z is a real function. If function z is complex, and hence is written as: \(z=x+\mathrm{i}y\) , its physical meaning is given by the product of \(z \) times its conjugate \(z^{*} =x-\mathrm{i}y\).The product \(zz^{*}=x^{2}+y^{2}\) is always a real number. Since the conjugate of a real function is its own function, in both cases their norm \( \parallel z \parallel \) is generally defined as the square root of the integral of the product \(zz^{*}\) over space.
- 15.
In this definition there is an analogy to modular arithmetic, often employed in periodic structures, where only positive numbers are used that are less than a chosen value, called modulus. In modular arithmetic, addition, subtraction and multiplication are like in regular mathematics, but there is no division. The notation used for equality expressions involving modular arithmetic is: \(x= y\) (mod \(m)\). This means that \(x\) and \(y\) leave the same remainder when divided by \(m\). For example, in time measurements a correspondence of 8 p.m. to 20 h can be expressed as \(8=20\) (mod 12).
- 16.
An adele \(x\) consists of an infinite sequence:
$$\begin{aligned} x = (x_{\infty } , x_{2}, \ldots . x_{p}, \ldots ) \end{aligned}$$where \(x_{\infty }\) is a real number, and \(x_{p} \) are \(p\)-adic integers except for a finite set of prime numbers, S. The adele ring A is hence given by:
$$\begin{aligned} A = {\bigcup \limits _{S} {A(S)}} \mathrm{} A(S) = R\times {\prod \limits _{p \in S} {Q_{p}} }\times {\prod \limits _{ \notin S} {Z_{p}}} \end{aligned}$$This ring possesses an ample generality, including real and \(p\)-adic numbers encompassing all values of \(p\).
- 17.
An excellent review of the current state of \(p\)-adic mathematical physics is given in Ref. [87]
- 18.
The integrals of the square of these functions extended to the whole space (their norm) must be equal to one, and the integral of the product of two different functions, extended to the whole space, is zero, These functions, suitably chosen, constitute a basis for space and correspond, in a certain manner, to the versors of the axes of reference in Euclidean space. Actually,we are dealing with a generalised concept of vector with an infinite number of components.
A basis can, for instance, be a class of simple orthonormal functions of type:
$$\begin{aligned} {\left\{ \text {sin} ( n\pi x), \in Z,n \ge 1 \right\} } {\qquad \mathrm{{and}}\qquad } {\left\{ { \,\text {cos} ( n \pi x), \in Z,n \ge 0} \right\} } \end{aligned}$$where Z represents the set of integer numbers. In fact, every wave function can be represented as a linear combination of these basis-functions, analogous to a vector, which can be represented by the sum of its projections on the reference axes. The space of wave functions (called Hilbertian space) is therefore defined by all possible linear combinations of the basis-functions.
- 19.
A quantistic correlation concerns two or more interacting objects, in whose inclusive wave function their states are, so to speak, “overlapped” and the objects have lost their individual identities.
- 20.
This is described by a wave function completely defined by the interaction potential of the two electrons.
- 21.
IBM has been conducting a research programme on this project whose main purpose is to guarantee that data transmitted across the networks of the financial world be undecipherable to possible criminal interceptors.
- 22.
It is strange that, in his strikingly innovative article of 1905, Einstein, at that time a simple, twenty-six year-old clerk in a patent office at Bern, though virtually unknown as a physicist, did not cite any bibliographical references, in contrast to the fundamental rules observed for scientific publications. Even stranger is the fact that the article was published without any objections in Annalen der Physik, a prestigious magazine edited by Paul Drude. The editor Drude, whose scrupulousness and scientific rigour were well known, was certainly informed of Lorentz’s work, but he did not participate in the controversy that in the following years involved Lorentz, Poincaré and Einstein because he committed suicide for unknown reasons in 1906. However, Lorentz thought highly of Einstein, and a few years later in one of his lectures at Columbia University honestly admitted: “Einstein’s general and fundamental principle ... besides the fascinating boldness of its starting point, has a marked advantage over mine”. Nonetheless, Lorentz still maintained his opinion that there was an aether, referred to which a resting observer is in a position of measuring “true time”. This delicate argument is considered by Edmund Whittaker , one of the most eminent historians of physics, in his fundamental work [71].
- 23.
“What it is now is the same as what it ever was, what is different is what is within it”.
- 24.
Aristotle (Physica IV, 11, 219b18) reported the opinion of the sophists who maintained that philosopher Coriscos is, in the two places, two different things. On the contrary, Aristotle’s view is that we have a common substrate with different attributes at both places. The existence of a distinct reality from which change originates is a necessary condition for the intelligibility of change.
- 25.
It is interesting that Newton used the word “flussion” to indicate the derivative of a function. From this usage we can imagine what importance Aristotelian speculation had on the historical development of infinitesimal analysis.
- 26.
Imaginary time is a generalisation of a non-interrupted series of successions of all possible motions considered as measuring all possible successions.
- 27.
Newton asserted that time and space were, respectively, eternity and the immensity of God. Their measure was then only a question of metrics.
- 28.
Different from random behaviour, deterministic chaos occurs where the very analysis of the deterministic equations predicts a divergence of uncertainty and non-observance of “strong causality.
- 29.
We can exempt motions caused by ultra-weak nuclear forces, but this does not eliminate the basic difficulty.
- 30.
As in biology, the age of a system can be referred to a process starting from initial conditions (birth) and evolving toward final equilibrium conditions where statistically relevant changes cease to occur (death). The sequence of changes from birth to death defines the scale of the (non-linear) lifetime of a system. For an equilibrium (i.e., dead) system there is no way to relate its properties to conventional physical time.
- 31.
Our example was inspired by a cybernetics monograph by Valentino Braitenberg, on what he calls “experiments of synthetic psychology” [38]. Though we do have some reservations on his fundamental thesis, we find Braitenberg’s ideas enlightening for the complex topics regarding the dependency of cerebral activity on the topological structure of neuronal networks.
- 32.
Gauss demonstrated that an assembly of equal spheres can fill the space with a maximum fractional density of \(\pi {/}(3\surd 2)=0.74\). The cube is actually the only regular polyhedron with which the space can be completely filled up. The other possible polyhedra that possess this property are not regular.
- 33.
Note that in lattices, for purely mathematical reasons, a rotation symmetry of order 5 does not exist, which in nature appears very frequently in non-periodic shapes (for example, in flowers).
- 34.
Note that the variation of free energy, \(\Delta G\), produced in a physical process occurring in a system at temperature \(T\) is given by: \(\Delta G=\Delta H-T\Delta S\), where \(\Delta H\) is the variation in mechanical energy and \(\Delta S\) that of the entropy. Every spontaneous process has as a product a negative value of \(\Delta G\). Since an ordering process in space produces a negative value of \(\Delta S\), it becomes only possible if the temperature, \(T\), is sufficiently low.
- 35.
The sequence can start from any point, beginning with the polyhedron of its nearest neighbours and then of the second-nearest ones, and so on, by always choosing as the next atom the nearest one.
- 36.
The speed of atoms and, in solids, their frequency of vibration around equilibrium positions, supply the foundation for defining continuum in thermodynamics, where measurements of quantities are obtained as averages of sufficiently large volumes and long periods of time.
- 37.
The definition has here been somewhat simplified with respect to that developed by Hausdorff.
- 38.
For instance, a quantity of primary importance in chemistry is the specific surface, on which the reactivity of a solid reagent depends. Reactivity and catalytic property of a surface depend on this quantity, which can, however, increase by orders of magnitude when the reagent surface approaches fractal structures.
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Ronchi, C. (2014). The Crisis of Growth. In: The Tree of Knowledge. Springer, Cham. https://doi.org/10.1007/978-3-319-01484-5_4
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DOI: https://doi.org/10.1007/978-3-319-01484-5_4
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