Читать книгу The Secret Harmony of Primes - Sam Vaseghi - Страница 9
Оглавление"To some extent the beauty of number theory seems to be related to the contradiction between the simplicity of the integers and the complicated structure of the primes, their building blocks. This has always attracted people." 4
Natural numbers are those numbers that have two main purposes: counting and ordering. In mathematical terms, they are cardinal and ordinal.
There is no agreement on whether the number ‘zero’ is a natural number too. This is why we still use two different mathematical notations for the set of all natural numbers; that excludes zero:
and that includes zero:
Figure 1.1: The sequence of natural numbers as a linear plot.
The set of natural numbers is infinite but countable. Generally, and including zero, one can apply a recursive addition to all natural numbers, beginning with and succeeding with:
that reads “ plus a successor equals the successor ”.
Through recursive addition every natural number is tied with an additive property in relation to other natural numbers: is a commutative monoid with identity element , the free monoid with one generator. Monoids are algebraic structures with a single associative binary operation and an identity element; they are categories with a single object.
This brings us closer to what is called a total order of the natural numbers. We can have if, and only if, there exists another natural number with:
But natural numbers are not only in total order, they are also well ordered: every non-empty set of natural numbers has a least element. There exists, at any time, a rank among the sets that can be expressed by an ordinal number .
Figure 1.2: Mayan numerals.
Given that with addition has been defined, a multiplication can also be defined, beginning with and succeeding with:
In this way, every natural number is tied to a multiplicative property in relation to other natural numbers: is a so-called free commutative monoid with identity element .
These properties of addition and multiplication mean that natural numbers emerge as an instance of a commutative semiring. They cannot be called a ring because is not closed under subtraction and lacks an additive inverse.
Although there is a defined procedure of division with remainder, it is not possible, in a generalised way, to divide a natural number by another natural number , and expect to get a natural number as the result.
In mathematical terms, the procedure of division with remainder can be expressed for any two numbers and with , by the equation:
where is the quotient and the remainder. Also the so-called operator helps:
means that divided by results in the remainder .
In the early 90s, documentarian George Csicsery created a documentary called ‘N Is a Number’ that captured the life and work of the grand mathematician Paul Erdös. Conversely, when I saw this movie for the first time in 2013, Csicsery’s beautiful title inspired me to look in exactly the opposite direction, to better understand the properties of natural numbers, namely to look at instead of . is not ‘a’ number but a totally ordered set, while on one hand, the cardinal and ordinal properties are tied with the numbers, and on the other they tie the numbers. It may sound simple or trivial, but herein lies the intuition to regard as a ‘natural phenomenon’, rather than a primitive mathematical sequence.
One crucial difference between a phenomenological mathematical approach and a pure mathematical approach is that a ‘natural phenomenon’ can be mathematically modelled as a phenomenon while a sequence in number theory is, as such, mathematically ‘defined’ by pure mathematics.
One challenge, however, that I encountered in my work was that studying as a ‘natural phenomenon’ – like any other natural physical, biological, linguistic phenomenon – must not deviate from the mathematical formal system and deductive reasoning, into vague, empirical or heuristic argumentations.
Figure 1.3: In the early 90s, documentarian George Csicsery created a documentary called ‘N Is a Number’ that captured the life and work of the grand mathematician Erdös.
In the early 90s, when I was a student at the University of Stuttgart and a scholar to Herman Haken, the renowned laser physicist and the founder of the theory of synergetics (the thory of self-organisation), I was fascinated with learning how to translate complex systems of different natural phenomena, independent of their disciplinary origin – physics, biology, sociology etc. – into comprehensive mathematical models. The theory of synergetics provides a fundamental paradigm that helps us to understand how patterns are formed and recognised in nature: the laser.
Synergetics tells us that, similar to laser light that can be regarded as a composition of fundamental waves, many patterns in nature can be scattered into interacting fundamental waves, known as elementary modes. When scattering a system into elementary modes, a spectrum appears that helps us to better understand the elementary structural properties of that pattern or system and how it behaves.
Figure 1.4: Weaving scheme for textile design. A two-dimensional pattern decomposed into vertical and horisontal elementary modes.
However, when we break a system into its elementary modes, like a child decomposing a Lego house into its basic elements, we lose the object as a whole, and we cannot identify a real state of the system as a whole because we have decomposed it: the toy is only present in pieces. This means that, at a certain point in time, we can EITHER have the toy house as decomposed into pieces (and observe/understand how it was composed), OR as one complete house (without an understanding of its details, but observing it as a whole).
Figure 1.5: Position x and momentum p wavefunctions corresponding to quantum particles. The opacity (%) of the particles corresponds to the probability density of finding the particle with position x or momentum component p. Top: If wavelength λ is unknown, so is momentum p, wave-vector k and energy E (de Broglie relations). As the particle is more localised in position space, Δx is smaller than for Δpx. Bottom: If λ is known, so are p, k, and E. As the particle is more localised in momentum space, Δp is smaller than Δx.5
Physicists have identified a general concept of ‘duality’ and ‘uncertainty’ through so called Fourier analysis, which is specifically known in the field of quantum physics as the Heisenberg Uncertainty Principle: when we want to fix the position and state of a specific particle, we cannot have it decomposed into waves – and vice versa: if we like it decomposed into waves, we cannot fix its position and state as a particle.
In physics, a particle can be understood as a probability wave. The amplitude of the wave roughly represents the probability of finding a particle in a given place, while the frequency of the wave roughly represents the particle’s state in terms of momentum (mass × velocity). If we imagine a localised particle as a short pulse, rather than a wave, the reason we cannot know its precise position and momentum is because we can’t make an infinitely short pulse out of a single frequency.
Figure 1.6: Katsushika Hokusai (1760-1849) created this extraordinary picture around 1831. It is known as ‘The Great Wave of Kanagawa.’ It is a fairly small colour woodcut. The foam of the wave is breaking into claws, which grasp for the fishermen.
Bearing in mind that can be phenomenologically regarded as a natural pattern, a system, a burning desire arises to establish a mathematical model for that allows us to decompose into a spectrum of fundamental waves or modes.
The more we study as a pattern, the more we can show that sets of natural numbers are similar to other physical phenomena (such as light or matter) in that they behave atomistically: natural numbers have properties of both waves and particles.
Indeed, as we will show later, one could deduce a similar type of duality as for particles and waves in physics: could be observed at the same time, EITHER as an ordered set of number particles (amplitudes) OR as an orchestrated symphony of waves (frequencies):
“Natural numbers act like waves”. – No they don’t exactly.
“They act like number particles”. – No they don’t exactly.6
One wide ranging implication of interpreting as a superposition of waves is that this interpretation would apply as universally as itself: all phenomena in the universe that can be counted by can be practically interpreted as a superposition of such waves.
In the next chapters of this book we will show, step-by-step, how natural numbers can be decomposed and (re-)composed as superposition of waves, what role the prime numbers play in this respect, and finally the order and rules the primes’ sequence follows in relation to this.
Figure 1.7: The first six longitudinal modes of a plane-parallel cavity.
A natural number greater than 1 is called a prime number if it has no positive divisors other than and itself. The property of being a prime is called primality.
