Editing The Road to Reality Study Notes (section)

== Chapter 3 Kinds of number in the physical world ==

===3.1 A Pythagorean catastrophe?===
We now switch over to the idea of ‘number’, and the layers of generality that lie beneath integers.  The Pythagoreans solved the question, by using proof by contradiction, of attempting to find a rational number (fraction) whose square is precisely 2.  There does not exist such a number within the confines are integers and rationals, which was troubling for them at the time since it was desired to have all of geometry be described by these types of physical numbers. 
 
Penrose gives the proof by contradiction for the Pythagorean question above and explains the necessity of identifying the precise assumptions that go into a proof.  He explains that there exist other generalities than that was originally used in the proof, whereby these precise assumptions must be used to judge the logic.  The Pythagoreans used integers and rationals to explain the existence of the real numbers.

===3.2 The real number system===

'''Natural Numbers:'''

‘Counting’ numbers from 1 to <math>\infty</math>.

'''Whole Numbers:'''

All counting numbers including 0, cannot be a fraction.

'''Integers:'''
 
All natural numbers and their negative counterparts and 0.  If  and  are integers, then their sum , their difference , and their product  are all integers (that is, the integers are closed under addition and multiplication), but their quotient  may or may not be an integer, depending on whether  can be divided by with no remainder.

'''Rational Numbers:'''

A number that can be expressed as the ratio a/b of two integers (or whole numbers) a and b, with b non-zero.  The decimal expansion is alyas ultimately periodic, at a certain point the infinite sequence of digits consists of some finite sequence repeated indefinitely.

'''Irrational Numbers:'''

A number that cannot be expressed as the ratio of two integers. When an irrational number is expressed in decimal notation it never terminates nor repeats. 

:'''Quadratic Irrational Numbers'''
:Arise in the solution of a general quadratic equation:

:<math>Ax^{2} + Bx + C = 0</math>

:With A non-zero, the solutions being (derived from the quadratic formula):

:<math>-\frac{B}{2A}\sqrt{\left(\frac{B}{2A}\right)^2}+\frac{C}{A}, \quad -\frac{B}{2A}\sqrt{\left(\frac{B}{2A}\right)^2}-\frac{C}{A}</math>

:where, to keep within the realm of real numbers, be must have B2 greater than 4AC.   When A, B, and C are integers or rational numbers, and where there is no rational solution to the equation, the solutions are quadratic irrationals. 

'''Real Numbers:'''

A number in the set of all numbers above that falls on the real number line. It can have any value.

:'''Algebraic Real Numbers:'''

:Any number that is the solution to a polynomial with rational coefficients.

:'''Transcendental Real Numbers'''

:Any number that is not the solution to a polynomial with rational coefficients.

===3.3 Real numbers in the physical world===
Penrose states that while the historical driving force for mathematical ideas were to find constructs that mimic the behavior of the physical world, it is normally not possible to examine the physical world in such precise detail to abstract clear cut mathematics from it. Instead, the mathematics provides its own ‘momentum’ within the subject itself whereby it may seem to diverge from what it had been set up to achieve, but ultimately leads us to a deeper meaning of the world.

The [https://en.wikipedia.org/wiki/Real_number real number] system is given as an example of this since no direct evidence from Nature exists to show that the physical notion of ‘distance’ extends from arbitrarily large scales to the indefinitely tiny.  Though, from cosmological scale to the quantum, to volumes and the metrics of spacetime, the ranges are increasing, and the fundamentals of calculus rely on the infinitesimal.

We may still ask whether the real number system is ‘correct’ for describing the physical universe or have we gotten lucky with extrapolation thus far.  Penrose describes quantum mechanics as not implying any discreteness to Nature, nevertheless this idea may persist and echoes a quote from Einstein’s last published work where he suggested ‘discretely based algebraic theory’ might be the way forward for future physics.  Penrose ends with stating that his idea of [https://en.wikipedia.org/wiki/Spin_network spin networks] are both discrete and foundational in quantum gravity, led to his [https://en.wikipedia.org/wiki/Twistor_theory Twistor Theory], but real numbers are still fundamental in our understanding of the physical world.

===3.4 Do natural numbers need the physical world?===
Progressing this philosophical train of thought to the natural numbers, Penrose notes that the operations of addition and multiplication of natural numbers are independent of the nature of geometry of the world.  Perhaps our notion of these numbers depends upon our universe specifically, but he stresses that it is hard to imagine that there would not be an important role for such fundamental entities.

To support this, Penrose introduces us to the idea of a set, which is an abstraction that does not appear to be tied to the specific structure of the universe.  He then describes a method by which an [https://en.wikipedia.org/wiki/Empty_set#:~:text=In%20mathematics%2C%20the%20empty%20set,its%20existence%20can%20be%20deduced. empty or null set] can be used to conjure the natural number system, out of nothing but the abstract idea of a set itself, and states that this technique can be used for the real number system as well.

Penrose ends by referring us back to Fig 1.3 (depiction of the mental, physical, and platonic worlds) and notes the mysterious nature of the fact that natural and real numbers having no reliance on the physical world for their existence, yet they seem to have direct relevance in describing the structure of the world.

===3.5 Discrete numbers in the physical world===
What might it mean to say that there are minus three cows in a field?  An interesting question to start us thinking about the nature of discrete numbers and negatives in our world as defined as scalar quantities. Penrose states that only in the last hundred years that the integers in the negative seem to have direct physical relevance.

Electric charge is quantified in terms of integral multiples, positive, negative, or zero.  Further, the three quarks within protons have charges 2/3, 2/3, -1/3.  Is this the fundamental entity, and as such, the basic unit of charge is 1/3?  This is just one example of an [https://en.wikipedia.org/wiki/Multiplicative_quantum_number#:~:text=A%20given%20quantum%20number%20q,electric%20charge%20is%20one%20example. additive quantum number] and Penrose notes that our present knowledge is that “all known additive quantum numbers are indeed quantified in terms of the system of integers…” Further, Dirac’s [https://en.wikipedia.org/wiki/Antiparticle antiparticle theory] shows us a physical and meaningful use for the negative numbers in which the additive quantum number of the antiparticle has precisely the negative of the value that it has for the original particle.

Penrose ends the chapter by stating that there are other kinds of number that appear to play a fundamental role in the universe, the most important and striking of which are the [https://en.wikipedia.org/wiki/Complex_number complex numbers].  While they are fundamental to mathematics, “it is an even more striking instance of the convergence between mathematical ideas and the deeper workings of the physical universe”.