I wrote thi essay during the1st_Semester at theUniKonstanz for the seminar on “Paradoxes: truth, infinity and time travel” by Sam Roberts.
Further related topics I want to check:
Abstract
A comparison of the real numbers with the Kantian noumenon will show important similarities between their roles in the epistemological debate. I’ll then analyse the implications of such a conclusion together with Naturalism by Quine. I’ll then connect this point with the concept of randomness and its possible forms in mathematics, this will then show how we should shape the concept we have in mind of real numbers.
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\title{Issues on Real Numbers}
\author{Simone Testino }
\date{January 2023}
\begin{document}
\maketitle
\begin{abstract}
A comparison of the real numbers with the Kantian noumenon will show important similarities between their roles in the epistemological debate. I’ll then analyse the implications of such a conclusion together with Naturalism by Quine. I’ll then connect this point with the concept of randomness and its possible forms in mathematics, this will then show how we should shape the concept we have in mind of real numbers.
\end{abstract}
\section{Introduction}
The question I’m going to answer in this essay is the following: given that we manage to acquire natural numbers, is it possible, thanks to that knowledge, to get to know real numbers? This question needs first a brief recap on what we are talking about when we say “natural numbers” in order to give a similar answer on reals. Notice that this question might be interpreted in two slightly different ways: epistemologically or on the reference fixing procedure. On one hand, if I would consider the question epistemologically then I would require a theory explaining the basic theory of how do we \emph{know} natural numbers, on the other hand, if I want to answer the reference fixing question, then I’d need a theory explaining what do we denote (what is the reference) of the expression “natural numbers”. The first section of this essay will primarily regard the epistemological question and the second section will answer the reference fixing question. In the second section, though, one should notice that something on the epistemological side is also said, since, if we would be able to have a reference of something thanks to nothing but what we believe to know already, then we would probably know that thing too.
\subsection{Context: on Natural Numbers}
As announced in the previous section, there is the need of giving a context to the epistemological and reference fixing side of the natural numbers. In order to do that I’ll mainly refer to \emph{Vom Zählen zu den Zahlen\footnote{Horsten, L. (2012). Vom Zahlen zu den Zahlen: On the Relation Between Computation and Arithmetical Structuralism. Philosophia Mathematica 20 (3):275-288.}}. The article concerns solely the natural numbers and therefore seems to fit perfectly with the needed context. Though the question that gets answered there is primarily the one concerning the reference fixing of natural numbers, anyway section three and four of the article surely contain an epistemological account too.\\
The natural numbers are said to be singled out (up to isomorphism) by Peano Arithmetic with an algorithm for \enquote{+} and first order induction. What that means is that, if we consider people (or the subject we’re taking in consideration) to be able to compute for any two numbers its sum and she is working in a Peano Arithmetic (namely, it satisfies the usual 8 axioms, comprehending first order induction) then she managed to single out what the natural numbers are and is now able to have a fixed reference. Similarly one might consider the case in which Peano Arithmetic with the second order induction is assumed but nothing particularly is said on computability of the operations. Both these cases allow the subject to talk on natural numbers up to isomorphism. As announced in the previous section, the question I’m going to answer here is if this reference fixing procedure on natural numbers enables us to fix the reference of real numbers too. This question on reference fixing will be answered in the second section with the aid of the concept of randomness.\\
The following section concerns the epistemological side and, on the natural number it should be enough to say that, as soon as a child has understood the mechanism ruling the successor function and realises that she can count up to any natural number, then she has acknowledged the natural numbers. To state it more precisely I distinguish what Kripke\footnote{Whitehead lecture 2} calls transitive and intransitive counting. The former is what we do when counting objects, the latter is when we are just saying the number with (e.g. in the decimal system) the usual successor function on numerals. It seems, at least at the first glance, that the first gives us the understanding of quantities, which is the actual semantic meaning of natural numbers and the second gives us nothing but the synctical rule of how to apply the successor function on numerals. For the sake of the argument, let’s consider that the child that epistemologically acknowledged natural numbers is able to do both transitive and intransitive counting.
\section{Intangibility of Real Numbers}
It's widely famous, even among non-mathematicians, that the discovery of irrational numbers has been a scandal in ancient Greece, since they represented something out of the rules, numbers that could not be represented as fractions, the Chaos arising even where Logos was mostly present. In this section I focus on one way one can have in reading the role of real numbers in mathematics, as opposed to rational numbers.
\subsection{Parallelism with the Kantian Noumenon}
When measuring the length of an object or the time needed for an event to occur, I won’t ever get anything but rational numbers. In the way I perceive numbers, as quantities in space and time, there will never be a way that will allow me to perceive a real number. So, if I take a pencil and I pretend to measure its length, no matter how precise I’ll try to be, I’ll always measure a rational number. But now the question is, is the length of the pencil \emph{actually} a rational number? It has not to be so, I don’t want here to investigate if the physical space is discrete or not, but certainly saying that I have no means to measure with perfect precision the length of the pencil is not a good reason for stating that the space needs to be not a continuum. Instead we could think that it is possible for the pencil to have a real numbered length, though no one of us will ever have access to it. Such relation between the intrinsic, essencial length of the pencil and its image accessible to us, will certainly remind to anyone who studied basic philosophy the distinction between the noumenon and the fenomenon by Kant. From so on, you’ll notice, I’ll talk about the real numbers as if they were a sort of noumenic objects and the rational numbers as their corresponding fenomenon.
\subsection{Connection with Quinean Naturalism}
Quine, briefly, believes that science and so empirical evidence, have the right to deny whatever we believe, since, as famously stated: \enquote{it is within science itself, and not in some prior philosophy, that reality is to be identified and described}.
By accepting a so described conception of mathematics\footnote{More on it here \emph{The Oxford Handbook of Philosophy of Mathematics and Logic}, S. Shapiro, OUP, C. 12-13} that fears empirical falsification as much as a scientific statement does (in principle, practically, as Quine admits, it’s way harder to falsify mathematical statements), one could consider the former exposed distinction. Does science work with real numbers? After forcing myself to not make any joke on $\pi$ and its approximations by scientists, one should consider, as previously noted, that we’ll never measure real numbers since there will never be means to do such a thing. Though, this is not the only way a scientist has to work with numbers, he also needs them to get results and make predictions, and obviously one can get real numbers as results of equations with rational numbers, as in the elementary problem of calculating the diagonal of a square of side one. There is hope for arguments in favour for the existence of real numbers in such a mathematics, though there are some further problems, like the fact that the only actual way we have to think about real numbers is as limits of sequences of rational numbers, this shows furthermore their noumenal nature. I won’t go deeper in this discourse for now, since there are other ways I want to investigate to show the chaotic nature of real numbers.
\section{On Para-Randomness}
What I mean with para-randomness, is the set of all the functions trying in some way to give random outputs in a system that doesn’t normally allow anything purely random. Computer programs, mathematical functions or also algorithms in classical logic might try to give very unrelated numbers such that they would appear random but though it’s hard to see if they will actually ever succeed. There are different ways to construct para-random functions, though all have one thing in common, they try to relate to something external, far away, apparently unrelated.
\subsection{Relation to Other Worlds}
The best way, as far as I know, classical computers have to generate random numbers is to measure something really unrelated, like background noise or, similarly, a very small fraction of a second of the present time. Such functions work perfectly for what is required by the user, since the input the functions are taking is something that is completely unrelated to the task for which the number is needed. Though one cannot say that such an output of the function is actually random since if we would be able to put the computer in the exact same situation the output would be the same. I find it fascinating that something that seems so easy like giving a random number seems to be that hard for a computer to do. Though, is it an easy job at least for us? I won’t even try to answer this question here, though I think we don’t have any purely random function either and I might claim that the para-random functions we use are even not that far away from the same principle I’m showing here, namely taking the input from something we believe unrelated. This is no more than speculation, though asking a bunch of people for a random number would probably show interesting patterns.\\
By correlating this section with the previous one, one might think that real numbers are a perfect source of chaos in mathematics, since their digits apparently are completely random.
\subsection{Existence of Some Real Numbers}
It is very tempting to say that in order to have randomness in mathematics one could take a real number and follow its digits, since they, by definition, will go on with no end or periodicity. By going far enough one will surely find \emph{purely} random digits. I strongly believe this is not the case and here I shall explain a good reason for believing so.\\
Here I investigate two questions: if the real numbers I can know are a source of randomness (next section) and if real numbers can be expressed as polynomials in $\mathbb{Q}$ (the section following the next).
\subsubsection{On the Rationality of Real Numbers}
Defining a number is surely a very hard job, and I think there could be two ways to do that, these are very close to the Russelian theory of denotation: one could define a number thanks to what one has defined already. Example of this would be any expansion from the natural numbers. If we consider $\mathbb{N}$ to be given (which is another big issue, though not strictly related to my point) then negative integers will be results of polynomials in $\mathbb{N}$, by subtraction of a bigger number to a smaller one. Similarly divisions in $\mathbb{Z}$ will give (all) numbers in $\mathbb{Q}$ and then polynomials in $\mathbb{Q}$ will give results in $\mathbb{R}$, which might cover it all. If we then take $\mathbb{R}$ as given, we do nothing but solve a simple polynomial like $x^2=-1$ to have all the numbers in $\mathbb{C}$.\\
If we look at numbers in this way, it would be very unnatural to expect something so unrelated to be random coming from numbers one had somehow access to at the beginning. For the same reason why in $0.025$ it is not random that the fourth digit is a $5$, then it is similarly not random that the thousandth digit of $\sqrt{2}$ is $1$.\\
Previously I announced a second way of defining numbers, and this might be a good way of getting something random. I might define a number by pointing at it. I might take the number line and put the tip of my pencil on a certain point, draw a little arrow and write, I call this $\alpha$. I don’t know if this would give me knowledge of the number $\alpha$\footnote{Kripke, \emph{Whitehead Lectures}, II, \emph{Logicism, Wittgenstein, and De Re Beliefs About Numbers}, on May 5, 1992} though it surely won’t let me know the infinite number of digits coming after the coma. As it’s obvious and as I explained in the first section, I would get no more than a rational approximation of it. So, there will be nothing random, but just something very unrelated like that piece of paper and some other calculations needing a random number.
\subsubsection{On the Polynomial Origin of Real Numbers}
In this section I’ll focus on a set of numbers that might represent (even if not cover fully) the core concept of definability I expressed in the previous section. We saw that, if naturals are given then it’s very easy to get definitions of all numbers up to the rational ones. From the title of this essay one might easily deduce that it should be at least harder if not problematic to define real numbers starting from the rational ones. But first I want to do something easier than getting to the real numbers, instead I’ll try to repeat the same process that took us from the naturals to the integers and then from the integers to the rational and see what we get if we apply it to the rationals. The process I’m referring to is the one of considering every polynomial we can build with the numbers we have already and then consider all the roots we can find of them. So, in symbols let's define what polynomials in $\mathbb{Q}$ are:
\[\mathbb{Q}[X] := \{\sum_{n=0}^{\infty} \lambda_n x^n | \textbf{ } n\in \mathbb{N}_0, (\lambda_n)_{n\in\mathbb{N}} \subseteq \mathbb{Q}\}\]
A set containing all the roots of such polynomials is simply defined as:
\[\mathbb{A}:= \{ x | \sum_{n=0}^{\infty} \lambda_n x^n = 0, n\in \mathbb{N}_0, (\lambda_n)_{n\in\mathbb{N}} \subseteq \mathbb{Q}\}\]
Let's see what numbers are in this set and sketch some properties in order to see the differences between $\mathbb{Q}$ and $\mathbb{R}$:\\
Evidently $\mathbb{Q} \subset \mathbb{A}$, for it consider just the case in which $\lambda_2, ...,\lambda_m, ... = 0$, $\lambda_1=-1$ $\lambda_0$ will cover every rational number since the polynomial will be of the form $\lambda_0 -x=0$, so $x = \lambda_0$.\\
$|\mathbb{A}|= |\mathbb{N}|$, consider that the sum runs on natural numbers, this makes already a bijection. From this we easily derive that $\mathbb{A} \not = \mathbb{R}$. So there must be some real numbers which aren't in $\mathbb{A}$, let's try some real numbers and check if they can be written as polynomials in $\mathbb{Q}$:\\
$\lambda_1, \lambda_3, ..., \lambda_m, ... = 0, \lambda_2=1$, then $x^2 + \lambda_0 = 0$, so for every $\lambda_0 \in \mathbb{Q}$, $\sqrt{\lambda_0} \in \mathbb{A}$, e.g. $\sqrt{2}$ but also $\sqrt{-2}$. We notice than that $i \in \mathbb{A}$ and so $\mathbb{Q}[i] \subset \mathbb{A}$.\\
Now one is surely asking if $\pi$ and $e$ are in $\mathbb{A}$ or not and the answer is sadly no, due to the following theorem that I'm not going to prove (remark: the set I denoted with $\mathbb{A}$ is a well known set called the algebraic numbers):
\begin{theorem}
If $\alpha_1, ..., \alpha_n$ are distinct algebraic numbers, then the exponentials $e^{\alpha_1}, ..., e^{\alpha_n}$ are linearly independent over the algebraic numbers.
\end{theorem}
This theorem then implies that $e$ is not an algebraic number and a quick look into Euler's formula shows that $\pi$ isn't one either.
\subsubsection{Implications on Randomness}
The previous paragraph should make the reader realise that there are two kinds of numbers that we have discovered: some are the numbers that come naturally from the natural numbers, which we took as given, the other numbers, namely those which we saw being not in $\mathbb{A}$ are somehow of a different sort. Those numbers, namely $\pi$ and $e$, which are very related ($\pi$ can easily be defined in terms of $e$, consider e.g. the definition of trigonometric functions), seem to be on one hand very external from mathematics, since they don’t come from the usual procedure we adopted to expand the range of numbers we consider. On the other hand, one could surely not imagine mathematics being without $e$ and $\pi$, and therefore saying that those numbers are random or that they can be regarded as being random in some mathematical discourse, is surely nonsense. In the philosophical notes\footnote{Kurt Gödel, \emph{Philosophisches Notizbücher}, Band 3, edited by Eva-Maria Engelen, translated by Merlin Carl, p. 90} by Gödel we notice a similar conclusion on the importance of such numbers and their origin in the \emph{empirical world}:
\begin{quote}
[...] The feeling of depth for $e$, $\pi$. These are numbers that can easily be characterised and thus must also correspond to fundamental things in the empirical world.
\end{quote}
The temptation exposed at the beginning that one could pick a real number, follow the tale of digits after the point and regard them as being completely random and independent from the rest of mathematics is now clearly to be avoided. The problem with such an intuition is the possibility of picking a random number in such a ruleless manner.
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