I wrote this essay during the1st_Semester at theUniKonstanz for the course on Gödel by Prof. Horsten. I couldn’t have this essay corrected by the professor since it was a Master course and, as I discovered after having written the essay and followed all lectures, I wasn’t actually allowed to be there.
Abstract
Starting from a quote by Gödel on the necessary coexistence of an a priori and an empirical access point on every notion, I underline the essential problems such a point of view implies. First I present reasons why one shouldn’t need any vagueness in defining a priori and a posteriori predicates, then I examine the single examples Gödel writes, conducting most of them onto one core concept, which I then analyse and deny, with a particular stress (and disagreement) on the example concerning mathematics.
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\title{A Priori's Division of Knowledge}
\author{Simone Testino}
\date{January 2023}
\begin{document}
\maketitle
\begin{abstract}
Starting from a quote by Gödel on the necessary coexistence of an a priori and an empirical access point on every notion, I underline the essential problems such a point of view implies. First I present reasons why one shouldn’t need any vagueness in defining a priori and a posteriori predicates, then I examine the single examples Gödel writes, conducting most of them onto one core concept, which I then analyse and deny, with a particular stress (and disagreement) on the example concerning mathematics.
\end{abstract}
\section{Introduction into a Priori and a Posteriori}
\subsection{Origin of the Words}
\emph{A priori} and \emph{a posteriori} (or empirical) have notably been since centuries a division of knowledge. It has most famously been presented by Kant in the \emph{Kritik der Reinen Vernunft}, but has a way older origin. Both Leibniz and Berkley used these terms before but their first use is to find in the Elements (\textgreek{Στοιχεῖα}) by Euclide\footnote{for a deeper analysis: Epstein, Peter Fisher (2018). A Priori Concepts in Euclidean Proof. Proceedings of the Aristotelian Society 118 (3):407-417.}. After Kant the terms became even more diffused and are today crucial terms in contemporary philosophy and even used commonly in different living languages.
\subsection{The Use of the Words}
A posteriori is traditionally used as the negative predicate of a priori which is a predicate on knowledge. Since it refers to knowledge of something and not to an object directly it is possible for an object to be known in both ways, once a priori and once a posteriori. Though, traditionally (Kant and Leibniz\footnote{Leibniz, Monadology, 1714, §34} are two clear examples), one tends to distinguish objects as knowbale by only a priori way or only an a posteriori one, few space (if any) is left for objects that are believed knowable in both ways.
\subsection{The Gödel’s Quote}
As written before, this paper will start with a comment to a quote by Gödel found in \enquote{Kurt Gödel, Philosophische Notizbücher, Band 3, p. 69}. Here is the quote in the translation by Merlin Carl.
\begin{quote}
Remark Psychology: For everything, there are two points of access: an empirical one and an a priori one. Examples:
\begin{enumerate}
\item Experienced people have knowledge that can also be acquired a priori (for example about right flat, right furniture, right conduct of life, etc.).
\item The simple laws of physics (law of inertia, sufficient cause, etc.). In the sense of a \enquote{weaker} a priori derivation from a physical theory in general.
\item Theology through revelation and natural theology.
\item In the realm of concepts, relation between formal and empirical concepts on the one hand and \enquote{perception} of an empirical concept on the other.\\
Perhaps also.
\item Seeing a theorem through intuition, and access with a formal proof (in particular recursive proofs, step-by-step mind).
\end{enumerate}
In principle, empirical access is always superfluous (as language is in principle superfluous for knowledge). It is a crutch for the weak (thus for humans in general).
\end{quote}
\subsection{The Restriction of Cases}
The first step when several examples of a general statement are presented is to try to reduce some of these examples to one case only by looking at their similarities, this will make a lot easier to understand what the statement and the examples actually mean and eventually help in the seek after a failure of the general statement.\\
In \textbf{1.} Gödel refers to the intuition one has when having so much experience in general or in a specific topic, that one doesn’t need to acquire anything new or to think deeply at all at something, but instead an intuition comes immediately to his mind and reveals what is the right choice. With the following writing I denote that the example 2 is of the same kind of the example 1 and therefore are essentially the same case, have the same structure.\\
\textbf{2. $\rightarrow$ 1.}, as noted when explaining 1., there is no particular need for experience to be very broad and general in order to enable one to have intuitions, one could be an expert in just one field and then have intuitions able to make him understand immediately, intuitively, simple derivations. One may also say that the derivation of such physical laws, since we are all very used to them in a not just theoretical point of view, makes them intuitively right to us all, and therefore, in a (weaker, as he writes) sense, a priori.\\
In \textbf{3.}, perceiving a religious feeling, a so-called revelation is, for Gödel, a certain intuition which, in this case, unlike before, is not caused by wisdom but instead that comes naturally into people. My doubts concerning this point are mainly in the fact that I can’t see any difference from the precise and single perception of the religious feeling and the perception of anything else. Shouldn’t any perception, in these terms, be something that we intuitively come to know, without any known previous cause (to our senses or reason, similarly to the Kantian fenomenon)? Because of my incapacity of finding any difference from the religious feeling and any other feeling, I say that \textbf{3 $\rightarrow$ \textgreek{φ}} where \textgreek{φ} stands for the case of perception in general which I just presented\footnote{The reason for the choice of the letter \textgreek{φ} is because of the verb \textgreek{φαινω} which literally means \enquote{to shine}, but has also the connotation to appear to someone, and is therefore the root word of phenomenon.}. The opposed a posteriori way would be any reasoning one could follow to get to the same conclusion that the feeling suggests. An example is hunger, one may feel it and then derive that he should eat or one might deduce it from other reasons, like counting how many hours passed from the last meal, the calories eaten in the day and so on.\\
In \textbf{4.} I notice that on one hand we have the empirical connection between formal and common terms which is, for example, learning physics in a very empirical way, like having the teacher naming objects we know using technical terms and stating laws on them. On the other hand we have the direct perception of an empirical concept, similar to the intuitive laws of physics I named while talking about 2. The fifth point, on mathematics, will be treated separately.
\section{First Confutation: on Kant}
If we are talking about knowledge it can either be a priori or a posteriori, but if we talk about an object it can be, at least in principle, knowable in both ways; even though, as motivated before, this goes against the traditional use of the terms.
Secondly, it is important to notice the order of the quantifiers in Gödel's examples: first an object is given, a subject and then one considers all possible ways that enable the subject to know the object, this observation will be particularly useful afterwards.
\subsection{Kantian Terms}
Kant most famously used the terms a priori and a posteriori and here I assume that the definition Gödel has in mind when stating the quoted note, is not far away from the Kantian definition. Here follows the piece in which Kant explains what he meant with the two words:
\begin{quote}
Our expositions accordingly teach the \emph{reality} (i.e., objective validity) of space in regard to everything that can come before us externally as an object, but at the same time the \emph{ideality} of space in regard to things when they are considered in themselves through reason, i.e., without taking account of the constitution of our sensibility. We therefore assert the \emph{empirical reality} of space (with respect to all possible outer experience), though to be sure its \emph{transcendental ideality}, i.e., that it is nothing as soon as we leave aside the condition of the possibility of all experience, and take it as something that grounds the things in themselves. (Kant, Critique of Pure Reason, 1787, B 44, emphasis in the text, Translated by P. Guyer, A. W. Wood)
\end{quote}
I underline how in this passage, among many, Kant uses a posteriori and empirical as synonyms, as Gödel does. It is crucial, in the Kantian system, to understand the relationship between a priori/a posteriori predicates on knowledge and analytical and synthetical predicates on knowledge. Here is again a brief passage in which Kant presents them:
\begin{quote}
In all judgments in which the relation of a subject to the predicate is thought (if I only consider affirmative judgments, since the application to negative ones is easy) this relation is possible in two different ways. Either the predicate B belongs to the subject A as something that is (covertly) contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case, I call the judgment analytic, in the second synthetic. (Ibidem, B10)
\end{quote}
\subsection{The Implications}
In Kant there are three possible combinations of these predicates, analytical a priori, synthetic a posteriori and the famous synthetic a priori; I use here these combinations to analyse all interpretations of the quoted Gödel's examples.\\
What Kant claims to be synthetic a priori are some very precise cases he chose carefully and justified each in \emph{\enquote{Systematische Vorstellung aller synthetischen Grundsätze desselben}}. I won’t answer the question if the quoted Gödel's sentences are actually synthetic a priori propositions, instead I examine the two cases and derive the very similar desidered conclusion. \\
If these statements are supposed to be in no way synthetic a priori, then we can directly infer that if and only if such statements are a posteriori then they need also to be synthetical and if and only if they are analytical then they also need to be a priori. Now, given a subject and an object, it can either be that the discovery of the object by the subject gives to it knowledge (it is then synthetic a posteriori) or it doesn’t (it is then analytic a priori), independently from the way he took to acknowledge the object. So, one notices that, if subject and object are given and we analyse the different ways the subject has to discover the object, all such ways together have to be either all analytic or all synthetic (and so either are all a priori or all a posteriori). The only different interpretation that would make the Gödel’s examples right is if we say that the object is given but the subject is not: in such a case he would affirm that an object can be discovered both analytically or synthetically by different subjects, which is an obvious statement (think at the subject that knows the object already or who can infer its knowledge from what it already knows). So, if the examples Gödel’s make are not synthetic a priori, then there’s no hope to make them make sense.\\
What if the examples Gödel does are actually examples of synthetic a priori knowledge? In such a case, one should first reject the \emph{\enquote{For everything}} at the very beginning of the quote, since, even if some among these examples might be synthetic a priori, surely this won’t hold for the discovery of every object. And also, the possibility for the discovery of a given object by a given subject to be (synthetic) a priori or (synthetic) a posteriori won’t be left open for a while, since there are, as precisely described by Kant, only few and very special objects that can be acknowledged synthetically a priori. So, given an object, either it falls in the very special case of synthetic a priori or it doesn’t and it will then be clear that all ways to acknowledge the object will be of the same kind.
\section{Second Confutation: on Mathematics}
First, Gödel's statement seems to easily refer also to logical proofs, since there we could, similarly to mathematics, have an intuition that shows us a certain theorem being true or we may get to the same through little steps of a proof. Therefore when reasoning on such a point, one should before examine precisely what a proof is \emph{to us}. The emphasis is due to the fact that here we are considering nothing but knowledge and therefore I shall not talk about what a proof is independently from us but instead, I shall talk on how we do proofs, and how they appear to us while completing them.
\subsection{On What a Proof Is, to Me}
Consider a proof from a statement $A$ to another statement $B$, then, as Gödel states, we have two different ways to do it, one is just by intuition, if the theorem is very easy or the subject very brilliant, otherwise we’ll proceed with a formal proof. A formal proof consists in finding consequences of $A$, thanks to axioms or previously proven theorems, until we get to $B$, when we get there, we can say that the proof is complete and we got the knowledge of $B$. But now, how do I know that a statement $C$ (which will then bring to $B$) is a consequence from $A$, even in the very simplest case? Consider that a certain proof needs the following arithmetical passage: if $x= 5+7$ then $x=12$. It is very obvious that $5 + 7=12$ but has no reason to be radically different from the complete and harder intuition from $A$ to $B$. If having an intuition on a simple statement would be different from having an intuition on a difficult theorem, where would be the boundary? It would then suggest that a priori and a posteriori knowledge is vague, but, because of what I affirmed on analytic and synthetic statements, we know that a statement either delivers or doesn’t deliver any knowledge and therefore is or isn’t a priori. This implies that, if we consider a priori and a posteriori as being not vague, then mathematics can either be all knowable a priori or all knowable a posteriori. In conclusion, I see proving as \emph{the process of simplifying an implication until the vague and subjective boundary of triviality is reached}.
\subsection{Doubts on Kantian Mathematics}
So, the truth of mathematical propositions are to be discovered either a posteriori, similarly to scientific discovery (this would particularly well fit with the Quinean thesis on mathematics), or be a priori as traditionally believed. Since what is written until now is more than enough to deny the Gödelian statement and reasoning on a priority or a posteriority of mathematics would be redundant to this scope, this question will need to remain unanswered in this essay\footnote{Notice the similarity of this unanswered question with the other unanswered question in \emph{The Frequency of Telling Stories}.}.
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