I wrote this essay during the1st_Semester at theUniKonstanz for the seminar on “Philosophical Method” by Sam Roberts.
Abstract
First I make clear what I mean with the codomain, field and type of a method, in particular using science as an example and considering some assertions by Quine and Popper. I’ll then compare it with what one could regard as the codomain of philosophy. I’ll focus on the reasons why one is reasonably confused when talking about the codomain of philosophy and propose a solution with particular emphasis on its limits and approximations.
code
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\title{The Method's Codomain}
\author{Simone Testino}
\date{January 2023}
\begin{document}
\maketitle
\begin{abstract}
First I make clear what I mean with the codomain, field and type of a method, in particular using science as an example and considering some assertions by Quine and Popper. I’ll then compare it with what one could regard as the codomain of philosophy. I’ll focus on the reasons why one is reasonably confused when talking about the codomain of philosophy and propose a solution with particular emphasis on its limits and approximations.
\end{abstract}
\section{Introduction}
This essay is an attempt to define some tools that will help us in understanding what a method is and, even more importantly, how it relates to other methods. In the next section I’ll explain more precisely what I mean with a method, though it’s also important to notice under which circumstances I am considering a method. I’m looking at a method at its earliest stage, when I’m trying to figure out what I \emph{want} such a method to be. This makes a difference since those tools that I’ll call the field and the type will be particularly useful in understanding how to develop a method in order to make it work how I want it. Particularly it’ll be easier to take aspects of different methods (like the field and the type) and mix them together to get a new one, and I’ll show some examples of how this happened already in some subjects.\\
The next section will then show more precisely what I mean with a method and make an example with science. I’ll then get deeper in that example recalling theories by Popper and Quine. The second section will instead focus on philosophy. That section will say little more than nothing on what I regard the method of philosophy to be, but instead it'll focus on how we can analyse it and compare it efficiently with other methods. I’ll then open a discussion on the relation between the mathematical and the philosophical method and take in consideration some methodological assertions by Wiliamson that will fit particularly well with the presented procedure for analysing methods.
\section{Codomain of a Method}
The key concept of this paper will be the codomain of a method and here I shall explain, as more precisely as possible, what I mean with this term. It comes from mathematics and there it denotes all the values a function can give as outputs. Though, what I mean here with codomain is a connected but slightly different concept, since I’m not referring to functions but instead to methods. If we think at a method as being like a complex function that takes some inputs (like observations for science) and gives some outputs (scientific laws) it becomes natural to think at the codomain of a method as being \emph{the set of all propositions a method can determine to be true or false}\footnote{we might differently see at a method as a function taking certain proposition and mapping them to truth values, though the idea of the function I presented fits better with the content of the essay.} This feature is one of the very key ones for describing methods. If we think about the main differences that the methods used in medicine, mathematics or science have, they all seem (at least at a first glance) to be caused by a substantial difference in the propositions they want to claim. So, reasoning on the domain of a given method is reasoning very deeply on what a certain subject is about, what it is able to assert as true and what we can actually pretend from studying the subject. For example, one cannot pretend from a scientist (who is actually doing science) to have a theorem proved necessarily as K. Popper famously showed, or, similarly, one cannot expect a phenomenological method (like the most modernly used in psychology or some fields of medicine) to give a scientific falsifiable law. These last examples show that, in examining different codomains of methods, we notice not just differences based on different \emph{fields} (like human body or necessary statements on numbers) but we also have different \emph{types} of propositions, like being merely probabilistic or eternally necessary.
\subsection{The Codomain of Science}
For such important and successful methods as the scientific one, it is not that easy to affirm which codomain it has. As shown in the previous section, we shall separately consider the field of science and then its type to have a more precise idea on its codomain. Science will here be used as an example to show more precisely what I mean with these terms, in the next section I’ll use them to talk about philosophy.
\subsubsection{The Field of Science}
In the most classical and popularly believed version of the method (here I may even refer to Galilei's version of it) the field is nothing but \enquote{physical objects}, in the relatively simple materialistic terms of the time. The core of this idea has remained the same in the later centuries, one never pretended from science to actually talk about things that are said to be not physical. An example of a proposition on which science should not give any truth values is $2+2=4$ or that god (considering it as being a non-physical entity) exists. On it two observations should be noted, first that it is legitimate to say that the existence of God or a mathematical statement are somehow seen to be more or less compatible with a given scientific theory, though it should not be counted as an implication, scientific method wouldn’t be capable of expressing a statement on it. It is also very relevant to note that the two examples (existence of god and $2+2=4$) are very different since the relation science has with mathematics is very different from the one it has with theology, mathematics is said to be the language of science, and therefore it is seen as being one of its grounds. A recent and deep investigation on the relation between scientific method and mathematical statements, conducted by W. V. Quine, changed radically the role science plays in mathematics. These thoughts are collected under the name of \emph{Naturalism} and, in the terms of this essay, it expands the field of science to (at least) the whole of mathematics. The most iconic and quoted formulation of naturalism is the following:
\begin{quote}
Naturalism is the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described.\footnote{https://plato.stanford.edu/entries/naturalism/}
\end{quote}
Without going deeper into the debate on this\footnote{More on it here The Oxford Handbook of Philosophy of Mathematics and Logic, S. Shapiro, OUP, C. 12-13}, it is just important to note, for the sake of argument, that what Quine argues is that the field of science should be expanded a lot.
\subsubsection{The Type of Science}
Reasoning on the type of science means to discuss which kind of truth values does science pretend to express. In previous introductory examples I showed already how it evidently differs from mathematical and phenomenological truth values, here I’ll go slightly deeper into the argument. The most iconic voice on this topic is surely the one of Popper which made very precise, clear and surprisingly intuitive what the type of a scientific proposition needs to be. In \emph{\enquote{The Logic of Scientific Research}} he exposes the idea that any scientific sentence needs to be falsifiable, meaning that it has to be possible for it to be proved false. Examples that Popper makes of theories that are not scientific are marxism or psychoanalysis, nor of them allows an experiment to make them undeniably false: marxism claims that a revolution is necessarily to come, without any way to know when it will happen; similarly psychoanalysis doesn’t foresee anything precisely on the behaviour of the patient, he could always be an exception. If a mathematical proposition is falsifiable, is also not a very easy question, since one could actually make an experiment that verifies if $2 + 2 = 4$, though it is a statement that is often seen as given, as explained in the previous section. These thoughts show how a change in the type of propositions of a method modifies the field of the method.\\
These last two sections should have made clear what I mean by field and type of a method through the example of science, now I’ll use these concepts to try to better understand the philosophical method(s).
\section{The Codomain of Philosophy}
Reasoning on the codomain of philosophy may be confusing and there are two main reasons for it. (i) There is no well known and fixed method like in science, therefore it’s way less clear what we mean with \enquote{philosophical method}. (ii) If we think of topics (field) on which philosophy pretendes to say something we find many, from the existence of numbers to determining what’s morally right or wrong, or studying the features of the soul and so on… These two points make not just talking of the codomain of philosophy very confusing but also talking about philosophy in general, since, for understanding what a method or even a subject is, one should surely define (or at least describe) its codomain as field and type. Since the codomain of philosophy seems to be so broad, a good technique may be to find something we are sure, philosophy cannot determine.
\subsection{The Negative Codomain}
The negative codomain is a useful tool when trying to find the codomain of a method that seems too vague, one should ask: what can’t this method determine at all? In this way one should find areas that are somehow occupied by other subjects and then find out if the subject we are considering comprehends them or not. Let’s do another comparison with science to make things clear: Heisenberg, in a book called \enquote{\emph{Physics and Philosophy}}, states that physics should regard whatever is physical (field) and its laws have to be formulated in a pure logico-mathematical system (type). Then he studies the overlaps of physics with other sciences and notices that chemistry has, from a century or so, a very similar type and a field that is just a bit smaller, concerning not all matter but just some properties of it. Then he can state that chemistry is a \enquote{submethod} of physics: every chemical proposition is also a physical one. When facing the same problem with biology, he realises, at least at the time, that the type of biology and of physics are different, biology doesn’t pretend (yet) to build axiomatic systems, therefore biology can’t be seen as a part of physics, even though its field is just a part of the physical one.\\
Now let’s come back to philosophy, which propositions are not philosophical at all and which overlaps do we notice between philosophy and other subjects? Let’s forget for a moment the field of philosophy, which we saw in (ii) being very confusing, so, what about the type? Type of philosophy is surely not the falsifiable scientific one, philosophers don’t do experiments and don’t love to hear arguments falsifying theories through experiments. Thought experiments appear often in philosophy, for example in Kripke, \emph{Naming and Necessity} or \emph{Locke Lectures}, though they seem to be radically different from scientific experiments, and this difference will be a key for distinguishing the philosophical and scientific type. We could, through this similarity, say that \emph{philosophy is the science that can be falsified by no more than thought experiments}. I regard this as a very interesting comparison that gives us two important informations about philosophy: (i) philosophy is not a science since thought experiments are intrinsically different from empirical experiments (more on it in the second point) and (ii) what is verifiable by no more than thought experiments needs no actual sensible experience and is therefore a knowledge a priori (unlike science). Saying that philosophy regards no more than the a priori is a strong statement, probably too strong to be actually true, though it seems to me that almost every discourse in philosophy, if is not completely talking on a priori assertions, is somehow considering the relationship between something a priori and something a posteriori. If we regard independence from sensual experience to be the key type feature of philosophical propositions, I shall say something more specific on this term, its ancestors and its actual meaning
\subsection{The Myth of Episteme}
Episteme is a word coming from the verb \textgreek{ἐπίσταμαι} (epistemai), the word can be splitted into two parts, \textgreek{ἐπί} (epi) meaning \enquote{above} and \textgreek{ἵστημι} (istemi) meaning standing, staying. The literal meaning of the word is then \enquote{what stands above} representing something that independently from all changeable things on the earth stays still unchanged in the sky and, at the time, meant \enquote{science}. This concept has been crucial in platonic philosophy but a very similar concept is also at the centre of the Meditationes by Descartes and many philosophical discourses regard directly or indirectly the episteme (more in analytical than in continental philosophy, since the latter became more focussed on the nature of man and culture). One example may be the mind-body problem, where we see how every argumentation, even if strictly related to experience, still focuses on the concept itself of perception which is a feature so deeply human that we can surely regard it as being a priori. What I mean then when stating that the type of philosophy is that its propositions have to be necessary is not more than an approximation, giving no particularly precise notion of necessity at all. Though this imprecision, this should clear much of the confusion present at the beginning of this section.
\subsection{The Relation between Mathematics and Philosophy}
As seen at the beginning of this section with the example of physics and chemistry, I call a method to be a submethod if and only if their type is the same and the field of the submethod is a subset of the field of the method. In the previous section I motivated why philosophy should have necessary propositions as a type, though, this could be said for mathematics as well. The kind of necessity that mathematics requires from every of its theorems and the one philosophy, in particular contemporary analytical philosophy, requires from its propositions, seems not, at least to me, to be that far apart. So, let’s say that the type of the two methods are very similar, is the field of mathematics a subfield of philosophy? Could philosophy be to mathematics what physics is to chemistry?\\
To answer this question we need to see if there are some boundaries on the codomain of philosophy and, secondly, if the codomain of mathematics is inside that codomain. It would be a long discourse to motivate in more detail why philosophy seems to have no boundary at all in its field, but here we may take it just as a traditional notion, it is so by tradition that philosophy can assert necessary propositions on every topic. So in these terms, mathematics, no matter its field, is said to be a submethod of philosophy.
\subsection{Example: an Application of the Tools}
We see a discussion between Wiliamson and Anton Kuznestov\footnote{Williamson, Timothy. The Philosophy of Philosophy, John Wiley & Sons, Incorporated, 2021.} in which they argue in different ways for the differences of the mathematical and philosophical method. While summarising the discussion, I’ll divide it into points that will be useful in order to analyse it through the mentioned tools afterwards. (I) First Wiliamsonargues that philosophy is different from science, mentioning its similarities with mathematics. (II) On the other side, Kuznestov argues for the intrinsic difference of these two methods, since philosophy misses the knowledge of the basic principles he claims that mathematics needs to have in order to proceed, and traces thanks to this an essential difference between the two methods. (III) Then Wiliamson gives an example of a basic principle of mathematics that is unknown to us, namely the Continuum Hypothesis, concluding that, even if philosophy and mathematics have differences, they have not to lie in the limits we have in them.\\
Before even starting to talk about the methodology and how to apply the tools I previously developed, I want to argue against step III. The notion of \enquote{basic law} is surely very vague, though a good argument against this step might be the following: if we consider basic laws of a system as being those that allow us to do computations and derive theorems from them, this is surely not the role of the CH. The process that brings us to question the CH is radically different from the one of the basic laws that make the mathematical practice computable.\\
Let’s now focus on the two arguments by Kuznestov and Wiliamsons. First we see that when Wiliamsons underlines in step I the similarities between mathematics and philosophy he concerns surely the similarity of type I also underlined in the previous section. In step II then the type again is the core of the discourse, since Kuznestov claims it to be founded by basic laws that philosophy doesn’t have. We see then that this discussion regards mainly the type of philosophy compared to the one of mathematics.
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