Why do many philosophers state their arguments without using mathematics or formal language?

Question

I am an amateur lover of philosophy and a researcher in physics and computer science. When reading a book of philosophy, I always find it frustrating that philosophers are so polysemous and ambiguous in their arguments.

Of course, I believe that there are philosophers, especially in analytic philosophy, who try to describe their theories by symbolic logic. However, I feel that the idea that philosophical claims must be described in a formal language such as mathematics is not mainstream.

Formal language such as mathematics is supposed to be a powerful tool for rigorously asserting claims. In fact, in theoretical physics, researchers are virtually obliged to describe their claims by mathematical formulas. Why are philosophers not compelled to make claims in formal language?

In my opinion, writing philosophy's claims by means of formal language contributes to removing ambiguity from the theory.

In physics and computer science, interpretations of previous studies are not considered academic research. Because mathematical statements remove the ambiguity from the claims, it does not occur in principle that a well-trained researcher would have trouble with interpretation. If Einstein had described general relativity only in natural language, it would have taken more time to decipher his ideas.

In philosophy, on the other hand, many researchers devote their lives to interpreting the arguments of past philosophers. Wouldn't posterity be pleased if contemporary philosophers made an effort to describe their own arguments using mathematics and formal language? At the very least, I think that a philosopher's effort to use formal language will contribute to clarifying the range between what she has a rigorous grasp of and what she only vaguely understands.

Moreover, I often experience that efforts to translate qualitative ideas of physics into mathematical formulas contribute to the refinement of my own immature ideas. For professional philosophers, does the effort to express their arguments in mathematical or formal language not contribute to the refinement of their theories?

My hypothesis as to why philosophers do not describe their theories in formal language or mathematics is as follows:

1. claims in philosophy cannot be expressed in mathematics. In particular, it seems difficult to define concepts that appear in philosophy using only mathematical language.
2. the community interested in philosophy is more familiar with natural language than with mathematics. Therefore, even if ambiguity remains in the theory, communication in natural language is superior in terms of convenience of communication.
3. expansion of theory through misinterpretation is also an important activity to enrich philosophy.
4. the claims of philosophy written in natural language are not ambiguous. The reason they look ambiguous is because readers (like me) are not well trained.
5. efforts to describe philosophy in formal language have already begun.

Answer 1

Main Answer
When I write something down and then show it to someone else, my intention is to communicate an idea I have to them. Therefore, the method of writing that communicates my idea the best is the method I should use. If natural language outperforms formal language in this regard, then I should be using natural language. The reason so many philosophers use natural language so heavily is simply because it communicates better. This is rather simply to demonstrate.

Let's say I wish to communicate an idea. In natural language, this idea is expressed as "Each human should act to maximize the fitness of all humans, where fitness is the number of descendants an individual has." Think about how you would express this idea in a formal language. Now, which do you think would communicate my idea better, the natural language version or the formal language version?

P.S.
A philosopher whose ideas could be best communicated with formal language would be called a mathematician. By dividing these two groups into distinct categories rather than having one a subset of the other, you unintentionally 'rigged the game'.

Answer 2

Your fifth point is in fact correct, for example Ed Zalta (of SEP fame) uses Isabelle ($\textit{inter alia}$ other interactive theorem provers) to show/verify theorems on abstract object models.

But your other points are not necessarily off either. You say you are a computer science researcher, so recall your first automata theory course. It is likely that the proofs regarding Turing machines were actually quite handwavy- not formal at all! Indeed, a quick look at all the modern first year texts (Kozen, Sispser,etc...) reveal this to be the case. Why was this so? The typical pedagogy is that the formality actually obscures the main ideas of the proof, and further that the formality would be (fairly easily) obtained by a competent user who understands the high level ideas. Further it is quite tedious to prove any of the many induction lemmas needed to establish one formal proof about Turing machines- it would slow down the course. In fact, math is just the same -many proofs in mathematics have yet to be formalized all the way back to axiomatic set theory (or your foundational language of choice). So it is with philosophy- with the added conundrum that there is no widely accepted foundational language, say ZFC.

Lastly, misinterpretation is arguably an important activity- perhaps Derrida would argue something along these lines.

Answer 3

The language of mathematics, physics and computer science is ordinary language and not numbers, or symbols. These are the tools of the trade and not the trade itself.

Although since the advent of category theory some infatuated mathematicians have said the language of mathematics is category theory, really all this means is that many constructions that had been used in abstract algebra and the like had been using techniques which at that point had not been adequately formalised.

Moreover, language is already symbolic. The word 'chair' is a symbol for the artifact chair. It is also formal, there are rules of grammar and spelling. In fact, the first fully developed axiomatic system was likely not Euclid's but Panini's formal grammar in ancient India. Mathematics is not more symbolic than language, they are at the same symbolic level. In many ways, less. How does one say 'chair' in mathematics?

Euclids treatise, written over two millenia ago, is written mostly with words. The famous paper of Einsteins introducing special relativity is mostly written in German.

As you mention analytic philosophy, I would remind you (supposing that you have read it) that Wittgenstein's Tractatus Logico-Philosophicus, a founding text of analytic philosophy and of model theory is written mostly in English apart from paragraph numbering and one very brief formula.

I would put it to you that your infatuation with mathematics and computer science has narrowed your outlook. In fact, Simone Weil, a Marxist and a philosopher, observed that many technical disciplines produced remarkably virtuosic works in their fields but their capacity of thinking more broadly and deeply appeared to have atrophied.

A case in point: A young woman of my acquaintance told me a couple of days ago that she was interested in logic. I thought this very funny as she has never displayed the slightest interest in logic or even philosophy before. So I asked her what's the difference between propositional and first order logic. And she replied, not that kind of mathematical logic but ordinary common-sense logic. I told her that's a better answer than most people on this site can manage who ostensibly have an interest in philosophy but appear instead to be infatuated by mathematics. She thought that very funny too.

Answer 4

However, I feel that the idea that philosophical claims must be described in a formal language such as mathematics is not mainstream.

My initial reaction to your question was to suggest something about Wittgenstein's Tractatus Logico Philosophicus and Philosophical Investigations. Formal languages, while useful tools for making the ontology of their users transparent, have an upper bound on their capacity to explain. For Wittgenstein, a purely formal philosophy must take a position of eliminativism with respect to those areas of life that are not within the bounds of formal explanation. Yet at the same time, these areas are ubiquitous within natural language use - if formal philosophy struggles with even foundational ideas like "how to recognize patterns and consistently apply rules", then at the very least, it cannot be the whole picture.

But on rereading the quoted section, a more salient point might simply be that to talk about the "mainstream" would render an answer to your question that isn't covered by any of your proposals - formal philosophy is incredibly dry, and while clarity is a key virtue in philosophical communication, so too is being able to write in a way that engages and motivates the reader. As Hume might have said about this, when it comes to the world of published writing, Logic is a servant of passion, not its master.

I don't think formal clarity is a detraction from engaging writing that develops the discussion of the issue at hand, but the activity of formal explication as a tool within the philosophical text is a diagrammatic component to serve the narrative of the piece, rather than the substantial core of the labour at hand; at least, certainly in "Mainstream" writing.