Is Philosophy formalisable?

Question

This is something that has irked me for quite some time, especially since I come from a mathematically oriented background.

Can the field of Philosophy be formalised in the sense that Mathematics is formalised? It doesn't necessarily have to be an axiomatic formalisation, but can we find some universal agreement? If not in the conceptual framework, perhaps in the methodology?

My main concern in raising this point is, if there is no logical structure holding the field together, how can we know for certain when progress has been made, or even what progress is?

Forgive me if this feels a bit off-kilter to the seasoned vets out there, I am relatively new to Philosophy and mostly do it as a hobby.

Answer 1

I think it depends a lot on what branch of philosophy you are talking about, but ultimately, the answer is pretty consistently no.

In epistemology, the study of knowledge itself, you're much more likely to get into the nitty gritty formal logic. A lot of this branch is studying what we can and can't know, and how different lines of reasoning work; this lends itself pretty easily to being formalized. However, even in the formalization there is no single universally accepted methodology - classic logic vs. intuitionism is one major point of debate, for example.

Metaphysics is the study of what exists (but not to be confused with ontology, which is much more focused). It can occasionally be formalized, as in Aristotle's Categories, but this does not much resemble formal logic. You're still likely to see reasoned arguments in metaphysics, but the methodology has been debated for basically the entire history of philosophy. This is a big part of why philosophers have such differing views on metaphysics.

Then, there's ethics, the study of morality, right and wrong. Here is where things get really squishy, with endless different views (all arrived at with different methodologies, which are very dependent upon the subject's view of metaphysics and epistemology). Thus, even on the most fundamental of questions, you're unlikely to be able to find a universal common ground. Ethics is also virtually impossible to formalize - realistically, how would you formalize moral arguments? Utilitarianism perhaps can be counted as an attempt at a mathematical formalization, but it's not exactly widely accepted.

So, in answer, philosophy (at the moment) does not seem formalizable. There is no universal agreement (cogito ergo sum being a point of debate among many, same with the crucial-to-many Law of the Excluded Middle), and you'd be hard put to establish one. Maybe some day...

EDIT: The following is a response to the asker's question:

Should a formalization not form a common ground for agreement?

My (fairly speculative) answer is yes, but only if the formalization itself is universal. The issue is that many have a fundamental disagreement as to how we should actually formalize things.

At the root of all formalization (I think) is a logical system. What everybody agrees on in our logic system really only amounts to symbols: you can find a list here, and you may recognize some if you've studied any number theory.

However, the universal agreement ends there, without any real progress having been made. Even the application of the symbols themselves are debated. I gave the example of classical logic versus intuitionism; this is a very important point, as it represents disagreement at the most fundamental level, where agreement is necessary to attempt to formalize things.

Classical logic holds that:

¬(¬A)=A

This is a very important principle: if you can disprove ¬A, you have proven A. However (my symbolism may be off, please do correct me if it's wrong), intuitionism does not hold the above statement to be true. Even if you have ¬(¬A), you cannot assume A.

It's small but very significant disagreements like these that have led to the massive scattering of schools of thought; if you were able to get them all to agree on a single formalization (a massive feat in itself), that would bring them to a universal agreement. However, I don't see it being possible to formalize without universal agreement to begin with. I would say that your question is valid, but not sound - we cannot first formalize without having agreement, but if we could, we would establish agreement itself. It would be like getting John Stuart Mill and Plato to agree on how to logically represent their systems of ethics, when the two are so different.

A more mathematical analogy would be agreeing to use the same formalization for binary and decimal: you would get 1+1=2 and 1+1=10, and without any unique formalization for the base ("system of thought") nothing would look right from the other's point of view.

Just speculation on my part.

Answer 2

I think commando has this covered nicely, but I do want to add that strictly speaking Western philosophy and its categories represent (this is contentious) only one culturally distinct view on how to go about philosophy. Other civilizations besides the Anglo-European had/have systems of ethics and ways of knowing that continue to vex Western categorization. Arguably, it is an act of colonialism to classify another culture's way of exploring, interpreting, and explaining reality by our (albeit flexible) standards. Parallels and easy groupings are found, but more often disparities and incommensurabilities.

Likewise, since no particular thinker from any particular culture is ever unanimously agreed upon to have been false or foolish, progress in philosophy is not terminal to predecessors in the way that it often is in science. Progress, for example, is being made right now in Aristotelian ethics - right alongside progress in utilitarian ethics.

If you are interested in the notion of progress as it pertains to western knowledge, I'd recommend delving into the philosophy of science. Thomas Kuhn's The Structure of Scientific Revolutions may be a good place to begin, as the concept of progress is central (and not just scientific progress).

Answer 3

It has been noted in the preceding answers that attempts at formalization were made in the first part of the 20th century. This isn't really news in philosophy, for centuries philosophers tried to import the rigor found in the mathematics of their time into philosophy by providing a mathematical treatment of philosophical problems. (Think of Spinoza's Ethica more geometrico demonstrata!).

There is currently a new attempt by philosopher and mathematician Hannes Leitgeb to reinvigorate this style of doing philosophy, without the reductionist pretenses of the past, called Mathematical Philosophy:

Mathematical Philosophy - the application of logical and mathematical methods in philosophy - is experiencing a tremendous boom in various areas of philosophy. At the new Munich Center for Mathematical Philosophy, which is funded mostly by the German Alexander von Humboldt Foundation, philosophical research is carried out mathematically, that is, by means of methods that are very close to those used by the scientists.

The purpose of doing philosophy in this way is not to reduce philosophy to mathematics or to natural science in any sense; rather mathematics is applied in order to derive philosophical conclusions from philosophical assumptions, just as in physics mathematical methods are used to derive physical predictions from physical laws. Nor is the idea of mathematical philosophy to dismiss any of the ancient questions of philosophy as irrelevant or senseless: although modern mathematical philosophy owes a lot to the heritage of the Vienna and Berlin Circles of Logical Empiricism, unlike the Logical Empiricists, most mathematical philosophers today are driven by the same traditional questions about truth, knowledge, rationality, the nature of objects, morality, and the like, which were driving the classical philosophers, and no area of traditional philosophy is taken to be intrinsically misguided or confused anymore. It is just that some of the traditional questions of philosophy can be made much clearer and much more precise in logical-mathematical terms, for some of these questions answers can be given by means of mathematical proofs or models, and on this basis new and more concrete philosophical questions emerge. This may then lead to philosophical progress, and ultimately that is the goal of the Center.

Have a look at the activities of the Munich Center for Mathematical Philosophy. Their work is really impressive.

Answer 4

Formalization of philosophy was a primary goal of Russell, the early Wittgenstein and the logical positivists. They applied formal systems of logic to a wide range of philosophical propositions and arguments and on the whole this led them in the direction of logical empiricism, a form of reductionism of sentences to a 'scientific' vocabulary (indeed, the logical positivists regarded themselves as scientists, not philosophers). From the 1940's onwards this approach to philosophy was widely abandoned as hopeless. Reductionism simply cannot be carried out as has been shown by Quine, Sellars, Hempel etc.

The tradition of the formalization of language continues however, in the works of Tarski, Quine and Davidson and their successors, and one may say some kind of progress has been made in formalizing large parts of natural language. However formalization does not lead to universal agreement. For how does one choose the 'correct' formalization among different logics, models and notations? Formalization is as underdetermined as any scientific theory in need of evidence. Hence the later Wittgenstein and others such as Austin and Ryle argued that formalization cannot be the ultimate goal of philosophy. Rather, clarification and perspicuity are its ongoing and legitimate concern.