“Formalizability” in the English Language

It is a peculiar feature of the current English language that the word “formalizable” is nearly non-existent. The word feels awkward. When looking up the equivalent of the German word “formalisierbar” in one of the online-dictionaries I am using, it turned out there was no corresponding English word in that dictionary. However, there was a discussion about it in the corresponding forum (see http://dict.leo.org/forum/viewUnsolvedquery.php?idThread=349924&idForum=1&lang=en&lp=ende ). There we read:


Why are the terms “formalizable” or “formalisable” virtually nonexistent in English?

There seems to be a tacit assumption in our culture that everything can be formalized, at least in principle. It might not be possible in practice to produce formal theories of everything, but in principle, it should be possible; or so the thinking goes. So there is no need for a word like “formalizable”. The “non-formalizable” seems to be viewed as an impossibility. The science-minded part of our culture approaches the world with the claim that everything can be formalized.

However, in mathematics we see that there are entities which can be exactly defined but for which a complete description in terms of a formal theory is impossible in principle. For example, the set of computable total functions is non-formalizable in this sense.

If such entities exist in mathematics then there is no a-priori reason why they should not also exist in physical reality (see https://creativisticphilosophy.wordpress.com/2014/03/09/laws-and-computability/). The claim that everything can be formalized turns out to be a mere hypothesis, but it is treated as a fact. Or worse, for many people it is not even expressed explicitly or thought about consciously. It is treated as an unconscious background axiom.

But I think it is wrong. I would argue (not within this article but within this blog) that human beings, human societies and human cultures actually are such entities for which complete descriptions in terms of formal theories are impossible in principle.

So the words “formalizable” or “formalisable”, “formalizability” or “formalisability” and “non-formalizable” or “non-formalisable” should be there in the English language. Excluding these words from the language creates a blind spot of thinking.

A closer look shows that these words actually do exist, but that they are exceedingly rare. A good tool to investigate this is the Google Ngram Viewer which allows us to plot statistics about the frequency of occurrence of words or phrases in a large corpus of books. Let us start with the diagram for “form” and “formal”:


This diagram is showing that about 0.04% of the single words in the corpus of Google books are occurrences of the word “form”. The word “formal” is rarer in comparison. The words “formalize” and “formalise” exist, but are so rare that their corresponding curves are practically invisible on this scale. If we leave out “form”, the scale changes. We can see now that the word “formalize” is just above the x-axis.

So let us zoom in further by leaving out the word “formal” as well. Instead, we are adding now the words “formalizable” and “formalisable”:


On this scale, the terms “formalizable” and “formalisable” start to become visible, although they are still in the “microscopic” zone just above the x-axis. Let us zoom in further by leaving out “formal”:


To see even more detail, I am starting the x-axis now at 1930 and I am leaving out the verbs “formalize” and “formalise”. Instead, I add the nouns derived from the adjectives: “formalizability” and “formalisability”, and the negated forms of the adjectives: “non-formalizable” and “non-formalisable”.


We find that all of these terms actually exist, though only in the fringes of the language, probably in some special mathematical publications. Leaving out the “most common” term “formalizable” allows us to zoom in even a little bit further:


Historically, the subject of formalizability came up during the 1030s and 1940 when it turned out that a complete formalization of mathematics is impossible. Before this time, there was still the hope in mathematics that this would be possible. The main proponent of this hope was David Hilbert in what is known as “Hilbert’s Program” (see http://plato.stanford.edu/archives/sum2015/entries/hilbert-program/). During the 1930s, starting with proofs devised by Kurt Gödel, Emil Post and some others, it turned out that such a complete formalization of mathematics is impossible.

It is astonishing, though, that the topic of formalizability was largely ignored by the majority of both scientists and philosophers (and by the research area of “artificial intelligence” that developed from the 1950s onwards). I am going to investigate the topic of “formalizability” (and the lack thereof in some areas) further in future articles since I think it lies at the very core of a large part of philosophy and deserves a more prominent status.


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