Galileo gave the metaphor that the natural world is written in the language of mathematics, but is mathematics even a language?
It is more than that. Even if we take the Galileo's metaphor literally, he is suggesting that there is a language of mathematics, specifically geometry, not that mathematics, as such, is a language:
"Philosophy is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it..."
There are languages of physics, art, jurisprudence, etc., but they are not themselves "languages". Whether we take the dictionary description of "language" as a method of communication, or as a system of symbols for it, mathematics is not just that. It is also a discipline, a structured practice in human communities, "a motley of techniques and proofs", as Wittgenstein put it. Even on purely linguistic aspects it is fair to say what would not have been apparent in Galileo's time, that there are multiple mathematical languages. Euclid summarized one of them, which dominated until 17th century, although Diophantus, Indians, and Islamic algebraists made significant additions to it before Cardano's and Vieta's transformation. Today, despite the efforts of Russell and Bourbaki, languages of mathematics as spoken are not parts of a single universal tongue, say the first order language of ZFC set theory, although large portions of them are translatable into it. There are alternative tongues too, like the category theory or constructivism.
But Galileo meant more than a language in his metaphor, invoking the medieval notion of enchanted nature, the Book of Nature, through which God is first known. He had in mind a particular philosophy of language and reality, where Nature was indeed written for human reading, and mathematical notions faithfully reflected their hidden counterparts in Nature, to be uncovered by "interrogating" it experimentally. This idea, albeit with a different method of discovery, shows already in the Pythagorean "everything is a number", and has modern supporters like Tegmark, who echos Galileo in telling us that "external reality is [not "is described by"!] mathematics (more specifically, a mathematical structure)", see How can the physical world be an abstract mathematical structure?
But there is also an alternative to this Pythagorean/Platonist version of Galileo's dictum, which developed since Kant, a phenomenalist one. While the idea that human, all too human, mathematics underwrites reality "in itself" was non-sensical to phenomenalists, that mathematical languages are uniquely suited for expressing our experience of it, because they line its very texture, seemed far more plausible. Peirce and Husserl, two turn of the 20th century philosophers who were mathematicians by training, and stood at the very origins of the modern analytic/continental divide, developed this idea largely independently of each other. According to Peirce, all reasoning is diagrammatic (broadly construed), and mathematics is the science of pure diagrams (the vast expansion of Kant's schemata), it thus becomes the first philosophy, and the source of scientific structures. Early Husserl held a similar view, of mathematizable a priori structures revealed in categorial intuition of experience, but he changed his mind between the two editions of Logical Investigations. As Stjernfelt comments in Diagrammatology:
"...in the first issue, it was seen as a major phenomenological task to describe vague forms in exact mathematical language – in the second issue, this task was given up in favor of the idea of ‘vague essences’ in experience which are assumed impossible to map mathematically. Why Husserl gave up that central idea is hard to say – it has not necessarily any inner connection to the well-known ‘transcendental’ turn..."
Even earlier life philosophers, like Dilthey and Bergson, already disputed that mathematical or natural scientific language is adequate for expressing lived experiences. Analytic tradition went with Peirce and neo-Kantians in affirming a phenomenalist version of Galileo's dictum, while continental tradition went with Husserl and life philosophers in restricting the scope of Galileo's Nature and its language, to exclude (at least) ethics, arts and humanities. It is interesting however that in his last book, Pursuit of Truth, Quine is reluctantly acquiescing to some such restriction:
"I conclude that the propositional attitudes de re resist annexation to scientific language, as propositional attitudes de dicto do not... Still the mentalistic predicates, for all their vagueness, have long interacted with one another, engendering age-old strategies for predicting and explaining human action. They complement natural science in their incommensurable way, and are indispensable both to the social sciences and to our everyday dealings."
To appreciate this passage fully one should keep in mind that to Quine "indispensable" signifies an ontological committment, that was why he admitted mathematical sets and numbers themselves into ontology after his early nominalist days, see Does Quine's dissolution of the Analytic/Synthetic distinction challenge mathematical realism?
In mathematical logic, we have a formal definition for what a language is:
A given formal language has the following primitive symbols.
Individual variables. A, B, C,..., Z, A', B', C',...,Z',A'',....
Logical symbols. I'm not sure how to format them on this site, but they are the logical operation symbols or, and, implies, not, if and only if, and the logical quantification symbols for all, there exists, as well as = and (,).
Operation symbols. These vary from one language to the next in number (perhaps there are even none in a particular language), shape, and rank. With each operation symbol there is an associated natural number 0, 1, 2, 3, 4 or 5 called its rank -- we do not need the properties of all of the natural numbers for this purpose, only the first 6. An operation symbol of rank 0 is called an individual constant.
Relation symbols. These vary once again from one language to the next, and each has a rank that is a positive natural number 1, 2, 3, 4 or 5.
No other symbols than these are allowed, and the specification of these symbols along with their ranks entirely determines a given formal language.
We do this primarily because lazy notions of language in combination with logical manipulation can give rise to paradoxes. Russel's paradox is a classic example of this -- in English (or any other language in the classical sense), I can ask you to 'consider the set of all sets which do not contain themselves'. This object doesn't really seem to 'exist' in any sense outside of my ability to ask you to consider it however, since it has properties about it that are 'true' and 'false' in the Tarskian sense at the same time (it is a member of itself and not a member of itself, and one implies the other).
Formalizing language and axiomatically developing our ability to 'form sets' (formalizing how we are allowed to collect together objects in our head and think about them), we avoid problems like this -- this was the original goal of set theory as set out by Zermelo. In my opinion, these languages are the really interesting ones and the ones that allow for robust and rigorous logical exploration -- any good definition of what a language is should include them.
This is a fairly common trope, however when we look at the history of physics we see that physics was first theorised without mathematics, ie Aristotles Physics.
So it's quite possible to do physics without mathematics, but perhaps not now given how much of physics is written in mathematics, but one should not then make the mistake that physics is somehow reducible to mathematics.
In the modern era, mathematics was introduced into physics by Galileo, so perhaps it's not so strange that Galileo would make such a remark.
As to whether mathematics is a language, well it should be obvious it is not, despite the fact that there such notions as a language in mathematics as well as syntax and semantics; they've borrowed these notions from linguistics, and in the act of borrowing transformed them, they have only a tangential relationship with the original idea, enough so that can see why the borrowing was made, and why it is useful; but not enough so that the original meaning can be retained.
So, No. Mathematics is not a language.
The definition of language is "a body of words and the systems for their use common to a people who are of the same community or nation, the same geographical area, or the same cultural tradition" according to dictionary.com.
Breaking this down, a list of requirements begins. There are four definitions, but each of them have similar requisites:
1) Symbols or words
2) System for using these symbols or words
3) Used by a group
Should the definition include the ability to communicate? Human language is used to pass thoughts. Programming language is used to pass electrical impulses. Body language can be used to interpret unspoken emotions and thoughts.
Language typed into Google does add communication to the mix.
Next, compare mathematics to the above requirements:
1) numbers are symbols
2) operators and rules are a system for using
4) does mathematics allow communication?
Another way to state the fourth requirement is this:
Does a comet communicate with humans using mathematics? Does a comet communicate it's trajectory with the Sun? Do humans communicate with other humans using only math? Do humans communicate with the comet using mathematics?
We can rule out the possibility that the comet is communicating, which leaves the question of human to human communication using only mathematics.
Since humans must express mathematics in a language, I find it difficult to suggest that this can be the case. Specifically, two is a word in the English language. I can not communicate even as simple an equation as 2+2=4 without using English words. Even if someone holds up fingers to express this, I must translate to the English language for this to make sense to me.
My opinion is that mathematics is not a language. I suggest Galileo was expressing the importance of using mathematics to explain the world.
It is indeed often argued that "mathematics is the (programming) language (the) God(s) used to describe the universe". While that view is definitely not without merit, three considerations need to be made:
- Mathematics is best described as the "study of topics such as quantity, structure, space, and change". While certain subsets of mathematics meet all the criteria for a language, the entire field of mathematics transcends it and we would be oversimplifying things by arguing that mathematics as a whole is a language.
- In spite of all its elegance and refinement, any scientific model built on top of mathematics will always remain "just" a model... which means that any such model will always represent "just" an approximation of the actual behavior of the universe itself. One can argue, however, that natural languages are not dissimilar in them being incapable of consistently and unambiguously describing every human thought or emotion.
- In light of phenomena like emergence, the notion of the universe as a "computer program" written in a "mathematical language" does not imply a "programmer" per se.
Yes, as human beings communicate with each other by means of natural language or as by manifesting and projecting their mental plane via language, nature or physical world interacts via a language which is called mathematics. Physical laws manifest/project in form of mathematics. Simply, mathematics is the language which encodes everything which is existing in material plane!