Is mathematics universal and can therefore be used as a generic form of communication?

Question

Let me preface this question with the following: I have a background in physics and thereby some knowledge on mathematics, but little knowledge about philosophy itself. The question in the title arose from a discussion I had with a friend the other day, which started with this thought: in physics, there is the so-called fine structure constant. This constant has the interesting property that it is just a number, it does not have any units. In other words, how we define a length, a time interval etc, i.e. our choice of e.g. meters and seconds as our units in which we measure the world (which is sort of arbitrary), does not influence the value of this constant.

Moreover, due to its status as a physical constant, i.e. a property of nature, it is universal (or rather, this is one of the fundamental assumptions of modern physics), meaning if we have a measuring apparatus for this constant, it does not matter whether we measure the constant's value here or in a galaxy billions of lightyears away. Let's assume that this universality is true.

The two factors above combined lead to statements such as "if we wanted to communicate with aliens, we should tell them about the fine structure constant". The thought process is that a sufficiently advanced alien civilization would have discovered the fine structure constant as we have, and found it to have the same value due to the above reasons. The question that we have arrived at is whether this is actually true. Is mathematics universal - or natural - in such a way that, even if such aliens were to operate mathematics under completely different axioms or in a completely different way from us (whatever that may look like), could we still "reduce" their and our maths to be "equivalent"? In other words, if they followed a completely different form of mathematics that we have not yet discovered, but their form is consistent and produces the same results, is it really "different" mathematics? (This goes somewhat into the direction of "If I have physical theory A and B, which operate differently but produce the same prediction of observations, are A and B really different?")

I realize that this is a very broad question, but maybe an answer could be a starting point for me to go deeper into this topic.

Answer 1

There was a recent SF film that explored this question called Arrival where humans were trying to establish communication with an alien species that had landed in several metropolitan cities around the world. The situation became urgent as the humans felt threatened when the word 'tool' was mistranslated as 'weapon' ... It's a great film, and I recommend it highly.

The basic structure of a language, as far as we know has a tripartite structure even when it is representing one thought: subject-verb-object. But even more basic in this is naming, the subject, the verb, and the object have to be named.

This is why in the Qu'ran, we have:

And He taught Adam the Names - all of them. Then he presented them to the angels and said: "Tell me the names of these, if you are sincere" 2:31

I also want to add that earliest place where I can see this tripartite structure of language has been recognised implicitly is in Aristotle in his philosophy of change where he noted that for change to occur, a agent is applied to an object and this by contact. He said this in the most general of terms and this is the root of Newtons notion of force in classical mechanics when this philosophy of change is specialised to physical phenomena; furthermore, when this philosiphy is specialised to language we get the tripartite structure I mentioned above. This specialisation is possible because language describes the phenomenal world, the world of change and so must be able to describe change and so references change.

Now, mathematics when it comes to naming only names numbers - one, two, three. These are all nouns. It does not names subjects like you or I or it or human or alien. Nor verbs like fly or kick or move or stop. It's a highly specialised fragment of human language which names a tiny fragment of all the nouns that humans know. It cannot be the basis of communication between aliens. This is just an SF fantasy as there, as one might expect, they fetishise mathematics like in Carl Sagan's Contact.

Answer 2

The foundation of mathematics is counting, and counting is a natural extension of the cognitive process of categorization: i.e., when we are cognizant of 'types' of things, it begins to make sense to ask how many of a given type exist. Whether the cognitive process of categorization is a universal of sentient intelligence and not just an aspect of human sentient intelligence is an open question. It seems reasonable enough, but then it would, wouldn't it?

So far as I'm aware, all theorems and practices in mathematics are abstractions of various counting problems, so in that sense mathematics as a whole might constitute a universal. I'm not sure I'd go with the fine structure constant when there are easier unit-less numbers to put forth: 1, √2, 𝛑, etc. And mathematics is an extremely limited language; I see no way in math to ask alien visitors if they would like a cup of tea. But math is usable, perhaps, for the single purpose of establishing that we are intelligent, sentient lifeforms, and not (say) snack food.