Stuff like this always brings to mind Borges’
Tlön, Uqbar, Orbis Tertius, in which, among many other strange things, we are encouraged to believe that nine equals one.
Suppose we lived in a world with radically different physical properties, such that, for example, when two objects came in close proximity, a third object materialized. Could it be said of such a world that one plus one equals three? If so, could a self-consistent arthiemtic and maths be derived from this?
This also brings to mind a
chapter from the book “Beyond Experience: Metaphysical Theories and Philosophical Constraints,” by the philosopher Norman Swartz. The relevant passages are here, the topic being that we should not assume, as is so often done, that we ourselves, and intelligent aliens, could use mathematics as a basis of communication:
Must one have a concept of mass, for example, to do Newtonian mechanics? We might at first think so, since that is the way it was taught to most of us. We have been taught that there were, at its outset, three ‘fundamental’ concepts of Newtonian mechanics: mass, length, and time. (A fourth, electric charge, was added in the nineteenth century.) But it is far from clear that there is anything sacrosanct, privileged, necessary, or inevitable about this particular starting point. Some physicists in the nineteenth century ‘revised’ the conceptual basis of Newtonian mechanics and ‘defined’ mass itself in terms of length alone (the French system), and others in terms of length together with time (the astronomical system).8 The more important point is that it is by no means obvious that we would recognize an alien’s version of ‘Newtonian mechanics’. It is entirely conceivable that aliens should have hit upon a radically different man- ner of calculating the acceleration of falling bodies, of calculating the path of projectiles, of calculating the orbits of planets, etc., without using our concepts of mass, length, and time, indeed without using any, or very many, concepts we ourselves use.
Their mathematics, too, may be unrecognizable. In the 1920s, two versions of quantum mechanics appeared: Schro ̈dinger’s wave me- chanics and Heisenberg’s matrix mechanics. These theories were each possible only because mathematicians had in previous generations invented algebras for dealing with wave equations and with matrices. But it is entirely possible that advanced civilizations on different planets might not invent both algebras: one might invent only an algebra for wave equations, the other only a matrix algebra. Were they to try to communicate their respective physics, one to the other, they would meet with incomprehension: the receiving civilization would not understand the mathematics, or even for that matter understand that it was mathematics which was being transmitted. (Remember, the plan in S E T I is to send mathematical and physical information before the communicating parties attempt to establish conversation through natural language.) Among our own intellectual accomplishments, we happen to find an actual example of two different algebras. Their very existence, however, points up the possibility of radically different ways of doing mathematics, and suggests (although does not of course prove) that there may be other ways, even countless other ways, of doing mathematics, ways which we have not even begun to imagine, which are at least as different as are wave mechanics and matrix me-chanics.