The Realm of the Determinate

4 min readOct 6, 2023

By a determinate, I mean a property of the world whose value is not dependent on chance or on human choice. I will start with determinacies of a mathematical nature, because of their stark nature, and then move on to physical determinacies.

Most people will agree that the result of multiplying two particular numbers is determinate, as is the set of factors of one particular number. More generally, programmers will agree that the output of any well defined algorithm, applied to particular inputs, and without calls to a random number generator, is determinate if the algorithm terminates. Alan Turing was able to formulate this assertion with mathematical precision. His machines have very simple and precise rules of operation. Yet no exception is known to the proposition that every algorithm can be implemented by a Turing machine; the many notions of computable function miraculously coincide. This conclusion is the Church-Turing thesis.

For each formal mathematical theory, such as Peano arithmetic, or ZFC set theory, there is a Turing machine which, given a proposition, will search for its formal proof. If there is such a proof, it will find it, though it might take many times the age of the universe at a gigahertz step rate to do so. If there is no such proof, the Turing machine will never halt. So if a proposition can be proved in a formal mathematical theory, this fact is determinate. Truth is another matter (as Gödel showed). For Platonists, mathematical truth is generally seen as determinate too, but this is not germane to this discussion.

The physical world has many properties that qualify as determinate, but the situation is complicated. Such properties are theory dependent, and theories change. We can assert that the atomic number and weight of a given element are determinate, but only after we have developed atomic theory. One interesting case is that of a physical system which is either built to implement mathematical determinacies (like a computer) or which has been observed to obey such determinacies (like the solar system). Such a system can be said to have determinacies that are derived from mathematics. There is always the caveat that physical theories (both formal and informal) are subject to possible conflict with future observations, and physical machines can break down. Full exclusion of randomness is impossible in most cases. Still, some conclusions can be drawn that, while fallible in principle, do not involve human choice or chance.

What is the status of the realm of the determinate? For example, does every million digit number, and its set of factors, exist in some sense? Does every billion long sequence of states of every run of a Turing machine exist? Is the number of stars with planets in a particular galaxy a billion light-years away determinate? Maybe not in all cases, but the realm of the determinate might as well exist, given the relationship that humans have to the realm. An element of the realm comes into ordinary existence for the people who happen look at the element (and who with rare exceptions come to the same conclusions about the element when working separately, as one would expect). One would have to say that the realm of the determinate takes a definite form in the world, but its elements are only visible under certain conditions.

Our epistemic relationship to the realm of the mathematically determinate is the same as it is to determinacies in the physical world: neither will ever be completely known, but both are always available for further exploration. Before there were lunar probes, it was believed that the locations and sizes of craters on the dark side of the moon were determinate at any moment in time. We believed that probes sent by different nations would all see the same thing, or nearly so given that new craters can appear at any time (and do so, but gradually; it takes 80,000 years to completely transfigure the moonscape ). This belief in crater determinacy has not been contradicted, or we would have heard about it. The dark side of the moon, and the digits of π from the 100,000th through the 200,000th, had the same status, epistemically, in 1950: unknown. Both have been explored since.

Mathematics and physical theory both evolve, and our idea of their determinacies with them. Mathematics evolves much more slowly than the physical theory. Euclid’s “Elements” (300 BC) still makes sense to us today, and does not contradict modern mathematics (although Euclid’s geometry is now embedded in a wider notion of the geometric). Set theory has stood firm for a century. On the whole, the mathematical sense of the determinate has greatly enlarged over the centuries, but seldom retreated.

All of the notions discussed here have arisen from human minds. Might alien minds have other notions of mathematics and of the determinate? How can we know until we encounter them?

However things may turn out with alien minds, the presence in the world of determinacy of the algorithmic kind — which includes mathematical provability- is not subject to doubt. Our phenomenal world does indeed contain the determinate and we too are parts of the wide world, therefore the wide world contains the determinate. Husserl formulates our membership in the wide world well: “Consciousness and things in general are thereby a combined whole, combined into individual psychophysical entities that we call ‘animalia’ and ultimately in the real unity of the entire world.” (Ideas 1, section 39)