Can we give a rational explanation for existence?

Can we give a rational explanation for existence?




How to get an explanation for existence



Why does anything at all exist? This question already makes an assumption. It just assumes that nothingness is more warranted than the existence of something. But what justifies this assumption? Without any assumptions what we can say is that it could be that something exists. Indeed it could turn out to be the case that the existence of some particular thing is logically inevitable, while nothingness is not.



Hypothesis: what ultimately exists is pure structure (mathematical objects)



I will not go in to this hypothesis except to say it is consistent with materialism.



By the hypothesis, we are after some mathematical structure that finds its own existence to be logically inevitable. It is sufficient to consider the case where the mathematical object is the web of theorems of some formal system T. Then we want a theory that, in some sense, asserts it’s own inevitability. (How a theory that talks about itself might be implemented is discussed below.) In other words, the mathematical theory T is such that



(1) T 



But even supposing we could find such a T, this condition is not enough to guarantee actual existence. All we can assert is that there could be some mathematical structure, such as T, that derives its own logical inevitability. But all we really know is that the existence of T is possible. In the many-worlds semantics of possibility this leads to two cases, worlds W for which T exists, and the disjoint worlds V where T does not exist.



We want to motivate the existence of T. In the worlds W where T exists, there is only the issue of finding a (the) T that satisfies (1). In the possible worlds V however, it is not obvious how to motivate existence. In these worlds T does not exist, so it is not around to assert anything about itself. In this case there is no way to get existence off the ground.



Happily, there are at least four promising ways out of this conclusion.



The first way out is to argue that the possible-worlds semantics of possibility is misleading here. In possible worlds semantics there are some worlds W in which T exists and some V where it does not. But this contains the trick of forcing an either-or decision upon us where it may be inappropriate. We do not want or else What we want is to consider the case that it is possible that T exists. If we give the notion of possibly existing this new status, it might be enough, in view of condition (1), to allow the theory to bootstrap itself into existence.



Returning to the possible worlds V and W, a second way out is to argue that, for the possible worlds V a theorem t may not exist , but t’s relations to other things might nevertheless exist. Thus there are second-order properties of t that claim existence in spite of t’s own non-existence. One example of this might be to say that if t is some assertion that is true, then that truth exists even if t does not (or only possibly exists).



A third way out, slightly different from the second, is to first show that T exists in the sense of T. Then, the argument goes, it is sufficient that T exists in the sense of T, because this existing-in-the-sense-of is all we need for a kind of absolute existence. It is not clear what it means to exist in the sense of something that does not (or only possibly) itself exist. What would it mean for a purple cat to exist in the sense of a flying pink elephant? Well, for all the flying pink elephant knows, the purple cat is real...



A fourth way out is to argue that all notions of existence are relative, and so there is no problem for the relative kind of existence in the third way out (above) because the initial existence predicate in is implicitly relative, and absolute existence is a priori relative existence anyway, etc.



My own preference is for argument (1) combined with argument (3).



Anyway, suppose we have a candidate theory T. T finds itself to be logically inevitable (and T does not find nothingness to be logically inevitable). According to the hypothesis, T does not merely represent the universe, it is the universe. If T is physical, then it would be a physical circumstance that the universe is logically inevitable.



Then, if we had good reason to believe T is our actual universe, we would be justified in claiming to have a rational explanation for existence.



Finding T



Let t be any theorem of T. Define t as



x such that x t



To express (1) and (2), T must contain 1st-order predicate calculus, equality and the modal operators. For T to find its own existence logically inevitable, (1) must be true of it. Since T is going to talk about itself, (1) is true in it. Therefore assume (1) as an axiom. Let T1 be the collection of resulting theorems. Now construct a model, in the sense of model theory, M2, such that the domain space of M2 is exactly the collection of theorems T1. Let T2 be the theory of M2. Construct a model M3 out of T2 in the same way, and continue this process indefinitely.



Conjecture: this process reaches a fixed point theory TFP



Claim: TFP satisfies (1).



TFP is our universe, and in being so asserts its own logical inevitability.





Paul