Towards a Rational Explanation for Existence
Is it possible to give a rational explanation for existence? Below is a web of suppositions which, if borne out, would allow us to claim to have a rational explanation for existence. Of course one is free to reject the suppositions. Nevertheless it is interesting to see how far we can get with the project.
The basic idea is to have a mathematical theory assert its own existence. Then argue this theory exists relative to itself. Next conclude this relative existence is sufficient for absolute existence. Finally, postulate that our universe is that theory.
Here is an attempt to lay bare the argument:
1) 2 + 2 = 4 is in some sense true whether or not the universe or anything at all exists. The interrelatedness of the definitions of 2, +, 4, and = in some sense precedes the question of existence.
2) Now define a consistent theory T that contains the statement:
“The existence of this statement is logically inevitable.”
(How this might be implemented is discussed below.)
3) T is always possible, it is reliable relative to itself, and actual relative to itself.
4) In some important sense, existence does not recognize the difference between T’s self-apparent existence and our actual existence. They are equally valid. Perhaps our actual existence is a kind of self-apparentness.
5) The universe is T, and it admits a rational explanation for its existence, namely 1-4.
Some considerations
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 making any assumptions all 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 and inconsistent with dualism.
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 sentences of some formal system T. Beyond that we want a theory that, in some sense, asserts it’s own inevitability. In other words, the mathematical theory T is such that
(1) T
But even supposing we could make sense of this and 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, so to speak.
Happily, there are at least four possible 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 about T’s existence where it may be inappropriate. We do not want or else What we want is to consider the possible worlds in which it is possible that T exists. If we give the notion of possibility 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.
We want to argue T’s self-apparent existence is sufficient because we can define T. We need the crucial supposition:
(2) definitions precede existence, and do not make an assumption about existence
Infinite numbers are definable (as in set theory), even though the physical universe may be only finite. Nevertheless we often “prove the existence” of this or that infinite number (such as a set). In this case we might be able to say that infinite numbers exist relative to the theory that defined it (e.g. set theory).
Then,
(3) if (1) is satisfied T (necessarily) exists relative to itself.
Now, it would be argued, this relative kind of existence is what we really mean by “existence”. This would have to apply to actual existence. And existence forms an equivalence class...
In this case, (3) is sufficient for actual existence.
What if things have “intrinsic” or “absolute” existence? In this case (3) would seem to be insufficient to generate actual existence…
But by the hypothesis, the universe is ultimately mathematical structure. Can’t parts of a mathematical structure exist relative to each other? Suppose we say
(4) the number 2 exists relative to the number 3
You might say: it may be true, but presupposes absolute existence. But how do you know? If absolute existence is, in fact, relative existence, we would not know the difference.
You might say: this is a false option, and (4) is neither true nor false. But it is either a theorem of the system that
(5)
or else it is not.
In the system T we want something like
(6)
which is not to be confused with
(7)
In (6) it says if there is an x then there also exists (relative to the formal system) the assertion that there is an x. This is plausible since x is itself an assertion of the formal system.
This requires some attention to the rule for - introduction. In addition to
(8)
a new rule is required, 2nd-order (nth-order) existence-introduction, or maybe some consistent implementation of (6).
To pick up where we left off with (9), we can say that relative to the definition, the theory necessarily exists. So relative to the definition, the definition itself exists, that is because the theory asserts it’s own logical inevitability. The interrelatedness of the definitions generates its own necessary existence, since the definitions are, in fact, so interrelated. Here is something (the interrelatedness of the definitions) from nothing (what the definitions are definitions of). The possibility of such a definition is all we need.
Then we would use the supposition
(9) absolute existence is relational existence
to conclude that absolute existence is given by the theory T.
( One might ask: where did the possibility for something from nothing come from? (It does not come from nothing, we do not assume nothing is more logically inevitable than the existence of some particular things. Instead of nothingness we take a stance of having no assumptions about the matter. I.e. why is the world so constructed as to allow for such a possibility? The short answer is that an answer to this question must lie outside of our capacity to reason (such as qualia do), since it is asking for a reasonable explanation of reason, i.e. the answer must involve reason, so the faculties to accommodate reason must already be accounted for.
The third way out of the conclusion above, slightly different from the second, is to first show that T exists in the sense of T. Then, the argument would go, 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. But it is not obvious what it means to exist in the sense of something that only possibly exists…
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 existence predicate in is implicitly relative, and absolute existence has really just been relative existence the whole time, etc.
My own preference is for argument (1) combined with argument (3).
Now 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 above, T does not merely represent the universe, it is the universe. If we chose to say that T has physical existence, then it would be a physical circumstance that the universe is logically inevitable.
Finally, if we knew 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. Make the definition
Now
(11)
We take (1) and (7) as axioms. To express these, the deductive system T must contain 1st-order predicate calculus, equality and the modal operators. We make no assumption about the model for T. For T to find its own existence logically inevitable, (1) must be true of it. Let T1 be the collection of resulting theorems. T1 is every theorem derivable from axioms (1) and (7) and 1st-order predicate calculus, equality, and the modal operators. Now construct a model, in the sense of model theory, M2, such that the universe 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)
The idea is that TFP is our universe, and in being so asserts its own logical inevitability.
Somewhere in there it should be added that the reason we want an ultimate explanation is that existence cannot be contingent on anything—is that if the explanation uses any assumptions we can always ask why state of affairs described by that assumption obtains.
Paul Merriam
Thanks to Jesse Folsum and John Mazetier.
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