State of the Art of the Attempt

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. It is interesting to see how far we can get with the project.

Some considerations

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 at all what we can reasonably expect is that something could exist. 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 (the relations between 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® á$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 may be at least four 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. It is true that in every possible world there is the possibility of T. If we give the notion of possibility this 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 sentence t may not exist[1], 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.

A 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 T 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).

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 actual relative to itself, it is “reliable” relative to itself, and is necessarily possible   

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 does not merely represent T), and it admits a rational explanation for its existence, namely 1-4.

Argument (1) Combined with Argument (3)

We want to argue T’s self-apparent existence is sufficient because we can define T. We use the supposition:

(2) the essence of structures is prior to existence

I tried to make this plausible in “1)” in the first section.

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 also apply to actual existence. The collection of things that exist forms an equivalence class. 

If relative existence is the only kind of existence there is, (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

One objection might be that 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.

Another objection could be that this is a false option, and (4) is neither true nor false. But it is either a sentence of the system that

(5)   

or else it is not.

In the system T we want something like

(6) 

This 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.

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 (necessary) possibility of such a definition is all we need.

Then we would use the supposition

(9)  absolute existence is relational existence

to conclude that T has an absolute existence.

(One might ask: where did the possibility for something being more logically inevitable than nothing come from? 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. it presumes reason.

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

We take (1) as an axiom. It is a reasonable, though probably not the best way to express logical inevitability. To express (1), the deductive system T must contain 1^st-order predicate calculus, equality and the modal operators. We make no assumption about models for T. Let T₁ be the collection of sentences closed under implication and use of the rules of inference. Now construct a model, in the sense of model theory, M₂, such that the universe of M₂ is exactly the collection of sentences 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 T_FP

Claim: T_FP satisfies (1)

The idea is that T_FP 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

Thanks John Mazetier and especially Jesse Folsom.

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[1] Recall that the sentences are what exists.