Previously I have shown that Quine’s paradox disappears once we take time into account. Computations and logical inferences neither happen instantaneously nor at the same time. For all practical purposes, when we engage in the application of math or logic, we always deal with a sequence of computations separated by time.

In this regard I would like to revisit the Quine’s paradox and rewrite it as a series of discrete computations as follows. We can rewrite the statement “this statement is false” as X = false, where “X” corresponds to “this statement”, and “=” corresponds to “is”.

We can further denote the statement “X = false” as “Y”, where “Y” is the result of equality operation “=”. In other words, (X = false) → Y. That is, Y is true if X was false and Y is false otherwise.

Therefore our comprehension of the statement “this statement is false” actually involves a computational process described above. Quine’s paradox does not arise in the context of this computational process since each logical operation is evaluated at a separate instance of time.

To illustrate this point better, we note that the term “statement” in the setntence “this statement” implies recursion. That is Y and X are referring to the same thing, but they are not exactly the same! We avoid the Quine’s paradox by noting that in the computational process framework each operation (recursive or not) is evaluated at a separate moment of time. This means that X is actually X(t = 0) and Y is X(t = 1). So, what we have is this:

“this statement is false” corresponds to X(t = 0) is false, which is our initial condition. So:

(X(t = 0) is false) → X(t = 1) is true;

(X(t = 1) is false) → X(t = 2) is false;

(X(t = 2) is false) → X(t = 3) is true;

etc.

Conclusion

As you can see, the X is not true and false at the same time, but rather it is true or false at different moments if time! The paradox arises only when we neglect the notion of time and are trying to think of logical operations not as a process (i.e. a series of steps that take finite time) but rather as something that happens instantaneously.

What Quine’s paradox is telling us is that math and logic are rules that govern computational processes. This means that math and logic rules make sense (i.e. are free of paradoxes!) only if we view them in a context of a computational process. After all, out thinking is a computational process as well.

Takeaway

Now I am convinced that the ‘mechanics’ of the Universe (i.e. ‘the works’ of the world, consciousness excluded) can be described as a computational process. In this regard it is not wrong to say that we live in a simulation, and Stephen Wolfram’s computational approach to the Universe actually makes sense.

The bottom line is that it is wrong to engage in math or logic without taking time into account! I think a great deal of problems in modern physics originate exactly from this error. Physics is meant to describe natural processes, and all processes in nature happen in time and are causal in nature. I.e. the cause precedes the effect in time. But this is not what our equations tell us! Most equations that we use are called equations because they relate entities to one another without regard for time! They do not separate causes from effect and imply that everything happens instantaneously and at the same time. I.e. the famous F = ma equation does not tell us if the acceleration causes force or of the force causes acceleration. This equation is telling us that the force and the acceleration are basically the same thing. And I have just shown that this exact thinking leads us to paradoxes because neither nature nor our thinking process works this way.

We clearly must switch our mathematical apparatus to a causal one, which describes natural processes as they unfold in time. Ironically, we already do that when we code computations on computers! This is exactly how we simulate things: by separating causes and effects in time.

Once again, we arrive at paradoxes only when we discard the notion of time and thus think of math and logic operations happening instantly in a singular moment in time. Fortunately nature does not work this way and neither our minds. Paradoxes do not happen when we bring time into picture. But this does not mean that computational results will converge. But who is to say that this is not natural? After all we believe that the Universe is fundamentally oscillatory at a quantum level…