When I was trying to work out my intuitions about causal paradoxes of infinity, which eventually led to my formulating the thesis of causal finitism (CF)—that nothing can have an infinite causal history—I toyed with views that involved information. I ended up largely abandoning that approach, partly because of my qualms about the concept of information and perhaps partly because of worries about physics that I will discuss below.
But I still think the alternative, which one might call information processing finitism, is something someone should work out in more detail.
- [IPF] Nothing with finite informational content can essentially causally depend on anything with infinite informational content.
Here, informational content is by definition contingent. The “essentially” excludes cases where finite informational content depends on a finite part of something with infinite informational content. How exactly the “essentially” is spelled out is one thing I am not clear on as yet.
The main difficulty with IPF is that our physics seems to violate it. The exact current temperature in Waco depends on the exact temperature, pressure and other facts around the world yesterday. Each of the latter facts involves infinite information—temperature is quantified with a real number, and a real number contains infinite information. Note that here IPF and CF may diverge. An advocate of CF can say that the exact current temperature in Waco depends on a finite number of past events such as “yesterday particle n has parameters P”, even if the parameters P involve real numbers that have infinite energy.
One way to escape this difficulty is to assume that our fundamental physics is actually discrete, and the real numbers in our equations are just an approximation. But I don’t want to stick my neck out so far.
Let’s see if we can make IPF work out with a continuous dynamics. We can suppose that metaphysically speaking, an entity’s having a real-valued parameter is constituted by the entity’s having an infinite sequence of discrete parameters, which parameters are more ontologically fundamental than the real-valued parameter.
For instance, by a one-to-one mapping we can assume our real number is strictly between zero and one, and then define it as an infinite decimal sequence 0.b1b2..., specified by an infinite sequence of digits. Unfortunately, then, we have some severe restrictions on what kind of dynamics we can have if we require that each digit of the output depend only on a finite number of digits of the input. For instance, multiplication by 3/4 cannot be defined, because to know whether f(x) starts with 0.24 or 0.25, you’d have to know whether x < 1/3 or x ≥ 1/3, and if the input is 0.333..., then you can’t tell from a finite number of digits which is the case. This kind of problem will occur with any other base.
It would be really nice to find some way of encoding a real number as an infinite sequence of discrete parameters each of which takes on a fixed finite range that escapes this kind of a problem. I am pretty sure this is impossible, but am too tired to prove it right now.
But there is another approach. We can have non-unique (many-to-one) encodings of reals. Here is one such approach, probably not the most natural one. Consider sequences of natural numbers n1, n2, ... such that for all k we have nk ≤ 2k and there exists a real number x between 0 and 1 inclusive with the property that |x−nk/2k| ≤ 1/k. Say that such a sequence encodes the real number x. In general, there will be more than one sequence encoding x by this rule.
Then if f is a function from [0,1] to [0,1], if we have a sequence n1, n2, ... encoding the real number x, to generate an acceptable kth term in a sequence encoding f(x), it suffices to know f(x) to within precision 1/2k, and if f is continuous, then we can do that by knowing a finite number of terms in a sequence encoding x (this is because every continuos function on [0,1] is uniformly continuous).
So any continuous dynamics from [0,1] to [0,1] can be handled in this way. The cost is that fundamental reality has degrees of freedom that are unimportant physically—for fundamental reality distinguishes between different sequences encoding the same real x, but the difference has no physical significance.
I don’t know if there is a way to do this with a unique encoding.
6 comments:
Isn't energy stored in quanta, which is finite? So, the thermal energy would be finite?
It might be helpful to refine the meaning of the word "content" in the IPF. A bit is finite, and a line segment that is 2pi long is also finite, so in both cases the actual content (not its measure) is finite. However, the precision needed to measure the bit is one, whereas in decomposing pi to digits we note that the digits are infinite.
One way to resolve your concern is to say that the current temperature, which depends on real numbers in our physics as you assert, is in fact also a real number, so the process of creating the temperature in Waco is actually a real numbers to real number function. These infinities are thus producing infinities, not finities (of digits in real numbers in their measure) until we approximate all that to a shortened decimal when we read the thermometer.
Additionally, I do not think that including real numbers in our calculations of a cause constitutes your kind of causal finitism, unless we tried to claim that we actually had to count infinite digits in the measurement, since such a count would never actually happen completely.
William: "These infinities are thus producing infinities, not finities (of digits in real numbers in their measure) until we approximate all that to a shortened decimal when we read the thermometer." Yes, but then we have that shortened decimal! And the problem is that the shortened decimal depends on ALL of the digits of the original data. (E.g., imagine that we know that the temperature goes down by a third of a degree. Then to know whether the first three digits of the temperature are 45.5, no finite collection of digits of 45.3333... is enough. This is assuming we're truncating rather than rounding, but a similar point applies when rounding.)
Don't all empirical measurements have limits in their precision, whether that leads to error or not?
Depending on all the digits but only using a finite number of digits just means there is an additional potential for error in the calculation, but as long as the measurement precision is sufficiently less than the number of digits chosen, the final precision should not worsen.
The limits in empirical measurements are themselves a problem for IPF. Assuming determinism, there will be a precise set of inputs that results in some specific measurement, say "12.3". Maybe we get that measurement whenever the inputs are in the range from 12.23 (exactly) to 12.34 (exactly), both inclusive. But then suppose the input is exactly 12.34. It seems you need infinite precision to know that the input is less than or equal to 12.34 rather than bigger than 12.34.
The problem is this. Plausibly, all our observations have only a finite number of possible outcomes. You look at a digital volt-meter, and there are only finitely many combinations of things the display can show. You look at an analog volt-meter, and there are (plausibly) only finitely many different needle positions your brain can distinguish. But whenever we have a mapping from a continuous input--a physical world described by real numbers--to a finite number of outcomes, there will be edge cases, where physical reality is right on the edge between the cases where the outcome is A and the cases where the outcome is B. In those edge cases, it seems, you need infinite precision.
I did say "seems" in some cases here deliberately. For if reality is set up very carefully, it is possible to avoid these problems. For instance, if our continuous reality represents real numbers as infinite decimals (surely it doesn't!), then one might have a hope that the intervals align neatly with the digits. Thus, perhaps, you feel a pain precisely when some specific thing in the brain has a voltage greater than or equal 0.50mV. In that case if physical reality represents voltages in decimal digits in millivolts (obviously it doesn't!), you only need two digits after the decimal point for reality to determine if you should have a pain.
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