Here is a thought experiment I have been thinking about concerning Free Will, Individuality and One-Way functions:

Assumptions:

Each conscious being in a world has material body which distincts it from each other conscious being with other material body in this world. Each conscious being is represented by two strings: Its current internal/individuality state string i_t and its current observable string o_t. The internal state string represents the internal state of this conscious being and corresponds to all knowledge, thoughts, feelings and expererience this consciousness has made up to time t. In a world where the consciousness being has a material body, those two strings are observable by the consciouss being themselve, but the internal state string is not observable by other consciouss beings with other material bodies. Each consciouss being can act on the world by modifying an observable string o_t to change it to o_(t+1). This string is observable by all consciouss beings in this world. Suppose there exists a one-way-function f on bitstrings (https://en.wikipedia.org/wiki/One-way_function). Suppose further, that for each consciouss being the same rules in the world apply. The free will might than be implemented through a one-way function f:

i_(t+1),o_(t+1) = f(i_t,o_t)

Other consciouss beings can observe only the choice between alternatives of one consciouss being: They observe only the string o_t,o_(t+1) etc.

This proposed procedure solves the free will dilemma:

1. To all consciousness beings the same rules f apply, hence it might be said that the world is deterministic.
2. Since f is a one-way function, it is easy for each consciouss being to apply f to (i_t,o_t) but it is practially impossible, for other consciouss beings to revert f, that is to say, they do not know the reasons (i_t,o_t) which led to o_(t+1). From their perspective the action of this consciouss being seem free in the choice how o_(t+1) looks like.
3. If the internal/individuality state strings i_t is for each time step equal to the empty string, then the consciouss being is merely a determinisctic function or machine and other consciouss beings can observe o_t, o_(t+1) and by applying f("",o_t) they get "",o_(t+1) and hence in retrospective they know that this consciouss being has no free will, since they know i_t = "".
4. On the other hand, if the internal/individuality state i_t is a very long bitstring of length N, then the free will of this consciouss being is very high \$2^N\$ and other consciouss beings with smaller free will \$2^n << 2^N\$ would have difficulties in understanding the choices/actions of this particular consciouss being.

Please if this thought experiment is discussed in literature, it would be nice to mention the reference-request, because I find it very interesting.

• Since the state of your beings at any moment depends only on their own state at the previous moment there is no interaction between them. There is no world, only a disjoint collection of independent automatons that can observe nothing of each other at all. And if they can interact, and change each others' strings, they can't assume that the changes are caused by your one-way function and draw conclusions about each others' internal strings. Complexity of interactions will obscure internal complexities. – Conifold Jan 21 at 13:05
• thank you for your insightful comment. @Conifold – stackExchangeUser Jan 21 at 13:13
• @Conifold: It is easy to extend the formalism to describe what is intended. For example the "world" might consists of all O = o_1,...,o_n concatenated strings at each time step t, where we have n beings and o_i corresponds to the observable string of the i-th being. Then f(i,O) = i*,O* where f can only be applied to the internal state and the observable state of the i-th being. This way every being can "observe" the observable states of each other beings. – stackExchangeUser Jan 22 at 9:52

These are good ideas and I am happy to see someone trying a mathematical approach to the mind like this. Some points:

• You are essentially saying that free will is when a person's true motivations are hidden and hard to figure out from their behavior.
• Are people's true motivations really that hard to figure out? In many cases it seems not. People usually have predictable, mundane desires. If someone buys a donut, it's probably because they're hungry, they think donuts taste good, and they habitually eat donuts. But many would still insist that buying the donut is an exercise of "free will."
• In theory, the whole brain is directly observable through technology. We can put someone in an fMRI machine and look at what areas are active. As technology advances, so does our ability to monitor what is happening in someone's brain. A brain is after all just a machine, if a very complex one.
• What about other systems whose internal state is difficult to deduce from their external behavior? Would you say that a simple cryptographic system has "free will"? How about a black box with a couple objects inside it that can't be seen?
• You may wish to see Stephen Wolfram's idea of free will. He suggests we could assign the term "free will" to systems whose evolution over time is irreducible - systems for which there is no way to compute the n'th time step, other than by simulating all n time steps one by one.
• Thanks for your thoughts in this question. I will have a look at the reference you gave. – stackExchangeUser May 21 at 21:40

You have a problem with your assumptions. In physical systems, matter, energy and momentum are conserved. So if part of a physical system changes, we can account for the changes in other parts of the system. "Observation" is just another word for interaction, and we know an interaction occurred when something changes. So when an observable body observes anything it must change according to physics and change anything it observed accordingly. So the internal state is also observable, contradicting your assumption that it was not observable.

• Thank your for your insight in this question. I mean a world where it is possible to change somebodys thoughts without having to be able to actually observe those thoughts. – stackExchangeUser May 21 at 19:16
• So I think there is much more needed in terms of foundational definitions before we can move on. Its like saying “what about math but sometimes 1+1 is 3 and sometimes 2”. Ok, but, it’s not clear any definition of math would allow that. So we just can’t presuppose we can have it on hand. – z5h May 22 at 20:38