Senses do gather accurate data about reality, because they are part of reality. It is our interpretation of that data into models about other parts of reality that may be partly or completely wrong.
Is this a screen which I see before me? I interpret what I see, in the context of my experience so far, as meaning there is a screen before me. So far it has been an extremely reliable predictor of other sensory data I might experience, such as the visual stimulus changing in familiar ways as I do some stuff which I interpret as typing on a keyboard connecting to a computer that connects to it, or tactile stimulus if I do some stuff that I interpret as reaching out to touch it.
The fact that I can predict anything whatsoever about my sensory experiences, even the fact that I appear to have a consistent enough identity to have a history of sensory experiences, suggests that I am part of an extremely lawful universe of some sort. Whether it is completely lawful is unknowable, and possibly not even a well-posed question.
I confronted some research claiming that senses of agents evolved under fitness pressure systematically diverges from reality, but in the abstract they state that the standard consensus between cognitive and perceptual scientists is the other way.
In any way, I think the answer to this question is not trivial, and the idea of using a mathematical model in which there's a universe with fixed set of laws and evolving agents to explore the possibilities seems appealing to me.
I think I have a more serious problem regarding these formulas. If a and b goes to 1, regardless of c, Pr(p) and Pr(q) goes to 1. So if p is the statement "q is true." and q is the statement "p is true." then p and q must be true, which I think is nonsense. But I cannot see where my mistake is. Could you help please?
If a=b=1 then Pr(p & q) / Pr(p) = P(p & q) / Pr(q) = 1 so that Pr(p) = Pr(q) = Pr(p & q). That doesn't require that Pr(p), Pr(q), or Pr(p & q) goes to 1. It just means that in a Venn diagram, p and q coincide (or in popular parlance, "are a circle").
How did you get Pr(p) = Pr(q) = 1?
I used the expression I derived in the post, . However I didn't notice that goes to 0 too, at least for the example I gave in my previous comment. So there seems to be no issue as long as goes to 0 since it causes the indeterminate form 0/0.
We do science using the data we collected through our senses and we use science to understand how our senses work. Although I lack any rigorous formulation of the problem, the following plan seems interesting and I want to share it with you.
p= "Human senses collect accurate data about reality."
q= "Reality is governed by laws of physics."
Now we want to know Pr(p) and Pr(q). I don't know any way to directly calculate these values. It seems to me that Pr(p|q) and Pr(q|p) are easier to get our hands dirty. The former seems to be a scientific question and I heard that there are research done already, and the latter could be addressed by something like Solomonoff's Induction. At this point, we should be able to calculate the ratio Pr(p)/Pr(q).
The last stage is to calculate Pr(q|¬p). Here for simplicity I assume that the trust in our senses is something binary, we either believe all data we collect is about reality or none (I hope that the scheme can be extended later). Now the term Pr(q|¬p) is simply our prior belief about the laws of physics before we analyzed any data, and that again could be addressed by the universal prior probabilities from Solomonoff's Induction.
Let's name the quantities we have so far. Pr(p|q)=a, Pr(q|p)=b, and Pr(q|¬p)=c.
Pr(p)Pr(q)=abPr(¬p)Pr(q)=Pr(¬p|q)Pr(q|¬p)=1−acPr(p)Pr(¬p)=acb(1−a)Pr(p)=acac−ab+b and Pr(q)=bcac−ab+bNow I'm aware that we still need to assume the statement "Reality is governed by some algorithm, some fixed set of rules.", because Solomonoff's Induction needs that assumption.
I'd be very happy to hear your thoughts and comments on this framework. Is it dumb in some obvious way, or does it remind you of some research you have already seen before?
Update: I think I changed my belief that there is a circularity here. I feel pretty confident accepting the statement "There are some data I receive" without needing any science. The interesting question seems to be how much of the reality should we expect to reach using our senses.