Where does probability theory come from anyway? Maybe I can find some clues that way? Well according to von Neumann and Morgenstern, it comes from decision theory.
I believe this is the step from where you started going astray. The next steps of your intellectual journey seem to be repeating the same mistake: attempting to reduce a less complex thing to a more complex one.
Probability Theory does not “come from” Decision Theory. Decision Theory is strictly more complicated domain of math as it involves all the apparatus of Probability Spaces but also utilities over events.
We can validate probability theoretic reasoning by appeals to decision theoretic processes such as iterated betting, but only if we already know which probability space corresponds to a particular experiment. And frankly, at this point this is redundant. We can just as well appeal to the Law of Large Numbers and simply count the frequencies of events on a repetition of the experiment, without thinking about utilities at all.
And if you want to know which probability space is appropriate, you need to go in the opposite direction and figure out when and how mathematical models in general correspond to reality. Logical Pinpointing gives the core insight:
“Whenever a part of reality behaves in a way that conforms to the number-axioms—for example, if putting apples into a bowl obeys rules, like no apple spontaneously appearing or vanishing, which yields the high-level behavior of numbers—then all the mathematical theorems we proved valid in the universe of numbers can be imported back into reality. The conclusion isn’t absolutely certain, because it’s not absolutely certain that nobody will sneak in and steal an apple and change the physical bowl’s behavior so that it doesn’t match the axioms any more. But so long as the premises are true, the conclusions are true; the conclusion can’t fail unless a premise also failed. You get four apples in reality, because those apples behaving numerically isn’t something you assume, it’s something that’s physically true. When two clouds collide and form a bigger cloud, on the other hand, they aren’t behaving like integers, whether you assume they are or not.”
But if the awesome hidden power of mathematical reasoning is to be imported into parts of reality that behave like math, why not reason about apples in the first place instead of these ethereal ‘numbers’?
“Because you can prove once and for all that in any process which behaves like integers, 2 thingies + 2 thingies = 4 thingies. You can store this general fact, and recall the resulting prediction, for many different places inside reality where physical things behave in accordance with the number-axioms. Moreover, so long as we believe that a calculator behaves like numbers, pressing ‘2 + 2’ on a calculator and getting ‘4’ tells us that 2 + 2 = 4 is true of numbers and then to expect four apples in the bowl. It’s not like anything fundamentally different from that is going on when we try to add 2 + 2 inside our own brains—all the information we get about these ‘logical models’ is coming from the observation of physical things that allegedly behave like their axioms, whether it’s our neurally-patterned thought processes, or a calculator, or apples in a bowl.”
I’m not sure what is left confusing about the source of probability theory after understanding that math is simply a generalized way to talk about some aspects reality in precise terms and truth preserving manner. On the other hand, I’ve figured it out myself, and the problem never appeared to me particularly mysterious in the first place. so I’m probably not modelling correctly people who have still questions about the matter. I would appreciate if you or anyone else, explicitly ask such questions here.
I believe this is the step from where you started going astray. The next steps of your intellectual journey seem to be repeating the same mistake: attempting to reduce a less complex thing to a more complex one.
Probability Theory does not “come from” Decision Theory. Decision Theory is strictly more complicated domain of math as it involves all the apparatus of Probability Spaces but also utilities over events.
We can validate probability theoretic reasoning by appeals to decision theoretic processes such as iterated betting, but only if we already know which probability space corresponds to a particular experiment. And frankly, at this point this is redundant. We can just as well appeal to the Law of Large Numbers and simply count the frequencies of events on a repetition of the experiment, without thinking about utilities at all.
And if you want to know which probability space is appropriate, you need to go in the opposite direction and figure out when and how mathematical models in general correspond to reality. Logical Pinpointing gives the core insight:
I’m not sure what is left confusing about the source of probability theory after understanding that math is simply a generalized way to talk about some aspects reality in precise terms and truth preserving manner. On the other hand, I’ve figured it out myself, and the problem never appeared to me particularly mysterious in the first place. so I’m probably not modelling correctly people who have still questions about the matter. I would appreciate if you or anyone else, explicitly ask such questions here.