Part 1: It is a mistake because the future does not resemble the past except in some vacuous senses. Why? Because stuff changes. For example an object in motion moves to a different place in the future. And human societies invent new technologies.

The object in motion moves according to the same laws in both the future and the past, in this sense the future resembles the past. You are right that the future does not resemble the past in all ways, but the ways in which it does themselves remain constant over time. Induction doesn't apply in all cases but we can use induction to determine which cases it applies in and which it doesn't. If this looks circular that's because it is, but it works.

Popper's approach is to improve our knowledge piecemeal by criticizing mistakes. The primary criticisms of this approach are that is it is incapable of offering guarantees, authority, justification, a way to force people to go against their biases, etc.. These criticisms are mistaken: no viable theory offers what they want. Setting aside those objections -- that Popper doesn't meet standard too high for anything to meet -- it works and is how we make progress.

As far as Bayesianism is concerned this is a straw man. Most Bayesians don't offer any guarantees in the sense of absolute certainty at all.

All justificationist epistemologies have connections to authority, and authority has nasty connections to politics. You hold a justificationist epistemology. When it comes down to it, justification generally consists of authority. And no amount of carefully deciding what is the right thing to set up as that authority changes that.

No Bayesian has ever proposed setting up some kind of Bayesian dictatorship. As far as I can tell the only governmental proposal based on Bayesianism thus far is Hanson's futarchy, which could hardly be further from Authoritarianism.

forcing them to follow rigorous rules.

See! I told you the authoritarian attitude was there!

You misunderstand me. What I meant was that as a Bayesian I force my own thoughts to follow certain rules. I don't force other people to do so. You are arguing from a superficial resemblance. Maths follows rigorous, unbreakable rules, does this mean that all mathematicians are evil fascists?

And there is no mathematical proof of Bayesian epistemology. Bayes' theorem itself is a bit of math/logic which everyone accepts (including Popper of course). But Bayesian epistemology is an application of it to certain philosophical questions, which leaves the domain of math/logic, and there is no proof that application is correct.

Incorrect. E.T. Jaynes book Probability Theory: The Logic of Science gives a proof in the first two chapters.

My mind is a universal knowledge creator. What design could be better? I agree with you that it wasn't designed for this in the sense that evolution doesn't have intentions, but I don't regard that as relevant.

Evolutionary psychology contains mistakes. I think discussion of universality is a way to skip past most of them (when universality is accepted, they become pretty irrelevant).

You obviously haven't read much of the heuristics and biases program. I can't describe it all very quickly here but I'll just give you a taster.

• Subjects asked to rank statements about a woman called Jill in order of probability of being true ranked "Jill is a feminist and a bank teller" as more probable than "Jill is a bank teller" despite this being logically impossible.

• U.N. diplomats, when asked to guess the probabilities of various international events occurring in the nest year gave a higher probability to "USSR invades Poland causing complete cessation of diplomatic activities between USA and USSR" than they did to "Complete cessation of diplomatic activities between USA and USSR".

• Subjects who are given a handful of evidence and arguments for both sides of some issue, and asked to weigh them up, will inevitably conclude that the weight of the evidence given is in favour of their side. Different subjects will interpret the same evidence to mean precisely opposite things.

• Employers can have their decision about whether to hire someone changed by whether they held a warm coffee or a cold coke in the elevator prior to the meeting.

• People can have their opinion on an issue like nuclear power changed by a single image of a smiley or frowny face, flashed to briefly for conscious attention.

• People's estimates of the number of countries in Africa can be changed simply by telling them a random number beforehand, even if it is explicitly stated that this number has nothing to do with the question.

• Students asked to estimate a day by which they are 99% confident their project will be finished, go past this day more than half the time.

• People are more like to move to a town if the town's name and their name begin with the same letter.

There's a lot more, most of which can't easily be explained in bullet form. Suffice to say these are not irrelevant to thinking, they are disastrous. It takes constant effort to keep them back, because they are so insidious you will not notice when they are influencing you.