Yes. That is the logical induction I was talking about.
If you're so interested in logical induction, aren't you already assuming that classical mathematics is The One True Logic? Why is that? Why not look at ordinary mathematics internal to a topos and then ask what logical induction looks like for that?
And as for reflection, a topos with a NNO has (higher order) primitive recursion so your claim about not having reflection is confusing.
And lastly, your title doesn't match your thesis. All you show is that you can't directly do probability in toposes. Category theory is extraordinarily useful for many areas of mathematics in general, and is more than just a language. See Beck's monadicity theorem, the adjoint functor theorem, the small object argument, Gabriel Ulmer duality, and so on for nontrivial results in category theory.Maybe you shouldn't base your entire identity around doing probability theory. At the very least, epistemology spills far beyond the purview of probability theory.
I think an important point here and with noticing a lot of things in general, is that the taste of lotus is somatically present - you can feel it in your body. I think that if one practices becoming attuned to the physical sensations in your body then a lot of things become available to Noticing. There is just *so much* that goes on in the body when you do things. This is a generalization of a key insight of Focusing - it's not just beliefs that you can get a handle on through your body. Action and decision are present there too (initiate the action of touching a hot stove or the soup in someone else's bowl and see what happens; compare "reach for that pen" with "stand up and do a backflip"; the Alexander Technique deals with objects in this layer too). In particular, the pull of the social web can be felt here, as well as a lot of social stuff in the moment like status (see Impro).
As someone who independently came across this, the generative seed for this idea was using Focusing on why I kept scrolling down on Facebook (which I'm happy is an example here).
Crushing what I say into some theory of bayesian epistemology is a great way of destroying the meaning of what I say.
But to try to fit it into your theory without losing as much information as your attempt: humans, by the evolved structure of our brains, especially by the nature of human perception and decision making, have a built in ontology - the way we cut out things in our perception as things, and the way we see them as being things which are relevant to our involvements in the world. You can't get rid of it, you can only build on top of it. Mistakenly taking reductionistic materialism as ontology (which is not an action you can take short of completely changing the fundamental structure of your brain) only adds its complexity on top of the ontology that is already there. It's like using a windows emulator to do everything instead of using the OS the emulator is running in.
If you tried to turn your statement into an actual mathematical statement, and tried to prove it, you would see that there is a large gap between the mathematics and the actual psychology of humans, such as yourself.
I think one of the biggest things Peterson has to offer is a way out of many of the fake frameworks that rationalists hold, by offering a fake framework which takes Being as primary, and actually being able to deal with Being directly (which becomes possible with a fake framework which permits the concept of being) is a pathway to Looking.
Parent commenter is doing some pretty serious cherry picking. 2) and 3) can basically be ignored. 2) comes from a 2013 deleted tweet which the parent commenter has pulled off of archive, and 3) from a 2011 debate which is anyways misrepresented by the parent commenter. He never lays out something that can unambiguously be taken to be quantum mysticism, even though he starts out talking about copenhagen. "consciousness creates reality" does actually correspond to a reasonable position which can be found by being a little charitable and spending some time trying to interpret what he says. 1 and 4 depend on his rather complex epistemology, "I really do believe this though it is complicated to explain," he prefaces the DNA comment with.
I would be much more concerned if something like 2) were something he repeated all the time rather than promptly deleted, and was central to some of his main theses.
When is something a misinterpretation of sensory input? When the interpretation is not rendered in terms of the laws of physics which your alternative implies or...?
A better hypothesis is "in a metaphysics which takes Being as primary, which is not in any way contrary to science (since science does not imply a metaphysics like scientific realism or reductive and eliminative materialism), mystical experience is permissible and not contrary to anything we know".
It's hard to get Peterson second-hand. I recommend actually watching some of his lectures
While Peterson is a bit sloppy when he talks about truth, the notion of truth that he is working with is not simply his own construction to write some bottom line. There is a lot of literature of pragmatist analyses of truth and belief that roughly align with what he is saying and I would consider closer to what is the nature of truth (truer about truth) than the correspondence theory of truth presented in the sequences.
I recommend Peirce's Making our Ideas Clear, Putnam's Corresponding with Reality, and James's The Will to Believe. Peirce and James can easily be found free online by searching and I can PM you Putnam if you want it.