Yes, I am confident. But not ambiguous that my confidence is backed up with some aesthetic factors, not just squeezing out the hand waving.
> FWIW you can actually just say that you accept infinite regress.
I think my claim that everything that has a specific value, has been constrained to it, is very strong. And the inverse, that where something isn't constrained, it will appear in all its forms.
In that sense, I admit brut facts. We don't need to explain a specific, if we know its alternatives play out disjointly. Exhaustion instead of determination.
(The possibilities in that survey didn't include exhaustion! A much simpler explanation than evolutionary universes, or inverting the arrow of the anthropic principle.)
Superposition and entanglement are an exact example of mutual determination/exhaustion. Along with cancellation as a direct expression of conservation.
But my view that finite regions are potentially always characterizable with an unbounded finite region it is embedded in, as apposed to characterized by an infinity of other independent specifics, is easily my weakest conjecture.
> You can also just reject the law of non-contradiction.
I think contradiction is a category error for primitives of reality. Conservation is the right view.
I make a clear distinction between descriptions, partial models, conjectured models, etc. Contradictions can occur in our descriptions. But I don't believe reality is constrained by logical primitives. Just conservation. And with primitives whose generative/reductive properties are not constrained by logic, because there are not even "virtual" contradictions to prune.
I expect logic to be necessary for us to reason about primitives. But that logic is the wrong way to think of basis elements. Those are very different things.
> A theory with no axioms would be desirable.
I think this is necessary. I am 100% confident in that, for what that is worth to anyone else. Realty is able to exist precisely because it requires no priors, and has structure because it creates nothing, destroys nothing (conservation). Tautological anti-axioms that become axioms.
> But also, like, we wanted that in mathematics and accepted that it was impossible too.
This is not nearly as well established as is assumed. Whether we look at Russell's Paradox, or Gödel/Turings incompleteness theorems, there is a repeated issue. The proofs are about machines (or sets) that return (or are defined by) true or false. But any logical system with open (cycle) expressions admits two more possibilities, undetermined and contradictory statements. Not for some deep reason, but because notation is not reality, and we can trivially say things with notation that are undetermined or contradictory. "x in {x}, is unknown". "x in { logical x = not(x) }" is a contradiction. A universal mathematical discriminator needs to be defined as first, determining whether a statement is unknown or a contradiction. Then decide true or false for remaining expressions.
Note that the problem is implicitly collapsing unknown into true (i.e. "satisfiable") and contradiction into false ("unsatisfiable"), but then still treating those values as if they were just primitive true and false, which they no longer are.
To be concrete: If M is the mathematic machine, and we defined it reduce expressions (any kind of reducable structure, including 4-valued logic, not just Boolean truth), and it is tasked with evaluating a copy of itself operating on a self-referencing contradiction "M will say this is false", it is trivial to show M(M("M will say this is false")) = M("M will say this is false") = <contradiction>. No inconsistency, and no window to rework the statement into a problematic alternate, because we have avoided a special dependency, inconsistent treatment (t/f vs. u/c), or premature limitation around logic.
M may still have limits, but Gödel's incompleteness is not one of them.
This is another indicator to me, that a fundamental description of math (as with reality), needs to be based on sub-logical primitives. Another big clue is that Boolean logic is not reversible. Logic will be easily created from the fundamental relationship, but it creates a premature deadend to start there.
There are a number of well accepted famous theorems with this limited scope problem. They are absolutely solid proofs. But they define something more restricted than it needs to be, then knock it down. Leaving trivial possibilities unrestricted.
Cantors diagonalization proof for cardinality of R being greater than for N, is also trivially defanged. But there are better reformulations of it, and I haven't had the time to work though them yet.
I find that mathematicians, like everyone else, have trouble truly seeing what they looking at. I could go on and on.
Well thanks for pushing me, I am sure I have taken up enough of your time. But, I am now aware of more philosophical viewpoints on the big questions!
> Yes, I am confident. But not ambiguous that my confidence is backed up with some aesthetic factors, not just squeezing out the hand waving.
I think that's fine. I'm actually sort of agnostic about it myself. I find nothing particularly striking about brute facts or contingencies, certainly it's not magical to me, but I wouldn't say that they're logically necessary. I find it a bit interesting to consider.
> I think contradiction is a category error for primitives of reality. Conservation is the right view.
It may be interesting as a thought exercise to wonder if non-contradiction could be an emergent property. It would certainly make for an interesting model for the beginning of the universe.
> I expect logic to be necessary for us to reason about primitives. But that logic is the wrong way to think of basis elements. Those are very different things.
I think this would probably be contentious, I think most people believe that logical values are basically "necessary". I don't really know though.
I will perhaps spend some time reflecting on the concepts you're referring to with regards to logic and primitives. I'm not really familiar enough with that sort of grounding or reduction.
> Well thanks for pushing me, I am sure I have taken up enough of your time. But, I am now aware of more philosophical viewpoints on the big questions!
I think that's a very good outcome, thank you for pursuing that, it's always interesting to reflect on such topics, to me at least.
> It may be interesting as a thought exercise to wonder if non-contradiction could be an emergent property. It would certainly make for an interesting model for the beginning of the universe.
Any theory of reality must have full closure. Meaning what happens when two constraints/structures collide depends on the two components, but that there is always a definite result, and it is the solution to conservation.
It isn't just that reality doesn't or can't ever "contradict" itself, but that contradiction dodging doesn't happen either. Bolted on contradiction avoidance, isn't needed when there is closure. Mixing up logic as tool, with logic as inherent reality is a category error. Things cancel, reduce, etc. All operations that happen locally where structure connects. They don't produce potential solutions, then winnow them down. We do that.
Thats what we do when we don't completely understand something, or we only have partial knowledge. Not scenarios that reality operates in. Entirely different.
> I think this would probably be contentious, I think most people believe that logical values are basically "necessary". I don't really know though.
Unknown is obviously a factor of notation, or our knowledge. Its dual is contradiction. Those concepts are foundational in terms of our ability to analyze things we only have partial knowledge of, or may be describing incompletely or inaccurately. Analysis needs that meta-level to handle all the provisionalism inherent in conjectural knowledge.
What is true, what is not?
What don't we know?
What have we incorrectly assumed?
We cannot operate on provisional/conjectural knowledge without all four of those concepts. (We rarely actually operate with just boolean true/false logic, even if unknown/contradiction are handled implicitly instead of explicitly, which only works in trivial systems. Which makes the major proofs of fundamental limitations' dependency on an excluded middle a bright red flag.)
Logic is how we analyze arbitrary structures. In that sense it is universal. Whether the artifacts we analyze have any resemblance to logic or not. We traverse them with logic as an external scaffolding that allows us to represent and operate directly on our understanding of the artifact.
Keeping those things separate sheds a lot more light on things like mathematical and computational limits. I.e. the weaknesses of starting with logic in Russell's Paradox and Gödel's theorems, instead of structure-agnostic reduction, are pretty stunning. Of course a system fundamentally restricted to true/false, without explicit "either/unknown/undetermined" or "both/contradiction/overdetermined", can't be consistent or complete.
Analysis without the four possible states of analysis inherently breaks. Whether that is a Russell set ("a set that includes all sets that don't include themselves", given sets are defined as boolean mappings), or a Gödel system (a Boolean mathematical system can't consistently handle something as simple as, "this system will say this statement is false"). Not how both of these examples pull the means of analysis into the contradictory statement loop. A set, in a set system. The mathematical system, into analysis of itself. So analysis as the subject. True, false, unknown and contradiction need to all be first class values in any study of analysis.
How would you feel if I sent you an email? I have a question but asking offline works better.
I don't really respond to email. I prefer to keep things scoped to single discussions on forums. Beyond that, I'm simply not an expert on these topics so I suspect we'll hit limits almost immediately to what I can meaningfully engage in before I either lead you entirely astray or simply have nothing to provide.
There are numerous academics in this area though who I suspect would be quite interested in discussion, and I'd point you towards the Stanford Encyclopedia of Philosophy as the defacto resource for these topics.
I'm actually quite sick with a fever currently so you'll have to excuse me for such a brief response, I'm barely awake as it is.
If you change your mind email me. I am collecting a few people for informal feedback on a series of essays and programming concepts. Noticed you have a Rust background which is relevant.
Yes, I am confident. But not ambiguous that my confidence is backed up with some aesthetic factors, not just squeezing out the hand waving.
> FWIW you can actually just say that you accept infinite regress.
I think my claim that everything that has a specific value, has been constrained to it, is very strong. And the inverse, that where something isn't constrained, it will appear in all its forms.
In that sense, I admit brut facts. We don't need to explain a specific, if we know its alternatives play out disjointly. Exhaustion instead of determination.
(The possibilities in that survey didn't include exhaustion! A much simpler explanation than evolutionary universes, or inverting the arrow of the anthropic principle.)
Superposition and entanglement are an exact example of mutual determination/exhaustion. Along with cancellation as a direct expression of conservation.
But my view that finite regions are potentially always characterizable with an unbounded finite region it is embedded in, as apposed to characterized by an infinity of other independent specifics, is easily my weakest conjecture.
> You can also just reject the law of non-contradiction.
I think contradiction is a category error for primitives of reality. Conservation is the right view.
I make a clear distinction between descriptions, partial models, conjectured models, etc. Contradictions can occur in our descriptions. But I don't believe reality is constrained by logical primitives. Just conservation. And with primitives whose generative/reductive properties are not constrained by logic, because there are not even "virtual" contradictions to prune.
I expect logic to be necessary for us to reason about primitives. But that logic is the wrong way to think of basis elements. Those are very different things.
> A theory with no axioms would be desirable.
I think this is necessary. I am 100% confident in that, for what that is worth to anyone else. Realty is able to exist precisely because it requires no priors, and has structure because it creates nothing, destroys nothing (conservation). Tautological anti-axioms that become axioms.
> But also, like, we wanted that in mathematics and accepted that it was impossible too.
This is not nearly as well established as is assumed. Whether we look at Russell's Paradox, or Gödel/Turings incompleteness theorems, there is a repeated issue. The proofs are about machines (or sets) that return (or are defined by) true or false. But any logical system with open (cycle) expressions admits two more possibilities, undetermined and contradictory statements. Not for some deep reason, but because notation is not reality, and we can trivially say things with notation that are undetermined or contradictory. "x in {x}, is unknown". "x in { logical x = not(x) }" is a contradiction. A universal mathematical discriminator needs to be defined as first, determining whether a statement is unknown or a contradiction. Then decide true or false for remaining expressions.
Note that the problem is implicitly collapsing unknown into true (i.e. "satisfiable") and contradiction into false ("unsatisfiable"), but then still treating those values as if they were just primitive true and false, which they no longer are.
To be concrete: If M is the mathematic machine, and we defined it reduce expressions (any kind of reducable structure, including 4-valued logic, not just Boolean truth), and it is tasked with evaluating a copy of itself operating on a self-referencing contradiction "M will say this is false", it is trivial to show M(M("M will say this is false")) = M("M will say this is false") = <contradiction>. No inconsistency, and no window to rework the statement into a problematic alternate, because we have avoided a special dependency, inconsistent treatment (t/f vs. u/c), or premature limitation around logic.
M may still have limits, but Gödel's incompleteness is not one of them.
This is another indicator to me, that a fundamental description of math (as with reality), needs to be based on sub-logical primitives. Another big clue is that Boolean logic is not reversible. Logic will be easily created from the fundamental relationship, but it creates a premature deadend to start there.
There are a number of well accepted famous theorems with this limited scope problem. They are absolutely solid proofs. But they define something more restricted than it needs to be, then knock it down. Leaving trivial possibilities unrestricted.
Cantors diagonalization proof for cardinality of R being greater than for N, is also trivially defanged. But there are better reformulations of it, and I haven't had the time to work though them yet.
I find that mathematicians, like everyone else, have trouble truly seeing what they looking at. I could go on and on.
Well thanks for pushing me, I am sure I have taken up enough of your time. But, I am now aware of more philosophical viewpoints on the big questions!