Thank you everyone for responding. Please keep in mind that I wrote that when my ideas about formal systems were more naive than they are now two years ago :D
On Mon, Jun 15, 2015 at 12:43 PM, Bruno Marchal <[email protected]> wrote: > > On 15 Jun 2015, at 17:15, Brian Tenneson wrote: > > I had forgotten I wrote this a while back, from my FB feed "on this day." > Seems relevant. > > Can truth ever be proven? > > That has no meaning. Truth about what? > Let's start with mathematical truth. Isn't it contingent? In other words, it seems to me that theorems are considered true but aren't they only true if the axioms are true and if the rules of inference are sound? Is logic infallible? If yes, can that be proven with logic? If no, what makes logic superior to other things like intuition and faith? (I know these are false dichotomies but I thought they might trigger some interesting discussions.) > > > > Here's something I wrote in a discussion I'm having. > > Structure does not cause something to be non-fictional, nor does lack of > structure cause something to be fictional. A theorem in one formal system > might be false in another, > > > Formal system = machine = 3p-person. yes, they have all theior on opinion, > but this does not mean that some are not true or false. > Ok, then, which theorems are absolutely true? (Of course the ones absolutely false would be negations of the former.) > > > > a lot like how different video games have different rules. Even if you > "prove" something about all formal systems, that "proof" has been carried > out in a larger formal system; > > > Not necessarily. Formal systems can prove a lot about themselves, > including their own incompleteness conditionalised on their consistency. > > > A la Godel's theorems, right? > > so there is an inherent circularity, > > > I think the one you allude to is the one solved by the diagonals of > Kleene. Not the time to say much more, but I have explained this. > > Sorry, I don't read every post in this group. Can you give me a brief tutorial on these results of Kleene? > > > or more accurately, an inherent interdependency. It's like being in a > video game trying to prove that something is true of all video games but > that meta-game proof is being conducted in one of the video games the proof > is about. Thus, the concept of proof needs to be anchored to something true > but by this rationale, proof is merely anchored to itself. > > > Anchoring proof on truth leads to the first person. > > That sounds reasonable. So how can you describe the connection(s) between proof of truth (in mathematics say) to the 1p? > > > Therefore, perhaps proof of truth is an unattainable goal in math. > > > Proof of which truth. In logic, truth is defined by a model. "A" is true > if it is the case that A in the intended model. > > This leads me back to the question earlier: is logic infallible? If so, is there a proof of that using logic? If not, then how can we use it with any confidence? > > > > > Perhaps proof of truth is an unattainable goal anywhere. > > ? > > Well, proof of Truth, with a big T is like the proof of the existence of > God: that does not exist, we have to start from something. But since Gödel > we know that "proof" means belief, and that it can't be taken for granted. > > > Isn't it true that once you lock in the definitions (of truth or god or Truth or God), the assumptions, what types of statements are grammatically correct, what the rules of inference are, and what constitutes proof, that conclusions are inescapable? If so, is THAT an inescapable or escapable theorem? > > > If I were to say that both confirming and denying the statement "there is > no such thing as truth" implies that there is truth, I am still formulating > that theorem "there is truth" within yet another formal system which, on > the surface of things, gets us nowhere. It is like inventing a two-player > game with, from an outside point of view, a bizarre set of rules, and > claiming that checkmating someone in that game amounts to producing not > just truth but proof of truth. The people outside our fishbowl looking in > on us must be very amused, just as are the people outside their fishbowl > looking in on them. > > Ok, that is Truth with a big "T", and so it needs faith. It is religion. > OK. > > > If I say, "there is no absolute truth," well, that is an absolutely true statement isn't it? IOW, if "there is no absolute truth" is false then there is absolute truth. If "there is no absolute truth" is true, then it is absolutely true. However, and perhaps you can explain this better than I, I think that quirky statements in language like "I always lie" don't prove much unless we start doing things like assuming logic is infallible which requires faith, does it not? > > Formal systems show us that our usual formal systems (the ones we use to > communicate, inform, and persuade in English for instance) have the same > relationship to truth that Earth does to the center of the universe. No > formal system is provably true and correct, though there are formal systems > that might conform to what we perceive. Formal systems can only be proved > relatively true compared to other formal systems. > > No, you can compared them to mathematical structure. That is why we have a > model theory. The situation is not that bad. We can use our intuition of > the finite to get the models. That is what we do when we talk about limit, > models, analysis, complex numbers, etc. > > > What has ontological primacy: sets or models? > > > At least until that anchor is found. > > That reduces math to a grand symphony. Grand symphonies aren't inherently > true or false and there is no hope in my mind of proving the grand symphony > that is math to be true. Another way to look at is is a grand poem that > makes up its own rules and even explicitly acknowledges that fact. > > The question of whether concepts referenced by the poem actually exist is > to open the door to many formal systems we might walk into in order to > answer the question. Moreover, it will be true in some but not others that > that concept exists. A really broad interpretation of existence would be > that something exists if it is referenced by a grammatically-correct > statement made in at least one formal system. > > > I think that we can agree on elementary arithmetic, then we can agree on > the fact that almost all universal numbers will disagree on what extends > arithmetic. > > Bruno > Why can't all 1p's agree that there are myriad and equally valid (or invalid) ways of extending arithmetic? -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
