Imagine you are an alien to this world. Your logic is different from the one we use everyday. Maybe the pigeonhole principle doesn't make sense to you. Maybe the excluded middle is a strange concept to you. Or non contradiction.
Given this, what would your mathematics look like?
I'm asking because of our limitations: mathematics seem like an adequate tool to model and describe the universe. But when we hit a wall like sheer randomness in quantum physics, some people have a hard time wrapping their heads around it. Could a different logic lead us to different mathematics and different descriptions of the universe?
Is dialetheism the future of our civilizations ?
Do you know of any mathematics developped in another "logic"?
Everyone in this universe follows the same logic, if they didn't we would have established alien contact some time ago.
Have a look at fuzzy logic user. A proposition is assigned a truth value which may be any real number on the interval from 0 to 1.
For example, suppose we can agree that a nice orange-red is halfway between those two colors. Then P(x) is the statement. "This color is red." P(x) is assigned a truth value of (about) 0.5, etc.
As for logic and simple math, I think that animals capable of abstract thought along the lines of what we're capable, of, have certain truths imposed on them from without (by the world), which they must then express as definite statements in that abstract thought. My point being that I would not expect alien logic and math to be wholly unrecognizable to us. For example, the pigeonhole principle is a physical fact of the world, imposed on us from without. So perhaps they didn't think of it, but that don't mean it ain't so.
Hi thanks for your response. I know about fuzzy logic but I'm not sure this is what I'm talking about. I feel like fuzzy logic is just some other mathematical tool, developped within the realm of our own mathematics. Does that make any sense to you? I don't think the pigeonhole principle is a fact of the world. That's exactly my point. It makes absolute sense to us because we cannot fathom a situation where it wouldn't be valid. But maybe such a situation exists but our limited minds can't understand it. We can't test it, we can only deduce it using our own logic.
[math](p \implies \neg p) \implies \neg p[/math]
>I don't think the pigeonhole principle is a fact of the world. That's exactly my point.
You are mistaken about this, and it's a source of confusion for you. If you persist in this view, the topic of your thread will not be able to pass beyond vague unscientific speculation and generalities.
how am I mistaken about this?
Don't paternalize me. Either bring something useful or don't say anything. I'm not here trolling or fresh out of undergrad. I want to explore the limits of the way we think.
[math]\displaystyle \left( \left( p \implies \neg p \right) \implies \neg p \right) \iff \left( p \implies \left( p \land \neg p \right) \right)[/math]
[math]\displaystyle \left( \left( p \implies \neg p \right) \implies \neg p \right) \iff \left( p \implies \left( p \land \right) \right)[/math]
The above equivalence is interesting in that while the LHS states a useful rule (i.e. "if p implies its own negation, then it must be false"), the RHS seems rather trivial and useless ("if p is true, then it's both true and true").
Another error. It should be
[math]\displaystyle \left( \left( p \implies \neg p \right) \implies \neg p \right) \iff \left( p \implies \left( p \land p \right) \right)[/math]
The problem with throwing out any of the classical laws of thought is that anything becomes provable, making logic completely useless as a tool.
>I don't think the pigeonhole principle is a fact of the world
implies that you don't think the law of excluded middle holds, and you can now use this new axiom (it's an axiom because it's not based on fact or argumentation, you're asking us to accept it at face value) to prove whatever you want.
Aren't qubits both 0 and 1?
why do I even bother asking this shit here. Maybe I secretely hope someone else will understand.
What? Dropping axioms makes less statements provable and (once other axioms are added that were in contradiction with the ones dropped) it opens up room for a broader class of statements to potentially be provable (but again, only once you added more axioms.)
I think that there are two main hurdles between our minds and the absolute truth (if the latter exists in the first place). Both of them happen to be physical. The first (and probably more important) is the limitations of our physical brains and the way they are "wired". The other is the physical world around us which is the only palpable thing we can observe in our life, and which greatly influences (and to a certain extent also limits) our reasoning.
The mind of each of us is trapped inside two boxes - one is our physical brain which the mind cannot step out of, the other is our physical world which we cannot step out of at all either.
Didn't read your post but here's a rare kot blini
You're not hard to understand.
But you also make a lot of explicit statements, the way you formulate your question. E.g. >Your logic is different from the one we use everyday. For one, this implies that the notion of "logic" (beyond mathematics) is valid. People came up with lots of what's called formal logics en.wikipedia.org/wiki/Template:Non-classical_logic and those, like classical propositional logic, say, try to capture modes of reasoning that became helpful. E.g. we feel statements like "I can't be a man and an elephant at the same time" express a truth about the world. And now there are some people here who think there is actually something like a logic that applies to this world, and they argue aliens must have the same logic because of it.
Give or take... you then ask >Given this, what would your mathematics look like? Well when you ask about mathematics, people have worked out a long array of formal logics. Pic one you like and study it. If there are aliens who are under the impression that there is a "true logic" and it were to coincide with that one, you have answered your question.
>Could a different logic lead us to different mathematics and different descriptions of the universe? In a sense no, because second order predicate logic (or first order logic and any theory of "collection") allows you to "construct" (or at least specify) "things" and it also let's you formalize other languages, and thus logic with modal operators or whatever can be modeled within them. That's often called semantics for this and that logic. E.g. en.wikipedia.org/wiki/Kripke_semantics for the logics of possible/believe/knowledge/...
everything you mention is proved and deduced from assumptions and logic we all agree upon like "a proposition can't be both true and false"
I'm not sure what you're saying.
Famously, the classical claim [math] P \lor \neg P [/math] "a proposition is either true or false" is not part of intuitionistic logic, while [math] \neg ( P \land \neg P ) [/math] "a proposition can't be both true and false" is.
Or, the classical claim [math] \exists t. \ B(t) \to (\forall s. \ B(s)) [/math] "There is something, so that if it's a bird, everything is a bird" (because if there is something that isn't a bird, then set it to t and false classically implies anything) is not part of relevance logic, which defines "implies" in a more complicated fashion.
There is a bunch of formal logics and claiming Frege got it right on first try in 18xx is more than fishy.
They are classically equivalent, yes. That's the logic you implicitly use there. But some of your \leftrightarrows are not provable in intuitionistic logic (pretty much by definition), ¬(¬p∧¬(¬p)) doesn't imply p∨¬p there
The statement (A ∧ (A=>B)) => B is true, because "if you have an argument for A true and a way of turning an argument for A to an argument for B, then together you have an argument for B." And ¬(¬p∧¬(¬p)) can be read as a special case of the above.
However, p∨¬p does, constructively thinking, not involve an implication (or a conversion of an argument) and just claims that you either have an argument for p or an argument against it. But what's that argument supposed to be?
So the DeMorgan laws don't work there? Only these (one of them in particular, namely [math](p \lor q) \iff \neg (\neg p \land \neg q)[/math]) are needed for the two to be equivalent.
It's not a law within this logic, yes, the mirroring operation actually takes you to another logic, and the related statements collapse classically.
I suppose these different logic systems must have different operators/truth tables (as all the tautologies are a direct consequence of the truth tables of operators)?
>But when we hit a wall like sheer randomness in quantum physics When will this meme die
Well, you can also just strip off axioms/deduction rules and you get a weaker logic.
Evaluation to truth values is a form of semantics. Some logical operators are not associated with truth values as such, and have different semantics (mentioned above) than what essentially amounts to Boolean logic.
Basically, second order logic (or first order logic+set theory), is an everything goes logic. That's of course convenient for math, but harder to justify.
Can everything be formalized?
say hello to nonary logic my friend
the big brother of ternary and imho the future of computing
Christ, those pancakes look delicious.
I have two integers between 2 and negative 2, inclusive.
The sum of these numbers is 2.
What two integers do I have?
Unless you believe there exists a form of logic that can provide the unique correct answer to this question with no other information, you accept the pigeonhole principle.
It must be either (0,2) (or also (2,0) if the order matters), or (1,1) (provided the numbers don't have to be different). How does it relate to the pidgeon hole principle.
I don't get it
Please explain how the (quite trivial) problem you posed relates to the pigeonhole principle.
The cat looks as if it means to say "lel, I didn't eat any of the cream", yet the white speck on its nose reveals otherwise.
I understand exactly what you're getting at. Explicitly stating it like you did is the best way to avoid thread derailment, but the concept you're talking about it somewhat murky.
I was looking further into the idea behind having mathematical axioms, and really, it's just completely arbitrary. Using arbitrary axioms based on logic that's familiar to us, has allowed us to map a good deal of all mathematics so far, but what if we developed our rules, our axioms, using an entirely different form of logic? The idea is fascinating, but the pillar of logic is consistency. Can a human consistently create and follow logic foreign to him as a human?
I mean, it appears obvious enough to me there are forms of logic we can never consolidate just do to our brains development, and use.
Our brains are capable of a lot of stuff, but most of that stuff isn't useful to survive or reproduce, so we can't do these things, and rely on the same neural pathways used since the dawn of mammals.
A good example to understand my digression would be a look at the new results from brain mapping individuals on LSD. They experience ego dissolution, which allows them to utilize different parts of their brains, in different quantities, communicating differently. There's a world of possibilities for what our brains can produce, but for pragmatic reasons we never see it.
Hopefully this wall of text will make some sense, but I left it sort of ambiguous. I personally believe the logic familiar to us is only familiar because it's a product of faculties necessary for survival, and different forms of logic exist within the rather extraneous functions of the brain that we can only induce with drugs.
It's unmarked turf user, but there could be fruit, good talk