-
We have updated our Community Code of Conduct. Please read through the new rules for the forum that are an integral part of Paradox Interactive’s User Agreement.
I mean, yes, I agree that having good hygiene and good health is a good thing. But, on the other hand, stereotypes that draw people with bad hygiene and bad health as undesirable and as jerks are unkind and judgmental and not good things.
Dun dun dun!
...ouch. But also, that is extremely exciting! How do you know that?
I agree with this to an extent, and this is why "born this way" narratives don't work very well.
But I also think that a lot of things, like subjective emotions, aren't controllable. Empirically, gay feelings don't seem to react well to attempts to change them.
It's also unclear in what sense you think that .9 repeating is _actually_ 1. Both are just different ways of writing the same thing.
How do you know this?
Anyway, I can see what you're saying here. The patterns described by mathematics on paper as ordered systems create thoughts and concepts in brains, which probably are somehow analogous to the maths patterns. Which... I definitely agree with.
Except that it completely contradicts your original point:
Everything we do is based in reality, including maths, because according to what you said above, even thoughts are real because they have physical representations in reality.
So I don't get why, on one hand, you maintain that our concepts must be applicable to reality, and that if a concept (which is a thought!) doesn't represent something that exists, then it must not exist; and then on the other claim that even thoughts exist physically. There is a major contradiction here.
Correction- it doesn't likely exist if nothing can detect it or observe its effects. The fact of your not comprehending something is a fact about you, not a fact about the world.
Also, your epistemic model of true or false isn't all-encompassing; it also fails to account for uncertainties, of which, as fallible beings, are always possible. So a probabilistic approach *might* work better at modeling degrees of truth.
See this:
https://en.wikipedia.org/wiki/Truth_value
And the wiki article on truth as well.
Truth is really complicated, and I disagree with your simplistic assessment here.
Ultimately, if we have no way of ever observing, conceiving, in any way knowing, something, then it doesn't exist.
That's a different claim than the one you were originally arguing, and I would still disagree. It's highly unlikely that that thing exists, but it's still possible that it does.
This is where the true/false dichotomy fails, because it limits your ability to observe nuance. Is there the possibility, however small, that there is an thing out there that humans are literally unable of ever observing, conceiving, or knowing? We can't eliminate this possibility completely, because we're not infallible in predicting things. And so it's not true and it's not false; it's somewhere in between (no, this is not Schrodinger's anything and it is, as far as I know, a different situation quantum relationships). It's possible that it's true, but highly probable that it's not. So it has an extremely small possibility of existence that is nonverifiable by definition.
What about really really big numbers that are finite? I'm talking about numbers that are like the number of every atom in the universe to the power of itself. We can't observe the number, as it's so big that there's nothing it can be used for, yet logic dictates it still exists if you agree the number 1 exists.
Arguably, you can never really observe the concept of a number, except as a concept in human minds.
Numbers are real though, right? For all intents and purposes they are, since we can use them to help get airplanes in the sky, something that doesn't happen by coincidence.
What about 1/0? What about a circle's circumference divided by its diameter(pi)? Do those exist too?
Of course they do. As concepts, even if we can't precisely express them.
So what's the problem with infinity existing and 0.999 repeating equalling 1?
Eh, that's mostly just me not liking their existence, to be honest, and disagreeing that they are 100% accurately expressed. Conceptually they exist, though. Rather, I should say my belief in the nonexistence infinity is whimsical, whereas my opinion that only things we can think of express, observe, etc. are real is not. It's a fundamental belief of mine.
As a mathematician, Kiwi knows that any system will contain at least one axiom that can't be proven by the system itself.
But I can't remember exactly how it is that they know that.
I don't think that you should disbelieve in things that you agree exist, but that you don't like. They still exist even if you don't like them?
And the expressions aren't the concepts themselves.
Godel's Incompleteness Theorem, but I was hoping that Kiwi could summarize the explanation.
I can't summarise Gödel well. While I studied aspects of computability and related topics like Turing's solution to the halting problem or the Church-Turing thesis, it's one of the most arcane areas of maths I've ever come across and I'd struggle to do it justice.
The gist is that if you accept a system of axioms and operations must be enumerable - you don't necessarily have to have all the axioms and operations in front of you, but you have to be able to decide if a rule is in the system and if it's an operation how to apply it using some algorithm - then it's possible to represent each component of the system as a unique number in a way that also lets you number the expressions created from those components uniquely. So any expression in the system has its own number, and you can take a number and turn it back into the one and only one logical expression that could generate that number with all of the syntax and ordering preserved. Because we can transform anything in this fashion it means we can apply the theorem to pretty much anything.
(Gödel numbering is extremely useful in many contexts, and easy to learn - if nothing else I suggest you look into it because it's one of the easiest and most powerful tools in mathematics)
Given this, in some semi-arbitrary system satisfying some fairly basic requirements we'll consider some number x which is a formula that proves the expression y. Assuming you can actually decide if an expression x really does prove y (which is an assumption but one you have to make for the system to have much usefulness), that means you must have an algorithm showing the relationship between xs and corresponding ys (the algorithm might just be a list pairing xs to ys, or some rules on whether a statement is true or false, it doesn't matter; it's whatever lets you know you're right). And that algorithm must be part of your system.
However it can also be shown, through some arcane wizadry, that it is always possible to create an expression such that a particular statement D is derivable within your system if and only if there exists some arbitrary function of it A(D) within your system. And if you let A() be the negation of your proof formula, that means D is derivable in your system if and only if its proof is not in the system. That sounds bad because it is bad. After making a few sacrifices to demons you can prove that if your system is consistent it cannot be capable of deciding whether that particular D is true or false. With a few exceptions in the most basic but occasionally interesting examples of axiom-operation combinations.
This doesn't mean that you can't decide on D in any framework. In fact there always is a system you can prove it in, as at the very least you can choose it as an axiom. But it means there's no one framework that can decide on everything; there will always be something that you need to add or change axioms for.
I think I've got the details right there, but it's on the edge of what I can handle. I'm not going to attempt to simplify the second theorem right now, but the summary is that any system as described above must actually contain both the acceptance and negation of a particular statement that can be generated by a similar sort of procedure. And so the system is not consistent.
Mathematics still exists because the statements that your axioms can't prove might not be particularly interesting to you at the time, and when they are interesting you choose something else. But there's much debate on the broader implication that there's no one system of logic that can describe everything, or even be consistent with itself in all cases.
Ooh, this is interesting. Thank you and I shall.
So I am not sure if I understand this bit. What proof formula is A() the negation of? Is it the negation of D's proof formula?
If A() is the negation of the proof formula for D, then would the proof formula for D end up not being part of the system anymore due to having been falsified?
Also, in order for the arcane ritual to happen, how many candles do I need to buy? Wax, or tallow?
Huh, this is really cool and I think I *mostly* get all of what you're saying in this quoted portion. Thank you, and I really like math!