/sci/ - Science & Math

>There is no way to know if a mathematical statement is actually true?
It really depends on how strong the proof must be. For example, the Riemann hypothesis is provable by inductive reasoning. However, this proof is not strong enough to win the Millennium Prize.

>>15225448
A formalized mathematical theorem is always one derived from axioms. Axioms are chosen. Derivation rules are chosen too.
Often they are well motivated. Pic related are a few nice axioms. Let's you prove the bulk of arithmetic statements of interest to you. The ones that this (first-order) theory doesn't prove are sparse or are already statements about analysis.

>There is no way to know if a mathematical statement is actually true?
With respect to accepted axioms and derivation rules, we can justify when something is provable. This gives one notion of "truth".
An capital T truth, independent of axiom choice, is more a philosophical idea.

>Then how do we know the incompleteness theorems are true?
It's "just" a non-existence statement in arithmetic that we can interpret to be about proofs in arithmetic.

>>15225448
>There is no way to know if a mathematical statement is actually true?
that's not even close to what the theorem means

>>15225448
No, there are axiom systems which are decidable.

>>15225448
> There is no way to know if a mathematical statement is actually true?
False. If a statement is proven then it is true. What you can't know is if given some question whether it is provable or not.

>>15225705
>What you can't know is if given some question whether it is provable or not.
it's not just "some question," it's a specific class of questions that can only be asked in a language which permits both recursive and complementary construction of terms.

>>15225456
>For example, the Riemann hypothesis is provable by inductive reasoning
Are you a social scientist?

>>15225837
more like a physicist

>>15225456
>Riemann hypothesis is provable by inductive reasoning.
Please show me this proof.

>>15225673
OP seems to be the anon who is always conflating incompleteness theorem with the undefineability of truth theorem.

>>15226516
His first assumption fails on the first number since zero and its negation are both the same as a positive zero value.

>>15226519
No zeros have been found on the critical line that contradict the Riemann hypothesis, despite intense computational efforts, and there is no reason to believe that such zeros exist. Therefore, the Riemann hypothesis is true. QED.

>>15226519
>riemman hypothesis unsolved even doe it would be bad for math for it to be unsolved

>>15226524
That just means being a negative number isn't the negation of being a positive number, even though it sounds so. He's not wrong, not there at least.

>>15226592
Then why does a negative number clearly negate its positive counterpart and why does his assumption hold up for every number except 0?

>>15225448
>So let me get this straight. There is no way to know if a mathematical statement is actually true?
False. He never said this.
>Then how do we know the incompleteness theorems are true?
Understanding.

>>15226553
This doesn't work like that in math. There's been "proofs" like yours that stood for centuries until we found one exception somewhere in the range of Graham's Number < x < Infinity

>>15226553
Scientific induction is evidential. It can't prove things. Using it in a proof is fallacious.

>>15226676
>He never said this.
Tarski did though and godel never refuted the undefineablity of truth, he contributed to the concept with his incompleteness theorem.

>>15226679
It doesn't work like that in science either. There have been theories, like geocentrism, that stood for thousands of years but turned out to be wrong. Confusing evidence with proof is fallacious.

>>15226685
Words don't need to be defined to be understood. People understand what the word apple means without referring to a dictionary. Definitions are useful for precision, but not every word can or needs to be defined.

>>15226689
>There have been theories, like geocentrism, that stood for thousands of years but turned out to be wrong.
In science a theory can be good enough, so that it's useable, and brings about some accurate ish predictions. Geocentric theory worked at predicting the movement of stars and planets for thousands of years and people navigatie using it, except at some special cases it didn't. And people wondered why. Then came heliocentric and these special cases were solved, but new knowledge helped us see that even this isn't accurate. There's actually barycenter around which all solar system bodies orbit, including the sun. Point is new knowledge doesn't completely supplants the new one, instead it complements it or extends it. Unlike math where proof has a certain weight that it's impossible to ignore.

>>15226697
>People understand what the word apple means without referring to a dictionary.
No, they understand what apple means because someone explains what an apple is for them at a young age, so it becomes intuitive when they are older, but they still might not understand the difference between an apple and another apple like fruit without someone else to explain.

>>15226685
The undefinability theorem just means that a formal language can never define what 'truth' is. It doesnt mean nothing is true, its just there is no sentence that is logically equivalent to truth. So you can have truth in a system, but the system itself cant tell you what it is. You kind of just have to have faith that the system is a good representation of reality and capital T truth.

>>15226741
Exactly like OP said, there is no way to know if a mathematical statement is actually true.

>>15226768
No thats not what it means. It means you have to assert that the systems truths correspond to whatever abstract idea of truth you have's truths. If you really think about it you cant actually define truth anyway, even in the capital T abstract sense.

>>15226792
You only have to do that because there is no way to know if a mathematical statement is actually true, so you have to look at it from a variety of ways to see if it is good enough to appear truthful from a different frame of reference.

>>15226795
You need to describe things as true within a given system, even if that system is just natural language. Otherwise you are just asserting some transcendental notion of truth.

I learnt that in my master's degree. Thats some... complicated math over there. ohmann@pptp.de

>>15226804
All I see is you doing is repeating OP's statement over and over that there is no way to know if a mathematical statement is actually true.

>"so let me get this straight"
>proceeds to utter the most wrong statement imaginable
why are /sci/ posters always like this?

>>15226812
You are just asserting the existence of this transcendental, system independent notion of truth, and then using the fact that you prove truths in systems and therefore that they may not be the same as this transcendental notion of truth to say that you can't know whether something is true. Try and define your idea of truth for us

>>15226834
>Try and define your idea of truth for us
In other words, you can't define truth and you are shifting the burden onto someone else because you basically agree with Tarski and OP.

>>15225448
Some mathematical statements can have an undefined truth value, e.g. Russell’s paradox.

https://www.youtube.com/watch?v=mhIkyqLDl9M

>>15226845
Tarski didnt say what OP is saying, I am asking you to define truth because you are using this system independent idea of truth as the real one, and then mathematics inability to capture whatever your idea of truth is somehow makes it unable to be true. Hence why I want you to define truth

>>15226858
I am not using any independent idea of truth, I am using the same definition Tarski is using which is arithmetic truth which is something that is undefinable within arithmetic.

>>15226880
Well then you should know that tarski just tells you that you can prove things are true in a system, but not what "truth" actually is. This doesnt mean you dont know if statements in a formal language are true, it just means that the language cant tell you what "true" actually is, so you just have to assume that the systems idea of truth is the same as your own transcendental one. This is not hard to understand

>>15226521
what is it about this board hat attracts such niche retards?

>>15226521
as it turns out tarskis undefinability theorem is a corollary, though they obviously say different things

>>15226902
>you can prove things you know are the same as a thing you don't actually know
No, you are wrong, you can't even define arithmetic truth using arithmetic statements.

>>15226905
You clearly just have to ask yourself what exactly brought you here if you want the answer to your question.

>>15226915
Thats what I said in my post. I feel like you should spend more time reading up on algebraic logic and model theory and less on pseuding out

>>15226922
Yes and that is what OP said in his post, so I don't see what you are arguing since you seem to agree that there is no way to know if a mathematical statement is actually true.

>>15226936
The fact that you cant tell the difference between " a formal language can't define truth in itself" and "you can't know if mathematical statements are true" means you need to do a whole lot more reading. There is a very large difference between those statements.

>>15226685
>Tarski did
no, he definitely didn't. it's entirely possible to weaken a language enough to defeat tarski's undefinability theorem, just as you can weaken arithmetic enough to defeat gödel's incompleteness theorems.
>>15226521
that's irrelevant because tarski doesn't say what anon thinks he says anyway

Not quite.
On /sci/ it is true that there is no way to know if a mathematical statement is actually true.
However, on /lit/ it is false.

>>15227407
>/lit/ can't into recursion and is thus un-Gödelable
checks out

>>15226676
>Understanding.
>>15226697
>Words don't need to be defined to be understood. People understand what the word apple means without referring to a dictionary. Definitions are useful for precision, but not every word can or needs to be defined.
Based and non-computational/algorithmic mind pilled

>>15227442
even worse
suppose children learn multiple, unrelated things using precisely the same grammatical constructions, to the point where it is obvious to the children that it is merely the case that words are being exchanged with no change in the grammatical structure of the context in which they are used
at what point do the children stop learning the things and start learning a "wordplay grammatical exchange game"
>>15227424
mathematicians think "wordplay grammatical exchange games" are objects of study
philosophers can see through this sophistry
mathematicians can (but generally don't) keep going, producing infinitely many theorems in infinitely many new symbolic formalisms
mathematicians aren't paid to understand the limits of symbolic formalisms...quite to the contrary, they are paid to produce more of them
the /sci/ mathematicians won't stop, always exploring the next layer, turn themselves into
>le RUSSIAN DOLL EXTREME MATH ENVIRONMENT HAZMAT SUITS
the /lit/ philosophers will stop...they will stop at the Russian doll hazmat suit clown act

>>15226516
Someone should catalog the amount of working memory needed to understand mathematical proofs like this.

>>15227424
On /lit/ all statements are true, anon.

>>15227467
>tfw you can't give the response you want to give without potentially doxxing yourself to anyone who's read your grad school applications

>>15226516
You can do the same thing with anything that is "the best". The famous example is the best paradise island. The best island would have to exist otherwise it wouldnt be the best as existing is a good thing and things that are the best have all of the good qualities, one of which is existence.

>>15227488
https://www.youtube.com/watch?v=t6c8ul8d2uw
>ASS
PARADISE

>>15227479
>implying we want to read your essay
We aren't going to read it.
We barely read each other's comments.
You're going to have to paraphrase it.
>I'm afraid of le Russian Dossier
go run for President, wise ass!!!

>>15227488
I want to assure you that my island has all of the best qualities. Really full, voluptuous qualities. Very smooth, very supple qualities.
I tell you, these qualities feel very good. Very very good. These are the finest qualities.
>LUXURY
QUALITIES

>>15226720
False. It's impossible to define every word without using circular definitions. Eventually you run out of words to use in definitions. Every system has primitives. People understand which essence a primitive refers to through example.

>>15227747
natural languages actually do use circular definitions, though. undefined primitives are a feature of axiomatic systems specifically. there's also the infinite regress approach of continuously creating new terms to define old terms, and natural languages include this method too.

>>15227766
natural languages don't use definitions
therefore natural languages don't use circular definitions
only mathematicians and philosophers use definitions

>>15227766
>there's also the infinite regress approach of continuously creating new terms to define old terms
that's a con, i.e. confidence scam

>>15226516
What is a positive property?

>>15226697
All words can and need to be defined and nothing can be understood without a definition.

>>15227863
This seems unlikely. There was a time before definitions even existed. Certainly certain phenomena must have been understood to imply that certain other phenomena would follow from the occurrence of the prior phenomena. So a dark and rumbling sky can be imagined to have been understood to imply impending rain to a primitive tribe, even before the advent of language.

>>15227863
yeah but people are constantly using wrong definitions
it was worse before 2006

>>15227804
the first dictionary was published long before any mathematician ever considered a formal language as an object of mathematical study
>>15227807
i'd wager your vocabulary includes words that didn't exist a hundred years ago

>>15227972
no no no, you don't understand
it's like saying
>Here, let me tell you the definition of snarklefabbits
>Okay, a snarklefabbits is a donkomohaire
>Ah, let met tell you what a donkomohaire is
>Fine, give me $10
>A donkomohaire is a trundleabra
>Oh, you aren't happy with that definition
>give me $20
>A trundleabra is the fruit of the darschnaps plant

>>15225448
Everything is made up. Some of our made up things are similar, but that's hard to measure (because that's also made up). Welcome to the fucking show.

>>15225448
Roger that.

>>15225678 Axioms are cool.

>>15228262
https://mp3.hardnrg.com/morgan/Morgan-Decepticon.mp3
>THIS
IS
>MY
HOUSE

>>15226516
>Gave into to jewish fairy tales

KEEEEK MY SIDES!!!

>>15227955
>a dark and rumbling sky
poet detected

>>15225448
>There is no way to know if a mathematical statement is actually true?

--- godel's theorems --->
[you head]

>>15226516
This results in modal collapse, axiom 4 is ridiculous; no explanation for what a positive property even is or why it is a priori.
Besides, all modal ontological arguments are flawed since the contrapositive is [math]\Diamond{\nexists{xG(x)}} \implies \Box{\nexists{xG(x)}}[/math] and these arguments don't give me any reason to believe that it's any more reasonable to say "it's possible there is a God" than "it's possible there is not a God".

>>15230409
Requesting a recommendation for a modal theory text.

Still seething about the vulnerability of the linux clock?

>>15230409
>no explanation for what a positive property
You just have to have faith

>>15225448
That's not what the Incompleteness theorems say

>>15227857
It's a property that some object has which makes that object "better with it than without." In other words, something he made up.

>>15226516
>but not both
>necessarily
Where did he pull these terms out of?

>>15231294
https://en.wikipedia.org/wiki/Ultrafilter
https://en.wikipedia.org/wiki/Ultrafilter_(set_theory)
>tfw you realize Gödel was right all along and God is an ultrafilter

>>15228890
That argument for a god has nothing to do with jewish fairy tales. It's based on a classical theistic concept of god
>Classical theism is associated with the tradition of writers like Plato, Aristotle, Philo of Alexandria, Plotinus, Proclus, Athenagoras of Athens, Clement of Alexandria, Basil of Caesarea, Augustine, Boethius, Cyril of Alexandria, John Damascene, Pseudo-Dionysius the Areopagite, Al-Kindi, Al-Farabi, Avicenna, Anselm of Canterbury, Maimonides, Averroes, Thomas Aquinas, Leibniz
The abrahamic religions are generally consistent with these features, but so were some non-abrahamic concepts of the deity. And many of the other non-formally logical arguments such as contingency and cosmological ones don't either have to be tied to strictly abrahamistic beliefs either. Gödel just was interested in anselmian and leibnitzian arguments for god and formalized them. He was just more of a general theist. Any highly intelligent person is going to eventually figure out that naturalism is a fake news philosophical belief.

>>15231319
>tfw you realize Gödel was right all along and God is an ultrafilter
true, as stated here in this kek blessed post
>>15232355

>>15231294
>In other words, something he made up
He didn't make up ontological arguments, idiot. He just formalised and strengthened versions based on his research of anselmian and leibnizian non formal ontological arguments. He would always try to formalize just about everything according to some who knew him. His efforts have been carried on as well.
>The overall idea of Gödel's proof is in the tradition of Anselm's argument, who defined God as some entity of which nothing greater can be conceived. Anselm argued that existence in the actual world would make such an assumed being even greater; hence, by definition God must exist. Gödel's ontological argument is clearly related to this reasoning pattern. However, it also tries to fix some fundamental weaknesses in Anselm's work. For example, Gödel explicitly proves that God's existence is possible, which has been a basic assumption of Anselm. Because of this, Anselm's argument has been criticized as incomplete by Leibniz. Leibniz instead claimed that the assumption should be derivable from the definition of God as a perfect being and from the notion of perfection. Gödel's proof addresses this critique, and it also addresses the critique of others, including Kant's objection that existence should not be treated as a predicate. On the other hand, Gödel's work still leaves room for criticism, in particular, his axioms are so strong that they imply modal collapse, that is, a situation where contingent truths and necessary truths coincide. More information on the philosophical debate on Gödel's proof is provided in

>>15231319
>>15232355
>>15232360
>>15232384
So, why did Gödel go up to the ultrafilter, sit down, meditate, and get hit on the head by one of these, that fell from the ultrafilter and hit him on the head, then he uttered
>I have found God!

>>15231170
https://www.google.com/books/edition/Handbook_of_Modal_Logic/urINMvvsT5MC

>>15232401
>>15227857
In this case, the positive property would be that of being a necessary being or entity, ie not/negation of contingent/contingency. Pic related speaks to this. So now there is an entrance into the realm of the principle of sufficient beings. So if
>Anything that exists has an explanation for it’s existence, either in the necessity of it’s own nature or in an external explanation
This the salient positive property is that of necessity. This is, by the way, also an important part of non-formally logical arguments (not YET formalized) such as cosmological arguments. The universe began, and so it is a contingent being. It could have NOT began, ie, the creator did not have to push the enter button to boot up the universe. Contingent beings demand an explanation. The explanation, ie the external cause can not ALSO be a contingent entity, or else you get an infinite regress of contingent entities, ie, 'well then, if the universe needs a cause, then who caused the cause which caused the universe', ect ad infinitum. The infinite regress can be stopped as well as the avoidance of the multiplication of entities beyond necessity by postulating A SINGLE, non-contingent, ie necessary, ie POSITIVE entity, namely 'god'. A single entity outside of temporal constraints (virtual constraints) such beginnings (bootings up) cycles ect.

>>15232401
whoops, this
>>15232617
was not to you
>>15227857
this here
>>15232617
> Pic related speaks to this. So now there is an entrance into the realm of the principle of sufficient beings.
should be
> Pic related speaks to this. So now there is an entrance into the realm of the principle of sufficient REASON.
principle of sufficient reason, psr

>>15232617
this is a philosophical argument and so doesnt belong on sci. But also this relies on the explicit proof the the universe is contingent but strictly not necessary. Your argument relies on a creator not pushing a button, but it is possible that the universe is necessary, that it is a self supporting web of law, or that our universe is just a small subset of a much larger, potentially eternal multiverse. In other words you can't prove that the universe is not necessary, or that any larger structure the universe is contained in is not. This is of course meaning that the universe is possibly necessary, which depending on your axiom system S5 imply it is necessary, or even if I am fucking something up here, you still have to show that it is ONLY possible and not necessary.

>>15232678
What do you want it to do exactly?

>>15225448
Proof: "here's how we know things"
Incompleteness: "there are sometimes things we can't know"
What's the problem? Instead of proving something true or false, you prove something is true, false, or incomplete. How exactly does this "end" math? Or even spell trouble for it?

>>15232700
Read dumb ass bitch. You will need to show that the existence of the universe, or any structure that the universe is contained in, is not necessary. This is not generally possible because if the universe happens to contain physical laws that lead to its own rebirth then there could have been an infinite number of universes before us. If you label us universe 0 and the previous one universe minus 1 etc it is possible that there may have been infinitely many universes before us, meaning that there is no first mover as there is now lowest integer. If the universe is contained in some eternal multiverse then the multiverse will need no creator as well, and so no first mover. There are possibilities that will have to be ruled out in order for you to definitively claim that the universe is only contingent.

>>15232678
>this is a philosophical argument and so doesnt belong on sci
eat shit. I will talk about whatever I want, whenever I want and you can't and won't do anything about it.
>but it is possible that the universe is necessary,
No, it is not, not unless you want to appeal to an infinite temporal regress and an actually infinite amount of prior events. Infinity is also a meaningless concept in terms of physics and what can be observed/measured. And in terms of observable data, all observable data points to a point to which it can be traced back to where both the space and time were initiated. As far as an infinite universe, this is a meaningless conjecture which could never be verified empirically. And logically, it is absurd in terms of infinite temporal regress.
>potentially eternal multiverse
There is no multiverse. There is no way to even empirically test this. This is a cope that atheists appeal to to try explain away fine tuning. We by definition have no access to these supposed universes, so this concept is just wank.
>This is of course meaning that the universe is possibly necessary
No, it isn't. You can't have an actually completed infinite past yet still be adding to it. This is is contradictory. You would also not have any initial conditions in such a situation so where are you getting your causality from?
>or even if I am fucking something up here
Yeah, you are fucking up everything. The first fuck up is trying to come up with giant coping mechanisms such as 'multiverses' and such instead of the simplest answer which avoids these ideas that multiply entities beyond necessity and just accept that there is one universe and this universe began, as all empirical data suggests, and that since this universe began, then this requires an external explanation, and this external entity can not ALSO have began, or else you get an infinite regress of contingent entities. So no postulation of infinite multiverses. This is the most parsimonious answer.

The incompleteness theorems are garbage.
Mathematicians aren't honest about the extent of the contamination.
Many many books of set theory and logic have rotten parts.

>>15234094
They patent this crap son

>>15226516
All this nonsense stuff ignores the simple obvious proof God exists. Existence and reality is so banal and evil that it could only possibly be designed. Something this needlessly cruel wouldn't come about from happenstance.

>>15225448
I don't get it myself

>>15237403
>obvious
Do those who use this or trivial have self awareness?

>>15225448
An incompleteness theorem says that there are true statements with no proof in Peano aritmetic. The Paris-Harrington Theorem cannot be proved in Peano arithemtic but it is still true, I beieve its proof uses a compactness theorem. That example is notable beacuse it's one of only a few facts that can be stated in the language of Peano arithmetic but not proved in it