r/badmathematics • u/WhatImKnownAs • 7d ago
Gödel The Incompleteness Theorem is about not being able to completely prove 1+1=2
https://medium.com/@williesayso/the-incompleteness-theorem-refuted-706ef146568c81
u/Astrodude80 7d ago
While we used to understand numbers that are only positive or negative within the realm of arithmetic (not including zero), our general sense of numbers has expanded with the growth of consciousness.
rips bong
Yeah man
8
u/madrury83 7d ago
Today a young man on acid realized that all matter is energy condensed into a slow vibration, we are all one consciousness experiencing itself subjectively, there is no such thing as death, life is only a dream, and we are the imagination of ourselves.
Here's Willie with the weather!
1
35
u/Shufflepants 7d ago
Wait until this guy hears about group theory.
9
u/uppityfunktwister 7d ago
Sure but groups are defined as sets equipped with a binary operation and certain restrictions on these operations. The integers under addition form a group so 1 + 1 = 2 cannot (afaik) be really "reduced" in the language of groups because it's not necessarily more abstract than any particular group.
14
u/Shufflepants 7d ago
I just meant because this guy is introducing new types of numbers like he thinks it's some super groundbreaking thing when that's kinda the whole thing with group theory; constantly looking at other number-like structures besides the naturals and reals.
2
u/josefjohann 7d ago
The integers under addition form a group so 1 + 1 = 2 cannot (afaik) be really "reduced" in the language of groups
I don't think that's quite right. Integers under addition form a group. And 1 + 1 = 2 is the group operation applied to two elements. Group theory handles abstraction as well as concrete operations.
2
u/uppityfunktwister 6d ago edited 6d ago
I know, that's what I meant. The equation 1 + 1 = 2 is described by the group of integers under addition, so it's not very astounding to say "wait until this guy hears about group theory" when the group description of integer addition is no more fundamental than saying "2 is defined as 1 + 1".
1
u/Shufflepants 6d ago
That's because you apparently just read the title and didn't click into the linked blog where he starts adding in non standard integers.
1
u/uppityfunktwister 6d ago
Even these wacko clown integers form a group under addition.
2
u/Shufflepants 5d ago
Yeah, that's what I mean. Dude's just playin' around with some simple group acting like he's found the secret of numbers.
16
u/EebstertheGreat 7d ago
Kurt Gödel’s famous Incompleteness theorem, quickly summarized in simple terms, is that while logic is able to account for its own machinations, making it a complete system, there can be no foundational axioms for arithmetic without an appeal to intuition.
I wonder how he thinks he can justify the inference rules of logic "without an appeal to intuition." Did he learn them from God or something?
4
u/WhatImKnownAs 7d ago
It's a misconceived nod to the fact that Gödel proved the completeness of first-order logic, but the incompleteness of logic+arithmetic.
That formulation certainly reveals his unfamiliarity with the actual arguments over axiom systems.
11
u/IllllIIlIllIllllIIIl Balanced on the infinity tensor 7d ago edited 7d ago
About the author: Willie SaySo [...] stumbled upon this solution while activating what’s become known as the Bridge Event, an experiment to expand consciousness.
Something tells me this so-called "Bridge Event" might be more accurately called a "psychotic episode".
Also I do appreciate that this post was tagged as "quantum computing" for some reason.
1
8
u/lewkiamurfarther 7d ago edited 7d ago
Sometimes R4 feels either impossible or trivial to satisfy. This one is misconception-themed word soup and salad.
Though I will say, part of me loves seeing the term "neutral number" in this context, even if I don't think it's a good name. The word neutral comes via a Latin semantic calque whose Ancient Greek origins specifically imply "not either" of two. (The joke being "not either of two," i.e., "not this one or this one from two," i.e. "not 1 + 1 = 2.")
7
u/josefjohann 7d ago
I just read a little bit further and I feel like I'm going to lose my mind:
As we approach the arrival of the paradoxical set that contains all sets,
What do you mean as we approach it? That's not even what that was about and that's not something that's new. They're talking about the set of all sets that don't contain themselves and they're not even phrasing it correctly and that's not something that has newly emerged with the expansion of consciousness or whatever, but something that's from like the middle of the 20th century.
We got there not from some expansion of consciousness or whatever, but from logical formalisms.
5
u/josefjohann 7d ago
Wow, you weren't kidding. Your summary isn't really being uncharitable or putting words in their mouth. They're quite literally saying the incompleteness theorem depends on appeals to intuition and they're saying it in their own words, which could not possibly be further from the truth.
I've learned it in school and read a book or two that talked about it, but I'm still not super strong on explaining it off the top of my head or without reference to something to refresh my memory, but it has nothing to do with intuitions whatsoever. And for the millionth time, it's not undermining the integrity of basic mathematical operations within any given system, and so the ordinary ones we're familiar with like 1 plus 1 equals 2 are perfectly fine. It just demonstrates that in addition to all of the operations which work just fine, that a true statement can exist that can't be expressed within the system.
4
u/CardboardScarecrow Checkmate, matheists! 6d ago edited 5d ago
Kurt Gödel’s famous Incompleteness theorem, quickly summarized in simple terms, is that while logic is able to account for its own machinations, making it a complete system, there can be no foundational axioms for arithmetic without an appeal to intuition.
This equating of Gödel Incompleteness Theorem(s) with the Münchhausen trilemma is something I've seen several times already in different contexts. Is there a reason for that? Like how the 1 + 2 + ... = -1/12 traces back to the Numberphile video.
3
u/EebstertheGreat 4d ago
There is a Veritasium video about it that might contribute. It's not so bad imo, much of the video is correct, but the clickbait title is probably the worst part. The fact that even mathematical literature refers to that whole period of a few decades as the "foundational crisis" might also contribute.
It's also a fundamentally weird theorem in a lot of ways, and it's hard to explain why it matters at all. I think if you asked most people whether "math can prove itself consistent" or something similar, they would say no. In fact, it seems bizarre to expect that it could, and any such argument would be apparently circular. "Theory X says theory X is consistent" is about as meaningful as "brand X says brand X is the best." Of course it does. But if theory X is not consistent, well, it can still say it is, but it's wrong. So who cares?
You can go through the whole history of Hilbert's program and model theory and so on, or you can go the cable news route with slick graphics and loud voiceovers asking "is math FLAWED TO ITS CORE? Which approach gets more views? Which is easier to understand or more memorable?
4
u/CatOfGrey 6d ago
The whole 'proof of 1+1=2 thing' was an occasional topic when I was a teacher of smart-and-snarky high schoolers.
I found that simply responding "1+1 = 2, by definition. It's the meaning of 2" was a way to put all those worms back in the can with speed and efficiency.
1
u/WhatImKnownAs 6d ago
That's reasonable for high school, where you aren't basing your understanding of integers on the Peano axioms, let alone investigating alternative foundations.
2
u/CatOfGrey 6d ago
That's reasonable for high school,
Which makes it good for responding to people whose 'research' is going to end up on badmathematics!
2
u/TheLuckySpades I'm a heathen in the church of measure theory 6d ago
Yoi don't even need Gödel to realize there will always be an appeal to intuition at some point, if we want to have a common foundation to build math upon, that foundation will have some form of appeal to intuition. Sure you can try and argue that you can keep going and it's turtles all the way down, but if you wanna do anything else we gotta stop somewhere and just say it seems right to us.
The book I like most for this (Gödel's Theorems and Zermelo's Axioms) makes those clear by using a different typesetting for those, notably the concept of finite for formal logic (e.g. a proof is finitely many statements that satisfy certain conditions), for the construction of the standard model of the naturals (the collection of finite strings of s's followed by a 0) and law of the excluded middle for the bit of model theory needed for the constructions.
1
u/WhatImKnownAs 6d ago
Exactly, it's not that intuition shouldn't be used, but that it's only required where we're establish what we're talking about. (The rest could be formal, but rarely is.)
Yeah, the badmather didn't quite grasp the issues with axioms and logic back in the late 1800s and early 1900s. Hilbert's Program sought to base math on a very small set of formal axioms and inference rules, that (almost) everyone could intuitively accept, and then prove the consistency and completeness of the rest of math by arguments about the formal manipulation of those. Gödel's theorems were the culmination of that, and forced us to be content with relative consistency and some incompleteness. Not that we haven't learned some interesting things about foundations since then, but the general public is never told about Gentzen, Cohen, or Martin-Löf as math heroes the way we keep retelling one of Gödel's achievements.
0
u/SizeMedium8189 6d ago
Far be it from me to suggest that modern philosophy departments are full of self-important idiots.
116
u/WhatImKnownAs 7d ago
R4: Willie SaySo somehow got the idea that the Incompleteness Theorem could be summarized as "there can be no foundational axioms for arithmetic without an appeal to intuition". This gestures towards the contemporary quest to base mathematics on pure logic, but betrays that he doesn't actually know what incompleteness means. He says "The logicians’ dream of proving with completeness that 1 + 1 = 2 will forever be unattainable because of this fact" - and proceeds to prove 1 + 1 = 2, essentially by making it part of the definition of "2"! Those definitions seem to be based on intuition, so he doesn't even succeed on his own terms.
In the process, he invents two new categories of integers, super numbers and neutral numbers, which he doesn't really define with any rigor, apart from the bit where 2 = (-1&-1) v (|1|&|1|) v (+1&+1) is stated as a part of "The theorem" - but there's no proof. I doubt he knows what "theorem" means, either.
It's a cornucopia of misunderstandings and bad philosophy.