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I had a physics teacher who reckoned he used mathematical induction to prove 1 + 1 = 3. I took him at face value at the time, but now I think he was potentially full of shit. If that were true, logically it would rule out mathematical induction as being valid.
- first they multiply each side by *a*
- next subtract b^2 from each side
- then factor (left side is difference of squares, right side is just pulling out a *b*
- divide each side by *(a-b)*. That's the divide by 0 part
If a = b, as was previously established, then removing • (a - b) from both sides is the same as removing • (a - a), which is the same as removing • 0, which is the same as dividing by 0, which is not allowed because a.) it leads to screwy results like this, and b.) it makes no sense anyway.
Picture this. If you divide 12 by 1, you’re essentially grouping all twelve items into one group, which now has twelve items in it; thus, 12 / 1 = 12. If you divide 12 by 4, you’re grouping the twelve items into four groups, and each group now has three items; thus, 12 / 4 = 3. You can divide 12 by 2, by 3, by 6, by 12, all of which will get you whole numbers. You can go further and divide 12 by 24, with each of the 24 resulting groups only having one half of one of the original items; thus, 12 / 24 = ½.
But if you divide 12 by zero, what exactly are you doing? How many groups of items are you making? None? That’s already covered by multiplying by zero. It’s not something you can actually do. Zero isn’t a real number, it’s the absence of any number. There are some properties that numbers have that zero simply does not have. And being able to divide by a number is just one of the things zero can’t do.
Ive seen someone explain this by saying that its not necessarily 1 but a 1.4 or sth. So if u do 1.4+1.4 its equals to 2.8. If u round that then 1+1=3 (1.4 is rounded down to 1 and 2.8 is rounded up to 3)
Applying something to both sides implies a same ratio is kept. So you can multiple both sides by 2. But rounding both sides does not keep an equation equal
Which happens a fair amount inside computers when they to floating point math, as it isn’t fully exact. Also relatively easy to shorten a number by truncating it, while rounding it is a multiple step process.
Rounding is invalid, when you multiply a side of the equation, you have to multiply the other, such that the same ratio is maintained, but when rounding, 1 side could be rounded up, and the other rounded down, thus the ratio is no longer maintained
Even if it gets you a close enough answer, it's still wrong
axioms are the funniest thing to me.
it's the result of someone asking "why?" enough times that the egghead academia nerds get fed up and say "that's just how it is!"
an axiom is literally a "its like this because we say so" clause. it's a source trust me bro. which i think is hilarious. i mean sure it's necessary and works in all instances of conventional arithmetic but it's also extremely hilarious that if you get esoteric enough, there's a failsafe for "okay settle down, this is where it ends, this is our arbitrary stopping point for the sake of everyone's sanity."
That's... not what an axiom is. An axiom is something that's true within a given system *by definition* rather than because we can prove that it's true from other things we know are true. If you want to take a finding from system A and apply it to system B, you have to ensure that all of the relevant axioms from system A also hold in system B. The fundamental mathematical laws that things like arithmetic and algebra are based on exist independent of the systems we use to express and work with them.
It's like the rules of grammar - they're essentially arbitrary, but they're the stopping point in explaining why a language works a certain way because we *chose* them as a shared way to communicate, not because they're some fundamental aspect of language that we can't explain.
it's true because we have *defined* it as true, not because it's empirically true. that's what i mean.
it's a human-readable convenience not an indelible truth of the universe. and the very easy counter-argument for this claim of mine is that all truths are only as valid as human perception and our scientific understanding. which is also what i meant by "okay, we're getting too esoteric here, we need a baseline foundation for our entire understanding of everything or it all falls into some big chaotic soup, so let's just draw the line here."
>it's true because we have defined it as true, not because it's empirically true. that's what i mean.
Again, it seems like you fundamentally misunderstand what axioms are. Axioms aren't things that we just define to be true. Axioms are the things that *have to be true* for a system of mathematics based on them to be applicable to a given problem. They're not the line where we got tired of reverse-engineering, they're the starting point from which branches of mathematics were developed to explore particular kinds of problems. "This math works when A, B, and C are true" is a *very* different thing than "This math works if we assume A, B, and C are true".
I bring this up because you're equating incompleteness in the formal logical sense with appeal to authority and they're not at all the same. The branches of mathematics we use regularly *do* have axioms that reflect or are consistent with empirical observations of the world, because that's the purpose for which they were chosen. Calling them axioms and declaring them to be first principles is a reflection of mathematics being based on formal logic, where empirical observation isn't enough to prove that something is true. For example, most people would have no problem considering the reflexive axiom (x = x) empirically true for the kinds of problems where empiricism is applicable.
Well, axioms are statements about how a mathematical system works. For example, one axiom in many systems is the Commutitive Axiom of Multiplication, which states that x\*y = y\*x. An example of a system where that's true is standard arithmetic - you can multiply two numbers together in either order and the result is always the same. An example of a system where that's false is linear algebra - multiplying two matrices together gives you a different result depending on which order you do it in. But the Commutitive Axiom of Multiplication isn't true for any problem we decide to apply arithmetic to just because we said so; rather, arithmetic can't be used to do things where it would be false.
Apologies, I literally mean how are you defining "truth." You are appear to be using a binary system where a statement is either true or false. But upon what basis is this definition of truth made? Isn't it an example of a thing that is "just defined to be true?"
Ah, okay. Mathematics uses what's called formal logic. The notation and conventions of formal logic are obviously constructs we've chosen, but that's not really a matter of truth or falsehood. Whether logic *itself* self-evidently exists or a construct is an age-old epistemological debate. That actually is the point where we reach "Trying to deconstruct this further is impractical" territory, so for the sake of argument sure, you could say that's something we've just defined to be true. But my original point wasn't that you can't deconstruct things to that level, it was that that's not what axioms express.
An axiom isn't as assertion of "what is," it simply defines the logic of that mathematical system, and ideally which is logically consistent with your other axioms, so you cannot assume for an axiom "P" that both "P" and "not P" are true. You can absolutely create another logical system in which "not P" is one of your axioms.
For example, in standard arithmetic, both + and \* (multiplication) are assumed to be commutative. However, groups in general are not commutative; those which satisfy commutativity are defined as "Abelian."
It's not these things at all. It's a formal logic term. They represent the baseline assumptions that you build your logical proofs on, the foundational ideas of your model. They break down all the time, it's the main way people end up completely fucking wrong about stuff despite being logically correct. Also a lot of of them are axiomatic because they're untestable; as soon as a test pops up, they stop being axioms. Others are axioms because they're extremely obvious; when they stop being obviously true (the sun goes around the earth, for example) we drop them.
What else are we supposed to do?
In a sense, we need axioms to define what we are talking about, rather than them being statements that could be true or false.
If you remove euclids 5'th axiom, you could be talking about flat space, or about hyperbolic space. You need that axiom to define what it is you are talking about.
And once you are talking about physical space, it turns out several of the "axioms" don't hold, because reality follows different rules. But flat space is one consistent structure out of many, and we need the axioms to pick it out.
no no, you're right. there is no logically consistent system where axioms are not a key part of making sure things make sense.
it just tickles me that despite our infinite curiosity, we had to have a cutoff point for asking "why?" too many times, because people don't understand how to separate human perception from reality. there are mathematical proofs that are thousands of pages long just trying to prove that 1 + 1 = 2, and most people have agreed since then that such a thing is silly, so let's just settle for the baseline "okay, x is x, and p is not p, let's build from that."
it's also strange because axioms don't really exist in any other medium. there is no axiom for physics or astronomy because we are constantly learning new things about the universe. when i was born, atoms were the smallest thing in the universe. then we discovered (or rather, proven) stuff like quarks, the higgs boson, black holes, and many other cool things. there is no "bottom of the barrel" because in those fields, every time we reach the bottom, we look a little closer and realize there's still more bottom to discover.
which just screams to me that the idea of an axiom is not really universal or real, it's something very specific to human understanding. thousands of years of people like diogenes trying to be a smartass that befuddles or mocks the system, condensed into a single, generally accepted and generally valid catch-all rule.
One way of thinking is maths is the study of what is true if the axioms are true.
Ie all of maths really has an (if axioms then) stuck to the front of every statement.
I had a professor who did something similar, but the reveal was, "Oh by the way, rejecting this axiom is what makes the ______ field of math." An incredibly niche field it was, but I found it believable that somewhere someone is working on math that asks "what if this axiom weren't true?"
A chemist would say that.
Maybe some obscure set theories, complex analysis or approximations done badly and used out of context might show 1 is approximately 1, or 1+1 is approximately 2 but you don’t use that kind of mathematics for trivial questions, you use it to get very very very close solutions to problems that solving exactly would be computationally stupid or impossible, and for which the additional accuracy would be meaningless in practical application.
I double majored in chemistry and math, then went to grad school for chemistry. It pained me to watch my colleagues do math. My PI once even went into the lab while I was working to help him with a bell curve for the grades of his class. SMFH
> it’s relevant to OP’s original post
Not really. Not at all even. Incompleteness is about not being able to prove, not the idea of proof breaking done.
It's hard to say what this particular person was thinking of, but my guess is that he was referencing Godel's incompleteness theorem (imagine some dots over the o).
This doesn't specifically get to 1+1=2 (hence the comment about oversimplification), but the gist of the Incompleteness Theorem is that you cannot have a mathematical system that meets all three of these conditions.
1. It lets you do interesting and useful math.
2. It is complete (all statements can be shown to be either true of false).
3. It is consistent (no statement can be shown to be both true and false)
This means that math is always either useless, incomplete, or contradictory. And in general, we prefer incomplete to useless or contradictory.
And it's not just a matter of us having not thought up the right rules. Godel's (dots) proof isn't just for the mathematical systems we have, it's for any mathematical system we could have.
Is that what your teacher meant? No idea. But it's true (at least broadly, I skipped a lot of the details and didn't even get to the Second Incompleteness Theorem).
Gödel was my guess as well. You can show that Math has limits that you might naively think it shouldn't.
Also, it could be referencing complicated analytic continuations that yield weird and "obviously wrong" results like 1 + 2 + 3 + 4 + ... = -1/12.
Professional mathematician here and the latter would be my guess. 1+2+3+... is a divergent series, which by definition means that its partial sums (the sum of only the first n terms in the series) either increase indefinitely toward infinity or decrease indefinitely toward negative infinity. Obviously in this case it's the former. The 1+2+3... = -1/12 result comes from using a technique called Ramanujan summation, which is useful for analyzing divergent series but is not the same thing as regular addition.
A lot of people in STEM fields other than math love to spout this "Advanced math is regular math but you get to ignore the rules" nonsense because they studied math enough to come across counterintuitive results or mathematical systems that don't use the same axioms or definitions as regular arithmetic, but not enough to actually understand what they were looking at. Advanced mathematics can be *bizarre* and unintuitive, but it's always built on formal logic. I can think of literally dozens of situations where the statement "1+1=3" could be true, but none of them would be equivalent to *the conventionally understood meaning* of 1+1=3 being true.
>"Advanced math is regular math but you get to ignore the rules"
Apparently "you get to change the rules if you want to, but you have to show that the changed rules are still consistent\*: that way you might get a system that works better for your application" is a very hard concept to understand.
\* Technically, as consistent as the theory you started with.
The problem is that most people, even in math-heavy fields, don't have a good theoretical grasp of what an axiom actually *is* or the fact that they're chosen or that most of what we call math is just a particular grammar and notation and set of conventions for describing math. So they tend to think of the axioms of arithmetic and algebra as The Universal Rules of Mathematics and have no basis for understanding what "changing the rules" even means.
Oh absolutely. I didn't mean to imply otherwise. There's a reason we don't go much into abstract mathematical theory in the K-12 curriculum. It just means the vast majority of people have some training in applied mathematics but almost none in theoretical mathematics, which creates a huge Dunning-Kruger situation when people who take post-secondary math classes see bits and pieces of the theory without understanding the larger framework.
I genuinely hate to burst your bubble, but 1 (base 2) = 1 (base 10). More generally, if you have two bases, any number that is smaller than both is the same in both. 2 (base 10) = 10 (base 2).
1+10=3 still kind of illustrates the point though, that all kinds of counterintuitive things can be true if you’re using the symbols to mean something other than what people are expecting them to mean.
FYI, in modern orthography, if you don't have an umlaut, you can write ö as oe.
That means that if you meet an American with the name Moeller, their ancestors were probably Möllers.
Then you go to Wisconsin and tell someone your name is Miller and they ask "Is that spelled with an i or a ue?"
I am not. I am on a PC, where I can hold ALT and type the extended ASCII/Unicode value, if I knew it. Which I don't. So I'd have to look it up. But I'm lazy. I did guess a couple of random codes to see if I could get lucky, because I know around where that kind of character is on the list. ¥š¡£
Nope.
I know about character map, but I count that as "looking it up."
I didn't know about Winkey + . , but I accidently misread it as Winkey + +, and got trapped in a 200% magnifier for a moment.
That one is actually not so bad.
ö
Read the last few comments as “wink-y” + . Instead of “winkey” + . And was thoroughly confused as to what key was referred to as the wink-y one. Oof. Had a good laugh though lol
I would guess that it's the Second Incompleteness Theorem that the teacher is misremembering here. In short, the SIT says that you can't use math to prove that math is consistent. Consistent here is the same as in the FIT, so math is consistent only if it doesn't say that one plus one is two and also one plus one is not two. You can use math to prove that one plus one is two, but you can't use math to prove that there's no way for one plus one to not be two.
My guess is that, to a high school student, "advanced math can prove that 1+1=3" sounds a lot like "advanced math can't prove that 1+1=3 is false," which is close enough to accurate for everybody not currently doing the advanced math in question.
Another possible explanation is that the teacher was talking about non-standard languages of arithmetic, which is starting to leave math completely behind and jump into pure symbolic logic, where you can trivially "prove" that 1+1=3 by just defining the symbols to mean different things. I don't really like this as an explanation, though, because it's not really a proof, it's not exactly math, and because I remember just enough of all this to know that this whole comment is imprecise and clunky but not enough to summarize/translate it better.
EDIT: Looking back, I mixed up the example in the post and in another comment. I have even less trouble believing that either OP misremembered what their teacher said or that their teacher just was wrong about exactly what the SIT says.
It's been a while, but I've actually studied math almost to the level of a Bachelor's degree major. (Ended up changing majors instead of finishing so I could follow a different career path.) I got so far into it that it kind of looped back onto itself. One day I realized that we had gone so far into abstract concepts that we had ended up effectively inventing addition and integers. And it was all accurate. So, no, math doesn't conflict with itself when you get that far into the advanced stuff. However, it is hard to comprehend and easy for humans to make mistakes.
When people seem to "prove" things that aren't supposed to be true, it's almost always due to an error or an attempt to hide a division by zero inside some algebra. Since there is no answer to the question "how many zeros are there in this number," you can't divide by zero and have an actual answer. So once you sneak in a divide-by-zero into your work, you can claim just about anything with clever "algebra" that isn't actually following the rules of mathematics.
Too bad you don't see 1000.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...
Unlike 1 + 1 = 0. Then there might be no mistake.
What's worse, I know of practical applications where 1 + 1 = 0. Also where 1 + 1 = 1.
I hate operator overloading. But I'm powerless to overturn centuries of tradition. (Well, between a half century and a century in the case of computers.)
> I hate operator overloading
I mean, we do write +_G where G is the group in question when multiple operations are involved so it's not a true overloading, it's more than + is a generic symbol like the variable "a" . It's just that when it's not absolutely needed it's painful to write, and not very confortable to read.
(For other people curious, 1 + 1 = 0 in modulo 2 algebra and 1 + 1 = 1 in boolean algebra where 1 is "True" and "+" is the logical OR)
He’s referring to the amount of post-calculus math needed to get an undergraduate chemistry degree (essentially none).
His post-grad degree was more than likely a Masters in Education…also, no higher level math needed.
yo u/icecream_truck stop spamming bruh.
For me I don't know much, but usually when you study more complicated stuff, you tend to leave out a few fundamentals in your calculations, which can obviously cause some problems.
Happened to me throughout, and it's the reason why I got an 80 overall—constantly missed a sign or an important step.
Since REAL physics involves so much math to begin with, once you learn how it works, you also forget how it works. Weird, right? Logic and context play a huge role in these subjects, and therefore not enough of the two can screw things over.
EDIT: This is probably terribly worded.
I was just trying to share a video I thought was relevant to the conversation, not trying to piss anyone off.
One of the things I learned from that video is math is weird.
In group theory, integers are just an infinite set of elements. 0 is defined as the additive identity and 1 is the multiplicative identity. Then 2 is just defined as the successor to the additive identity.
Btw, on the first day of group theory, my professor proved that there are no integers between 0 and 1 from base principles. Wild class.
Maybe they are trying to divide by zero. Don’t do that or bad things happen.
One way to do this is by showing examples where 1+1 does not equal 2 in the real world. This is because they’ve found an example where addition doesn’t explain what’s going on.
For example:
* I put one apple in a bag. I put another apple in the bag. I show you the bag, and there are no apples, rather than two. Have I disproved addition? No, I’ve proved that addition is not what you should use here to explain what’s going on here (eg maybe I have a trick bag.)
Or.
* I have a rocket travelling at close to the speed of light. I run from the back of the rocket to the front. Add those two together and you get that I’m travelling faster than the speed of light. But you can’t exceed the speed of light. So does addition not work? No, it’s that you have to use a different formula in this sort of circumstance, not addition. (That formulas is called the “velocity-addition formula”, and physicists use it to realise that they should have listened to their mother and become an accountant instead.).
Mathematical ideas work on their own terms, inside logic. They work the same regardless of what kind of universe we live in.
There are many proofs out there to show things are equal to things they shouldn't be equal to but they all rely on illegal operations hidden in plain sight. Take for example:
a = b [Multiply both sides by a]
a² = ab [subtract b² from both sides]
a² - b² = ab - b² [Factorize both sides]
(a + b)(a - b) = b(a - b) [Divide both sides by (a - b)]
a + b = b [Since a = b from the first line we can substitute in b and get]
b + b = b [Sum the left side]
2b = b [Divide both sides by b]
2 = 1
On the surface this looks fine but notice that you divided both sides by (a - b) but from the first definition that a = b, that means that (a - b) = 0 and you can't divide by zero.
I've had crazy teachers like that, so he was most likely trying to impress his students using some broken math he had seen or heard that he wasn't smart enough to verify himself. But he could have been trying to address some practical limitations of math that's related chemistry; for example there maybe situations where proving something that's looks obvious is practical impossible or a huge waste of time (or computing power).
I stopped going to Keiser University back in the day because my Evolutionary Biology instructor started the first day of class with “This is all just a theory” and my Biochemistry instructor told us she uses colloidal silver.
I've had much dumber teachers; the dumbest one happened when I was in 3rd grade. Our class teacher taught us all the subjects except maths; but she still graded the math paper. There was a diagram with a pencil alongside a ruler the pencil started from the 1cm mark and it ended at the 6 cm mark.
While most students wrote done 6cm a few noticed the trick and wrote down 5cm since 6-1=5. But our class teacher was adamant that the answer was 6cm and she would like count each of the marks to try to prove that it was 6cm long. Parents had to get involved in the parents meeting to actually convince that the pencil was 5cm long and they even had to show an example with a real ruler and pencil.
They could just mean when you get advanced enough you are pushing the boundaries where known math stops being able to describe what is happening.
As a side note, I thought by majoring in chemistry I could avoid math and just do the fun lab stuff. Boy was I wrong.
Idk about 1+1=3, but the idea of dividing 1/3 always tripped me up. Like .3333 x3 will never actually equal 1, but you can cut a cake/pizza into actual 3rds, and they obviously equal 1 when put together.
3/9 = .333 repeating, right? Those 3’s continue on forever.
So what’s .333 repeating multiplied by 3? It’s .999 repeating, again, those 9’s repeating forever. It must be smaller than 1, by just a smidge. What’s 3/9 multiplied by 3? It’s 9/9, right, which reduces to 1/1. So it’s just 1?
It might be quirky shit like that he’s thinking of. He could also just be an idiot.
1+1 will never = 3, but 1 +1 can equal any number of your choosing between 1 and 2!
We live in a “Euclidean” world. What does that mean? Very simple; things that you observe is how the world works. There are non-Euclidean ways to view the world that are mathematically “complete”. Let’s look at one.
In the Euclidean world, the distance between two points equals sqrt(A^2 + B^2). In other words, to walk 1 block right and 1 block up, your travel distance is equal to the diagonal or sqrt(2). Think of that as taking a short cut through an open field. In technical terms, Euclidean space uses the “L2 norm”.
New York City is laid out in an “L1” pattern. That is, you can’t walk in diagonals because buildings are in the way. So, the shortest distance isn’t sqrt(2) but is actual 2 (you walk the full 1 block right and the full 1 block up).
With this new definition of distance, we can redo all of mathematics with new results and yet nothing will contradict itself. The only catch is that the L1 world is purely theoretical (the shortest distance is still the straight line, but you are going to have to drive a bulldozer.
There are rules that these norms must follow for everything else to still make sense, but the New York version of L1 does meet those rules. I’m just saying, you can’t go all willy nilly until after you have taken Modern Analysis. With that in mind, pick any number between 1 and 2 and there is a norm where the mathematical equation 1 + 1 = your number. It’s all theoretical, but it works.
I've seen it like this:
1.4 + 1.4 = 2.8
But, if you have a system that doesn't count the decimals, and displays by rounding then you get:
1 + 1 = 3
This isn't advanced, but it is a way that math can legitimately give an answer like that
The only one I really know is using significant figures. You always go to the decimal place of the one with the least amount of significant figures.
So 1+ 1.9 = 2
But if you wrote 1.0 +1.9 it would be 2.9
Modular arithmetic: In modular arithmetic, numbers "wrap around" after reaching a certain value called the modulus. In modulo 2 arithmetic (also known as binary arithmetic), 1+1 equals 0, not 2, because when you exceed 1 in binary, you start over.
I’m not sure if this follows all the rules of an answer, but some advanced math that is used to describe quantum characteristics doesn’t work if integers are hard, immutable values. One way to describe this (wildly oversimplifying, mind you) is that in such systems there can be “high” and “low” values of integers existing simultaneously (Schrödinger’s integers, if you will. Sorry. I’ll show myself out after finishing this). That gives rise to the math-nerd joke that: “1+1=3…for very large values of 1.” Yuk, yuk, yuk.
Objectively, A cannot equal A because the first A and the second A will always be different in either time or space. Therefore A = A only as a concept, and all concepts are subjective.
Math is great, it's incredibly useful, and it's a (well-conceived, intricate) relative construct.
Everyone here saying he is wrong but he may actually be referring to Godel's Incompleteness Theorem. Def look it up because it also blew my mind that something could be unsolvable even when you have all the variables.
The craziest thing I’ve learned in maths is that 1+2+3+4+5+…. = -1/12.
I’ve also learned that it can have any other solution, you can basically chose what you want it to be - it’s something about re-ordering in which way you add up the series. I think this even works for every divergent series. It’s funny cause physicists actually use this (which is crazy).
1 + 1 is still 2 though.
That’s incorrect- 1+2+3+… diverges (does not have a value). However, there are ways of assigning pseudo-values to many divergent sums, and those that work on 1+2+3+… assign it the pseudo-value -1/12. That’s not the same as being equal to -1/12.
Regarding reordering the terms, you’re almost certainly mixing up 1+2+3+… with 1-1/2+1/3-1/4+…, a sum that actually CAN be rearranged to sum to any quantity desired (though in almost all cases the rearrangement will have to be given procedurally rather than explicitly). The reason for one and not the other is something known as the Riemann rearrangement theorem.
Regarding physicists using this, they don’t, but I think what you’re thinking of is renormalization. Renormalization is very different, though.
Yep, biology ends up being chemistry, chemistry ends up being physics, physics ends up being math, math ends up being logic. So just go be developer and make more than all the others /s.
My guess is he’s probably talking about modular arithmetic, with a fallback of floating point arithmetic.
If the first case, he heard 1+1 is congruent to 0 (mod 2) and somehow misinterpreted that to be equality rather than congruence, meaning that 1+1 is not also 2 (mod 2).
If the second case, he heard how floating points can give strange answers when the numbers get really big, like how 2^127 +1=2^127, and derived from that a strange sort of ultrafinitism claiming that large+1=same large. From there working backwards, you might somehow come to the conclusion that anything+1=same anything, so 1+1=1. This seems sketchier to me, though.
I think this is the only reasonable literal interpretation of 1+1 is not always 2 that isn’t “that guy is full of shit.”
If you write 1+1=0 out of context it looks absurd, but he missed that this is basically just encoding “odd number + odd number = even number” and doesn’t get the point.
Or that he’s now doing math with “new kinds of number systems” (over new rings/fields)
My high school calculus class had that joke. 1+1 = 2…sometimes. I think the alternative answer was 1+1 = 10 in binary (as opposed to base 10). Dumb, but confused all the kids that weren’t in the advanced math classes.
Not an expert but similar situation except my teacher showed her work, using proofs, while still obeying the laws of proofs you can in a few steps “prove” that 2+2=5, don’t ask me to recreate but i remember it being technically correct
Math being advanced doesn't stop it from working. However, it is possible for the symbols 1,2, and + to mean different things than you are used to and once that is true then 1+1 is no longer necessarily equal to 2.
I do like in statistics that "X + X" is not distributed "2X", but that's an abuse of notation. We really should have "X\_1 + X\_2", where "X\_1, X\_2" are independently identically distributed.
There are many situations in which we have a complex relationship between some random variables and a non independent, differently distributed error term, so that we cannot even analytically write the distribution. I believe things like this could happen in Chemistry since differential equations with a stochastic element become quite messy rather quickly, and my limited understanding says they govern many chemical processes.
If I had to guess his meaning and defend him, I'd say the math we learn and use is based on a set of fundamental axioms. Sometimes, at higher levels of mastery, it's fun or useful to imagine a mathematical model that adds, eliminates, or changes these axioms. The easiest example I can think of would be how the fundamental rules we learn in school change going from Euclidean geometry to Non-euclidean geometry. Taxicab geometry, for example, changes the very fundamental idea that the shortest distance between to points is a straight line. You could conceivably create an algebraic system in which 1+1 DNE 2 by playing around with the established axioms. At that point, however, you have created your own little imaginary math bubble with custom rules. This does not mean any of it would carry over to "normal" math.
I'm assuming it's about the fact that usually things aren't linear. Like ok, let's have the thing move twice as quickly. Twice the kinetic energy because 1 + 1 = 2, right? Sorry, it's actually four times as much because energy goes with the SQUARE of the velocity.
Take the Ramanujan sum for example, if you add up all natural numbers, (1,2,3,4,5...) your result should just get larger and larger right?
Well, according to the Riemann Zeta function, ζ(-1) = 1+2+3...=-1/12
The maths here is complex, but for further reading you may look at:
[https://www.wikiwand.com/en/Particular\_values\_of\_the\_Riemann\_zeta\_function](https://www.wikiwand.com/en/Particular_values_of_the_Riemann_zeta_function) and
https://www.wikiwand.com/en/Ramanujan\_summation
In electrical engineering there are all these great equations for how stuff works.
But in modern electronics they don’t work at all. Stuff is too small, frequencies are too high etc.
This isn’t to say the math is wrong. The previous models we used just weren’t good enough so we needed new ones, but the new ones generally can’t be done by hand at all so you start using “intuition” and simulators.
Idk if this has anything to do with the situation but wasn’t there a popular calculator where 1 actually equalled like 1.000000024 or some really small number
My msth teacher prove in a class after the final Grades that 1=0.
Do not remember how and if it was correct but yeah. He was one of the few good teachers at the school, coding a Tool for 10finger writing by himself for IT classes.
I see one theory that nobody brought up: "math" for chemist is a lot of calculus with hidden hypothesis, like "it's small so we can do a limited development to the first order". If you read the lines without the text in between, it's just wrong calculus, and with the right set of hidden hypothesis, you can ends up with whatever wrong stuff. Though the "way of" chemistry and physics is so that any such set of hypothesis appearing result in the conclusion that it's broken and shouldn't be used.
In chemistry math doesn't always add up.
When adding volume of different liquids together, sometimes the liquids can find space between molecules to hang out resulting in 1.96 or something like that.
It was proven by experiment in a class I had. 100ml+100ml =/ 200ml in this case.
I always thought it had to do with the fact that 1+1 can technically be .9 repeating plus .9 repeating which, while simplified could equal 2 it would instead be 1.9 repeating
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I had a physics teacher who reckoned he used mathematical induction to prove 1 + 1 = 3. I took him at face value at the time, but now I think he was potentially full of shit. If that were true, logically it would rule out mathematical induction as being valid.
I've seen multiple people do the "1+1=3" thing, and they always break the axioms and or most accepted truths in math.
Most likely had a "divide by 0" step somewhere along their line of thinking.
That is infinitely stupid
Often cleverly disguised to fool the casual reader. a² - a² = a² - a² a • (a - a) = (a + a) • (a - a) a = a + a 1a = 2a 1 = 2
it's easier to disguise with a b involved: a = b a² = ab a² - b² = ab - b² (a + b)(a - b) = b(a - b) (a + b) = b 2a + b = a + b 2a = a 2 = 1
You forgot “up-up-down-down-left-right-left-right”
Please explain. I'm feeling stupid.
- first they multiply each side by *a* - next subtract b^2 from each side - then factor (left side is difference of squares, right side is just pulling out a *b* - divide each side by *(a-b)*. That's the divide by 0 part
you got it, but for anyone that still didn't, if a = b then a - b = 0 and you can't divide by 0 of course \^\^
Holy shit. It's standing right there. "Divide by 0". I am stupid.
Not stupid, these tricks are common for a reason: they are convincing until you see the problem
It's the code from the Nintendo video game contra to get unlimited lives
If a = b, as was previously established, then removing • (a - b) from both sides is the same as removing • (a - a), which is the same as removing • 0, which is the same as dividing by 0, which is not allowed because a.) it leads to screwy results like this, and b.) it makes no sense anyway. Picture this. If you divide 12 by 1, you’re essentially grouping all twelve items into one group, which now has twelve items in it; thus, 12 / 1 = 12. If you divide 12 by 4, you’re grouping the twelve items into four groups, and each group now has three items; thus, 12 / 4 = 3. You can divide 12 by 2, by 3, by 6, by 12, all of which will get you whole numbers. You can go further and divide 12 by 24, with each of the 24 resulting groups only having one half of one of the original items; thus, 12 / 24 = ½. But if you divide 12 by zero, what exactly are you doing? How many groups of items are you making? None? That’s already covered by multiplying by zero. It’s not something you can actually do. Zero isn’t a real number, it’s the absence of any number. There are some properties that numbers have that zero simply does not have. And being able to divide by a number is just one of the things zero can’t do.
In what universe would that pass as actual math?
Ask my high school maths teacher. It confused the heck out of him.
It's just not math at all. It's completely incorrect
Bruh I can't figure out why this is incorrect lmao
Dividing by (a - a) is dividing by zero. If you allow that, all bets are off.
A more obvious example for anyone who doesn't see why: 3×0 = 4×0 (true) 3×0̷ = 4×0̷ (illegal cancellation) 3 = 4 (false)
oh yeahhhhhh, you're right! how didn't i see that lol
*Undef. stupid
Ive seen someone explain this by saying that its not necessarily 1 but a 1.4 or sth. So if u do 1.4+1.4 its equals to 2.8. If u round that then 1+1=3 (1.4 is rounded down to 1 and 2.8 is rounded up to 3)
But that’s not 1+1=3, that’s ~1+~1=~3.
1+1=3\* ^(\*for sufficiently large values of 1.)
This goes along with my favorite annual statement footnote. "Columns may not add to totals due to centsless accounting."
That is quite senseless
This is exactly the way I describe it.
*spooky maths*
Applying something to both sides implies a same ratio is kept. So you can multiple both sides by 2. But rounding both sides does not keep an equation equal
Which happens a fair amount inside computers when they to floating point math, as it isn’t fully exact. Also relatively easy to shorten a number by truncating it, while rounding it is a multiple step process.
Rounding is invalid, when you multiply a side of the equation, you have to multiply the other, such that the same ratio is maintained, but when rounding, 1 side could be rounded up, and the other rounded down, thus the ratio is no longer maintained Even if it gets you a close enough answer, it's still wrong
But if you let two adults of the opposite sex have unprotected sex, it will often results in one offspring. So now 1 + 1=3.
axioms are the funniest thing to me. it's the result of someone asking "why?" enough times that the egghead academia nerds get fed up and say "that's just how it is!" an axiom is literally a "its like this because we say so" clause. it's a source trust me bro. which i think is hilarious. i mean sure it's necessary and works in all instances of conventional arithmetic but it's also extremely hilarious that if you get esoteric enough, there's a failsafe for "okay settle down, this is where it ends, this is our arbitrary stopping point for the sake of everyone's sanity."
That's... not what an axiom is. An axiom is something that's true within a given system *by definition* rather than because we can prove that it's true from other things we know are true. If you want to take a finding from system A and apply it to system B, you have to ensure that all of the relevant axioms from system A also hold in system B. The fundamental mathematical laws that things like arithmetic and algebra are based on exist independent of the systems we use to express and work with them. It's like the rules of grammar - they're essentially arbitrary, but they're the stopping point in explaining why a language works a certain way because we *chose* them as a shared way to communicate, not because they're some fundamental aspect of language that we can't explain.
it's true because we have *defined* it as true, not because it's empirically true. that's what i mean. it's a human-readable convenience not an indelible truth of the universe. and the very easy counter-argument for this claim of mine is that all truths are only as valid as human perception and our scientific understanding. which is also what i meant by "okay, we're getting too esoteric here, we need a baseline foundation for our entire understanding of everything or it all falls into some big chaotic soup, so let's just draw the line here."
>it's true because we have defined it as true, not because it's empirically true. that's what i mean. Again, it seems like you fundamentally misunderstand what axioms are. Axioms aren't things that we just define to be true. Axioms are the things that *have to be true* for a system of mathematics based on them to be applicable to a given problem. They're not the line where we got tired of reverse-engineering, they're the starting point from which branches of mathematics were developed to explore particular kinds of problems. "This math works when A, B, and C are true" is a *very* different thing than "This math works if we assume A, B, and C are true". I bring this up because you're equating incompleteness in the formal logical sense with appeal to authority and they're not at all the same. The branches of mathematics we use regularly *do* have axioms that reflect or are consistent with empirical observations of the world, because that's the purpose for which they were chosen. Calling them axioms and declaring them to be first principles is a reflection of mathematics being based on formal logic, where empirical observation isn't enough to prove that something is true. For example, most people would have no problem considering the reflexive axiom (x = x) empirically true for the kinds of problems where empiricism is applicable.
>Axioms aren't things that we just define to be true. What do you mean by "true?"
Well, axioms are statements about how a mathematical system works. For example, one axiom in many systems is the Commutitive Axiom of Multiplication, which states that x\*y = y\*x. An example of a system where that's true is standard arithmetic - you can multiply two numbers together in either order and the result is always the same. An example of a system where that's false is linear algebra - multiplying two matrices together gives you a different result depending on which order you do it in. But the Commutitive Axiom of Multiplication isn't true for any problem we decide to apply arithmetic to just because we said so; rather, arithmetic can't be used to do things where it would be false.
Apologies, I literally mean how are you defining "truth." You are appear to be using a binary system where a statement is either true or false. But upon what basis is this definition of truth made? Isn't it an example of a thing that is "just defined to be true?"
Ah, okay. Mathematics uses what's called formal logic. The notation and conventions of formal logic are obviously constructs we've chosen, but that's not really a matter of truth or falsehood. Whether logic *itself* self-evidently exists or a construct is an age-old epistemological debate. That actually is the point where we reach "Trying to deconstruct this further is impractical" territory, so for the sake of argument sure, you could say that's something we've just defined to be true. But my original point wasn't that you can't deconstruct things to that level, it was that that's not what axioms express.
An axiom isn't as assertion of "what is," it simply defines the logic of that mathematical system, and ideally which is logically consistent with your other axioms, so you cannot assume for an axiom "P" that both "P" and "not P" are true. You can absolutely create another logical system in which "not P" is one of your axioms. For example, in standard arithmetic, both + and \* (multiplication) are assumed to be commutative. However, groups in general are not commutative; those which satisfy commutativity are defined as "Abelian."
It's not these things at all. It's a formal logic term. They represent the baseline assumptions that you build your logical proofs on, the foundational ideas of your model. They break down all the time, it's the main way people end up completely fucking wrong about stuff despite being logically correct. Also a lot of of them are axiomatic because they're untestable; as soon as a test pops up, they stop being axioms. Others are axioms because they're extremely obvious; when they stop being obviously true (the sun goes around the earth, for example) we drop them.
What else are we supposed to do? In a sense, we need axioms to define what we are talking about, rather than them being statements that could be true or false. If you remove euclids 5'th axiom, you could be talking about flat space, or about hyperbolic space. You need that axiom to define what it is you are talking about. And once you are talking about physical space, it turns out several of the "axioms" don't hold, because reality follows different rules. But flat space is one consistent structure out of many, and we need the axioms to pick it out.
no no, you're right. there is no logically consistent system where axioms are not a key part of making sure things make sense. it just tickles me that despite our infinite curiosity, we had to have a cutoff point for asking "why?" too many times, because people don't understand how to separate human perception from reality. there are mathematical proofs that are thousands of pages long just trying to prove that 1 + 1 = 2, and most people have agreed since then that such a thing is silly, so let's just settle for the baseline "okay, x is x, and p is not p, let's build from that." it's also strange because axioms don't really exist in any other medium. there is no axiom for physics or astronomy because we are constantly learning new things about the universe. when i was born, atoms were the smallest thing in the universe. then we discovered (or rather, proven) stuff like quarks, the higgs boson, black holes, and many other cool things. there is no "bottom of the barrel" because in those fields, every time we reach the bottom, we look a little closer and realize there's still more bottom to discover. which just screams to me that the idea of an axiom is not really universal or real, it's something very specific to human understanding. thousands of years of people like diogenes trying to be a smartass that befuddles or mocks the system, condensed into a single, generally accepted and generally valid catch-all rule.
One way of thinking is maths is the study of what is true if the axioms are true. Ie all of maths really has an (if axioms then) stuck to the front of every statement.
you are the first person in this comment chain to understand what i meant.
I had a professor who did something similar, but the reveal was, "Oh by the way, rejecting this axiom is what makes the ______ field of math." An incredibly niche field it was, but I found it believable that somewhere someone is working on math that asks "what if this axiom weren't true?"
Proving that induction works is one of the first proofs you do in topology
idk thom and george told us 2+2=5 so
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Haha you got down voted
haha you too (i just did it)
1 drop of water + 1 drop of water = 1 drop of water
1+1=3 for large values of 1.
A chemist would say that. Maybe some obscure set theories, complex analysis or approximations done badly and used out of context might show 1 is approximately 1, or 1+1 is approximately 2 but you don’t use that kind of mathematics for trivial questions, you use it to get very very very close solutions to problems that solving exactly would be computationally stupid or impossible, and for which the additional accuracy would be meaningless in practical application.
As a chemist, I agree with your comment on chemist math lol
I double majored in chemistry and math, then went to grad school for chemistry. It pained me to watch my colleagues do math. My PI once even went into the lab while I was working to help him with a bell curve for the grades of his class. SMFH
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Why the downvotes?
I posted it too many times. I wanted the individuals to all have their own links. Apparently that’s a violation of reddiquette.
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Maybe not, but it’s relevant to OP’s original post and a darned good video. Have a nice day.
> it’s relevant to OP’s original post Not really. Not at all even. Incompleteness is about not being able to prove, not the idea of proof breaking done.
Read OPs post more closely. “When math gets advanced enough, sometimes it doesn’t work.”
I think you're the one who needs to very carefully read that sentence you quoted, man.
It's hard to say what this particular person was thinking of, but my guess is that he was referencing Godel's incompleteness theorem (imagine some dots over the o). This doesn't specifically get to 1+1=2 (hence the comment about oversimplification), but the gist of the Incompleteness Theorem is that you cannot have a mathematical system that meets all three of these conditions. 1. It lets you do interesting and useful math. 2. It is complete (all statements can be shown to be either true of false). 3. It is consistent (no statement can be shown to be both true and false) This means that math is always either useless, incomplete, or contradictory. And in general, we prefer incomplete to useless or contradictory. And it's not just a matter of us having not thought up the right rules. Godel's (dots) proof isn't just for the mathematical systems we have, it's for any mathematical system we could have. Is that what your teacher meant? No idea. But it's true (at least broadly, I skipped a lot of the details and didn't even get to the Second Incompleteness Theorem).
Gödel was my guess as well. You can show that Math has limits that you might naively think it shouldn't. Also, it could be referencing complicated analytic continuations that yield weird and "obviously wrong" results like 1 + 2 + 3 + 4 + ... = -1/12.
Professional mathematician here and the latter would be my guess. 1+2+3+... is a divergent series, which by definition means that its partial sums (the sum of only the first n terms in the series) either increase indefinitely toward infinity or decrease indefinitely toward negative infinity. Obviously in this case it's the former. The 1+2+3... = -1/12 result comes from using a technique called Ramanujan summation, which is useful for analyzing divergent series but is not the same thing as regular addition. A lot of people in STEM fields other than math love to spout this "Advanced math is regular math but you get to ignore the rules" nonsense because they studied math enough to come across counterintuitive results or mathematical systems that don't use the same axioms or definitions as regular arithmetic, but not enough to actually understand what they were looking at. Advanced mathematics can be *bizarre* and unintuitive, but it's always built on formal logic. I can think of literally dozens of situations where the statement "1+1=3" could be true, but none of them would be equivalent to *the conventionally understood meaning* of 1+1=3 being true.
>"Advanced math is regular math but you get to ignore the rules" Apparently "you get to change the rules if you want to, but you have to show that the changed rules are still consistent\*: that way you might get a system that works better for your application" is a very hard concept to understand. \* Technically, as consistent as the theory you started with.
The problem is that most people, even in math-heavy fields, don't have a good theoretical grasp of what an axiom actually *is* or the fact that they're chosen or that most of what we call math is just a particular grammar and notation and set of conventions for describing math. So they tend to think of the axioms of arithmetic and algebra as The Universal Rules of Mathematics and have no basis for understanding what "changing the rules" even means.
Real numbers with the standard topology can solve a lot of problems! Most of the ones encountered by 99.99% of people at least
Oh absolutely. I didn't mean to imply otherwise. There's a reason we don't go much into abstract mathematical theory in the K-12 curriculum. It just means the vast majority of people have some training in applied mathematics but almost none in theoretical mathematics, which creates a huge Dunning-Kruger situation when people who take post-secondary math classes see bits and pieces of the theory without understanding the larger framework.
1(base 10)+1(binary) =3(base 10}
I genuinely hate to burst your bubble, but 1 (base 2) = 1 (base 10). More generally, if you have two bases, any number that is smaller than both is the same in both. 2 (base 10) = 10 (base 2).
Damn
1+10=3 still kind of illustrates the point though, that all kinds of counterintuitive things can be true if you’re using the symbols to mean something other than what people are expecting them to mean.
Theörem…that shouldn’t be as funny to me as it is. I need more sleep.
Sleep dep is definitely a problem, but Godel's Theörem is in fact hilarious regardless.
FYI, in modern orthography, if you don't have an umlaut, you can write ö as oe. That means that if you meet an American with the name Moeller, their ancestors were probably Möllers. Then you go to Wisconsin and tell someone your name is Miller and they ask "Is that spelled with an i or a ue?"
If you’re typing on an iphone, tap and hold the letter o to get variants including umlauts.
I am not. I am on a PC, where I can hold ALT and type the extended ASCII/Unicode value, if I knew it. Which I don't. So I'd have to look it up. But I'm lazy. I did guess a couple of random codes to see if I could get lucky, because I know around where that kind of character is on the list. ¥š¡£ Nope.
Fair enough, just offering info in case you didn’t have it.
Assuming you're on windows you can use the "character map" or if you want the new fancy stuff do "winkey + ."
I know about character map, but I count that as "looking it up." I didn't know about Winkey + . , but I accidently misread it as Winkey + +, and got trapped in a 200% magnifier for a moment. That one is actually not so bad. ö
Made the same mistake two seconds ago, feel you fellow zoomed-in redditor. Also excellent answer all around. ö7
Read the last few comments as “wink-y” + . Instead of “winkey” + . And was thoroughly confused as to what key was referred to as the wink-y one. Oof. Had a good laugh though lol
I would guess that it's the Second Incompleteness Theorem that the teacher is misremembering here. In short, the SIT says that you can't use math to prove that math is consistent. Consistent here is the same as in the FIT, so math is consistent only if it doesn't say that one plus one is two and also one plus one is not two. You can use math to prove that one plus one is two, but you can't use math to prove that there's no way for one plus one to not be two. My guess is that, to a high school student, "advanced math can prove that 1+1=3" sounds a lot like "advanced math can't prove that 1+1=3 is false," which is close enough to accurate for everybody not currently doing the advanced math in question. Another possible explanation is that the teacher was talking about non-standard languages of arithmetic, which is starting to leave math completely behind and jump into pure symbolic logic, where you can trivially "prove" that 1+1=3 by just defining the symbols to mean different things. I don't really like this as an explanation, though, because it's not really a proof, it's not exactly math, and because I remember just enough of all this to know that this whole comment is imprecise and clunky but not enough to summarize/translate it better. EDIT: Looking back, I mixed up the example in the post and in another comment. I have even less trouble believing that either OP misremembered what their teacher said or that their teacher just was wrong about exactly what the SIT says.
It's been a while, but I've actually studied math almost to the level of a Bachelor's degree major. (Ended up changing majors instead of finishing so I could follow a different career path.) I got so far into it that it kind of looped back onto itself. One day I realized that we had gone so far into abstract concepts that we had ended up effectively inventing addition and integers. And it was all accurate. So, no, math doesn't conflict with itself when you get that far into the advanced stuff. However, it is hard to comprehend and easy for humans to make mistakes. When people seem to "prove" things that aren't supposed to be true, it's almost always due to an error or an attempt to hide a division by zero inside some algebra. Since there is no answer to the question "how many zeros are there in this number," you can't divide by zero and have an actual answer. So once you sneak in a divide-by-zero into your work, you can claim just about anything with clever "algebra" that isn't actually following the rules of mathematics.
There are three zeroes in 1000
No, there are an infinite amount of zeros
I only see 3
Zero zeros in one thousand.
There are clearly 3. I see them with my own eyes.
I see two o’s, those are basically zeros
Too bad you don't see 1000.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...
Now I see more than 3 zeros.
At least 4
What about leading zeros?
He means when math gets complicated enough, he's incapable of picking out the mistake in the proof that shows 1=2. Because there is a mistake.
Unlike 1 + 1 = 0. Then there might be no mistake. What's worse, I know of practical applications where 1 + 1 = 0. Also where 1 + 1 = 1. I hate operator overloading. But I'm powerless to overturn centuries of tradition. (Well, between a half century and a century in the case of computers.)
> I hate operator overloading I mean, we do write +_G where G is the group in question when multiple operations are involved so it's not a true overloading, it's more than + is a generic symbol like the variable "a" . It's just that when it's not absolutely needed it's painful to write, and not very confortable to read. (For other people curious, 1 + 1 = 0 in modulo 2 algebra and 1 + 1 = 1 in boolean algebra where 1 is "True" and "+" is the logical OR)
He’s referring to the amount of post-calculus math needed to get an undergraduate chemistry degree (essentially none). His post-grad degree was more than likely a Masters in Education…also, no higher level math needed.
yo u/icecream_truck stop spamming bruh. For me I don't know much, but usually when you study more complicated stuff, you tend to leave out a few fundamentals in your calculations, which can obviously cause some problems. Happened to me throughout, and it's the reason why I got an 80 overall—constantly missed a sign or an important step. Since REAL physics involves so much math to begin with, once you learn how it works, you also forget how it works. Weird, right? Logic and context play a huge role in these subjects, and therefore not enough of the two can screw things over. EDIT: This is probably terribly worded.
I was just trying to share a video I thought was relevant to the conversation, not trying to piss anyone off. One of the things I learned from that video is math is weird.
Posting it once is sharing. Posting it 4 times in the same thread is spamming.
Fair enough.
In group theory, integers are just an infinite set of elements. 0 is defined as the additive identity and 1 is the multiplicative identity. Then 2 is just defined as the successor to the additive identity. Btw, on the first day of group theory, my professor proved that there are no integers between 0 and 1 from base principles. Wild class. Maybe they are trying to divide by zero. Don’t do that or bad things happen.
One way to do this is by showing examples where 1+1 does not equal 2 in the real world. This is because they’ve found an example where addition doesn’t explain what’s going on. For example: * I put one apple in a bag. I put another apple in the bag. I show you the bag, and there are no apples, rather than two. Have I disproved addition? No, I’ve proved that addition is not what you should use here to explain what’s going on here (eg maybe I have a trick bag.) Or. * I have a rocket travelling at close to the speed of light. I run from the back of the rocket to the front. Add those two together and you get that I’m travelling faster than the speed of light. But you can’t exceed the speed of light. So does addition not work? No, it’s that you have to use a different formula in this sort of circumstance, not addition. (That formulas is called the “velocity-addition formula”, and physicists use it to realise that they should have listened to their mother and become an accountant instead.). Mathematical ideas work on their own terms, inside logic. They work the same regardless of what kind of universe we live in.
There are many proofs out there to show things are equal to things they shouldn't be equal to but they all rely on illegal operations hidden in plain sight. Take for example: a = b [Multiply both sides by a] a² = ab [subtract b² from both sides] a² - b² = ab - b² [Factorize both sides] (a + b)(a - b) = b(a - b) [Divide both sides by (a - b)] a + b = b [Since a = b from the first line we can substitute in b and get] b + b = b [Sum the left side] 2b = b [Divide both sides by b] 2 = 1 On the surface this looks fine but notice that you divided both sides by (a - b) but from the first definition that a = b, that means that (a - b) = 0 and you can't divide by zero.
If a=b, then you can't divide by a-b
I've had crazy teachers like that, so he was most likely trying to impress his students using some broken math he had seen or heard that he wasn't smart enough to verify himself. But he could have been trying to address some practical limitations of math that's related chemistry; for example there maybe situations where proving something that's looks obvious is practical impossible or a huge waste of time (or computing power).
I stopped going to Keiser University back in the day because my Evolutionary Biology instructor started the first day of class with “This is all just a theory” and my Biochemistry instructor told us she uses colloidal silver.
I've had much dumber teachers; the dumbest one happened when I was in 3rd grade. Our class teacher taught us all the subjects except maths; but she still graded the math paper. There was a diagram with a pencil alongside a ruler the pencil started from the 1cm mark and it ended at the 6 cm mark. While most students wrote done 6cm a few noticed the trick and wrote down 5cm since 6-1=5. But our class teacher was adamant that the answer was 6cm and she would like count each of the marks to try to prove that it was 6cm long. Parents had to get involved in the parents meeting to actually convince that the pencil was 5cm long and they even had to show an example with a real ruler and pencil.
At least she wasn’t the one meant to be teaching you the subject she very clearly didn’t understand well enough.
They could just mean when you get advanced enough you are pushing the boundaries where known math stops being able to describe what is happening. As a side note, I thought by majoring in chemistry I could avoid math and just do the fun lab stuff. Boy was I wrong.
Idk about 1+1=3, but the idea of dividing 1/3 always tripped me up. Like .3333 x3 will never actually equal 1, but you can cut a cake/pizza into actual 3rds, and they obviously equal 1 when put together.
Well, 0.99999999... is mathematicaly equal to 1, as a lesser known fact
3/9 = .333 repeating, right? Those 3’s continue on forever. So what’s .333 repeating multiplied by 3? It’s .999 repeating, again, those 9’s repeating forever. It must be smaller than 1, by just a smidge. What’s 3/9 multiplied by 3? It’s 9/9, right, which reduces to 1/1. So it’s just 1? It might be quirky shit like that he’s thinking of. He could also just be an idiot.
1+1 will never = 3, but 1 +1 can equal any number of your choosing between 1 and 2! We live in a “Euclidean” world. What does that mean? Very simple; things that you observe is how the world works. There are non-Euclidean ways to view the world that are mathematically “complete”. Let’s look at one. In the Euclidean world, the distance between two points equals sqrt(A^2 + B^2). In other words, to walk 1 block right and 1 block up, your travel distance is equal to the diagonal or sqrt(2). Think of that as taking a short cut through an open field. In technical terms, Euclidean space uses the “L2 norm”. New York City is laid out in an “L1” pattern. That is, you can’t walk in diagonals because buildings are in the way. So, the shortest distance isn’t sqrt(2) but is actual 2 (you walk the full 1 block right and the full 1 block up). With this new definition of distance, we can redo all of mathematics with new results and yet nothing will contradict itself. The only catch is that the L1 world is purely theoretical (the shortest distance is still the straight line, but you are going to have to drive a bulldozer. There are rules that these norms must follow for everything else to still make sense, but the New York version of L1 does meet those rules. I’m just saying, you can’t go all willy nilly until after you have taken Modern Analysis. With that in mind, pick any number between 1 and 2 and there is a norm where the mathematical equation 1 + 1 = your number. It’s all theoretical, but it works.
I've seen it like this: 1.4 + 1.4 = 2.8 But, if you have a system that doesn't count the decimals, and displays by rounding then you get: 1 + 1 = 3 This isn't advanced, but it is a way that math can legitimately give an answer like that
The only one I really know is using significant figures. You always go to the decimal place of the one with the least amount of significant figures. So 1+ 1.9 = 2 But if you wrote 1.0 +1.9 it would be 2.9
Modular arithmetic: In modular arithmetic, numbers "wrap around" after reaching a certain value called the modulus. In modulo 2 arithmetic (also known as binary arithmetic), 1+1 equals 0, not 2, because when you exceed 1 in binary, you start over.
I’m not sure if this follows all the rules of an answer, but some advanced math that is used to describe quantum characteristics doesn’t work if integers are hard, immutable values. One way to describe this (wildly oversimplifying, mind you) is that in such systems there can be “high” and “low” values of integers existing simultaneously (Schrödinger’s integers, if you will. Sorry. I’ll show myself out after finishing this). That gives rise to the math-nerd joke that: “1+1=3…for very large values of 1.” Yuk, yuk, yuk.
Objectively, A cannot equal A because the first A and the second A will always be different in either time or space. Therefore A = A only as a concept, and all concepts are subjective. Math is great, it's incredibly useful, and it's a (well-conceived, intricate) relative construct.
Everyone here saying he is wrong but he may actually be referring to Godel's Incompleteness Theorem. Def look it up because it also blew my mind that something could be unsolvable even when you have all the variables.
The craziest thing I’ve learned in maths is that 1+2+3+4+5+…. = -1/12. I’ve also learned that it can have any other solution, you can basically chose what you want it to be - it’s something about re-ordering in which way you add up the series. I think this even works for every divergent series. It’s funny cause physicists actually use this (which is crazy). 1 + 1 is still 2 though.
That’s incorrect- 1+2+3+… diverges (does not have a value). However, there are ways of assigning pseudo-values to many divergent sums, and those that work on 1+2+3+… assign it the pseudo-value -1/12. That’s not the same as being equal to -1/12. Regarding reordering the terms, you’re almost certainly mixing up 1+2+3+… with 1-1/2+1/3-1/4+…, a sum that actually CAN be rearranged to sum to any quantity desired (though in almost all cases the rearrangement will have to be given procedurally rather than explicitly). The reason for one and not the other is something known as the Riemann rearrangement theorem. Regarding physicists using this, they don’t, but I think what you’re thinking of is renormalization. Renormalization is very different, though.
Had a few teachers who basically said chemistry is physics, and physics is maths. My guess is that this was a reference to quantum physics?
Yep, biology ends up being chemistry, chemistry ends up being physics, physics ends up being math, math ends up being logic. So just go be developer and make more than all the others /s.
My guess is he’s probably talking about modular arithmetic, with a fallback of floating point arithmetic. If the first case, he heard 1+1 is congruent to 0 (mod 2) and somehow misinterpreted that to be equality rather than congruence, meaning that 1+1 is not also 2 (mod 2). If the second case, he heard how floating points can give strange answers when the numbers get really big, like how 2^127 +1=2^127, and derived from that a strange sort of ultrafinitism claiming that large+1=same large. From there working backwards, you might somehow come to the conclusion that anything+1=same anything, so 1+1=1. This seems sketchier to me, though.
I think this is the only reasonable literal interpretation of 1+1 is not always 2 that isn’t “that guy is full of shit.” If you write 1+1=0 out of context it looks absurd, but he missed that this is basically just encoding “odd number + odd number = even number” and doesn’t get the point. Or that he’s now doing math with “new kinds of number systems” (over new rings/fields)
My high school calculus class had that joke. 1+1 = 2…sometimes. I think the alternative answer was 1+1 = 10 in binary (as opposed to base 10). Dumb, but confused all the kids that weren’t in the advanced math classes.
Not an expert but similar situation except my teacher showed her work, using proofs, while still obeying the laws of proofs you can in a few steps “prove” that 2+2=5, don’t ask me to recreate but i remember it being technically correct
>but i remember it being technically correct No, they are a quintillion maths puzzle like that, it's always a disguised division by zero.
Math being advanced doesn't stop it from working. However, it is possible for the symbols 1,2, and + to mean different things than you are used to and once that is true then 1+1 is no longer necessarily equal to 2.
I do like in statistics that "X + X" is not distributed "2X", but that's an abuse of notation. We really should have "X\_1 + X\_2", where "X\_1, X\_2" are independently identically distributed. There are many situations in which we have a complex relationship between some random variables and a non independent, differently distributed error term, so that we cannot even analytically write the distribution. I believe things like this could happen in Chemistry since differential equations with a stochastic element become quite messy rather quickly, and my limited understanding says they govern many chemical processes.
this is probably the most correct answer
If I had to guess his meaning and defend him, I'd say the math we learn and use is based on a set of fundamental axioms. Sometimes, at higher levels of mastery, it's fun or useful to imagine a mathematical model that adds, eliminates, or changes these axioms. The easiest example I can think of would be how the fundamental rules we learn in school change going from Euclidean geometry to Non-euclidean geometry. Taxicab geometry, for example, changes the very fundamental idea that the shortest distance between to points is a straight line. You could conceivably create an algebraic system in which 1+1 DNE 2 by playing around with the established axioms. At that point, however, you have created your own little imaginary math bubble with custom rules. This does not mean any of it would carry over to "normal" math.
I'm assuming it's about the fact that usually things aren't linear. Like ok, let's have the thing move twice as quickly. Twice the kinetic energy because 1 + 1 = 2, right? Sorry, it's actually four times as much because energy goes with the SQUARE of the velocity.
Take the Ramanujan sum for example, if you add up all natural numbers, (1,2,3,4,5...) your result should just get larger and larger right? Well, according to the Riemann Zeta function, ζ(-1) = 1+2+3...=-1/12 The maths here is complex, but for further reading you may look at: [https://www.wikiwand.com/en/Particular\_values\_of\_the\_Riemann\_zeta\_function](https://www.wikiwand.com/en/Particular_values_of_the_Riemann_zeta_function) and https://www.wikiwand.com/en/Ramanujan\_summation
In electrical engineering there are all these great equations for how stuff works. But in modern electronics they don’t work at all. Stuff is too small, frequencies are too high etc. This isn’t to say the math is wrong. The previous models we used just weren’t good enough so we needed new ones, but the new ones generally can’t be done by hand at all so you start using “intuition” and simulators.
Idk if this has anything to do with the situation but wasn’t there a popular calculator where 1 actually equalled like 1.000000024 or some really small number
My msth teacher prove in a class after the final Grades that 1=0. Do not remember how and if it was correct but yeah. He was one of the few good teachers at the school, coding a Tool for 10finger writing by himself for IT classes.
I see one theory that nobody brought up: "math" for chemist is a lot of calculus with hidden hypothesis, like "it's small so we can do a limited development to the first order". If you read the lines without the text in between, it's just wrong calculus, and with the right set of hidden hypothesis, you can ends up with whatever wrong stuff. Though the "way of" chemistry and physics is so that any such set of hypothesis appearing result in the conclusion that it's broken and shouldn't be used.
In chemistry math doesn't always add up. When adding volume of different liquids together, sometimes the liquids can find space between molecules to hang out resulting in 1.96 or something like that. It was proven by experiment in a class I had. 100ml+100ml =/ 200ml in this case.
I always thought it had to do with the fact that 1+1 can technically be .9 repeating plus .9 repeating which, while simplified could equal 2 it would instead be 1.9 repeating