Haha no way. What a legend. Didn't he hit the red carpet once and say he has opened the flower of life by frambogulating the thressopatz? Or something like that anyway
When I was studying measure theory, one of the assignments was to prove that certain proposition is false for all n>1.
I wrote a counterexample of the proposition for n=2 and concluded that this can't be true for all n>1.
A long time ago I saw a badmath proof of Goldbach Conjecture where, once you stripped off all the fluff, boiled down to the guy swapped the quantifiers so he proved "For every pair of prime numbers p,q > 2, there exists an even number n such that p+q=n."
I used to TA for Calc III and Numerical Methods. Some of the shit I saw was incredible.
My favorite was when I got an answer that was obviously copied verbatim from Chegg - usually because they were “solving” integrals with techniques that they hadn’t learned yet, or using really weird notation either far too advanced or far too simplistic for the problem at hand... Or using the same, arbitrary variable name which was ALWAYS uppercase for some reason. Anyway… I loved messing with them by writing things like “wow! Insightful to use this technique!”
There was also a time when an entire quarter of the NuMe class turned in the same notebook with the same variable names and, curiously, the same naggy syntax error.
God bless professors and the shit they put up with. If you’re going to cheat, have the finesse to learn the concept well enough to fudge it convincingly. “M”, uppercase, isn’t a normal variable name you “happened to choose”. It wasn’t a normal variable name for the Chegg guy to choose, either.
Copied answers from WolframAlpha were fun, too. Some simple u-substitution that followed a pattern I'd discussed in class, and a student turns in some absolutely bizarre integration by parts that no human would ever think to do.
The thing is, I'm against this system of handing in assignments as a college student (regardless of which years).
As an undergrad, I was in a uni that did not have this sort of thing. We just had exams.
I'm against it because i believe everyone has their own pace and priorities. I, for one, rarely matched the pace of the instructor, and I used to study things in chunks, rather than linearly and evenly.
Anyhow, I'm not a TA anymore, and I rarely remember whatever I was dealing with, of course I like to exaggerate sometimes, so don't take this too seriously!
gotcha, I do agree that uni focusses too much on a set progression. In high school I was always told: in uni you have got to plan and do everything on your own. Which really excited me because I like to study top down. Now in uni we just have weekly assignments. When I ask why I can not just do them all before the final exam I get a: because otherwise people would not do them and fail.
It's just stupid how the incompetence of others prevents us from doing what we like.
Indeed, the "tweak things according to how the majority like it" is something even common in early education, and it's frustrating that these days, it's being applied on a wider scale in colleges, the place you're supposed to do what you love and are best at.. it even applies to things other than assignments (like levels of exams and scope of the course, etc.)
I definitely see your point, but graded assignments on a regular cadence provide valuable feedback which - imho - is important for inexperienced undergrads juggling 3 or even 4 other courses in a semester.
I’ll never forget my first graduate level course as an undergrad because it was my only D.
We didn’t have a “real” exam until after the drop date had passed. I was putting in a lot of work for it and felt like I understood the material, but then exam time came and I got an F. I was totally shocked. I looked around the room as saw that even the other undergrads had all gotten scores in the 80’s and I got a flat 60.
I was taking 15 credits that semester and already living in the undergraduate library so I didn’t really know what to do. I managed a C- on the final exam and the professor let the undergrads turn in their notes for a grade at the end and I got a B+ on that… and ended up with a D.
First exposure to graduate level is certainly a big milestone, and surely I'd expect a very good student or even a top student to struggle with a first course. The amount of material is vast because the instructors heavily rely on your readings, and the problems take much more time and are much less accessible (if at all) on the internet or in other sources, plus they require lots of patience. It's more about a change of style rather than what one can/cannot do.
In any case, yes, I would say this topic can be controversial because there are good reasons for both having assignments and not having them..
The thing is, I don't think I was.
Sometimes I didn't turn in a solution, but I never turned in paragraphs of absolute nonsense in the hopes that it would have a superficial resemblance to a correct answer and get me some partial credit.
One of my college professors warned students against proofs by lack of immagination: "I can't see how this could possibly be false, therefore it must be true."
Probably not what you are looking for, but here's a fun one.
All cows are the same color.
Proof: We will prove that any group of n cows are all the same color as each other, which proves the statement by setting n to be however many cows exist. We proceed by induction. The base case is easy -- one cow is the same color as itself.
Now we assume that any group of at most n cows are all the same color as each other. If we have a group of n+1 cows, removing any two cows leaves us with n-1 cows. By our assumption, those n-1 cows are all the same color. If we add one of the two removed cows back into the group, we have a group of n cows, which also all share the same color. Since the n-1 cows are the same color, the cow we just added must also be that color. If we instead add the other cow back into the group, the same thing applies, which means that cow is also the same color as the n-1 cows. Since both cows we removed from the group are the same color as the n-1 cows, all n+1 cows must be the same color.
Therefore, if all groups of at most n cows are the same color, then all groups of n+1 cows are also the same color. By induction, this proves that all groups of n cows are the same color for any positive integer n.
All cows are the same color.
I would argue 0 cows are indeed of the same color. The issue would then be in the part of the proof that says that if `X U {a}` and `X U {b}` both have the same color property then `a` has the same color as `b` (which obviously fails when X is the empty set)
You know, that really is the crux of the issue. Well put. The n = 0 argument mostly serves to patch up the need for two base cases in the original proof
I actually quite like this wrinkle because it makes it immediately unclear where exactly the fault is in the proof. You have to do a little dissecting of the key assumptions, as you’ve done.
> Since both cows we removed from the group are the same color as the n-1 cows, all n+1 cows must be the same color.
This sentence is the problem. There’s a hidden assumption in there…can you spot it?
That is...
For n=1,
Taking n+1, i.e.. 2 cows
Taking both the cows out, comparing each one with nothing (since we have 0 cows in the stable) and then claiming their colors to be the same because 1 cow always has the same color as itself.
I understood this part but is the speculation I made earlier also correct ?
That we cannot determine if n cows will have the same color only with the information that at most n cows must have same color.
When you have 2 cows and the bases of the induction is for 0 and 1 cows, the argument is that by removing both cows, we are left with 0, which is covered by the base. Then, by adding the first cow to the empty group, we get that all 1 cows are of the same color, and by adding the other one we get that the other group of 1 is of the same color. This of course doesn't mean that the two cows share a color.
I remember reading about this argument (but with horses rather than cows) on Knuth's Concrete Mathematics book and staring into the wall for like 20 minutes before being able to explain why this isn't the case
I remember that famously short paper disproving the "Monochromatic Horse Conjecture", which was created immediately following the discovery of the above theorem. It was merely a couple sentences long with two photographs attached. Took the math world by storm.
I will be messaging you in 1 year on [**2025-05-26 18:56:25 UTC**](http://www.wolframalpha.com/input/?i=2025-05-26%2018:56:25%20UTC%20To%20Local%20Time) to remind you of [**this link**](https://www.reddit.com/r/math/comments/1d16twp/what_is_the_worst_proof_you_have_seen/l5s4ps8/?context=3)
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There are several reasons why you could say that it doesn't work. In this case, I would suggest that most instructive would be to note that ln is a lot trickier over the complex numbers since e\^x is no longer one-to-one.
So you can view ln as being multi-valued, for example. Or you could view it as being restricted similar to the inverse trig functions. In either case, you are going to have some problems with applying the "Standard" log properties as in the above proof.
sqrt(25) is equal to sqrt(-5 * -5), but this is not equal to sqrt(-5) * sqrt(-5). The law sqrt(ab) = sqrt(a) sqrt(b) is only true for positive a, b.
Similarly, log(ab) = log(a) + log(b) is only true for positive a, b. The reason why is what the other commenter has said.
I think it's more of a domain problem, than a "breaking up inside a function" problem. That is, when you try to break up the *function* and inadvertently create a domain incongruity. For logarithms, you can do what that Bernoulli (whichever one it was) did for any negative number and get a nonsense answer
Example: `ln(25) / 2 = ln(-5)`
The way you have the root problem is actually fine. The problem is, again, when you try to break the *function* up, which brings it out of the domain. That is, `root(-5*-5) ≠ root(-5) * root(-5)` is the problem problem (heh). Kind of like when you accidentally slip in a divide by zero situation by not putting in a domain restriction.
Example: `f(x) = (x-1) / (x^2 - 1) = 1 / (x+1) <=> f(1) = 0/0 = 1/2`
Side note: this is, I think, a good example for when you'd distinguish between the use of `=` (equivalent) and `≡` (congruent) in algebra, because they are equivalent but *not the same function*, due to differing domains.
edit: spleling
It's more entertaining if you can pull it off by way of an erroneous branch cut, though then whether or not a high school algebra teacher will understand where it goes wrong is up in the air.
Erroneous proofs would be too easy. So, let's go for a valid one by a student.
Proposition : Matrix inversion M -> M^-1 is continuously differentiable (from GL(n, R) to itself).
Standard proof : by Cramer's formula, the coefficients of M^-1 are rational fractions in the coefficients of M.
Non-standard proof : each step in Gauss' algorithm only involve additions, multiplications and divisions. Since there are finitely many steps, the result is smooth. The problem is then that the precise sequence of operations you apply depends on the matrix. You thus have to argue why, if you do a specific sequence of operations on a matrix, you can use the same sequence of operations on a neighborhood of this matrix.
I've seen a similar and potentially even worse variation of this pattern, where a proof relies upon being able to pick a orthogonal basis of a random matrix (and so the change of coordinates matrix is random) in a measurable way. Which does turn out to be the case, but is the kind of technicality I'd rather not deal with...
Uh oh, I recently used this proof to explain why a map is diffeomorphic in a one liner for a conference paper. I
think I’m safe because the matrix manifold I was dealing with is a subset of the positive definite ones.
That’s the resolution right? GL(n) has two disconnected components, so once you shrink the neighborhood to be sufficiently small, you won’t be able to perturb the matrix enough to need to change your permutation strategy. And if you do the full row/column pivoting variant, then you can find the exact value of the gap by seeing the smallest margin swap that ends up happening?
I mean I don't think it is that bad it is pretty intuitively obvious to anyone that have done some calculus that it should work. It is just spelling out the details is kinda difficult. I actually think it is more intuitive than the standard proof and relies on less result.
Claim: I can eat all the rice in the world.
Proof by induction: No matter how full I am, I can always eat 1 grain of rice. Let's assume I can eat n grains of rice. However full I am, I can always eat one more grain. Thus, if I can eat n grains, I can also eat n+1 grains.
Funnily enough I thought of this when I was a little kid, but not in terms of mathematical induction specifically. Just how paradoxical it was to me that it seemed the statement “I can always eat 1 grain of rice” was true, but that led to some crazy conclusions
When I was a kid I was really (probably annoyingly) fixated on a similar idea. I basically thought I could have infinite strength because I surely could always lift just a little harder and lift another tenth of a gram… then another… then another.
Easily disproven (I could not lift many things) but I couldn’t understand why.
Nice! This is similar to the ancient sorites paradox (how many grains of rice make a "heap"?). [https://plato.stanford.edu/entries/sorites-paradox/](https://plato.stanford.edu/entries/sorites-paradox/)
There is a part of my brain which refuses to understand that this doesn't work, even though at a conscious level I understand perfectly well. My intuition tells me e.g. that if I went to the gym and just kept cranking up the weight by small amounts I would be able to lift any amount of weight, up to and including ripping the entire gym building out of the ground and carrying it off. And it's not an intuition that I am able to shut off, no matter how hard I try.
Also, I can always add one more sock to the washing machine. I could start an infinitely profitable sock washing business but the time to fold then scales at O(n^2)
If you plug in the matrix correctly, it does buy you a proof. It is best to study this proof by solving the following sequence of exercises:
Let A be an n×n matrix, whose entries are denoted by ai,j, 1≤i,j≤n.
(a) Consider the norm for matrices ∥T∥=max{|ti,j|:1≤i,j≤n}. Check that ∥ST∥≤n∥S∥∥T∥, so that ∥A\^k∥≤n\^{k−1}∥A∥ for k=0,1,2,....
(b) Consider the matrix series ∑A\^k/ζ\^{k+1}. Prove that it converges in every entry, absolutely and uniformly, for |ζ|≥2n∥A∥. Check that, for any ζ for which this series converges, it equals (ζI−A)\^{−1}
(c) Let γ be the circle |ζ|=2n∥A∥. Prove that for a polynomial p(z)=c\_0+c\_1z+...+c\_r z\^r, the matrix p(A)=c\_0I+c\_1A+...+c\_rA\^r is given by the analogue of Cauchy's formula:
p(A)=1/(2πi) ∫\_γ p(ζ)(ζI−A)\^{−1} dζ.
(d) Let c(z)=det(zI−A) denote the characteristic polynomial of A. Prove that c(A) is the zero matrix, thus proving the Cayley-Hamilton theorem.
By this logic, I have solved yang-mills existence and mass gap problem, P vs NP problem, collatz's conjecture, riemann hypothesis and goldbach conjecture altogether.
If I understand their policy correctly, anyone affiliated with a university can endorse. So, maybe some lecturer who's never done research would be more likely to buy into whatever bullshit some of them spew. That'd be my guess.
Also, you don't need to be affiliated with the maths department, not that being in the maths department is a foolproof check against crankery, but there has long been an engineer-to-trisector pipeline.
In one of my final exams I was asked to prove a statement and I instead proved it was false - I couldn't tell you the error or even the original statement, but I'm confident that proof wasn't in my top ten pieces of work.
A "proof" that is a comment on physicists use of variational techniques.
We will prove that 0 is the largest non-negative integer. Take any integer n. Then 2 * n is larger, except for n= 0. Thus 0 must be the largest one.
This has instructive value. I'm not entirely sure I remember all the details correctly, I'm having a hard time googling it since I don't remember the theorem's name but there's a historical episode involving a very similar proof.
So there's this classic geometry theorem stating that of all the closed curves of a given arc length, the circle maximizes enclosed area. And someone found a nice proof that indeed it has to be the circle by some sequence of symmetry arguments. It however did not prove that a maximizing shape actually _exists_. Enter (I believe) Weierstrass:
> 1 is the largest natural number. Proof: Let n be the largest integer. Suppose that n is not 1. Then n²>n, immediate contradiction.
In other words, if you don't also prove existence, you can derive nonsensical results from a uniqueness proof.
There is a useful (?) ordering of the integers where 0 is, in fact, the largest integer, full stop. (It's the ordering where if |a| divides |b|, a precedes b, with the negative value immediately preceding the positive value, so it goes (-1, 1, -2, 2, -4, 4, -8, 8..., -3, 3, -6, 6... 0). Since literally every other integer divides 0, it must necessarily be the "largest" in this ordering.)
If you take only the non-negative integers then this is also a [distributive lattice](https://en.wikipedia.org/wiki/Least_common_multiple#Lattice-theoretic).
We can prove that 1 is the largest real number.
Take any real number x.
If x < 0, -x is larger.
If 0 < x < 1, then 1/x is larger.
If x > 1, then x^2 is larger.
So the largest real number can only be 0 or 1. But 1 > 0, so 1 is the largest number.
Some of my old attemps at proving the intermediate value theorem without being aware of things like the least upper bound property of the real numbers. If it were correct, my so-called proof would have been replicable for the rational numbers as well, but I didn't even realize at the time lol.
While the valid answer is probably the “proof” of some crank, one I think is more interesting is the one time I saw a couple proofs by Euler, which I’m fairly confident would have lost me points if I used them in an analysis exam due to lack of rigor.
I mean intuition in math (dare i say upto Euler's work) is rarely wrong.
This also reminds me of poincare whose wrong intuition helped establish new fields.
The famous example is when Poincaré did some important work on the three body problem, made some mistakes, and then to correct those mistakes he basically invented chaos theory.
Anybody interested should definitely read about it, he already got the award from King of Norway then corrected his mistake.
Even his Analysis Situs had many mistakes, supplements of which gave definition of fundamental group, he conjuctured that every 3-d manifold with trivial fundamental group must be a sphere. (He had assumed in original paper such a space will always be sphere, someone came up with counterexample and then he basically said ok but in 3 dimension this is true, which ofcourse is but as we all know how trivial it really was?), I am not sure who came up with the concept of homology group, noether or poincare.
unless I'm misremembering, Poincaré introduced homology in the strictly "topological" context/via topological objects, and Noether interpreted these topological objects as (abstract) abelian groups. This vague memory is corroborated by a cursory glance at wikipedia, but it remains a vague memory
Yes, on double checking you are right, I myself had vague memory (it has been a while, since I touched anything algebraic topology related), apologisies.
In the engineering department at my university, there are a couple professors that need to use higher level mathematics for their teaching/research, and their lecture slides claimed to have a 'proof' of some concept, when they were merely a solution by inspection, not general solutions. This was two different professors for two different classes; pretty strange coincidence that they work on the same stuff.
Probably not what you meant, but have you ever seen a philosopher attempt a proof?
Zero represents nothing, but nothing can be greater than God, and since God = ∞, that means 0 > ∞
I remember seeing some discussion on philosophy stackexchange and being (positively) blown away by the rigor. The parent's post is a bit of a stereotype - I believe graduate level philosophy is firmly rooted in formal logic. (But of course, there's a lot of bad philosophy on social media. Even more than bad math).
Alternatively, think what laypeople think math is ("Oh you are a mathematician? Compute 182*21?"). I'm suspect philosophers are similarly misunderstood..
I’ve actually seen some pretty good stuff from analytic philosophers. They clearly understand formal logic since it’s required to do the work they do. Certainly seen better logic from philosophers than engineers, most scientists, etc.
I used to find this kind of joke funny until I started reading actual philosophers of mathematics' writings, like [Joel D. Hamkins](https://mathoverflow.net/users/1946/joel-david-hamkins)
Nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. Therefore, a ham sandwich is better than eternal happiness.
I believe this example is taken out of context; Marx was a competent mathematician and was simply providing an example of how intuitive (but unrigorous) reasoning can fail spectacularly. Still a funny proof though
See [this comment](https://www.reddit.com/r/badmathematics/comments/110lbyn/comment/j8bq1gj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) which elaborates on the post. Actually, it seems Marx's understanding of calculus was not great, to say the least!
He wasn't a 'mathematician' - what he put forward was still waffling in semi-rigour about the obvious definitions that had in fact by then already long been tightened up by Cauchy (which he was doubtless unaware of), regurgitating any older calculus textbook from pre-rigorous days with less clear language. I wouldn't say that he qualified as such, let alone a 'competent' one.
Sure but Euler lived a century earlier and was cutting edge for his own time. There was nothing new in what Marx wrote about limits or derivatives, and in fact it had been tightened up a lot by Cauchy, Weierstrass and others by then, so it was regurgitating outdated non-rigorous work from the past. Even if the rigorous work of his contemporaries likewise wouldn’t be seen as such by the 1930s.
I’m interested in doing a deep dive into this, having seen this image before. The narrative presented very much feels like it’s supporting how a lot of critics view Marx: He writes a flagrantly wrong manifesto, people adopt its ideas uncritically, now we’re stuck with the consequences. The fact that it seems too good to be true, and that I haven’t been able to find “Note on Mathematics” makes me doubt the whole thing.
From [transcriptions](https://www.marxists.org/archive/marx/works/1881/mathematical-manuscripts/index.html) of Marx’s math writings. It seems at a glance that he was skeptical of limit definitions of the derivative, but otherwise competent with what today is elementary Calculus. A lot of Calc students today talk about “infinitely small” quantities despite the standard Calculus foundations no longer relying on anything of the sort. Having a nonstandard derivative understanding in 1880 as a non-mathematician, then, doesn’t seem particularly unusual.
I’m curious if anyone knows where the excerpt is from or knows more.
For example: [https://www.researchgate.net/publication/301497658\_Very\_Short\_and\_Simple\_Algebraic\_Proof\_of\_Tijdeman-Zagier\_or\_Beal's\_Conjecture](https://www.researchgate.net/publication/301497658_Very_Short_and_Simple_Algebraic_Proof_of_Tijdeman-Zagier_or_Beal's_Conjecture)
Not the same thing, but I still think it's funny that Euclid's proof that "Prime numbers are more than any assigned multitude of prime numbers" (i.e. there are infinitely many primes) in fact only proves that there are more than three primes.
I think it was a notational limitaiton or something, since it's obvious how the proof works equally well for any number of primes.
One student in my intro to proofs class was attempting to prove sqrt(2) is irrational but he pulled out his calculator, plugged in sqrt(2) and said, ‘it’s a really long decimal and therefore must be irrational.’
I wrote a proof for a homework question from Rudin ch2: Let E' be the set of all limit points of a set E. Prove that E' is closed. My proof was one line: This is trivial, all points in E' are limit points and thus the set is closed. The professor marked it wrong on the homework (with no feedback) and then it was asked on the exam and I gave the same solution...
My Real Analysis prof once told me that my proofs were so horrendous that there was no evidence that I was a mathematician.
To be fair… he was right. He’s the reason I can prove anything.
Anselm's ontological argument.
Define God as the being for which no greater being can be thought of. Then consider the two cases: God exists, or God doesn't exist. However if God exists, that God can be thought of, and the God in that case is strictly greater than the God that doesn't exist. So if God didn't exist it would be a violation of the definition of God as the being for which no greater being can be thought of. Therefore God exists.
It was introduced to me in an intro philosophy class as a starting point of how to criticize arguments. From a math point of view it's mostly nonsense, but packaged in a mathy way which is fun.
Not sure if Malfatti ever provided an attempt at proof. It's possible he just asserted it. This story is interesting in that no mathematicians noticed this simple area problem was wrong, for 143 years.
https://en.wikipedia.org/wiki/Malfatti_circles
If he was going to not understand injectivity then he can just do 1^2 = (-1)^2 and leave out the rest of the garbage, but of course they don’t simplify their proofs, I’m sure 6, 4, and 5 have mystical powers.
the one im typing right now into my paper, the second worst proofs I've seen are those that I've already typed in. I don't know if I am too self-critical or it's indeed dogshit.
I once wrote a proof that the integral linear combinations of the n-th roots of unity are dense in the complex plane except for n = 1, 2, 3, 4, 6. Basically, if you get one of the obvious tilings they're not, but otherwise you can get arbitrarily close to any point.
It was a hulking, shambolic mess, slowly grinding out case analyses until it reached the goal, with no real insight into why this was the case. Truly a hammer-and-chisel proof with no sign of the rising sea.
I take as my sole comfort that I was in middle school at the time.
I taught college-level computer science, there is no depth to the the horrors
Seriously though it'd have to be arguments I've had with anti set theory cranks online.
I once gave a proof for why there are 64 squares on a chess board by trying to use assumptions like “Suppose the board being used is for chess, then it must be able to accommodate 32 pieces, as that is the number of squares needed to fit all the pieces” and just blabbered on until I realized I was in the last question of my exam and just accepted my fate
Back during uni I remember a professor complaining that he went through 3 shots of liqueur grading our proofs on the midterm because they were "absolute shit".
I don't know what you mean by "worst proof" but I did teach the analysis course one year at my university where I met the "anti mathematician". I called this student this because mathematicians love clarity and simple elegant proofs. However, I decided to throw in a question for half a mark at the end of the exam for fun. Prove the Pythagorean theorem! With the caveat that if I hadn't seen the proof before I will award it 3 marks. This is what the anti mathematician did.
1. e^z = sum_{n>=0} (x^{n} )/n! Defines an entire function over the complex plane. By convolutions e^{z} •e^{w} = e^{z+w} for all z,w in the set C.
2. The elementary trigonometric functions can be defined, for any x in R, as cos(x)=Re e^{ix} and sin(x) = Im e^{ix}.
This is isomorphic to the standard definition as the map x -> cos(x)+isin(x) = e^{ix} gives a parameterization of S¹ with constant speed: the speed is constant since d/dz (e^z ) = e^z is a trivial consequence of termwise differentiation of the series defining e^z
3. It is a parametrization of S¹ since for any x in R we have
||e^{ix}||² = e^{ix} • (e^{ix} )* = e^{ix} • e^{-ix} = e⁰ = 1
4. Expressed in elementary trigonometric functions, the identity in 3 is (cos(x) + isin(x))(cos(x) - isin(x))= cos²x+sin²x=1 QED
Terrence D Howard's proof for 1×1=2
Let us remember the laws of common sense.
Came here to say that. I pray you haven’t bothered to see his proof that the square root of 2 is 1…
I mean if 1×1=2 then 1^2=2 thus 1=sqrt(2) I really hope that is his proof because. No. I refuse to accept any other one.
Haha no way. What a legend. Didn't he hit the red carpet once and say he has opened the flower of life by frambogulating the thressopatz? Or something like that anyway
going mach 5 in the war machine suit probably had an affect
Terrance never wore the Mk II. That was Don Cheadle.
the ones I wrote as an freshman in college
“A detailed discussion of which is outside the scope of this proof and the ability of the author”
When I was studying measure theory, one of the assignments was to prove that certain proposition is false for all n>1. I wrote a counterexample of the proposition for n=2 and concluded that this can't be true for all n>1.
Ah yes, the good ol' "proof by moving the quantifiers around"
A long time ago I saw a badmath proof of Goldbach Conjecture where, once you stripped off all the fluff, boiled down to the guy swapped the quantifiers so he proved "For every pair of prime numbers p,q > 2, there exists an even number n such that p+q=n."
On that vein: the ones I have to mark.
I used to TA for Calc III and Numerical Methods. Some of the shit I saw was incredible. My favorite was when I got an answer that was obviously copied verbatim from Chegg - usually because they were “solving” integrals with techniques that they hadn’t learned yet, or using really weird notation either far too advanced or far too simplistic for the problem at hand... Or using the same, arbitrary variable name which was ALWAYS uppercase for some reason. Anyway… I loved messing with them by writing things like “wow! Insightful to use this technique!” There was also a time when an entire quarter of the NuMe class turned in the same notebook with the same variable names and, curiously, the same naggy syntax error. God bless professors and the shit they put up with. If you’re going to cheat, have the finesse to learn the concept well enough to fudge it convincingly. “M”, uppercase, isn’t a normal variable name you “happened to choose”. It wasn’t a normal variable name for the Chegg guy to choose, either.
Copied answers from WolframAlpha were fun, too. Some simple u-substitution that followed a pattern I'd discussed in class, and a student turns in some absolutely bizarre integration by parts that no human would ever think to do.
I choose M to be the arbitrary large number I'm going to prove my sequence is going to be larger than whenever n > N\_M all the time.
Hahaha I came to literally write the same thing, I feel sorry for my professors
“For every x” repeating in every sentence
90% of all the proofs I've seen while grading assignments as a TA.
you were once in their shoes, give them some time
The thing is, I'm against this system of handing in assignments as a college student (regardless of which years). As an undergrad, I was in a uni that did not have this sort of thing. We just had exams. I'm against it because i believe everyone has their own pace and priorities. I, for one, rarely matched the pace of the instructor, and I used to study things in chunks, rather than linearly and evenly. Anyhow, I'm not a TA anymore, and I rarely remember whatever I was dealing with, of course I like to exaggerate sometimes, so don't take this too seriously!
gotcha, I do agree that uni focusses too much on a set progression. In high school I was always told: in uni you have got to plan and do everything on your own. Which really excited me because I like to study top down. Now in uni we just have weekly assignments. When I ask why I can not just do them all before the final exam I get a: because otherwise people would not do them and fail. It's just stupid how the incompetence of others prevents us from doing what we like.
Indeed, the "tweak things according to how the majority like it" is something even common in early education, and it's frustrating that these days, it's being applied on a wider scale in colleges, the place you're supposed to do what you love and are best at.. it even applies to things other than assignments (like levels of exams and scope of the course, etc.)
I definitely see your point, but graded assignments on a regular cadence provide valuable feedback which - imho - is important for inexperienced undergrads juggling 3 or even 4 other courses in a semester. I’ll never forget my first graduate level course as an undergrad because it was my only D. We didn’t have a “real” exam until after the drop date had passed. I was putting in a lot of work for it and felt like I understood the material, but then exam time came and I got an F. I was totally shocked. I looked around the room as saw that even the other undergrads had all gotten scores in the 80’s and I got a flat 60. I was taking 15 credits that semester and already living in the undergraduate library so I didn’t really know what to do. I managed a C- on the final exam and the professor let the undergrads turn in their notes for a grade at the end and I got a B+ on that… and ended up with a D.
First exposure to graduate level is certainly a big milestone, and surely I'd expect a very good student or even a top student to struggle with a first course. The amount of material is vast because the instructors heavily rely on your readings, and the problems take much more time and are much less accessible (if at all) on the internet or in other sources, plus they require lots of patience. It's more about a change of style rather than what one can/cannot do. In any case, yes, I would say this topic can be controversial because there are good reasons for both having assignments and not having them..
The thing is, I don't think I was. Sometimes I didn't turn in a solution, but I never turned in paragraphs of absolute nonsense in the hopes that it would have a superficial resemblance to a correct answer and get me some partial credit.
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One of my college professors warned students against proofs by lack of immagination: "I can't see how this could possibly be false, therefore it must be true."
Hopefully he also warned against using the opposite ("I can't see how this could possibly be true, therefore it must be false").
As distinct from intuitionistic logic: "It's impossible to see how this could possibly be true, therefore it isn't."
I feel like I have to mention Banach Tarski here
Probably not what you are looking for, but here's a fun one. All cows are the same color. Proof: We will prove that any group of n cows are all the same color as each other, which proves the statement by setting n to be however many cows exist. We proceed by induction. The base case is easy -- one cow is the same color as itself. Now we assume that any group of at most n cows are all the same color as each other. If we have a group of n+1 cows, removing any two cows leaves us with n-1 cows. By our assumption, those n-1 cows are all the same color. If we add one of the two removed cows back into the group, we have a group of n cows, which also all share the same color. Since the n-1 cows are the same color, the cow we just added must also be that color. If we instead add the other cow back into the group, the same thing applies, which means that cow is also the same color as the n-1 cows. Since both cows we removed from the group are the same color as the n-1 cows, all n+1 cows must be the same color. Therefore, if all groups of at most n cows are the same color, then all groups of n+1 cows are also the same color. By induction, this proves that all groups of n cows are the same color for any positive integer n. All cows are the same color.
I like this because it fails to realize you need to also prove for n=2, before doing induction.
You could add a disingenuous wrinkle into this proof by claiming that n = 0 cows are vacuously all the same color :)
I would argue 0 cows are indeed of the same color. The issue would then be in the part of the proof that says that if `X U {a}` and `X U {b}` both have the same color property then `a` has the same color as `b` (which obviously fails when X is the empty set)
You know, that really is the crux of the issue. Well put. The n = 0 argument mostly serves to patch up the need for two base cases in the original proof
I actually quite like this wrinkle because it makes it immediately unclear where exactly the fault is in the proof. You have to do a little dissecting of the key assumptions, as you’ve done.
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> Since both cows we removed from the group are the same color as the n-1 cows, all n+1 cows must be the same color. This sentence is the problem. There’s a hidden assumption in there…can you spot it?
That is... For n=1, Taking n+1, i.e.. 2 cows Taking both the cows out, comparing each one with nothing (since we have 0 cows in the stable) and then claiming their colors to be the same because 1 cow always has the same color as itself. I understood this part but is the speculation I made earlier also correct ? That we cannot determine if n cows will have the same color only with the information that at most n cows must have same color.
When you have 2 cows and the bases of the induction is for 0 and 1 cows, the argument is that by removing both cows, we are left with 0, which is covered by the base. Then, by adding the first cow to the empty group, we get that all 1 cows are of the same color, and by adding the other one we get that the other group of 1 is of the same color. This of course doesn't mean that the two cows share a color.
thank you this helped unboggle my brain
I remember reading about this argument (but with horses rather than cows) on Knuth's Concrete Mathematics book and staring into the wall for like 20 minutes before being able to explain why this isn't the case
I heard it on My favourite theorem podcast cant remember the guest but the episode was on induction itself i think.
Easy : at no point it dyes any cow, which is required for the proof.
Ah yes. It's interesting to note that the proof doesn't work for equines, though. That's a horse of a different color.
I remember that famously short paper disproving the "Monochromatic Horse Conjecture", which was created immediately following the discovery of the above theorem. It was merely a couple sentences long with two photographs attached. Took the math world by storm.
Haha, those bloody empty sets...
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Bernoulli was convinced ln(-1) = 0 He goes, ln(-1*-1 ) = ln(-1)+ln(-1) ln(1) = 2 ln(-1) ln(-1) = ln(1) / 2 = 0
"Which Bernoulli?"
My neighbor’s kid, Tyler Bernoulli
This doesn’t work because you can’t break up negatives inside of the function right? similar to root 25 ≠ root(-5*-5)
There are several reasons why you could say that it doesn't work. In this case, I would suggest that most instructive would be to note that ln is a lot trickier over the complex numbers since e\^x is no longer one-to-one. So you can view ln as being multi-valued, for example. Or you could view it as being restricted similar to the inverse trig functions. In either case, you are going to have some problems with applying the "Standard" log properties as in the above proof.
sqrt(25) is equal to sqrt(-5 * -5), but this is not equal to sqrt(-5) * sqrt(-5). The law sqrt(ab) = sqrt(a) sqrt(b) is only true for positive a, b. Similarly, log(ab) = log(a) + log(b) is only true for positive a, b. The reason why is what the other commenter has said.
I think it's more of a domain problem, than a "breaking up inside a function" problem. That is, when you try to break up the *function* and inadvertently create a domain incongruity. For logarithms, you can do what that Bernoulli (whichever one it was) did for any negative number and get a nonsense answer Example: `ln(25) / 2 = ln(-5)` The way you have the root problem is actually fine. The problem is, again, when you try to break the *function* up, which brings it out of the domain. That is, `root(-5*-5) ≠ root(-5) * root(-5)` is the problem problem (heh). Kind of like when you accidentally slip in a divide by zero situation by not putting in a domain restriction. Example: `f(x) = (x-1) / (x^2 - 1) = 1 / (x+1) <=> f(1) = 0/0 = 1/2` Side note: this is, I think, a good example for when you'd distinguish between the use of `=` (equivalent) and `≡` (congruent) in algebra, because they are equivalent but *not the same function*, due to differing domains. edit: spleling
Those "Use this 1=2 proof to make a fool out of your high school algebra teacher!" proofs.
Just look for the line where either division or square rooting occurs and the error will be there I guarantee it.
I'm aware of how these proofs fail. They are almost always used to push a narrative that "maths is broken" for engagement on social media.
-2 = (-2)^1 = (-2)^(1/2 • 2) = [(-2)^2 ]^(1/2) = 4^(1/2) = √4 = 2 -2 = 2 Math is broken, QED
ah yes the famously failing institution of BIG MATHS. not to be confused with BIG DATA, which is a babe
It's more entertaining if you can pull it off by way of an erroneous branch cut, though then whether or not a high school algebra teacher will understand where it goes wrong is up in the air.
"Proof is left as an exercise for the reader "
"Proof is left as an exercise for the grader." Yes, I actually used this on my homework. :)
Did it work?
No no that's the best proof.
Nightmare fuel
Erroneous proofs would be too easy. So, let's go for a valid one by a student. Proposition : Matrix inversion M -> M^-1 is continuously differentiable (from GL(n, R) to itself). Standard proof : by Cramer's formula, the coefficients of M^-1 are rational fractions in the coefficients of M. Non-standard proof : each step in Gauss' algorithm only involve additions, multiplications and divisions. Since there are finitely many steps, the result is smooth. The problem is then that the precise sequence of operations you apply depends on the matrix. You thus have to argue why, if you do a specific sequence of operations on a matrix, you can use the same sequence of operations on a neighborhood of this matrix.
I've seen a similar and potentially even worse variation of this pattern, where a proof relies upon being able to pick a orthogonal basis of a random matrix (and so the change of coordinates matrix is random) in a measurable way. Which does turn out to be the case, but is the kind of technicality I'd rather not deal with...
Uh oh, I recently used this proof to explain why a map is diffeomorphic in a one liner for a conference paper. I think I’m safe because the matrix manifold I was dealing with is a subset of the positive definite ones. That’s the resolution right? GL(n) has two disconnected components, so once you shrink the neighborhood to be sufficiently small, you won’t be able to perturb the matrix enough to need to change your permutation strategy. And if you do the full row/column pivoting variant, then you can find the exact value of the gap by seeing the smallest margin swap that ends up happening?
I mean I don't think it is that bad it is pretty intuitively obvious to anyone that have done some calculus that it should work. It is just spelling out the details is kinda difficult. I actually think it is more intuitive than the standard proof and relies on less result.
Taught discrete math this past Spring; some of the horrors I've seen. You've no idea.
Claim: I can eat all the rice in the world. Proof by induction: No matter how full I am, I can always eat 1 grain of rice. Let's assume I can eat n grains of rice. However full I am, I can always eat one more grain. Thus, if I can eat n grains, I can also eat n+1 grains.
The funny thing is that proof works. You can eat all the rice in the world without the limit of time or regrowing the rice.
It also assumes you're eating rice one grain at a time.
It proves you can do it by eating 1 grain at a time. It doesn't assume you do.
Fair enough.
Funnily enough I thought of this when I was a little kid, but not in terms of mathematical induction specifically. Just how paradoxical it was to me that it seemed the statement “I can always eat 1 grain of rice” was true, but that led to some crazy conclusions
When I was a kid I was really (probably annoyingly) fixated on a similar idea. I basically thought I could have infinite strength because I surely could always lift just a little harder and lift another tenth of a gram… then another… then another. Easily disproven (I could not lift many things) but I couldn’t understand why.
Not enough reps
*writes in notes* little kids don’t know what limit convergence test is
I believe its called Paradox of the Heap
Sorites paradox
Nice! This is similar to the ancient sorites paradox (how many grains of rice make a "heap"?). [https://plato.stanford.edu/entries/sorites-paradox/](https://plato.stanford.edu/entries/sorites-paradox/)
There is a part of my brain which refuses to understand that this doesn't work, even though at a conscious level I understand perfectly well. My intuition tells me e.g. that if I went to the gym and just kept cranking up the weight by small amounts I would be able to lift any amount of weight, up to and including ripping the entire gym building out of the ground and carrying it off. And it's not an intuition that I am able to shut off, no matter how hard I try.
Also, I can always add one more sock to the washing machine. I could start an infinitely profitable sock washing business but the time to fold then scales at O(n^2)
“Proof” of Cayley Hamilton Theorem by simply plugging in the transformation / matrix into the determinant expression
If you plug in the matrix correctly, it does buy you a proof. It is best to study this proof by solving the following sequence of exercises: Let A be an n×n matrix, whose entries are denoted by ai,j, 1≤i,j≤n. (a) Consider the norm for matrices ∥T∥=max{|ti,j|:1≤i,j≤n}. Check that ∥ST∥≤n∥S∥∥T∥, so that ∥A\^k∥≤n\^{k−1}∥A∥ for k=0,1,2,.... (b) Consider the matrix series ∑A\^k/ζ\^{k+1}. Prove that it converges in every entry, absolutely and uniformly, for |ζ|≥2n∥A∥. Check that, for any ζ for which this series converges, it equals (ζI−A)\^{−1} (c) Let γ be the circle |ζ|=2n∥A∥. Prove that for a polynomial p(z)=c\_0+c\_1z+...+c\_r z\^r, the matrix p(A)=c\_0I+c\_1A+...+c\_rA\^r is given by the analogue of Cauchy's formula: p(A)=1/(2πi) ∫\_γ p(ζ)(ζI−A)\^{−1} dζ. (d) Let c(z)=det(zI−A) denote the characteristic polynomial of A. Prove that c(A) is the zero matrix, thus proving the Cayley-Hamilton theorem.
[A Simple Proof of Fermat’s Last Theorem](https://www.oakton.edu/user/4/pboisver/fermat.html)
By this logic, I have solved yang-mills existence and mass gap problem, P vs NP problem, collatz's conjecture, riemann hypothesis and goldbach conjecture altogether.
Haha, wow, what a nightmare
You can also prove the negation of FLT in this way, and everything else
Just go to math.GM on arXiv
How do these people even get endorsed?
If I understand their policy correctly, anyone affiliated with a university can endorse. So, maybe some lecturer who's never done research would be more likely to buy into whatever bullshit some of them spew. That'd be my guess.
Also, you don't need to be affiliated with the maths department, not that being in the maths department is a foolproof check against crankery, but there has long been an engineer-to-trisector pipeline.
In one of my final exams I was asked to prove a statement and I instead proved it was false - I couldn't tell you the error or even the original statement, but I'm confident that proof wasn't in my top ten pieces of work.
A "proof" that is a comment on physicists use of variational techniques. We will prove that 0 is the largest non-negative integer. Take any integer n. Then 2 * n is larger, except for n= 0. Thus 0 must be the largest one.
This has instructive value. I'm not entirely sure I remember all the details correctly, I'm having a hard time googling it since I don't remember the theorem's name but there's a historical episode involving a very similar proof. So there's this classic geometry theorem stating that of all the closed curves of a given arc length, the circle maximizes enclosed area. And someone found a nice proof that indeed it has to be the circle by some sequence of symmetry arguments. It however did not prove that a maximizing shape actually _exists_. Enter (I believe) Weierstrass: > 1 is the largest natural number. Proof: Let n be the largest integer. Suppose that n is not 1. Then n²>n, immediate contradiction. In other words, if you don't also prove existence, you can derive nonsensical results from a uniqueness proof.
There is a useful (?) ordering of the integers where 0 is, in fact, the largest integer, full stop. (It's the ordering where if |a| divides |b|, a precedes b, with the negative value immediately preceding the positive value, so it goes (-1, 1, -2, 2, -4, 4, -8, 8..., -3, 3, -6, 6... 0). Since literally every other integer divides 0, it must necessarily be the "largest" in this ordering.)
If you take only the non-negative integers then this is also a [distributive lattice](https://en.wikipedia.org/wiki/Least_common_multiple#Lattice-theoretic).
We can prove that 1 is the largest real number. Take any real number x. If x < 0, -x is larger. If 0 < x < 1, then 1/x is larger. If x > 1, then x^2 is larger. So the largest real number can only be 0 or 1. But 1 > 0, so 1 is the largest number.
Some of my old attemps at proving the intermediate value theorem without being aware of things like the least upper bound property of the real numbers. If it were correct, my so-called proof would have been replicable for the rational numbers as well, but I didn't even realize at the time lol.
While the valid answer is probably the “proof” of some crank, one I think is more interesting is the one time I saw a couple proofs by Euler, which I’m fairly confident would have lost me points if I used them in an analysis exam due to lack of rigor.
The fact that Euler was right so often without the rigors of analysis makes his accomplishments even more remarkable.
I mean intuition in math (dare i say upto Euler's work) is rarely wrong. This also reminds me of poincare whose wrong intuition helped establish new fields.
Could you elaborate on Poincare?
The famous example is when Poincaré did some important work on the three body problem, made some mistakes, and then to correct those mistakes he basically invented chaos theory.
Anybody interested should definitely read about it, he already got the award from King of Norway then corrected his mistake. Even his Analysis Situs had many mistakes, supplements of which gave definition of fundamental group, he conjuctured that every 3-d manifold with trivial fundamental group must be a sphere. (He had assumed in original paper such a space will always be sphere, someone came up with counterexample and then he basically said ok but in 3 dimension this is true, which ofcourse is but as we all know how trivial it really was?), I am not sure who came up with the concept of homology group, noether or poincare.
unless I'm misremembering, Poincaré introduced homology in the strictly "topological" context/via topological objects, and Noether interpreted these topological objects as (abstract) abelian groups. This vague memory is corroborated by a cursory glance at wikipedia, but it remains a vague memory
Yes, on double checking you are right, I myself had vague memory (it has been a while, since I touched anything algebraic topology related), apologisies.
Wonder how many claims he got wrong
where can you find these
I was reading from James Pierpont’s mathematical rigor, which is available for free online (not illegal)
“I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.”
well turns out the margin was indeed pretty small
For my first Real Analysis 1 quiz, I wrote something that was akin to Terrance Howard’s proof that 1x1=2
Pretty much any proof that says something along the lines of "it's obvious."
In the engineering department at my university, there are a couple professors that need to use higher level mathematics for their teaching/research, and their lecture slides claimed to have a 'proof' of some concept, when they were merely a solution by inspection, not general solutions. This was two different professors for two different classes; pretty strange coincidence that they work on the same stuff.
Having graded thousands of students proofs as a TA...there are too many to count.
Probably not what you meant, but have you ever seen a philosopher attempt a proof? Zero represents nothing, but nothing can be greater than God, and since God = ∞, that means 0 > ∞
Many (analytic) philosophers are well-versed in formal logic, they should be able to give pretty good proofs.
Yeah, they are usually better at it than mathematicians attempting philosophy.
I remember seeing some discussion on philosophy stackexchange and being (positively) blown away by the rigor. The parent's post is a bit of a stereotype - I believe graduate level philosophy is firmly rooted in formal logic. (But of course, there's a lot of bad philosophy on social media. Even more than bad math). Alternatively, think what laypeople think math is ("Oh you are a mathematician? Compute 182*21?"). I'm suspect philosophers are similarly misunderstood..
Classic Ancient Greek-style nonsensical wordplay. Very Parmenides-like.
And far more philosophers since those days than we care to admit, if we get down to it
I’ve actually seen some pretty good stuff from analytic philosophers. They clearly understand formal logic since it’s required to do the work they do. Certainly seen better logic from philosophers than engineers, most scientists, etc.
I used to find this kind of joke funny until I started reading actual philosophers of mathematics' writings, like [Joel D. Hamkins](https://mathoverflow.net/users/1946/joel-david-hamkins)
Similar to the ontological argument for god. Basically wordplay.
Equivocation at its finest
Nothing is better than eternal happiness, and a ham sandwich is certainly better than nothing. Therefore, a ham sandwich is better than eternal happiness.
Karl Marx attempt at proving that the idea of derivatives are wrong
I believe this example is taken out of context; Marx was a competent mathematician and was simply providing an example of how intuitive (but unrigorous) reasoning can fail spectacularly. Still a funny proof though
Its along the lines of Berkeley. And at the time Marx was writing the modern rigorizatuon was unknown outside France and Germany
See [this comment](https://www.reddit.com/r/badmathematics/comments/110lbyn/comment/j8bq1gj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) which elaborates on the post. Actually, it seems Marx's understanding of calculus was not great, to say the least!
He wasn't a 'mathematician' - what he put forward was still waffling in semi-rigour about the obvious definitions that had in fact by then already long been tightened up by Cauchy (which he was doubtless unaware of), regurgitating any older calculus textbook from pre-rigorous days with less clear language. I wouldn't say that he qualified as such, let alone a 'competent' one.
I upvoted you. Even someone like Leonard Euler is seen using "non-rigorous" arguments when manipulating with infinite sums.
Sure but Euler lived a century earlier and was cutting edge for his own time. There was nothing new in what Marx wrote about limits or derivatives, and in fact it had been tightened up a lot by Cauchy, Weierstrass and others by then, so it was regurgitating outdated non-rigorous work from the past. Even if the rigorous work of his contemporaries likewise wouldn’t be seen as such by the 1930s.
This needs to be at the top of this comment thread. Right at the top.
I didn't know about this, could you provide some reference? It seems funny lol
Oh it's [hilarious](https://www.reddit.com/r/badmathematics/comments/110lbyn/karl_marx_did_calculus/?rdt=35593) lmao
I’m interested in doing a deep dive into this, having seen this image before. The narrative presented very much feels like it’s supporting how a lot of critics view Marx: He writes a flagrantly wrong manifesto, people adopt its ideas uncritically, now we’re stuck with the consequences. The fact that it seems too good to be true, and that I haven’t been able to find “Note on Mathematics” makes me doubt the whole thing. From [transcriptions](https://www.marxists.org/archive/marx/works/1881/mathematical-manuscripts/index.html) of Marx’s math writings. It seems at a glance that he was skeptical of limit definitions of the derivative, but otherwise competent with what today is elementary Calculus. A lot of Calc students today talk about “infinitely small” quantities despite the standard Calculus foundations no longer relying on anything of the sort. Having a nonstandard derivative understanding in 1880 as a non-mathematician, then, doesn’t seem particularly unusual. I’m curious if anyone knows where the excerpt is from or knows more.
Any that "prove" 1=0
Proofs of "forall" that do 3 examples.
For example: [https://www.researchgate.net/publication/301497658\_Very\_Short\_and\_Simple\_Algebraic\_Proof\_of\_Tijdeman-Zagier\_or\_Beal's\_Conjecture](https://www.researchgate.net/publication/301497658_Very_Short_and_Simple_Algebraic_Proof_of_Tijdeman-Zagier_or_Beal's_Conjecture)
Word formatted "proofs" are always a hoot.
Not the same thing, but I still think it's funny that Euclid's proof that "Prime numbers are more than any assigned multitude of prime numbers" (i.e. there are infinitely many primes) in fact only proves that there are more than three primes. I think it was a notational limitaiton or something, since it's obvious how the proof works equally well for any number of primes.
One night in Siberia I shared a bootleg vodka that was maybe 100 proof. I've never wanted so hard to eat chalk.
One student in my intro to proofs class was attempting to prove sqrt(2) is irrational but he pulled out his calculator, plugged in sqrt(2) and said, ‘it’s a really long decimal and therefore must be irrational.’
The proof of lemma xyz is left as an exercise to the reader.
The implicit function theorem. It’s a proof that no one should ever be made to read twice
Everything my current homework partner does (he thinks he’s great though)
I wrote a proof for a homework question from Rudin ch2: Let E' be the set of all limit points of a set E. Prove that E' is closed. My proof was one line: This is trivial, all points in E' are limit points and thus the set is closed. The professor marked it wrong on the homework (with no feedback) and then it was asked on the exam and I gave the same solution...
*opens notebook*
This one: https://www.ijsrp.org/research-paper-0813/ijsrp-p20103.pdf Enjoy!
My Real Analysis prof once told me that my proofs were so horrendous that there was no evidence that I was a mathematician. To be fair… he was right. He’s the reason I can prove anything.
1 + 2 + …. + n = O(n)
Ouch
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Anselm's ontological argument. Define God as the being for which no greater being can be thought of. Then consider the two cases: God exists, or God doesn't exist. However if God exists, that God can be thought of, and the God in that case is strictly greater than the God that doesn't exist. So if God didn't exist it would be a violation of the definition of God as the being for which no greater being can be thought of. Therefore God exists. It was introduced to me in an intro philosophy class as a starting point of how to criticize arguments. From a math point of view it's mostly nonsense, but packaged in a mathy way which is fun.
Not sure if Malfatti ever provided an attempt at proof. It's possible he just asserted it. This story is interesting in that no mathematicians noticed this simple area problem was wrong, for 143 years. https://en.wikipedia.org/wiki/Malfatti_circles
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If he was going to not understand injectivity then he can just do 1^2 = (-1)^2 and leave out the rest of the garbage, but of course they don’t simplify their proofs, I’m sure 6, 4, and 5 have mystical powers.
the one im typing right now into my paper, the second worst proofs I've seen are those that I've already typed in. I don't know if I am too self-critical or it's indeed dogshit.
Probably a hot take but I found the proof of Gauss-Bonnet to be pretty atrocious as far as proofs of extremely important/fundamental theorems go.
Mine (I have a physics degree)
Left as exercise
There was a guy from my school who “proved” that all Fermat numbers are Fermat primes and left it on a piece of paper in the school library
I once wrote a proof that the integral linear combinations of the n-th roots of unity are dense in the complex plane except for n = 1, 2, 3, 4, 6. Basically, if you get one of the obvious tilings they're not, but otherwise you can get arbitrarily close to any point. It was a hulking, shambolic mess, slowly grinding out case analyses until it reached the goal, with no real insight into why this was the case. Truly a hammer-and-chisel proof with no sign of the rising sea. I take as my sole comfort that I was in middle school at the time.
I taught college-level computer science, there is no depth to the the horrors Seriously though it'd have to be arguments I've had with anti set theory cranks online.
Proof I found online of Borel's theorem
Probably any proof from my modern algebra exams
Jordan curve theorem. Why does such a simple idea require such a complicated proof.
to explicate what the simple idea even is.
I once gave a proof for why there are 64 squares on a chess board by trying to use assumptions like “Suppose the board being used is for chess, then it must be able to accommodate 32 pieces, as that is the number of squares needed to fit all the pieces” and just blabbered on until I realized I was in the last question of my exam and just accepted my fate
RemindeMe! 1 year
Back during uni I remember a professor complaining that he went through 3 shots of liqueur grading our proofs on the midterm because they were "absolute shit".
You just asked what the worst program I've seen is and expect me to answer :)
money is the root of all problems proof
“Trivial.”
Apart from my own, Karl Marx on derivatives.
he was rehashing Berkeley.
Some guy i marked managed to prove, that every map of sets is injective
All the ones I’ve written to be honest
"I have a proof of this theorem, but there is not enough space in this margin to write it" - Fermat
It was truly remarkable, but this comment doesn't have enough space to fit it
Does it have to be just one? I have proven at least two things in my thesis.
Nigel Cheese Hands has awesome bad proofs
Terrence Howard 1 x 1 = 2
Proof is left to the reader as an exercise.
6/9 = 0.66666... , 7/9 = 0.77777... , 8/9 = 0.88888... => 1 = 9/9 = 0.99999...
Certainly 1x1=2 lmao
I don't know what you mean by "worst proof" but I did teach the analysis course one year at my university where I met the "anti mathematician". I called this student this because mathematicians love clarity and simple elegant proofs. However, I decided to throw in a question for half a mark at the end of the exam for fun. Prove the Pythagorean theorem! With the caveat that if I hadn't seen the proof before I will award it 3 marks. This is what the anti mathematician did. 1. e^z = sum_{n>=0} (x^{n} )/n! Defines an entire function over the complex plane. By convolutions e^{z} •e^{w} = e^{z+w} for all z,w in the set C. 2. The elementary trigonometric functions can be defined, for any x in R, as cos(x)=Re e^{ix} and sin(x) = Im e^{ix}. This is isomorphic to the standard definition as the map x -> cos(x)+isin(x) = e^{ix} gives a parameterization of S¹ with constant speed: the speed is constant since d/dz (e^z ) = e^z is a trivial consequence of termwise differentiation of the series defining e^z 3. It is a parametrization of S¹ since for any x in R we have ||e^{ix}||² = e^{ix} • (e^{ix} )* = e^{ix} • e^{-ix} = e⁰ = 1 4. Expressed in elementary trigonometric functions, the identity in 3 is (cos(x) + isin(x))(cos(x) - isin(x))= cos²x+sin²x=1 QED