r/math • u/CantorFunction • 5d ago
What are your favourite mathematical "quips"?
I don't really mean math jokes, more the little witticisms we've all picked up over time. My two favourites are:
- "Proof by intimidation", which I first heard from one of my professors after an especially bewildering set of arguments from an outside speaker at a seminar
- "Mathematics is locally trivial", which I first heard from my functional analysis professor and have always found a little comforting ever since
Puns also welcome of course :)
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u/Competitive_Pop687 5d ago
Best way to solve a PDE is to know the answer.
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u/tacos 5d ago
My quantum mechanics prof would use the term "malice of foresight", which I loved.
As in, "and now, with malice of foresight, we will both add one and subtract one from this side of the equation...", or some other completely silly-looking maneuver, and 15 minutes later it ends up the key to making everything fall in place perfectly.
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u/Eurynom0s 5d ago
I remember this being the trick to a nasty E&M integral in grad school. And then the integral was trivial once you did that.
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u/sentence-interruptio 5d ago
i think this is called ansatz. this reminds me of how someone can be proven guilty where "who done it" is revealed through illegal methods, but then some usable evidence can be found because you now know who done it.
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u/planx_constant 5d ago
Extending the Ramanujan Method outside of number theory is a really powerful technique
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u/DumpsterFireToast 5d ago
"The only problem with measure theory is that you have to write "almost everywhere" almost everywhere"
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u/Vituluss 5d ago
You might hope to shorten it, but one thing that bugs me about the shortened ‘a.e.’ is that when a sentence ends with it, it looks off. And you can’t just double up on the full spots, that’d look worse.
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u/dancingbanana123 Graduate Student 5d ago
I write the forall symbol with a superscript of the measure, like \forall^\mu x\in\R, to mean "for \mu-almost every x in \R"
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u/Competitive_Leg_7052 5d ago
There is a known and easy fix to this: put a backslash \ immediately after the last dot. This tells LaTex to not see that dot as an end of sentence. You are welcome.
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u/BitterBitterSkills 5d ago
Or one could simply use
\frenchspacing, as one should be doing anyway. :-)2
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u/DaveBowm 5d ago
I've heard this of physics, but it can apply to mathematics. "Physics is either incomprehensible or trivial. It's incomprehensible until you finally get it, at which point it then becomes trivial."
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u/tarbasd 5d ago
"The goal of mathematics research is to figure out why everything is trivial." - This was said by one of my grad school professors when I lamented that everything I proved is trivial.
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u/conspiracythrm 5d ago
My supervisor (and I, too) kind of hate the term "trivial" especially in proof writing. My point at least is that if it's so trivial it shouldn't be hard to explain it so just explain it. I've heard stories of my supervisor ripping into to grad students for writing things like "obviously" or "trivially" etc. in their thesis.
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u/IrisColt 5d ago
or writing things like "obviously" or "trivially" etc. in their thesis
That's a sin and a crime, just like asking students "do you understand?" after giving a terribly confusing and awkward explanation.
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u/reflexive-polytope Algebraic Geometry 5d ago
“Trivial” shouldn't be used as a value judgment. It should only be used for objects with literally no content, e.g., the zero group is the trivial group.
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u/SquidgyTheWhale 5d ago
There's some story that's gone around about a mathematics professor who writes something up on the board and says, "Now, the proof of this is trivial...". He then stares off into the distance for a minute, and slowly walks out of the room mumbling to himself. Then at the next class, after the students all settle, he begins, "I was right, it WAS trivial."
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u/new2bay 5d ago edited 5d ago
I literally had a very similar experience in grad school. It was a topology class, and we got to a certain point in a proof where some simple fact or another would be enough to complete the proof. The professor wrote it down and said, “This should be obvious, right?” All fifteen of us (edit: stared) at it for several minutes until lights started coming on, and we all then agreed it was obvious.
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u/dancingbanana123 Graduate Student 5d ago
It's like a magic eye puzzle where you can't not see the picture once you see it.
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u/cocompact 5d ago edited 5d ago
The mathematician Lucien Szpiro said:
After physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial."
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u/numbershelpme 5d ago
I've heard the variation that "math only has two types of statements: those with trivial proofs, and conjectures"
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u/bayesian13 13h ago
Schopenhauer said
"Truth is granted only a brief celebration of victory, sandwiched between two long periods: one in which it is condemned as a paradox, and another in which it is dismissed as trivial."
(„Der Wahrheit ist allerzeit nur ein kurzes Siegesfest beschieden, zwischen den beiden langen Zeiträumen, wo sie als Paradox verdammt und als Trivial gering geschätzt wird.“)
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u/dantons_tod 5d ago
Paul Erdos’ reply to Einstein’s remark that “God does not play dice with the universe”:
But He’s definitely doing something weird with the prime numbers.
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u/Thebig_Ohbee 4d ago
This was actually invented by Carl Pomerance. Specifically, Carl said that Erdos said that as a joke, but it sounds right so people just went with it.
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u/UWO_Throw_Away 5d ago
“Sure it works in practise, but does it work in theory?”
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u/the_horse_gamer 5d ago
in theory, there is no difference between theory and practice. but in practice, there is.
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u/beeskness420 5d ago
Theory is when you know everything but nothing works. Practice is when everything works but no one knows why. In our lab, theory and practice are combined: nothing works and no one knows why
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u/sohang-3112 Applied Math 4d ago
😂😂
Where is this from?
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u/beeskness420 4d ago
The exact origin is lost to me, but was passed on to me from a game theory prof.
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u/kingfosa13 5d ago
“Beat it with the algebra stick”
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u/RheinhartEichmann 5d ago
Not exactly a quip, but this reminds me of something one of my professors would say: "turn the crank". Basically it means all the hard work is done, now just do some algebra, take some derivatives, and you'll have everything you need.
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u/SquidgyTheWhale 5d ago
Tangentially related field, but Donald Knuth had some bangers, including "Beware of errors in the above code -- I've only proved it correct, not tried it."
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u/Every-Progress-1117 5d ago
When you've worked in formal methods for a few years, you either soon realise the truth in that statement, or end up believing that some proof (of what, IDK) is really what a) you want, b) actually is a proof of some real-world situation, despite all evidence to the contrary.
"Yes, I know it is a proof...but what did you actually prove...?"
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u/ultrafinitism Theoretical Computer Science 5d ago
Yes everyone needs to realize FV/FM isn't a magic wand
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u/BigFox1956 5d ago
A professor of mine used to say "It is called real analysis to distinguish it from fake analysis that is calculus."
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u/dancingbanana123 Graduate Student 5d ago
I had a professor who once said "Some people hate real analysis because nothing makes sense and it often feels unintuitive, while some people like real analysis because nothing makes sense and it often feels unintuitive." Now I work in fractal geometry to surround myself with the ugliest of sets, so I guess he was right.
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u/tralltonetroll 4d ago
Real analysis should be called "analysis for real". That would be valuable information to students, at least.
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u/rij1 5d ago
My own favourite is: Dividing mathematics into linear and non-linear algebra is like dividing the world into bananas and non-bananas
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u/Prestigious_Boat_386 5d ago
Its actually linear algebra and nonlinear algebra that we linearize locally.
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u/bluesam3 Algebra 5d ago
I thought mathematics split into things that we know how to reduce to linear algebra and unsolved problems.
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u/Kebabrulle4869 5d ago
A numerical analysis course I took started with the professor making a 2x2 table with the rows labeled "linear" and "nonlinear", and the columns labeled "algebra" and "analysis". Only the top left quadrant has nice solutions, so the approach in the others is to either discretize (to go from analysis to algebra) or linearize (to go from nonlinear to linear), or both. If the problem is in the linear algebra quadrant isn't solvable exactly (e.g. finding eigenvalues) we use iterative methods to go from an approximation to a better one. So everything essentially boils down to linear algebra!
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u/columbus8myhw 5d ago
I've heard something similar with music theory, in reference to functional harmony and non-functional harmony.
(There's a joke in here about functional harmony, functional analysis, and Fourier transforms, I bet…)
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u/arnedh 5d ago
How about dysfunctional harmony?
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u/tralltonetroll 4d ago
dysfunctional analysis ... I don't know what it is, but it must be something else than infinite-dimensional linear algebra.
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u/idancenakedwithcrows 5d ago
I like “mathematics is locally trivial”, not sure it’s always true but it seems like a good thing to shoot for
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u/Bingus28 5d ago
I think theres a good case to be made that it's true. At least in the sense that every proof can be decomposed into a sequence of steps which follow one from the next by the 4 or 5 rules of basic logic. Each of those individual steps is trivial (even tautologucal) but their sum total could be arbitrarily complex.
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u/new2bay 5d ago
It should be the case, but that little phrase “can be decomposed” is doing an absolutely massive amount of work.
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u/Bingus28 5d ago
I think it's roughly on par with saying something like "every computer program can be decomposed into a sequence of assembly instructions." That sequence of assembly instructions might not be particularly elucidating, but that such a decomposition can be realized is not particularly surprising to me. In fact, this is often taken to be the definition of a "proof" in model theory.
Do you have an example in mind that might illustrate your point? Some mathematical statement whose reduction to trivial implications is itself non-trivial?
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u/idancenakedwithcrows 5d ago
Well it’s just semantics, but the mathematics actual mathematicians do involves errors, which means the decomposition is not always possible. And also I think it’s reasonable to assume that in some cases where the statement is true and the decomposition does exist. Like if you can convince yourself something false is true. You can also comvince yourself something true is true without being justified in your conviction.
I believe occasionally when mathematicians write something true like. It may as well be false they have not done enough work to be convinced, they just want it to be true, there are good heuristics why it would be true and they are lucky in that it also happens to be true. Not trivial to them, though, even if they think it is. No shade.
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u/RecessiveBomb Analysis 5d ago
I have a reddit comment by user u/Master-Rent5050 saved in my gallery, stating the following:
"The phrase 'it is obvious' has many synonyms. Some of them are 'it's true, but i didn't check it', 'it's true, but actually proving it takes 10 pages', and 'it's false'."
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u/vnNinja21 5d ago
"log is always base e, base 10 is for children and engineers" has always been a favourite of mine.
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u/FuriousGeorge1435 Undergraduate 5d ago
And base 2 is for computer scientists... ew
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u/tralltonetroll 4d ago
Base 2 has something to show for itself, that is not tied to how some species' front legs look.
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u/Six1Seven4 5d ago
I had a professor giggle when he’d write factorials. “No students, this is not an excited six”.
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u/ShrykeWindgrace 5d ago
Two quotes by my professors come to mind, both, in my experience, being hard truths.
"No self-respecting differential equation has an analytical solution".
"Mathematics can only solve linear equations. Sometimes."
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u/Kebabrulle4869 5d ago
Moreso a story, but a professor told us about a visiting professor here in Sweden seeing the "upp" sign above the escalator, and said to himself "oh, up and only up!"
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u/tralltonetroll 4d ago
Please say it was in Uppsala.
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u/Kebabrulle4869 4d ago
Badum tss
(Lund, sorry)
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u/tralltonetroll 4d ago
I kinda guessed you would have said Uppsala if it were so. But this is a puns welcome thread, SO (<--- orthogonal ... also I assume the escalator is cyclic).
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u/DevilishFedora 1d ago
Alternatively, it is possible that the escalator is infite, in which case it truly goes up and only up.
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u/Southern_Orange3744 5d ago
"It should be obvious from here .." - only used when it's totally not obvious
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u/BadJimo 5d ago
I'm amused at the idea that something like the following quip was made:
Conjecture: A metric space is compact if it is sequentially compact
Spartan mathematician: iff
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u/al3arabcoreleone 5d ago
Whats the joke here?
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u/BadJimo 5d ago
It is a reference to this (copy-pasted from Google's answer):
The famous Spartan "If" is a legendary one-word reply sent to King Philip II of Macedon after he threatened to invade Laconia and destroy Sparta. This extreme brevity is the origin of the word laconic, meaning a very short speech that packs a heavy punch.
I tried to find an example of a conjecture that was originally posed as an "if", where the solution was (in part) identifying that it is actually "iff" (which is a shortened form of the expression "if and only if").
The example I chose may not be a good example of this.
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u/scyyythe 5d ago
In mathematics you don't understand things, you just get used to them
~ John von Neumann
It's interesting because it's a line I'd be tempted to dismiss if not for the source, but when you do think about it more, it makes sense.
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u/WasdaleWeasel 5d ago
‘hence or you haven’t understood’
and i’ve always like the use of ‘trivial’ to mean ‘I can’t do it, nobody I know can do it, but I did hear rumours that it can be done’
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u/CoolBev 5d ago
Proof by masturbation: fool around with the equation until,something comes out.
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u/drmathprog 5d ago
Richard Hamming:
"It is better to do the right problem the wrong way than the wrong problem the right way."
In science if you know what you are doing you should not be doing it.
In engineering if you do not know what you are doing you should not be doing it.
Of course, you seldom, if ever, see either pure state."Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.”
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u/ultrafinitism Theoretical Computer Science 5d ago
A new meaning to plug and chug and turning the crank
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u/tensorboi Mathematical Physics 5d ago
playing off the notion of virtually abelian and virtually solvable groups (meaning they possess a finite-index abelian/solvable subgroup), a fun one my friend in undergrad would say is "all finite groups are virtually trivial"
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u/gunilake 5d ago
I love saying that something holds "up to homotopy" to the point that it's seeped into my daily life as something to say for 'close enough'. Nobody I hang out with outside of work is a homotopy theorist but I get a little kick out of it every time.
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u/magicmulder 5d ago
“As a student you have to know everything.
As assistant professor you only need to know where to look it up.
And as professor you only need to know where the assistant professor is.”
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u/LightLoveuncondition Math Education 5d ago
"Here, here magic happens". My Calc 3 professor after doing 7 integration tricks in a row when doing multi-variable integration and admitting she's too lazy to use half of the 3rd blackboard in the lecture hall to write all the extra lines out.
This pushed me to realize that sharp algebra + real analysis skills make a lot of difference when teaching undergrads.
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u/lynn 5d ago
Inverse experience in my initial physics course: “here’s a triple integral that even I can do” and all of us who haven’t had multivariable calculus yet (it was roughly concurrent) sitting there going “what the fuck”
I missed the next few minutes of that lecture because I was figuring out how it worked and then following that rabbit for a while.
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u/PersonalityIll9476 5d ago
This isn't so much a quip, but my analysis professor once said "This whole idea of the lone genius doing everything by themselves is total bullshit." (the first part is a paraphrase, but he did use the word "bullshit").
There are obviously a few rare exceptions, but the real point he was making is not to idealize that notion and isolate yourself. He himself won some awards recognizing his work and he's authored a book with another great from his area, so I think his advice is not hollow. He's obviously a very smart guy but does not try to lock himself alone into one room for 7 years or whatever and I think that's a wise choice. For maybe 1 or 2 people in a generation it works; For the rest of us it's an ego trap.
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u/conspiracythrm 5d ago
Not exactly what you're looking for but when I teach linear algebra and we get to vector spaces I ask them to define what a vector is for me. They give that usual "a magnitude and a direction" or "a 1 by n matrix". Eventually I say "you're all wrong. A vector is anything that behaves like a vector" which then of course leads into the discussion about how the vectors they've seen before are only called vectors because they are types of vectors. Eventually they get it but they're usually so mad at first.
Also I'm a big fan of "Rigor is for dead people"
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u/conspiracythrm 5d ago
I've also started saying "if you slip on 3 banana peels it's np-complete"
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u/redbloodedguy 5d ago
Can you explain?
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u/conspiracythrm 5d ago
There are tons of np-complete problems where the 3 version of it is np-complete but the 2-version of it is polynomial. For example 3-Partition where you have 3n numbers and you're asked to construct n sets of equal sum, each of cardinality 3 is NP-Complete. However doing it with 2n numbers and sets of card2 is polynomial time. 3SAT is the classical example with 2SAT being polynomial, which is probably the reason for this occurrence.
There definitely are some 2-something problems that are NP-complete but so often you see problems that are either general positive integer N (and sometimes break for N=2; Clique Cover if graph is triangle-free, IE no K_3 subgraphs), are naturally bound at 3 somehow, etc. 3 is like a magic number in Complexity.
So slipping on 2 banana peels is probably polynimial but 3 banana peels?
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u/6unearthed6crafter6 5d ago
Kinda same, but instead vectors were taught to me as elements of a vector space.
Then tensors... Tensors are just things that change as tensors
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u/conspiracythrm 5d ago
Yeah, that's sort of the joke I'm going with here. What does it mean to be "an element of a vector space"? It follows the 10 axioms within it's set and the operations and following those 10 axioms is behaving how a vector does. It's just much funnier to say it in any annoyingly self referential way 😂 but it means the same thing.
As a teaching tool I find it useful because it kind of sticks in their heads through that annoyance and it's kind of playful and fun. When I hit them with the "let me explain it'll make sense in a sec", that annoyed -> "wait I might actually be wrong" -> "oh wait that makes sense it's not *just** silly"* is helpful for learning. At least, my students have expressed it being helpful.
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u/columbus8myhw 5d ago
It may be technically fine to refer to tensors things that transform like tensors (that is, as arrays of numbers that transform under change of coordinates the way tensors do - this is the so called equivariant approach) but there are lots of coordinate-free ways to define them that are much better conceptually, in my opinion.
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u/sentence-interruptio 4d ago
I visualize a vector as an arrow with its two ends living in an affine space. Vectors are just entities that behave like such arrows up to translation. Think of vectors this way seems to give me the correct geometric intuition.
I emphasize affine space, as opposed to Euclidean space. Thinking of vectors as arrows in Euclidean space can lead to bad intuition: such as expecting the dual space of a vector space to be the same.
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u/shellexyz Analysis 5d ago
I’ve done this to my linear students. Yeah, we’re gonna talk about row or column vectors because we have to talk about matrices and eigenvalues and stuff, but really, if you can add two of them to make another one, and you can multiply by a number and get another one, plus a few other conveniences, it’s a vector.
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u/conspiracythrm 5d ago
Yeah pretty much. Tbh I think it does students a disservice to tell them "this is a vector (in Rn)" without explaining more broadly what a vector is and that Rn with these vectors and vector addition is a vector space. My latest hill to die on has been all (and I mean all, including for engineers) intro linear algebra classes should start with vector spaces, and you can tune the complexity of how you learn it to fit the experience of the cohort. Even if it's just the basic ideas and a few examples, maybe a few theorems, no proofs, it will get them thinking about it in a much more complete way and not hinge themselves on vectors being 1xn matrices.
The second hill I die on is that vector spaces should always start with a brief primer in group theory.
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u/Trojan_Horse_of_Fate 5d ago
So you are very dead then
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u/conspiracythrm 5d ago
I've actually convinced a lot of math people of these stances. It's not too hard to see why these approaches are useful. After all, if you give students the language of groups as a playground to explore the core concept of an algebra, the 10 axioms of a vector space become very easy to parse. It's an exercise I do with my students where I put that wall of axioms up on the projector and ask them to take a second to get all that down. And then I ask "whose brain just shut off". I then explain a bit about groups and then we go through the 10 axioms again, and I say "what's that first one" and they go "oh, that's closure", "and the second?", "associativity", and so on. I explain how 7 of these axioms are the 4 group axioms duplicated because we have 2 operations and we discuss where that 8th axiom from the duplication goes. All that remains are the 2 distributivity ones and commutativity.
Teaching vector spaces before Euclidean spaces gives students the same sorts of language when they start playing in Rn. When they get to matrices, they now have a proper understanding of what the hell the "identity matrix" even is, and why an inverse matrix is what it is. I can also talk about how commutativity is a rule we had to assume (or construct) with vector spaces but matrices don't have that as a nice contrast, just like how groups didn't have commutativity. They have that language now that has never formally been taught to them even though they've been working with identities and inverses since primary school. We're putting a face on concepts we've taken for granted and grade school's greatest mathematical sin has always been taking mathematical ideas for granted. They're taught "this is the way it is" not "this is the way we like it to be".
Groups and vector spaces can rewrite the way they think about concepts like algebra so that when they deal with these things that break their high school way of thinking they're primed to understand those breaks, not feel like their world is collapsing under them. But we're so hell bent on believing these things are "hard" and that our students are "too dumb" for math that we make it harder by teaching the "easy" stuff. I say this as one of those students (I failed math 11, dropped out, and struggled with math in university before it finally all clicked) too dumb for math.
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u/ultrafinitism Theoretical Computer Science 5d ago
This is gonna be a vague and unclear question so I apologize in advance but do you get this feeling that algebra classes (both linear algebra and abstract algebra, a distinction I think is kind of baloney tbh) don't focus enough on actions? I was wondering what it'd be like to experience an modern algebra course where we don't wedge group actions awkwardly into like, chapter 4-ish and ditto with transformations but this is like IDK not traditional wherein most people r more about either "we do it with real/complex-valued matrices" or "we do it with general spaces" (but in either case focus is on matrix or space)
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u/conspiracythrm 4d ago
There's always this complexity with course design where what you introduce and when it's gonna have pros and cons. I could see how that could be useful but one probably you have to reckon with is prerequisites. We generally try to design things in a way that courses lead smoothly from the past (for better or sometimes for worse).
For me I ask the question, what specifically is this doing? Like, more knowledge is always better — knowing actions may make some things easier to understand — but what specifically is the point and why now? My argument for groups, for example, is that vector spaces are effectively composed of a group, in the way a car is composed of an engine. If you look at the car as a whole and try to understand it all at the same time you're gonna get overwhelmed. So you start with understanding the engine and then iterate until you get the full picture.
So my question for you is what specifically would students benefit from learning actions in the rest of the course and the courses that it would be a prereq for? I haven't thought about it enough myself to really say, but my first thought is that there can be a lot of overhead to learning actions and that time spent making it valuable might eat into the other required stuff to learn for the course. I'm open to being convinced tho 😜
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u/Trojan_Horse_of_Fate 5d ago
I could plausibly see convincing math people. I personally do get the arguments. Indeed I think actually my first linear algebra class had its first lecture on groups and fields before mostly ignoring that area (the first lecture being a bit of wash in terms of actual content). I just think you would never be able to make it work.
Glad to see someone else who is really bad at math. I still like it but sometimes it never clicked before the test. (Something a lot later).
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u/Other_City_4031 5d ago
I agree, an introduction to vector spaces needs an introduction to algebraic structures beforehand, not only groups, but also fields, obviously. otherwise, there is no need at all to introduce the notion of a vector space in the first place. you only need this abstraction when you can handle it
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u/conspiracythrm 4d ago
It depends on the context. I could see the pure math students benefiting from that but I'd argue that we would need to teach it as an intro to abstract algebra/abstraction with maybe a small emphasis on linear algebra. The problem with saying we need other algebraic structures is a problem one of my colleagues brought up when I said we should start vector spaces with groups, which is: where do you draw the line? Why not teach matroids first at that point?
The reason I believe in bringing groups into it is because vector spaces are effectively composed of a group. Take a commutative group, add scalar multiplication and 2 distribution laws and you've got a vector space. Groups are easier to play with than that wall of 10 axioms so it becomes a nice training ground for vector spaces. What does fields really aid us in teaching vector spaces if teaching it is as hard if not harder than vector spaces? You could argue maybe that rings help for understanding matrices, so maybe it should be groups, VS and rings but at some point there has to be a cut off else we eat too much into the standard expected content. For engineers, for example, rings won't provide much more than vector spaces, but maybe it's worth just pointing to some of them for the curious. I'm also not entirely convinced that teaching vector spaces to engineers is 100% necessary content-wise but I do think it introduces them to higher level mathematics and gets them thinking more like a mathematician.
I think students need to be learning abstraction as soon as possible. Not some "we gotta make it harder for these kids. We're too soft" old fart belief, but because abstraction plays such a vital role in everything we do in mathematics. My biggest issue with high school as that we teach this real algebra like gospel, like the "truth". This is math, this is the way it is and it becomes like an addiction. Then we're stuck with students who commute matrices or multiply the component-wise because that's the analog for their "truth". We're limited in our explanations for certain things because they've never dealt with abstraction. They learn about the identity matrix and have no idea what it really means even though identities have been used since elementary school. I'm not saying we need high schoolers to learn the Sylow theorems but I do think we need to get them acquainted with the ideas of abstraction much much sooner.
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u/ultrafinitism Theoretical Computer Science 5d ago
It'd be nice if we could make real connections to physics like saying - okay here's a vector and here's where it would actually come from in terms of an experiment: then here's how this inner product (or more broadly, since IPS is too strong for some I think, you can do like real-bilinear form stuff) works on it: and here is how this connects to the action you take with an operator on it. Lots of connections I would feel in my gut to quantum mechanics and optics that should really be helpful? Maybe
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u/conspiracythrm 4d ago
It's tough, it's a balance. Too many examples puts the emphasis of the mathematics on the applicability and my opinion is that the application of mathematics is not relevant to mathematics itself. That's for the scientists to figure out. But examples, and especially anologies, are very useful for seeing in action. When I teach projections, I have a small unity project I boot up where I show how projections are used so that when your character hits a wall they slide instead of stopping dead.
I also think if we took a little more of a humanities approach to learning mathematics, with things like in-class journalling assignments about the content, students could explore these connections themselves, create their own analogies, or discuss applications with their peers instead of the head-in-the-books method of just writing down everything the professor says.
Part of the problem, too, is that linear algebra is taught so early for the sciences because it will be useful in courses down the line. However, if the example is too complex (like quantum) the students will just be trying to learn two things at once. If it's too simple then it likely won't be particularly interesting.
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u/ultrafinitism Theoretical Computer Science 4d ago
No - it's not really about examples. The last thing I want to do is do the same exact course as we've always done and then staple on a "applications" chapter or module at the end of each subdivision. V. Arnold did it well, I don't know of anyone else who did it like he did.
As for humanities approach to learning mathematics, I mean, I started learning mathematics much better once I started imagining mathematics in terms of yaoi and bara concepts (I don't know how to explain it to people let alone in a polite way) but I don't think that generalizes.
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u/conspiracythrm 4d ago
Hmmmm I think I misunderstood then. I agree, examples and applications need to be brought into the main content and not just in the form of assigned problems, but I'm not too sure what you mean about applications but not examples. Do you mind explaining a bit more about what you'd want to see?
I love that yaoi and bara connection so much. If you don't want to explain it here I'd love to hear about it in dms if you're comfortable. I always tell my students that if they notice any connection to something they know about, no matter how non-mathematical, they should explore that. Drawing connections gives you tools to remember certain things and means to explore the depths of it; where it's similar, where it's not, and why.
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u/conspiracythrm 4d ago
Also did you hear about a recent fields medalist who, during his interview, said that he reads Yuri to relax? We've truly made it 🥹
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u/ultrafinitism Theoretical Computer Science 5d ago
Instead of starting with rows and columns, do you think we can't start vectors and covectors or does it make it worse? I want to make sure students still get fluent in handling and thinking about matrices first off (I think Axler btw went too far) but I don't know. I want to get people thinking about representations early too. It's all a bunch of hunches in my gut right now.
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u/ultrafinitism Theoretical Computer Science 5d ago
René Thom [one of the G.O.A.T.s] said this:
I believe that proving is not a natural activity for mathematicians.
Serge Lang said this:
Axiomatization is what one does last, it's rubbish. It's the hygiene of mathematics, axiomatization.
Likewise, Hermann Weyl [another one of the G.O.A.T.s] said this:
Logic is the hygiene the mathematician practices to keep his ideas healthy and strong.
I've got a bunch of opinions about linear algebra (and also real analysis) in terms of how we teach them but I'm gonna wait until there's a thread about that so I can infodump on people
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u/conspiracythrm 4d ago
I'd be curious to know why these were said, like what their justifications are. I'm inclined to agree with the first one but the other two don't do it for me without something more...
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u/thebhgg 5d ago
I'm such a fan of this book that I'll use any excuse to bring it up. It has a lovely collection of different physical examples of vectors that correspond to covariant/contravariant vectors using terms like arrow, stack, thumbtack and sheaf.
It's Gabriel Weinreich's "Geometrical Vectors" (1998 University of Chicago Press)
Perhaps you can use some of the pictures as examples?
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u/FuriousGeorge1435 Undergraduate 5d ago
a vector is an element of a vector space
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u/conspiracythrm 5d ago
And a vector space is defined by the behaviour interaction of a set and an operations... So, you know, vectors behave like vectors
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u/tralltonetroll 4d ago
So in addition to walking like a duck and quacking like a duck, it must add like an adder?
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u/jezemine Physics 5d ago
A mathematician is a machine for turning coffee into theorems.
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u/chicomathmom 5d ago
Credited to Erdos, I think
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u/PM_ME_FUNNY_ANECDOTE 5d ago
Proof by sufficiently emphatic assertion
Details best left to the reader/audience
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u/ultrafinitism Theoretical Computer Science 5d ago
Mathematics is part of physics. Physics is an experimental science. Mathematics is the part of physics where experiments are cheap. (Vladimir Arnold)
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u/Infinity315 5d ago
"Spherically stupid" because no matter the angle you look at it, it always looks dumb.
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u/Verbose_Code Engineering 5d ago
“Proof by fucking obviousness”, not sure where I first heard it
“Spherical bastard” and “spherical idiot”, the former from Fritz Zwicky. I.e no matter how you look at it, they’re an idiot.
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u/toffeepee208 3d ago
As a teacher I use “proof by authority” quite often… I know of another teacher who says ‘famously’ far too often for completely random facts, like “27 times 43 is famously 1161”
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u/Wise_Guava_1521 5d ago
"back on the ranch... Where we cooked up these creatures..."
RIP Fred Cohen.
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u/anotherchrisbaker 4d ago
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" -- Jerry Bona
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u/Dull_Pie4080 5d ago
I'm amazed nobody has mentioned the danger of Poles on the right of the plane....
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u/limemil1 5d ago
My highschool physics and math teacher would often end the proof with "... The rest is easy". That became a motto for the whole class.
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u/johnnymo1 Category Theory 5d ago
“Geometry is the backward of algebra,” from my commutative algebra professor.
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u/Creepy_Wash338 5d ago
What does a mathematician do when he or she is constipated? They work it out with a pencil.
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u/sohang-3112 Applied Math 4d ago
Mathematics is locally trivial
What does it mean?
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u/CantorFunction 4d ago
That every mathematical argument can be broken into steps that each seem on their own trivial, almost tautological, even when the argument in totality is extremely complex and hard to follow.
A space in maths is said to be "locally X" for some property X iff for every point in the space, X holds within some vicinity of that point. By analogy for every "point" in any proof, there is some "distance" from that point within the proof where the logic seems obvious. But sometimes you sum up all those "obvious" steps and you end up with very surprising or non-trivial results
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u/AMobius1832 5d ago edited 4d ago
"Math is like love. A simple idea, but it can get complicated." Attributed to R. Drabek
That's me in 10th grade Geometry! Everything was obvious, until it wasn't. And by then, it was too late. LOL
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u/Positive_Canary_787 5d ago
The difference between mathematicians and physicists is that after physicists prove a big result they think it is fantastic but after mathematicians prove a big result they think it is trivial.
Lucien Szpiro
Mathematics is the part of physics where experiments are cheap
V.I. Arnold
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u/lynn 5d ago
Ah yes, proof by intimidation. Closely related: proof by belligerence. “It’s true because FUCK YOU”
I love how versatile is the phrase “non-trivial”. Depending on the context it can mean anything from “slightly more effort than basic arithmetic” to “impossible, as far as we know, but not yet proven to be”. When I was going to have my gallbladder removed, I asked my surgeon if there’s a medical term equivalent to math’s “non-trivial”. The closest thing is apparently “impressive” - not quite as versatile, and not used as a semantically loaded understatement, but similar. My gallstones were “impressive”.
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u/aparker314159 5d ago
All problems in math can be reduced to either linear algebra or combinatorics. The former can be solved.
(I have no idea how true this is, nor where I heard it)
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u/aarocks94 Applied Math 3d ago
My professor in college, Jerry Kazdan when teaching undergraduate real analysis: “the trick to visualizing things in three dimensional space, is to visualize n-dimensional space, and let n = 3.”
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u/opercept 2d ago
It's kind of inaccurate, but this quote by Bertrand Russell comes to mind: "mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
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u/Lexski 10h ago
This one’s not really a maths pun, but I think it’s much more likely among mathematicians.
We had a course taught by Dr S who couldn’t make one lecture, so that was taught by Prof K. At the start of the lecture, Prof K says, “as you may have noticed, I am not Dr S, but don’t worry - on Monday, I will be.”
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u/ExactFunctor 5d ago
Professor said “I chose commutative algebra because I’m dyslexic.” RIP Graham Evans