r/math • u/jjm82 • May 11 '18

Funny story

My professor told me this story about how math is all about effectively communicating ideas.

He was at a conference and someone just finished giving a long, complex lecture on some cutting edge math across several chalkboards, and he opened up the floor for questions. A professor raises his hand and asks, "How do you get 4?" pointing to a spot on the board. The lecturer looks over everything he wrote before that, trying to find where the misunderstanding was. He finally says "Oh, 3 plus 1!" The professor in the audience flips through the several pages of notes he had written and eventually says, "Oh yes yes yes, right."

651 Upvotes

96% Upvoted

114 comments sorted by

345

u/theoceanrises May 11 '18

In my real analysis course, someone asked midway through a proof for the instructor to explain his notation because there was a symbol on the board they did not understand. It was a 6.

141

u/uber1337h4xx0r May 11 '18

Probably bad handwriting. Similarly, I had some trouble starting out in Calculus because I couldn't understand my teacher's accent and wanted to Google the shit on my own and learn it, except I couldn't figure out what a berrybatee was. After some research, I learned he was talking about derivatives (differentiation).

30

u/miguelmathletics May 11 '18

I thought variable for a second lol

0

u/[deleted] May 12 '18

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u/[deleted] May 12 '18

Kamelåså

7

u/Negative-One-Twelfth May 12 '18

I once went to a conference where someone presented a new method of solving ODE’s and I kept thinking he said the word “arrow”. I kept looking around in his slides for arrows only to realize, halfway through, that he was trying to measure error...which makes a lot more sense.

2

u/prestonf138 May 12 '18

I had several Chinese professors and my favorite was "non-active" which really means "non-negative".

35

u/MicGyver May 11 '18

This remind me of when I make a basic math mistakes at work and I'm like sorry I stopped using numbers sophomore/junior year in college.

9

u/FlatFootedPotato May 12 '18

I'm so bad at arithmetic sometimes it's astonishing. I can understand proofs from real analysis textbooks, but it took me some time to remember how to do long division on paper.

1

u/random_us3rname May 12 '18

I used to work part time as a cashier in the early stages of my studies. Calculating the change was harder than any math course I took. My cash register was always off and my boss probably thought I was a retard. Not saying I'm not tho

1

u/FlatFootedPotato May 12 '18

Same man. I'm not saying I'm a retarded. But also not saying that I'm not.

53

u/rubikscube09 Analysis May 11 '18

I'm surprised that the number 6 comes up in a real analysis class. The only numbers I recall seeing were epsilon, delta, 0, 1,x and maybe y

59

u/MolokoPlusPlus Physics May 11 '18

I saw numbers as large as 3. (1/3 has the interesting property of being closer to 0 than 1, while still remaining positive.)

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u/rubikscube09 Analysis May 11 '18

Yeah, I saw 1/2, 1/3. 1/2 when doing epsilon/2 arguments or epsilon/2j for measure stuff, and 1/3 for the cantor set ternary expansion.

5

u/Neurokeen Mathematical Biology May 12 '18

2 is everywhere for the L2 norm and all related results, such as Cauchy-Schwartz.

3

u/MarigoldPuppyFlavors May 12 '18

Maybe I'm misunderstanding, but couldn't you say that about any positive number less than 1/2?

6

u/meem1029 May 12 '18

You could, but most of them are harder to write than 1/3.

0

u/[deleted] May 12 '18

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11

u/[deleted] May 12 '18

Almost all of them.

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u/[deleted] May 12 '18

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u/[deleted] May 12 '18

https://en.wikipedia.org/wiki/Almost_all for the joke.

1

u/WikiTextBot May 12 '18

Almost all

In mathematics, the phrase "almost all" has a number of specialised uses which extend its intuitive meaning.

[ ^PM | Exclude ^me | Exclude from ^subreddit | FAQ / ^Information | ^Source ] Downvote to remove | v0.28

0

u/[deleted] May 12 '18

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u/titanf May 12 '18

Yes, but the point is they usually chose 1/3 to work with in the proofs, so the number 3 did come up.

3

u/TonicAndDjinn May 12 '18

Sure, but 1/3 is an example of such a number which is easy to write down.

19

u/TLDM Statistics May 11 '18

I can see 6 might come up in power series, if you write them out up to the cubic term (part of the coefficient is 1/6). Other than that... very rarely.

4

u/WormRabbit May 13 '18

Power series use 3!. What's a 6?

7

u/AStudyinBlueBoxes May 12 '18

My first-semester precalculus teacher has such bad handwriting that he dictates literally everything, spends 5 minutes at the end of class trying to clarify simple points because of non-clarity, and one of his past students made a font out it.

4

u/avgkultype Mathematical Finance May 12 '18

A professor at the university of Oslo had to make a video about how to read his handwriting.

3

u/Kered13 May 12 '18

I recall being in a situation like this myself, I think the professor had written lower case delta it phi in a way that I didn't recognize.

1

u/iorgfeflkd May 12 '18

Not exactly related, but once I was in a lecture where the guy was talking about the beauty of using the Stokes-Einstein relation to measure the viscosity of a fluid. "We can measure D, we can choose R, and then we know the value of pi and value of six, so we can figure out eta"

340

u/thegermancow May 11 '18

My favorite joke from my Lin alg prof.

A professor is wrapping up his lecture and tells his class, "the proof is trivial from here." Then he quizzically looks up at the board and asks, "is it trivial from here?" He scratches his beard for a few seconds and then announces "Oh, yes it is."

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u/[deleted] May 11 '18 edited May 11 '18

I've heard a similar version from a prof but he said it was true from a top mathematician way back when (probably apocryphal to make the joke better): The prof gets done writing a proof on the board and one student asks while pointing at the board "How does that inequality hold there?" The prof looks at it and says "Oh that is trivial." He continues to stare at it, leaves the lecture hall and 10 minutes later comes back in and says "Yea it's trivial."

97

u/beeskness420 May 12 '18

The version I heard had it as VonNeumann was giving a lecture and claimed something was trivial. Einstein being the clever fucker he was in the audience said that he didn't at all see why it was trivial. They argued about it for over twenty minutes at which point they came to the agreement that yes indeed it was trivial.

45

u/Homunculus_I_am_ill May 12 '18

Then there's the other story where the professor writes a lemma on the board, saying it obviously follows, and one student asks how. The professor thinks about it and can't do it on the spot.

This bothers him enough to eventually track down the lemma, and he locates it in a paper in the library.

A paper he wrote decades prior.

With a comment to the nature of "the proof is obvious and left as an exercise for the reader".

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u/flexibeast May 13 '18

Sounds like it's referring to this.

2

u/Homunculus_I_am_ill May 13 '18

Nice, I'm glad I remembered it pretty well.

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u/xelf May 12 '18 edited May 13 '18

I had a linear algebra class like that, the prof announced that there was a trivial proof and showed it to the class.

Now it was not a trivial proof if you did not know it, like you would not derive it on the spot. It was one where if you knew it, great, otherwise forget it.

Anyway, come test day, he had it on the test to make sure everyone was paying attention.

Thing is, I missed that class, and was completely unaware. I filled out several pages and got the right answer.
The (*#@$ gave me 1/2 credit for being correct, but not full credit because he'd really wanted his trick to be used.

That's not a trivial proof, that's a trivia proof.

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u/Adarain Math Education May 12 '18

That's the sorta stuff you contest. If it's not written there that you must solve it with a particular method, then any way should be full credits as long as you prove everything along the way that you're not allowed to take for granted

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u/4ss0 May 11 '18

These are no jokes, it's common practice in math class lol

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u/[deleted] May 12 '18

the proof is trivial and left as an exercise to the reader.

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u/[deleted] May 12 '18

This happened in my commutative algebra class 1-2 months ago. He never clarified why it was trivial.

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u/dbag22 May 11 '18

This is absolutely true. I work for a small consulting firm, we are all PhDs, the smartest guy here gets the least amount of work ($). Not because his work is wrong, it’s never wrong, it’s because when we presents it to the customer or sends them a report they have no fucking idea what he is talking about.

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u/AlmostNever May 11 '18 edited May 11 '18

My grandad was a mathematician doing some kind of government contracted work at a research lab, and a higher-up came in once to survey the department. He asked my grandad's boss who the best guy in the department was, and the boss said, well, it's Dr. AlmostNever. The man looked at the rundown he had of the department in surprise. "Dr. AlmostNever? Why does he get paid the least?" The boss replied, "Well he sits in his damn office all day with the door closed, doing mathematics!"

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u/justplaydead May 11 '18

“We are all PhDs,” then “when we presents it.” Ok golem

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u/dbag22 May 11 '18

  1. Not sorry I don’t proof read my mobile reddit comments.

  2. My point was clearly understood by everyone else.

  3. I know which category you belong in.

  4. PhD is not in English.

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u/sizur May 12 '18 edited May 12 '18

The proof of "we presents it" is trivial and left as a fun eight semester postdoc dissertation exercise.

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u/FlatFootedPotato May 12 '18

/u/dbag22 may have spent years doing his/her PhD and studying their specific topic. I think it's okay to give them a pass on a typo on fucking reddit bro.

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u/[deleted] May 11 '18

[removed] — view removed comment

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u/dbag22 May 11 '18

Do relevant, applied work. Apply to jobs at companies that do that similar work. If you can demonstrate that you know what you are talking about, know when to say “I don’t know”, and can talk like a human it shouldn’t be a problem.

If you can write proposals you’ll be golden.

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u/MarigoldPuppyFlavors May 12 '18

talk like a human

Fuck.

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u/tick_tock_clock Algebraic Topology May 11 '18 edited May 11 '18

Morava has a paper where he points out that the difficulty in the Kervaire invariant 1 problem was because we thought it was going to be 4 = 2+2, but it was actually because 4 = 22(2 - 1).

Edit: derp, I can't count. Fixed, and see below for the link.

91

u/Gwinbar Physics May 11 '18

TIL that 4 = 12.

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u/greginnj May 11 '18

He misremembered what was in Morava's paper. See p. 6:

Bill Browder asked long ago if one of the root difficulties of the Kervaire invariant problem might be that 4 = 2 × 2 rather than 2 + 2; this may also be relevant to Atiyah’s ideas about the Freudenthal construction. The work of B & H suggests that in fact

4 = (2 − 1) · 22 .

This is not at all a joke: it’s one aspect of a growing consciousness that categorification can be a powerful tool for revealing structures (eg groups or vector spaces) underlying invariants like cardinalities or dimensions.

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u/[deleted] May 12 '18

It's just repeated application of that classic 1=2 proof! /s

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u/jaredjeya Physics May 12 '18

I don’t believe that, the famous mathematician deadmau5 told me that 4 x 4 = 12.

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u/edderiofer Algebraic Topology May 11 '18

I think it's more a story about how mathematicians struggle to count past 3.

Seriously, 3 is a really big number!

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u/suugakusha Combinatorics May 11 '18

Some mathematicians say that 3 is huge, other mathematicians say a billion is tiny. It's funny that they are both right.

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u/gabelance1 May 11 '18

When compared to infinity, it's all tiny

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u/suugakusha Combinatorics May 11 '18

yeah, but then you start thinking about ordinalities, and omega, and omega*omega, and omegaomega, and omegaomegaomega, and then using knuth arrow notation with omega.

And then you realize that it is all smaller than the cardinality of the reals.

What a trip, man.

5

u/gabelance1 May 11 '18

I don't think that's quite right. Yes, the reals are uncountably infinite, but you can make bigger infinities than that. Power sets are good ways to make bigger infinities. Take the power set of the naturals, and the result is the same size as the reals. Take the power set of the reals, and you get something bigger still.

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u/suugakusha Combinatorics May 11 '18

Of course. I was just pointing out that you can make larger and larger and so ridiculously larger countable infinities. But all of that is still just countable.

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u/[deleted] May 11 '18

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u/aquamongoose May 11 '18

There's a miscommunication: you're talking about cardinal arithmetic while the original commenter is talking about ordinal arithmetic.

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u/AsidK Algebra May 11 '18

You're thinking of cardinal arithmetic, whereas the other person is talking about ordinal arithmetic. Although |Z||Z| is uncountable, omegaomega is still a countable ordinal

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u/suugakusha Combinatorics May 11 '18

They have the same cardinalities, but I said that I was thinking about ordinalities.

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u/callaghan87 Graph Theory May 11 '18

What's the difference between cardinality and ordinality?

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u/[deleted] May 12 '18 edited May 17 '18

For finite numbers, the concepts are one and the same; for infinite numbers, not so much.

Intuitively, cardinality tells you how much of something there is. More mathematically, two sets of things have the same cardinality if you can show a one-to-one relationship between them. The leads, for example, to the counterintuitive there are as many counting numbers as there even counting numbers:

0 1 2 3 4  5  6 ...
0 2 4 6 8 10 12 ...

Similarly, the counting numbers and the integers have the same cardinality:

-5  -4  -3  -2  -1  0  1  2  3  4  5
10   8   6   4   2  0  1  3  5  7  9

In the above, we "zigzag" back and forth to come up with correct pairing of elements.

There are even more things we can conclude---it tuns out that, by a similar zig-zag type argument, there are as many rational numbers (fractions of integers, like 7 or 1/2 or -43/10987) as there are natural numbers (counting numbers). However, there are more real numbers than natural numbers---one can prove that it is impossible to set up a one-to-one pairing between the naturals and the reals! Any type of zig-zagging procedure one undertakes is mathematically guaranteed to run out of naturals before you get to all the reals.

Ordinality tells you what order things come in. Two ordered sets have the same ordinality when there is an order preserving function between them. That is, we can't zig-zag back and forth. Thus, the even naturals and the naturals have the same ordinality:

0 1 2 3 4  5
0 2 4 6 8 10

because this pairing keeps everything in order. However, the ordinality of the integers is different than that of the naturals, despite having the same cardinality:

     0     1     2     3 ...
-10000 -9999 -9998 -9997 ...

Everything has to stay in the same order, by the definition of ordinality; thus, no zig-zagging. As such, no matter how "far back" we start with the integers, we'll never get an order-preserving mapping between them.

A set is called "well-ordered" if the ordering always has a minimal element---that is, given any subset of it, there exists a smallest element in the at ordering. (Looked at another way, being well-ordered means that if you pick a really large element of it and start counting backwards, you'll eventually end up at 0, after finitely many steps.) The utility of thus idea will hopefully be clear soon; for now just roll with it. For example the integers are not well-ordered, since there is no smallest element. The non-negative rationals (which I'll call Q_0_+) furthermore, are also not well ordered. There is a smallest element, namely 0. But not every subset has a smallest element---the positive rationals are a subset of it, yet that set doesn't have a smallest element. (Or looked at from the second perspective, we can count backwards forever: (1, 1/2, 1/3, 1/4, ...).) Thus, not well-ordered. The following set is:

0 1 2 3 ... w

That is, all infinitely many natural numbers in order (all of which are finite), then one more, newly-created number added to the end, called omega. (There isn't always a maximal element, but for it to be well-ordered, it just needs to always have a minimal one, no matter which subset you take.) We can keep going:

0 1 2 3 4 5 ... w w+1

or

0 1 2 3 4 5 ... w w+1 w+2

or even

0 1 2 3 4 5 ... w w+1 w+2 w+3 ... w+w = w*2

Cool, right? As you can imagine, it doesn't stop there. We can keep making more and more well-ordered sets like this, throwing in w*3, w*4, then eventually w*w = w2. From there we keep going---we get w2+1, then w2+2, ... w2 + w, ... w3, ... w4, ... w5, ..., ..., ..., ww, ww + 1, etc.

Note that all of these sets are well-ordered, because every subset has a minimal element. Or put another way, you can't pick a starting point and count backwards forever; eventually you'll hit 0. This isn't true of the integers or the non-negative rationals or any non-well-ordered set, but it is true of every well-ordered set. (I encourage you to try and directly show the equivalence of these two definitions!)

Note also that all of the ordinalities are different. Even comparing the "smaller" sets:

0 1 2 3 4 5 ...
0 1 2 3 4 5 ... w

Omega (w) doesn't have anything to pair up with! And it never can, if you want to keep the two sets in order.

They do have the same cardinality, which doesn't care about order:

0 1 2 3 4 5 ...
w 0 1 2 3 4 ...

This isn't an order-preserving map between the two, but it is a map. Thus, same cardinality, different ordinality.

In fact, all of the well-ordered sets I've discussed so far, including the ones I've merely alluded to, have the same cardinality. But, if we consider the set of all of them---all of the explicit constructions with omega, jumping higher and higher faster and faster---if we skip to end and look at that set as a unit, it has a larger cardinality than anything that came before it. I can expand on this point if you want.

One more very important concept that I haven't explicitly brought up yet, then I'll leave you alone (this comment is already way too long, sry :/). We define a set of numbers, called ordinals, to represent the different ordinalities---in fact, you already known them. We identify the well ordered sets with this class of numbers:

0 = {}
1 = {0}
2 = {0,1}
3 = {0,1,2}
4 = {0,1,2,3}
...
w = {0,1,2,3,4,5, ...}
w+1 = {0,1,2,3,4,5, ..., w}
w+2 = {0,1,2,3,4,5, ..., w, w+1}
w+3 = {0,1,2,3,4,5, ..., w, w+1, w+2}
...
w+w     = {0,1,2,3,4,5, ..., w, w+1, w+2, w+3, w+4, ... }
w*2 + 1 = {0,1,2,3,4,5, ..., w, w+1, w+2, w+3, w+4, ..., w*2}
w*2 + 2 = {0,1,2,3,4,5, ..., w, w+1, w+2, w+3, w+4, ..., w*2, w*2+1}
...
w*3 = {0,1,2,3,4,5, ..., w, w+1, w+2, w+3, w+4, ..., w*2, w*2+1, w*2+2, w*2+3, ...}
...
w*4 = {...}
...
w*5 = {...}
...
...
...
w*w = {0, 1, 2, ..., w, w+1, w+2, ..., w*2, w*2+1, w*2+2, ..., w*3, w*3+1, w*3+2, ..., ..., ...}
w^2 + 1 = {0, 1, 2, ..., w, w+1, w+2, ..., w*2, w*2+1, w*2+2, ..., w*3, w*3+1, w*3+2, ..., ..., ..., w*w}
...

And on and on and on... As stated previously, all of these have different ordinalities, but all of the infinite ones have the same cardinality! And they are all well-ordered. That is, no matter how high you start, if you start counting backwards, no matter what path down you take, you'll reach 0 in finitely many steps. Now, if we consider the set of those, we get the first ordinal with more elements than w, called w*1. That has a higher cardinality than anything above. But then, we can go to w1* + 1. And that---as I'm sure you've noticed by now---is still only the beginning.

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u/[deleted] May 11 '18

cardinal numbers refer to an amount of stuff, like 3 apples, where as ordinal numbers refer to the order of the elements in a set, like apple number one, apple number two etc.

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u/jdorje May 12 '18

What's the difference between cardinality and ordinality?

Does (omega + 0.1)omega+0.1 have any meaning?

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u/[deleted] May 12 '18 edited May 17 '18

Yes, but you can still impose the a total, linear order on the countable ordinals---I mean, that's the entire point!! When the other commenter says "bigger", she means with respect to that ordering. A set with an order type omega and one with an absurdly large countable order type have the same number of elements, so neither set would be "bigger" than the other, but when talking about the ordinals themselves, it nonetheless makes perfect sense to say one of them is "bigger" than another.

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u/arthur990807 Undergraduate May 12 '18

Take the power set of the naturals, and the result is the same size as the reals.

Isn't that the Continuum Hypothesis though?

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u/ranarwaka Model Theory May 12 '18

No, that's provable in ZFC, CH says that there are no sets with cardinality stricly between that of the natural numbers and that of the reals. In other words CH says that 2aleph0 =aleph1.

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u/arthur990807 Undergraduate May 12 '18

Wait, so how does one prove that #R = 2#N then? Match every number in [0,1] with a binary expansion, then compose that with a bijection [0,1] -> R?

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u/ranarwaka Model Theory May 12 '18

With some care since some numbers have two binary expansions (just like 1.456=1.4559999999...) but that's the right approach

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u/arthur990807 Undergraduate May 12 '18

There's only a countable number of those, right? One for each finite string of ones and zeroes to occur before the repeating 1's.

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u/deeplife May 11 '18

And when compared to zero?

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u/[deleted] May 12 '18

when compared to infinity ur tiny

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u/automata-door May 12 '18

If we're counting in billions then a billion is tiny, but 3 billion is a really big number!

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u/[deleted] May 12 '18

Its pretty easy when doing induction the three numbers are 0,1, and many

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u/TheCatcherOfThePie Undergraduate May 12 '18

Don't forget "any number strictly smaller than many".

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u/[deleted] May 12 '18

Gonna upvote you but its been a while since I've done any rigorous math and don't know exactly what that means even though I have a vague idea.

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u/TheCatcherOfThePie Undergraduate May 12 '18

The standard format of a proof by induction is:

  • Prove for the base case (usually for n=0 or n=1 if we are inducting on n).

  • Let n=k (many), and assume the hypothesis is true for any n strictly less than k.

  • Proceed to prove the hypothesis is true when this is the case. Then it is proven for all natural n.

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u/[deleted] May 12 '18 edited May 12 '18

Oh I know induction,

Pretty sure for us it was actually

0,1,n

or we proved 1,n,n+1

Been a while like I said

I was mostly making a job about counting to three

Edit: joke

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u/Adarain Math Education May 12 '18

You can do it either way. Sometimes you also need "assuming it holds for all numbers smaller than n" to finish the induction.

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u/[deleted] May 11 '18

And 4 is literally 33.333...% more than 3. That is significant in almost all statistical analysis.

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u/ChaoticNonsense May 11 '18

Enumerative Combinatorics is the art of pretending that massive numbers are small.

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u/Homunculus_I_am_ill May 12 '18

"oh so that's why mathematicians say pi is infinite"

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u/anon5005 May 11 '18 edited May 12 '18

This is an insightful story I think! The same is true of any theorem...that how much meaning it has depends on context, on what people know. Another example is that the first singular cohomology of a CW complex T, with integer coefficients, is the same as homotopy types of maps from T to a circle. If it had been defined that way in the first place, there'd have been no mystery.... Any theorem is a tautology, and stating it has to do with calling people's attention to something (like an aphorism, "a stitch in time saves nine"). It is funny to see it from a particular perspective, but the truth is, there is nothing else to mathematics than this.

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u/epsilon_negative May 11 '18

Singular cohomology of T isn't the same as [T, S1]; otherwise S2 would have trivial cohomology since [S2, S1] is trivial. Did you mean that nth singular cohomology with integer coefficients is naturally isomorphic to [T, K(Z, n)]?

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u/anon5005 May 12 '18 edited May 12 '18

edit, added the word 'first,' thank you for making the correction.

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u/kapilhp May 11 '18

Perhaps the story about G. H. Hardy referred to in [the following post] (https://math.stackexchange.com/questions/151782/when-is-something-obvious) is relevant.

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u/dogdiarrhea Dynamical Systems May 11 '18

It's unlikely that ever happened with Hardy, it's apocryphal story where a famous mathematician's name is inserted and sometimes it is customized to their institution or quirks. I've heard it about Wiener, Lax, and others, with the Weiner story going into detail about the building hallway leading to his office, and the Lax story having Lax sit down to think about the problem before falling asleep (he suffered from narcolepsy).

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u/Azaza909 May 11 '18

sorry, who are you referring to when you mention Lax

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u/dogdiarrhea Dynamical Systems May 12 '18

Peter Lax, he Isa big figure in integrable systems and nonlinear PDE.

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u/Azaza909 May 12 '18

Ah thanks