→ 3 IX 2021

To all the people wondering how to do proofs: A good place to start is to read "Book of Proof" by Richard Hammack. Just Google it, it's completely free and available online!

Yes! Also, for people just curious about mathematical proofs, who want to kind of see what they're like I suggest 'Proofs from the book' by Martin Aigner and Günter M. Ziegler. A short summary of this book is some of the most beautiful mathematical proofs from a range of mathematical fields. You may not understand it fully as a layman but it can be an interesting look into proofs.

Here is a free link to the 5th edition of the book

I am so fucking normal right now. *stands perpendicular to the tangent of the plane*

oh and there is the dual thing: sometimes you just know that the professor hates the subject. like when I was taking one of the analysis courses, where the lecture was with one professor and the tutorials were with a different one

at the lectures we were two months into measure theory while at the tutorials haven't even started doing exercises on that topic, but oh it was fine, still plenty of time, he knows what he's doing – we thought, like fools. then the midterm was announced, two weeks left, we still haven't started measure theory. then it was one week left, so the professor tried to solve some lebesgue integrals with us, but he got so bored with each example that he hasn't finished a single one. at this point we just hoped that maybe measure theory just won't be on the midterm, it was too late to do anything. well, unfortunately, the midterm consisted mostly of measure theory problems, it made sense because that was the main content of the course

the professor was clearly very passionate about hating measure theory

One of the really amusing things about college is that if you pay attention you sometimes can discern some of your professor's favorite pet concepts.

For instance, in my Topology course this semester, the Zariski topology has come up at least once in every single homework set so far, and in multiple lectures.

And okay, that's not that weird. The Zariski topology is a really important object in a LOT of fields, especially algebraic geometry. And discussing it at length is a really pedagogically sound move because the Zariski topology is a good example of a topology with a very well motivated structure (the closed sets are the algebraic sets!) that still very naturally gives rise to a lot of strange features, like the way all open sets in the standard topology are Zariski-dense. It was quite effective at startling me out of the complacency of unconsciously basing my intuition of how topologies behave entirely on the standard topology on the reals. So my professor bringing up Zariski so often doesn't necessarily mean he has any special affection for it.

except...

My professor writes many of the homework problems himself. Not all of them - the less interesting ones he lifts from the textbook- but some. Well, every single Zariski topology question I've encountered so far is an original from this guy. I know because the all the questions he writes personally have paragraphs of commentary contextualizing why he thinks the problem is interesting and where the ideas in the problem are going later in the course. And well- let's just say the asides on the Zariski topology have been copious indeed

AND THEN there's the way he talks about the Zariski topology in class! It's with this blend of enthusiasm and fascination only comparable to the way I've seen tumblrites talk about their blorbos. Like hey! Come behold this sgrungy little guy! Isn't he fucked up? Isn't he marvelous? And I look and I can only conclude YEAH that is indeed a spectacular specimen, he's so strange, I want to put him in a terrarium and study him (and then I get to! In my homeworks!)

Anyways. It makes me really happy picking up on how excited my professor is to share this topology with us. I'm kind of baffled that people assume math is a boring field full of boring people when there exist folks like my professor who get this passionate about a topology!

hairy ball theorem, stokes theorem, poincaré duality, nullstellensatz, idk too much to choose one

What is your favourite mathematical theorem? I'm personally torn between the compactness theorem for first-order logic, and the fundamental theorem of Galois theory.

Right. So. A Tarot sequence of three cards, A -> B -> C is exact if everything you take from A as part of B is all that you leave behind when you interpret B as part of C.

For example let's look at a relationship spread:

Self -> Other -> Dynamic

Start with the Self, then identify the self with aspects of the Other; those aspects are precisely the parts of the Other that you ignore when interpreting the Other in the Dynamic. With me so far?

Let's add another link to the sequence:

Self -> Other -> Dynamic -> void

"void" has no card. It has no interpretation, consumes all, and yields nothing. All aspects of the Dynamic are consumed by the void, but when we know this sequence to be exact this tells us much:

The aspects of the Self that we see in the Other are those parts we leave behind when we see the Other in the Dynamic. The aspects of the Other that we see in the Dynamic are those parts we leave behind in the void (which is everything). So for this sequence to be exact we know that the Dynamic is fully explored by those parts of the Other than we cannot identify with the Self.

Pick a point inside a triangle and drop perpendicular projections onto the sides. These define another triangle. Repeat, with the same point but within the new triangle. Do the same thing once more. The fourth triangle now has the same angles as the first one, although it’s much smaller and it’s rotated.

Please fund my research in finding fewer applications of mathematics. I'm going to start my project with trying to find fewer uses of trigonometry, so that ideally we can eliminate the need for remembering trigonometric identities. Then I'm going to move on to researching fewer uses for integration by parts, because that tends to get real tedious real fast. With your unending financial support, I believe I can return mathematics to the purity and simplicity it has always yearned for.

When banned from using "trivially" in a proof...

“Hello all. In a fellow mathposter's topology class they were not allowed to use the word "trivially" or any synonym thereof his proofs. The person presenting his work then crossed out "trivially" and wrote instead "indubitably." This inspired him to write a program that will insert condescending adverbial phrases before any statement in a math proof. Trivially, this is a repost. Below is the list--please come up with more if you can!

Obviously

Clearly

Anyone can see that

Trivially

Indubitably

It follows that

Evidently

By basic applications of previously proven lemmas,

The proof is left to the reader that

It goes without saying that

Consequently

By immediate consequence,

Of course

But then again

By symmetry

Without loss of generality,

Anyone with a fifth grade education can see that

I would wager 5 dollars that

By the contrapositive

We need not waste ink in proving that

By Euler

By Fermat

By a simple diagonalization argument,

We all agree that

It would be absurd to deny that

Unquestionably,

Indisputably,

It is plain to see that

It would be embarrassing to miss the fact that

It would be an insult to my time and yours to prove that

Any cretin with half a brain could see that

By Fermat’s Last Theorem,

By the Axiom of Choice,

It is equivalent to the Riemann Hypothesis that

By a simple counting argument,

Simply put,

One’s mind immediately leaps to the conclusion that

By contradiction,

I shudder to think of the poor soul who denies that

It is readily apparent to the casual observer that

With p < 5% we conclude that

It follows from the Zermelo-Fraenkel axioms that

Set theory tells us that

Divine inspiration reveals to us that

Patently,

Needless to say,

By logic

By the Laws of Mathematics

By all means,

With probability 1,

Who could deny that

Assuming the Continuum Hypothesis,

Galois died in order to show us that

There is a marvellous proof (which is too long to write here) that

We proved in class that

Our friends over at Harvard recently discovered that

It is straightforward to show that

By definition,

By a simple assumption,

It is easy to see that

Even you would be able to see that

Everybody knows that

I don’t know why anybody would ask, but

Between you and me,

Unless you accept Gödel’s Incompleteness Theorem,

A reliable source has told me

It is a matter of simple arithmetic to show that

Beyond a shadow of a doubt,

When we view this problem as an undecidable residue class whose elements are universal DAGs, we see that

You and I both know that

And there you have it,

And as easy as ABC,

And then as quick as a wink,

If you’ve been paying attention you’d realize that

By the Pigeonhole Principle

By circular reasoning we see that

When we make the necessary and sufficient assumptions,

It is beyond the scope of this course to prove that

Only idealogues and sycophants would debate whether

It is an unfortunately common misconception to doubt that

By petitio principii, we assert that

We may take for granted that

For legal reasons I am required to disclose that

It is elementary to show that

I don’t remember why, but you’ll have to trust me that

Following the logical steps, we might conclude

We are all but forced to see that

By the same logic,

I’m not even going to bother to prove that

By Kant’s Categorical imperative,

Everyone and their mother can see that

A child could tell you that

It baffles me that you haven’t already realized that

Notice then that

Just this once I will admit to you that

Using the proper mindset one sees that

Remember the basic laws of common sense:

There is a lovely little argument that shows that

Figure 2 (not shown here) makes it clear that

Alas, would that it were not true that

If I’m being honest with you,

According to the pointy-headed theorists sitting in their Ivory Towers in academia,

We will take as an axiom that

Accept for the moment that

These are your words, not mine, but

A little birdie told me that

I heard through the grapevine that

In the realm of constructive mathematics,

It is a theorem from classical analysis that

Life is too short to prove that

A consequence of IUT is that

As practitioners are generally aware,

It is commonly understood that

As the reader is no doubt cognizant,

As an exercise for the reader, show that

All the cool kids know that

It is not difficult to see that

Terry Tao told me in a personal email that

Behold,

Verify that

In particular,

Moreover,

Yea verily

By inspection,

A trivial but tedious calculation shows that

Suppose by way of contradiction that

By a known theorem,

Henceforth

Recall that

Wherefore said He unto them,

It is the will of the Gods that

It transpires that

We find

As must be obvious to the meanest intellect,

It pleases the symmetry of the world that

Accordingly,

If there be any justice in the world,

It is a matter of fact that

It can be shown that

Implicitly, then

Ipso facto

Which leads us to the conclusion that

Which is to say

That is,

The force of deductive logic then drives one to the conclusion that

Whereafter we find

Assuming the reader’s intellect approaches that of the writer, it should be obvious that

Ergo

With God as my witness,

As a great man once told me,

One would be hard-pressed to disprove that

Even an applied mathematician would concede that

One sees in a trice that

You can convince yourself that

Mama always told me

I know it, you know it, everybody knows that

Even the most incompetent T.A. could see,

This won't be on the test, but

Take it from me,

Axiomatically,

Naturally,

A cursory glance reveals that

As luck would have it,

Through the careful use of common sense,

By the standard argument,

I hope I don’t need to explain that

According to prophecy,

Only a fool would deny that

It is almost obvious that

By method of thinking,

Through sheer force of will,

Intuitively,

I’m sure I don’t need to tell you that

You of all people should realize that

The Math Gods demand that

The clever student will notice

An astute reader will have noticed that

It was once revealed to me in a dream that

Even my grandma knows that

Unless something is horribly wrong,

And now we have all we need to show that

If you use math, you can see that

It holds vacuously that

Now check this out:

Barring causality breakdown, clearly

We don't want to deprive the reader of the joy of discovering for themselves why

One of the Bernoullis probably showed that

Somebody once told me

By extrapolation,

Categorically,

If the reader is sufficiently alert, they will notice that

It’s hard not to prove that

The sophisticated reader will realize that

In this context,

It was Lebesque who first asked whether

As is tradition,

According to local folklore,

We hold these truths to be self-evident that

By simple induction,

In case you weren’t paying attention,

A poor student or a particularly clever dog will realize immediately that

Every student brought up in the American education system is told that

Most experts agree that

Sober readers see that

And would you look at that:

And lo!

By abstract nonsense,

I leave the proof to the suspicious reader that

When one stares at the equations they immediately rearrange themselves to show that

This behooves you to state that

Therefore

The heralds shall sing for generations hence that

If I’ve said it once I’ve said it a thousand times,

Our forefathers built this country on the proposition that

My father told me, and his father before that, and his before that, that

As sure as the sun will rise again tomorrow morning,

The burden of proof is on my opponents to disprove that

If you ask me,

I didn’t think I would have to spell this out, but

For all we know,

Promise me you won’t tell mom, but

It would be a disservice to human intelligence to deny that

Proof of the following has been intentially omitted:

here isn’t enough space in the footnote section to prove that

Someone of your status would understand that

It would stand to reason that

Ostensibly,

The hatred of 10,000 years ensures that

There isn’t enough space in the footnote section to prove that

Simple deduction from peano’s axioms shows

By a careful change of basis we see that

Using Conway’s notation we see that

The TL;DR is that

Certainly,

Surely

An early theorem of Gauss shows that

An English major could deduce that

And Jesus said to his Apostles,

This fact may follow obviously from a theorem, but it's not obvious which theorem you're using:

Word on the streets is that

Assuming an arbitrary alignment of planets, astrology tells us

The voices insist that

Someone whispered to me on the subway yesterday that

For surely all cases,

Indeed,

(To be continued)

[ID: a figure in a textbook that has curved arrows to look like vectors in a field. The figure caption reads, "Is this a vector field? No. It's a picture" /end]