Mothematician

If you are scared of something, do it anyway. If you feel anxious, you must do things that make you scared. Get out of your comfort zone. Tell your fears they are wrong. Act as though you are not afraid. Ignore, ignore, ignore, silence, silence, silence.

It hurt me-- it is a horrible psychological weight to carry for a child to be certain that she will suffer unbearably over and over and that she will never deserve sympathy or compassion for it-- but it is also fundamentally incurious and disconnected.

If your body expresses something that is inconvenient or hard to understand, just silence and ignore it, because the things the body wants are wrong and the things the body communicates are false.

Look, I got to thinking about this when reading scientific articles about nutrition.

So much research is conducted about why people eat foods that are Wrong and Bad. But the research is conducted around an already-known truth, like a tree that has grown around a metal fence: people eat wrong and bad food because people like pleasure and avoid discomfort, and "bad" foods are pleasurable whereas healthy foods are not.

I feel a hole big enough for the wind to howl through: the joyful table, the raw ecstasy of staining my fingers with raspberries in the thicket, the peaceful bubbling of soup on the stove, salsa canned from vegetables in our garden. Stir-fried wild mushrooms, pawpaws messily devoured in the woods, the fragrance of soil and green and growing things. Curry powder. Smoked paprika. Ginger. Allspice. Garlic and onions hitting a hot pan. Nourishment. Connection. Caretaking. Safety. Pleasure. Pleasure.

Why does nobody ask, What is the goodness of food? What makes food good? Why does nobody say, Let's explore and study that goodness. Let's understand it deeply. Let's investigate the pleasure we feel, the condition of satisfaction of the things our bodies crave and need, the sense of belonging and interconnectedness that is present when good food is shared among friends. What does it mean to be nourished? To be satisfied? To feel peacefulness and comfort in the act of eating?

Comfort must be one of the least understood things in the world. No one is curious about the secrets it may hold.

Why was I burdened with the obligation to get over my fear and never encouraged to explore what would it mean to feel safe?

The goal of the therapy and medications was clear, to get my fear to a manageable enough level that I could "function" "normally." Safety was not part of it, the feeling or the reality.

The physiological functions and maladaptive thought patterns of fear were exhaustively discussed and explained to me. They only talked to me about the fear. How to ignore it. How to dominate it. How to force the physiological process of it to stop. How to manage it. How to understand and confound its patterns.

No one talked to me about safety. How it unfolded in the body. What it felt like. How to recognize when I was feeling it.

It was an attitude of profound incuriosity. I was never prompted or encouraged to ask, and no one else in the world seemed to ask: What does it mean for a person to feel safe? What does it feel like when I am safe? What things create that condition of safety? What are my safety needs? How is safety felt in my body? What can my body tell me about what I need to feel safe?

It is this flat, dull insistence that forcing oneself into what causes pain and discomfort automatically orients one in the direction of growth, whereas comfort and pleasure provide no information or guidance.

I'm a gardener and so are the majority of people I spend time around. If you are mowing 3+ times a week and regularly spending money on fertilizer, soil tests, herbicides, fungicides, and insecticides, you have chosen the most expensive, time consuming thing you could possibly do with your yard. Unless you are a farmer as your livelihood, NOTHING else you could grow is that high maintenance. Nothing.

Most turfgrasses are invasive species. I said it.

The practice of "nuking" your lawn (killing everything in it and "starting over")...If you have a so-called "weed problem" this is probably the worst thing you can do.

Listen to me very carefully: "Weed" seeds are everywhere. There is, at all times, a supply of seeds lying dormant in the soil, waiting for the right conditions to sprout. (It's called the "soil seed bank" and you can look it up.) They are capable of "waiting" for years, even decades. Furthermore, most "weed" species spread by wind, meaning you can't physically eliminate them from an outdoor area unless you...surround your entire yard with an incredibly fine mesh netting and never leave, I guess.

Heavy management will make your "weed" problem progressively worse and worse because those plants are specifically adapted to colonize barren areas that recently underwent disastrous events that killed off most life.

Basically all plants are adapted to live in the company of other organisms, and suffer when there are no other plants around. "Weeds" with deep taproots penetrate into and aerate the soil. Clover puts nitrogen in the ground that other plants need. Low ground covers keep the soil moist and stop the sun from baking your grass to a crisp.

The plant "taking over" your lawn is probably not killing your grass. Your grass is dying and it's being replaced by something more suited to the environment. This is supposed to happen.

Monocultures are notoriously susceptible to disease and mass die-offs. "Oh no a big patch of my lawn is dying!" Yeah, that happens when you plant monocultures. You set yourself up for this.

"Why is there a bare patch in my yard/why won't grass grow well here?" Because in nature, each plant has a relatively narrow range of conditions it likes to grow in, so other plants it might otherwise compete with can stick to their preferred conditions and nobody has to compete directly. Win-win. Not all parts of your yard have the exact same amount of sun, moisture, etc. Expecting the plant life to look the same is unrealistic.

We do a lot of proofs in mathematics. So it's relevant to have some general intuition for the informal notions of how we talk about proofs, so that there is intuition that more precise notions can crystallise around.

While there are more developed and formalised systems of proofs capable of proving things more clearly and precisely that are able to be studied in their own right, the goal here is to introduce some basic ideas and intuitions for proofs. These don't tend to be used individually, but mixed and matched and represent a more general notion of things.

Constructive Proofs

These are their own whole rabbit hole on their own, and there are whole branches of math devoted to this. Most notably, type theory.

One simple and intuitive way you might want to try to prove something is to just provide an example of the thing.

Let's say you want to prove that 4 is even. Well a number is even if it is a multiple of 2. So if we provide "2*2=4", then we have a proof that 4 is even.

"Any even number plus 2 is even."

So it takes an even number x. Since we know that x is even, we have an element y such that 2y=x.

2y+2=x+2

2(y+1)=x+2

Therefore, we found our candidate. If x is a number, and it has a proof y that it is even, x+2 has a proof that it is even, namely y+1.

How can you prove that there exists a number such that the sum of its factors equals its squares? By providing a number: 6, and checking the computation.

This kind of proof is called a Constructive Proof. Proof by construction can be surprisingly strong at times. But it has a handful of large weaknesses.

We often call the provided proof in a constructive proof a witness.

For example: Can you proved a witness to the claim that 64 is a perfect square?

Proof by Exhaustion

This is just a proof where you brute force the thing. For example, you can prove that every number less than ten thousand that ends in 0 is a multiple of ten by manually checking each one. That's a waste of time in that specific case but technically possible.

If you have reduced something to a finite list of things, you can just try all of them. Very often people don't manually do proof by exhaustion, but rather have a computer check a bunch of cases. An example of an exhaustion step was how all the base cases were checked for the four-colouring problem.

There isn't much to say on this one.

Proof by Cases

This is kind of the same thing as proof by exhaustion, where you prove it when one thing is true, and again if it was false. I have listed it separately because it's a special case of proof by exhaustion. You can nest these to get more cases, and thus do arbitrary proof by exhaustion. The vibe is sort of different, though.

An example is the proof that n²+n is always even. There are two cases, n is even, and n is odd. If n is even, it is 2m, and thus n²+n=4m²+2m=2(2m²+m)

And if it is not even, then it is 2m+1, and hence

n²+n=4m²+4m+1 + 2m+ 1= 4m²+6m+2=2(2m²+3m+1)

And hence is even.

Proof by Assumption

Let's say you want to prove that whenever A is true, B is also true. In that case you begin by assuming A, and then you can see what else is true.

For example, if a divides k, then a divides bk for every b.

First you assume that a divides k. That means there is some n such that na=k

And then that means that for every b such:

bna=bk

Hence an also divides bk.

Proof by Contrapositive

This is a special case of proof by assumption. Instead of assuming the premise, you assume the end goal is false, and show the premise must be false.

For example, if n² is even, then n is even.

If n is not even then n=2m+1 for some integer m. And thus n²=4m²+4m+1, which is an even number plus one. n²=2(2m²+2m)+1 hence if n is not even, n² is not even.

Therefore, by contrapositive, n² is even means that n is even.

Proof by Contradiction

Proof by contradiction is where you begin by assuming that something is not true, and you show that that doesn't make any sense.

For example, √2 is irrational:

Let's begin by assuming the opposite, that it is rational. In that case

a/b=√2

And thus there is some smallest pair a and b such that

a²/b²=2

Where this is as simplified as you can make it. But the goal is to show it can still be simplified, because it doesn't make sense if either number is odd.

Or:

a²=2b²

This means a² is even.

And an odd number squared is odd, so a² being even implies a is even. Let's call hald of a as n.

Notably, this means a²=(2n)²=4n², hence

4n²=2b²

And thus

2n²=b²

But that means b is even, or equal to some 2m.

But if a/b is made of two even numbers, then n/m is the same as a/b. n/m is thus a more simplified version of a/b. Which contradicts where we said a/b was the most simplified version. Therefore, there is no such rational a/b.

Hence the square root of 2 is irrational.

Inductive Proof

Proof by induction! This is where I come in! (Ordinals are the structure that induction works on.)

This is where you prove something by showing it is true for some smallest case, and then if it is true for all the smaller cases, then it must be true for the next case.

For example, every natural number is either 2m or 2m+1.

Base case: 0

This follows by construction by the witness m=0

Inductive step: n+1

Now we prove it for any n+1.

If you already know it's true for every number less than n+1, then that means you know n is equal to some 2k or some 2k+1.

Proof by case:

Case: n=2k

Then you have the witness m=k, that shows n+1 equals 2m+1.

Case: n=2k+1

n+1=2k+1+1=2k+2

And thus the witness m=k+1 shows that n+1=2m for some m.

Formal Proof

Writing 和風細雨 (he2feng1xi4yu3; gentle breeze and light rain) with a cat's tail on a water-writing mat.

Water-writing mats turn black when wet and return to their original colour once dry, so they are often used to practise calligraphy with water. The black colour allows you to see your writing clearly, and the mat can be reused as soon as the writing dries and disappears.

bypass paywall

bwah :3

I've been diving down a [graph theory + topology] rabbit hole lately; here's some of the fruits of my labor.

("Rabbit hole? Fruit? Mix metaphors much?" Not at all, this is a peanut.)

As you likely know, K_n is the "complete graph" with n nodes, i.e. an edge between every pair of nodes. K_1 is a dot, K_2 is a line segment, K_3 a triangle; you can get K_4 by putting a dot in the middle of a triangle and adding spokes.

K_5 is famously one of the simplest graphs you can't draw without two edges crossing. ...on a plane, that is! But if you're drawing on a donut, you're in luck:

The obvious next question is, can we draw K_6 on a donut? Yes! K_7? Yes! K_n? ...turns out no, 7 is as high as you can go before you need to get more topologically interesting.

...and of course, given that, it makes perfect sense...