Frostgage

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

> 55-hour week
> 40-hour previous week
> supervising 20-meter boreholes being drilled from daybreak until dusk every day
> gashes head on corner of open window as i arrive home after finishing the last one
> aggressively sleep-deprived, blood clotting in hair, dirt coating all exposed skin and sweat coating all clothed skin
> still gets valentine's day card

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

hi sir i have a question

the earth is inside the sky

but if we are living earth

and breath is living sky

is the sky inside the earth too

on a scale of glabrescent to scabrous how would you rate your skin

Fun factoid! Not maths-related for once, either!

This stuff is called methane clathrate, a.k.a. fire ice! It's formed naturally when methane gas molecules are trapped within the crystal structure of ice as it freezes, thus permeating it with flammable gas! You can actually set it on fire and it will burn under its own heat! Russians discovered it in the '60s (because of course Russia discovered flammable ice during the Cold War) and these days it's even mined commercially as a source of natural gas!

Are there any movies or tv shows that you love but are also so sad that you struggle to rewatch them?

Friday...that anime.

Friday...that anime.

Hey hey hey and welcome to another episode of How To Survive a Maths Degree! I'm your disquietingly dapper droll, Duality!

I'm sure all of you already know the objectives of attempting such a bohemian subject. Learn numbers, enhance your mental ability, write groundbreaking papers on the nature of logic itself, survive the tenurewolves, all textbook stuff. However, it is few indeed who escape out the other side of university with their mental and physical integrity, mostly due to the terrible dearth of terrible-death-avoiding knowledge among the undergraduate rabble. Fortuitously for your fine selves, Dr Duality is here to help!

First rule of all maths degrees is coffee. Drink it with a teabag and tiny marshmallows. Anything less will provoke the Elder Ones. Exsanguination-flavoured bubble tea is also acceptable if you don't mind the premonitory hallucinations.

Second rule of maths degrees: You may define exponentiation however you choose. Multiplication may also be tampered with if suitable precautions are taken. Messing with the formal basis of addition, on the other hand, invariably enrages Causality herself and results in a janitor's nightmare worth of shredded organ puddles. This is generally not advised due to the resultant slip hazard.

Third rule of maths degrees: Do not let the number 17 out of your sight. It will escape.

Fourth rule of maths degrees: You just let it out of your sight. You incompetent urelement. Go and get a butterfly net and catch it before the number line caves in.

Fifth rule of maths degrees: You may look down on all other fields of study for 'being too applied' and 'falling short of the heart of reality' and 'is dum n stupd'. Philosophy does not count as a field of study due to not involving numbers. Numbers are perfection. Numbers are eternal. Numbers are the highest object of thought. It is legal to murder heretics.

Sixth rule of maths degrees: Don't put anything in the empty set or set theory will break again.

Seventh rule of maths degrees: Primes are evil. Infinities are evil. Diophantine equations are extra evil.

Eighth rule of maths degrees: 'Have you ever heard a pickup line about pickup lines? I've never meta girl who had.' is the only acceptable means of securing meiotic reproduction among mathematicians. Mitosis is hence the conventional method of propagating our genes.

Ninth rule of maths degrees: This is left as an exercise to the reader.

Tenth rule of maths degrees: The clocks go to thirteen in the maths department. Use the extra hours wisely.

Sixteenth rule of maths degrees: Rigorous ordinal succession requirements on rulesets are optional.

Seventeenth rule of maths degrees: IT ESCAPED AGAIN YOU FOOL GO GET IT

Eighteenth rule of maths degrees: The contents of these rules are 73% arbitrarily generated, 24% somewhat nonspecific, and have 59% less apparent ground in empirical reality than linguistics. This is either universal in mathematics or not at all universal in mathematics, depending on your philosophy. If you have a philosophy it is legal for us to murder you as a heretic.

Anyway welcome to university if you wish to escape you must either pay off a student debt larger than the national deficit or solve maths forever and become as a god among mortals

So how do y'all cope after mini panic attacks. Asking for a friend (that was a lie; actually asking for me)

Welcome to the latest installation of the hottest show in showbiz, Explanations of Things You Never Thought to Question Because You Never Cared!

This episode: we separate the concept of pure number from the primitive quantitative concept generated by counting things!

Once upon a time we only used numbers for counting things. Then, we had the shocking realisation that numbers also applied to creating lengths and thus areas and thus volumes and thus shapes. Then, we had the further shocking realisation that the only way to reliably figure out how shapes and quantity worked without using gross approximations was by logical proof. A subsequent breakthrough was that some guy demonstrated that the square root of two couldn't be written down in decimal form without an infinite chain of whole numbers. This was naturally very distressing to some randoms who worshiped numbers and thought counting numbers were the best numbers because they're nice and pretty and simple, so they killed him.

Anyway, further shocking realisations and terrible mistakes led us to realise that numbers are really weird and apply to literally every field of study in existence except flat-earthery if properly applied (that is, with extensive chains of logical reasoning to prove exactly how and why the numbers work this way in this given field of study). When I say every field of study, I mean that if you're thinking of sociology and being like 'what how' remember that statistics is also technically a wimpy type of maths. These fields naturally couldn't all be based on counting numbers, so such a vast intellectual advancement (read: we didn't execute the number heretics nearly as often) required us to discover all of the following types of perfectly valid and consistent numbers:

1. The counting numbers (e.g., 1, 2, 18, 57, 1376) - used for counting
2. The rational numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67) - used for fractions and dividing things up
3. The real numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π) - used for measuring lengths and volumes and times and other continuous quantities
4. The complex numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π, 17 + 5i, 12i, 5555 - 4343i, where i^2 = -1) - used for quantities that mess around in two directions at the same time, like impedance in AC electrical circuits
5. The quaternions (e.g.,  1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π, 17 + 5i, 12i, 5555 - 4343i, 2 + 3j, 9 - 57k, 5 - 343i + 21j + 14k, where i^2 = j^2 = k^2 = -1) - used for quantities that mess around in four directions at the same time, like forces rotating across 3 dimensions
6. The cardinal and ordinal hierarchy (e.g., [all the real numbers plus a bunch of conglomerated symbols in three or four different languages that crash my computer's keyboard to use]) - used in analysing the ability of a mathematical system to prove various theorems
7. A PILE OF OTHERS LIKE OCTONIONS THAT MESS AROUND IN EIGHT DIRECTIONS AT THE SAME TIME AND HYPERREALS WHICH ARE HYPER + REALS AND TRANSFINITES WHICH ARE BASICALLY INFINITE BUT LIKE A WEIRDLY SPECIFIC TYPE OF INFINITY AND THE UBERSUPLEX NUMBER TREE WHICH I MADE UP JUST NOW

Remember, folks, all formally constructed number classes are Nice and Good and Applicable to Some Domain of Study so if you discriminate against weird and abstract numbers because they're weird and abstract I'll cry and use your sleeve as a hanky.

Now, with that aside, note how the first few types of numbers all include the previous types of numbers but are also more extended and abstract than all previous ones - countings are a subset of rationals are a subset of reals are a subset of complexes are a subset of quaternions are a subset of octonions. Furthermore, reals are also a subset of hyperreals which are in turn a subset of the ordinal hierarchy which is in turn a subset of the UBERSUPLEX NUMBER TREE which I made up just now but which also kinda exists in the form of the SURREAL NUMBERS WHICH ARE GREAT AND IDEAL AND TWILIGHT SPARKLE'S BEST FAVOURITE

SO THERE WAS THIS GUY CALLED CONWAY WHO WAS SO SMART IT'S ILLEGAL IN SWITZERLAND TO TELL HIS STORY IN LOWERCASE (i don't live there but i'm lawful good don't judge me)

HE DECIDED NUMBERS WEREN'T COOL ENOUGH

HE DECIDED NUMBERS WEREN'T SIMPLE ENOUGH TO CONSTRUCT USING LOGIC

HE DECIDED MATHS USED WAY TOO MANY BORING AXIOMS

SO YOU KNOW WHAT THIS LEGEND DID

WHAT THIS LEGENDARY MAN DECIDED TO DO

HE CREATED ALL THE NUMBERS

ALL THE REAL NUMBERS

PLUS EVERY INFINITY KNOWN TO MAN

PLUS A BUNCH OF INFINITESIMALS THAT NOBODY ELSE HAD REALLY USED BEFORE BECAUSE THEY WERE SO SMALL NOBODY COULD AIM A MICROSCOPE AT THEM PROPERLY

WITH ONLY TWO AXIOMS

TWO BASIC MATHS STATEMENTS

Quote

Definition 1. A surreal number is a pair of sets of previously created surreal numbers. The sets are known as the “left set” and the “right set”. No member of the right set may be less than or equal to any member of the left set.

Definition 2. A surreal number x is less than or equal to a surreal number y if and only if y is less than or equal to no member of x’s left set, and no member of y’s right set is less than or equal to x.

Note how neither of these definitions state that any number actually exists. The vast majority of other maths systems are like '1 exists' and 'addition gives you bigger numbers' and then '1 + 1 = the next bigger number' and continues on up to prove the existence of all other numbers. This system says 'a surreal number is a pair of sets of previously created surreal numbers' and starts with 0 being the number with both sets empty, since no other numbers have been created yet. The definitions aren't based on addition or multiplication or even an equals sign - it's less than or equal right from creating 0 to creating the transfinite hierarchy. Nothing more is needed: addition and multiplication and so on can be defined later as convenient shorthand for meaningful ways to get from two numbers to another number in the already-fully-created class of surreal numbers.

conway was magic guys

tune in next week for more incoherent raving about infinities

Welcome to the latest installation of the hottest show in showbiz, Explanations of Things You Never Thought to Question Because You Never Cared!

This episode: we separate the concept of pure number from the primitive quantitative concept generated by counting things!

Once upon a time we only used numbers for counting things. Then, we had the shocking realisation that numbers also applied to creating lengths and thus areas and thus volumes and thus shapes. Then, we had the further shocking realisation that the only way to reliably figure out how shapes and quantity worked without using gross approximations was by logical proof. A subsequent breakthrough was that some guy demonstrated that the square root of two couldn't be written down in decimal form without an infinite chain of whole numbers. This was naturally very distressing to some randoms who worshiped numbers and thought counting numbers were the best numbers because they're nice and pretty and simple, so they killed him.

Anyway, further shocking realisations and terrible mistakes led us to realise that numbers are really weird and apply to literally every field of study in existence except flat-earthery if properly applied (that is, with extensive chains of logical reasoning to prove exactly how and why the numbers work this way in this given field of study). When I say every field of study, I mean that if you're thinking of sociology and being like 'what how' remember that statistics is also technically a wimpy type of maths. These fields naturally couldn't all be based on counting numbers, so such a vast intellectual advancement (read: we didn't execute the number heretics nearly as often) required us to discover all of the following types of perfectly valid and consistent numbers:

1. The counting numbers (e.g., 1, 2, 18, 57, 1376) - used for counting
2. The rational numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67) - used for fractions and dividing things up
3. The real numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π) - used for measuring lengths and volumes and times and other continuous quantities
4. The complex numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π, 17 + 5i, 12i, 5555 - 4343i, where i^2 = -1) - used for quantities that mess around in two directions at the same time, like impedance in AC electrical circuits
5. The quaternions (e.g.,  1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π, 17 + 5i, 12i, 5555 - 4343i, 2 + 3j, 9 - 57k, 5 - 343i + 21j + 14k, where i^2 = j^2 = k^2 = -1) - used for quantities that mess around in four directions at the same time, like forces rotating across 3 dimensions
6. The cardinal and ordinal hierarchy (e.g., [all the real numbers plus a bunch of conglomerated symbols in three or four different languages that crash my computer's keyboard to use]) - used in analysing the ability of a mathematical system to prove various theorems
7. A PILE OF OTHERS LIKE OCTONIONS THAT MESS AROUND IN EIGHT DIRECTIONS AT THE SAME TIME AND HYPERREALS WHICH ARE HYPER + REALS AND TRANSFINITES WHICH ARE BASICALLY INFINITE BUT LIKE A WEIRDLY SPECIFIC TYPE OF INFINITY AND THE UBERSUPLEX NUMBER TREE WHICH I MADE UP JUST NOW

Remember, folks, all formally constructed number classes are Nice and Good and Applicable to Some Domain of Study so if you discriminate against weird and abstract numbers because they're weird and abstract I'll cry and use your sleeve as a hanky.

Now, with that aside, note how the first few types of numbers all include the previous types of numbers but are also more extended and abstract than all previous ones - countings are a subset of rationals are a subset of reals are a subset of complexes are a subset of quaternions are a subset of octonions. Furthermore, reals are also a subset of hyperreals which are in turn a subset of the ordinal hierarchy which is in turn a subset of the UBERSUPLEX NUMBER TREE which I made up just now but which also kinda exists in the form of the SURREAL NUMBERS WHICH ARE GREAT AND IDEAL AND TWILIGHT SPARKLE'S BEST FAVOURITE

SO THERE WAS THIS GUY CALLED CONWAY WHO WAS SO SMART IT'S ILLEGAL IN SWITZERLAND TO TELL HIS STORY IN LOWERCASE (i don't live there but i'm lawful good don't judge me)

HE DECIDED NUMBERS WEREN'T COOL ENOUGH

HE DECIDED NUMBERS WEREN'T SIMPLE ENOUGH TO CONSTRUCT USING LOGIC

HE DECIDED MATHS USED WAY TOO MANY BORING AXIOMS

SO YOU KNOW WHAT THIS LEGEND DID

WHAT THIS LEGENDARY MAN DECIDED TO DO

HE CREATED ALL THE NUMBERS

ALL THE REAL NUMBERS

PLUS EVERY INFINITY KNOWN TO MAN

PLUS A BUNCH OF INFINITESIMALS THAT NOBODY ELSE HAD REALLY USED BEFORE BECAUSE THEY WERE SO SMALL NOBODY COULD AIM A MICROSCOPE AT THEM PROPERLY

WITH ONLY TWO AXIOMS

TWO BASIC MATHS STATEMENTS

Quote

Definition 1. A surreal number is a pair of sets of previously created surreal numbers. The sets are known as the “left set” and the “right set”. No member of the right set may be less than or equal to any member of the left set.

Definition 2. A surreal number x is less than or equal to a surreal number y if and only if y is less than or equal to no member of x’s left set, and no member of y’s right set is less than or equal to x.

Note how neither of these definitions state that any number actually exists. The vast majority of other maths systems are like '1 exists' and 'addition gives you bigger numbers' and then '1 + 1 = the next bigger number' and continues on up to prove the existence of all other numbers. This system says 'a surreal number is a pair of sets of previously created surreal numbers' and starts with 0 being the number with both sets empty, since no other numbers have been created yet. The definitions aren't based on addition or multiplication or even an equals sign - it's less than or equal right from creating 0 to creating the transfinite hierarchy. Nothing more is needed: addition and multiplication and so on can be defined later as convenient shorthand for meaningful ways to get from two numbers to another number in the already-fully-created class of surreal numbers.

conway was magic guys

tune in next week for more incoherent raving about infinities

Hi! How's everyone doing?

This got me very excited https://getpocket.com/explore/item/let-s-colonize-titan

Also this passage in particular was fascinating to learn. I had no idea this was possible

Welcome to the latest installation of the hottest show in showbiz, Explanations of Things You Never Thought to Question Because You Never Cared!

This episode: we separate the concept of pure number from the primitive quantitative concept generated by counting things!

Once upon a time we only used numbers for counting things. Then, we had the shocking realisation that numbers also applied to creating lengths and thus areas and thus volumes and thus shapes. Then, we had the further shocking realisation that the only way to reliably figure out how shapes and quantity worked without using gross approximations was by logical proof. A subsequent breakthrough was that some guy demonstrated that the square root of two couldn't be written down in decimal form without an infinite chain of whole numbers. This was naturally very distressing to some randoms who worshiped numbers and thought counting numbers were the best numbers because they're nice and pretty and simple, so they killed him.

Anyway, further shocking realisations and terrible mistakes led us to realise that numbers are really weird and apply to literally every field of study in existence except flat-earthery if properly applied (that is, with extensive chains of logical reasoning to prove exactly how and why the numbers work this way in this given field of study). When I say every field of study, I mean that if you're thinking of sociology and being like 'what how' remember that statistics is also technically a wimpy type of maths. These fields naturally couldn't all be based on counting numbers, so such a vast intellectual advancement (read: we didn't execute the number heretics nearly as often) required us to discover all of the following types of perfectly valid and consistent numbers:

1. The counting numbers (e.g., 1, 2, 18, 57, 1376) - used for counting
2. The rational numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67) - used for fractions and dividing things up
3. The real numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π) - used for measuring lengths and volumes and times and other continuous quantities
4. The complex numbers (e.g., 1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π, 17 + 5i, 12i, 5555 - 4343i, where i^2 = -1) - used for quantities that mess around in two directions at the same time, like impedance in AC electrical circuits
5. The quaternions (e.g.,  1, 2, 18, 57, 1376, 1/2, 34/6, 23222/67, √2, 87.34343677, π, 17 + 5i, 12i, 5555 - 4343i, 2 + 3j, 9 - 57k, 5 - 343i + 21j + 14k, where i^2 = j^2 = k^2 = -1) - used for quantities that mess around in four directions at the same time, like forces rotating across 3 dimensions
6. The cardinal and ordinal hierarchy (e.g., [all the real numbers plus a bunch of conglomerated symbols in three or four different languages that crash my computer's keyboard to use]) - used in analysing the ability of a mathematical system to prove various theorems
7. A PILE OF OTHERS LIKE OCTONIONS THAT MESS AROUND IN EIGHT DIRECTIONS AT THE SAME TIME AND HYPERREALS WHICH ARE HYPER + REALS AND TRANSFINITES WHICH ARE BASICALLY INFINITE BUT LIKE A WEIRDLY SPECIFIC TYPE OF INFINITY AND THE UBERSUPLEX NUMBER TREE WHICH I MADE UP JUST NOW

Remember, folks, all formally constructed number classes are Nice and Good and Applicable to Some Domain of Study so if you discriminate against weird and abstract numbers because they're weird and abstract I'll cry and use your sleeve as a hanky.

Now, with that aside, note how the first few types of numbers all include the previous types of numbers but are also more extended and abstract than all previous ones - countings are a subset of rationals are a subset of reals are a subset of complexes are a subset of quaternions are a subset of octonions. Furthermore, reals are also a subset of hyperreals which are in turn a subset of the ordinal hierarchy which is in turn a subset of the UBERSUPLEX NUMBER TREE which I made up just now but which also kinda exists in the form of the SURREAL NUMBERS WHICH ARE GREAT AND IDEAL AND TWILIGHT SPARKLE'S BEST FAVOURITE

SO THERE WAS THIS GUY CALLED CONWAY WHO WAS SO SMART IT'S ILLEGAL IN SWITZERLAND TO TELL HIS STORY IN LOWERCASE (i don't live there but i'm lawful good don't judge me)

HE DECIDED NUMBERS WEREN'T COOL ENOUGH

HE DECIDED NUMBERS WEREN'T SIMPLE ENOUGH TO CONSTRUCT USING LOGIC

HE DECIDED MATHS USED WAY TOO MANY BORING AXIOMS

SO YOU KNOW WHAT THIS LEGEND DID

WHAT THIS LEGENDARY MAN DECIDED TO DO

HE CREATED ALL THE NUMBERS

ALL THE REAL NUMBERS

PLUS EVERY INFINITY KNOWN TO MAN

PLUS A BUNCH OF INFINITESIMALS THAT NOBODY ELSE HAD REALLY USED BEFORE BECAUSE THEY WERE SO SMALL NOBODY COULD AIM A MICROSCOPE AT THEM PROPERLY

WITH ONLY TWO AXIOMS

TWO BASIC MATHS STATEMENTS

Quote

Definition 1. A surreal number is a pair of sets of previously created surreal numbers. The sets are known as the “left set” and the “right set”. No member of the right set may be less than or equal to any member of the left set.

Definition 2. A surreal number x is less than or equal to a surreal number y if and only if y is less than or equal to no member of x’s left set, and no member of y’s right set is less than or equal to x.

Note how neither of these definitions state that any number actually exists. The vast majority of other maths systems are like '1 exists' and 'addition gives you bigger numbers' and then '1 + 1 = the next bigger number' and continues on up to prove the existence of all other numbers. This system says 'a surreal number is a pair of sets of previously created surreal numbers' and starts with 0 being the number with both sets empty, since no other numbers have been created yet. The definitions aren't based on addition or multiplication or even an equals sign - it's less than or equal right from creating 0 to creating the transfinite hierarchy. Nothing more is needed: addition and multiplication and so on can be defined later as convenient shorthand for meaningful ways to get from two numbers to another number in the already-fully-created class of surreal numbers.

conway was magic guys

tune in next week for more incoherent raving about infinities