(My sister teaches math at a University - in 2016 she asked if I could send over some examples for her students of how I'd used math in real world applications in the tech industry. This was my answer. Specifically written for students entering college math classes. Still useful for others interested in thinking about what Math is.)
You won't use "Math" very much at all in your life. I promise you you’ll never find the volume of a cylinder after college. That’s what you’ve wanted someone to come clean about, and now that I’ve done it I hope I’ve earned your trust to keep reading something longer than a tweet.
I will refer to "Math" with quotations, and Math without quotations as two different things.
So let’s start over: you won’t use “Math” very much at all…but you will use Math every single day of your life.
“Math” is what you were taught in elementary and high school to believe the subject is. That was a white lie. Math without quotations is a different subject. If you want to stay sane in college, you need to strip the quotations off.
White lie is generous. In fact, many of your past and current teachers actually think Math is “Math."
Below are “6 Impossible Questions” that you’ve posed to your Math teachers. You’ve never received adequate answers to them, and that inadequacy has ruined STEM education for decades. A correct definition of Math should be able to answer these questions, and that’s what the rest of this essay will do. The worthless answers that “Math” provides are included below each question.
The utter lack of logic in these answers is the number one, undisputed reason that humans age 0-22 spend as much time as they can avoiding Math classes, bragging about how few Math classes they can take while still meeting their degree requirements, and "dropping out" of STEM majors. "Dropping out" is the media’s term…the accurate description is freeing themselves of the utter lack of logic provided to them every time they asked "what's the f***ing point of this?" between ages 9 and 18.
Ironically, this means the only people who stay in STEM ignore rational logic. This turns out to be absolutely true. And they become great Mathematicians and engineers. That's because, contrary to what you’ve been told up to this point, Math isn't defined as facts and rational figures. "Math" is defined as facts and figures, but not Math. From this point forward, you should realize what you’re learning is a different subject than you thought it was.
Here are the two definitions…
"Math"
Numbers, symbols, charts, graphs, properties, laws, facts, indisputable reason, unquestioned procedures, rote memorization, one way to do things.
Math
A way to communicate using less physical and verbal space than communication using words would require. By condensing communication among the voices in your head (your thoughts) it allows more thoughts to be held in your head at one time than using other forms of communication. Condensing communication between two people allows more thoughts to pass between the two of you than using other forms of communication.
I’ll give logical answers to the first four impossible questions now, simply by using the correct definition of Math above - that it is a way to communicate concisely.
1. When will I use this again?
You will use the technique of summarizing 10 thoughts into 1 thought as a building block to create systems of large thoughts. For example, you'll be able to hold 100 thoughts in your head as just 10 thoughts. And when you can do that, you'll be ready to learn the next-level skill of how to concisely turn those 10 thoughts into 1, which will allow you the wizard-like skill of holding 100 basic thoughts in your head as 1. This goes on as long as you want to keep learning, there are all sorts of masters and grandmasters who are up to 10,000 thoughts in 1 thought. This is most obviously useful for building any large and complex system like an engine, a building, or a computer. It also turns out to be useful in other fields that you’ve been told aren’t technical / “math” related fields.
2. When will I use this in real life?
Any time you encounter a problem that requires you or a group of people to think about more than 10 things at once, you will be able to use this magical technique to put all those things in your head at the same time, hold them there long enough to think about them all at the same time - and then come up with the solution. Whether you are an engineer, a nurse, a salesperson, or a construction worker, you will use this technique.
3. Why do I have to show my work?
The other day, my 2-year old niece wanted ice cream. I told her, "I don't know if there's going TO BE any tonight." When her mom came home, the 2-year old took my phrasing and posed the following, very logical question to her mom: "To be or not to be?" We all know Shakespeare was was saying a lot more than "do I get ice cream?" He was actually summarizing hundreds of pages worth of thoughts in one concise phrase. If you understand the hundreds of pages of communication behind a concise statement, you can “show” that you do by writing a lot more than the line itself.
Your “Math” to this point was like spelling bees and vocab books - the goal was “just know the symbols.” You were 100% rational to claim there was no need to show your work. From now onward, realize you’ve moved from vocab words to studying Shakespeare where the goal is “explain what he meant when he used these symbols.”4. How can my 3-hour Final only have one problem?
"Math" students entering college love to brag to friends and family that they had a "Math" test and it only had 1 problem! Can you believe that?! Their friends and family gasp in disbelief, wondering what mysterious type of numbers and symbols are you learning about that would take so long to calculate. Everyone is confused because they are picturing "Math" where the amount of lines on the page are roughly equal to the time it should take to do the test. Complete a list of 50 multiplication problems? It should take approximately a linear amount of time.
So, if one problem takes 3 hours that sounds like it must be some crazy calculation.
However, if you told friends and family - "I went to an English final, they gave us a one line essay prompt and we had 3 hours to write the essay” - there would be no gasps of disbelief. That is actually what is really happening on your Math tests from now on. Math is a concise way to write something long. A good Math teacher is really giving you a prompt for a five paragraph essay, and then wants you to show them how you have now learned how to write that in one paragraph, or better yet, 2 sentences. Same thing. No disbelief or fake genius stuff at work. Just communication.Taylor Swift and Jay-Z can help us here. They are two of the most unbelievable Mathematicians of our time, but very few people realize it.
“Our Song”….
I don't know how old you kids were when this song came out, but most people my age read those two words and know that it means "the slammin' screen door, sneaking out late tapping on your window," and a myriad of other things. In fact, in 2 words, Taylor Swift summarized and communicated something that would have taken 300 pages in a 16-yr old's diary to describe otherwise.
How can you describe the inside jokes, the emotions, the feelings, all the little things, and all the unique oddities that you think about when you think about the immenseness of a long and winding relationship with your high school crush? She thought about all those things, and then she concisely defined it and communicated it all in two words as: “our song.”
Concisely creating a way to communicate 300 pages in 2 words is remarkable. Now, anyone who has heard the definition can speak the same concise language to describe a very abstract, previously long-winded concept. Instead of stopping mid-conversation and trying to explain a myriad of feelings about all the little things that make up your unique relationship, you could say to a significant other "this is part of our song," and they would know what you meant. You meant “this is part of all the little unique things that make up everything special about our path, dot, dot, dot."
"Our Song" and "To be or not to be" are very similar to the phrase "Energy is mass times the speed of light squared." Or in symbols: E = mc 2
E = mc 2
Einstein took thousands of pages of concepts and thoughts...held them in his head for a really long time, and came up with a way to concisely communicate all of those thoughts in 9 english words, or 5 symbols. He started by taking 1,000 pages and turning them into 1. Then thousands of those single pages of thoughts and turning them into 1. To the point where in one line, he is concisely communicating literally millions of thoughts.
By doing so, he makes it a building block that others can use to communicate concisely, just like Taylor did for couples everywhere.
The simplicity of E, m, and c (all easily defined individually: Energy, mass, speed of light), is just like the simplicity of the minuscule words Shakespeare used. It is very easy to trick yourself into thinking, "if I understand the symbols, then I understand what is being communicated." Just remember E = mc 2 the next time you think this. The symbols are not the concept. Just like “our song” - the two words alone are not the abstract concept.
We’ll get to Jay-Z, and the genius of the two words “Forever Young” shortly, but first, let’s show that your Math homework has been concise communication all along.
Zero. Books have been written about that concept. Giving nothingness a quantifiable definition is something that requires high levels of cognition. Once it has been done, it can be used and built upon.
Infinity. That is a mind-blowing concept that takes a long time to comprehend. Imagine if every time you wanted to write zero to infinity you had to write 500 pages worth of definitions of those two topics. That would not allow us to continue communicating about other things.
Notice I wrote zero to infinity out in words. The great thing about Math is that you can say it in words. It might take a lot of words, but the fact you can talk about Math in words shows it's just a way to communicate. This sounds confusing when you're stuck thinking "Math" means numbers and symbols, but that's just a remnant from your elementary and high school days when you had to learn the numbers and symbols first.
It’s like grammar, letters, and spelling. There was a time when you were learning the letters B, E, N, O, R, T and how to say them. Then you learned what the words TO, NOT, OR, and BE meant and how to spell them. Then you learned how to read and how to place them in a line such as "To be or not to be."
Next came a whole different type of understanding. When Shakespeare wrote the sentence it was a concise way of communicating the eternal human question of whether living life is worth the suffering.
If asked what made Shakespeare one of the greats, no one would say “it was probably because he was the best speller.”
We use words everyday, so we get past the "learning symbols" part really quick. However, it takes a longer part of our lives (basically through high school) to build up the basic symbols part of Math.
By that time, you can't blame anyone for thinking that’s “the subject of Math.”
If you maintain this mindset: you will incorrectly believe that understanding the symbols means you understand the concept.
Bonus impossible question: “I’ve been really good at numbers and doing math in my head my whole life, so why am I now struggling in college Math classes?”
A: Same reason being a good speller doesn’t make you Shakespeare. This also explains what some people find odd which is that your Math professors and Ph.D TA’s will often tell you they’re not actually good at “numbers” at all. They’re often terrible at quick multiplication of the tip at dinner or what people in their everyday life think of as “Math.”
In the early stage of "Math" the length of your homework is roughly equivalent to how many lines you will end up writing on the paper. We spend about 12 years learning a lot of ways to write 1 sentence as 1 symbol, or sometimes 2 lines in 1 line. That means you can roughly estimate how long a homework assignment will be based on the questions or the number of lines it will take up on the page.
If you maintain this mindset: you will incorrectly forecast how long every homework assignment and test should take you.
Starting now, think of a one-liner Math question as the equivalent of an essay prompt in English class. When you get a prompt for an essay, you are really good at estimating how long it will take. You think "I can probably write a 3 page essay quickly in about 2 hours, and I know that if I do it super quick it won't be that good so I'll take another hour to revise it a few times so I need a full afternoon to do this paper right."
In high school, the speed you could do "Math" was proportional to how good you were at it. You were proving that you could memorize the symbols, or write one line as one word. Finish 15 lines of questions the fastest? That means you're good at it. But, when you wrote an essay for English class you KNEW the OPPOSITE was true. “If I write this paper REALLY fast, it means by definition I didn't think about it that much, and I know it won't be that good.” So, you made sure to give yourself more time to finish it.
You’ve seen every symbol that Einstein used. The symbol game is over. The Math you are learning is "how to write a paragraph of thoughts in one line" and "how to write 5 pages in only one paragraph."
Start thinking like you do with essays and use that same time-forecasting that you use for writing papers. This is not a "trick" to help you forecast better…it will work because you're actually learning how to think about concepts that are paragraphs and pages in length, just like essays.
If you're told you’ll have one problem set per week, you should interpret that as, "you will be required to write one 5-page essay each week.”
College math lectures are absurd. No need to sugar coat it. Professors will admit this privately, students will admit this privately, but everyone acts like a happy family and sweeps it under the rug publicly.
You walk in and there is an equation or topic written on the board that the lecturer plans to “show” you or “prove” for you. They might even make some comment like “this is such an amazing, eloquent proof, and it really summarizes everything that you need to know about the fundamentals of calculus.” You stare blankly thinking they’re nuts and hoping they’ll just tell you the example that will be on the Final.
They proceed to write 10 whiteboards worth of notes that you copy down in your notebook.
If you think “Math” is about getting to the next level of “symbols” you will be confused about which of those whiteboards was important.
You will see some of the whiteboards that use really simple symbols, and think “I get that one.” You will give those less attention than the chapters or lectures in which you're seeing some slightly “new" looking equations. Don't fall for this. In reality, your lecturer started with one sentence, like “to be or not to be” and then spent the lecture writing 5 paragraphs about what it really means, what it summarizes, how it came to be. The symbol days are over, you’ve seen basically all of them. Think about the concept behind those 10 whiteboards. This is where Jay-Z the Math teacher comes in.
The term “show your work” has been ruined to the point of being basically useless now. Better to think of it as “explain the concept behind the lyrics.”
If the two words “our song” were written on the board at the start of lecture, and then your lecturer wrote 10 white boards about the concept behind the lyric…they were explaining the concept.
On a test, if you were asked what the chorus was, you wouldn’t simply write “our song,” or even the chorus verbatim. That would be the version where you don’t show your work. The correct answer is to write a page worth of symbols that show you understand the full concept.
From now on, “showing your work” is not the cliched phrase that was ruined for you by elementary and high school teachers everywhere. It’s not "part of turning in your assignment"...it is the entire point of your classes from here onward. It’s the difference between a rapper who steals lyrics, and one who can create their own.
Imagine you're trying to become a rapper, and you get to learn from Jay-Z. He walks you through his hit "Forever Young" and tells you to study it.
You come into the studio office hours and he says, “do you see why I used the line ‘Forever Young’?" You would never be so childish to say to his face: "ya it rhymes with the line above it, so I'll throw those two words in my own raps whenever it fits."
He'd toss you out of the studio for literally saying to his face that you'd plagiarize his work. With him, you’d actually respect his office hours and answer his question by 'showing your work' and trying to explain the paragraph-long concept behind those two words.
Maybe you tell him: "I think you're trying to convey the idea that you will always have your youthful spirit, and that you never want to forget your youthfulness as you age. It was also a clever riff on an old classic song and common cliche that we regularly use."
This is what you should do when you go to office hours. Respect your professor or TA like he or she is Jay-Z. Don't just say "tell me the right lyric and I'll copy it down." That's how you get your ass banned from Jay-Z's recording studio. Go in and tell them what you think the lyric means. That's the only way they can make you a good rapper!
When you show that you're trying to understand the paragraph, you'll learn. Like the fact that "Forever Young" is even deeper than what you thought. (following interpretation credit goes to the aptly named RapGenius)…
Jay-Z, in office hours, will correct you and tell you there's even more layers and more pages of communication behind those words. One of his nicknames is "Young" so actually what he is saying is that he wants his legacy to carry on forever. He (Young) wants to be listened to and passed down for generations. One of his ultimate goals as a rapper has been to reach the echelon of a classic and to know that what he wrote will carry on after he’s gone. It's actually not about him wishing for the old days, or lamenting his aging body, the song is about his legacy and how it has the chance to last forever.
Damn that’s deep. Taylor Swift level communication skills. Einstein level.
So you just finished 8 weeks of studying daily with Jay-Z, and you go in for your Final. The one question on the page is:
"Using the concepts of Forever Young, craft a verse that describes your path leaving home and coming to college."
Student 1: who copies people and never shows work writes:
The last bell of high school has finally rung,
I'm in college now, but I'll be forever young.
Completely wrong. Poser.
Booed off the stage in Eight Mile as a lyric-stealing plagiarizer.
Your Math teacher can't even guess what you were thinking other than to know you have no clue that there is more to those two words than those two words.
Student 2: the student who is wrong but shows their work writes:
The concept behind Forever Young is that Jay-Z wanted to talk about the universal human condition that involves aging and losing ones faculties. He is struggling with this, and he wishes that he could go back to the good times when he was younger, and vibrant, and active. He also uses the phrase "Forever Young" which is common in our every day language. Because of that, my verse is the following:
Although I was dumb in my high school ways,
I'll always yearn for those Good Ol' Days.
Wrong, but the Math teacher knows that you understood there was some deeper concept behind the two words. They also respect that you tried to match your verse with the concept. You have proven you are actually trying to think harder about this than just what the two words are.
Student 3: the student who understood the actual pages-long concept behind the line writes:
Jay-Z takes a personal nickname, Young, to embody his own legacy and his life's work. He puts that into a clever, cliche phrase and is communicating that he hopes what he has done will never be forgotten and that it will be remembered for generations. Back home, my Dad used to call me Sonny Boy. My verse is about leaving home for college this Fall, but always wanting to remember what my parents taught me and to convey that I will take it with me as I grow up.
I drove away from home, we parted ways,
But my future will bring more Sonny days.
Correct on the interpretation of the concept. Correct on applying it to your specific instance. Would have made no sense at all to the teacher if you didn't SHOW YOUR WORK and explain that your nickname was Sonny. They would've thought you just mis-spelled the word sunny.
It’s common to find college students who are Student 1.
It’s tempting to remain content being Student 2.
It’s rare to find Student 3.
Bonus impossible question: "How come I can keep passing classes without ever really knowing what it was I learned, or while still feeling no confidence about it?"
It's because colleges would never fail all the Student 1’s, even if hundreds of students don’t understand a concept. They need your tuition. So colleges brush over the issue by curving grades and making the test questions easier. Then, you learn that they'll curve the grades to help the hundreds who don’t understand the concept pass the next class…so when you get the problem wrong but still pass you don't go talk to your teacher. You’re just content knowing you can do the same thing next semester and keep getting by.
I promise you: You absolutely can pass every college class as Student 1.
But just remember: YOU WOULD NEVER DO THAT IF JAY-Z GRADED YOUR TEST.
There’s a problem if you think it’s “cool” to pass classes without knowing what it was about. To write a paper without reading the book. You think you’re clever and beat the system. That’s funny, because the exact same behavior in the rap game would get you labeled a poser. Just remember, the next time you brag about ‘sneaking by’ you’re bragging about being a poser.
A common way colleges help posers is they use "practice tests" or they feed you example questions during the week and then on the Final they give you the exact same question just with the numbers changed. You should not feel a sense of achievement when you beat them at this game. It's an easy game, and it also does not teach you how to be a rapper.
Replacing example questions with new numbers is like giving you a question that says…
Q: “Fill in the blanks of Jay-Z's final verse with new words….”
A: Instead of Forever Young, you'd write 'Always Youthful'
They'd give you credit for the right answer, and claim that you "understand the concept.”
Another common way your colleges create Posers is by accepting verbatim answers. If your professor was Dr. Taylor Swift, she might ask you to describe your high school crush and what “your song” with them was. You’ve memorized the concept from the book, all the lyrics, and you say “it was sneaking out late, and talking quietly on the phone in my room.”
Wrong. That’s HER story behind concept, you’re supposed to take the concept and then apply it to YOUR story. What were the little things in your experience with your high school crush? If you try to claim that you had the exact same love story as Taylor, that sounds like a great way to get kicked out of Taylor's crew.
An exam that accepts fill in the blanks or verbatim quotes doesn't actually test if you understood the story and the concept behind the lyrics, or the concept behind the Math equation. It is stuck in the world of "Math" testing.
On your "Math" tests in elementary school, if you got the symbols right, or if you filled in the blanks right, it means you understood. That's because when you're only working with turning one line into a shorter line, this testing method works. If you can answer 2 x 4 by filling in the final blank with 8, then it is reasonable for everyone to assume you understand the concepts behind multiplication. Spelling bees test spelling, they don’t test Shakespearean interpretation of the word “be.”
When colleges create poser factories it is lazy, unfortunate, and dangerous. Given how much money you (or your parents) are paying, it's also unethical.
Verbatim definition questions and “straight from the example” questions teach you how to memorize and how to use synonyms. Always Youthful drastically misses the real point. Even if you wrote Forever Hova (an alternative Jay-Z nickname), it just proves someone once told you the concept was about a nickname, but you didn't understand how that concept was actually created. Just know if you go up on stage and all you have is synonyms for things you've been told before, you’ll be a bad rapper.
You may have had a Math teacher try to answer impossible question #2: "when will I use this in real life?" This is a trap question because the worst thing they can do is try to answer it with a real world example.
You: “When will I need to use integrals to take the volume of a curved cylinder.”
Them: "Well, imagine you're at your first job and you have to design an oil tanker…”
You win the argument every time. ”I'm not going to go into the oil tanker design business, it’s not on my short list of dream jobs, so I guess I can skip this unit and I just proved the irrationality of this whole semester.”
Any examples you are shown in class are equivalent to listening to "Our Song" or "Forever Young" while learning from Taylor or Jay-Z…. You should NEVER expect (nor want) to copy them verbatim.
You listen to the songs to learn the concept behind the songs. You would never expect to plagiarize the exact lyric in the real world, but you should always expect the need to absorb the concepts behind the lyric.
Volume of a cylinder? WORTHLESS as a standalone thing. You'll spend weeks on this problem in Calculus and you will never take the volume of a curved cylinder in your life.
The concepts behind volume of a cylinder? The story behind the lyric? USED EVERYWHERE in your life. In almost every computer program you use everyday. The facebook news feed algorithm? Google's search results? Yep. Volume of cylinders.
Impossible, you say, there's no cylinders on my screen! That's like saying: “Hey, Mr. Carter, my nickname isn't Young, so this example is worthless, let's move on to the important stuff."
The cylinder concept teaches you to take something with more than 2-dimensions and concisely perform a mathematical operation on it. In the field of artificial intelligence and machine learning, things with 20, 100, even infinite-dimensions are concisely handled and have operations performed on them. Some of those operations tell you more than the 'spatial volume between the points' - some of them tell you a ‘unique relational distance between abstract features.' The results of the operation can communicate the order of your search results, or your news feed, or which way a self-driving car should turn.
There may have been some concepts in that paragraph that you don't understand (like infinite dimensions). Guess what is necessary to be able to concisely hold all the thoughts contained in the above paragraph in your head? First you have to go from understanding Calculus operations on 2-dimensions to Calculus operations on 3. It’s actually one of the biggest leaps in the magic thought technique. It’s ingrained in the vocabulary of anyone making self driving cars.
Learning how a 3-(or more)-dimensional thing can have Calculus performed on it concisely sums up thousands of pages of thoughts and can be used in millions of ways. That's why so much time is spent on volume of a cylinder even though you'll never use the example specifically…just like you’ll never actually say the words “Forever Young” in your own rap.
Now that I brought ‘real world’ examples into the discussion, it’s time to break down “real world” and “first job” confusion that is being fed to you in college.
It's not popular to tell kids in college Math classes that they have freedom to do any major these days. There’s an outsized emphasis on STEM. This is dangerous, so I’ll address it: Do not feel trapped in your major, you can switch to anything you want. Never tie your intelligence to whether you stay in Math despite how many articles tell you to do so. In any profession you choose, you can use the concepts of concise communication. You do not need to do it in tech or engineering. In fact, these days it's critical that many of you think concisely in other fields because "Math" is attacking every profession.
My mother was a nurse - she learned how to concisely communicate the expertise of caring for someone. She could take 10 pages of "what someone needs" and distill it down into the 1 most important thing they need. Let's say she knew to bring someone a home cooked meal at precisely the right time. Much like simple lyrics or common math terms, this would look "obvious" to a non-expert. They'd say, "I know how to cook, I know that's important" but they wouldn't have known precisely why that one thing would take care of hundreds of the others, or understand the thousands of pages of context and years of experience behind that choice. They could make that choice after it was shown to them, but they wouldn't have made it from base concepts alone. Learning and refining the concepts over a career in nursing is Math.
My father was the Athletic Director who made the football field at Boise State blue. This was one of the most Mathematically effective marketing decisions ever made in the history of college sports. He was given a limited athletic budget and told to fulfill the requirements of an entire athletic department, all of its coaching salaries, all of its marketing needs, and at the same time told he had to replace the aged green turf field they had within the year.
Today, with “data" and “math” a consulting firm would measure every number they could find around this assignment. They'd look at the number of likes on facebook that other college fields receive, they’d count the retweets of each coach’s twitter feed to gauge the fanbase support of each team in the athletic department, they'd analyze the financial impact of prior decisions at every athletic department in the country and use a sophisticated algorithm to determine the best moves for a given level of athletic budget in Boise State’s conference and geography. They'd come up with 50 pages of marketing plans and strategies that fit the budget, the coaching salary needs, and how to replace the field.
Not one of those 'data science' or 'mathematical' models would have said that making the turf blue was the single most cost effective way you could condense 50 pages of marketing into 1 decision while hitting every other budget requirement.
The "facts" of the consulting firm would have been indisputable, they would have had "numbers" and "proof" and "data." My Dad had something better, he knew the base concepts and had the ability to think through them on his own - he had Math.
You don’t get blue football fields with “Math,” you get a numerical comparison of light green vs. dark green fields.
You don't get "Our Song" or "Forever Young" by taking the average of every prior number one song and merging them together. Data science has brought “Math” to the music industry today, and has pushed out many Math songwriters. “Math” in the music industry means tracking things such as the exact moment you turn up or down a digitally streamed song on Spotify, or when you hit next, so that the industry can use that “Math” to make ‘better’ songs. It is the start of a lost generation of music. Songs produced today are more alike than ever before, using the same “data driven” beats and hooks more often. It’s literally “Math.”
Like the “data driven record studios,” the consulting firm gave all the “right” answers to Boise State. It was only hard numbers and facts. “No decision is more optimal” they say. Factually and “Mathematically” they had numerical proof of the optimal decision. You can’t win this flawed argument by “Math” adherents….because you’ll never get someone to believe the world missed out on something amazing if they are under the impression that blue football fields do not exist.
I’ll call this the “Jimmy Stewart Problem.” Stewart being the lead actor in It’s a Wonderful Life. An angel takes Stewart around in “It’s a Wonderful Life” and shows him the world that didn’t exist due to him being alive. The problem with “Math” isn’t what is there, it’s what’s not there. There’s nothing wrong with the music of today, or the news you read today, or the football fields of today….as long as you don’t know what could have existed if all those creators didn’t get stopped in their tracks by “Math.”
If you’re forcing yourself to stay in a STEM major that you despise, just so you can get a degree and get a job as a data scientist or a programmer or someone who works at a company disrupting some industry with "mathematical" thinking…don’t. You will get by simply on “Math” like I promised you that you can. And the people with Bachelors, Masters, and Ph.Ds in STEM who graduated simply by learning the “Math” will do more harm in those industries than if they had dropped out as Freshmen, and gone directly into the field they were interested in to learn its base concepts from the ground up. That’s where they would have brought Math to the world. Don’t propagate the Jimmy Stewart Problem by romanticizing STEM. There are intelligent people everywhere.
Taylor Swift was the solo songwriter of “Our Song” over a decade ago. It was before streaming apps existed. It was deep and Mathematical for a 16-year old. The multiple songwriters and co-producers on her later albums use a lot of data: the indisputable result… "Look What You Made Me Do” and “Ready For It” sound like the average of every hit song you’ve heard the last 5 years. We’d think nothing of it….if we hadn’t heard that Mathematical communicator a decade ago and wondered what songs aren’t being written by that mind today.
Math examples are harder to find than you think. The places people are told to look for the best STEM careers, like facebook, are actually full of “Math.” Consider the fake news arguments happening inside a facebook conference room:
They involve STEM majors using 7th and 8th grade level “Math”…
'what’s the probability this article is more truthful than another'
…arguing over publishers using 4th and 5th grade level “Math”…
'the number of likes on this article if we put the word boobs in the title will be greater than the number of likes if we use breasts'
…to keep users on the site who are using 1st grade “Math”…
'if something costs $0 dollars that is a good thing, and if something distracts me I will pursue it.'
The Math thinker in you will pick out the one word that holds thousands of pages behind it in the above example - truthful. You might be able to claim something is “not wrong” with “Math” but to get to truthful, you need Math. Which means the facebook argument is hilariously entertaining to anyone watching it and looking at it through the lens of Math. Because the only outcome is that the 7th and 8th graders will avoid defining truth by saying they’ll let it be defined fairly by “data” and “numbers”…the data will necessarily come from the 5th graders who settled on the word “boob” because it got the optimal “Mathematical” reaction from the 1st graders.
Arguing over a trivial piece of your homework or Final grade, or an extra partial point with your Math professor is the equivalent of arguing over whether “Look What You Made Me Do” is better than “Ready For It”….or if “Article Boobs” on facebook is realer than “Article Breasts”….or if “Light Green Fields” are better than “Dark Green Fields.” The real conversation should be about what’s happening in the Jimmy Stewart World….what’s the great song that wasn’t written, what investigative journalist left the profession and isn’t writing articles at all, and what blue fields don’t exist due to the fact that everyone is focused on “Math” rather than on real concepts.
I’ll close this “future careers” part of the discussion by giving hope to any of you Taylor Swift fans. I think she’s actually upped her Math game. There’s an interpretation of some ancient literature that claims the authors would write two stories in one. Since their books wouldn’t be published if they held opposing views as the King, the writers would put those philosophies in the mouths of the villains in the story so that the rulers would think the story was in line with their beliefs, but the reader of the story would still be able to hear those philosophies.
I’d like to believe Taylor was given a list of “Mathematical” songs to choose from by her recording studio…the ones the data showed as “most likely to increase the credit card transactions recorded at the bar within 10 minutes of playing it.” From the list, she at least chose one with the words “Look What You Made Me Do” - a line that the “Math” thinkers would think was perfectly gossip-level provocative, but which was really her way of saying to those of us still listening for Math: look what the 1st graders choosing their music based on “Math” made her do.
In a fascinating demonstration of “It’s a Wonderful Life” in the real world - around the same time she recorded that album she emailed a song she wrote to Little Big Town and they produced it without revealing who wrote it. It won CMA Song of the Year with the single writing credit of an anonymous Mathematician (the song was Better Man).
If numbers are communication, and if we all accept that within any communication there can be opinions, assumptions, and lies, then the axiom "numbers don't lie" doesn't tell the whole story. This is the great reveal that might leave you uneasy going down this rabbit hole.
It's why I said earlier that it's actually just fine that the only people who stay in Math are the ones who can accept irrational logic.
As you look at Math as a way to communicate, you'll start to realize that a chart is really just a concise way to summarize 4 paragraphs of written words. If I wrote 4 paragraphs of words, you would know that there were some 'opinions' baked into those words. If you believe a chart is a fact, and always true, then you actually don't understand the interpretation of the chart.
A great test of whether you understand a mathematical equation, graph, or statistic is to ask yourself if you can describe it fully in words. Ask yourself if you can 'show the work' behind the “math.” As said earlier, this is really what your Math lectures are in college. 50 minutes of your lecturer writing so much that it fills 10 pages of notes. All to show you the work, assumptions, and full story behind one equation.
Every great Mathematician has understood that there are as many assumptions and opinions in a Math concept as there are in a written document or verbal statement. In fact, Einstein is the quintessential example. He knew that there were opinions and white lies in the "facts" of Newton's laws. Many of us are taught Newtonian physics and shown the laws. Rarely do we "read the full 10,000 pages" behind the one liners that Newton so ingeniously put forth. Einstein did, and when he did, he understood the exact places where some really important opinions were built into the essay that Newton wrote.
One was that the observable physical nature of objects was all that mattered in the concise formulation of his laws, and that the atomic and quantum nature could be assumed to be minimal enough to be ignored. This is true almost all of the time. However, if you ever want to write an even more concise communication explaining the large system that is the universe, it would include quantum physics.
To do so, Einstein had to start by understanding that every single communication using numbers and symbols has assumptions in its full story. It's really hard to read through the whole story to figure out what those are. It's even harder to then formulate your own new, concise statement of the enormous story. It's hardest to then communicate that to a society who believes fundamentally that you can't be on to something because if you're on to something than it means "Math" isn't what they were told it was.
Claiming Einstein didn’t believe in “Math” might fly in the face of everything you have been told. It probably even flies in the face of why you want to study Math. Especially today, the media and others who talk about "Math" hold it up in their untouchably indisputable fact-checks that they use to make someone "right." You may have even used "Math" in arguments to prove your intelligence.
The number 85 is a factual piece of “Math." It is as true as saying the letters E and B exist. Is 85 high or low? Good or bad? That depends on the longer essay behind the number. As a percentage? 85% is high. On a scale of 1 to 10,000, it's low. If 85% is your team's winning percentage it's good. If it’s USC’s win percentage against Colorado and you’re a Colorado fan, then it's bad.
If you say what that percentage is in words and realize it is the "percentage of times their school has beaten our school over the last 50 years, but none of the players or coaches from years 1-46 are actually still on the team, and in the last 4 years the winning percentage is actually not 85%" Well, now you've actually read the full story behind the "indisputable fact" and you have the ability to decide whether that indisputable fact actually communicates anything useful to you, or if the opinions in it are not important to you.
Nodding along with this essay that Math is a concise way to communicate probably sounded good to you up until this point where I said that it also means parting with some of the most tempting societal uses of "Math." Telling someone that you'd like to understand the opinions and assumptions behind a Math statement will inevitably lead them to call you a conspiracy theorist who "doesn't believe in math.” Telling someone you believe there is a Jimmy Stewart Problem that arises online when people track all of your data will definitely lead them to call you paranoid. Math be damned.
Ironically, the only progress ever made in the history of Math comes from exactly those people who approached the field with the attitude that it is simply communication. Every one of them decided that if they understood the entire essay, they might be able to come up with a way to communicate it even better. Einstein didn’t believe in “Math.” If he did, he would have just taken Newton at his word and re-used Newtons lyrics in every Einstein rap battle for the rest of his life. Instead, he believed in Math.
Now we have the foundation built to answer impossible questions 5 and 6.
5. If Pythagoras and Newton were geniuses, how come it only took 50 min for you to teach what they took a lifetime to figure out?
Pythagoras and Newton took their entire lives to concisely express huge concepts. In class, your teacher writes 10 whiteboards worth of equations in a lecture about one concept. What they are really doing is writing 10 pages that at the end of the class you should be able to carry forward and communicate in 1 paragraph. It's easy when you've seen it. Just like understanding "Our Song" is easy once it's been written down. The writing of it, however, was not trivial.
Pythagoras and Newton were geniuses. Their greatest genius was not in numbers but rather in their ability to convey their genius in such an understandable way that future generations could build upon it. That takes having genius of thought, genius of courage, and genius-level communication skills. So take what they’ve given to you, and build upon it! Don't take it as trivial simply because you understood the Pythagorean theorem in 45 minutes in 5th grade. Conversely, don't think that something is impossible for you simply because it kept Newton up at night.
6. What the hell does a Ph.D student in Math do all day in the basement of this building?
Math Ph.D's are not sitting in basements of buildings seeing how high they can count. They are taking hundreds of concepts and determining whether they can write those as one concept. As long as there are things to communicate, and systems to think about, there will be the opportunity to communicate those things better, more concisely, or to combine multiple previously uncombined concise statements. There will always be Math research to be done.
The media does not understand this, which is why hilariously it will make the news if Mathematicians discover a new prime number. While this is a breakthrough, it is not bigger than hundreds of other breakthroughs that happen in Math each year. It is, however, one of the only ones that "sounds" like "Math" to the majority of the public. And thus, you are led to believe it's what your TA's spend their time doing.
You’ve entered college at a time where Math is under attack by “Math" creating an “It’s a Wonderful Life” dual reality. The phenomenon of data and data science is mostly one of "Math" and it has caused Mathematical thinking to dwindle and honest people to be questioned or vilified as flakes.
The view of Math I just offered you is not comfortable. It will make things harder for you than the easier “Math” definition. However, if one model of the world does not answer the six most difficult questions about your quest, and another one does, which one do you believe?
An important parting note: this essay heavily used analogies, which are good for explanation but which are not that good for motivation. By definition, an analogy is not something else. I say this because when you are up at 2am struggling with your Math homework, you definitely will not feel like Jay-Z or Taylor Swift. This may have gotten you more interested in reading about what Math is, but to stick with it yourself, you’re going to have to come up with your own motivation to do so.
It will be hard. Because until the end of time, it will always be simple to sing along with "Our Song" and "Forever Young” in your car proving to everyone that you “got the lyrics right, verbatim, 100%, A+”….but it will forever remain difficult to think hard about them or to create lyrics yourself. That's what Math is, and you will always have the choice to use it.