Comment due Sunday, March 3
Contents
Background
Gracie’s questions
Gracie Cunningham was a student who went viral on Twitter a few years ago when someone tweeted her Tik Tok and said, “this is the dumbest video ive ever seen.” (That person’s Twitter account was soon suspended btw.)
After catching a lot of hate on Twitter, Gracie made a follow-up video and tweeted it herself. (I like both videos but I think I prefer the first one tbh.)
Alongside all the critical comments calling her dumb, Gracie got a huge outpouring of support from mathematicians, physicists, philosophers, and teachers, who loved her videos. Like, really, really loved them.
Dr. Cheng’s answers
Eugenia Cheng is a mathematician who is very good at explaining math to non-mathematicians. She’s published a few popular books about math for a general audience and is very active on social media. She’s even appeared on the Late Show with Stephen Colbert!
Gracie’s questions are mostly about the history and philosophy of math (which is a real academic discipline that people can study and get PhD’s in!). Some of her questions have concrete answers and some of her questions just lead to more questions. When Dr. Cheng saw Gracie’s videos and the critical comments on them, she tried answering Gracie’s questions one-by-one on her blog.
Assignment instructions
None of us are experts in the history of math or in the philosophy of math (unless there’s something you’re not telling us in your OpenLab introduction post from last week!). But we all have studied math and encountered math in some form in our day-to-day lives…which means that we’ve all spent time thinking about math, so we can ask questions about it.
For this week’s assignment, think about math in the big picture of the human experience, not just the math you see in your math classes.
- Watch both of Gracie’s videos above and read Eugenia Cheng’s blog post.
- Choose one of the following prompts:
- What are you curious about? Have you ever had any questions like “Is math real?” or like Gracie’s questions that you’ve thought about before? What is one of your questions and what have your thoughts been about it? Was there something in particular that made you have question? Was there something that changed your mind about how you think about it? Do you have any possible answers for your question, even if they contradict each other?
- Which of Dr. Cheng’s answers is the most interesting to you? Why? Did you agree with everything she said or do you have a different idea? How would you have answered this question?
- Imagine Gracie is your friend. What would your answers to her questions be? Which of her questions would you ask her more questions about? What would you ask her? How would you engage with her ideas in a supportive way?
- What’s something mathematical that you have encountered in your life that had nothing to do with the math you learned in school? Was there a problem you had to solve on your own? Did you have to look up how to solve it or did you figure out a way to solve it yourself? Did you use algebra without realizing you used algebra? How do you know what you did counts as math instead of as something that’s not math?
- Look up the history of a mathematical fact, formula, or idea. What problem were people trying to solve when they discovered it? How did it solve the problem for them? How did they know they were right and how did they use it? What is the story of this fact, formula, or idea? (I’m not sure how reliable it is, but the website the Story of Mathematics might be a good place to start).
- Make your own video (on Tik Tok or anywhere else that’s public) asking your own questions about the history and philosophy of math.
- In a comment below, respond to the prompt you chose in at least 5 sentences. Make sure to tell us which prompt you chose so we know what you’re responding to! If you are making your own video, include a link to it in your comment.
You will receive participation credit for your comment.
Euler’s formula creates a connection between exponential functions and trigonometry. This formula originated from Euler’s investigation of the Basel problem in 1734, in which he found a closed form for the infinite sum of inverse squares. Euler’s contemporaries acknowledged the formula through mathematical proofs. The impact of this equation was huge, particularly on fields such as physics and engineering, where it simplifies many complex mathematical concepts. The formula finds extensive application in diverse domains, including signal processing, quantum mechanics, and electrical engineering, making it one of the most useful and renowned formulas in both physics and mathematics.
https://medium.com/swlh/eulers-identity-990b52f0d8fc
e. The Pythagorean Theorem, credited to ancient Greek mathematician Pythagoras, addresses the relationships within right-angled triangles. It originated from the quest to establish a link between their side lengths. The theorem posits that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Initially proven geometrically, subsequent algebraic validations solidified its acceptance. Practical applications in architecture and navigation highlighted its utility. Rigorous proofs and empirical testing cemented confidence in its accuracy, and today, the theorem remains a foundational concept in geometry with broad applications across disciplines.
Prompt 1:
One thing I wonder about is consciousness. What exactly is it, and how does it happen in our brains? It’s a question that has puzzled philosophers, scientists, and thinkers for centuries. We don’t have a clear answer yet, but there are different ideas. Some think it comes from how our brain cells interact, while others think it’s more mystical. Exploring this question leads to fascinating discussions about neuroscience, the philosophy of mind, and even artificial intelligence. What gets me about this question isn’t just how hard it is, but also how much it matters. It makes us think deeply about who we are and how we fit into the world. It’s a topic that keeps us guessing, arguing, and learning, and that’s super interesting to me.
Prompt 1:
One time when I doubled if math’s real is when I learned about imaginary numbers and complex numbers. What I learned is that imaginary numbers are the square root of negative numbers, the basic unit “i” is the square root of -1. The complex numbers are the addition of real numbers and imaginary numbers, such as (1+i) and (2-3i). It also mentioned the complex plane, which is a Cartesian coordinate system with all the real numbers at the x-axis, all the imaginary numbers at the y-axis, and all the complex numbers are away from x-axis and y-axis. That brought me so many questions: is imaginary numbers exist, since they are not “real numbers”? How they prove the complex plane is correct, like are real numbers and imaginary numbers really perpendicular to each other? why the square root of -1 is on top of 0 instead of 1 or -1? They created new theory so that the square root of negative numbers exist, will they going to create theories to make a number divided by 0 exist?
c. Under Plato’s philosophy, math belongs to an unchanging, eternal truth; it exists independently of the changing world. Even if we had not invented math (for example numerical notations, formulas, etc.), 1+1 would equal 2 because it’s a universal truth. One can believe 1+1=3, but it does not change math. So nobody invented math, humans just came up with ways to write it down and further make use of it. Some theories or formulas we think are right can prove to be wrong (Newtonian Physics vs Einstein Physics). So, to see if the formulas or theories are right, we can look at our universe and recognize patterns.
When I look at the Pythagorean Theorem, what really stands out to me is how long it’s been around and how we’re still using it today. It was this guy, Pythagoras, who figured out this rule for right-angled triangles a really long time ago. Basically, he found out that if you take the longest side of the triangle and square it, it’s the same as if you square the other two sides and add them up. People first proved this with shapes, and then later with numbers and formulas, which just made everyone even more sure it was right. What’s cool is that people started using this in real life, like when they were building things or finding their way around. Reading through Eugenia Cheng’s blog, she gets how math isn’t just numbers and rules; it’s a way of solving problems and seeing the world that’s been with us for ages. The Pythagorean Theorem isn’t just a math thing; it’s a human thing that shows how smart we can be and how we use that smarts to do awesome stuff.
Prompt e.
In ancient Greece, mathematicians were primarily concerned with geometric constructions using straightedges and compasses. One such problem that puzzled mathematicians was the impossibility of constructing a square with the same area as a given circle using only these tools. This problem led to attempts to express the ratio of the circumference of a circle to its diameter, which we now know as pi (π), as a rational number. The ancient Greek mathematician Hippasus of Metapontum is often credited with the discovery of irrational numbers. He supposedly proved that the square root of 2 is irrational, which contradicted the Pythagorean belief that all numbers could be expressed as ratios of integers. Mathematicians later developed more rigorous proofs of the irrationality of certain numbers, such as the square root of 2, using techniques from number theory and algebra.
Q: Which of Dr. Cheng’s answers is the most interesting to you? Why? Did you agree with everything she said or do you have a different idea? How would you have answered this question?
A:
I found Dr. Cheng’s answer to Gracie’s first question (How did people know what they were looking for when they started theorising about formulas because I wouldn’t know what to look for?) the most interesting because it is really an insightful question which is something that I have personally joked about, but never really gave it much thought. I found Dr. Cheng’s answer so interesting because when you put the answer in perspective it seems very simple and straight forward, like something you could have come up with yourself. However, in reality answers to questions like these are easy to understand, but difficult to find. It makes me wonder how many of the difficult questions that we do not have an answer have answers as straight forward as, “spotting patterns and seeings things in common between different situations and wondering why that is going on”. It also makes me wonder about how many different “difficult” questions and problems humans can solve if we only change our perspectives about situations and let our knowledge be fluid and keep our minds open to a variety of possibilities.
Prompt a:
I’ve always been curious to how math begun. Like Gracie, I questioned how someone came up with theories from scratch and how others have continued those theories with. Certain topics I can understand how the clock was invented because the physical shadows the sun creates as it rotates. But I cannot try to understand the origins of the string theory. How could they have made a theory about particles when you can’t see them, if that makes sense?
I’ve never really tried to answer any of these, mainly because it just hurts my brain to think about but this prompt has really made me curious to find out what I can understand and what will remain questionable to me.
A) I always wonder why math works so well for us. Like, we as humans are the ones who made numbers so how is it that there are so many unique patterns, formulas, and other things we can make with it, and numbers are a universal language. How did we make all 10 fingers from our hands into numbers we can use for prices, time, measurement, and everything else. It just seems so, weird to me. I also wonder if math is real since it’s a made-up concept and yet it is so solidified in our society. The way pi, calculus, algebra, trig, Pythagorean theorem, matrices, and the way all those other math concepts work will always feel surreal to me, and that’s why I think math is very cool!
I have had to ask myself if “math” or specific math branches are real. One is Calculus and how Newtown created it in the 17th century. It’s astounding to think about how Newton, in the 17th century, developed an entirely new branch of mathematics just because he was trying to understand the laws of motion and universal gravitation, which required him to calculate continually changing quantities. Thinking about changing something that is ever-changing into something measurable and constant is just foreign. It is an alien language, even after having seen Calculus 2. I believe it is because I still wrestle with the fact that this is not something created recently but 5 centuries ago, which echoes what Gracie argues.