sexuallyfrustratedfowlandshort:
This is beautiful, and I wish to understand it more.
Can I just minor in math? Because look at this gorgeous thing. Oh my God.
A collection of random borderline-mediocre insights inspired by the ordinary everyday world.
-
-
It has been proven…
How many times have you gotten into a debate with someone on a topic that may obviously have no actual answer (e.g. a philosophical question, or simply a vague one), and one side smacks down the words, “well, it’s been scientifically proven that…”
With emphasis, NO. There is something deeply, perhaps morally wrong saying that. Why? Merely because of the limitations of science. See, science (and I will make the distinction henceforth between science and mathematics) is an almost purely empirical subject. One observes the physical universe, hypothesizes an explanation for a phenomenon and then attempts to rigorously test it via a series of experiments to hold against the validity of the claim. The more success an experiment has, the more evidence that is now acquired to support such the hypothesis. Science is powerful!
However, therein lies the limit. Science cannot prove— it merely entirely convincing. The scientific method gives us the power to create better successive approximations to what is happening in nature, a sort of fine-tuning precision that allows us to understand more and more of the world around us. It does not give us absolute, universal truth though, merely because no matter how much evidence you have on a subject, as long as you haven’t proved it it cannot be absolute. For example, we have loads of evidence supporting evolutionary theory, and indeed it does give us the most powerful interpretation of the diversity of life on the planet, but it is not proven. We have not scientifically proven it. Because we simply cannot. Science does not have that power.
What of mathematics? What do you say is absolute truth in mathematics then? The unique aspect of mathematics that other scientific fields do not enjoy is the power of proof. Mathematicians can prove statements using formal logical systems and a system of assumed axioms, giving rise to our theorems, universal truths— it is difficult to argue with a mathematical statement that has been proven, as any attack on it will be an attack on logic itself.
To illustrate the power of a proof, we will consider Euclid’s classic proof on the infinitude of the primes. Recall that a prime number \(p\) is a number with its only divisors being \(1\) and itself. The first few primes are \[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, …\]
How do you know there’s an infinite amount of them? We have evidence to suggest that there’s a lot of them— we have computers currently finding larger and larger primes, the largest one currently millions of digits long. However, even with such vast numbers, they are finite. And any given number, no matter how many digits long, is such an immense distance away from the infinite. However, we can prove it.
Theorem: (Euclid) There are an infinite number of primes.
Proof) Suppose not, and we have a list of all of them, \({2, 3, 5, … , p_n}\). Consider the number \[N= 2\cdot3\cdot5\cdot\cdot\cdot p_n + 1\]. Then none of our primes divide \(N\), which is impossible since every number can be divided by a prime number. So there must be some prime that divides it, which we don’t have in our list! However, we claimed to have all of them… that must have been a wrong assumption. Hence, we can never have all of them in a finite list, and so there are an infinite number of primes.
Mathematics allows us to express the infinite, and prove things about it, giving it a sense of certainty and structure. There is something seemingly magical about proof— in a way, it is philosophically remarkable that our system of logic and construe something as complex as the mathematical universe, and yet still be consistent even as mathematics continues to be built and researched every day. It is a limitless science, a statement the other sciences would be hard-pressed to say.
-
-
Selected work by Andy Gilmore.
About his work:
A master of color and geometric composition, Andy Gilmore’s work is often characterized as kaleidoscopic and hypnotic, though it could just as well be described as visually acoustic, his often complex arrangements referencing the scales and melodies in music.
More of the awesomeness that is Andy Gilmore.
-
The silence is deafening.
I’m not really sure why it happened today as opposed to other days, but one way or another, I woke up with an incredible feeling of depression. It’s difficult to explain— there were no tears, no bitterness or spite. Just a feeling of loneliness and isolation.
Why though? I have my closest friends a flight of stairs away— hell, they even invited me to go food shopping with them. But today I couldn’t find that energy, that zeal I thought I always had. Instead, I holed myself up inside the library and drowned myself in mathematics— hours upon hours of trying to learn algebraic geometry, of understanding the semidirect product \(H\rtimes K\) and split exact sequences,
\[0 \longrightarrow H \longrightarrow G \longrightarrow J \longrightarrow 0\]
watching Edward Witten’s most recent IAS lecture and trying my hand on understanding the RSA cryptosystem. It’s in math that I feel the most comfortable, perhaps safe in the little hermetic bubble in my head. Part of me wants to scream it out to the world, to share and revel with like-minded people and have deep discussions about the nature of mathematics and the crude approximation is has of the real world.
But the silence is deafening. It may be stupid and irrational to get sad over a subject, but I’m depressed today because there is no one who can hear those screams. Ever since my best friend graduated, I have found no one to talk about math with— to confide a secret love for a theorem, shared appreciation for a proof, debate over the validity of unsolved conjectures. Sure, there are those who will listen— but beyond what they need to hear for their courses or for the passing grade, they turn deaf. Why do I feel like I’m the only one here who cares?
It’s dumb, yes, but I can’t help it. I love math (or maff, whatever you want to call it). It’s beautiful, oh god it’s beautiful. To hold the universe in an equation, to express the dynamics of so much in a single theorem— how could my colleagues turn away? Whenever someone asks with a tinge of scorn, why I get so worked up over math, I just die a little bit each day. Why do people spend hours with their loved ones, fixated on a game or a wonderful movie? Because they want to— they love to.
Every step into the mathematics department these days seems more and more like a lonely journey. Talking to my thesis advisor is probably what I look forward to the most each week— those random asides into a different field, a new problem he’s working on, just makes me feel like my interests are validated, as if they are accepted. But beyond it, any chance to talk to a fellow senior about even anything more than their homework is lethal treading ground.
I’m tired. Everyday I’m more and more fearful of never getting into graduate school, of never meeting what I believe are people like me, those quirky people who only want to talk math and their adoration of it all. Perhaps that’s it— maybe my usual paranoia is kicking in the form of a strong depression. I know my friends will always be there for me— that hasn’t been ripped out of me. But still, no one here will hear the excitement I feel every time I read a proof, prove a theorem, solve a problem.
And sometimes, it can make me feel a bit lonely.
-
-
View high resolution
One of the things that really got me into mathematics was a conversation with my friend Justin about prime numbers, back in middle school. Despite the random structure that comes with being the irreducible elements of \(\mathbb{Z}\) (or more generally, the prime ideals of the ring), they remarkably have a hidden pattern— a sort of emergent complexity coming out of looking at the “limiting lens”.
Above is a quadratic spiral with the integers marked off at periodic ticks. The dots shown in the picture are the prime numbers, and immediately (even if you don’t know much about mathematics) one can see patterns in the spiral. It’s as if the primes lie on parabolic curves, opening and stretching into one direction, a sort of regular periodicity forming amidst the randomness.
…Kinda makes you fall in love with math, eh?
primes on a quadratic spiral
-
Typing notes in class… is kinda hard.
Especially when it’s all in LaTeX, and the equations are huge. But hey, PDEs don’t write themselves, right? Also, FOURIER SERIES.
\[\int_{-L}^x f(t) dt = \int_{-L}^x a_0 dt + \sum_{n=1}^\infty a_n \int_{-L}^x \cos(\frac{n\pi t}{L}) dt + \sum_{n=1}^\infty b_n \int_{-L}^x \sin(\frac{n\pi t}{L}) dt\]
-
-
My life. All day, everyday.
Theme: Linear by Peter Vidani
