What Is That, a Bossa Nova Beat?
Posted: 01 April 2021
Imported from substack so the formatting is messed up, it looks better on substack.
I’m going to try to write less about music, because after some thinking about what I like to read, it’s not music reviews. It’s never been music reviews. The only good music review is Lou Reed’s review of Yeezus: [link]. Mine have been more of an exercise in writing, generally, than in writing something I think anybody wants to read, so I’m cutting back.
———————
Index
A * means I can see myself listening to the album again at some point in the future (for fun). Everything else can be anywhere between great and not so great. Music is more condensed and less of a focus now.
Art
Portraits I Like pt. 2
Epistemology
What Makes a Proof Valid?: On Mochizuki and the ABC Conjecture
Music
Remo Drive (Emo)
Foreigner (Arena Rock)
Algernon Cadwallader (Emo)
Father John Misty (Indie Folk)
Hawkwind (Psych Rock)
* Electric Wizard (Stoner Metal)
* Tenue (Screamo)
Anzo (Electronica)
Art
———————
Portraits I Like pt. 2
Here are some more figurative paintings/portraits I like by two contemporary painters. These have some similarity in their minimal expression and realist approach, and are in a style I like but am incapable of replicating.
I’m compelled by Mark Tennant’s work. It looks to me like he works from photos primarily (I didn’t think about this too much looking at his work on instagram, but it’s very obvious looking at his work on his website), but he does a really really interesting job flattening everything in them out and removing details to bring the figures into full prominence. I don’t think I would ever be able to paint like this, but there’s something very human and engaging about his work, despite, or maybe because of, anything identifying being pushed out of the way.
Another artist, Kirsten Valentine, also flattens a lot in her figurative paintings, but does a bit more sculpting and has more fleshed out figures. I am interested in her figures that look like they’re taken out of photos floating around in decontextualized space.
And to round us out for today, Phil Hale, whose paintings look like ghost possessions in progress. It’s incredible work with enchanting colors and movement.
Math
———————
What Makes a Proof Valid?: On Mochizuki and the ABC Conjecture
What makes something true? Some combination of observation, belief, and consensus. Everyone has their own individual notion of truth, personal ratios of these three things thrown in a big witches pot and stewed with cattails and newts and frog eyes. It’s a tremendous question. We find a definition of truth we’re comfortable with and go on with our lives. Questioning reality tends to lead down a neverending windy road.
Generally, in the sciences, we conduct experiments to find out something close to the truth. We start with a hypothesis, we test it, observe and record our test, and figure out what our observations show about our hypothesis. This is pretty straightforward. How about math? What is a math proof? What is a math theorem? How do we know that they’re true? You can’t do experiments on math.
I’m going to sidestep the larger, unanswerable question of “what is math” and just deal with how math theorems are verified. A math theorem is a mathematical statement that can be proven true by taking commonly accepted axioms (a.k.a. starting points. For example, if x=y, then y=x. You can’t really prove this, it’s just something that has to be true based on how we define and use math) and performing various transformations to demonstrate, in one way or another, that some final mathematical statement either must or cannot be true. What is the purpose of this, if you can just derive the resulting theorems yourself from the base axioms? Why do we need theorems? Well, they serve as a shorthand and allow the building of larger and more complicated mathematical concepts and structures, which find use in all sorts of fields from logistics to robotics to astrophysics.
The catch is, a lot of conjectures in mathematics are very difficult to prove, and many people spend a lot of time and effort finding a way to combine these axioms and other accepted theories to create new proofs. Let me emphasize: very difficult. Andrew Wiles spent seven years and won awards in an undertaking many failed at, proving Fermat’s Last Theorem, which reads:
There are no three distinct positive integers a, b, or c such that
Is true for any integer n > 2.
So, there are no three distinct positive integers for a, b, or c where, say, a^3 + b^3 = c^3.
Very cool! But ok, so a math theorem takes a bunch of axioms, things that we accept as true to be able to do any math at all, and other theorems that are built up from these axioms, in order to demonstrate new mathematical concepts, like Fermat’s Last Theorem above. It’s going from point a to point b with all of the intermediate steps laid out. If we accept some definition of math as a starting point, and all the steps are correct, the final theorem must be capital T True. To verify it, you would just need to follow along with the proof and check for any mistakes in the math. Anybody can do that, right?
Well, let me tell you, modern math is wild. Check out the conclusion of part 2 of Wiles’ proof of Fermat’s Last Theorem:
If the geometric Galois representation ρ(E,p) of a semi-stable elliptic curve E is irreducible and modular (for some prime number p > 2), then subject to some technical conditions, E is modular.
Do you know what a Galois representation of a semi-stable elliptic curve is? I don’t. So we find ourselves in the position where we have to rely on people who have a background in particular fields of mathematics, and understand things like Galois representations, to verify a proof and let us know if something is True or not.
But what happens if this isn’t possible? What happens when a well-esteemed mathematician publishes a 500 page proof that nobody seems to understand?
Enter Shinichi Mochizuki, an award-winning mathematician who, in 2012, published a paper claiming to prove the ABC conjecture. To accomplish this task, he has invented an entirely new field of mathematics called “Inter-universal Teichmüller theory.” He has refused to go on tour, as is the norm in math, to give talks at Universities and explain his theory. He has deflected almost all criticisms by saying that his critics are overlooking or do not understand basic parts of his theory, without any further elaboration. An inner circle of mathematicians he is close with have slowly been voicing that they believe the proof is valid, but they also are not helping explain any of the parts that other mathematicians don’t understand. There are [a] [number] [of] [articles] on the topic that I think are very interesting, and get into the details better than I can. For what it’s worth, I don’t understand the ABC conjecture or number theory at all.
But we’re left in a situation where a small number of mathematicians claim to understand the proof, and thus accept the ABC conjecture as a theorem and as true, and a large number don’t, and for them the ABC conjecture is still a conjecture. They don’t know if it’s true or not. The argument isn’t about whether the ABC conjecture holds, it’s an argument about whether the instructions to prove the ABC conjecture are correct. And a lot of people don’t know because they can’t understand them.[2] I think there’s something really fascinating about that.
There’s a Borges story about a society that has existed for so long, and has such a deep well of writing and knowledge, that people devote their entire lives to studying what has already been written. Only at the very end of their lives, after learning everything there is to learn, do they have any hope of trying to use their knowledge to make new discoveries or pursue new research.
Mochizuki’s proof seems like a real life, microcosmic example of this fictional society. What does truth mean when it requires a lifetime of study to understand? What happens when nobody but the writer of a proof has enough background knowledge to be able to understand it? We’re stuck in a situation where we can only hope that the progenitor will try to make their explanation simpler.
Will there be a day where it becomes impossible to verify any further advancements in any art or science? Where the knowledge being built on becomes so dense it requires a lifetime of study? Mochizuki’s proof shows some of these cracks. Truth might have very human limitations.
Music
———————
Remo Drive - Greatest Hits (Emo)
Very tricky of Remo Drive to title their debut album Greatest Hits. “Art School” has some cool Death From Above (who I have learned are quite problematic,,) vibes, “Yer Killin’ Me” is fun, outside of those songs I don’t love their brand of indie-rockish emo.
Foreigner - Double Vision (Arena Rock)
The eponymous track “Double Vision” rules but otherwise it’s arena rock I’ve heard 10000 times on 106.7 Light FM and in every single 90’s coming of age movie. Not interested.
Algernon Cadwallader - Parrot Flies (Emo)
It’s 4th wave emo, alright.
Father John Misty - Fear Fun (Indie Folk? I guess?)
Did he change the band’s name recently? Was it Father Saint John Misty before? Why do I think that. Regardless, I really like “Hollywood Forever Cemetery Sings,” and it’s a fine album outside of that.
Hawkwind - Space Ritual (Psych Rock)
I like this! I like psych rock. It sounds a little rough around the edges but I’ll take it.
* Electric Wizard - Dopethrone (Stoner Metal)
This album inspired me to buy a fuzz pedal a few years ago. Every riff in “Funeralopolis” is embedded in my brain. Outstanding stoner metal. “Mind Transferal” is such a monumentous track to end on. Love it. I listen to this album all the time. I pop it on often when I can’t decide what to listen to.
* Tenue - Territorios (Screamo)
I wasn’t sure about this but after a few minutes I was hooked. Outstanding new screamo out of the crust punk scene in Spain.
Anzo - Moonbound (Electronica)
Beep boops for your workday background music. Beep boop.
———————
[1]
Some stuff about the scientific method I cut from the math essay:
What makes something true? I don’t think it’s possible to answer with any certainty. Some combination of observation, belief, and consensus. Everyone has their own individual notion of truth combined from these three things, give or take, thrown in a big witches pot and stewed with cattails and newts and frog eyes. It’s a tremendous question. We find a definition of truth we’re comfortable with and go on with our lives. Questioning reality tends to lead down a neverending windy road.
As an inquisitive species, we have come up with systems for cataloguing truths as they are learned or discovered or invented. The sciences use the scientific method to put forth hypotheses, gather data, and report on their observations. A common high school physics experiment involves calculating the expected acceleration due to gravity on Earth (~9.8 meters per second squared), dropping an object from some set height and measuring how long it takes to fall, and validating that, yes, the object accelerates at a rate of roughly 9.8 meters per second squared. It is a simple experiment, but it lays out a foundation for a process used in pretty much all of the sciences.
If somebody does an experiment and reports on their hypothesis, it moves us in a direction that looks something like truth. This isn’t a flawless process, however! It’s possible for people to fake data or for results to be unreliable or have hidden confounding factors, so there is generally a review these studies and experiemtns undergo that add scrutiny and check for errors and mistakes. Peer review isn’t foolproof, and problematic studies and experiments occasionally squeak by, but it’s an added layer of security to make sure that the original hypothesis and the conclusion of it’s validity are, in as much as they can be, true.
For simpler experiments like measuring the acceleration of gravity on Earth, the results are really straightforward and easy to accept. Anybody could recreate it, and you don’t need much specialized knowledge to understand, perform, or verify the experiment yourself. This gets tricker as experiments become more arcane. Here are the title and abstract of the most recent molecular biology study I could find in Nature [link]:
Inefficient splicing curbs noncoding RNA transcription
Pervasive genome-wide transcription initiation by RNA polymerase II (Pol II) necessitates mechanisms that restrain the quantity and length of the transcripts. A new study investigates a mechanism for inducing early transcription termination, employed primarily at genomic regions producing noncoding RNAs.
I have a rudimentary understanding of what RNA is, but there is no chance that I or any other layperson would be able to validate this study in any way. We can either reject or accept it for reasons other than its observations, or we can surrender some of our understanding of truth to the scientists conducting the study and other qualified scientists in the field reviewing it. We accept the process they use to conduct their experiment, we accept the qualifications of their peers conducting a review, and we accept that their results seem to be true.
So truth becomes a little different in topics we don’t ourselves understand: we have to rely on other people, who have qualifications that we accept, to verify whether an experiment or study seems correct.
I know I’m being long-winded here, but hopefully you will bear with me for a little longer.
[[ end of stuff I cut from the essay ]]
I wrote all of the above and then got rid of it because I felt like it was just too much background. I still feel like I wrote too much preamble, the real meat of all of this is Mochizuki’s mysterious proof that nobody understands, but I feel like truth and math proofs need more context. I recurse into the same well of requiring background knowledge to explain things that the proof does.
[2]
It actually goes a little further than that: After 8 years in limbo, the paper was accepted for publication in 2020. Publication is usually the last step a theorem goes through after it has been accepted by the mathematical community. So there’s a large community of mathematicians who don’t understand the proof, a small community who think they do and reject it, a small community that think the proof holds, and that the people who think it has problems are fools, and there’s going to be an official publication of the proof, ipso facto making the ABC conjecture a theorem, despite multiple people disagreeing. It’s a hot mess.
———————
That’s it for this week. Thanks for reading, leave some feedback, tell your friends.