# hashtag math isn't real

August 29, 2020 11:22 AM Subscribe

Earlier this week, teenager Gracie Cunningham posted a minute long TikTok asking questions like "how do mathematicians know where to look for formulas?" and "how do they know that their formulas are right?" After being roundly mocked for a day or so by trolls both anonymous and famous, she posted a new TikTok, clarifying her original questions and adding a final one: "Why are the only people disagreeing with me the ones who are dumb, while the physicists and mathematicians are agreeing with me?"

I used to be a middle/high school math tutor and I'd love to have a student as inquisitive as she is. I think history and philosophy of science is so rarely and poorly taught even at the college level.

Or maybe I'm just sensitive because I ask "dumb" questions all the time even though I'm 36 and have a bachelors of science degree...

posted by muddgirl at 11:30 AM on August 29, 2020 [21 favorites]

Or maybe I'm just sensitive because I ask "dumb" questions all the time even though I'm 36 and have a bachelors of science degree...

posted by muddgirl at 11:30 AM on August 29, 2020 [21 favorites]

Like let's get *in* to the golden age of Islam! But maybe not in a way that blames a high school girl for not knowing about ongoing European erasure of Islamic culture...

posted by muddgirl at 11:32 AM on August 29, 2020 [18 favorites]

posted by muddgirl at 11:32 AM on August 29, 2020 [18 favorites]

Because non pure math heavy disciplines think the answer is trivial

posted by polymodus at 11:35 AM on August 29, 2020 [6 favorites]

posted by polymodus at 11:35 AM on August 29, 2020 [6 favorites]

“Where Mathematics Comes From” was published in 2000, and was an attempt at answering pretty much the questions she’s asking, written by some extremely smart, dedicated people and critiqued in various ways their equally smart, committed peers. For me, at least, that book was a really heavy lift. The questions she’s asking are not easy questions at all, and a lot of powerhouse thinkers have been working on them for a long time without uniform or elegant success.

posted by mhoye at 11:41 AM on August 29, 2020 [30 favorites]

posted by mhoye at 11:41 AM on August 29, 2020 [30 favorites]

I got a C minus in Math 101 my freshman year in college, in 1988, and I haven't engaged meaningfully with math since then, other than the normal daily checking-account-balancing and bill-paying stuff. The questions she asked fascinate me, because I have literally no idea how math works or where it comes from, and I'm glad she asked them and it's gross that she's taking all kinds of crap for doing so.

posted by pdb at 11:42 AM on August 29, 2020 [14 favorites]

posted by pdb at 11:42 AM on August 29, 2020 [14 favorites]

Combined curriculum world history and math though, yes. We should have that.

posted by mhoye at 11:42 AM on August 29, 2020 [4 favorites]

posted by mhoye at 11:42 AM on August 29, 2020 [4 favorites]

Math is hard, let's go chopping.

posted by flabdablet at 11:43 AM on August 29, 2020

posted by flabdablet at 11:43 AM on August 29, 2020

I think when an American teenage girl says something people interpret it differently to the same words coming from a British man in his thirties.

Also damn, the courage and strength of that second video. To film the same material again and be able to laugh off all the misogynist bullshit that was tossed her way.

posted by Nelson at 11:49 AM on August 29, 2020 [75 favorites]

Also damn, the courage and strength of that second video. To film the same material again and be able to laugh off all the misogynist bullshit that was tossed her way.

posted by Nelson at 11:49 AM on August 29, 2020 [75 favorites]

Not too long ago I was at my very first job after getting my shiny engineering degrees, having lunch with my coworkers, a couple of whom have math PhDs, and I just blurted out, "What do mathematicians

posted by btfreek at 11:49 AM on August 29, 2020 [9 favorites]

*do? How do you math?*" and the answer there seemed to be "drink a lot of coffee and grouch about politics". I am glad I have these answers now too.posted by btfreek at 11:49 AM on August 29, 2020 [9 favorites]

How dare this youth bring the humanities into STEM?

posted by NoMich at 11:52 AM on August 29, 2020 [25 favorites]

posted by NoMich at 11:52 AM on August 29, 2020 [25 favorites]

An old boyfriend told me once about a college course he took called "Math For Poets", where it was more about the history and development of math as opposed to being about how to prove theorems. He said it was fascinating, and I actually wish I'd had something like that.

posted by EmpressCallipygos at 12:01 PM on August 29, 2020 [16 favorites]

posted by EmpressCallipygos at 12:01 PM on August 29, 2020 [16 favorites]

*I have literally no idea how math works or how it comes from*

A lot of people get taught in school how to manipulate numbers, and also get told that this set of skills is called Math, and form the impression that

*numbers*are the thing that math is about. But that's not even

*close*to true. Math is

*way*bigger than just numbers.

When it comes right down to it, what math is about is every way to apply logic to the creation, generalization and analysis of

*patterns*.

*Any kind*of pattern. Even patterns that are only perceptible once you're looking deep inside

*other*patterns; perhaps even inside patterns that you made yourself via creative application of sound logical rules to some interesting starting point to find out where they took you.

If thing A is like thing B to an extent that's useful to know about, mathematics will more than likely be able to provide ways to describe the ways in which that vague idea of "likeness" can be nailed down and made more precisely communicable.

Numbers are just an example of this. Here's thing A:

X X

and here's thing B:

😺

😺

and here's thing C:

*

*

and one way that thing A is like thing B and thing C is that each of them is made of similar pieces; and when we care more about what we can

*do*with those pieces than what those pieces

*are*, we can summarize

*all*of these patterns as instances of the number 2.

posted by flabdablet at 12:02 PM on August 29, 2020 [53 favorites]

the only bummer about the second video is that, in the first, asking these utterly legit metaquestions while working on her makeup is such a [chefkiss] perfect way to drag her detractors before they've even opened their mouths.

posted by qbject at 12:11 PM on August 29, 2020 [17 favorites]

posted by qbject at 12:11 PM on August 29, 2020 [17 favorites]

Anyone who makes fun of others for critically thinking about why we think what we think has much greater problems than math phobia.

Faith in science or math is no better (or worse) than faith in invisible sky fairies. Prove it to yourself and ideally others, or it can only ever be faith. And faith is not sufficient for progress.

posted by apathy at 12:12 PM on August 29, 2020 [4 favorites]

Faith in science or math is no better (or worse) than faith in invisible sky fairies. Prove it to yourself and ideally others, or it can only ever be faith. And faith is not sufficient for progress.

posted by apathy at 12:12 PM on August 29, 2020 [4 favorites]

A lot of males identify as naturally good at math in the same way they identify as skilled drivers, superior athletes, or talented comedians. People have told them over and over that they’re inherently brilliant at those things by virtue of being male. Then you watch them try to calculate a 20% tip, apply brakes in a rainstorm, finish a road race they didn’t bother to train for, or give a wedding toast, and it’s so damn painful.

Those kinds of guys, and perhaps the desperate self-hating women who try to impress them, were the trolls. Guaranteed. I’m glad she called that out.

posted by armeowda at 12:13 PM on August 29, 2020 [51 favorites]

Those kinds of guys, and perhaps the desperate self-hating women who try to impress them, were the trolls. Guaranteed. I’m glad she called that out.

posted by armeowda at 12:13 PM on August 29, 2020 [51 favorites]

*Then you watch them try to calculate a 20% tip, apply brakes in a rainstorm, finish a road race they didn’t bother to train for, or give a wedding toast, and it’s so damn painful.*

To be fair, watching them fail to do

*all those things at once*can get pretty damn amusing.

posted by flabdablet at 12:15 PM on August 29, 2020 [15 favorites]

the profoundly stupid part of this story is that somehow the whole thing spiraled into her getting accused of being a holocaust denier and having to post an apology video. (i think because she also expressed some kind of philosophical skepticism about historical as well as mathematical knowledge, which people spun in the worst possible way to justify ganging up on a teenage girl.)

posted by vogon_poet at 12:17 PM on August 29, 2020 [2 favorites]

posted by vogon_poet at 12:17 PM on August 29, 2020 [2 favorites]

*desperate self-hating women*

84.6% of women hold at least one bias against women. Everything is terrible.

posted by aramaic at 12:19 PM on August 29, 2020 [5 favorites]

*“Where Mathematics Comes From” was published in 2000, and was an attempt at answering pretty much the questions she’s asking, written by some extremely smart, dedicated people and critiqued in various ways their equally smart, committed peers. For me, at least, that book was a really heavy lift. The questions she’s asking are not easy questions at all, and a lot of powerhouse thinkers have been working on them for a long time without uniform or elegant success.*

Jordan Ellenberg's "How Not to Be Wrong" is a very approachable book that illustrates mathematical concepts that the average layperson would likely be unfamiliar with, in a very engaging way, while also giving a history of how these questions and concepts came about. I highly recommend it (

*nb: my wife is a mathematician and friends with Ellenberg.*)

posted by Navelgazer at 12:29 PM on August 29, 2020 [17 favorites]

Agreed. Everything is terrible but that teenage girl was able to respond with grace to a very challenging situation. Good for her and good for the smart and kind people who came to her defense.

posted by Bella Donna at 12:29 PM on August 29, 2020 [8 favorites]

posted by Bella Donna at 12:29 PM on August 29, 2020 [8 favorites]

Now that you mention it, I don't know what the difference is between algebra and what the ancient Greek mathematicians were doing.

posted by Nancy Lebovitz at 12:42 PM on August 29, 2020

posted by Nancy Lebovitz at 12:42 PM on August 29, 2020

Parents, don't let your children have unrestricted social media, nothing good can come from it.

(And because someone is sure to misinterpret: I am not in any way, shape or form blaming her for the idiocy that followed her post)

posted by madajb at 12:48 PM on August 29, 2020

(And because someone is sure to misinterpret: I am not in any way, shape or form blaming her for the idiocy that followed her post)

posted by madajb at 12:48 PM on August 29, 2020

This topic is very near and dear to my heart as an educator, and often heart breaking as I hear women (and men to a lesser extent) say things like, "I was very good at math until high school (or college)."

Math is soooo poorly taught, for the most part, especially in later grades.

posted by haiku warrior at 1:02 PM on August 29, 2020 [11 favorites]

Math is soooo poorly taught, for the most part, especially in later grades.

posted by haiku warrior at 1:02 PM on August 29, 2020 [11 favorites]

*84.6% of women hold at least one bias against women. Everything is terrible.*

Well, culture from every angle has repeatedly told us we only have value if we’re the only one in the world.

I always hesitate to bring it up, though, because it feels like once again we’re putting all the responsibility to fix it on the people it hurts most. We wouldn’t internalize misogyny if the more-powerful half of the population didn’t accept it as currency.

I hope the TikTok generation is smarter about this than mine has been, let alone those who came before us.

posted by armeowda at 1:05 PM on August 29, 2020 [4 favorites]

Now, pick your favorite divisive political issue, and imagine someone on the other side of it saying, "I have no idea why anyone would think differently." Sometimes they are intelligent people who just haven't ever had a good explanation.

Actually, I believe it's more than sometimes.

posted by amtho at 1:08 PM on August 29, 2020 [1 favorite]

Actually, I believe it's more than sometimes.

posted by amtho at 1:08 PM on August 29, 2020 [1 favorite]

Hello, I work in a math-heavy field. I have a physics PhD. I have done a certain amount of math teaching. I would

I also found it intensely sad that she can quote (eg) "Y=mX+b" off the top of her head but can't connect it to the concept of linear relationship, when surely the priorities of a mathematical education should be exactly the reverse order. Otherwise it's like, I dunno, being taught to read music but never being allowed to listen to it.

posted by doop at 1:09 PM on August 29, 2020 [46 favorites]

*crawl over broken glass*to have students asking questions like this.I also found it intensely sad that she can quote (eg) "Y=mX+b" off the top of her head but can't connect it to the concept of linear relationship, when surely the priorities of a mathematical education should be exactly the reverse order. Otherwise it's like, I dunno, being taught to read music but never being allowed to listen to it.

posted by doop at 1:09 PM on August 29, 2020 [46 favorites]

*But that's not even close to true. Math is way bigger than just numbers.*

My middle-school math teacher said Mathematics is the Queen of Science.

posted by Rash at 1:14 PM on August 29, 2020 [1 favorite]

Your middle school math teacher was biting Gauss, who said "Mathematics is the queen of the sciences and number theory is the queen of mathematics."

I studied math in college, and I loved it but, it's

In college is where the focus became proofs and building things up from first principles. Tests stopped being "evaluate these 30 integrals" and became 5 questions that all began with "prove or disprove:" which is a really good way to force you to think but is also much harder. I don't for a second regret majoring in it but it's daunting and I imagine most people would rather do the 30 integrals. There's a reason why the Putnam Exam has an average score in the single digits out of 120.

posted by axiom at 1:35 PM on August 29, 2020 [15 favorites]

I studied math in college, and I loved it but, it's

*hard*(not helped by being forced to take the aforementioned number theory class at 8:30am). I had taken AP Calc and Physics in high school, and what advanced math taught me is that those other subjects are taught in a fairly backward way. You do a lot of turn-crank rote mechanical stuff without really understanding why. I could solve a bunch of integral problems, say, but couldn't connect that process to the reasons why anyone would want to. I think this is a major failing of how most Americans learn math, further exacerbated by things like No Child Left Behind and a focus on testing instead of understanding.In college is where the focus became proofs and building things up from first principles. Tests stopped being "evaluate these 30 integrals" and became 5 questions that all began with "prove or disprove:" which is a really good way to force you to think but is also much harder. I don't for a second regret majoring in it but it's daunting and I imagine most people would rather do the 30 integrals. There's a reason why the Putnam Exam has an average score in the single digits out of 120.

posted by axiom at 1:35 PM on August 29, 2020 [15 favorites]

Proofs and Refutations is probably my favorite book on the question of 'how does maths?' (Here's a pdf of the first part of the book.) It's a fictionalized dialog that tracks the early history of topology. The book ends with the invention of homology, which more-or-less turns questions about arbitrary shapes into a collection of questions in linear algebra.

The important part here is that the math is a dialog, with different sides discovering bits that add to the conversation. People build up intuitions based on what they've seen, and over time new proofs and counter examples come along which sharpen that understanding.

posted by kaibutsu at 1:47 PM on August 29, 2020 [12 favorites]

The important part here is that the math is a dialog, with different sides discovering bits that add to the conversation. People build up intuitions based on what they've seen, and over time new proofs and counter examples come along which sharpen that understanding.

posted by kaibutsu at 1:47 PM on August 29, 2020 [12 favorites]

Any Middle school (math) teacher would/should love to have Grace as a student.

With so many schools starting remotely these weeks, her questions and the many thoughtful answers will be 'viral' in many Google Classroom/Zoom math sessions on Monday.

posted by TDIpod at 2:05 PM on August 29, 2020 [3 favorites]

With so many schools starting remotely these weeks, her questions and the many thoughtful answers will be 'viral' in many Google Classroom/Zoom math sessions on Monday.

posted by TDIpod at 2:05 PM on August 29, 2020 [3 favorites]

I have done quite a bit of expert witness consulting, mostly for intellectual property cases. My technical background gets me the cases, but what makes me an effective expert witness is my skill at explaining what are (or seem to be) complex/scary terms and/or concepts in a non-threatening, accessible or familiar way. Here's an example.

I have started telling students to "think like an expert witness," by explaining concepts is about 30 seconds using an example. I ask, "how would you describe algebra to a person who doesn't know math beyond simple arithmetic?" The answers are I get things like "Find x." "Using letters for variables." Here is my "expert witness" answer:

You rock, Gracie!

posted by haiku warrior at 2:16 PM on August 29, 2020 [46 favorites]

For a personal injury case I was asked by own side's (plaintiff's) lawyer during direct examination, "And Professor, do you have all the information needed to determine if [the defendant's] Electronic Stability Program caused this accident?" "No, I don't have the computer program. I don't have the specification. I don't have the algorithm that implements it." Then he asked a question for which he had not prepared me. "And just what is an algorithm, doctor?" I earned my fee by not missing a beat:The point is that many mathematical concepts are familiar, but have unfamiliar names. Furthermore, the concept are *useful*. However, they are rarely taught with a motivating application from the beginning.

"An algorithm is a procedure for doing a calculation, like your taxes. The specification is the tax law passed by Congress. The IRS then creates a form with instructions that the tax payer fills out with information from various sources. At the end there is a number which is either how much one owes, or how much the government will refund. That is a algorithm.

"In the Electronic Stability Program, there is a computer program, which receives information from sensors. The computer program calculates numbers which determines which brakes to apply and by how much."

That lawyer was the happiest attorney on the planet after that!

I have started telling students to "think like an expert witness," by explaining concepts is about 30 seconds using an example. I ask, "how would you describe algebra to a person who doesn't know math beyond simple arithmetic?" The answers are I get things like "Find x." "Using letters for variables." Here is my "expert witness" answer:

Algebra is a way of representing the relationships between known and unknown quantities and then using the relationships to determine those unknown quantities.So much more to say about this topic. But what I'd really like to say is that Gracie Cunningham is so aptly named. She responded with true *grace* to the hateful comments on Twitter. (That forum is also aptly name for all the twits that post to it.)

For adults my example is: Buying a house requires mortgage of $200,000. The interest rate is 3.5%, and the term of the mortgage is 15 years. (Known quantities) Determining the monthly payment (unknown quantity) is an algebra problem.

For teens my example is: A used car cost $1,400. You make $12/hour and work 15 hours per week. How long before you can buy the car?

You rock, Gracie!

posted by haiku warrior at 2:16 PM on August 29, 2020 [46 favorites]

>Faith in science or math is no better (or worse) than faith in invisible sky fairies. Prove it to yourself and ideally others, or it can only ever be faith. And faith is not sufficient for progress.

Nobody can prove that ZFC is consistent, but it has to be taken on faith because you can't prove it's consistent

posted by BungaDunga at 2:16 PM on August 29, 2020 [6 favorites]

Nobody can prove that ZFC is consistent, but it has to be taken on faith because you can't prove it's consistent

*within ZFC*. So mathematicians still do math with ZFC, not knowing for sure that it won't eventually produce a contradiction (it probably won't, but strictly speaking we don't*know*).posted by BungaDunga at 2:16 PM on August 29, 2020 [6 favorites]

*"Why are the only people disagreeing with me the ones who are dumb, while the physicists and mathematicians are agreeing with me?"*

I think this is also a rough translation of Socrates' last words.

posted by rollick at 2:21 PM on August 29, 2020 [57 favorites]

Also, mathematicians don't prove every theorem to themselves. There's too many. So some theorems end up used because sufficient

posted by BungaDunga at 2:21 PM on August 29, 2020 [4 favorites]

*other*mathematicians have vouched for them. You can*use*a theorem without actually reading and understanding the proof and the proofs of all the theorems it relies on. That's part of the weird power of mathematics: if a sufficiently large number of mathematicians agree that something has been proved, it's*vanishingly rare*that they turn out to be wrong. This is quite different from other fields of inquiry.posted by BungaDunga at 2:21 PM on August 29, 2020 [4 favorites]

Is it different? It seems to me that a great number of things we take as true, we do so without first-hand verification. I believe Seattle exists but I've never seen it. I believe the earth is round even though I've never done the stick/shadow experiment or blasted off into space. Requiring anyone to only believe what they'd verified for themselves would result in a pretty small knowledge base. I think our ability to accept things as fact based on other people's verification is a major building block of all science and human inquiry in general, and that's a feature not a bug.

posted by axiom at 2:31 PM on August 29, 2020 [7 favorites]

posted by axiom at 2:31 PM on August 29, 2020 [7 favorites]

Now that we're both back home, my wife says that Francis Su's "Mathematics for Human Fourishing" would be better than "How Not to Be Wrong" for Grace's sorts of questions. But again I should mention that Prof. Su (who did a great job of answering Grace directly, as seen in the Mary Sue link) was a prof at Harvey Mudd while my wife was in undergrad there.

posted by Navelgazer at 2:32 PM on August 29, 2020 [7 favorites]

posted by Navelgazer at 2:32 PM on August 29, 2020 [7 favorites]

*"What do mathematicians do? How do you math?"*

I spent five years training to be a professional mathematician, and have been a professional mathematician for 13 years since then, and I still don't have a very good answer to this. Somehow the math gets done by me and by people like me, but the process remains somewhat inexplicable. There are lots of tedious bits and a lot of frustrating bits and moments of sheer awe but very little that actually feels like any sort of systematic process. Publishing math is, AFAICT, all about concealing this messy bit and making it look straightforward, so reading math ends up being a very poor proxy for doing math.

I guess that's all to say, it's not a dumb question. If it was it'd probably have a simple answer.

*Jordan Ellenberg's "How Not to Be Wrong"*

Seconding this as an approachable and enjoyable work which gives a decent sense of the "big picture" of what kinds of things grab mathematicians' attention and why. Also, by MetaFilter's own.

posted by jackbishop at 2:33 PM on August 29, 2020 [14 favorites]

*Nobody can prove that ZFC is consistent, but it has to be taken on faith because you can't prove it's consistent within ZFC. So mathematicians still do math with ZFC, not knowing for sure that it won't eventually produce a contradiction (it probably won't, but strictly speaking we don't know).*

I think it's key here that mathematicians still understand that this is an "Article of faith" to an extent. They understand the extent to which it's not fully knowable. The existence of sizes of infinity between the understood alephs is another example of something that can (seemingly provably) neither be proven nor disproved. Advanced math gets weird but it builds on a foundation of knowing what we know.

posted by Navelgazer at 2:37 PM on August 29, 2020 [3 favorites]

Some of this is high-level metaphysics. The question of whether we ought to be platonists, nominalists, or some other camp is not an easy one and it comes with significant metaphysical consequences.

These questions are neither dumb nor trivial.

The question of how math works or why it works is a different set of concerns.

posted by oddman at 2:39 PM on August 29, 2020 [4 favorites]

These questions are neither dumb nor trivial.

The question of how math works or why it works is a different set of concerns.

posted by oddman at 2:39 PM on August 29, 2020 [4 favorites]

I disliked (and wasn't good at) maths at school. I still remember "ladders learning against frictionless walls" and wondering what the heck that was all about. None of it seemed in the slightest bit useful, especially the calculus and algebra, where I could turn the handle and crank out the answer as well as the next lad, but with no understanding.

Fast forward fifty years and I'm fascinated by mathematics, even though I still can't understand it. I'm a developer by trade and have been looking into category theory, and it's wonderful to see something that is (to me, at least) so abstruse, but which helps explain what I'm doing every day.

My biggest gripe is with the papers and books I read that say they assume "no mathematical background", when what they actually mean is "only undergraduate degree level mathematics". It's frustrating to find yourself coming to a halt on the first equation because you don't know what some (presumably very common) symbol means!

posted by 43rdAnd9th at 2:55 PM on August 29, 2020 [3 favorites]

Fast forward fifty years and I'm fascinated by mathematics, even though I still can't understand it. I'm a developer by trade and have been looking into category theory, and it's wonderful to see something that is (to me, at least) so abstruse, but which helps explain what I'm doing every day.

My biggest gripe is with the papers and books I read that say they assume "no mathematical background", when what they actually mean is "only undergraduate degree level mathematics". It's frustrating to find yourself coming to a halt on the first equation because you don't know what some (presumably very common) symbol means!

posted by 43rdAnd9th at 2:55 PM on August 29, 2020 [3 favorites]

I had never ever thought about math as being related to philosophy before Gracie's video and the conversation it sparked.

posted by Stoof at 2:59 PM on August 29, 2020 [2 favorites]

posted by Stoof at 2:59 PM on August 29, 2020 [2 favorites]

What do mathematicians do? How do you math?I'm definitely not a mathematician. But, I've shared a hallway with mathematicians on more than one occasion and count among my close friends several people on the mathy edge of physics. . . and I have absolutely no idea what a mathematician does. To be fair, I would also have a hard time describing what an experimentalist in my own field does, despite being one and advising others. (Except for soldering. That part I understand. Or, I kind of understand it. Except at the atomic level, where I don't actually understand it at all and can only make vague analogies to things that are true about crystaline materials and perhaps not entirely untrue about solder. But, I can describe how to do it. That's something.)

Cheers to Ms. Cunningham. Looks like she's going to be just fine, despite the world's dedicated effort to be awful.

posted by eotvos at 3:00 PM on August 29, 2020 [3 favorites]

*My middle-school math teacher said Mathematics is the Queen of Science.*

They weren't wrong. I like to think that Math is how we try to express what we think we know about all the other sciences. As our knowledge of math gets deeper, so does our knowledge of the other sciences, and vice-versa.

When you try to teach math without the context of why we're doing this math, that it expresses THIS concept from another science, you're gonna have a bad semester.

posted by mikelieman at 3:19 PM on August 29, 2020

*My middle-school math teacher said Mathematics is the Queen of Science.*

He (or she, on edit) was in good company:

Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics."But these days I'd be surprised if even a lot of number theorists wouldn't award the Math crown to algebra.

posted by jamjam at 3:30 PM on August 29, 2020 [2 favorites]

Perhaps this woman, Dr. Hannah Fry, British Mathematician, has some answers. Her BBC series Magic Numbers: Hannah Fry's Mysterious World of Maths is described as a "Documentary series in which Dr Hannah Fry explores the mystery of maths.

posted by bz at 4:00 PM on August 29, 2020 [11 favorites]

*Is it invented like a language or is it discovered and part of the fabric of the universe?*"posted by bz at 4:00 PM on August 29, 2020 [11 favorites]

*Now that we're both back home, my wife says that Francis Su's "Mathematics for Human Flourishing" would be better than "How Not to Be Wrong" for Grace's sorts of questions.*

Francis Su is a treasure. I read his essay The Lesson of Grace in Teaching at least once a year. (This essay might be too Christian for some. Am personally atheist Jew, but anyone who takes the lessons Dr. Su takes from his religion is A-OK by me.)

And I don't know what to love more, Gracie Cunningham's original questions or her based defiance of the trolls. I suspect she is going to perceive more wonders in the world than they are.

posted by aws17576 at 4:07 PM on August 29, 2020 [14 favorites]

These are great questions.

posted by carter at 4:35 PM on August 29, 2020 [1 favorite]

posted by carter at 4:35 PM on August 29, 2020 [1 favorite]

It's a legit question with no good answer because (applied) mathematics is essentially magical. And a lot of people don't think magic is real. So why math?

posted by grog at 5:13 PM on August 29, 2020

posted by grog at 5:13 PM on August 29, 2020

**aws17576:**That essay is astouding, and yes, he mentions Jesus once, to describe where his own concept of "Grace" originates, but he is in no way trying to indoctrinate or prosthelytize. He's just imparting a truly inspiring message that our human worth is something totally separate from our achievements, academic or otherwise, and that there are ways to base one's teaching in that concept. That essay knocked me flat.

posted by Navelgazer at 5:37 PM on August 29, 2020 [4 favorites]

I like flabdablet’s comment about noticing patterns and being able to apply patterns predictively.

Humans have a fundamental capacity and understanding of math concepts, long before we’re taught about numbers. In fact, it’s evident even in lower primates. It manifests, at least in one way, as a sense of fairness and relative quantity. We (and other primates) innately understand the difference between “more” and “less”.

We innately understand certain additive and subtractive properties of physical items. I pick up one apple, then another. I eat an apple and have one less than I did. My neighbor has three apples, and even if I cannot count, I have enough primal sense to resent him for having more than me. And so forth.

Math, at its most fundamental level, seems to be about developing and articulating an awareness of these patterns and properties of the physical world. Mathematicians are the engineers and linguists developing the tools and language to be able to elucidate and articulate those patterns.

posted by darkstar at 5:40 PM on August 29, 2020 [4 favorites]

Humans have a fundamental capacity and understanding of math concepts, long before we’re taught about numbers. In fact, it’s evident even in lower primates. It manifests, at least in one way, as a sense of fairness and relative quantity. We (and other primates) innately understand the difference between “more” and “less”.

We innately understand certain additive and subtractive properties of physical items. I pick up one apple, then another. I eat an apple and have one less than I did. My neighbor has three apples, and even if I cannot count, I have enough primal sense to resent him for having more than me. And so forth.

Math, at its most fundamental level, seems to be about developing and articulating an awareness of these patterns and properties of the physical world. Mathematicians are the engineers and linguists developing the tools and language to be able to elucidate and articulate those patterns.

posted by darkstar at 5:40 PM on August 29, 2020 [4 favorites]

Also, I remember a time, in my mid-twenties, among a bunch of dudes I had only recently become friends with, when I asked a Chinese-born dude something about, I think, explaining to me what the culture was likely like in China in the run-up to the Beijing Olympics (or something of that nature) and the group became kinda quiet, and I got a little embarrassed, and one of them just said how rare and refreshing it was for any of us to sincerely ask for knowledge rather than to just jump in trying to prove we knew things.

It's a lesson I still have trouble internalizing, but that was some grace, right there.

posted by Navelgazer at 5:44 PM on August 29, 2020 [6 favorites]

It's a lesson I still have trouble internalizing, but that was some grace, right there.

posted by Navelgazer at 5:44 PM on August 29, 2020 [6 favorites]

*Otherwise it's like, I dunno, being taught to read music but never being allowed to listen to it.*

That is a perfect encapsulation of all the years of math classes I took.

posted by Dip Flash at 5:49 PM on August 29, 2020 [7 favorites]

For fear of sounding dumb (boy, that really sticks with you, huh), but I have only passed *one* math class in my entire life, in summer school. I have to admit for the first time, I still secretly count on my fingers sometimes.

Half of my family is a loooong line of engineers, physicists, rocket scientists, data scientists, and software engineers. Hell, my cousin teaches at MIT. (*Cough* they’re all men.)

And growing up, they all nearly tore their hair out with me, because I’d be all, “But WHY! How do you know? What’s the point of learning this?”

Nobody ever taught me that you can literally explain how parts of the universe work, or why certain shells are shaped the way they are, or that we can actually start to understand the sheer force of a black hole, or that you can work to slow viruses that grow exponentially, that when things get small they get mysteriously very weird, or... just... how the study of math is a kind of magical drive to Understand and Prove how amazing and unlikely it is to just be alive and sentient. To explain a CONCEPT.

As an adult, I’ve found that my brain grasps and can formulate nebulous concepts with relative ease— but with words! I can see a piece of an idea and be all, “oh yeah, here’s how the rest goes.” A lot of my career entirely rests on that skill, and bosses come to me for it. But it has to be in words. For Pete’s sake, I read The Theory of Relativity by choice as a teen and didn’t find the higher level concepts actually too hard. (But I was just a girl, *cough*)

This is all to say (and maybe some math teachers can chime in, this is a real question): why don’t we teach from the conceptual top and why this particular Thing is cool and important, and then how to “do the math?” Because my entire education was “memorize this thing, here’s a pull-quote about Pythagoras maybe,” then grandfathers pulling their hair out and failure.

Anyway, I also stan this kid, and wish my education had been different. Because I suspect I’m not *actually* bad at math.

posted by functionequalsform at 5:52 PM on August 29, 2020 [16 favorites]

Half of my family is a loooong line of engineers, physicists, rocket scientists, data scientists, and software engineers. Hell, my cousin teaches at MIT. (*Cough* they’re all men.)

And growing up, they all nearly tore their hair out with me, because I’d be all, “But WHY! How do you know? What’s the point of learning this?”

Nobody ever taught me that you can literally explain how parts of the universe work, or why certain shells are shaped the way they are, or that we can actually start to understand the sheer force of a black hole, or that you can work to slow viruses that grow exponentially, that when things get small they get mysteriously very weird, or... just... how the study of math is a kind of magical drive to Understand and Prove how amazing and unlikely it is to just be alive and sentient. To explain a CONCEPT.

As an adult, I’ve found that my brain grasps and can formulate nebulous concepts with relative ease— but with words! I can see a piece of an idea and be all, “oh yeah, here’s how the rest goes.” A lot of my career entirely rests on that skill, and bosses come to me for it. But it has to be in words. For Pete’s sake, I read The Theory of Relativity by choice as a teen and didn’t find the higher level concepts actually too hard. (But I was just a girl, *cough*)

This is all to say (and maybe some math teachers can chime in, this is a real question): why don’t we teach from the conceptual top and why this particular Thing is cool and important, and then how to “do the math?” Because my entire education was “memorize this thing, here’s a pull-quote about Pythagoras maybe,” then grandfathers pulling their hair out and failure.

Anyway, I also stan this kid, and wish my education had been different. Because I suspect I’m not *actually* bad at math.

posted by functionequalsform at 5:52 PM on August 29, 2020 [16 favorites]

*Otherwise it's like, I dunno, being taught to read music but never being allowed to listen to it.*

Yes, this.

posted by functionequalsform at 5:52 PM on August 29, 2020 [2 favorites]

These are excellent questions.

The question of whether math can be invented or only discovered is one of those questions that starts to actually kind of freak me out if I think about it too long. That means it's a really good question, I think.

posted by lazaruslong at 5:59 PM on August 29, 2020 [2 favorites]

The question of whether math can be invented or only discovered is one of those questions that starts to actually kind of freak me out if I think about it too long. That means it's a really good question, I think.

posted by lazaruslong at 5:59 PM on August 29, 2020 [2 favorites]

*Otherwise it's like, I dunno, being taught to read music but never being allowed to listen to it.*

See A Mathematician's Lament.

posted by samw at 6:02 PM on August 29, 2020 [2 favorites]

When I went to CEGEP (college kind of in Quebec), in the Pure & Applied Science program, we learned Physics and Calculus at the same time - same teacher, same class. That was I think the best way. The same group of students took Chemistry and that teacher also brought in Calculus for enthalpy and entropy calculations.

We did not learn formulas, derivations and integrations by rote. We learned what was behind what they allowed you to solve for. On our end of term exams, we did not need to memorize the different equations, we had to use math to derive them. It was illuminating.

I was always strong in math, but this type of learning really made everything click. I'm not sure if other schools do this, but more definitely should. And this type of approach could easily be applied to other types of math and fields of study, at at different levels of schooling.

posted by mephisjo at 6:31 PM on August 29, 2020 [3 favorites]

We did not learn formulas, derivations and integrations by rote. We learned what was behind what they allowed you to solve for. On our end of term exams, we did not need to memorize the different equations, we had to use math to derive them. It was illuminating.

I was always strong in math, but this type of learning really made everything click. I'm not sure if other schools do this, but more definitely should. And this type of approach could easily be applied to other types of math and fields of study, at at different levels of schooling.

posted by mephisjo at 6:31 PM on August 29, 2020 [3 favorites]

I'd just like to point out that it's not that social media is itself terrible, but rather that people really really suck, and social media just lets them suck more publicly and at scale.

posted by signal at 6:34 PM on August 29, 2020 [11 favorites]

posted by signal at 6:34 PM on August 29, 2020 [11 favorites]

And a counterpoint to the above, I also took a linear algebra class at the same time, but it was a regular type of class. It did not click. The teacher taught linear algebra but did not once say what it would be used for. The matrices were fun, like little puzzles. But to this day I have no idea what they solved.

posted by mephisjo at 6:36 PM on August 29, 2020

posted by mephisjo at 6:36 PM on August 29, 2020

My point being that when I say 'twitter is a dumpster fire with another, worse dumpster fire burning inside of it' I'm not being a Luddite or technophobe, I'm being a misanthrope.

posted by signal at 6:37 PM on August 29, 2020 [4 favorites]

posted by signal at 6:37 PM on August 29, 2020 [4 favorites]

*why don’t we teach from the conceptual top and why this particular Thing is cool and important, and then how to “do the math?”*

Short , sad answer: It's hard. And it's hard to fit that kind of knowledge into a standardized test. And teachers (many of whom would love to teach this way) are already spread far too thin and hamstrung by far too many ill-thought and badly implemented requirements. I remember math classes in grade school, high school, and the couple years of college I did - it was all taught as thought they wanted to torture any love of the beauty in mathematics out of you.

I got way more knowledge and way more enjoyment out of math once I was out of those environments and just taught myself.

Incidentally, functionequalsform, I'm an ok-successful software developer, and I can't do arithmetic in my head, like *at all*. I have difficulty making change, or calculating a tip (not that I can't, but it takes me long enough that other people look at me funny). I suspect you're right that you're much better at math than you've been lead to believe.

Music can't be taught by theory without songs, writing can't be taught by a dictionary without stories, painting can't be taught by a color wheel without art. You couldn't teach a sport with a rulebook and no play time. The really high-level abstract stuff doesn't always really have context readily available, but by the time you get there you should have enough understanding from your previous learning that it's not as much a problem - like a composer working in complicated atonalities and unconventional scales, or Joyce writing Finnegan's Wake.

posted by mrgoat at 6:52 PM on August 29, 2020 [14 favorites]

Your experience is all too common,

And that was and is exactly the issue—“puzzles” aren’t not for most people. The way math is taught is like teaching people to read by doing the New York Times Crossword. If you happen to like that kind of mental exercise learning to read would be fun. But reading is a really useful skill for all kinds of important things as well as for entertainment.

If math were taught as a useful skill, and there happened to be some “entertaining” aspects, then I think mathematics education would be far more successful, and far fewer people would be frustrated, confused, and end up hating the field.

posted by haiku warrior at 7:07 PM on August 29, 2020 [4 favorites]

**functionequalsform**. I was someone who excelled at math, past the sixth grade. After doing a bazillion long division problems, finally we got to algebra, which seemed like grown-up math, and the manipulations were fun. Like solving puzzles, I suppose.And that was and is exactly the issue—“puzzles” aren’t not for most people. The way math is taught is like teaching people to read by doing the New York Times Crossword. If you happen to like that kind of mental exercise learning to read would be fun. But reading is a really useful skill for all kinds of important things as well as for entertainment.

If math were taught as a useful skill, and there happened to be some “entertaining” aspects, then I think mathematics education would be far more successful, and far fewer people would be frustrated, confused, and end up hating the field.

posted by haiku warrior at 7:07 PM on August 29, 2020 [4 favorites]

Another reason we don't teach things this way is because we're, as a society, doing education wrong. We don't spend enough money on it, and we put people in charge who don't know what they're doing or worse are actively bad (look at DeVos or this recent article about school administrators engaging in human trafficking to get teachers from the Philippines). On top of that we have created perverse incentives for the people who are supposed to be teaching to care too much about tests (there's a reason the phrase "teaching to the test" exists).

All that being said, I don't know that rote turn-cranking is the worst place to start. I think it goes on too long, but much like one might plink around a piano before learning about fifths, it makes sense as a starting point. On the other hand, unless they have math degrees, I don't know that many school teachers would be capable of teaching much theory because they probably don't know it. Hoisted on our own petards there, it seems.

posted by axiom at 7:08 PM on August 29, 2020 [6 favorites]

All that being said, I don't know that rote turn-cranking is the worst place to start. I think it goes on too long, but much like one might plink around a piano before learning about fifths, it makes sense as a starting point. On the other hand, unless they have math degrees, I don't know that many school teachers would be capable of teaching much theory because they probably don't know it. Hoisted on our own petards there, it seems.

posted by axiom at 7:08 PM on August 29, 2020 [6 favorites]

So if I wanted to do high level math stuff, but for fun without having to do a graduate degree, where/how would I do that?

posted by Cozybee at 7:24 PM on August 29, 2020 [2 favorites]

posted by Cozybee at 7:24 PM on August 29, 2020 [2 favorites]

Thank you for your thoughtful answers,

posted by functionequalsform at 7:34 PM on August 29, 2020

**mrgoat**and**haiku warrior**. I’m not terribly surprised, but man.posted by functionequalsform at 7:34 PM on August 29, 2020

Can someone help this old millennial out and tell me how to watch the actual vids everyone is talking about? All I can see is content describing it.

posted by rebent at 8:18 PM on August 29, 2020

posted by rebent at 8:18 PM on August 29, 2020

So I've mentioned my wife (who originally pointed me to this story this morning) a few times already, but it's important, I think, to tell her story (or at least my filtered version of understanding it.)

She is the daughter of very, very intelligent parents, neither of whom have college degrees. Her dad has used his intelligence getting deep into studying things like jazz history and new-age religion in his spare time, while her mom started a home business making children's clothes that actually, you know, work for how children act in the world. Similarly, her stepfather is an independent contractor who is very curious about the world, very very knowledgable about his field (which means my wife is way better at knowing how to fix things than I am, and fixing things is something I often enjoy). None of them has never made much money, but mostly because they never really aimed to do so. In short, she grew up in a family that was smart beyond their socioeconomic class, and discriminated against by their intellectual peers.

So my wife grew up in a college town feeling like an outcast with social limits on her future. She got into theater, art, and music from a young age—well into high school her plan was to become a drama teacher. My own beloved drama teacher from HS was a lot like my wife, in a lot of ways. She would have been great.

But one of her older friends invited her to come down to Harvey Mudd at some point, at which point she saw, for the first time, that math and science, the way that dedicated mathematicians and scientists do, is about collaboration and creativity. You explore, you find patterns and interesting questions, and you work with other people to figure them out.

Math, as presented in America, through papers, classes, or media, is NEVER presented this way.

And she had fallen in love with the idea. While she did ok in school, math in particular was an annoyingly opaque subject. She thought that because she disdained and failed at all things rote memory, that math was totally beyond her. When she turned on a dime, and decided to apply to a prestigious STEM college, admission counselors tried politely to suss out whether she was lost, and even one of her most beloved teachers tried to convinced her that she was “ruining her life” by running headlong into a program in which she would fail. But somehow, even as a capricious teenage girl, she powered through that rejection.

And reader: she got herself into Harvey Mudd, maybe the toughest undergraduate math program to get into (up there with MIT at least.) She rocked it, got her PhD at Wisconsin, and now teaches as a professor in New York, while obviously continuing her research.

It's hard. Math is hard. Research is hard. But the thing that drives her crazier than anything, even living with me, is how many people respond to her with "Oh I hate math" or "I can't do math" because, as she knows better than anyone, the issue there isn't math, or anyone's capability with it, but that it's culturally understood in such a way as to scare people away from a fundamental branch of knowledge that should be accessible to everyone. She didn't know she could be good at math until a friend showed her what it was like outside of the shitty American framing of math as a whole.

I, myself, was never good at math. It was the one course I didn't AP in. It took falling in love with a mathematician who could show me the beauty of it to make me fascinated. Would that all children could have that experience early enough for it to make a difference.

posted by Navelgazer at 8:32 PM on August 29, 2020 [22 favorites]

She is the daughter of very, very intelligent parents, neither of whom have college degrees. Her dad has used his intelligence getting deep into studying things like jazz history and new-age religion in his spare time, while her mom started a home business making children's clothes that actually, you know, work for how children act in the world. Similarly, her stepfather is an independent contractor who is very curious about the world, very very knowledgable about his field (which means my wife is way better at knowing how to fix things than I am, and fixing things is something I often enjoy). None of them has never made much money, but mostly because they never really aimed to do so. In short, she grew up in a family that was smart beyond their socioeconomic class, and discriminated against by their intellectual peers.

So my wife grew up in a college town feeling like an outcast with social limits on her future. She got into theater, art, and music from a young age—well into high school her plan was to become a drama teacher. My own beloved drama teacher from HS was a lot like my wife, in a lot of ways. She would have been great.

But one of her older friends invited her to come down to Harvey Mudd at some point, at which point she saw, for the first time, that math and science, the way that dedicated mathematicians and scientists do, is about collaboration and creativity. You explore, you find patterns and interesting questions, and you work with other people to figure them out.

Math, as presented in America, through papers, classes, or media, is NEVER presented this way.

And she had fallen in love with the idea. While she did ok in school, math in particular was an annoyingly opaque subject. She thought that because she disdained and failed at all things rote memory, that math was totally beyond her. When she turned on a dime, and decided to apply to a prestigious STEM college, admission counselors tried politely to suss out whether she was lost, and even one of her most beloved teachers tried to convinced her that she was “ruining her life” by running headlong into a program in which she would fail. But somehow, even as a capricious teenage girl, she powered through that rejection.

And reader: she got herself into Harvey Mudd, maybe the toughest undergraduate math program to get into (up there with MIT at least.) She rocked it, got her PhD at Wisconsin, and now teaches as a professor in New York, while obviously continuing her research.

It's hard. Math is hard. Research is hard. But the thing that drives her crazier than anything, even living with me, is how many people respond to her with "Oh I hate math" or "I can't do math" because, as she knows better than anyone, the issue there isn't math, or anyone's capability with it, but that it's culturally understood in such a way as to scare people away from a fundamental branch of knowledge that should be accessible to everyone. She didn't know she could be good at math until a friend showed her what it was like outside of the shitty American framing of math as a whole.

I, myself, was never good at math. It was the one course I didn't AP in. It took falling in love with a mathematician who could show me the beauty of it to make me fascinated. Would that all children could have that experience early enough for it to make a difference.

posted by Navelgazer at 8:32 PM on August 29, 2020 [22 favorites]

*Is it different? It seems to me that a great number of things we take as true, we do so without first-hand verification.*

The relative solidity of that foundational bedrock varies wildly from one field to another, though. Psychology is in the grips of an ongoing reproducibility crisis, for example, and economics as a field can reasonably be described as an epistemological tire fire. The same hasn’t even been alleged of mathematics.

posted by mhoye at 8:32 PM on August 29, 2020

Rebent: I think these should do it for you:

First video

Second video

posted by Reverend John at 8:36 PM on August 29, 2020 [1 favorite]

First video

Second video

posted by Reverend John at 8:36 PM on August 29, 2020 [1 favorite]

*She didn't know she could be good at math until a friend showed her what it was like outside of the shitty American framing of math as a whole*

This pattern generalizes.

posted by flabdablet at 8:44 PM on August 29, 2020 [9 favorites]

I went to college planning to be a math major. Loved looking at an equation and trying to figure out how to get the derivative. But in college, math was all about WHY the derivative. Didn't help that the math profs there were some of the worst teachers, and a few of them, some the worst human beings I had ever encountered...

Got a degree in Geology, and then found my problem solving niche in programming. My son, the mentat, finally graduated with a degree in Physics. I'd ask him about his classes when he came home for holidays. Took about 30 seconds before he had flown over my head. Pretty happy about that...

posted by Windopaene at 9:30 PM on August 29, 2020 [2 favorites]

Got a degree in Geology, and then found my problem solving niche in programming. My son, the mentat, finally graduated with a degree in Physics. I'd ask him about his classes when he came home for holidays. Took about 30 seconds before he had flown over my head. Pretty happy about that...

posted by Windopaene at 9:30 PM on August 29, 2020 [2 favorites]

what’s really freaky about mathematics is from time to time an accepted proof will turn out to be incorrect due to an error everyone missed. but usually, the theorem still turns out to be true anyway, after the proof is repaired.

posted by vogon_poet at 10:01 PM on August 29, 2020 [2 favorites]

posted by vogon_poet at 10:01 PM on August 29, 2020 [2 favorites]

I studied electrical and electronic engineering at uni. In many ways, engineering is applied mathematics. We had a big bag of mathematical tools, and we had to learn how to pick which ones we needed to get from what we started with to where we wanted to go - to turn chaos into order, fundamentally. And how we could use real world components to apply those models, and where we could 'cheat' by throwing unimportant bits away, and where they introduced their own changes.

I found this use of maths far more interesting than the dry, rote learning at school. We didn't dive very deeply into where these tools came from, but the idea that you could use them to extract useful patterns, or change them to your own ends was fascinating - maths didn't exist solely in the pages of a textbook, but could be used to explain and manipulate the real world, to draw meaning from the raw chaos, and do stuff with it. Maths at school was like learning chess by learning the moves of the pieces, and being asked to use those moves to get to a specific square - without ever being told about the rest of the rules, or why you might choose to make those moves, or that it was even a game.

I have nothing but respect for the people that can start with a blank piece of paper and some colleagues and come up with ways of finding the patterns in this vast universe, then write them down.

posted by Absolutely No You-Know-What at 11:42 PM on August 29, 2020

I found this use of maths far more interesting than the dry, rote learning at school. We didn't dive very deeply into where these tools came from, but the idea that you could use them to extract useful patterns, or change them to your own ends was fascinating - maths didn't exist solely in the pages of a textbook, but could be used to explain and manipulate the real world, to draw meaning from the raw chaos, and do stuff with it. Maths at school was like learning chess by learning the moves of the pieces, and being asked to use those moves to get to a specific square - without ever being told about the rest of the rules, or why you might choose to make those moves, or that it was even a game.

I have nothing but respect for the people that can start with a blank piece of paper and some colleagues and come up with ways of finding the patterns in this vast universe, then write them down.

posted by Absolutely No You-Know-What at 11:42 PM on August 29, 2020

Well, that was heart-wrenching... and the twitter thread once again showed the very best and worst of humanity. I would not be able to answer any of these and applaud her for asking.

posted by greenhornet at 12:09 AM on August 30, 2020

posted by greenhornet at 12:09 AM on August 30, 2020

*>I had never ever thought about math as being related to philosophy before Gracie's video and the conversation it sparked.*

With apologies to my friends who are philosophers, to a large degree mathematics is

*actually doing*what philosophy

*wishes*it could do. That is to say, building really, really large, self-consistent, logical edifices that seem to help us understand quite a bit about how the world works and how logic works and how language works and how human thought works.

Part of the secret, I think, is they start by shooting really low instead of really high. For example, if you start out by asking "What is the nature of reality" or "What is God" it is going to be a good 25 millennia before you can even get a good start at defining a few of the terms that allow you to partially understand what the question is that you are trying to answer.

On the flip side, if you start with, "We have this really basic thing, addition. How does that really, really work, in detail?"

"We have these really basic ideas, lines, and the fact that they can intersect. How does that really, really work, in detail?"

Work out those simple, basic, concepts in real detail for a few millennia and you start to understand things in depth. Like to the degree you can actually

*do things*with your knowledge. Say, you can build a microscope or a telescope (lines -> geometry -> optics -> lenses) or navigate around the world or calculate the positions of the planets or a space probe (numbers -> addition/subtraction -> algebra + geometry -> calculus -> understanding physical movement & space).

But it comes from starting with the small and seemingly very simple things, but then spending time to dive deep down within them until you really understand them at a very refined level.

Anyway, that's one reason most people have such a hard time figuring out why, when studying a summary of the mathematical edifice that has been painstakingly worked out and refined over the past 10 millennia, any particular littl thing you are studying is worth the bother.

Yes, every little thing you spend time on is simultaneously small and seemingly insignificant and definitely unrelated to anything important in your real life and at the same time,

*really, really hard*to fully understand.

But really, really understanding small, basic, insignificant things--at a very, very deep level--holds a surprising lot of power.

Simple things like, what do we

*really*mean by the numbers 1, 2, 3,... or by a simple concept like a straight line.

There is amazing power in truly deep understanding.

posted by flug at 2:09 AM on August 30, 2020 [10 favorites]

*I think this is also a rough translation of Socrates' last words.*

Socrates' last words were to ask his friend to sacrifice a chicken.

posted by Cardinal Fang at 2:11 AM on August 30, 2020 [3 favorites]

*>I also took a linear algebra class at the same time, but it was a regular type of class. It did not click. The teacher taught linear algebra but did not once say what it would be used for. The matrices were fun, like little puzzles. But to this day I have no idea what they solved.*

This is particularly sad, because linear algebra is quite literally, the key to the universe.

Anything involving geometry is most usefully understood--and calculated--using matrices and matrix transformations. That includes everything from rotations and translations and perspective--so, for example, viewing a particular scene from any given viewpoint or angle--which are most easily understood and calculated as matrix transformations, to all of modern physics.

Just for example, every model of the physical universe, such as Einstein's Special Relativity or General Relativity, is built in a particular type of linear algebra system. If you want to understand say aerodynamics, you're talking about calculating everything you do using linear algebra and different matrix operations.

And--to bring things really back to home base--the fancy graphics card in your computer is nothing but a really, really fancy and hugely optimized gadget to complete bazillions of linear algebra operations per second.

When you play Mortal Kombat 11 or Splatoon or Minecraft or (maybe the ultimate example) the recently released Microsoft Flight Simulator literally every pixel you see on the screen is produced by several bucketloads of matrix operations and without them you could do literally

*none*of it.

Having said all that--which is just barely starting to scratch the surface of the power of linear algrebra and what it's about--it is so essential and so fundamental and so basic that is can be really, really hard in teaching a class of it to come up with examples that really speak to the majority of a classroom.

It's a little like when the kid asks, "What real USE is addition, teacher?" or "Multiplication tables" or whatever.

The teacher usually trots out the same old "balancing your checkbook" or whatever and the kids immediately start rolling their eyes.

The

*real*answer is that--like addition and subtraction and multiplication--matrix operations and linear algebra are literally baked into the fabric of the universe. But understanding even one little corner of that is a subject for several semesters, at least, not just a 45 second class discussion.

Just for example, in a typical class you're going to have a hard time getting people to crank through a couple dozen 3X3 matrix operations or inverse matrix calculations. Whereas generating a tenth of a second of MS Flight Simulator visuals (and audio--let's not even

*go*there) is going to take, minimally, a few

*trillion*such operations.

(nVidia's current fastest video card does something like a trillion operations every 1/50th of a second and

*can't keep up*with the current version of MS Flight Sim.)

Point is, you're going to have a real hard time working out anything right there in class that most people would be able to grasp at all or that would really make them go "wow!" Because you're dealing with very, very basic building blocks of things that are, also, themselves very, very difficult to understand.

posted by flug at 2:55 AM on August 30, 2020 [7 favorites]

flug:

/me gives up on life

posted by Too-Ticky at 3:16 AM on August 30, 2020 [1 favorite]

*Yes, every little thing you spend time on is simultaneously small and seemingly insignificant and definitely unrelated to anything important in your real life and at the same time, really, really hard to fully understand.*/me gives up on life

posted by Too-Ticky at 3:16 AM on August 30, 2020 [1 favorite]

I think it is important in some cases to distinguish between arithmetic and math. Arithmetic was pretty easy for me, as was basic math like simple algebra and euclidean geometry. I always got poor grades, but that was a lack of motivation to apply the knowledge. Rote memorization was never an issue for me, hence being the "trivia* specialist on the quiz bowl team.

My ability to understand totally went off the rails when it came to more advanced algebra and especially calculus. I still can't sit down and compute an answer. However, the past few years I've watched a lot of physics lectures, which along with basic explainers of notation have allowed me to develop enough of an intuitive sense of the relationships involved that I can understand what the hell they're talking about when explaining quantum mechanics or GR or whatever.

While it's nowhere near enough to solve problems, it has given me the tools to better understand than I ever could through analogy. Or at least to be able to understand the limits of the analogies physicists and cosmologists use to communicate with us mathematically challenged folk.

Sean Carroll's ongoing "Biggest Ideas in the Universe" series is amazing. He simplifies the math quite a bit, but uses it in a way that is relatively easily understandable to explain both the what and the why of much of the fundamentals of various parts of our scientific understanding. Even simplified, it shows why some things must be true far better than prose ever can.

posted by wierdo at 3:41 AM on August 30, 2020

My ability to understand totally went off the rails when it came to more advanced algebra and especially calculus. I still can't sit down and compute an answer. However, the past few years I've watched a lot of physics lectures, which along with basic explainers of notation have allowed me to develop enough of an intuitive sense of the relationships involved that I can understand what the hell they're talking about when explaining quantum mechanics or GR or whatever.

While it's nowhere near enough to solve problems, it has given me the tools to better understand than I ever could through analogy. Or at least to be able to understand the limits of the analogies physicists and cosmologists use to communicate with us mathematically challenged folk.

Sean Carroll's ongoing "Biggest Ideas in the Universe" series is amazing. He simplifies the math quite a bit, but uses it in a way that is relatively easily understandable to explain both the what and the why of much of the fundamentals of various parts of our scientific understanding. Even simplified, it shows why some things must be true far better than prose ever can.

posted by wierdo at 3:41 AM on August 30, 2020

To sum up my last comment in terms that might be easier for many to understand, an industry built on the foundations of linear algebra--just one of many hundreds to thousands of potential interesting applications of linear algebra--is currently a $20 billion/year industry and poised to do $200 billion business annually within a few years.

So yeah, linear algebra is easily a worth a cool few tens to hundreds of billions a year.

posted by flug at 4:26 AM on August 30, 2020 [1 favorite]

So yeah, linear algebra is easily a worth a cool few tens to hundreds of billions a year.

posted by flug at 4:26 AM on August 30, 2020 [1 favorite]

Another vote for students/people asking such questions, on anything but in particular on the nature of mathematics. Mathematicians can have fun playing with extremely abstract concepts, much in the same way one could play a game with some arbitrary rules, but with a lot more freedom to change the rules or make new ones, and then it turns out that sometimes the rules they came up with are useful to model the real world. It is such an amazing thing.

Also flabdablet's example is really good. I use it when I teach a class which is an elective and basically "maths for non-mathematicians" (hard science students are not allowed to take it). I illustrate it with the probably apocryphal story of some antique civilization, with no notion of writing or even of numbers, which could still know if some of their sheep were missing when bringing them back in in the evening: for each sheep coming out in the morning, put a stone in a pouch, in the evening take one stone out for each sheep coming back, at the end see if any stone is left in the pouch. From this you get the natural numbers as cardinals, i.e. as a notion of count, and mathematically as the equivalence classes for the bijection relation, which is flabdablet's example: the common thing between two apples, or two birds, or two stones is, in a very fundamental way, the number 2, but you do not even need to have a word or a sign for the number 2 to notice it. But it can bring you to choose some representative for this common thing, say two scratches on a piece of wood, or two fingers on your hand, and from there to invent signs for numbers.

posted by anzen-dai-ichi at 5:39 AM on August 30, 2020 [2 favorites]

Also flabdablet's example is really good. I use it when I teach a class which is an elective and basically "maths for non-mathematicians" (hard science students are not allowed to take it). I illustrate it with the probably apocryphal story of some antique civilization, with no notion of writing or even of numbers, which could still know if some of their sheep were missing when bringing them back in in the evening: for each sheep coming out in the morning, put a stone in a pouch, in the evening take one stone out for each sheep coming back, at the end see if any stone is left in the pouch. From this you get the natural numbers as cardinals, i.e. as a notion of count, and mathematically as the equivalence classes for the bijection relation, which is flabdablet's example: the common thing between two apples, or two birds, or two stones is, in a very fundamental way, the number 2, but you do not even need to have a word or a sign for the number 2 to notice it. But it can bring you to choose some representative for this common thing, say two scratches on a piece of wood, or two fingers on your hand, and from there to invent signs for numbers.

posted by anzen-dai-ichi at 5:39 AM on August 30, 2020 [2 favorites]

I dunno. I always did fine in higher math classes, but I never really found them relevant, and I stopped taking math after Calc BC (18 years ago). The rest of my family are all engineers, so I am definitely the black sheep.

Our problem sets used to try to be relevant by having "story problems," e.g. "You are filling a bucket at x rate but meanwhile the water leaks out through a hole of y diameter. How long until you fill the bucket?" I was always tempted to write in "First, you plug the damn hole." Another perennial favorite was two trains approaching each other on parallel tracks, with a bird flying between them; how much distance does the bird cover before the trains meet, does anyone care? My physics final had one about being charged by a bull elephant and how much time do you have to load and fire your elephant gun, and who exactly wrote these problems, anyhow??

The fact that I remember the worthlessly worded problem and my snarky response, but not how to solve it, says a lot. Arithmetic always felt extremely practical to me -- adjusting recipes, calculating tip, even trying to figure sales tax so I could decide, in the pre-credit card days, if I had enough cash on me to buy that thing I wanted. Calculus was just a thing I had to pass so I could go to medical school.

We did revisit calculus briefly in pharmacology, because of the ways drugs are absorbed/metabolized/excreted (i.e. the actual real-world version of the holey bucket from high school), and the concept of a half-life. But again, on a day-to-day level, I do think about the concepts (this sine wave in particular*) but I'm not sitting at my desk doing integrals or derivatives or whichever you would be doing to solve that problem. I don't know if thinking about the concept counts as "doing math," because in my head, "doing math" is so closely linked to those problem sets from high school.

* I will say that whenever I've drawn that graph for people to try and explain why their medications work/don't work, I get a lot of confused looks, but when I just wave my hand in the air to imitate the sine wave peaks/troughs, people get it intuitively. I think the graphiness intimidates people who, like me, spent their childhood and early adulthood telling themselves they "aren't good at math."

posted by basalganglia at 5:44 AM on August 30, 2020

Our problem sets used to try to be relevant by having "story problems," e.g. "You are filling a bucket at x rate but meanwhile the water leaks out through a hole of y diameter. How long until you fill the bucket?" I was always tempted to write in "First, you plug the damn hole." Another perennial favorite was two trains approaching each other on parallel tracks, with a bird flying between them; how much distance does the bird cover before the trains meet, does anyone care? My physics final had one about being charged by a bull elephant and how much time do you have to load and fire your elephant gun, and who exactly wrote these problems, anyhow??

The fact that I remember the worthlessly worded problem and my snarky response, but not how to solve it, says a lot. Arithmetic always felt extremely practical to me -- adjusting recipes, calculating tip, even trying to figure sales tax so I could decide, in the pre-credit card days, if I had enough cash on me to buy that thing I wanted. Calculus was just a thing I had to pass so I could go to medical school.

We did revisit calculus briefly in pharmacology, because of the ways drugs are absorbed/metabolized/excreted (i.e. the actual real-world version of the holey bucket from high school), and the concept of a half-life. But again, on a day-to-day level, I do think about the concepts (this sine wave in particular*) but I'm not sitting at my desk doing integrals or derivatives or whichever you would be doing to solve that problem. I don't know if thinking about the concept counts as "doing math," because in my head, "doing math" is so closely linked to those problem sets from high school.

* I will say that whenever I've drawn that graph for people to try and explain why their medications work/don't work, I get a lot of confused looks, but when I just wave my hand in the air to imitate the sine wave peaks/troughs, people get it intuitively. I think the graphiness intimidates people who, like me, spent their childhood and early adulthood telling themselves they "aren't good at math."

posted by basalganglia at 5:44 AM on August 30, 2020

tl;dr Grace Cunningham is not alone

A couple of years ago, for Science Week, we got our graduate students to make posters to populate an Alphabet of Eponyms. Brief: max 150 words and 3 pictures. Because

E for Eurillas [Pat Hall]

M for Meitnerium [Lisa M]

S for Score [Virginia Apgar]

V for Virus [Yvonne Barr]

W for Witch [Maria Agnesi]

helped. We also had a competition to give our high-school visitors a go at inventing a new eponym. There was one rowdy engaged group from a DEIS (Delivering Equality of Opportunity in Schools) school. Two of the girls were quietly occupying the 1 table and 1 chair and working on the Eponym competition while their class-mates were doing Practical Acoustics 101. Despite much pencil sucking, the girls were coming up empty so I asked if they could name any female scientists.

"Mmmble Crmmble", one of them replied [the shouty acoustics were really woeful and I’m quite deaf]

"Say that again"

"Marie Curie"

"That's okay, but she already has something named after her; do you know any others?"

I was distracted by something else but when I came back to them, they had an answer.

"A Gabriela!"

"Ace! which scientist is called Gabriela?"

"Me!"

Here was a 15 y.o. young woman who had a degree of self-belief to say that a) she was a scientist [I blame two personable committed young men who were their teachers] b) she was, or had the potential to be, every bit as good as Marie Curie [I blame herself for that, helped by her pals and her family].

We weren't finished because the name had yet to be pegged on something. After a couple of minutes the girls had decided that a Gabriela would be a measure of stress. Not engineering stress, but medical stress. With a little help from their teacher and a suggestion from me about salivary cortisol we finished up with a new Eponym:

Gabriela [Gb] n. A measure of internal stress. 1 gabriela = 1μg/dl cortisol in the blood.

posted by BobTheScientist at 6:04 AM on August 30, 2020 [6 favorites]

A couple of years ago, for Science Week, we got our graduate students to make posters to populate an Alphabet of Eponyms. Brief: max 150 words and 3 pictures. Because

*300 years of patriarchy*we had to work quite hard to get anything like gender balance. IncludingE for Eurillas [Pat Hall]

M for Meitnerium [Lisa M]

S for Score [Virginia Apgar]

V for Virus [Yvonne Barr]

W for Witch [Maria Agnesi]

helped. We also had a competition to give our high-school visitors a go at inventing a new eponym. There was one rowdy engaged group from a DEIS (Delivering Equality of Opportunity in Schools) school. Two of the girls were quietly occupying the 1 table and 1 chair and working on the Eponym competition while their class-mates were doing Practical Acoustics 101. Despite much pencil sucking, the girls were coming up empty so I asked if they could name any female scientists.

"Mmmble Crmmble", one of them replied [the shouty acoustics were really woeful and I’m quite deaf]

"Say that again"

"Marie Curie"

"That's okay, but she already has something named after her; do you know any others?"

I was distracted by something else but when I came back to them, they had an answer.

"A Gabriela!"

"Ace! which scientist is called Gabriela?"

"Me!"

Here was a 15 y.o. young woman who had a degree of self-belief to say that a) she was a scientist [I blame two personable committed young men who were their teachers] b) she was, or had the potential to be, every bit as good as Marie Curie [I blame herself for that, helped by her pals and her family].

We weren't finished because the name had yet to be pegged on something. After a couple of minutes the girls had decided that a Gabriela would be a measure of stress. Not engineering stress, but medical stress. With a little help from their teacher and a suggestion from me about salivary cortisol we finished up with a new Eponym:

Gabriela [Gb] n. A measure of internal stress. 1 gabriela = 1μg/dl cortisol in the blood.

posted by BobTheScientist at 6:04 AM on August 30, 2020 [6 favorites]

*Psychology is in the grips of an ongoing reproducibility crisis, for example, and economics as a field can reasonably be described as an epistemological tire fire. The same hasn’t even been alleged of mathematics.*

This is a bit unfair. The set of things we can do logical proofs of that we don't point at and call "math" is very small, and the discipline called "math" is relatively unconcerned with direct applications of its techniques and patterns. When it starts getting applied to actual phenomena in the real world, that's when you get econ's epistemological tire fire, or physicists proving that climate change is because of God.

posted by GCU Sweet and Full of Grace at 6:15 AM on August 30, 2020 [1 favorite]

*or physicists proving that climate change is because of God.*

Never send a physicist to do the work of a meteorologist or climate scientist.

posted by Dysk at 7:19 AM on August 30, 2020 [1 favorite]

It's often been observed that the ability of mathematics to facilitate models of the way the real world behaves - models with apparently very high explanatory power as well as

Personally, I think that what this says is more about

We notice patterns. It's what we do. And the things we build our patterns from are correlations: things that seem

And having done that - having invented those signs, or been taught them - if you're of a certain cast of mind you do the

And once we start playing with symbols - things that

The reason mathematics seems so unreasonably good at describing Nature, it seems to me, is right there. Mathematics provides a way for human minds to generate and share this vast internal Cambrian explosion of patterns - patterns of things in categories

And given the sheer number of those patterns that we as a species have collectively generated, and the literally infinite complexity of the world we find ourselves occupying (it doesn't matter what questions you ask of Nature; each answer you get prompts ten more questions, and this is a process that could clearly go on forever to at least the same extent as anything else we routinely conceive of as infinite), what would be astonishing to me is if we

But there is not likely ever to exist, in my view, a mathematical Theory Of Everything. I can understand the burning drive that so many physicists feel toward finding one, and I'm in no way saying they're wasting their time by doing so; but the idea that some particularly refined piece of math, like the Dirac Equation for example, actually says anything

When I was in third grade, it occurred to me that I didn't have to go to school any more because I already knew everything. And I could prove it! Everything I knew, I

But of course I did keep going, and I did keep learning more stuff, because there's

I don't care if you

And sometimes it's just going to be the case that you're going to have to invent some completely

The danger comes, as GCU Sweet and Full of Grace suggests, when we make the fundamental mistake of thinking that the world is in some way compelled to conform to our notions of it rather than the other way around. And yes, I too have observed that economists

Yes there is. There is indeed. And it's long been a matter of considerable sorrow for me that there isn't, in general, a similarly amazing amount of responsibility that accompanies that power.

Now I'm off to drink a lot of coffee.

posted by flabdablet at 8:28 AM on August 30, 2020 [8 favorites]

*spookily*high predictive power - is just flat-out weird, and says all kinds of mystical things about mathematics, or the world, or both. I've seen quite respectable scientists wax quite poetic*at length*on this topic.Personally, I think that what this says is more about

*us*than about the rest of the universe of which we're relatively sub-yoctoscopic but oh so self-important parts.We notice patterns. It's what we do. And the things we build our patterns from are correlations: things that seem

*like*other things to us.*the common thing between two apples, or two birds, or two stones is, in a very fundamental way, the number 2, but you do not even need to have a word or a sign for the number 2 to notice it. But it can bring you to choose some representative for this common thing, say two scratches on a piece of wood, or two fingers on your hand, and from there to invent signs for numbers.*And having done that - having invented those signs, or been taught them - if you're of a certain cast of mind you do the

*other*thing we do as humans, which is*play*with them.And once we start playing with symbols - things that

*represent*other things - the complexity of the kinds of correlation it's possible to make, especially if we approach the thing with enough rigour to keep our use of those symbols consistent and*precisely*communicable, just*explodes.*The reason mathematics seems so unreasonably good at describing Nature, it seems to me, is right there. Mathematics provides a way for human minds to generate and share this vast internal Cambrian explosion of patterns - patterns of things in categories

*literally unthinkable*without these symbolic tools.And given the sheer number of those patterns that we as a species have collectively generated, and the literally infinite complexity of the world we find ourselves occupying (it doesn't matter what questions you ask of Nature; each answer you get prompts ten more questions, and this is a process that could clearly go on forever to at least the same extent as anything else we routinely conceive of as infinite), what would be astonishing to me is if we

*didn't*find large numbers of correlations between parts of one and parts of the other.But there is not likely ever to exist, in my view, a mathematical Theory Of Everything. I can understand the burning drive that so many physicists feel toward finding one, and I'm in no way saying they're wasting their time by doing so; but the idea that some particularly refined piece of math, like the Dirac Equation for example, actually says anything

*profound*about the nature of reality as a whole strikes me as wishful thinking. If there's profundity to be found in an equation like that, it's the profundity of our ability to fuck ourselves and everything else up by acting as if we're capable of anticipating way*way*more of the possible consequences of what we choose to do than we actually*can*anticipate.When I was in third grade, it occurred to me that I didn't have to go to school any more because I already knew everything. And I could prove it! Everything I knew, I

*knew*that I knew that. It was*all right there!*All of it! I checked!But of course I did keep going, and I did keep learning more stuff, because there's

*always more stuff to notice*if we just allow ourselves to notice ourselves noticing it.I don't care if you

*do*believe you*do*grasp the ultimate nature of what it is to be an embodied conscious being in this world we all share: that's not going to change the fact that tomorrow, and the day after, and the day after that, and on and on and on for the rest of your life, you're going to have those moments where you notice something you've not noticed before and you go hmm, that's weird.And sometimes it's just going to be the case that you're going to have to invent some completely

*new*kind of math to un-weird that thing.The danger comes, as GCU Sweet and Full of Grace suggests, when we make the fundamental mistake of thinking that the world is in some way compelled to conform to our notions of it rather than the other way around. And yes, I too have observed that economists

*do*seem particularly prone to making that kind of error.*There is amazing power in truly deep understanding.*Yes there is. There is indeed. And it's long been a matter of considerable sorrow for me that there isn't, in general, a similarly amazing amount of responsibility that accompanies that power.

Now I'm off to drink a lot of coffee.

posted by flabdablet at 8:28 AM on August 30, 2020 [8 favorites]

*Part of the secret, I think, is they start by shooting really low instead of really high. For example, if you start out by asking "What is the nature of reality" or "What is God" it is going to be a good 25 millennia before you can even get a good start at defining a few of the terms that allow you to partially understand what the question is that you are trying to answer.*

On the flip side, if you start with, "We have this really basic thing, addition. How does that really, really work, in detail?"

"We have these really basic ideas, lines, and the fact that they can intersect. How does that really, really work, in detail?"

Work out those simple, basic, concepts in real detail for a few millennia and you start to understand things in depth. Like to the degree you can actually do things with your knowledge. Say, you can build a microscope or a telescope (lines -> geometry -> optics -> lenses) or navigate around the world or calculate the positions of the planets or a space probe (numbers -> addition/subtraction -> algebra + geometry -> calculus -> understanding physical movement & space).

On the flip side, if you start with, "We have this really basic thing, addition. How does that really, really work, in detail?"

"We have these really basic ideas, lines, and the fact that they can intersect. How does that really, really work, in detail?"

Work out those simple, basic, concepts in real detail for a few millennia and you start to understand things in depth. Like to the degree you can actually do things with your knowledge. Say, you can build a microscope or a telescope (lines -> geometry -> optics -> lenses) or navigate around the world or calculate the positions of the planets or a space probe (numbers -> addition/subtraction -> algebra + geometry -> calculus -> understanding physical movement & space).

To be fair to philosophers, this is basically what Descartes (who was both philosopher and mathematician, crucially) tried to do with his skepticism, and precisely where "cogito ergo sum" ("I think, therefor I am") came from. Basically, "If I start by doubting literally everything, what can I prove,

*a priori*?" And it turned out that Descartes could prove his own existence, in at least the most ephemeral, undefined form, based on his own thought process about it.

But then everything next required him leaning on religious teachings at least a little bit. Philosophy and math intersect way more than how they are usually taught (except in logic classes, which everyone should take, seriously) but they are different fields with different needs and honestly, different goals. Philosophy is sometimes like math, mining through the mountains of knowledge about the universe to find "truth," but it's just as often like poetry or fiction, shining light on those mountains in creative ways to just have us look at things differently.

And then, and this is what really freaked my bean from meeting my wife and learning about combinatorics (one of her her fields of speciality): Sometimes math is like poetry or fiction in that way too.

posted by Navelgazer at 8:32 AM on August 30, 2020 [1 favorite]

e

posted by flabdablet at 8:35 AM on August 30, 2020 [4 favorites]

^{iπ}+ 1 = 0*is*a poem. No two ways about it.posted by flabdablet at 8:35 AM on August 30, 2020 [4 favorites]

*it turned out that Descartes could prove his own existence, in at least the most ephemeral, undefined form, based on his own thought process about it.*

To my way of thinking, that proof was Descartes' ego taking credit for stuff that wasn't its due. I'm with these fine thinkers to some extent on this point.

That I exist is a

*given*. It doesn't

*require*a proof. The question worth pondering, it seems to me, is how (and perhaps where, if spacetime happens to be an applicable framework right now) I can usefully be understood to stop and everything else can usefully be understood to start.

The best answer I've found to that, so far, begins "it depends".

Sometimes making that distinction is simply not possible.

posted by flabdablet at 8:52 AM on August 30, 2020 [1 favorite]

Technical competence or preferential enjoyment of math is a red herring for serious discussion of math education, the real issue is that studying math is a privileged activity, one largely still a young boys' club, and compartmentalized from social, political needs of real people. The number of professors who demonstrated in some way or other being out of touch with the social realities of the students, I can't begin to count. Developmental psychology is relevant here; meet the psychosocial needs of the human being, only then can any person autonomously flourish and learn, in general.

posted by polymodus at 9:09 AM on August 30, 2020 [3 favorites]

posted by polymodus at 9:09 AM on August 30, 2020 [3 favorites]

I was asked to talk to a bunch of eighth graders about being a scientist and had to wait in the principal's lobby for the program to begin. On the shelf was a book on the history of math, and I picked it up to kill time and was immediately engrossed because it was so well-written and I knew so little about the subject. I would love to keep reading it, but I didn't write the author's name down. I think it started with a "B".

The students weren't interested in a career in science. They wanted to be rock stars and football players.

posted by acrasis at 9:46 AM on August 30, 2020 [1 favorite]

The students weren't interested in a career in science. They wanted to be rock stars and football players.

posted by acrasis at 9:46 AM on August 30, 2020 [1 favorite]

acrasis, might it have been Eric Temple Bell's (unfortunately titled) Men of Mathematics?

posted by aws17576 at 10:34 AM on August 30, 2020 [1 favorite]

posted by aws17576 at 10:34 AM on August 30, 2020 [1 favorite]

*That I exist is a given. It doesn't require a proof.*

OK, Bomb #20... what concrete evidence do you have of your own existence?

posted by Cardinal Fang at 2:11 PM on August 30, 2020 [2 favorites]

*When I went to CEGEP (college kind of in Quebec), in the Pure & Applied Science program, we learned Physics and Calculus at the same time - same teacher, same class. That was I think the best way. The same group of students took Chemistry and that teacher also brought in Calculus for enthalpy and entropy calculations.*

When I took high school physics, I had an amazing teacher. Like, amazing enough that our senior class chose him as our graduation speaker even though the majority never had a class with him.

He began his first class by building a hierarchy of sciences on the blackboard, not disrespecting any field but showing how each one sprung from the knowledge of a higher one, and at the top, he put, of course, Physics.

And then, in the space he had to reach real high for above that, he put, in huge all-caps "MATH."

But he was dealing with students who were both taking calc concurrently or else weren't. (I wasn't.) And so, when explaining concepts and how to measure them, he was explaining things both in calc terms and otherwise, and giving students like me just enough calc understanding when necessary to get the material.

It was the best math class I ever had.

posted by Navelgazer at 2:33 PM on August 30, 2020 [1 favorite]

I'm also remembering now two separate instances I've had of recognizing a pattern, getting curious, and then a mathematician being able to explain the pattern I recognized.

The first was in my early years on MeFi, recognizing that if you tightly pack bottles, cans, what have you of equal size, that six of them will perfectly fit around a central one. Why, I wondered, and Steven C. den Beste popped in to explain that it wasn't really about circles, except inasmuch as they were allowing for the six equilateral triangles that form a hexagon. That blew my mind.

The second was about a year after meeting my wife. I was bored at work and counting the minutes until I was done, so much that at a hundred minutes to go I created a 10x10 grid and just started filling the spaces in from the outside each minute that passed, and it struck me: each square whole number can be constructed by adding consecutive odd numbers.

Think about it!: 1 is square. +3=4, +5=9, +7=16, +9=25, +11= 36, etc. This was outstanding to me!

And my wife then explained to me that, yes, this is awesome, and one of the first things profs have students prove in upper-level math courses, because there are a LOT of different proofs for it. But she treated my discovery with love and respect. Because discovery is what makes math great.

posted by Navelgazer at 2:54 PM on August 30, 2020 [3 favorites]

The first was in my early years on MeFi, recognizing that if you tightly pack bottles, cans, what have you of equal size, that six of them will perfectly fit around a central one. Why, I wondered, and Steven C. den Beste popped in to explain that it wasn't really about circles, except inasmuch as they were allowing for the six equilateral triangles that form a hexagon. That blew my mind.

The second was about a year after meeting my wife. I was bored at work and counting the minutes until I was done, so much that at a hundred minutes to go I created a 10x10 grid and just started filling the spaces in from the outside each minute that passed, and it struck me: each square whole number can be constructed by adding consecutive odd numbers.

Think about it!: 1 is square. +3=4, +5=9, +7=16, +9=25, +11= 36, etc. This was outstanding to me!

And my wife then explained to me that, yes, this is awesome, and one of the first things profs have students prove in upper-level math courses, because there are a LOT of different proofs for it. But she treated my discovery with love and respect. Because discovery is what makes math great.

posted by Navelgazer at 2:54 PM on August 30, 2020 [3 favorites]

*OK, Bomb #20... what concrete evidence do you have of your own existence?*

Cogito ergo sum. It's literally

*a priori*proof. I can't prove your existence, or anyone else's with it, but if I can take an action, even just thinking, then I must exist in a form capable of taking that action.

posted by Navelgazer at 3:06 PM on August 30, 2020

Navelgazer's recognition that "each square whole number can be constructed by adding consecutive whole numbers", is I think, a great example of how bad math education is at the middle school and high school level, because it's just the concrete realization of a formula that you will have to have memorized in order to pass high school algebra:

(x+1)^2 = x^2 + 2x + 1

In other words:

1^2 = 1

2^2 = 4 = 1 + 3

3^2 = 9 = 4 + 5

4^2 = 16 = 9 + 7

...and so on. You can measure how bad math education is because lots of people, including myself, who are able to pass math classes with high marks, are still so ignorant of the underlying pattern that can "discover" it independently and find it genuinely thrilling. If you happen to be absent minded, you can even

posted by Ipsifendus at 3:40 PM on August 30, 2020 [7 favorites]

(x+1)^2 = x^2 + 2x + 1

In other words:

1^2 = 1

2^2 = 4 = 1 + 3

3^2 = 9 = 4 + 5

4^2 = 16 = 9 + 7

...and so on. You can measure how bad math education is because lots of people, including myself, who are able to pass math classes with high marks, are still so ignorant of the underlying pattern that can "discover" it independently and find it genuinely thrilling. If you happen to be absent minded, you can even

*forget*having discovered it and discover it a couple of times more when bored during work meetings. And then remember that it's not a new thing you're discovering...it's just the formula you already memorized for Algebra I.posted by Ipsifendus at 3:40 PM on August 30, 2020 [7 favorites]

Like, I was barely able to function with math in school, and not because of bad teachers (frankly, I think any teachers tackling math as taught in America are unsung heroes, but the ones I had were particularly diligent, and I still just barely hung on.)

But finding patterns and pondering over them is one of my literal favorite things in life. I just never connected the two things in school.

posted by Navelgazer at 3:56 PM on August 30, 2020 [2 favorites]

But finding patterns and pondering over them is one of my literal favorite things in life. I just never connected the two things in school.

posted by Navelgazer at 3:56 PM on August 30, 2020 [2 favorites]

@mhoye, Mathematics went through its crisis about a century ago.

About two decades ago I was browsing the Highway Book Shop near Temagami, and in it's eclectic collection was

So for "how do they know that their formulas are right?" mathematicians just prove them in some mathematical system, but for "how do they know their mathematical systems are right?", they don't, not really. Now there are some signs you can look for that your mathematical system has gone wrong: proving both "P" and "not P" for any proposition P is a very bad sign; a typical mathematical system will fall apart completely if this happens. But not being able to find such contradictions doesn't really mean that all is good.

posted by mscibing at 5:54 PM on August 30, 2020 [2 favorites]

About two decades ago I was browsing the Highway Book Shop near Temagami, and in it's eclectic collection was

*Foundations of Mathematical Logic*by Haskell B. Curry. It gets very technical very fast but the introduction is accessible and it goes through some of the history of resolving paradoxes in mathematics, and then he says:*Using formalistic conceptions to explain what a theory is, we accept a theory so long as it is useful, satisfies such conditions of naturalness and simplicity as are reasonable at that time, and is not known to lead us into error. We must keep our theories under surveillance to see that these conditions are fulfilled and to get all the presumptive evidence of adequacy that we can. The Gödel theorem suggest that this is all we can do; an empirical philosophy of science suggests it is all we should do.*So for "how do they know that their formulas are right?" mathematicians just prove them in some mathematical system, but for "how do they know their mathematical systems are right?", they don't, not really. Now there are some signs you can look for that your mathematical system has gone wrong: proving both "P" and "not P" for any proposition P is a very bad sign; a typical mathematical system will fall apart completely if this happens. But not being able to find such contradictions doesn't really mean that all is good.

posted by mscibing at 5:54 PM on August 30, 2020 [2 favorites]

Another well-spoken answer to some of Gracie's questions.

posted by Navelgazer at 6:17 PM on August 30, 2020

posted by Navelgazer at 6:17 PM on August 30, 2020

*So for "how do they know that their formulas are right?" mathematicians just prove them in some mathematical system, but for "how do they know their mathematical systems are right?", they don't, not really. Now there are some signs you can look for that your mathematical system has gone wrong: proving both "P" and "not P" for any proposition P is a very bad sign; a typical mathematical system will fall apart completely if this happens. But not being able to find such contradictions doesn't really mean that all is good.*

And I remember in my philosophy of science class at NYU, how much time we spent on paradigm shifts (the real, monumental kinds in scientific understanding, not taking a website 2.0) and how, like, particle physics made a lot of Newtonian physic obsolete or just-plain incorrect, but Newtonian physics are better for building bridges, and how, going further back, geocentric theory is clearly wrong, but makes seafaring easier. Etc.

posted by Navelgazer at 6:21 PM on August 30, 2020 [1 favorite]

You can measure how bad math education is because lots of people, including myself, who are able to pass math classes with high marks, are still so ignorant of the underlying pattern that can "discover" it independently and find it genuinely thrilling.

I am a mathematician and I have that kind of experience often -- maybe not about things quite as elementary as (x+1)

The thrill you're talking about is the thrill of finding a new road that leads, unexpectedly, to a familiar destination. Sometimes you don't recognize the territory right away because of the new angle you're viewing it from.

One of the things I love about mathematics is that there are so many paths to the same truths that you're almost guaranteed to find interesting stuff if you explore long enough. There are also

posted by aws17576 at 6:43 PM on August 30, 2020 [11 favorites]

I am a mathematician and I have that kind of experience often -- maybe not about things quite as elementary as (x+1)

^{2}= x^{2}+2x+1, but certainly about things I learned in my grad school classes, or even undergrad. It's not necessarily a sign that those classes were badly taught (nor that I was an inattentive student).The thrill you're talking about is the thrill of finding a new road that leads, unexpectedly, to a familiar destination. Sometimes you don't recognize the territory right away because of the new angle you're viewing it from.

One of the things I love about mathematics is that there are so many paths to the same truths that you're almost guaranteed to find interesting stuff if you explore long enough. There are also

*so many*paths that the most talented teacher can't show you all of them, they can only make you want to look for them (and help you get better at finding them). If someone is playing around with 1+3+5+7+... in an office meeting, well, at the very least their math teachers didn't kill off their interest in the subject. That's closer to a win than a loss in my book.posted by aws17576 at 6:43 PM on August 30, 2020 [11 favorites]

1. I don't remember ever learning (x+1)2 = x2+2x+1 in my algebra classes, but it's possible someone tried to teach it to me.

2. I wasn't even in a meeting. I was just sitting at my desk waiting for the day to be done. And noticing "17 minutes 'til the next clean square... 15 minutes... 13 minutes... 11 minutes... hey!" was my inspiration. But those are exactly the kinds of patterns that drive innovation!

posted by Navelgazer at 6:58 PM on August 30, 2020 [2 favorites]

2. I wasn't even in a meeting. I was just sitting at my desk waiting for the day to be done. And noticing "17 minutes 'til the next clean square... 15 minutes... 13 minutes... 11 minutes... hey!" was my inspiration. But those are exactly the kinds of patterns that drive innovation!

posted by Navelgazer at 6:58 PM on August 30, 2020 [2 favorites]

This is why Archimedes last words were "There's sheet cake in the break room"

posted by thelonius at 7:15 PM on August 30, 2020 [6 favorites]

posted by thelonius at 7:15 PM on August 30, 2020 [6 favorites]

**Navelgazer**, I applaud you. Yes, noticing those patterns as you did is

*exactly*how mathematics (and other fields) advance. The next step would be representing the successive squares and the successive odd numbers to demonstrate that the pattern continues indefinitely.

First square: x^2

Next square: (x + 1)^2

Odd number: 2*x + 1

You would have to notice that the odd number is one plus twice the base of the first square. But I think you might already have seen that.

The pattern you identified is the next square is the first square plus twice the base of the first square plus one. Symbolically this is

(x + 1)^2 = x^2 + 2*x + 1

So without realizing it you discovered the relationship that

**Ipsifendus**stated that many learn in early algebra. Likely long ago someone did something similar to what you did to arrive at the formula in the first place.

Nice work!

posted by haiku warrior at 7:26 PM on August 30, 2020 [3 favorites]

*Algebra is a way of representing the relationships between known and unknown quantities and then using the relationships to determine those unknown quantities.*

I'd quibble with this slightly, or at least add an additional important detail, that algebra is a way of representing relationships between

*categories of quantities*, and using that relationship to determine one quantity given a value for the other. That is, if there's only one variable in your equation (eg. because the value of the other variable has been fixed) then solving for its value is an arithmetic problem. Algebra is about solving all of the similar arithmetic problems in one go. That is, arithmetic asks the question, "how do I find this specific unknown quantity given this specific known quantity?" while algebra asks the question, "how would I go about finding the unknown quantity no mater what value the known but variable quantity has?" You can solve an arithmetic problem with a basic calculator, but algebra tells you what formula to put in a spreadsheet that will be copied down the entire column so you don't have to plug the calculation into your calculator for every single row.

To answer Cozybee's question, I'd recommend the book Measurement by Paul Lockhart.

I think this would also be an even better resource to answer the questions that Grace asks than some of the other recommendations that have been tossed out, though those essays and books are interesting in their own rights. But I don't think that Grace is asking so much about the Platonic versus discovery issue. Instead she seems to be asking more about how mathematical proof works. (Except that she doesn't yet have the language to explain that - which is one of the impressing things about her video, that she's thinking about these issues despite not seeming to have had any introduction to the idea of mathematical proof).

posted by eviemath at 8:05 PM on August 30, 2020 [3 favorites]

“Where Mathematics Comes From” was published in 2000Another good book on this subject is Pi in the Sky: Counting, Thinking and Being.

posted by L.P. Hatecraft at 8:15 PM on August 30, 2020 [2 favorites]

*I don't remember ever learning (x+1)2 = x2+2x+1 in my algebra classes, but it's possible someone tried to teach it to me.*

Does "foil method" ring a bell, maybe? This is just applying the distributive property a couple times:

(x+1)*(x+1)

= x*(x+1) + 1*(x+1)

= x*x + x*1 + 1*x + 1*1

= x

^{2}+ 2*x + 1

and, to the previous discussion, then noticing that you can group things slightly differently going from the second-to-last line to the last line, as

= x

^{2}+ x + (x+1)

Of course, x is just (x-1) + 1, so you can do the same thing to x

^{2}in relation to x-1 as you did to (x+1)

^{2}in relation to x:

x

^{2}= ((x-1) + 1)

^{2}

= (x-1)

^{2}+ x + (x-1)

and so, plugging this in for x

^{2}from the first equation:

(x+1)

^{2}= x

^{2}+ x + (x+1)

= ((x-1)

^{2}+ (x-1) + x) + x + (x+1)

= (x-1)

^{2}+ (x-1) + x + x + (x+1)

You could apply the same reasoning to re-write (x-1)

^{2}, and so on, and keep working backwards to eventually get

x

^{2}= 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + ... + (x-2) + (x-2) + (x-1) + (x-1) + x

= 1 + (1 + 2) + (2 + 3) + (3 + 4) + ... + ((x-2) + (x-1)) + ((x-1) + x)

= 1 + 3 + 5 + 7 + ... + (2x - 3) + (2x - 1)

which is the sum of odd numbers pattern folks were talking about.

Eg. when x = 1, 2x - 1 = 2*1 - 1 = 2 - 1 = 1, and so 1

^{2}= 1.

When x = 2, 2x - 1 = 2*2 - 1 = 4 - 1 = 3, and so 2

^{2}= 1 + 3 = 4.

When x = 3, 2x - 1 = 2*3 - 1 = 6 - 1 = 5, and so 3

^{2}= 1 + 3 + 5 = 9.

When x = 4, 2x - 1 = 2*4 - 1 = 8 - 1 = 7, and so 4

^{2}= 1 + 3 + 5 + 7 = 16.

etc.

posted by eviemath at 8:37 PM on August 30, 2020 [3 favorites]

And on failing to preview before posting,

posted by eviemath at 8:38 PM on August 30, 2020 [2 favorites]

**haiku warrior**already explained that :)posted by eviemath at 8:38 PM on August 30, 2020 [2 favorites]

*> a great example of how bad math education is*

More of an example of how hard it is to make connections between different ways of thinking about the same thing, even if you have had good education and training and a natural aptitude on top of it.

Your math education has been a smashing success if you can even

*understand*your explanation of how Navelgazer's discovery relates to the formula

*--even if it takes a while of thinking about it. Or if you can come up with explanations and connections of that sort on your own every once in awhile, even if it takes while of thinking about it.*

(x+1)^2 = x^2 + 2x + 1

(x+1)^2 = x^2 + 2x + 1

The goal here isn't to get people to memorize a bunch of stuff so they'll recognize it later in life on sight. It's to get people comfortable with and used to the mode of thinking and problem solving where you have strategies and you can think things through and you aren't intimidated and scared off just because you don't know right off the bat exactly how to tackle something.

And if you end up re-discovering something the Chinese discovered 1700 years ago or the Bablyonians 3900 years ago or some French dude 300 years ago or you yourself 23 years ago in 9th grade--then, hooray! That's success, not failure.

posted by flug at 9:19 PM on August 30, 2020 [6 favorites]

My parallel experience to Navelgazer's discovery was noticing that if you take a square number - say sixty-four, eight squared, the number of squares on a chessboard - and then multiply the numbers one either side of the square root - seven and nine, in this case - you always seem to end up with a result one less than the original square (sixty-three).

And I'd had this pattern in my mind for

I agree with flug - failure to discern this kind of connection immediately is about neither a lack of mathematical chops nor a poverty of mathematical education. In fact it's not a failure at all, because connecting perceived patterns with strings of symbols is

If it

But mathematics

posted by flabdablet at 4:06 AM on August 31, 2020 [6 favorites]

And I'd had this pattern in my mind for

*months*before it occurred to me that it was a special case of a^{2}- b^{2}= (a + b)(a - b), a standard factorization I'd known about since being taught it in third form and always really liked the tidiness of.I agree with flug - failure to discern this kind of connection immediately is about neither a lack of mathematical chops nor a poverty of mathematical education. In fact it's not a failure at all, because connecting perceived patterns with strings of symbols is

*not an obvious and natural thing for a human to do*.If it

*were*obvious and natural, mathematics would never need to have been invented to make it possible.But mathematics

*has*been invented, and when a connection like this actually does happen, it feels really really nice.posted by flabdablet at 4:06 AM on August 31, 2020 [6 favorites]

*algebra is a way of representing relationships between categories of quantities, and using that relationship to determine one quantity given a value for the other.*

I love this and am cribbing it.

posted by aspersioncast at 8:06 AM on August 31, 2020

The relationships that algebra can manipulate don't even need to involve categories of

If those rules can be defined in ways that share a few vital features with those of the algebra that schools generally teach, that same kind of algebra can work with these new categories of thing every bit as well as it works with quantities, meaning that

That struck me as a

It finds practical application in e.g. digital signal communication and processing, where you can use pretty much all of arithmetic to derive truths about groups of symbols that

posted by flabdablet at 9:08 AM on August 31, 2020

*quantities*. They can involve categories of damn near anything, as long as those categories are well-defined enough to allow for the creation of similarly well-defined rules that apply to the objects they categorize.If those rules can be defined in ways that share a few vital features with those of the algebra that schools generally teach, that same kind of algebra can work with these new categories of thing every bit as well as it works with quantities, meaning that

*the same theorems that work for quantities will work for the new categories as well.*That struck me as a

*deeply*weird and beautiful idea when I first encountered it.It finds practical application in e.g. digital signal communication and processing, where you can use pretty much all of arithmetic to derive truths about groups of symbols that

*are not*numbers, in that they don't represent quantities or measurements, even though they look*just like*numbers when we write them down. If this didn't work, neither would CD drives or hard disks or phone systems or*anything*that relies on error detection and correction codes. But it does, and they do. And so does modern cryptography, which is built on the same foundations. It's all rather gorgeous.posted by flabdablet at 9:08 AM on August 31, 2020

Category theory shows up in a fun way in software. Consider the following example. You ask a student:

"Did you do your science homework?"

"Yes!"

"Did you do your math homework?"

"Yes!"

"Did you do your English homework?"

*five seconds goes by* "Yes!"

Same replies, word for word. But there's not just what, there's also when. Of course, spoken, they are not the same replies, pressure wave for pressure wave. Not how speech works.

So the question of whether two things are equal is of practical concern. AI needs to know multiple expressions of the word Yes are the same word, security needs to *keep* you from knowing multiple expressions of the word Yes are *different* words (kid's lying!).

That's what category theory is trying to systematize.

posted by effugas at 12:31 PM on August 31, 2020

"Did you do your science homework?"

"Yes!"

"Did you do your math homework?"

"Yes!"

"Did you do your English homework?"

*five seconds goes by* "Yes!"

Same replies, word for word. But there's not just what, there's also when. Of course, spoken, they are not the same replies, pressure wave for pressure wave. Not how speech works.

So the question of whether two things are equal is of practical concern. AI needs to know multiple expressions of the word Yes are the same word, security needs to *keep* you from knowing multiple expressions of the word Yes are *different* words (kid's lying!).

That's what category theory is trying to systematize.

posted by effugas at 12:31 PM on August 31, 2020

Roughly also on the topic of categories, and swinging that back around to where mathematics comes from:

It's long seemed to me that we use what most people consider to be perfectly ordinary numbers in two quite different ways, and it's a difference that's usually not made explicit and therefore often leads to confused thinking.

On the one hand we use numbers to

On the other hand we use numbers to

And since in general there's no guarantee of

Which gives rise immediately to the rationals, and then - when we start wanting to measure stuff like the diagonals of squares or the circumferences of circles against the standard of their own widths, which we eventually work out we

I love the idea that we invented "reals" in order to deal with technical issues arising from things that

And having invented these things we then tie ourselves up in conceptual knots with issues like how many of them there must be compared to the ordinary counting numbers, and just what we think the property of being "continuous" might imply. At which point we're forced to abandon

Theologists arguing about angels and pinheads have

posted by flabdablet at 1:11 PM on August 31, 2020 [2 favorites]

It's long seemed to me that we use what most people consider to be perfectly ordinary numbers in two quite different ways, and it's a difference that's usually not made explicit and therefore often leads to confused thinking.

On the one hand we use numbers to

*count*things. One potato, two potato, three potato... The lineage of numbers descended from this use case is the integers.On the other hand we use numbers to

*measure*things. Measuring is kind of like counting but it has an extra step: instead of counting things directly, we're counting how many of some predetermined measuring standard - that is, some*other*thing that's not necessarily even*here*when we're doing a measurement -*would be*required to match whatever aspect of reality we're measuring.And since in general there's no guarantee of

*any*tidy relationship between the things that we measure and the standards we measure them against, we start to need the kind of numbers that can let us account for*parts*of things.Which gives rise immediately to the rationals, and then - when we start wanting to measure stuff like the diagonals of squares or the circumferences of circles against the standard of their own widths, which we eventually work out we

*can't do*with rationals - the reals.I love the idea that we invented "reals" in order to deal with technical issues arising from things that

*are not*real. Nature*contains no*perfect circles, much to the disappointment of Platonists; and when conducting any*actual*measurement we always*can*and always*do*end up using a rational that's close enough for all practical purposes.And having invented these things we then tie ourselves up in conceptual knots with issues like how many of them there must be compared to the ordinary counting numbers, and just what we think the property of being "continuous" might imply. At which point we're forced to abandon

*any*attempt to take inspiration from the world of the senses and have to start creating and publishing entirely new rules, in an attempt to keep our self-generated paradoxes as contained as we can; and mathematics gently pushes off from its berth in the real world and sails out into the infinite oceans of creativity like the art form it undoubtedly is.Theologists arguing about angels and pinheads have

*nothing*on Cantor.posted by flabdablet at 1:11 PM on August 31, 2020 [2 favorites]

How cirtcular does a thing have to be, to be a circle?

If you say absolutely perfect, then your math can say nothing about nature (which arguably does have circular probability distributions, but let's stipulate here).

That makes your math unable to demonstrate things considered true, unable to analyze and decompose systems that can in fact be modeled. Means there are true things that cannot be proven within your system...

Paging Dr. Godel...

posted by effugas at 1:50 PM on August 31, 2020

If you say absolutely perfect, then your math can say nothing about nature (which arguably does have circular probability distributions, but let's stipulate here).

That makes your math unable to demonstrate things considered true, unable to analyze and decompose systems that can in fact be modeled. Means there are true things that cannot be proven within your system...

Paging Dr. Godel...

posted by effugas at 1:50 PM on August 31, 2020

*That's what category theory is trying to systematize.*

Category theory is a bit more general than that. In various branches of math, we look at collections of mathematical objects of some type or other, and look at the relationships (of which equivalency or sameness is one possibility) between them. Category theory then takes those and says, how does this set of relationships between these mathematical objects relate to this other set of relationships between those different mathematical objects? It's kind of metamathematics, in some sense - a third or higher level generalization beyond the ways that various branches of math already generalize mathematical objects or relationships.

So a better example might be as follows. First ask if you figure out how to get the computer to distinguish between those different "Yes!"es, is that the same or different from the process that would distinguish between different "No!" answers; or between different "That's what she said." answers, etc. Then ask how the process for distinguishing between different plant species relates to the process for distinguishing between different animal species, different fungus species, etc. Now category theory would say: how does comparing the taxonomic identification processes for different kingdoms of organisms relate to comparing the taxonomic processes for distinguishing between different uses of the same word or phrase in response to questions?

posted by eviemath at 3:18 PM on August 31, 2020

evie,

Oh, of course. Equality (complete or partial) is but one relationship, and it doesn't precisely define inequality (because "no data" exists too). This shows up pretty frequently now, as much of machine learning is determining metric spaces in which similarities can be recognized and manipulated. The common example is word vector spaces in which King - Man + Woman = Queen, ***numerically***. This emerges, probably biologically as well.

You might have been joking about That's What She Said, but it is one of my favorite papers.

The main thing for me is that cateogories are contextual. Things can be equal, or not, depending on the context or scale in which they're analyzed. Corporations are fairly predictable on the scale of months, but workers come and go all the time. Their heart rates are pretty predictable on that timescale, but individual cells of the heart might well die or go irregular.

The same exact point in spacetime, over even the same period of time, can only be measured for randomness or unpredictability given a definition of context. What category are we going to consider this point in? It can be in any of them.

posted by effugas at 4:50 PM on August 31, 2020

Oh, of course. Equality (complete or partial) is but one relationship, and it doesn't precisely define inequality (because "no data" exists too). This shows up pretty frequently now, as much of machine learning is determining metric spaces in which similarities can be recognized and manipulated. The common example is word vector spaces in which King - Man + Woman = Queen, ***numerically***. This emerges, probably biologically as well.

You might have been joking about That's What She Said, but it is one of my favorite papers.

The main thing for me is that cateogories are contextual. Things can be equal, or not, depending on the context or scale in which they're analyzed. Corporations are fairly predictable on the scale of months, but workers come and go all the time. Their heart rates are pretty predictable on that timescale, but individual cells of the heart might well die or go irregular.

The same exact point in spacetime, over even the same period of time, can only be measured for randomness or unpredictability given a definition of context. What category are we going to consider this point in? It can be in any of them.

posted by effugas at 4:50 PM on August 31, 2020

*>How cirtcular does a thing have to be, to be a circle?*

If you say absolutely perfect, then your math can say nothing about nature (which arguably does have circular probability distributions, but let's stipulate here).

That makes your math unable to demonstrate things considered true, unable to analyze and decompose systems that can in fact be modeled.

If you say absolutely perfect, then your math can say nothing about nature (which arguably does have circular probability distributions, but let's stipulate here).

That makes your math unable to demonstrate things considered true, unable to analyze and decompose systems that can in fact be modeled.

Actually, it's the opposite of what you are saying.

The systems that math has built up to deal with this kind of approximation or error or uncertainty are in fact made to deal

*exactly*with all the various degrees of approximation that you are talking about.

So just for example, the whole machinery of calculus, which is designed to deal with exactly the type of approximation you are talking about, will work whether your measurements and your needed degree of accuracy are within a light year, an astronomical unit, a million miles, a thousand miles, a league, a kilometer, a meter, a yard, a foot, an inch, a millimeter, a nanometer, an angstrom, an attometer, a yoctometer, a planck length, or whatever uncertainty or approximation of length or value you need to work with--from the very largest to the very smallest.

And that is exactly the consideration that ends up bringing in the nasty old "real" numbers and other such complications. Because you don't want a system that can deal with just a few selected cases out of the infinite number of possibilities.

You want to be able to deal with each and

*every one of them*, from the largest to the smallest.

And that is exactly what the real numbers do.

(And interestingly, if what you are talking about is being able to handle

*every*fraction and size of thing from the largest to the smallest, and a few other handy-dandy properties like keeping the usual ordering of the fractions, being able to take the square root of every number, and complete other useful, everyday operations of that sort, and not have any obviously "holes" or "gaps,"

*unique*set of numbers that completes and fills out in the rational numbers in a sensible way.)

In some sense it is the need (or wish) to make universal system that can indeed handle many, many different scenarios--both anticipated and unanticipated--that forces you to go beyond integers and a few basic fractions all the way to the "real" numbers.

You can make do without the real numbers--just use some of the most common integers and a few well-selected fractional measurements--and in fact civilizations did so for many thousands of years.

But a system that can deal with

*all*such eventualities is far, far more powerful. And finding exactly such "universal" solutions is in fact much of the drive behind mathematics.

You can say this is a bunch of irrelevant hoo-haw but in fact the ability to understand and calculation with and communicate with and about numbers and measurements and fractions and concepts of a much, much wider range than ever before is one of the things that makes the modern endeavor of science and technology

*possible*.

This isn't just theoretical. Take a look at this page outlining what numbers and values can be written using Roman Numerals--integers as high as a few hundred thousand and certain selected fractions as small as 1/2304. And that's it.

Now imagine trying to do any kind of modern science, technology, or engineering using

*only*those numbers. Now try to imagine making a computer that would somehow work using

*only*those numbers.

You literally couldn't.

posted by flug at 7:12 PM on August 31, 2020 [3 favorites]

My name is not evie. My metafilter nickname is eviemath. All one word. Thanks.

posted by eviemath at 8:22 PM on August 31, 2020 [3 favorites]

posted by eviemath at 8:22 PM on August 31, 2020 [3 favorites]

There also seems to be some use of "categories" or "category theory" in programming that is different from the category theory of mathematics, which may be where the disconnect in our descriptions is coming from,

posted by eviemath at 8:27 PM on August 31, 2020

**effugas**? Or there is a whole bunch of detail that I'm not familiar with that connects the two but that is being left out here because it's metafilter not a math conference and we're really going off on a derail from the main topic by now:Pposted by eviemath at 8:27 PM on August 31, 2020

*the nasty old "real" numbers*

Just want to clarify that I in no way think of the reals as nasty. The idea of them is just magnificent. But that doesn't change the fact that the reason they were invented (and they

*were*invented, not discovered) was to deal with apparent paradoxes inside mathematics rather than any difficulty with making useful numeric measurements of real

*things*. Which is why I take so much pleasure in their name.

*finding exactly such "universal" solutions is in fact much of the drive behind mathematics*

Mathematics is one of the very few primarily technical disciplines in which it's absolutely a

*good*thing to be an Architecture Astronaut.

posted by flabdablet at 8:47 PM on August 31, 2020

*it's metafilter not a math conference*

☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕☕

no, wait.

☕...

posted by flabdablet at 8:49 PM on August 31, 2020

*It's kind of metamathematics, in some sense - a third or higher level generalization beyond the ways that various branches of math already generalize mathematical objects or relationships.*

I'm pretty convinced that at this point in the human timeline

*most*of mathematics is some kind of metamathematics, and that it probably pays to stay a bit aware of that so as not to end up experiencing anything more troublesome than genuine delight when encountering this kind of paradox.

posted by flabdablet at 9:01 PM on August 31, 2020

I'd love to have a time machine to send the tiktok in this post and tons of youtube videos (3blue1brown, numberphile, etc...) to my younger self before I got disillusioned by math (or the conventional way to study math).

Math education does not only suck in the USA, it sucks almost everywhere. Unless you are lucky enough to find a passionate mentor.

Among other things, I dropped out of engineering school when I started asking this type of questions and got this type of reactions not only from peers, but from professors too.

The last Electrical Engineering class I took, Circuits 2, I got told by my professor that my "stupid questions" were just an excuse to "buy time", and I should focus my efforts in just "memorizing the damned formulas and doing your homework". I switched to Chemical Engineering believing that it would be closer to "the truth" and not full of arbitrary conventions made up by rich dudes in the 19th century (pause to allow the chemists to laugh or cry at the naiveté of the young) and did pretty well, until I started getting the same reactions to basic questions. One of the last times I went to office hours the professor said "Just visualize the reactions, leave the questions to the physicist" and the assistant looked at me (long hair, long beard, wearing huaraches) and said "Or to the metaphysicists, the look more your type".

I did well on math and science, I loved them, but I was too scared to go into Math or Physics, they are reserved for geniuses you know. Some of my peers did, and ended up with PhDs and stuff like that.

I quit STEM and got a degree in graphic design.

It took me a long time to overcome the inferiority I felt as a graphic designer when talking to my topologist friend or my particle physicist neighbor.

Now I realize that I did well and enjoyed design because I was always looking for patterns and relationships, solving for unknown quantities, categorizing, abstracting, generalizing, doing math. But as a designer I felt that was cheating, those were my secret crutches I had to use because I did not have the innate sensibility and artistic talent of my peers (95% of graphic design problems can be solved to the satisfaction of 95% of clients by following some really simple mathematical relationships, it is the remainder that really really exercises the mind of the designer).

Every 5 or so years I remember that I used to be able to do calculus and differential equations, that I could do linear algebra and really get why e^iπ + 1 = 0 is a poem. I go reread some books, skip the really hard parts, and rest easy for another 5 years knowing that if I really wanted, I could still do it.

The funny twist in the story for me is that for the last few years I have been leading a data science and engineering team. Now I have a bunch of really smart people in whose best interest it is to explain somewhat advanced math in simple and engaging terms to their graphic designer boss. The number 1 rule in the team is that there are no stupid questions.

posted by Dr. Curare at 9:04 PM on August 31, 2020 [7 favorites]

Math education does not only suck in the USA, it sucks almost everywhere. Unless you are lucky enough to find a passionate mentor.

Among other things, I dropped out of engineering school when I started asking this type of questions and got this type of reactions not only from peers, but from professors too.

The last Electrical Engineering class I took, Circuits 2, I got told by my professor that my "stupid questions" were just an excuse to "buy time", and I should focus my efforts in just "memorizing the damned formulas and doing your homework". I switched to Chemical Engineering believing that it would be closer to "the truth" and not full of arbitrary conventions made up by rich dudes in the 19th century (pause to allow the chemists to laugh or cry at the naiveté of the young) and did pretty well, until I started getting the same reactions to basic questions. One of the last times I went to office hours the professor said "Just visualize the reactions, leave the questions to the physicist" and the assistant looked at me (long hair, long beard, wearing huaraches) and said "Or to the metaphysicists, the look more your type".

I did well on math and science, I loved them, but I was too scared to go into Math or Physics, they are reserved for geniuses you know. Some of my peers did, and ended up with PhDs and stuff like that.

I quit STEM and got a degree in graphic design.

It took me a long time to overcome the inferiority I felt as a graphic designer when talking to my topologist friend or my particle physicist neighbor.

Now I realize that I did well and enjoyed design because I was always looking for patterns and relationships, solving for unknown quantities, categorizing, abstracting, generalizing, doing math. But as a designer I felt that was cheating, those were my secret crutches I had to use because I did not have the innate sensibility and artistic talent of my peers (95% of graphic design problems can be solved to the satisfaction of 95% of clients by following some really simple mathematical relationships, it is the remainder that really really exercises the mind of the designer).

Every 5 or so years I remember that I used to be able to do calculus and differential equations, that I could do linear algebra and really get why e^iπ + 1 = 0 is a poem. I go reread some books, skip the really hard parts, and rest easy for another 5 years knowing that if I really wanted, I could still do it.

The funny twist in the story for me is that for the last few years I have been leading a data science and engineering team. Now I have a bunch of really smart people in whose best interest it is to explain somewhat advanced math in simple and engaging terms to their graphic designer boss. The number 1 rule in the team is that there are no stupid questions.

posted by Dr. Curare at 9:04 PM on August 31, 2020 [7 favorites]

*If you say absolutely perfect, then your math can say nothing about nature (which arguably does have circular probability distributions, but let's stipulate here).*

That makes your math unable to demonstrate things considered true, unable to analyze and decompose systems that can in fact be modeled. Means there are true things that cannot be proven within your system...

That makes your math unable to demonstrate things considered true, unable to analyze and decompose systems that can in fact be modeled. Means there are true things that cannot be proven within your system...

Again, I blame Descartes and his ilk (and he definitely

*had*an ilk, make no mistake about that; in fact even though he's long gone, his ilk thrives to this day, running in thundering herds through the groves of academe) for any distress that people experience on finding out that this is so.

In a healthy mind it

*should*be axiomatic that reason is not the only tool in the drawer, and that attempting to use it

*instead*of intuition and empathy and exercise and love, rather than alongside those things in mutually supportive ways, should be

*expected*to yield limited results.

Having experienced being forcibly hospitalized as a result of an attempt to do that very thing myself, I can attest to the fact that my mind was absolutely

*not*healthy at the time.

*Way*too much being lost in self-generated beauty, not enough grounding. It's a hazard.

posted by flabdablet at 9:48 PM on August 31, 2020 [1 favorite]

*Anyway, I also stan this kid, and wish my education had been different. Because I suspect I’m not *actually* bad at math.*

I'm

*really*late to this thread, but my personal philosophy is that no one is bad at math, they were just taught badly. Being naturally good at math can help you make it through a year with a bad teacher, but even that won't save you sometimes.

*"Mathematics is the queen of the sciences and number theory is the queen of mathematics."*

But these days I'd be surprised if even a lot of number theorists wouldn't award the Math crown to algebra.

But these days I'd be surprised if even a lot of number theorists wouldn't award the Math crown to algebra.

Another personal philosophy of mine was that all math is algebra. Even the highest order proofs or the most complicated integrals are all manipulated with algebra. Calculus is algebra taken to higher levels. It is the heart of all existence.

I have my undergraduate degree in mathematics and my very favorite of all my classes was Number Theory. So much fun! (My second favorite was intro to real analysis: we got to prove that 1 + 1 = 2! Super hard but super fun). I have a Bachelor of Arts instead of Science because I didn't want to take a programming class (that I'd already failed once) but I like that. Math is an art that describes the universe.

If anyone is looking for a movie version of the history of math, I recommend The Story of 1 (narrated by Terry Jones!) that was my first introduction to the history of numbers...4 years after I finished my education. Really blew my mind and I loved it!

posted by LizBoBiz at 5:10 AM on September 2, 2020 [2 favorites]

LizBoBiz:

I have dyscalculia. It's a thing.

I was relieved to find out that it existed at all: apparently I was not just lazy or just stupid. But, yes, I am innately bad at math. There's a good chance I would have done a bit better if I'd had better teachers, but most likely I would still have been bad at math.

posted by Too-Ticky at 11:19 PM on September 3, 2020 [1 favorite]

*my personal philosophy is that no one is bad at math, they were just taught badly.*I have dyscalculia. It's a thing.

I was relieved to find out that it existed at all: apparently I was not just lazy or just stupid. But, yes, I am innately bad at math. There's a good chance I would have done a bit better if I'd had better teachers, but most likely I would still have been bad at math.

posted by Too-Ticky at 11:19 PM on September 3, 2020 [1 favorite]

Thank you for your comment, Too-Ticky. I don’t want to derail this thread but also want to acknowledge how much I learn from fellow MeFites. I did not know about Dyscalculia until you posted. Much appreciated.

posted by Bella Donna at 12:28 AM on September 4, 2020 [2 favorites]

posted by Bella Donna at 12:28 AM on September 4, 2020 [2 favorites]

I loved math as a kid, right up until the calculus switched from "this is how this works from first principles" to "remember this formula for solving this form of the equation, no I can't explain why, it's too complex". Engineering math killed it outright for a good 5-6 years until I joined the games industry, where I am now a very happy professional applied mathematician.

I had a couple of very good teachers, and a very fortunate route through some experimental coursework, that became utterly derailed when I hit academia's idea of what industry wanted of me.

posted by inpHilltr8r at 7:05 AM on September 5, 2020 [1 favorite]

I had a couple of very good teachers, and a very fortunate route through some experimental coursework, that became utterly derailed when I hit academia's idea of what industry wanted of me.

posted by inpHilltr8r at 7:05 AM on September 5, 2020 [1 favorite]

« Older Rewriting Country Music’s Racist History | District 9 was Packed Newer »

This thread has been archived and is closed to new comments

posted by Sokka shot first at 11:30 AM on August 29, 2020 [9 favorites]