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1: Forbidden Formulas (Elitism in Math)
3.8k Plays8 years ago
From Pythagoras to Einstein, from the banks of the Nile to the streamlined curves of the Large Hadron Collider, math has shown itself again and again to be fundamental to the way that humans interact with the world. Then why is math such a pain for so many people? Our answer is simple: math is, and always has been, in one way or another, guarded as an elite skill. We visit the worlds that were shaped by math, the secrets people died for, the false gods created through this noble science, and the gradual chipping away of this knowledge by a people who have always yearned for this magical skill. So what is it? And how can we make it better?
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[Featuring: Sofía Baca, Gabriel Hesch; Amy Lynn, Ian McLaughlin]
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Transcript
Introduction to Breaking Math
00:00:00
Speaker
Hello, my name is Jonathan. And my name is Gabriel. And you're listening to Breaking Math. We don't want to make a big deal out of the first episode, but we thought it'd be nice to explain why we're doing what we're doing here. And we think you'll be pleasantly surprised that you find something great. Here at Breaking Math, the goal for the podcast is to make mathematics, even higher concepts in mathematics, so enjoyable that you can literally bring it up in a conversation at a party. And yeah, we know how that sounds.
00:00:29
Speaker
It works though. I brought up math while flirting and it actually worked. Wow. That's amazing. And if you're the kind of person who doesn't have anything to talk about on a date or at parties, why not make it math?
Math's Ubiquity and Future Topics
00:00:42
Speaker
Now, the Platonic ideal that has been embodied perhaps most powerfully since the Enlightenment, with the advent of the Mathematicization of physical laws, is something that we assert exists in every topic. That's to say, math is not only in everything, but is everything, and if you understand anything, you understand math without even knowing it.
00:01:01
Speaker
Every other week, we're going to be covering an advanced mathematical topic in a way that anyone who can spare 45 minutes can begin to grasp. That is our aim. In the next few episodes, we intend to cover Shannon's information theory, which is a really cool way to measure the amount of interesting things in the universe. We plan to talk a lot about the history of computation. Now, you might think that this is just computers in the last 100 years or 200 years or so.
00:01:28
Speaker
This actually goes back 40,000 years. We also plan to talk about chaos theory. And yes, as we all know, that's where butterflies' missing flaps may change where tornadoes happen. And we also intend to talk about the unreasonable effectiveness of mathematics in physics. Or, why is math often invented before it has a physical use and vice versa?
00:01:51
Speaker
we would actually like to take on the daunting task of bringing all of these things together into a topic all about human consciousness. And we know this is going to get a little bit weird, but we will still teach you things that are all firmly rooted in mathematics. And we hope that anyone in eighth grade or above will be able to enjoy these episodes.
Guest Appearances and Amy Lynn's Introduction
00:02:11
Speaker
And those of you who are more technically minded, we will have a write-up in other multimedia online or on Facebook. We'll give you the link at the end of the episode.
00:02:19
Speaker
And in every episode, we will have a guest or two joining us. Some will be mathematicians, some will be engineers, some will be artists, and some will be your everyday laymen. We'd like to hear from everyday folks who may not have a mathematical background. Exactly. Today, however, we'll be talking about the history of elitism in math. And joining us today, will you introduce yourself, Amy?
00:02:41
Speaker
My name is Amy Lynn. I actually got my Bachelor of Science in Mathematics at the New Mexico Institute of Mining and Technology, New Mexico Tech, down in Socorro. That's where I know Gabriel. That's how I got roped into this. We're Facebook friends. I am a math teacher, so I went to grad school for two years before realizing that I was here because I wanted to teach. I went back and got certified in secondary education.
00:03:07
Speaker
And I teach at a public charter school here in Albuquerque, the public academy for performing arts. So I know lots of artists who are afraid of math and that's why I'm here. I want to help people to understand math better so that they don't have to be afraid of it. Now, having been acquainted with the concept of the show, what do you hope people will get out of this?
00:03:27
Speaker
I'm hoping that people will appreciate mathematics the way, not necessarily exactly the way I do, that I understand it at a level that I think most people don't, or at least some branches of mathematics, but I think math is really beautiful.
The Beauty and History of Math
00:03:40
Speaker
And when people are afraid of it, they don't ever get to experience that beauty. And for some of our listeners who may be new to the concept of mathematical beauty, can you give us a real quick rundown?
00:03:52
Speaker
Well, to me, math is beautiful because it's perfect that it is a closed system, depending on the branch you're working in. But it's a beautiful system that theoretically is perfect, right? Real world is messy.
00:04:08
Speaker
Nice, nice. You know, I'm very excited about your expertise and your perspective, and this brings back the whole idea of how it is that we're going to make math accessible to the crowds. One thing I did want to mention, obviously, one goal is to talk about it on this podcast by varying topics and varying perspectives. The hope, perhaps someday, we can actually have a YouTube channel where we've got videos as well as visuals in general that will also help with that concept.
00:04:34
Speaker
In the meantime, I'm really, really excited that we have math educators with us. And anything you hate about math? I am not a big fan of probability and statistics. I took one class, and that was plenty. There are so many different branches of mathematics. That's another misconception that people have, that math is this one specific topic, and it's multitudes. And so I was never very good at probability and statistics, and so I didn't like it. It's the grammar of math in my mind.
00:05:01
Speaker
Now without further ado, the history of elitism in math. From Pythagoras to Einstein, from the banks of the Nile to the streamline curves of the LHC, math has shown itself again and again to be fundamental to the way that we interact with the world. Then why is math such a pain for so many people? Our answer is simple.
00:05:26
Speaker
Math is, and always has been, in one way or another, guarded as an elite skill. We will visit the worlds that were shaped by math, the secrets people died for, the false gods created through this noble science, and the gradual chipping away by a public who, knowingly or not, has always yearned for this magical skill. So what is it, and how can we make it better? All of this and more on this episode of Breaking Math.
00:06:02
Speaker
Hello. Once again, I'm Gabriel. And I'm Jonathan. And you're listening to Breaking Math. Our topic today is elitism in math. In short, with elitism, it's this concept that something is not accessible to the masses. It's only accessible to a select group of people. There are many, many ways that elitism manifests itself in mathematics and in many other topics, of course.
00:06:27
Speaker
And basically the reason why is because it's useful. People guard things jealously, it's a resource and if you can restrict it, then it's an elite skill and it doesn't have to be.
00:06:39
Speaker
And of course, in our current podcast, we have a lot we plan to talk about that support the idea of an elitism with mathematics. One example nowadays that we have mentioned before is when you research an article in Wikipedia in mathematics or in other topics, the elitism is that the jargon is so, so thick that you really have to have a firm background in mathematics to make any sense of it. And therefore, it's very inaccessible to many people. Wouldn't you all say that's a modern example?
00:07:09
Speaker
Yes. Yeah. And in fact, again, our goal is to actually combat that. So I really look forward to introducing a concept that may be jargon heavy, but then saying in other words, and as we said earlier, I think our mission is to really bring this down to an eighth grade level so that our listeners by and large should be able to understand these concepts and maybe then they can even look at the Wikipedia article and make more sense of it.
Math in Ancient Civilizations
00:07:35
Speaker
Yeah, we're going all the way back 100 centuries to the advent of agriculture and the flooding of the Nile. Floods in certain types of rivers occur periodically, so periodically that they can be predicted by the movement of the stars.
00:07:49
Speaker
And this was one of the first things that people noticed. It was a boon to agricultural business. One crucial discovery that led to widespread agriculture was just having walls on the banks of the Nile that tracked the comings and goings of the water. And with just the ability to count and be patient, mathematics was in a way born. Some people say it had more to do with fortune telling, but that's a topic for another episode.
00:08:18
Speaker
Wow, interesting. So agriculture and flooding, and really it just comes down to counting. So there were those who basically wised up and they were able to count, was it calendar days? Is that right? Yes. It was the ascension, I believe, of Sirius that was the sign of the flooding of the Nile that would deposit silt.
00:08:38
Speaker
Okay. And then basically through people who were able to keep track of the days. So really, if it comes down to establishing a calendar, you could predict when the floods would happen. And that obviously gave you an awesome sense of power. If you knew a flood was going to happen on whatever day, you could then be, how should I phrase this? You could threaten people with punishment from the gods if they don't comply.
00:09:02
Speaker
and then say, you know, the gods are going to flood this whole area. And then obviously, you knew the flood was going to happen, so you had that insider knowledge, and then the people were forced to believe you. And they say, wow, this person has a direct line to God. He knows the future. He can predict floods. But of course, the reality is that floods are annual and they're predictable. So that
00:09:23
Speaker
That's a ferocious power. And it's simple too. It's only counting. They don't actually have any direct line to God. They're not better. They're not elite. All they know is how to count. So that's amazing. But it was this sort of patience that really lay the foundation for mathematics.
00:09:41
Speaker
Right across a Mediterranean from the Kingdom of Egypt lay a very different sort of place, where ideas rather than dynasties distinguish subculture from subculture. But not all was rosy in such a place. People suffered, fought, and even died to defend ideas which would seem innocuous to the modern mind.
00:09:59
Speaker
And of course, what we're talking about is the kingdom of ancient Greece. And I think that it's very, very hard to do ancient Greek mathematics justice in a single podcast. Our entire podcast, all of the episodes could be about ancient Greek math, and we would still not touch the surface. No, Amy, any views on ancient Greek math?
00:10:20
Speaker
Well, what I find really interesting is that the transfer from Egyptian mathematics to Greek mathematics was a transfer from application to theory, that mathematics really took a leap, that Egyptians were about, you know, I need to predict the flood and then once there was a flood I need to be able to draw borderlines on whose land was whose, that they used
00:10:45
Speaker
geometry very effectively in ancient Egypt and it was the Greeks who kind of took that learning and then from there went from application to the more advanced mathematics that we attribute to the Greeks of theory and knowledge for the sake of knowledge.
00:11:02
Speaker
One thing that that immediately reminds me of is, in the 19th century, how number theory was purely a game, and now we use it every day for credit card transactions, for bank statements. Anytime you do anything private online, you're using mathematics from the 1800s that was just done by people who appreciated the beauty of math.
00:11:24
Speaker
That's so cool, that's so cool. I just saw a recent TED Talk, in fact, where they talked about that we have no idea the applicability for the sake of knowledge. That is, if you do research, you can't always say, this is for this purpose. And in the case of mathematics, oh my gosh, I mean, not to jump topics here, but there's so many examples of discoveries in mathematics that have an amazing relevancy later on, as you had said.
00:11:51
Speaker
And I think this is very true when you're trying to prove something. It's like trying to find your way to the store in a car where your first stop, you end up at the park. Second stop, you're on the moon. Third stop. Oh, you finally at the store. I like it. Now we have a short story about Meno Slave by guest Ian McLaughlin. Gabriel met him on the campus of UNM.
00:12:16
Speaker
Yes, I was very, very excited to run into Ian. I literally was walking around UNM and I was asking folks who I ran into if they'd be interested in being on a podcast about mathematics. Lo and behold, I run into the student Ian McLaughlin, and he is a philosophy and English major, but he was very, very happy to share with us the story of Socrates and Mino's slave.
00:12:39
Speaker
Here we are talking with Ian McLaughlin again, the English and Philosophy major, and he said in my conversation, especially for our podcast on math and elitism, that it would be great to talk about the story of Socrates and how he taught geometry to a slave. I'd love to hear your story. Okay, so in the Mino dialogue, Socrates is talking with Mino.
00:13:00
Speaker
And Mino says that slaves can't learn math because it just has something inherent to do with them being a slave. And Socrates, as usual, plays devil's advocate, just keeps asking
Socrates and the Accessibility of Math
00:13:13
Speaker
questions, and eventually he calls one of Mino's slaves over and he tries to teach him a little bit of geometry.
00:13:20
Speaker
So he draws a square in the dirt on the ground, and he tells the slave each side of this square is one unit. And that means that the square has an area of one square unit. How would we make a square with an area of two square units? Well, the slave, not having any math training, did what
00:13:38
Speaker
most would do, and drew another three squares around it, giving it two units on each side. Well Socrates says, that's a good try, but that has an area of four square units because it's got four one-unit squares. And so he says to the slave, okay, so I'll show you how to double the area. And he draws a square using the diagonal of each of those four squares that the slave had drawn.
00:14:05
Speaker
And he says, see how this is exactly half the size of the one that you drew because it cuts each one in half on the diagonal? And the slave says, yeah. And he says, well, what does that mean? He says, it means it has half the area, which means that the area is two.
00:14:23
Speaker
And so and I think it's funny that they don't have a way to measure the length of that size in Greek in ancient Greek mathematics. But but there we go. And then the slave was able to repeat back the theory behind it.
00:14:36
Speaker
proving that slaves could, in fact, learn, uh, mathematics. Oh, man, gosh, that is so cool. I was doing it in my head as well, and I, uh, uh, I'm not sure if I should fully disclose this, but in my head I thought, well, you know, just, sure, I'll just double the side. Duh! I'm an electrical engineer and graduate student, and just in that story. Okay, wow, that's very cool, very cool. Awesome, dude, awesome.
00:14:56
Speaker
Wow, that was an amazing story. I really enjoyed that story. Now, for our listeners, I don't know if you're doing what I was doing, that is, following along with Ian's story, I tried drawing the square, and I tried to double it. I'm almost embarrassed to say this. I made the exact same mistake that Mino did the first time. Now, full disclosure, I am an electrical engineering graduate student who has a firm, robust background in math, and I made that same mistake.
00:15:24
Speaker
No, Amy, as an educator, I'd love to hear your impression of this.
00:15:30
Speaker
Well, the first time I heard the story, I actually went back to my school, and with my calculus class, I introduced this problem. I was like, you guys will think this is cool. How do you make a square that has two square units? And their immediate thought was the one that I immediately had. Well, obviously, the side length has to be the square root of 2. But how do you do that? How would you manage to do that? Square root of 2 is an irrational number, which cannot be measured. And so how do we create
00:15:57
Speaker
an irrational side length. I was actually interested to see that my students went about it a slightly different way, that they took two square units and then cut those in half and reshaped them into a two by two unit square. So I was pretty impressed that they were able to think outside the box because we get so restricted and I know advanced mathematics that it's going to be the root too, but we don't know how to get there because we don't have to.
00:16:21
Speaker
Now, one thing that strikes me is that what you've described, thinking that the square root of 2 should be immediately the side, is the difference between knowing and understanding. You can know what a diffeomorphism is, but if you don't have a firm grasp on moving surfaces, then there's no way that you can understand certain theorems, you can parrot them.
00:16:44
Speaker
basically. You know, aside from the mathematics itself, which is a really, really interesting topic, I think this is a phenomenal example of elitism in mathematics, because clearly Mino was under the impression that a slave did not have the intellect to understand mathematical concepts. And as Socrates often does, he simply proved him wrong. He proved that a slave absolutely has the ability to understand these mathematical concepts. I mean, that's
00:17:14
Speaker
That's also a great commentary on how strangely we as humans treat other humans and the stories we make up in our mind about other humans in order to feel elite. What a social commentary.
00:17:27
Speaker
Back in the early 1800s, mid-1800s, when psychiatry was in its infancy, but slavery hadn't gone away yet, they invented a disease that apparently only slaves can get, drabitomania, the disease of being lazy, which is just awful in like 8,000 different ways.
00:17:49
Speaker
One thing though, Socrates, the guy from the story, we know that he didn't write anything down and everything that we know about him is secondhand, and it was written in an era where putting words into other people's mouths was seen simply as a rhetorical device rather than plagiarism, and it's a liberty that some argue that Plato took.
00:18:11
Speaker
An interesting takeaway, however, is that whether or not this was a true story, or it was just Socrates fan fiction, it was written in a society that valued knowledge, and it hid it from the lower classes. That continued until the Library of Alexandria and beyond, and we argue today. With information, this is nothing new, obviously. With something like mathematics, there's no doubt that this takes concerted efforts. Concerning elitism, there's even examples of that with regards to the Cult of Pythagoras.
00:18:42
Speaker
Yeah, he was afraid of beans because of flatulence. He was a vegan and a cult leader. Cults, if you don't really know how they operate, quick tangent. They deprive humans of sleeping calories. Now we're going to tell you real quick about the story of Hippasus. Hippasus was a Pythagorean and he was drawn to them like so many others were because they had knowledge and humans are drawn to knowledge.
00:19:08
Speaker
But unlike the others, he felt the need to divulge a deadly secret. The ratio of most diagonals of rational numbers are irrational.
00:19:17
Speaker
We know from school, from elementary and middle school, that there is the Pythagorean theorem. Which, of course, that's when you're finding the length of the hypotenuse of a triangle, the length of the long side of a right triangle. The Pythagorean theorem, as we all know, is the square root of a squared plus b squared is the length of c, or the hypotenuse. That obviously is a tribute to Pythagoras. There's other things that Pythagoras was known for as
Pythagoras and Greek Math Ideals
00:19:42
Speaker
well.
00:19:42
Speaker
Music was his main contribution for many years. He discovered the thing about if you take great shows of small integers, they tend to sound good together.
00:19:56
Speaker
very little doubt that he did not invent the Pythagorean theorem. Interesting. Yeah, actually I have heard varying stories about the Pythagorean theorem. I guess the point is he was firmly established among the Greek mathematicians, which is, you know, quite, that's quite a feat. So even as a person who made contributions to mathematics, he still had a lot of quirks.
00:20:18
Speaker
Oh, he was a kook. Oh, completely. They slaughtered him because he wouldn't cross a field of beans. He was trying to run away and he refused. And that was the end of Pythagoras. Wow, that I did not know. And Hippasus, who we mentioned earlier, was killed by the Pythagoreans, according to stories. Some people say he was thrown off a boat, stabbed. There's many different interpretations. Tell me again why it is that Hippasus met his fate. He revealed the secret of irrational numbers.
00:20:46
Speaker
We mentioned briefly that obviously, as you said earlier, we know that a lot of cults actually can operate by depriving people of both sleep and calories. Now, this is interesting. This actually relates directly to human behavior. That is to say when you're robbed of sleep and when you're robbed of calories, you will be submissive. There's more to it than that.
00:21:06
Speaker
It's almost a survival mechanism. If your caloric intake and if your sleep are controlled, and forgive me, I don't quite know by exactly what degree, but you will be submissive, it's been shown. Yes, which makes sense, evolutionarily speaking, if you're in a situation where there's very few calories, you have to work with your fellow human beings to survive.
00:21:29
Speaker
So that's a way of manipulating, that's a way of controlling people. And that's been shown to be successful in cult. I'd love to have more examples of what we would call cults using that method to control. Pythagoras was in a position of power. As a cult leader, he had many ways of manipulating information. Of course, our main topic today is essentially that, the manipulation of information.
00:21:54
Speaker
That's what elitism is. And Pythagoras, even someone who is now sort of revered as much as he, was at his heart a manipulator, a person who was an elite and a person who would kill people to control the information that he wanted to still control. Elitism was not far from him. But why is it that people were persecuted for revealing the knowledge of irrationals?
00:22:24
Speaker
It was because of the ancient Greek sort of philosophy worldview that these Pythagoreans had that whole numbers were perfect. They believed that the sum of four plus three plus two plus one being 10 was a sacred thing. What an irrational number is is
00:22:49
Speaker
suppose you have two integers, an integer is just a whole number, and you divide one by the other, you'll get another number. You can't do that with every single number. There are certain numbers like the square root of two, where there's no two integers that you can divide that equal it. Okay, very good. And then of course, you know, there's just real briefly, there's many examples of irrational numbers, as we said, the square root of two also pi
00:23:15
Speaker
it's provable that there's more rational numbers than rational numbers. That's fascinating, actually. Correct? Correct. You better make sure of that. So what I wanted to mention about Pythagoras and his followers, he used knowledge as his carrot, right? That he kept people following him by offering them enlightenment, which is not an unknown way to get followers, I think. But it was interesting to me that
00:23:42
Speaker
The Greeks were so enamored of mathematics that they were willing to sacrifice food, sacrifice health and freedom in order to learn.
00:23:53
Speaker
It really does show the essence of mathematical beauty, how it's basically intoxicating. And I really do believe that that's why elitism shows up again and again in mathematics for the reason that you mentioned, that it's such an attractive thing. And that's why it's such a shame that so many people fear it.
00:24:15
Speaker
Yeah, you know, I gotta say, I think for all of us here, we certainly share that Greek idea that the mathematics is beautiful and enchanting. And then again, based on how someone presents it, it could very easily be brought up to an occult-like status, like many things, many things, I know.
00:24:31
Speaker
As a human species, we of course are a pattern-seeking people, and we're always looking for meaning, and if someone can spin a story and use mathematics to back it up, I can very well see how people can be led to elevate people like Pythagoras to a cult-like figure, to a god-like status.
00:24:51
Speaker
Now spanning the bridge of time from ancient Greece to late medieval Europe were a great many empires. They came and they fell. And in that time, much information was lost to the Western world preserved by the Islamic world. During this time, the Aristotelian view of science dominated and was backed up by the Christian church.
00:25:18
Speaker
The quote that we have is from a dialogue concerning the two chief world systems. It's that book that Galileo got persecuted for, and that in itself is an example of elitism. However, I think that you'll find that the content of this quote demonstrates another form of elitism.
00:25:34
Speaker
One day I was at the home of a very famous doctor in Venice where many persons came on account of their studies and others occasionally came out of curiosity to see some anatomical dissection performed by a man who was truly no less learned than he was a careful and expert anatomist. It happened on this day that he was investigating the source and origin of the nerves about which there exists a notorious controversy between the Galenist and Aristotelian doctors.
00:25:58
Speaker
The anatomist showed that the great trunk of nerves leaving the brain and passing through the nape extended on down the spine and then branched out through the whole body and that only a single strand as fine as a thread arrived at the heart. Turning to a gentleman whom he knew to be a Aristotelian philosopher and on whose account he had been exhibiting and demonstrating everything with unusual care, he asked this man whether he was at last satisfied and convinced that the nerves originated in the brain and not in the heart.
00:26:24
Speaker
The philosopher, after considering for a while, answered, you have made me see this matter so plainly and so palpably that if Aristotle's texts were not contrary to it, stating clearly that the nerves originate in the heart, I should be forced to admit it to be true.
00:26:39
Speaker
Wow, that's absolutely amazing. I'm rereading that again as I'm thinking about that. So you literally had an individual here who was clearly shown a dissection and we clearly see that the nerves do not originate in the heart. And yet he still says if it were not for Aristotle's text saying that they do originate in the heart, then I'd be forced to admit that they don't.
00:27:05
Speaker
And I think what this shows is a big problem with elitism at all. Proponents of elitism say that it preserves the knowledge, it keeps the knowledge away from those who would harm it. However, because elitism has such an intricate power structure,
00:27:22
Speaker
It can defy reason. I mean, look at all the ancient mathematicians. They were all in very high positions of power. Archimedes was friends with the king of Syracuse. So we've discussed this before, that as an educator, my perception of that
00:27:37
Speaker
is that there are many who are not able to pursue mathematics or to pursue higher level thinking merely because they are focused on survival. By construction, mathematics and education can be very elitist because only people who have money and who have extra time are able to commit themselves wholly to that education.
00:27:59
Speaker
So there is some intrinsic ability that elite have that maybe working class do not both in ancient times and now that they don't really have the room in their brain or the room in their lives to make mathematics and education a priority. It almost reminds me of the problem of how
00:28:19
Speaker
Poor people can't afford to buy in bulk, so they're stuck in a vicious cycle. It seems like you're suggesting that elitism itself is somewhat self-perpetuating. That's why it's lasted all these millennia.
00:28:32
Speaker
You know, when I hear this story, there's so many things that I'm thinking about right now. And, you know, among what we said here, there's this amazing reluctance to let go of an old worldview. But that's also a huge commentary on how exactly we work, psychologically speaking.
00:28:49
Speaker
How is it that we're so comfortable with a worldview? You know, I mean, you would think that if we're presented with obvious evidence to the contrary, we would give up a worldview, but this is a clear indication that's just not the case.
00:29:05
Speaker
This is something that I find deeply profound. I think of my own biases. This makes me think, do I have my own biases that I simply will not let go of, even when shown evidence to the contrary? It's a phenomenal question. The question this also raises for me is, how do we go about being able to change our worldviews?
00:29:29
Speaker
And sometimes what it needs is a random component not to get too off topic. Socrates, who we keep coming back to, defied the educational system of his time by simply asking questions instead of directly lecturing. Oh, interesting. So maybe in your formative years, when you're introduced to who the authority is, maybe, I mean, this could be an interesting experiment.
00:29:54
Speaker
In psychology, the age of 10 is known as age of mastery. It's where the brain is still very neuroplastic, but it hasn't gone through the metamorphosis of puberty. Okay. So then the stated goal is to raise kids or raise a generation that's able to critically think and to evaluate based on logic. Just like in this quote here, a suggestion would be to have a small element of chaos or a small element of, oh gosh, um, I'm, I'm struggling with the words right here. Um, a little bit of a stochastic input.
00:30:24
Speaker
Can you can we clarify stochastic input stochastic just means disordered random. It's like jostling a bucket of sand to get it level. OK. So then how would that be implemented in an education system such that individuals do not get to where they're susceptible to this elitism to where they see you know one person as infallible.
00:30:48
Speaker
Well, that itself is a difficult question to answer because any biases that a system will have can blind the system to the biases. I mean, by definition, the system would be blind to biases. Even the randomness itself could be randomness that's targeted in a certain direction. If you have the bucket of sand and you're trying to make it level, but you have the bucket of sand not level, it's never going to be level. Wow.
00:31:15
Speaker
As to education and how we can open children up to be able to accept knowledge and not be married to it, to be able to think analytically and make their own decisions, we teach math as a fact. We teach it as a known. We never actually teach children
00:31:38
Speaker
anywhere really I think that math was developed. It was to some extent invented. There's always the argument of whether math was invented or discovered. And I think there's an element of both. But we never teach them that someone had to decide what numbers to use, what these symbols meant, right? Somebody had to decide that a number was its own concept rather than having to be specifically, you know, three represents three apples, right?
00:32:04
Speaker
And if you look at a lot of very old languages, you can see evidence of this. Just in English, we have a herd of sheep, but we have a flock of geese. Different words for plurals get actually to the point where there's different words for two or three objects in Proto-Indo-European. I believe that's correct. Along with the centuries of time between ancient Greece and medieval Europe came a deep deficit of knowledge.
00:32:32
Speaker
For centuries, engineering, cultural, and mathematical knowledge was lost to the west, rare books were bleached and used to reprint common ones, and all knowledge of curatorship was preserved by a loose network of monasteries. It was on the latter half of that history that the peripatetic, that is to say the Aristotelian view, dominated and was supported by states as well as the monolithic medieval church.
00:32:55
Speaker
It was in this environment that thinkers such as Gerbert of Orlock, who was accused of convening with the devil, and Galileo, who was accused of heresy, struggled against elitism.
00:33:06
Speaker
Galileo, of course, was from Italy, and Italy was the heart of the Catholic Church. He had many friends high up in the church, and it was because of this that he felt safe publishing his treatise, the view concerning the two chief world systems. And we're going to read you a little excerpt of that right now, and it demonstrates something very deep about elitism at that time, and perhaps today.
00:33:32
Speaker
This this treatise, by the way, this is this is pretty deep stuff. It's a pretty thick quote here. What's fascinating about this treatise is, as you'll see, it is a treatise between I'm sorry, is it a doctor? And it's well, the book itself is a dialogue. But the excerpt is about a. A person who's a medical curiosity seeker refusing to believe what's right in front of him.
00:34:02
Speaker
Yeah, this is a fascinating quote without further ado. So you could kind of see the situation that Galileo found himself in during that time. It was not pleasant. I believe they put him under house arrest for that treatise. It didn't help that he cast the pope in the simplicio, the simpleton character. He did it as a method of sort of endearment, but that could be taken very badly.
00:34:27
Speaker
I had no idea about that. I mean, I knew the standard. Well, I knew a little bit about Galileo, but I had no idea that he actually cast the pope as a simple thing. That's not going to bode well, especially during that time period when the pope is obviously well-respected. Yeah. The simpleton character, of course, in these old treatises was the one who does the most learning. So it wasn't an unambiguously bad role.
00:34:50
Speaker
Oh, okay, okay, okay. I was not aware of that either. Now, of course, that's one form of elitism. Galileo was very well-connected, and we're gonna look at another very well-connected person who was born about 500 years earlier, Gerbert of Orlock. He was born in the heart of the Dark Ages.
Arabic Numerals and Mathematical Progress
00:35:08
Speaker
He was a clergyman who studied in Moorish, Spain. During this time, he became familiar with Arabic numerals, far superior to the Roman numerals with which his peers were acquainted. The speed with which he was able to compute using these foreign symbols earned him a reputation for convening with the devil. He was even rumored to have a robotic head, which whispered him answers. Now, Amy, what do you think about the ramifications that elitism has for xenophobia with respect to the story?
00:35:38
Speaker
Well, specifically, the fear of the Arabic numerals because they were foreign, because they were different, they didn't come from what was known or what was established is hugely indicative of the elitism, that we don't want anything that isn't ours, that we didn't create, right? And it's a very close-minded approach to learning.
00:36:00
Speaker
which perhaps is why it occurred in the Dark Ages or why the Dark Ages were so dark, that we were so close-minded to other opportunities or other ways of looking at the world.
00:36:09
Speaker
The attitude, of course, seems to be something, unfortunately, human. It continues to this day. People don't like views that they view as foreign. It wasn't really until what Leonardo of Pisa, Fibonacci, more commonly known as, was able to really popularize the Arabic numerals. Now, one interesting thing about Gerbert Oralak is that he did attain a great amount of power. He became Pope Sylvester II.
00:36:39
Speaker
Yet, people were so set in their ways that they still viewed him as convening with the devil. Of course, when the church had more political power, that was more of a common view. So wow, once people peg you as something, it's very hard to change their mind.
00:36:58
Speaker
Of course, this was a time where you had, just a little later, you had Pope Innocent with their incredible amount of debauchery. So it was a different time, a different understanding politically altogether.
00:37:12
Speaker
So what would you say the repercussions are of Gerbert? I'm not sure if it's so much repercussion as it is sort of a barometer for that time because it was so in the heart of the Dark Ages that information really could not be disseminated. He didn't have much of an impact on the way that the Western world viewed Arabic numerals. There are that set in their ways that even the Pope couldn't change that considerably.
00:37:38
Speaker
So when were Arabic numerals come into place? About two to 300 years later with Fibonacci. Okay. You popularize them. Actually Florence, Italy banned them because they thought that shopkeepers would use them to falsify records because I don't know why they didn't think that that could be done with Roman numerals, but there you go.
00:38:03
Speaker
Wow, wow, again, kind of like maybe the American system and the metric system. You know, there's just a lot of resistance to changing to something that can be shown to be more efficient. Well, you could take my gallons of milk from my cold, dead hands.
00:38:19
Speaker
Okay, I've understood. But yeah, I believe it's very similar. So why do you think it took 200 years? Was it just exposure to the Arabic numerals and more and more convincing? Because we discussed the Galileo quote that even though we can see very clearly the evidence that something is true, that something is better, we still accept what was before, what changed that allowed us to take on Arabic numerals.
00:38:43
Speaker
See, I'm not sure if it was economic forces or what, because when the economy was not growing during the Middle Ages, when it was in fact completely stagnant, this was a time where guilds would burn down houses of anybody who was more efficient than the average person. You don't have a real need for growth. You don't have a need for improvement. So there's no real effect, something that can have an improvement in that fashion will make.
00:39:14
Speaker
I think it's sort of like, like Psy, the guy from Korea, he made a bunch of other songs and then all of a sudden for whatever reason some trends and forces happened and then Gangnam Style got billions of views.
00:39:28
Speaker
Well, does it correlate with the Renaissance, perhaps, relinquishing of control of the church over development that was Galileo the tipping point of allowing knowledge to flourish again? If you asked me, I do think that the Renaissance was, it does seem to correlate with that. Yeah, that's when all the merchants started using the Arabic numerals. That's when Leonardo da Vinci's boyfriend invented modern accounting.
00:39:59
Speaker
Moving just a few hundred years forward, we come to Galois. Galois lived in politically tumultuous early nineteenth-century France. Dead by twenty in war, but immortalized by his ideas, one ignorant of elitism would believe that this was inevitable. However, because he lacked much formal education, his notes were often ignored by the Academy because they were too messy. This is the form that we argue that medieval elitism metamorphosed into.
00:40:26
Speaker
Now it's true that during this time there was a lot of people reformulating mathematics, creating a solid foundation for it that had to do with axioms, some which would come crashing down in the late 19th century and early 20th century. I have seen it argued that elitism survives through the overdoctrinization of axiomatic systems.
00:40:51
Speaker
Real quick, just for those who, again, are less familiar with mathematical vernacular, can we talk about what exactly axioms are and what their place is in mathematics? What an axiom is, is a rule that you go by. Let me give you an example.
00:41:08
Speaker
is that a straight line can be continued indefinitely. That's to say that if you draw a line, you can make it longer. That's a really simple, very clear and intuitive idea that's basically an axiom. Would you like to expand on that, Amy? Well, as I recall, that is the one axiom that has yet to be proven. Of his axioms, that is the one that we cannot prove.
00:41:30
Speaker
Yeah, I was just curious for the sake of our listener, you know in fact we could even just mean my perception of an axiom is it's our foundation that we We suppose these things to be true that much of the time these things are obvious And so supposing them to be true does not require a leap in thought But it's from these axioms we can claim other things we can prove other things but the axioms are the foundation and
00:41:54
Speaker
the Google definition of an axiom, a statement or proposition that is regarded as being established. It's accepted or is self-evidently true. And again, I think you said a line goes on forever or two parallel lines will never intersect.
00:42:07
Speaker
An example of a modern axiom is a piano's axioms. They're about whole numbers, and one of the axioms is that if two numbers' successors, five is four's successor, 100 is 99's successor, if they're equal, then the numbers are equal.
00:42:26
Speaker
see it's very obvious and use it to prove other things. You could use that and five other axioms, which I won't go into, but they'll be on the website to prove that one plus one equals two. It's actually kind of fun.
00:42:38
Speaker
Nice, nice. I look forward to that. I look forward to seeing those on the website. Just for some familiar... Just as a fun little side note here, the inductive proof is something that philosophers, my brother is a philosopher and he gets very frustrated with inductive proofs and that's where you, you know, you assume it's true a finite number of times and then show that it's true one more time and therefore it's always true. You have to remember I went to a tech school, so a nerdy place. I went to the bar one night
00:43:02
Speaker
And I won a beer for proving that an odd number squared minus another odd number squared will always be divisible by four. Oh my gosh. Yeah. Amazing. Okay, so that's obviously a challenge for the listeners then.
00:43:15
Speaker
And maybe an explanation of induction that's very obvious is one of my professors called it the domino analogy. If you hit the first domino and prove that it hits the second domino, and then you prove that any domino hits the domino after that, you can prove that all the dominoes fall. That's induction. Nice, nice. I like that analogy.
00:43:35
Speaker
It works. Okay. And then I realized that was a rather lengthy aside to explain what an axiom is. However, I think that's part of our mission is to make mathematics clear. So we know now that an axiom is a statement that we regard as true for the purpose of our proof.
00:43:49
Speaker
And if you don't understand what an axiom is, you don't truly understand what it has to do with elitism.
Controversies and Evolution in Math Education
00:43:55
Speaker
Also, I think we can say that you can't just say, well, that axiom isn't true. Or in other words, how often do people challenge axioms in mathematics? There is a big problem with axioms in the late 1800s. Basically, somebody said,
00:44:10
Speaker
What if there's a group of stuff where the definition of that group of stuff was that it couldn't have itself in it? And because they love playing these games, that's actually Lewis Carroll from Alice in Wonderland hated this kind of math. Oh my gosh. But basically, you can prove that there's a contradiction from some very, very reasonable sounding axioms. Like if you have several groups of things, you could pick one out of everything. That's one of the most controversial ones is the axiom of choice.
00:44:40
Speaker
Oh wow. You know what? That's something that we should also put on the website. This is really, really cool Axiom games. And again, what you said earlier that it's fascinating. These axioms seem self-evident, but when taken together, they fall apart in certain cases. That's when the foundation of math has difficulties, which happens every once in a great while.
00:44:58
Speaker
Well, and it's led to other branches of mathematics that set theory came from that issue with axioms that we had to introduce new axioms to develop set theory. We kept coming up with paradoxical issues that you have the set of all sets that do not contain themselves doesn't contain itself. Yeah. So set, you can think of it as a bucket that you put things in, you could put other buckets in this magical bucket.
00:45:21
Speaker
Oh, and look at this. We even have a definition here. This one is provided by Wikipedia. In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, you know, 2, 4, and 6 are distinct objects when they are considered separately, but when they are considered collectively, they form a single set of size 3, written in braces 2, 4, 6. There you go. There's for our listeners. So we've even delved into what an axiom is and what a set is.
00:45:50
Speaker
And of course, you could tell by the fact that we're talking about the late 19th, early 20th centuries that we're getting into modern elitism. Now, modern elitism has to do somewhat with classical education, how frustration and simple things like childhood ADHD are not tolerated.
00:46:11
Speaker
Do you have anything to add on this, Amy, as an educator yourself? Well, from the transition of the, you know, you talk about Socratic seminar, that knowledge was something that was shared, that was questioned, that the whole Socratic seminar is about asking questions in order to delve deeper into a topic.
00:46:26
Speaker
that we transitioned from that style of learning, of collaborating and questioning into the more recent version of the classical education of being lectured, right? I give you information and you absorb it. That there was very little room for discussion, very little room for any kind of analysis that it was, I'm telling you what is true and you will learn it.
00:46:54
Speaker
Yeah, and of course that doesn't foster critical thinking, except amongst, like you were mentioning earlier, the people who can afford an educational environment that can foster that sort of thing. Well, even now, in modern times, we lecture, we give notes, we tell something that is a fact, and we
00:47:15
Speaker
Certainly in mathematics and to some extent as well in science, we don't question anymore, right? We tell students that they need to think analytically, but we don't ever give them an experience on how to do that or give them the opportunity to do that. We give them an outline of the scientific method and have them fill in the blanks, that they never have to ponder what it is to prove something.
00:47:39
Speaker
And the fact that you brought up the scientific method that's almost completely antithetical to the spirit of the scientific method is to not question it.
00:47:47
Speaker
I actually wanted to talk specifically about how that's currently being addressed. Now, I was a teacher for five years before I became an engineer, and I was a teacher in two charter schools, actually. And one major push, one buzzword that you're going to hear in charter schools is project-based learning. This is the attempt. This is the attempt to give students a position where they don't only take the knowledge that was told to them, but they're given a project with which they then solve themselves and hopefully can see a practical application
00:48:13
Speaker
where they have to make decisions that will then help them to discover the
Relevancy and Techniques in Modern Math Education
00:48:18
Speaker
concepts in math. One of them involved, actually it involves simulating a race in MATLAB of all things. That was pretty cool. I don't have the actual code right now, but do you all do much project-based learning in math at your school Amy?
00:48:31
Speaker
No, there's no time that we have so much we have to get through in a single year. And that is another aspect of elitism that we are expected to teach. And all the research says that relevancy is important to understanding. And then we are required to teach things that are really not relevant until you are able to apply them.
00:48:54
Speaker
Oh, that's so frustrating. It is because all of this is the core of breaking math of our podcast. We want to make math relevant. So for, you know, again, for all the listeners, if we had a call in line, we could have listeners call in and rant about how much they hate math. Well, one thing I wanted to bring up real quick.
00:49:10
Speaker
is Tractenburg Speedmath. This was developed by a man who was persecuted by the Nazis in concentration camp to keep him sane. I taught this to students who don't think that they could do math. I tutor. And they are amazed when they're able to multiply a 30-digit number by 11 in their head. It's a way that has been proven to work. It's been proven to work well. But because of the inertia of elitism,
00:49:38
Speaker
it has not been adopted and improved upon. To multiply a number by 11, write down the number. If you have a pencil ready, write down 1, 2, 4, 7, 5, 1, 1. Okay, got it. I'm just exactly what you said, that random array, right?
00:50:00
Speaker
Yep, now the digits on the right, you just write it down right below it. Now the next one, you just add the neighbor. So you add one plus one, you get two. You add the five in the one, you get six, seven in the five, you get 12 to carry the one.
00:50:17
Speaker
four plus seven, add one, you get 12, carry the one, two plus four, six, seven with the one carried, three, and then one. And that's all there is to it. And it's a list of tricks like this and students love learning them. There's a group of remedial students that Trachtenberg taught that became more successful than the gifted students. We just learned a fabulous method of multiplying very, very quickly. I wish I knew this.
00:50:42
Speaker
Yes. And, um, if you'd like to learn more about it, there's going to be information on the paper for this episode online. The answer that you should have gotten for multiplying 1,247,511 by 11 is 13,722,621. Boom. That's so cool. So look at that. Look at that. Now our listeners, those who didn't know it before are now empowered with that.
00:51:08
Speaker
I'm going to teach you how to square a two-digit number pretty much in your head. Right now, use some paper if you'd like. Is this another one from the same gentleman? Exactly. Trachtenberg speed math. Outstanding. So write down 23, but put the two and the three sort of far away from one another. Okay. Oh, I think I've seen this one. Now, above the three, you write three squared, which is nine, but write zero nine.
00:51:31
Speaker
Can you clarify, you said write, oh, as in write zero nine as the nine, like a two digit nine. Yes. Okay. Okay. I understand. I understand. And then write zero four above the two. So you should have zero four zero nine above the two and the three. Okay. Got it. Now multiply two by three and you get six, double that and you get 12. Okay. Put the one above the four and the two above the zero. The one above the four. Which zero? The zero right next to the nine and the four.
00:52:01
Speaker
Okay, the one above the four, and then the two above the zero that is next to the nine. Yes, to the left of the nine. It's a little tricky. It's going to be very helpful when people can see a file that has this. Oh, yeah. But then you just take the pile of numbers that you made and add it up. You get 925, 529 is 23 squared. When you say add it up, I'm just following your directions as a column. As columns, correct. Okay. Very good. Let me add these up real quick then.
00:52:29
Speaker
So you drag down the nine. Okay. Then you add the two and zero. Okay. Then you add the one and the four. Okay. And then you bring down the last zero. Okay. And you can do this with other numbers for 97. I'll do this one quickly. You could, uh, re-listen if it's too quick. Let's see what the final number should have been.
00:52:50
Speaker
you write eight one above the nine, four nine above the seven, multiply nine by seven and you get 63, multiply 63 by two and it's 126. So you write 126 above the eight one and the four, add that together and you get nine zero, carry the one,
00:53:15
Speaker
three plus one is four and then nine. So 97 squared is 9,409. Wow. 97 squared. That's a really cool technique. Look at that. I just, I just did it on my calculator. You're right. It's exactly 9,409.
00:53:34
Speaker
And on the paper, you'll see how to add together a column of numbers 13 wide and 13 high in just a couple of minutes using the method of casting out 11s.
Changing Perspectives on Math's Role
00:53:48
Speaker
There will be several examples. Oh, this is exciting. This is really cool. Well, what gets me about this? We never get to play in math. It's always work. It's always a job that, oh, man, I got to go to math class, unless it's my class, in which case they're excited because my class is awesome.
00:54:02
Speaker
That's why I was so grateful for having a math teacher as a mother. I never viewed math as a chore. I thought it was fun the whole time. We'd multiply and add and subtract fractions in the car. I got a book on mathematical games, future episode. And it was just part of art for me. Your mom was a math teacher. How cool. So tell me what she taught.
00:54:26
Speaker
She's taught everything up to trigonometry. We should have your mom on an episode, I think. I think the listeners would love that. Meet Jonathan's mother and she could tell us all kinds of embarrassing stories. Okay, I'm done. Well, it's true in education that we don't really encourage students
00:54:44
Speaker
to try things on their own. We give them assignments. We give them tasks. We teach them that this is how it's done. And it really isn't until grad school that they're encouraged to think about it their own way. Even undergrad, you're learning the foundations. You're still learning how to
00:55:03
Speaker
communicate in your field that you never really get an opportunity to discover anything for yourself until suddenly in grad school you're dropped in and you have to come up with this whole project on your own and you have no idea where to start because you're used to someone else telling you what to do. Wow. I know of many students who feel that way when they're doing their PhD dissertation that terrifying freedom of what do I do? Nobody is holding their hand anymore.
00:55:30
Speaker
We never get the opportunity to explore, and so we don't even know how to when we finally are given that opportunity. I mentioned it once or twice, the opportunity aspect of being that nowadays it's middle class and up, that you have the opportunity to explore these, I mean STEM really, that impoverished people just don't.
00:55:57
Speaker
There's a systemic issue. And then beyond that, we have the, if you are not taught the classical way, if you don't have the piece of paper that you're never going to be taken seriously, that that is not so much systemic as it is the elitism that we've inherited.
00:56:13
Speaker
Yeah, you got to have your degree, which I mean, I don't know. So here's my question, though. I understand the frustrations with got to have a degree. And it's also inaccessible when colleges do cost a lot of money. I guess the question is, is it bad to require a degree? Is it a successful measure of some base competence?
00:56:33
Speaker
Well, there's a little bit of movement away from that with places like, I want to say like Google, that if you can prove yourself that you are capable and motivated, you will get a job, especially in the technological age where information is so readily available without having to go through formal education. But I think that mathematics specifically is still pretty separate from that phenomena. Really, the only exception I can think of in recent history is Ramanujan. And he had to have somebody vouch for him who was in the elite.
00:57:03
Speaker
before he was ever taken seriously. And so that, I mean, really is what it's come to. And I think for the most part, it is required that you know the, I guess the language that you're trying to communicate in, that you do need to know the formal classical education version of mathematics in order to be able to communicate whatever it is you've discovered or whatever it is you've theorized.
00:57:27
Speaker
to the rest of the populace. So having that kind of common language is important, and I think going to school to learn that is valid. And maybe the piece of paper as well that you need that piece of paper that says, I know how to talk to you. But it also, I think, can be restricting. Why is elitism so prevalent?
00:57:50
Speaker
and what can we learn from its role in mathematics to minimize its damage? These are questions that circumstance forces us to ask ourselves, and it is up to us as a society to understand the role that freedom should have in education, if freedom is indeed antithetical to elitist corruption. We have explored the businessmen of the Nile, the mental explorations of Athens, the revolutionary thinkers of the Renaissance, and indeed modern students and how they are all influenced by elitism.
00:58:15
Speaker
As much as we have improved upon the barbarism of the past and established a system where at least middle-class people have leisure for thought, we still have a significant way to go. But with every improvement that rids the world of an inefficiency comes new modalities of thought, and new rich histories are written. I am Jonathan. I am Gabriel. And I am Amy. And this has been Breaking Math. A simple model of the weather, populations of prey and predators, leaky buckets on a ferris wheel, and magnets on a pendulum.
00:58:45
Speaker
What do these all have in common? You've heard of the butterfly effect. Bean may not know as much as you think about chaotic systems, and neither do mathematicians. Prepare to learn about one of the most philosophically revolutionary branches of mathematics in the past 500 years. Next on Breaking Math. Chaos and fractals.
00:59:03
Speaker
Well, it's exciting because we don't actually know the repercussions yet that we develop. That's often the case that we develop mathematics before we understand what they're useful for. That's what physicists are for. So we don't know what we're getting into. Like what if by talking about it, what if we make a big black hole in the studio? Are we sure we want to go there? We probably shouldn't do that. That's why it's called breaking math because guys, we broke math.