Mathematics and the Formation of the American Mind

Lee Smith welcomes educator and author Jake Tawney to examine the true nature of mathematics and its role in shaping both the individual mind and the American founding. Challenging the modern view of math as mere computation, Tawney argues that mathematics is fundamentally about truth, discovery, and the formation of human reason; training the mind to think clearly, pursue what is true, and communicate it effectively.

Drawing on the intellectual traditions of the Founding Fathers, the conversation highlights how mathematical reasoning played a critical role in early American governance, including the debate that led to President George Washington’s first veto. As modern education moves away from these foundations, Tawney makes the case that recovering mathematics as a formative discipline is essential not only for intellectual development, but for the preservation of a free and self-governing society.

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Transcript

SPEAKER_02 0:05

To the unstoppable force of American ingenuity, to the sacred inheritance of freedom we must protect. This is our legacy. Join investigative journalist, Lee Smith, on Roots, Rights, and Reason. Powered by America's future.

SPEAKER_00 0:21

Hi, I'm Lee Smith. Welcome and thanks for joining us for this new episode of Roots, Rights, and Reason. This week we're discussing one of the pillars of our educational system, mathematics, and we'll be talking about how it shaped our American political system. In a recent article titled A Seat Divided Cannot Stand, today's guest, Jake Tawney, explains how a difficult math problem divided the founding fathers over a fundamental question about representative government. How exactly do you apportion the number of representatives? The Constitution established that representation should be based on population, but it did not explain how to make that idea work in practice. Should all states have equal power or should representation reflect population size? The 1787 Constitutional Convention determined that in the Senate, each state would have equal representation, but in the House of Representatives, representation would be based on population, without specifying how many total seats the House should have or how to divide them. Populations do not divide neatly into whole numbers. For example, if a state's population entitles it to 13.4 representatives, it cannot actually have a fraction of a representative. So should it receive 13 seats or 14? At first glance, this might seem like a simple rounding problem. However, basic rounding methods can lead to inconsistencies. If each state's number of representatives is rounded individually, the total number of seats in the House may end up being too high or too low. This means that even a straightforward mathematical solution could produce unfair or impractical results. Alexander Hamilton proposed that the total size of the House should first be decided. Then each state would receive a share of seats proportional to its share of the total population. Congress narrowly passed a bill using Hamilton's method in 1792, but Thomas Jefferson opposed it. After all, the system would have reduced representation for Virginia, his home state, and also home state of the President George Washington. Jefferson argued that Hamilton's method relied on fractional calculations that were not clearly authorized by the Constitution. He also claimed that it could allow some states to have more representatives than permitted by constitutional limits. Washington sided with Jefferson and vetoed the bill in 1792, the first presidential veto in American history. This decision reflected his concern for constitutional principles, even in the face of political pressure and regional division. After the veto, Congress passed a new apportionment bill based on Jefferson's method, setting the House at 105 members. However, this hardly resolved the issue permanently. Over time, different methods of apportionment continued to be proposed, debated, and revised. Each method attempted to balance fairness, mathematical consistency, and political realities, but none proved perfect, not even the system we have today. Today's guest is Jake Tawney, author of Another Sort of Mathematics: Selected Proofs Necessary to Acquire a True Education in Mathematics. Jake Tawney, welcome to Roots, Rights, and Reason. Thanks so much for being here with us in this episode to discuss one of the pillars of our educational system. And you argue in your uh in your recently published your great book, Another Sort of Mathematics, that what we're doing right now in school in our educational system isn't really mathematics. Can you explain what you mean by that?

SPEAKER_01 4:15

Yeah, I really appreciate this. And thank you for having me. Yeah, it's helpful to sort of note the origins of the book. Uh, it was inspired by another book called Another Sort of Learning by Father James Shaw years ago. Um, it's a book that I think is now back in print, but for a while, I think me and my friends were driving the price on Amazon up because we kept buying all the used copies of it. And Shaw's thesis was that he was he was writing really a book of book lists. Uh, because his thesis was that these are the things that you should read that you probably were never made to read in all of your education. Um, but these are the things that provide a real education. Now, with sort of the renaissance of classical education, um, in a delightful way, there are lots of schools now reading the things that Shaw says that we should read. Um, but it got me thinking years ago as somebody with a math degree that I think we have a same problem in math, right? There's there's this way in which students go through at least 13 years of math education from kindergarten through 12th grade. Uh more, of course, if you took some things in college. Uh, and and I think, and this is probably the best way to describe this, I think most people cannot describe what an actual professional mathematician does, right? Um, and yeah, usually when I ask this question, uh, I'll get one of sort of three correct, three incorrect answers. You know, one is something that resembles uh someone sitting behind a desk for large numbers of hours finding creative ways to multiply big numbers together. Um and and we we chuckle at that one, but that might actually be the closest to the I'm pretty sure that's what I would think.

SPEAKER_00 5:41

If I was asked that question, that's that that would probably be my answer. That's why I'm laughing. All right, you caught me out right away right away.

SPEAKER_01 5:48

Yeah, for sure. Uh, but but that might be the closest lead to the correct answer. Uh I also get answers that resemble an engineer or a statistician, which of course are great and noble careers, but they're not mathematicians. Um so I'm of course taking for granted that a mathematician does mathematics. And so then this sort of begs the question: if people go through 13, 13 plus years of math courses and they can't describe what a mathematician does, what is it that they're studying? Now, to be fair, uh, it's not that they're not studying math, it's that they're studying sort of a reductive view of it. Um, I think if if your math classes were like mine growing up, it seems to me that what we get is something that blends together computation and application, right? Computation meaning um uh memorize the algorithms, be able to solve these problems, solve these sorts of equations that all look alike. And all of that is good. Like I wanna be clear, all of that is good. There are ways in which every discipline has skills and facts that you need to memorize. Uh, but then the other side of this is application, right? How do we take these problems and apply them to the real world? Now, that's pretty interesting to me because if you kind of think back to all of your math classes and think of those word problems, um, I'm not sure how many of them are actually real world problems. They tend to be sort of overly idealized. Um, I'm I'm not advocating necessarily for better real world problems, and I'm not even advocating that we take those word problems out. But I am saying that that if that's our view of mathematics, it's no wonder that we go through the system and we come to think of this as somehow algorithmic thinking mixed with the ability to apply this to the world. And so then we we tend to justify math by saying, look, you should study this because it's useful. And and my bold claim in the book is that we we don't study math because it's useful. It is useful, and that's an amazing thing, right? It's an amazing thing that in math we study the pure ideals of the heavens, the pure triangles that we've never seen in down here on earth. Um, and yet the earth seems to have been created in a way that speaks out mathematically. So again, hear me, I'm not opposed to applications, but I am saying uh we don't study math because of those things. We don't study because it's useful. Do you know why we study math? Lee? Why we study it because there is something unique in the human soul that can only be satisfied by thinking mathematically. And the audacious thing is, I actually believe that, right? And so it's it's audacious because it that also means the reverse is true. That if we don't learn to think mathematically, then there's something in our soul that's crying out, that's that's missing that ability, right? And so I want to get back to your question then about what is- No, this is this is this is very moving.

SPEAKER_00 8:34

This is this is very moving. This is deeply moving. And, you know, it reminds me the the little bit that I I know almost nothing about mathematics, but I'm deeply moved by what you're saying. But what I I've met different mathematicians before, and of course, you know, reading up on how a lot of these people understand, they know that they can do their gifted mathematicians at in their teens, right? And also I understand that people by the time they're in college, I met one guy, I remember one guy who'd gone to Princeton as a mathematician, and he was extremely gifted, but he understood being surrounded by all these other guys who were geniuses, he understood that he was never going to be one of the top mathematicians. So he m he moved to the world of finance. So what you're talking about, what you're talking about, this, this, the a vision of heaven, uh this mathematical view. How do people come to this understanding at such a young age of who they are and their relationship to mathematics? I mean, like really gifted mathematicians, which you clearly are too, to be able to talk about it like this, it's it's it's beautiful.

SPEAKER_01 9:40

Well, I'm not sure I'm a gifted mathematician. I certainly appreciate that, that moniker. Um, but I have thought, you know, quite a bit about the nature of the subject, uh, you know, particularly being in schools, uh, teaching at the high school level, uh, teaching a little bit at the college level. Uh so it's something near and dear to my heart. Um, I I think people come to understand their relationship with mathematics wherein mathematics is presented for what it is, right? Which really, in some ways, gets back to your original question. What is the discipline? First and foremost, math has real objects, right? It's there are things of study. It's it's not, as I said, uh an application for things of the world, right? That's a secondary thing that's important, but it's secondary. Math has real objects. A triangle is a real thing, a number is a real thing. Um, a structure, a mathematical abstract structure is a real thing. So that's first and foremost. It's not different than science in that way. And then the mathematician's job is to walk around that structure, getting this view and that view to discover true things about it, right? We we walk around the triangle, we look at it from this view and we realize its angles will always add to a straight line. We look at it from another view and we say, oh, if it's a right triangle, then it satisfies this Pythagorean relationship that has something to do with squares. And we come to discover these true things. And then here's the important thing: at the end of it all, we prove our results. A professional mathematician is a discoverer and a professional proof writer. And so the closest thing that most of us had in our K-12 education to math is actually the study of geometry, right? Where we get the, I mean, at least back when I was in school, we were still doing some proofs. Now, I'm not sure we taught those proofs exactly correctly, but at least we were getting the inklings of what a mathematician does. So, so in this way, and I don't mean to blur the two, because I there is an important conversation to have about inductive and deductive reasoning and the difference between math and science, but there is a way in which we can see them in some ways as the same thing. They both have objects of study. Those objects of study are wonderful and have amazing properties to them, and our soul yearns to know, just as the historian's soul yearns to know facts of history. Uh, and then once we discover that truth, it is incumbent upon us to share it with others. And there comes the mathematical proof. The ability to say once and for all with certainty that this fact about triangles is true. And here's the argument because once I've discovered it, it's incumbent upon me to share it with others, because truth is effusive. That's the beautiful vision of mathematics. And I think people who who who had that vision at one point, because of this teacher or that teacher who really got it, they're the ones that end up being inspired to then pursue mathematics. I'm not sure that it's necessarily just the intellectual level that gets people to aspire to be mathematicians. I think it's the it's the the view of what the discipline is, right? Uh, and and that's a view that should be given to all students from kindergarten through calculus and beyond.

SPEAKER_00 12:43

Your student, your students must love you. I wish that I'd had you as a math teacher. I mean, not that I would have had a career in it, but just the way you talk about it is so is it it's it's very moving. And I like the way that you talk about and the book when you talk about the proof. It's like that that is your once you come upon something that's a true thing, it's sort of your duty to share it with other people, to share it with the rest of the world. Say, look, I found something. And that's true with other things too, right? Like if this is why people publish poems, not just to get jobs and universities, but to say, I've I've I've put something beautiful into the world. I want to share it with you. I want other people to know about it.

SPEAKER_01 13:23

Yeah, it's also why rhetoric is the end of the arts of language, right? In the classic seven liberal arts, the first three are the arts of language, the trivium, uh, the grammar, logic, and rhetoric. And it's really interesting if you think about it. Why would the arts of language not stop at logic? Sort of like grammar is our way of sort of using putting together, this is oversimplifying, but putting together uh words in a in an organized way to make a coherent thought, right? And logic is the way in which we string thoughts together to make a coherent argument. And kind of at that point, we have truth. But why did why did the the ancients and the medieval say, no, no, that's not good enough. It's not good enough just for us to have truth. The arts of language end in an act of rhetoric. And rhetoric is a term that needs redeemed in our culture, right? We think of the the rhetoric in a bad way, right? We use the phrase mirror rhetoric. If you use words to convince people of false things, that's not rhetoric. That's not even mere rhetoric. That's that's what I would call anti-rhetoric. And so that's as true in political speech as it is in academic historical research as it is in the act of mathematics itself. That the end of all of those things is the communication of the things that you have found to be true. It's it's like at the end of the Mino when Socrates uh says, says to to Mino, you've now discovered some things that are true. Go off and convince others of the things that you have found.

SPEAKER_00 14:49

That is that that is greatly moving. Look, I I I want to talk about uh a generation of Americans who did understand this was a part of their education, this was a part of their classical education, uh, as well as their study of the ancients. Mathematics was something that the founding fathers understood. And as you write, uh, as you write in a great new article, uh a seat divided cannot stand, you write about a very important mathematical problem that was presented to our founding fathers. And this was, and I mean, this is the the principle of our political system, representative government. But how do you apportion representation across the population? So can you talk about that, the, the, the, the exact mathematical problem that the founding fathers faced and how they tried to solve it?

SPEAKER_01 15:44

Yeah, Lee, this is this is wild. And it's wild that our generation doesn't even know about the problem. And and there's good reasons for that. It was sort of in some ways at least temporarily put to rest in in the 1940s. But at the at the dawn of the country, as we know, the House of Representatives was to be the people's house, right? Before the the popular election of senators, the senators actually represented the states, right? They were appointed by the state legislators. But it was the it was the House of Representatives that was the paragon of what it means to have a representative democracy. Um but uh and so they they kind of knew how they wanted, the founding fathers knew how they wanted this to go, right? The the idea is that the larger states get get more representatives, the smaller states get smaller representatives, so smaller and larger, of course, meaning in terms of uh the number, the the size of the population of those states. Um so so they they wrote this into the Constitution, right? They wrote this into Article I, Section Two. Uh, and here's where we read that uh representatives and direct taxes shall be apportioned among several states which may be included in the Union according to their respective numbers. So that's the language of proportionality. Now, they said later on in that in that section, the actual enumeration shall be made within three years of the first meeting of Congress. So they kind of punted until they could get an accurate census, right? And and so the rest of that section says, look, until then, here's the number of representatives each state gets, right? But it ultimately becomes up to the Congress to do this and to solve this problem. Now, there are a couple other restrictions in that in that same section. The section says that the number of representatives shall not exceed one for every 30,000. Um, and this is meant to limit the size of the house. We we didn't want a house that started as the country grew to have, you know, 3,000, 10,000, 20,000 members. That's just not helpful. Um, and then of course it says each state must also get at least one representative. So no, no matter how you do this, you can't do this in proportion where somehow some small state loses representation altogether. Now, now look, that that's a pretty, pretty, in some ways, I think, simple problem to understand. But but here's the issue. If you look at the population at the time, um, you know, from 1790, it's it's about 3.6 million, right? So if you use the number 30,000, you get kind of a maximum house size of 120. So we'll just play with that number for a little bit. So when you think about proportionality, what this means is a state like the Virginia that has about 17% of the nation's population, they therefore, if you're going to legislate a house size of 120, they should get uh 17% of the the the 120 seats. The problem is that number comes out to be about 20.8, 20.9. And so now what do you do? Now that may be an easier case. Most people would say it's 20.9, give them 21 states. But but what about another state that's entitled to 14.6 or 14.4? What do you do with this? And and so our our natural tendency, usually if I have more time or kind of with a group of teachers or students, um, I kind of have them play with this. And almost everybody comes up with the same first answer, which is let's just conventionally round, right? If it's greater, if it's 0.5 or greater, round up. If it's 0.5 or if or if it's lower than 0.5, round down. The problem is when you do that, you almost never get the hunt, the size of the house, which is 120. Now keep in mind the size of the house is legislated, right? So the the house is going to say, here's the house size and here's how to split this up. So you could say, okay, fine, maybe the total comes to 119. Let's just change the number uh to 119. The problem is if you then reapportion, you don't get 119 anymore. And so this is a real problem, right? And like now people think about apportionment in terms of drawing congressional districts, right? But that the drawing of districts was not the original problem. The original problem was how do you even decide how many seats each state gets? And so this was a problem at the very foundation of our government, at the at the on the very first moments.

SPEAKER_00 19:57

Well, how how did they solve it? I mean, I the there were there were two, I mean, there were there was Alexander Hamilton who came up with one proposal, and then you talked about Virginia. Virginia was the home state of the then the our first president, George Washington, as well as the very powerful Thomas Jefferson. And I I know that they they that but in your article, you explained how how they opposed Hamilton's proposal. Not surprisingly, that Hamilton and Jefferson were in opposition again. Yeah.

SPEAKER_01 20:25

I mean, it turns out even in something seemingly as objective as mathematics, Hamilton and Jefferson were at odds. Um, yeah, I mean, we we probably don't have time to go through the whole story, but in uh, but it's it's a wild story where Congress is kind of in a deadlock of how to do this. The House and the Senate have passed different versions using some method of how to deal with this. Um, they're not able to come to any reconciliation. Hamilton comes into the to the rescue. He proposes an entirely different method for getting this done. Um, and and of course, it it ends up like favoring his home state over others. Politics is missed in this. Um, and so so that both houses of Congress end up going with the Hamilton plan. And this this bill ends up getting up to Washington's desk. Now, Washington, of course, has a certain amount of allotted time for the Constitution to decide to either sign this bill into law, directly veto it, or just let it become law by letting it sit there long enough through through sort of a pocket veto. Um and Washington uh, you know, doesn't know what to do. There, there are other issues at play here. Some methods favor the South, some favor the North. And so we already see those tensions in the early days of the country. Washington was worried about a highly fragile country in its youth and didn't want to fracture it. And so he sends out uh this this Hamilton bill, which is not really a Hamilton bill, it was the congressional bill that passed. He sends this out uh to his entire cabinets, right? Now, the cabinet at that time uh was small. It really only had uh four people. And he says, What do you think of this? And and they ended up dividing on on uh lines as well. But but Jefferson's response. Is crazy. He looks at Hamilton's method and says, this is repugnant to the spirit of the Constitution. It involves a difficult and inobvious doctrine of fractions that are left unprovided for in the Constitution. Jefferson is making a textual argument. And so he makes an argument in writing to Washington that this is an unconstitutional argument and therefore you should veto it. What's crazy is that Hamilton writes his own support for his own method, right? I mean, everybody knows Hamilton was behind this, even though it's the bill that came up through Congress. And Hamilton says, no, no, no, it is constitutional, and he gives his own arguments for that. And therefore you should respect the will of the people. Now, this is crazy if you think about presidential vetoes now, right? Uh both his cabinet members were making arguments to him that the veto was there in order to strike down a past law because of its lack of constitutionality. That's not how we think of the veto now. We think of the presidential veto as satisfying what, you know, in an ideal way, kind of the will of the people. Ultimately, it's the will of his own party, right? And we think of it as the court who's the one that strikes down laws because they're unconstitutionality. Of course, that's only true because of Marbury versus Madison. That's a whole nother conversation for another time. Um, but that's really interesting to me. And so it's it's the day.

SPEAKER_00 23:41

This was the first veto, wasn't it? This was this was right. Yes. All right. So another historical That's right.

SPEAKER_01 23:47

It's it's completely historical. So the day before this is this is going to be passed into law, uh, the the the president, Washington, reassembles uh members of a cabinet. He he brings in Jefferson, he brings in the attorney general, uh, and and he says to them, Go off and confer with other Virginians and and and come back. Uh and and the story goes that as the attorney general was there advocating that morning to Washington, we we've drafted the veto, we think this is what you should do. Washington, as if to hesitate one more time, says, and do you agree, sir? And and he says, and the attorney general says, Yes, sir, upon my honor, I do. And so as you as you mentioned, at that moment, uh, President Washington signs the first veto in presidential history. So all of you mathematicians and non-mathematicians alike out there, just note that the very first veto in presidential history was over a mathematical controversy.

SPEAKER_00 24:45

Jake Tawney, um I I I I don't I I have a whole bunch of questions, and also I have no questions. I'm just stunned by by how great that story is. Uh, I want to give you the last word. Anything else that we should know about mathematics going forward? About what, yeah, how should we be thinking about mathematics in our daily lives? Not just, you know, not just counting apples or oranges, but how should we see the world? As as if we can't do mathematics, what I loved about what you're talking about is shapes and ideas that are real things, how should we see it? How do you how do mathematicians want uh laymen to go forth in the world?

SPEAKER_01 25:27

Yeah, let me start with that. Uh, mathematicians do want laymen to go forth into the world. That's that's in the essence the point of the story of the apportionment of the house is that these statesmen, these founding fathers, donned the hats, yes, of politicians, but they also donned the hats of mathematicians. They were educated enough in the art of mathematics to be able to think through a real problem of fair division and what that means, right? Uh, so mathematics is not simply for the mathematician. It is you it is something that satisfies something that's unique in all of our souls that cries out, that can only be satisfied by thinking mathematically. And so my final message is if you have never read a math proof, a real math proof, pick up Euclid's elements. Just read the first proposition, just read the construction of an equilateral triangle. And and it's not that long. I'm not saying it's easy, but but all almost all things worth doing are difficult, right?

SPEAKER_00 26:33

Well, you talk about this in the book, you talk about how much Lincoln uh treasured Euclid. And I I that that was uh again, that that was terrific and and and very moving.

SPEAKER_01 26:43

Yeah, he carried a copy of the elements with him. He realized as a lawyer, a young lawyer, that he didn't know what it meant to demonstrate. And so he went home and read uh the first six books of Euclid, so he could learn what it means to demonstrate. And he would end up quoting Euclid in several of his his Lincoln Douglas debates. Um so that's that's my closing message, Lee. We are all mathematicians. Embrace the mathematician within you, choose to wonder about math, and and very specifically, if you've never read Euclid, pick up that first proposition. It's it's only half a page long. And if you have to spend three months on it, spend three months on it. That in itself will be rewarding because you will have discovered something that was true before you were born and will be true long after you are dead, so that it can be written on your tombstone. This person proved Euclid's first proposition.

SPEAKER_00 27:33

Wow. Well, that's an assignment. I I'm gonna do it. So the next time we speak, I'm gonna have lots of questions on that. And I hope everyone who's been watching today, uh, this great episode of Roots, Rights, and Reason with our friend Jake Tawney. Just it's fantastic, Jake. Thank you so much for being with us on Roots, Rights, and Reason. And uh, we'll see all of you in our next episode. And and you bet we're gonna be talking to Jake again very soon.