“Think about the knife tip. That is where you are. Now feel with it, very gently. You’re looking for a gap so small you could never see it with your eyes, but the knife tip will find it, if you put your mind there. Feel along the air till you sense the smallest little gap in the world…”

– Philip Pullman, “The Subtle Knife”

The modern era of mathematics began on November 24, 1858, when mathematician Richard Dedekind proposed the first firm foundation for the real number system and, even more importantly, established a new way to think about – or maybe I should say a new way to avoid thinking about – the ultimate nature of mathematical reality.

You may remember from last month’s essay that Dedekind had come up with an axiom called the Completeness Axiom which, in combination with the rules of algebra, will let you prove all the important properties of the real number system that calculus needs. This is the postulational approach: list the properties you want numbers (or points or lines or whatever) to have, postulate that those properties hold, and see what consequences follow. But there’s a problem with this method: how do we know that the postulates are true? As Bertrand Russell wrote, “The method of ‘postulating’ what we want has many advantages; they are the same as the advantages of theft over honest toil.”

For instance, consider the assertion that twice the cube root of 2 equals the cube root of 16. If we postulate the existence of a number x that satisfies x³ = 2 while also obeying the ordinary rules of algebra, it’s easy to show that (2x)³ = (2³)(x³) = (8)(2) = 16. But how do we know that such a number x exists to begin with? It seems a bit like circular reasoning.

Dedekind found a way out of the circle, relying on the gimmick that you can sometimes build a new number system by hitching a ride on an old one.

Longtime readers of this blog will already have seen several examples of this kind of bootstrapping. For instance, the complex number a + b sqrt(–1) can be replaced by the ordered pair of real numbers (a,b) and we can operate on such pairs with new operations ⊕ and ⊗ given by the definitions (a, b) ⊕ (c, d) = (a+c, b+d) and (a, b) ⊗ (c, d) = (ac−bd, ad+bc), and then the question “But what does sqrt(–1) mean?” drops out of view; we’ve got the ordered pair (0,1) sitting in front of us, and when we square it (that is, when we ⊗-multiply it by itself in accordance with our definition of ⊗) we get (−1, 0), which acts just like the ordinary number −1 in a whole lot of ways. So then we switch to writing ⊕ as +, writing ⊗ as ×, writing (−1, 0) as −1, and (why not) writing (0, 1) as i, and suddenly, we’ve returned from our sojourn in Oz only to find that Kansas has changed: now −1 has a square root.

Putting the gimmick differently, to work with the complex number a + b sqrt(–1) compounded of the real numbers a and b along with their imaginary friend sqrt(–1), all you need to know is what a and b are, not what sort of magical creature sqrt(–1) is. So we can banish “sqrt(–1)”, and with it all ontological qualms about imaginary numbers, by avoiding the expression sqrt(–1) entirely and just writing (a, b). This surrogate can be taught how to do everything we could ask a + b sqrt(–1) to do, so it’s all we need.

Dedekind realized that in an analogous way, questions about real numbers can be reduced to questions about rational numbers. But before we get to that, let’s restate the quandary that Dedekind was in on November 23, 1858, the day before he invented Dedekind cuts.

HOLES EVERYWHERE

The rational number system is full of holes that are individually too small to see but are sprinkled everywhere. For instance, there are rational numbers x satisfying x³ < 2 sitting on the number line and to the right of them are the rational numbers x satisfying x³ > 2, and the two sets nestle together as intimately as two disjoint sets can,¹ but in between there is no rational number x satisfying x³ = 2. That’s an example of hole in the rational number line – it’s the hole in the rational number line where the cube root of 2 should be – and it’s one of infinitely many such holes. Each hole in the rational numbers is associated with a way of cutting the set of rational numbers into a left half that has no largest element and a right half that has no smallest element.

Dedekind’s great idea, phrased one way, was: If we want numbers to fill the holes, and if each hole is specified by a cut, why not operate directly on those cuts? Each cut contains all the information about which hole we’re trying to fill and so can serve as a surrogate for the irrational number we want to fill that hole with. For instance, let L be the set of rational numbers x satisfying x³ < 2 and let R be the set of rational numbers x satisfying x³ > 2. If we let L′ be the set you get by doubling every element of L and R′ be the set you get by doubling every element of R, then L′ and R′ determine another cut of the rational numbers, splitting the rational numbers into those that satisfy x³ < 16 and those that satisfy x³ > 16.

So, doubling the cut associated with the missing number cube-root-of-2 gives us the cut associated with the missing number cube-root-of-16. We don’t have to postulate the existence of the cube root of 2; we actually construct a surrogate for it in the form of a cut, a way of splitting the rational numbers into two sets. And this changeling does just what we want the cube root of 2 to do, so who cares if it’s really what the cube root of 2 is?

Perhaps you think that what the cube root of 2 really is, what it really must be, is 1.2599…. Never mind that mathematicians have been talking about the cube root of 2 for millennia and have only been using decimals for centuries; maybe you feel comfortable with decimals and want to base the real-number concept on them. You aren’t alone in wishing to do this; in the past century some mathematicians have given constructions of the real number systems that use decimal expansions as their starting point.² But these approaches, in contrast to Dedekind’s sleek method, are cumbersome to work with. Even specifying what it means to double an infinite decimal is a bit awkward because of carries. The usual method of adding decimals says to start at the rightmost position, but with infinite decimals there is no rightmost position! Multiplying two infinite decimals together is even worse than adding one infinite decimal to itself or to another, and proving the theorems of calculus using infinite decimals would be a nightmare. If you want to focus on concepts rather than calculations, Dedekind cuts provide a much better foundation for real numbers than infinite decimals.

Another way to look at what Dedekind did is to think of the Dedekind cut associated with cube-root-of-2 as being something like a question, namely the riddle “What number is bigger than every x satisfying x³ < 2 but smaller than every x satisfying x³ > 2?” In the rational numbers, this is a question without an answer; but we can treat the unanswerable question as a thing in itself, and even as a sort of an answer, much as the question “What number gives −1 when you square it?” gives rise to the symbol sqrt(−1) (which, when you think about it, is just a symbolic statement of the question), and thence to the symbol i (which hides what the original question was), and helps birth a new kind of number.

I’m a big fan of Dedekind, and he’s more or less the hero of my book-in-progress on how the number-concept has evolved to become more inclusive, especially over the past few centuries. But before I tell you why Dedekind matters to me, I should mention a few ways in which the innovation of Dedekind cuts didn’t have all the impact Dedekind may have hoped for. If you haven’t heard of Dedekind cuts before now, the next section will help you understand why that might be.

WHAT CUTS AREN’T

You have questions? Permit me to ask some for you.

What sorts of problems are easier to solve with cuts than with, say, decimals?

I already mentioned that Dedekind’s setup makes it easy to prove the general theorems about real numbers. For instance, if you define surrogate notions of addition, subtraction, multiplication, and division for cuts, as well as a surrogate notion of the relation is-less-than, then you can show that Dedekind’s new number system does indeed satisfy his Completeness Axiom. This is the kind of “honest toil” Bertrand Russell would have approved of, and it’s actually not that hard.³

But for the first few centuries of its existence, calculus wasn’t about proving general propositions like the completeness property of the real number system or theorems like “the sum of two differentiable functions is differentiable”; calculus was about solving problems, and cuts don’t provide any traction for problem-solving. If you want to know the volume of some crazy solid of revolution or get a good approximation to cube-root-of-2 or π, Dedekind cuts won’t help you. The tidiness of his approach comes at a cost. Cuts may give esthetic pleasure to pure mathematicians, but they have little value for an applied mathematician or scientist.

Okay, but they’re pedagogically useful, right?

Given that Dedekind’s original motivation for constructing the real numbers via cuts arose from his teaching, it’s ironic that hardly any teachers use cuts when they teach calculus, at least when they’re teaching it to calculus neophytes. A typical college student is familiar with infinite decimals and feels fairly comfortable working with irrational numbers — rewriting the cube root of 16 as twice the cube root of 2 and computing it to one decimal place, for instance. A teacher who socratically asks a class “But what is an irrational number, really?” is likely to get blank stares in reply.

Graduate students are more receptive to such a question. For them, exploring Dedekind cuts can be a fun digression from evaluating tricky integrals. But at the end of the day, cuts are intellectual scaffolding that gets thrown away, much like writing (a, b) instead of a + bi.

Hmm. Well, at least they’re less philosophically troubling than infinite decimals, right?

Maybe. I agree that there is something troublingly obscure about the decimal expansions of irrational numbers when you consider that for instance no intelligent being in this universe is likely to ever know the googolplexeth digit of π. And googolplex isn’t even that big a counting number, when you consider that there are infinitely many counting numbers that are bigger than googleplex and only finitely many that are smaller.

But if infinite strings of digits trouble you, you should notice that Dedekind’s cuts don’t really avoid infinity either, since the two sets that constitute a cut (the left side and the right side) are both infinite sets of rational numbers. When Dedekind proposed his approach in the middle of the 19th century, set theory hadn’t been invented yet. In fact, his definition of cuts was one of the proto-set-theoretic constructions that inspired the birth of the subject. Not everyone was happy with the carefree way in which Dedekind treated infinite sets; Leopold Kronecker, in particular, found Dedekind’s “new math” suspect. Back then, it was common to distinguish between “potential infinity” and “actual infinity”; the former could be tolerated but the latter was anathema in some quarters. And the left set and right set in a Dedekind cut are both examples of the actually infinite.

I see. Still, Dedekind’s way of constructing the real numbers inspired a lot of constructions of other number systems, right?

You can use a slight extension of Dedekind cuts to construct the extended real numbers — the ordinary real numbers with two bonus numbers +∞ and −∞ thrown in — but that’s not very exciting.⁴ Some people might argue that John Conway’s construction of the surreal numbers by way of left sets and right sets, a century later, drew inspiration from Dedekind, but in fact Conway was led to his construction by thinking deeply about two-player games; it was only when he got deeper into the theory of numerical values of games, and began to think of games as generalized numbers, that he realized that his work could be construed as an extension of Dedekind’s. (For more on surreal numbers see my essay “The Life of Games“.)

In contrast, Georg Cantor’s construction of the real numbers (discussed in my essay “Things, Names, and Numbers“) was enormously fruitful. Cantor considered all possible infinite sequences of rational numbers, winnowed out the ones that diverged in unpromising ways, collected the promising sequences into bunches that seemed to be heading to the same hole, and used these bunches as surrogates for real numbers. This is a special case of the trick mathematicians call completing a metric space, and it’s all over the place in mathematics; it lurked behind the scenes in our discussion of p-adic numbers in my essay “Marvelous Arithmetics of Distance“, and it can be modified to yield Abraham Robinson’s definition of the nonstandard real numbers.⁵

In any case, Cantor’s construction of real numbers as bunches of infinite sequences draws much of its appeal from the fact that infinite sequences are the strawberry jam of the calculus kitchen, to go along with the bread of integrals and the butter of derivatives. In contrast, cuts are not a mainstay of calculus, so bringing them into the foundations of the subject feels a bit strange.

Wow. Well, at least Dedekind’s idea was stunningly original, right?

That’s debatable. One could argue that the core of Dedekind’s insight was already two thousand years old.

BEFORE PI WAS PI

Given that π is both a Greek letter and a number – namely the funky number you get when you divide the circumference of a circle by its diameter – and given that the ancient Greeks studied circles, you might think it was the Greeks who decided to let that letter represent that funky number. But you’d be wrong on two counts. First, the Greeks didn’t use the symbol π to represent that number,⁶ and second, the Greeks didn’t think that what we now call π was a number at all, funky or otherwise.

Back then, numbers were for counting. Fractions weren’t considered true numbers, and certainly things like cube-root-of-2 and π weren’t. The Greeks had two kinds of things loosely corresponding to what we call real numbers: magnitudes and ratios. Magnitudes had units attached to them, such as units of length, so you could add two magnitudes of like kinds but you could only multiply them under special circumstances; for instance, you could multiply two lengths together to obtain an area, but you couldn’t multiply two areas together because what would that even mean? You could sort of divide one length by another, canceling out the units to get something sort of like a fraction, and that’s where ratios came in.

If two lengths A and B are in mathematical lingo “commensurable” – that is, if there’s some fundamental length-unit U such that the first magnitude can be written as m times U while the second magnitude can be written as n times U – then we can just say that the ratio A : B is the whole number ratio m : n.

But what if there’s no such unit? Or, putting it in math lingo, what if A and B are incommensurable? In this case, it’s harder to say what we might mean by the ratio A : B.

We know nowadays that the diameter and circumference of a circle are incommensurable (that’s another way of saying that π is irrational, which Johann Heinrich Lambert proved in the 1760s). The Greeks didn’t know that, but they did know that the side and diagonal of a square are incommensurable (or as we would say, that sqrt(2) is irrational). So if this ratio can’t be written as a ratio of integers, what exactly is it?

The mathematician Eudoxos found an answer that’s curiously similar to Dedekind cuts. First notice that the unit U can be pushed into the background in the case where A and B are commensurable: instead of saying A = mU and B = nU (where mU means a sum of m copies of U and nU means a sum of n copies of U), we can equivalently assert mB = nA (check: mB = mnU = nmU = nA).

In the case where A and B are incommensurable, there are no counting numbers m and n such that mB = nA; instead, we find that for every pair of counting numbers m and n, either mB < nA or mB > nA. For instance, suppose A is the circumference of a circle and B is its diameter. Rather than try to pin down A/B, which would have made no sense to Eudoxos because A and B are magnitudes, not numbers, and so cannot be divided by one another, Eudoxos would compare mB with nA for different pairs m,n. For instance, in place of saying that A/B lies between 3 and 22/7, Eudoxos would say that 3B < A but 22B > 7A.

Bringing the commensurable and incommensurable cases together, Eudoxos proposed that, for any fixed pair of lengths A and B, we can divide all possible pairs m, n into three classes (the second of which might be empty): pairs satisfying mB < nA, pairs satisfying mB = nA, and pairs satisfying mB > nA. Eudoxos said that when you know which pairs m,n fall into those three respective categories, you’ve pinned down which ratio you’re talking about.⁷

In the case where A and B are incommensurable, there are no pairs m, n satisfying mB = nA, so the three-way classification of Eudoxos becomes a two-way classification; for each pair m, n we need to know whether mB < nA or mB > nA. If we now permit ourselves to think of the A and B of Eudoxos not as lengths but as real numbers and so allow ourselves to divide one by the other, then the two cases mB < nA and mB > nA can be rewritten as m/n < A/B and m/n > A/B. So in a certain (anachronistic) sense, Eudoxos is saying that you know which ratio you’re talking about if you know which fractions m/n are less than it and which fractions m/n are greater than it. This dichotomy is just Dedekind’s, dressed in a chiton.

For a visual explanation of how the π of Eudoxos compares to the π of Dedekind, say we sketch all the points (x,y) where x and y are positive integers, coloring the point blue if xD < Cy and red if xD > Cy (where C and D are respectively the circumference and diameter of a circle), as in the dichotomy of Eudoxos. Now draw a line from each colored point (x,y) to the origin (0, 0) and color the point (x/y, 1) (where that line crosses the line y = 1) the same color as (x,y). You’ll find that the rational number x/y is colored red or blue according to whether x/y < π or x/y > π, as in the dichotomy of Dedekind.

Now I’ll leave out the points (x,y) and just show the colored points on the line y = 1.

And lastly I’ll show what we would get if we let x and y vary over all positive integers: a Dedekind cut in which every rational number (at least every positive one) gets colored blue or red according to whether the number is less than π or greater than π.

The work of Eudoxos appears in Euclid’s Elements, written over two thousand years ago. So you could say that if Dedekind’s revolutionary idea was new, it had been new for a long, long time.

WAIT, SO WHY THE FUSS THEN?

What’s missing from Eudoxos is Dedekind’s brashness. Where Eudoxos says “Here’s how you can recognize when A is to B as C is to D”, Dedekind says “Here’s what an irrational number is.” The essay in which Dedekind publicized his approach to irrational numbers appeared as one part of a book he called “What are numbers, and what should they be?”⁸

Dedekind displayed a similar esprit when he made the ideal divisors of Kummer less metaphysical (see my essay “When Five Isn’t Prime” for more about Kummer and Dedekind). Kummer’s divisors were sort of like numbers (you could multiply them by each other) but also rather different (you couldn’t sensibly add them to each other). What were they? Dedekind realized that if you want to compare two ideal divisors a and b in a number ring, what you need to know is which elements of the ring are divisible by a and which elements of the ring are divisible by b. More specifically, the ideal divisor a is equal to the ideal divisor b precisely when the set I, defined as the set of ring-elements divisible by a, is equal to the set J, defined as the set of ring-elements divisible by b. So Dedekind took the step of taking the sets I and J as surrogates for a and b.

This led to a broader concept of ideals in rings. A ring (the concept is more or less due to Dedekind) is, loosely speaking, an algebraic system with a notion of addition and a notion of multiplication, and a subset I of a ring R is nowadays called an ideal if the sum of any two elements of I is in I and if every multiple of an element of I is in I. The set derived from an ideal divisor a has those two properties, but Dedekind’s abstract concept is far more general. The influence of this conceptual framework can hardly be overstated; the work of Emmy Noether and Alexandre Grothendieck, to name only two influential 20th century mathematicians, is hard to imagine without rings and ideals.

One offshoot of ideals that has been important in my own mathematical research is the concept of an “order ideal”. These were so-named by the 20th century mathematician Marshall Stone in tribute to the notion of ideals in ring theory. Stone was working in Boolean algebras (for more on Boole see my essay “When 1+1 Equals 1“), which nowadays is seen as part of the theory of partially ordered sets. Just as a ring is an abstraction of some of the features of + and ×, a partially ordered set is an abstraction of some of the features of <. A subset I of a partially ordered set S is an order ideal if, for each element a that belongs to I, every element of S that’s less than a is also in I. In the case where our ordered set S is the set of rational numbers, order ideals should smell familiar: an order ideal in the set of rational numbers is nothing other than the left half of a Dedekind cut.

Yet as important as cuts and ideals a la Dedekind are, I think Dedekind’s greatest legacy was a whole style of doing math, as embodied in Dedekind’s slogan “Numbers are free creations of the human mind.” We have the freedom to collect rational numbers into pairs of sets, to call those pairs “cuts”, and to introduce operations on cuts; or to collect algebraic numbers into other kinds of sets, to call those sets “ideals”, and to introduce operations on ideals; or design other assemblages, introduce yet more operations on them, and study those operations. These creations may draw inspiration from our intuitions and from the real world, much as our perceptions of magnitude guided the Greeks to invent ratios and later guided Dedekind to invent cuts; or they may draw inspiration from constructs in pure mathematics itself, much as Dedekind invented ideals as a surrogate for Kummer’s ideal divisors; or they may be whimsies, invented for purely esthetic reasons and investigated in a spirit of unfettered curiosity. At the moment we conceive of these structures, they are born into the world of ideas, and all their latent properties are born in that same instant, unknown to us as those properties may be.

David Hilbert wrote, apropos of set-theory, “No one shall expel us from the paradise which Cantor has created for us.” But set theory in its early days was as much Dedekind’s as Cantor’s. Cantor’s work was extravagant and mind-bending and showed that the concept of sets could lead to wondrous and terrifying conclusions; Dedekind’s use of sets was more pedestrian, and was confined to more infrastructural concerns such as “What are counting numbers?” and “What are irrational numbers?” If mathematicians like Hilbert came to see set-theory as a paradise, it wasn’t because set theory, taken to extremes, was stranger than any math that had come before; it was because the language of set theory was profoundly useful in clearing away metaphysical cobwebs. Thanks to Dedekind’s kind of set theory, you don’t need to know what irrational numbers “really” are, whatever that would mean; it’s enough to build surrogates for them, and set theory provides us with an ample set of tools for building not just irrational numbers but p-adic numbers and surreal numbers and all kinds of other things we haven’t dreamed of yet.

Just as importantly, Dedekind’s work helped bring about the structuralist approach to mathematics, shifting the focus away from individual objects — “What is the cube root of 2?”, “What is π?” — and toward networks of objects and the relations among them — “Is the cube root of 2 less than π?”.

Is a real number an infinite decimal, as most students today think, or is it a cut,⁹ as Dedekind taught his students, or is it an equivalence class of Cauchy sequences, as Cantor described, or is it something else entirely? For that matter, is a complex number an ordered pair of real numbers, or is it a point in the plane, or is it a 2-by-2 matrix?¹⁰ Most modern mathematicians are structuralists, and the structuralist answer is, “Who cares?” That is, it doesn’t matter what numbers “are”; what matters is what numbers do and what is done to them, and since those doings involve other numbers, what matters isn’t the inner nature of individual numbers in isolation but the intricate relationships between numbers, however we may choose to represent them. Structuralism frees us mathematicians from worrying about the fundamental nature of mathematical objects so we can spend more quality time playing with our creations.

Near the start of this essay, I asked the semi-rhetorical question, “How do we know the cube root of 2 even exists?” In the modern setting – in the outlook that Dedekind bequeathed to us – this question is incoherent. No numbers exist in the real world or even exist within our minds except in relation to other numbers. Putting it differently, the question of whether a certain sort of number exists only makes sense within the context of a specified number system. Does the cube root of 2 exist? It depends on what number system you’re working in. Are you working in the rational numbers? Then no, 2 has no cube root. Are you working in the real numbers? Then yes, 2 has a cube root.

But what’s really going on is that questions about existence have been deepened. Does the rational number system exist? Does the real number system exist? The modern answer is, a number system “exists” if it is logically self-consistent. By showing that you can bootstrap a construction of the real number system from the rational number system, Dedekind showed that the former is just as consistent, and hence just as existent, as the latter.

One of the bylines of my blog is “Adventures in fantastic realms you can build inside your head”. Even a mere two centuries ago, this would have been an odd way for a mathematician to describe mathematics. “What is math” (so mathematicians back then might have thought) “but a tool we have developed for better understanding our world?” But somewhere along the way there was a shift. Math is still what it used to be, but it’s also something more. Things we can measure with numbers may exist in the physical world, but cuts in the rational numbers live in our minds, and once we make up rules for how to add and multiply them, or how to add or multiply other residents of our minds, the consequences of our choices live in our minds too. This is the shared inner paradise Dedekind helped create, and no one can expel us from it.

This essay is a supplement to chapter 4 (“Holes Too Small to See”) of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds cool and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!

ENDNOTES

#1. Is the gap between the two sets of size 1/100? No, it’s smaller, since 1.25³ < 2 while 1.26³ > 2. Is it of size 1/1,000,000? No, it’s smaller, since 1.259921³ < 2 while 1.259922³ > 2. And you can replace 1/1,000,000 by as small a positive number as you like, and you will still find two numbers whose difference is smaller still, with one number having a cube that’s less than 2 and the other number having a cube that’s greater than 2.

#2. One such treatment is “The real numbers as a wreath product” by F. Faltin, N. Metropolis, B. Ross, and G.-C. Rota, Advances in Mathematics Volume 16, Issue 3, June 1975, pages 278-304.

#3. For instance, if one irrational number lies in the hole between the left-set L and the right-set R, while another lies in the hole between the left-set L′ and the right-set R′, and we want to say what it means for the first irrational number to be less than the second, we can say

provided that every number in L is in L′ and every number in R′ is in R. This is typical of the way we can take all the familiar operations of algebra and carry them over to our new, mind-made world of cuts. Then, when cuts come to seem like numbers to us – when Dedekind’s mind-toys become real to us – the circle around the “<” can fade away, and we’re back in a new Kansas that has irrational numbers in it.

#4. If the left set L contains all the rational numbers and the right set R contains none of them, then the cut given by L and R has many of the properties we’d want +∞ to have; let L be the empty one and you’ve got a surrogate for −∞.

#5. Nonstandard real numbers, along with surreal numbers and many other things, are topics I plan to talk about in a second book, possibly called “What Can Infinity Be?” But first I have to write my first book!

#6. British mathematician William Jones came up with that symbol in 1706. Later in the century Euler used it and made it popular.

#7. Here’s the definition Eudoxos gave: “Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.” That is, A and B are in the same ratio as A′ and B′ when, for all choices of m and n, the relationship between nA and mB — bigger, smaller, or equal? — is the same as the corresponding relationship between nA′ and mB′.

#8. Given that Cantor’s definition strikes me as being more or less as good as Dedekind’s, and given that other definitions have been proposed, I imagine that a more apt title for the book would’ve been the more open-ended “What could numbers be?” – which is in fact one of the titles I’m considering for my own book.

#9. Technically Dedekind cuts as I’ve described them give us only the irrational numbers. If we want rational numbers to be represented by cuts, then we represent the rational number r by the set of rational numbers that are strictly less than r. In this set-up, we only use left-sets, not right-sets; the left-sets are called “lower cuts”, and the criteria that characterize a lower cut L are (1) L contains some but not all rational numbers, (2) if the rational number a is in L then every rational number less than a is in L, and (3) L contains no greatest element.

#10. In my essay “What is a Matrix?” I mentioned that “complex numbers correspond to certain special 2-by-2 matrices”; what I neglected to spell out is that if we represent the complex number a + bi by the matrix

then matrix addition and matrix multiplication give us a model of addition and multiplication of complex numbers.