This essay chronicles the liberation of plus and times from narrow notions about what sorts of things can be added and multiplied — and, relatedly, what sorts of things numbers should be. With the advent of abstract notions of fields and rings in the late 19th and early 20th centuries, algebraists took addition and multiplication far beyond the borders of Number and in so doing showed that the border was artificial and an obstacle to progress.
One way I’ll bring a 20th century perspective into focus is to show how an abstract point of view can unify many of the disparate number systems (and not-quite-number systems) we’ve looked at so far, using the simple but powerful idea of modding out, freed from number-centric bias.
As a warm-up to all that, I want to tell you about an under-publicized paradox of mathematical history: The French mathematician Augustin-Louis Cauchy (1789–1857), the person who did more than anyone else to bring complex numbers into calculus, didn’t believe in that most famous of complex numbers, sqrt(−1). (See my essay “Twisty numbers for a screwy universe” if you need a brief reminder about the twisty ways of imaginary and complex numbers.) Cauchy couldn’t have been clearer; he wrote “We completely repudiate the symbol sqrt(−1), abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.”
And yet, even as he rejected sqrt(−1), Cauchy accepted the symbol i that Euler had introduced for sqrt(−1). In fact, Cauchy didn’t just accept i; he glorified it. So what exactly did Cauchy think complex numbers were, or could be?
RECKONING WITH REMAINDERS
Remember modular arithmetic, from my essay “The Triumphs of Sisyphus”? The idea there is that you choose some positive integer n called the modulus, and you look at all the different remainders you can get when you divide an integer by n; for instance if n is 10, the remainders you can get are 0 through 9. Then you can define remainder-ish operations on those remainders. To perform mod n addition on two remainders, first add them as ordinary numbers, and then divide the resulting sum by n and take the remainder. Likewise for multiplication. So for instance in mod 10 arithmetic, 8 times 9 is 2, because the ordinary product is 72 and when you divide 72 by 10 the remainder is 2. (We have no interest here in the quotient; the remainder is where the action is.1)
Cauchy proposed something similar as a way to make complex numbers kosher, except that instead of working with integers he worked with polynomials in the variable x, and instead of dividing by the number 10 he divided by the polynomial x2+1. For instance, in “mod x2+1 polynomial arithmetic”, x+2 times x+3 is 5x+5, because the ordinary product of x+2 and x+3 is x2+5x+6, and when you divide x2+5x+6 by x2+1 the remainder is 5x+5. It’s not a coincidence that this is the same answer you would get from multiplying complex numbers in the usual way; Cauchy showed that this always happens. Consequently, he claimed that the statement about remainders is what we really mean when we say that i+2 times i+3 is 5i+5 (or, as many teachers would prefer that you say, 2+i times 3+i is 5+5i).
Of course, Cauchy’s method wouldn’t be much good if it weren’t compatible with the familiar formula i2 = −1; in fact, his method tells us that the formula is true.2 So what was Cauchy’s problem with defining i to be the square root of −1?
Well for one thing, i isn’t the only complex number that gives −1 when you square it; the complex number −i has the property too. Why does one of them deserve to be called the square root of −1 more than the other? But even more importantly, Cauchy objects to our taking “the square root of −1” as a definition of i. After all, suppose I said “Let ε be a nonzero square root of zero” (even though the only real number whose square is zero is zero itself). What would such a definition mean? It’s possible we can create a number system in which zero has a nonzero square root (in fact I’ll do that later in this essay), but you can’t just wish it into existence; you have to construct it.
The same constructive impulse can be found in Kronecker’s way of handling sqrt(2) and other irrational quantities. Kronecker did important work in algebraic number theory, which is all about irrational numbers that satisfy algebraic equations like x2 = 2, and the dream of his youth was to gain a deeper understanding of these numbers using transcendental functions (like trig functions, only weirder). But Kronecker was uneasy with irrational numbers, even the very ones he was putatively proving theorems about. He never became comfortable with infinite processes such as the systematic but unending process that gives us (ever-better but never-perfect) rational approximations to the square root of 2, and he didn’t want to base his mathematics on things like that, let alone the unruly chaos of the decimal expansion of sqrt(2). The only numbers he thought of as being truly solid were the integers, and he went so far as to say “God made the whole numbers; all the rest is Man’s work.”3
So what did Kronecker think sqrt(2) was? He thought it was a remainder. If you divide a polynomial in x by the degree-two polynomial x2−2, the remainder will be a degree-one polynomial in the variable x, and if that remainder is ax+b, then for Kronecker it signifies a sqrt(2) + b. So like Cauchy’s i, Kronecker’s sqrt(2) was the polynomial x viewed as a remainder, except now the divisor-polynomial was x2−2 instead of x2+1.4 In this way, Kronecker was able to do arithmetic with the square root of 2 without having to worry about where it comes from, and without worrying about whether it’s legitimate to bring sqrt(2) into math by fiat.
Here a caveat is necessary: even though these are the preachings of Cauchy and Kronecker, urging us all to view i and sqrt(2) as remainders under polynomial division, Cauchy and Kronecker did no such thing in their actual practice as researchers. Once Cauchy had proved using remainder-multiplication that i times i was −1, and once he’d proved that his way of adding and multiplying remainders satisfied the distributive property, there was no reason not to use these properties as a way to avoid the tedious process of polynomial division. Likewise for Kronecker; polynomial division was a proof-method of last resort. So their theories about what i and sqrt(2) “really were” didn’t have much of an impact on how they worked with these expressions in practice.
Meanwhile, most mathematicians (including Cauchy and Kronecker on most days) were all too happy to work with numbers like i and sqrt(2) as formal expressions and not worry about what they really were and whether they deserved to be called numbers. They had other puzzles to keep them busy. For instance: as we saw in my essay “When Five Isn’t Prime”, Kummer tried to invent a new kind of number to rescue unique factorization for certain systems of complex numbers, and Dedekind figured out that a better way to study Kummer’s “ideal divisors” was to look at certain nicely-structured collections of complex numbers (like the two shown in the picture) that Dedekind dubbed ideals.
It was unclear back then that Dedekind’s ideals had anything to do with Cauchy and Kronecker’s remainders, but in the 20th century that would change.
NUMBER FIELDS, FIELDS, NUMBER RINGS, …
In 1871, Dedekind employed the German word “Körper” to denote any set of real or complex numbers whose elements we can add, subtract, multiply, or divide at will (as long as we don’t divide by zero) and obtain a number that still belongs to the set. The important examples are the rational numbers, the real numbers, and the complex numbers. The literal meaning of “Körper” is “body” and the word “corpse” is a close linguistic cousin, but I suspect that what Dedekind had in mind was something more like an organization of individuals (as in the phrase “governing body” or indeed the word “corporation”). In 1893 the American mathematician Eliakim Hastings Moore introduced the term “field” as an English-language counterpart.5
The mathematician Heinrich Weber took a momentous step in the liberation of plus and times in 1895 when he defined a field as a collection of elements (not necessarily numbers in the ordinary sense) that satisfy certain axioms. For instance, rational functions of the variable x (that is, expressions like (x+2)/(2x2−3x+7)) form a field in Weber’s sense. In 1910, mathematician Ernst Steinitz went on to make a systematic study of fields of all kinds, including the p-adic fields I wrote about in my essay “Marvelous Arithmetics of Distance”, and he proved some non-obvious constraints on what a field could be (for instance, he showed that the number of elements in a finite field can only be a power of a prime).
You may have noticed that one important number system – the integers – isn’t an example of a field, because a divided by b (with b nonzero) isn’t always an integer. Sets of numbers that we can add, subtract, and multiply at will, but can’t necessarily divide, needed nomenclature of their own. Dedekind had dubbed them “orders”, but his countryman David Hilbert (1862–1943) chose the phrase “number ring” (in German, the compound word “Zahlring”6) in 1892, and introduced the word in his published writings in 1897. But Hilbert restricted himself to collections of complex numbers, so Cauchy’s polynomial-remainders didn’t qualify.
In a way it’s strange that Hilbert, one of the most broadly knowledgeable mathematicians of his day, didn’t see a need for a notion that would encompass all these examples. After all, he was responsible for freeing the words “point”, “line”, and “plane” from fixed meanings in geometry,7 and the compound word “Zahlring” practically begged to be de-compounded.
It was the logician Abraham Fraenkel who first proposed, in 1914, the modern, abstract notion of a ring (as opposed to a number ring). Fraenkel saw no reason to assume that the elements of the ring are numbers; all one needs to know about these assemblages is that the elements satisfy various axioms (listed below).8 But Fraenkel didn’t develop his concept in any detail. What was needed was someone to explore what a ring could be, much as Steinitz had explored what a field could be – someone to complete the arc of abstraction that Hilbert had applied to geometry but not to algebra. That person was Emmy Noether.
EMMY NOETHER
The physicist Albert Einstein called Emmy Noether (1882–1935) a mathematical genius and the greatest female mathematical genius of all time, on account of work she did that was of great interest to physicists (which I’ll say nothing more about since it doesn’t relate to numbers). But Einstein was more of a consumer than a producer of mathematics, so let’s hear from some mathematicians.
The mathematician and physicist Hermann Weyl, speaking at her funeral, credited her with “transforming axiomatic thinking”. The topologist Pavel Alexandroff said, in a 1935 address to the Moscow Mathematical Society, “She taught us just to think in simple, and thus general, terms.” And her disciple Bartel Leendert van der Waerden wrote that one of her outstanding characteristics was “the ability to tirelessly and consistently pursue conceptual penetration of her subject matter in order to achieve utmost methodological clarity.”
Her own assessment of her career as a researcher and teacher, made near the end of her life, was crisp: “I always went my own way.” But this makes her sound like a hermit, and she was anything but: one of her outstanding characteristics was the way she gathered people into community.
Amalie Emmy Noether was part of a small mathematical community from the start; her father was the mathematician Max Noether, and her younger brother Fritz became a mathematician too. Later in life Emmy and Fritz would have friendly arguments over the nature of mathematics; Fritz, an applied mathematician who studied phenomena like turbulence, thought mathematicians were scientists, whereas Emmy thought they were primarily artists.
Emmy’s being a woman placed obstacles in her career path, as did her being Jewish (the former earlier in her career, the latter later). To allow her to continue her schooling beyond the tenth grade, her father had to write to the Bavarian Ministry, obtaining permission for her to audit classes at Erlangen University where he taught. Once her talent and passion for mathematics became clear, her family arranged for her to receive private lessons. It helped to have a slightly younger brother who shared her interest in math and who (being a boy) was allowed to take classes at the Erlangen Gymnasium; they often studied together. After passing her university entrance exams she decided to attend Göttingen University where the mathematical mogul Felix Klein (1849–1925) presided. Enthusiasm led to overwork and exhaustion, so she moved back home and enrolled at Erlangen University in 1904, finishing her degree in 1908. But her time spent at Göttingen, and the opportunity to hear lectures given by the likes of Hermann Minkowski, Otto Blumenthal, Felix Klein, and David Hilbert, had been inspirational.
As Noether continued to learn more cutting-edge mathematics, she began to see affinities between topics that others thought of as quite disparate. For instance, she saw, more clearly and deeply than those around her, that the game Cauchy had played with remainders to make sense of complex numbers was closely related to Dedekind’s way of making sense of Kummer’s ideal divisors. In both cases, you have an algebraic system containing a smaller subsystem. With Dedekind, you had (for example) the set of Gaussian integers a + bi (see my essay “When Five Isn’t Prime”) and contained within it the set of Gaussian integers divisible by 2; with Cauchy, you had the set of polynomials in x and within it the set of polynomials divisible by x2+1. So what if polynomials aren’t “numbers”? They still satisfy most of the properties of numbers vis-a-vis addition and multiplication, so Noether followed Fraenkel in thinking of both the set of Gaussian integers and the set of polynomials in x as being different species belonging to the same genus: both were rings.
A ring is a set R with two operations written as + and ×, and two special elements written as 0 and 1,9 satisfying the following10 axioms:
• For all a in R, a+0 = a = 0+a.
• For all a and b in R, a+b = b+a.
• For all a, b, and c in R, (a+b)+c = a+(b+c).
• For each a in R there exists some b in R such that a+b = 0.
• For all a in R, a×1 = a = 1×a.
• For all a, b, and c in R, (a×b)×c = a×(b×c).
• For all a, b, and c in R, a×(b+c) = a×b + a×c.
• For all a, b, and c in R, (a+b)×c = a×c + b×c.
Of course these axioms are all true if we assume that the elements of R are numbers. But we do not assume that they are numbers. The “0” need not be the number zero; the “1” need not be the number one; and “+” and “×” need not be numerical addition and multiplication. It’s just more convenient and comforting to use those familiar symbols when we’re exploring uncharted territory.
MODULAR ARITHMETIC, GENERALIZED
A key feature of Noether’s approach was the algebraist’s ability to form new rings from old. If you’ve got a ring, and sitting inside it you have a smaller ring of a special kind – the kind Dedekind called an ideal – then you can bring new mathematical entities into being when you mod out the ring by the ideal, or as we say, “form the quotient”.
One bit of 19th century mathematics that we can recognize after-the-fact as an example of Noether’s quotient construction (much as we did with Cauchy and Kronecker’s way of dealing with i and sqrt(2)) is the way Cantor built the real numbers, which I wrote about in my essay “Things, Names, and Numbers”. Here the big system is the set R consisting of all the infinite sequences of rational numbers that have the Cauchy property (these are the sequences of rational numbers that seem to be trying to converge to something) and the small system inside it is the set I consisting of all the sequences of rational numbers that converge to 0. Notice that
(1) Every sum of two elements of I is an element of I.
(2) Every multiple of an element of I is an element of I.
(Here “multiple of I” means “element of I multiplied by an element of R”.) If those two conditions look familiar, they should; they constituted my definition of an “ideal” in “When Five Isn’t Prime”. There, I was a set of numbers; now it’s a set of infinite sequences of numbers. But don’t panic; that set of infinite sequences of numbers is, from Noether’s perspective, just another ring! That is, if you add two or multiply infinite sequences term by term, the eight ring axioms are satisfied. And in that ring R, I is an ideal.
Now, you can’t “divide” R by I in any obvious sense, but remember what we did when we constructed the reals Cantor’s way: we formed “bags”, where two Cauchy sequences go into the same bag when their difference is a sequence whose terms converge to zero. So for instance the sequence .9, .99, .999, … and the sequence 1, 1, 1, … both go into the same bag because the terms of the difference-sequence .1, .01, .001, … go to zero.
And while we’re exploring the “bags” viewpoint, we should notice that one way to think about modular arithmetic is that, instead of throwing away all the integers that aren’t between 0 and n−1, we’re keeping all the integers and putting them into bags according to the remainder they leave when we divide them by n. When we do this, we put two integers into the same bag provided they differ by a multiple of n. That is (phrasing things in terms of rings and ideals), we put two elements of a ring (the set of integers) into the same bag provided they differ by an element of an ideal (the set of multiples of n).
The mathematician and historian David E. Rowe, who wrote the book on Noether (two books, actually), put it well when he described Noether’s mathematical superpower as the ability “to strip mathematical objects down to their bare essentials in order to recognize deeper underlying relationships among them.”
THE ALGEBRAIST’S MAGIC WAND
The quotient construction allows us to make lots of new number systems.
For instance, what if, like Cauchy and Kronecker, we start with the ring of polynomials in x, but then, instead of modding out by x2+1 or x2−2, we mod out by x2−1? We get an obscure but respectable number system called the split complex numbers, also called bireal numbers, double numbers, perplex numbers, or spacetime numbers; they crop up in certain approaches to special relativity. Putting it differently, if we say “Hmm, I wish there were a square root of 1 different from boring old +1 and −1!”, the quotient construction grants our wish. Traditionally the new “number” is called j (not to be confused with Hamilton’s j).
What if instead of modding out by x2−1 we mod out by x2? Then we get the obscure-but-respectable dual numbers, characterized by the property that x isn’t zero but its square is (I promised you we’d devise such a number system). Dual numbers have been applied to rigid body mechanics and to statics. And if you’ve studied calculus, you may have caught a glimpse of the motivation behind dual numbers when your teacher told you “Okay, epsilon is small but epsilon-squared is really small, so even though we’re going to keep track of terms equal to epsilon we’re going to ignore all higher powers, like epsilon squared.” Once again, the quotient construction is like a magic wand that lets algebraists make their wishes come true.11
One big idea of this essay is that the quotient construction unifies many of the different number systems we’ve met in my earlier essays, and this construction has deeper, earlier roots that are worth taking note of. If we step back and view the 20th century not in isolation but as the culmination of half a millennium of progress, we can see that the crucial step in the development of abstract algebra was the acceptance of an indeterminate, x, on an equal footing with ordinary numbers. Once we let x in the door, treating it not as a concession to our lack of knowledge about the value of some quantity but as a strange kind of number in its own right, lots of x’s even stranger friends can follow.
But the central idea of this essay is that at some point mathematicians gave up policing the border between numbers and non-numbers, and it was a happy day for algebra, for mathematics, and for the broader math-using world. If there are specific ways to add and multiply elements, and those operations satisfy certain properties (such as a + b = b + a), then who cares whether we call the elements numbers or not? And the modern notions of rings and fields – rigid as to the properties operations-on-elements must satisfy, but permissive as to the nature of the elements being operated on – carried us into this new way of thinking about math.
When I was in high school and learning from an algebra textbook written by Mary Dolciani, there was a brief passage on “fantasy numbers” right after the section on complex numbers. If you haven’t heard of fantasy numbers, that’s because they’re not really a thing; Dolciani presented them as a short cautionary fable, showing what can happen when a mathematical apprentice misuses the sorcerer’s algebraic wand. She defined the fantasy number f to have the property that f+1 = f, much as she had defined the imaginary number i to have the property that i2 = −1, and then invited us to notice that, subtracting f from both sides of the equation f+1 = f+0 we derive the nonsensical equation 1 = 0. This was supposed to make the student realize that there’s more to inventing complex numbers than wishful thinking; how do we know that allowing i to play alongside real numbers won’t cause the same sort of havoc that ensues when we allow f to play alongside real numbers?
But now we can look at fantasy numbers through the lens of rings and ideals. When we mod out the ring of polynomials in x by the set of multiples of (x+1) − x, we’re just modding out by 1. What does that mean? It means that two polynomials in x should be viewed as referring to the same fantasy number if they differ by a polynomial that’s a multiple of 1. But every polynomial is a multiple of 1! So all fantasy numbers are equal to each other: in particular, the fantasy number 0f + 0 (aka 0) is the same as the fantasy number 0f + 1 (aka 1). But this vexing conclusion is seen to be no more troublesome than the observation that 0 and 1 are equal to each other in mod 1 arithmetic; the contagion of triviality is confined to the quotient ring, where it can’t do any harm. What happens in the quotient ring stays in the quotient ring.
The fantasy-number quotient ring is called the “zero ring”. (It’s also called the “trivial ring”, no disrespect intended.) The zero ring is a necessary triviality, like the number 0. You wouldn’t want to base a whole theory centered on it, but trivialities are a necessary element of any theory that tries to span the gamut from simplicity to complexity, as ring theory does.
I’m not saying you can do everything with modding out (for instance, we don’t get the quaternions or p-adic numbers this way). Still, to borrow a phrase from social media, it’s pretty amazing how much This One Weird Trick can do.
EMMY, INTERRUPTED
If Noether had been a man, we know how her story would have continued; she would have become a full professor at a major German university. But such opportunities were not available to women. After she obtained her doctorate in 1907, she spent seven years working without pay at the Mathematical Institute of Erlangen. In 1915, Felix Klein and David Hilbert invited her to join the stellar mathematics department at Göttingen as a Privatdozent (in modern American terms, an associate professor). Unfortunately, Göttingen was a university divided: the mathematicians and scientists were politically progressive, but the linguists and historians were conservative and opposed the entry of women into university life. Despite Hilbert’s energetic advocacy (in which he colorfully declared “I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, we are a university and not a bath house”), Noether was denied the position; the humanities faculty at Göttingen felt that women were “altogether unsuitable for regular instruction of our students because of the phenomena connected with the female organism.” Also, one of Hilbert’s colleagues demanded to know: “What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?” It took four years before Hilbert and Klein were able to push her appointment through, and a few more years before she was able to begin to collect a small salary as a lecturer in algebra.
In 1920 Noether wrote her now-classic paper “Ideal Theory in Ring Domains”, establishing the modern theory of commutative rings. Soon a community of scholars sprang up around her, and she became known as a champion of the use of algebraic methods in other branches of mathematics. For example, back in the mid-1920s, when algebraists tended not to know the latest results in topology and topologists tended not to know the latest results in algebra, topologists Pavel Alexandroff and Heinz Hopf passed through town and gave talks about Betti numbers (numbers that, informally speaking, count how many whacks it takes to break a topological space into pieces). Noether pointed out that these numbers have algebraic meaning12 and the subject nowadays called “algebraic topology” clicked into gear.
If Noether had been of a different ethnicity, she might have been able to continue to teach math at Göttingen indefinitely (though her progressive political views would almost certainly have gotten her into trouble with the authorities sooner or later), but she was Jewish, and as a mathematics grad student named Oswald Teichmüller wrote at the time, “A German student should not be trained by a Jewish teacher.” (Never mind that Teichmüller had taken a course with Noether and that he ended up basing some of his own work on hers.)
When Adolf Hitler became Chancellor in January of 1933, one of his first actions was a mass expulsion of Jews and other undesirables from government jobs, which included university jobs. By April, Noether had lost her position at Göttingen. Fortunately, by the end of the year she had a position at Bryn Mawr College, and the following year she was invited to lecture at the Institute for Advanced Study in Princeton. But this idyll was short-lived; in April of 1935 doctors found she had developed a large ovarian cyst, and she died during surgery a week or two later.
Noether did not get to see the full flowering of ring theory in the 1940s and 1950s, and its applications to topology, as homology groups begat cohomology groups which in turn became organized into cohomology rings. Rings ceased to be a frontier topic and became bread-and-butter mathematics, part of the material that all graduate students are expected to learn about. In a graduate algebra class you will hear many questions about rings in general and specific types of rings (such as rings of complex numbers, or rings of polynomials and the quotient-rings they give rise to), but you’re unlikely to hear a student ask “Wait, do these things really exist?” and “Are these things really numbers?” Such questions are seen as being part of philosophy, not mathematics, and as having limited relevance to the practice of mathematics. Conversely, Benjamin Peirce’s 1882 theory of associative algebras, seen by many in his time as being more akin to philosophy than mathematics, is now seen as an early flowering of abstract algebra. But whereas Peirce set up his algebras symbolically, Noether made no assumptions about what the elements of algebra had to be. Do you have a collection of objects in mind? Do you have two operations on that collection that satisfy the eight axioms I gave above? Then Noether’s theory has things of value to tell you.
Mathematicians continue to find new uses of ring theory, including many that Noether never dreamed of. An abstract term like “ring” may be initially forbidding (especially if you haven’t seen enough examples of rings) because of its stark generality. In this regard the ring-concept reminds me of the symbol x, which can at first baffle students because of its very flexibility. Variables arose to serve as placeholders; in a similar way, abstract constructs like “ring” and “field” give us a way to set aside a place today for the math we may dream up tomorrow.
Thanks to Jeremy Cote, Michael Gilson, Sandi Gubin, Evan Romer, and Glen Whitney.
This essay is a draft of chapter 15 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. Although I wrote that the modulus should be a positive integer, there’s no reason you can’t let it be zero. When you try to divide a number by zero, the quotient is problematic, but the remainder isn’t: it’s the very number you tried to divide by zero. So “mod zero arithmetic” is just ordinary integer arithmetic.
#2. We replace the symbol i by the variable x, so i times i becomes x times x, or x . Then we have to divide x2 by x2+1. Now you might think that, because x2+1 is bigger than x2, the quotient will be 0 and the remainder will be x2, but polynomial division isn’t like ordinary division in that way. We have to choose c so that subtracting c(x2+1) from x2 causes the x2 term to go away, and the winning choice is c = 1. When we subtract x2+1 from x2, the x2’s cancel and we’re left with the remainder −1.
#3. The original German is “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” This is sometimes translated as “God made the integers, . . . ”, but by “ganzen Zahlen” Kronecker meant just the non-negative integers (or maybe just the positive integers). In an 1887 article Kronecker constructed the negative integers via a quotient construction; specifically, he took polynomials with nonnegative integer coefficients and then divided them by x+1, using the resulting remainders as stand-ins for the integers (so that 3x for instance stood in for −3). Why would Kronecker have constructed the negative integers in this fashion if all the integers were already God-given?
#4. Let’s look at sqrt(2) Kronecker’s way and try squaring it to make sure we really get 2. First we replace sqrt(2) by x; then we square x to get x2; and lastly we divide x2 by x2−2, obtaining a quotient of 1 and a remainder of (x2)−(x2−2) = 2, just as we expected. (There’s a last step that I glossed over, where we must replace x by sqrt(2) wherever it appears in the remainder; but since our remainder in this case didn’t have an x in it, no replacement was necessary.)
You might worry that in dividing polynomials by x2−2 we’re dividing by zero, since x is supposed to be a surrogate for sqrt(2). But we don’t replace x by sqrt(2) until after we’ve done the polynomial division and taken the remainder. A similar issue sometimes crops up in calculus classes, where the derivative is introduced as limh→0 (f(a+h)−f(a))/h; it’s important that we take the quotient before we replace h by 0.
#5. The terms were not completely synonymous; Moore used the term more broadly than Dedekind had, applying it to finite fields as well (as treated in my essay “Numbers Far Afield”). But why did Moore introduce the word “field” instead of sticking with the word “body”? A fun though probably false theory is that Moore thought that the word “body” would be too racy for his straitlaced Victorian-era English-speaking counterparts across the Atlantic ocean.
#6. The German words for both orders and rings can be applied to collections of people, as in the English phrases “monastic order” and “crime ring”, so it’s possible that the coinages were meant to suggest assemblages of individuals.
#7. In the generation before Hilbert, the geometer Moritz Pasch had proclaimed “If geometry is to be truly deductive, the process of inference must be entirely independent of the meaning of the geometrical terms.” In the last years of the 19th century Hilbert completed Pasch’s mission, and mischievously and memorably rephrased Pasch’s proclamation as “One should always be able to say, instead of ‘points, lines, and planes’, ‘tables, chairs, and beer mugs’.”
#8. It would have been fun if Fraenkel, to stress that his new notion allowed elements to be unspecified things (Dingen) as opposed to numbers (Zahlen), had called his assemblages Dingringen.
#9. Noether’s original definition of rings differs in one small detail from the modern notion: she did not require that there be a multiplicative identity element, a “1”. So she would have for instance considered the set of even integers to be a ring as well. Her lenience in this matter was prescient, since as time passed it became increasingly urgent to algebraists to develop an understanding of “rings-without-a-multiplicative-identity-element”. So nowadays we have the awkward word “rng”, pronounced “rung”, to signify an algebraic system that satisfies all the ring axioms except the existence of a multiplicative identity element.
#10. A friend of mine once complained that I used too many phrases like “Consider the following scenario: …” in my spoken language. I’m sure I fell into this nerdy and pompous-seeming habit from all the math I’ve read!
#11. If you want to bring a new number (call it ν) into the real numbers as a special guest, first extend the real number system by throwing in an indeterminate x. This gives us a ring R containing all polynomials in x. Now take all the polynomial expressions in ν that would equal zero if ν had all the properties you want it to have, put them into an ideal called I (with ν replaced by x), and mod out R by I. The quotient is just like the ring you wanted to construct (that is, the real numbers with bonus number ν thrown in), except for the fact that the bonus “number” is called x instead of ν.
#12. More specifically, Betti numbers are the ranks of certain abelian groups, where an abelian group is a kind of algebraic structure and the rank of an abelian group is a way of measuring how big it is.
REFERENCES
Clark Kimberling, ”Emmy Noether”, The American Mathematical Monthly, Feb. 1972, Vol. 79, No. 2, pp. 136–149.
Israel Kleiner, ”From Numbers to Rings: The Early History of Ring Theory”, Elemente der Mathematik 53 (1998), 18–35.
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