From my Third A-to-Z: Zermelo-Fraenkel Axioms

The close of my End 2016 A-to-Z let me show off one of my favorite modes, that of amateur historian of mathematics who doesn’t check his primary references enough. So far as I know I don’t have any serious errors here, but then, how would I know? … But keep in mind that the full story is more complicated and more ambiguous than presented. (This is true of all histories.) That I could fit some personal history in was also a delight.

I don’t know why Thoralf Skolem’s name does not attach to the Zermelo-Fraenkel Axioms. Mathematical things are named with a shocking degree of arbitrariness. Skolem did well enough for himself.

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gaurish gave me a choice for the Z-term to finish off the End 2016 A To Z. I appreciate it. I’m picking the more abstract thing because I’m not sure that I can explain zero briefly. The foundations of mathematics are a lot easier.

Zermelo-Fraenkel Axioms

I remember the look on my father’s face when I asked if he’d tell me what he knew about sets. He misheard what I was asking about. When we had that straightened out my father admitted that he didn’t know anything particular. I thanked him and went off disappointed. In hindsight, I kind of understand why everyone treated me like that in middle school.

My father’s always quick to dismiss how much mathematics he knows, or could understand. It’s a common habit. But in this case he was probably right. I knew a bit about set theory as a kid because I came to mathematics late in the “New Math” wave. Sets were seen as fundamental to why mathematics worked without being so exotic that kids couldn’t understand them. Perhaps so; both my love and I delighted in what we got of set theory as kids. But if you grew up before that stuff was popular you probably had a vague, intuitive, and imprecise idea of what sets were. Mathematicians had only a vague, intuitive, and imprecise idea of what sets were through to the late 19th century.

And then came what mathematics majors hear of as the Crisis of Foundations. (Or a similar name, like Foundational Crisis. I suspect there are dialect differences here.) It reflected mathematics taking seriously one of its ideals: that everything in it could be deduced from clearly stated axioms and definitions using logically rigorous arguments. As often happens, taking one’s ideals seriously produces great turmoil and strife.

Before about 1900 we could get away with saying that a set was a bunch of things which all satisfied some description. That’s how I would describe it to a new acquaintance if I didn’t want to be treated like I was in middle school. The definition is fine if we don’t look at it too hard. “The set of all roots of this polynomial”. “The set of all rectangles with area 2”. “The set of all animals with four-fingered front paws”. “The set of all houses in Central New Jersey that are yellow”. That’s all fine.

And then if we try to be logically rigorous we get problems. We always did, though. They’re embodied by ancient jokes like the person from Crete who declared that all Cretans always lie; is the statement true? Or the slightly less ancient joke about the barber who shaves only the men who do not shave themselves; does he shave himself? If not jokes these should at least be puzzles faced in fairy-tale quests. Logicians dressed this up some. Bertrand Russell gave us the quite respectable “The set consisting of all sets which are not members of themselves”, and asked us to stare hard into that set. To this we have only one logical response, which is to shout, “Look at that big, distracting thing!” and run away. This satisfies the problem only for a while.

The while ended in — well, that took a while too. But between 1908 and the early 1920s Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem paused from arguing whose name would also be the best indie rock band name long enough to put set theory right. Their structure is known as Zermelo-Fraenkel Set Theory, or ZF. It gives us a reliable base for set theory that avoids any contradictions or catastrophic pitfalls. Or does so far as we have found in a century of work.

It’s built on a set of axioms, of course. Most of them are uncontroversial, things like declaring two sets are equivalent if they have the same elements. Declaring that the union of sets is itself a set. Obvious, sure, but it’s the obvious things that we have to make axioms. Maybe you could start an argument about whether we should just assume there exists some infinitely large set. But if we’re aware sets probably have something to teach us about numbers, and that numbers can get infinitely large, then it seems fair to suppose that there must be some infinitely large set. The axioms that aren’t simple obvious things like that are too useful to do without. They assume stuff like that no set is an element of itself. Or that every set has a “power set”, a new set comprising all the subsets of the original set. Good stuff to know.

There is one axiom that’s controversial. Not controversial the way Euclid’s Parallel Postulate was. That’s the ugly one about lines crossing another line meeting on the same side they make angles smaller than something something or other. That axiom was controversial because it read so weird, so needlessly complicated. (It isn’t; it’s exactly as complicated as it must be. Or for a more instructive view, it’s as simple as it could be and still be useful.) The controversial axiom of Zermelo-Fraenkel Set Theory is known as the Axiom of Choice. It says if we have a collection of mutually disjoint sets, each with at least one thing in them, then it’s possible to pick exactly one item from each of the sets.

It’s impossible to dispute this is what we have axioms for. It’s about something that feels like it should be obvious: we can always pick something from a set. How could this not be true?

If it is true, though, we get some unsavory conclusions. For example, it becomes possible to take a ball the size of an orange and slice it up. We slice using mathematical blades. They’re not halted by something as petty as the desire not to slice atoms down the middle. We can reassemble the pieces. Into two balls. And worse, it doesn’t require we do something like cut the orange into infinitely many pieces. We expect crazy things to happen when we let infinities get involved. No, though, we can do this cut-and-duplicate thing by cutting the orange into five pieces. When you hear that it’s hard to know whether to point to the big, distracting thing and run away. If we dump the Axiom of Choice we don’t have that problem. But can we do anything useful without the ability to make a choice like that?

And we’ve learned that we can. If we want to use the Zermelo-Fraenkel Set Theory with the Axiom of Choice we say we were working in “ZFC”, Zermelo-Fraenkel-with-Choice. We don’t have to. If we don’t want to make any assumption about choices we say we’re working in “ZF”. Which to use depends on what one wants to use.

Either way Zermelo and Fraenkel and Skolem established set theory on the foundation we use to this day. We’re not required to use them, no; there’s a construction called von Neumann-Bernays-Gödel Set Theory that’s supposed to be more elegant. They didn’t mention it in my logic classes that I remember, though.

And still there’s important stuff we would like to know which even ZFC can’t answer. The most famous of these is the continuum hypothesis. Everyone knows — excuse me. That’s wrong. Everyone who would be reading a pop mathematics blog knows there are different-sized infinitely-large sets. And knows that the set of integers is smaller than the set of real numbers. The question is: is there a set bigger than the integers yet smaller than the real numbers? The Continuum Hypothesis says there is not.

Zermelo-Fraenkel Set Theory, even though it’s all about the properties of sets, can’t tell us if the Continuum Hypothesis is true. But that’s all right; it can’t tell us if it’s false, either. Whether the Continuum Hypothesis is true or false stands independent of the rest of the theory. We can assume whichever state is more useful for our work.

Back to the ideals of mathematics. One question that produced the Crisis of Foundations was consistency. How do we know our axioms don’t contain a contradiction? It’s hard to say. Typically a set of axioms we can prove consistent are also a set too boring to do anything useful in. Zermelo-Fraenkel Set Theory, with or without the Axiom of Choice, has a lot of interesting results. Do we know the axioms are consistent?

No, not yet. We know some of the axioms are mutually consistent, at least. And we have some results which, if true, would prove the axioms to be consistent. We don’t know if they’re true. Mathematicians are generally confident that these axioms are consistent. Mostly on the grounds that if there were a problem something would have turned up by now. It’s withstood all the obvious faults. But the universe is vaster than we imagine. We could be wrong.

It’s hard to live up to our ideals. After a generation of valiant struggling we settle into hoping we’re doing good enough. And waiting for some brilliant mind that can get us a bit closer to what we ought to be.

In Our Time podcast has episode on Pierre-Simon Laplace

I have another mathematics-themed podcast to share. It’s again from the BBC’s In Our Time, a 50-minute program in which three experts discuss a topic. Here they came back around to mathematics and physics. And along the way chemistry and mensuration. The topic here was Pierre-Simon Laplace, who’s one of those people whose name you learn well as a mathematics or physics major. He doesn’t quite reach the levels of Euler — who does? — but he’s up there.

Laplace might be best known for his work in celestial mechanics. He (independently of Immanuel Kant) developed the nebular hypothesis, that the solar system formed from the contraction of a great cloud of dust. We today accept a modified version of this. And for studying the question of whether the solar system is stable. That is, whether the perturbations every planet has on one another average out to nothing, or to something catastrophic. And studying probability, which has more to do with these questions than one might imagine. And then there’s general mechanics, and differential equations, and if that weren’t enough, his role in establishing the Metric system. This and more gets discussion.

My All 2020 Mathematics A to Z: Statistics

I owe Mr Wu, author of the Singapore Maths Tuition blog, thanks for another topic for this A-to-Z. Statistics is a big field of mathematics, and so I won’t try to give you a course’s worth in 1500 words. But I have to start with a question. I seem to have ended at two thousand words.

Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Statistics.

Is statistics mathematics?

The answer seems obvious at first. Look at a statistics textbook. It’s full of algebra. And graphs of great sloped mounds. There’s tables full of four-digit numbers in back. The first couple chapters are about probability. They’re full of questions about rolling dice and dealing cards and guessing whether the sibling who just entered is the younger.

But then, why does Rutgers University have a Department of Mathematics and also a Department of Statistics? And considered so distinct as to have an interdisciplinary mathematics-and-statistics track? It’s not an idiosyncrasy of Rutgers. Many schools have the same division between mathematics and statistics. Some join them into a Department of Mathematics and Statistics. But the name hints at something just different about the field. Not too different, though. Physics and Chemistry and important threads of Economics and History are full of mathematics. But you never see a Department of Mathematics and History.

Thinking of the field’s history, though, and its use, tell us more. Some of the earliest work we now recognize as statistics was Arab mathematicians deciphering messages. This cryptanalysis is the observation that (in English) a three-letter word is very likely to be ‘the’, mildly likely to be ‘one’, and not likely to be ‘pyx’. A more modern forerunner is the Republic of Venice supposedly calculating that war with Milan would not be worth the winning. Or the gatherings of mortality tables, recording how many people of what age can be expected to die any year, and what from. (Mortality tables are another of Edmond Halley’s claims to fame, though it won’t displace his comet work.) Florence Nightingale’s charts explaining how more soldiers die of disease than in fighting the Crimean War. William Sealy Gosset sharing sample-testing methods developed at the Guinness brewery.

You see a difference in kind to a mathematical question like finding a square with the same area as this trapezoid. It’s not that mathematics is not practical; it’s always been. And it’s not that statistics lacks abstraction and pure mathematics content. But statistics wears practicality in a way that number theory won’t.

Practical about what? History and etymology tip us off. The early uses of things we now see as statistics are about things of interest to the State. Decoding messages. Counting the population. Following — in the study of annuities — the flow of money between peoples. With the industrial revolution, statistics sneaks into the factory. To have an economy of scale you need a reliable product. How do you know whether the product is reliable, without testing every piece? How can you test every beer brewed without drinking it all?

One great leg of statistics — it’s tempting to call it the first leg, but the history is not so neat as to make that work — is descriptive. This gives us things like mean and median and mode and standard deviation and quartiles and quintiles. These try to let us represent more data than we can really understand in a few words. We lose information in doing so. But if we are careful to remember the difference between the descriptive statistics we have and the original population? (nb, a word of the State) We might not do ourselves much harm.

Another great leg is inferential statistics. This uses tools with names like z-score and the Student t distribution. And talk about things like p-values and confidence intervals. Terms like correlation and regression and such. This is about looking for causes in complex scenarios. We want to believe there is a cause to, say, a person’s lung cancer. But there is no tracking down what that is; there are too many things that could start a cancer, and too many of them will go unobserved. But we can notice that people who smoke have lung cancer more often than those who don’t. We can’t say why a person recovered from the influenza in five days. But we can say people who were vaccinated got fewer influenzas, and ones that passed quicker, than those who did not. We can get the dire warning that “correlation is not causation”, uttered by people who don’t like what the correlation suggests may be a cause.

Also by people being honest, though. In the 1980s geologists wondered if the sun might have a not-yet-noticed companion star. Its orbit would explain an apparent periodicity in meteor bombardments of the Earth. But completely random bombardments would produce apparent periodicity sometimes. It’s much the same way trees in a forest will sometimes seem to line up. Or imagine finding there is a neighborhood in your city with a high number of arrests. Is this because it has the highest rate of street crime? Or is the rate of street crime the same as any other spot and there are simply more cops here? But then why are there more cops to be found here? Perhaps they’re attracted by the neighborhood’s reputation for high crime. It is difficult to see through randomness, to untangle complex causes, and to root out biases.

The tools of statistics, as we recognize them, largely came together in the 19th and early 20th century. Adolphe Quetelet, a Flemish scientist, set out much early work, including introducing the concept of the “average man”. He studied the crime statistics of Paris for five years and noticed how regular the numbers were. The implication, to Quetelet — who introduced the idea of the “average man”, representative of societal matters — was that crime is a societal problem. It’s something we can control by mindfully organizing society, without infringing anyone’s autonomy. Put like that, the study of statistics seems an obvious and indisputable good, a way for governments to better serve their public.

So here is the dispute. It’s something mathematicians understate when sharing the stories of important pioneers like Francis Galton or Karl Pearson. They were eugenicists. Part of what drove their interest in studying human populations was to find out which populations were the best. And how to help them overcome their more-populous lessers.

I don’t have the space, or depth of knowledge, to fully recount the 19th century’s racial politics, popular scientific understanding, and international relations. Please accept this as a loose cartoon of the situation. Do not forget the full story is more complex and more ambiguous than I write.

One of the 19th century’s greatest scientific discoveries was evolution. That populations change in time, in size and in characteristics, even budding off new species, is breathtaking. Another of the great discoveries was entropy. This incorporated into science the nostalgic romantic notion that things used to be better. I write that figuratively, but to express the way the notion is felt.

There are implications. If the Sun itself will someday wear out, how long can the Tories last? It was easy for the aristocracy to feel that everything was quite excellent as it was now and dread the inevitable change. This is true for the aristocracy of any country, although the United Kingdom had a special position here. The United Kingdom enjoyed a privileged position among the Great Powers and the Imperial Powers through the 19th century. Note we still call it the Victorian era, when Louis Napoleon or Giuseppe Garibaldi or Otto von Bismarck are more significant European figures. (Granting Victoria had the longer presence on the world stage; “the 19th century” had a longer presence still.) But it could rarely feel secure, always aware that France or Germany or Russia was ready to displace it.

And even internally: if Darwin was right and reproductive success all that matters in the long run, what does it say that so many poor people breed so much? How long could the world hold good things? Would the eternal famines and poverty of the “overpopulated” Irish or Indian colonial populations become all that was left? During the Crimean War, the British military found a shocking number of recruits from the cities were physically unfit for service. In the 1850s this was only an inconvenience; there were plenty of strong young farm workers to recruit. But the British population was already majority-urban, and becoming more so. What would happen by 1880? 1910?

One can follow the reasoning, even if we freeze at the racist conclusions. And we have the advantage of a century-plus hindsight. We can see how the eugenic attitude leads quickly to horrors. And also that it turns out “overpopulated” Ireland and India stopped having famines once they evicted their colonizers.

Does this origin of statistics matter? The utility of a hammer does not depend on the moral standing of its maker. The Central Limit Theorem has an even stronger pretense to objectivity. Why not build as best we can with the crooked timbers of mathematics?

It is in my lifetime that a popular racist book claimed science proved that Black people were intellectual inferiors to White people. This on the basis of supposedly significant differences in the populations’ IQ scores. It proposed that racism wasn’t a thing, or at least nothing to do anything about. It would be mere “realism”. Intelligence Quotients, incidentally, are another idea we can trace to Francis Galton. But an IQ test is not objective. The best we can say is it might be standardized. This says nothing about the biases built into the test, though, or of the people evaluating the results.

So what if some publisher 25 years ago got suckered into publishing a bad book? And racist chumps bought it because they liked its conclusion?

The past is never fully past. In the modern environment of surveillance capitalism we have abundant data on any person. We have abundant computing power. We can find many correlations. This gives people wild ideas for “artificial intelligence”. Something to make predictions. Who will lose a job soon? Who will get sick, and from what? Who will commit a crime? Who will fail their A-levels? At least, who is most likely to?

These seem like answerable questions. One can imagine an algorithm that would answer them fairly. And make for a better world, one which concentrates support around the people most likely to need it. If we were wise, we would ask our friends in the philosophy department about how to do this. Or we might just plunge ahead and trust that since an algorithm runs automatically it must be fair. Our friends in the philosophy department might have some advice there too.

Consider, for example, the body mass index. It was developed by our friend Adolphe Quetelet, as he tried to understand the kinds of bodies in the population. It is now used to judge whether someone is overweight. Weight is treated as though it were a greater threat to health than actual illnesses are. Your diagnosis for the same condition with the same symptoms will be different — and on average worse — if your number says 25.2 rather than 24.8.

We must do better. We can hope that learning how tools were used to injure people will teach us to use them better, to reduce or to avoid harm. We must fight our tendency to latch on to simple ideas as the things we can understand in the world. We must not mistake the greater understanding we have from the statistics for complete understanding. To do this we must have empathy, and we must have humility, and we must understand what we have done badly in the past. We must catch ourselves when we repeat the patterns that brought us to past evils. We must do more than only calculate.

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This and the rest of the 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be gathered at this link. And I am looking for V, W, and X topics to write about. Thanks for your thoughts, and thank you for reading.

My All 2020 Mathematics A to Z: Hilbert’s Problems

Beth, author of the popular inspiration blog I Didn’t Have My Glasses On …. proposed this topic. Hilbert’s problems are a famous set of questions. I couldn’t hope to summarize them all in an essay of reasonable length. I’d have trouble to do them justice in a short book. But there are still things to say about them.

Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Hilbert’s Problems.

It’s easy to describe what Hilbert’s Problems are. David Hilbert, at the 1900 International Congress of Mathematicians, listed ten important problems of the field. In print he expanded this to 23 problems. They covered topics like number theory, group theory, physics, geometry, differential equations, and more. One of the problems was solved that year. Eight of them have been resolved fully. Another nine have been partially answered. Four remain unanswered. Two have generally been regarded as too vague to resolve.

Everyone in mathematics agrees they were big, important questions. Things that represented the things mathematicians of 1900 would most want to know. Things that guided mathematical research for, so far, 120 years.

It does present us with a dilemma. Were Hilbert’s problems listed because he understood what mathematicians would find important? Or did mathematicians find them important because Hilbert listed them? Sadly, mathematicians know of no professionals who have studied questions like this and could offer insight.

There is reason to say that Hilbert’s judgement was good. He listed, for example, the Riemann hypothesis. The hypothesis is still unanswered. Many interesting results would follow from it being proved true, or proved false, or proved unanswerable. Hilbert did not list Fermat’s Last Theorem, unresolved then. Any mathematician would have liked an answer. But nothing of consequence depends on it. But then he also listed making advances in the calculus of variations. A good goal, but not one that requires particular insight to want.

So here is a related problem. Why hasn’t anyone else made such a list? A concise summary of the problems that guides mathematical research?

It’s not because no one tried. At the 1912 International Conference of Mathematicians, Edmund Landau identified four problems in number theory worth solving. None of them have been solved yet. Yutaka Taniyama listed three dozen problems in 1955. William Thurston put forth 24 questions in 1982. Stephen Smale, famous for work in chaos theory, gathered a list of 18 questions in 1998. Barry Simon offered fifteen of them in 2000. Also in 2000 the Clay Mathematics Institute put up seven problems, with a million-dollar bounty on each. Jair Minoro Abe and Shotaro Tanaka gathered 22 questions for a list for 2001. The United States Defense Advanced Research Projects Agency put out a list of 23 of them in 2007.

Apart from Smale’s and the Clay Mathematics lists I never heard of any of them either. Why not? What was special about Hilbert’s list?

For one, he was David Hilbert. Hilbert was a great mathematician, held in high esteem then and now. Besides his list of problems he’s known for the axiomatization of geometry. This built not just logical rigor but a new, formalist, perspective. Also, he’s known for the formalist approach to mathematics. In this, for example, we give up the annoyingly hard task of saying exactly what we mean by a point and a line and a plane. We instead talk about how points and lines and planes relate to each other, definitions we can give. He’s also known for general relativity: Hilbert and Albert Einstein developed its field equations at the same time. We have Hilbert spaces and Hilbert curves and Hilbert metrics and Hilbert polynomials. Fans of pop mathematics speak of the Hilbert Hotel, a structure with infinitely many rooms and used to explore infinitely large sets.

So he was a great mind, well-versed in many fields. And he was in an enviable position, professor of mathematics at the University of Göttingen. At the time, German mathematics was held in particularly high renown. When you see, for example, mathematicians using ‘Z’ as shorthand for ‘integers’? You are seeing a thing that makes sense in German. (It’s for “Zahlen”, meaning the counting numbers.) Göttingen was at the top of German mathematics, and would be until the Nazi purges of academia. It would be hard to find a more renowned position.

And he was speaking at a great moment. The transition from one century to another is a good one for ambitious projects and declarations to be remembered. But the International Congress of Mathematicians was of particular importance. This was only the second meeting of the International Congress of Mathematicians. International Congresses of anything were new in the late 19th century. Many fields — not only mathematics — were asserting their professionalism at the time. It’s when we start to see professional organizations for specific subjects, not just “Science”. It’s when (American) colleges begin offering elective majors for their undergraduates. When they begin offering PhD degrees.

So it was a field when mathematics, like many fields (and nations), hoped to define its institutional prestige. Having an ambitious goal is one way to define that.

It was also an era when mathematicians were thinking seriously about what the field was about. The results were mixed. In the last decades of the 19th century, mathematicians had put differential calculus on a sound logical footing. But then found strange things in, for example, mathematical physics. Boltzmann’s H-theorem (1872) tells us that entropy in a system of particles always increases. Poincaré’s recurrence theorem (1890) tells us a system of particles has to, eventually, return to its original condition. (Or to something close enough.) And therefore it returns to its original entropy, undoing any increase. Both are sound theorems; how can they not conflict?

Even ancient mathematics had new uncertainty. In 1882 Moritz Pasch discovered that Euclid, and everyone doing plane geometry since then, had been using an axiom no one had acknowledged. (If a line that doesn’t pass through any vertex of a triangle intersects one leg of the triangle, then it also meets one other leg of the triangle.) It’s a small and obvious thing. But if everyone had missed it for thousands of years, what else might be overlooked?

I wish now to share my interpretation of this background. And with it my speculations about why we care about Hilbert’s Problems and not about Thurston’s. And I wish to emphasize that, whatever my pretensions, I am not a professional historian of mathematics. I am an amateur and my training consists of “have read some books about a subject of interest”.

By 1900 mathematicians wanted the prestige and credibility and status of professional organizations. Who would not? But they were also aware the foundation of mathematics was not as rigorous as they had thought. It was not yet the “crisis of foundations” that would drive the philosophy of mathematics in the early 20th century. But the prelude to the crisis was there. And here was a universally respected figure, from the most prestigious mathematical institution. He spoke to all the best mathematicians in a way they could never have been addressed before. And presented a compelling list of tasks to do. These were good tasks, challenging tasks. Many of these tasks seemed doable. One was even done almost right away.

And they covered a broad spectrum of mathematics of the time. Everyone saw at least one problem relevant to their field, or to something close to their field. Landau’s problems, posed twelve years later, were all about number theory. Not even all number theory; about prime numbers. That’s nice, but it will only briefly stir the ambitions of the geometer or the mathematical physicist or the logician.

By the time of Taniyama, though? 1955? Times are changed. Taniyama is no inconsiderable figure. The Taniyama-Shimura theorem is a major piece of elliptic functions. It’s how we have a proof of Fermat’s last theorem. But by then, too, mathematics is not so insecure. We have several good ideas of what mathematics is and why it should work. It has prestige and institutional authority. It has enough Congresses and Associations and Meetings that no one can attend them all. It’s moreso by 1982, when William Thurston set up questions. I know that I’m aware of Stephen Smale’s list because I was a teenager during the great fractals boom of the 80s and knew Smale’s name. Also that he published his list near the time I finished my quals. Quals are an important step in pursuing a doctorate. After them you look for a specific thesis problem. I was primed to hear about great ambitious projects I could not possibly complete.

Only the Clay Mathematics Institute’s list has stood out, aided by its catchy name of Millennium Prizes and its offer of quite a lot of money. That’s a good memory aid. Any lay reader can understand that motivation. Two of the Millennium Prize problems were also Hilbert’s problems. One in whole (the Riemann hypothesis again). One in part (one about solutions to elliptic curves). And as the name states, it came out in 2000. It was a year when many organizations were trying to declare bold and fresh new starts for a century they hoped would be happier than the one before. This, too, helps the memory. Who has any strong associations with 1982 who wasn’t born or got their driver’s license that year?

These are my suppositions, though. I could be giving a too-complicated answer. It’s easy to remember that United States President John F Kennedy challenged the nation to land a man on the moon by the end of the decade. Space enthusiasts, wanting something they respect to happen in space, sometimes long for a president to make a similar strong declaration of an ambitious goal and specific deadline. President Ronald Reagan in 1984 declared there would be a United States space station by 1992. In 1986 he declared there would be by 2000 a National Aerospace Plane, capable of flying from Washington to Tokyo in two hours. President George H W Bush in 1989 declared there would be humans on the Moon “to stay” by 2010 and to Mars thereafter. President George W Bush in 2004 declared the Vision for Space Exploration, bringing humans to the moon again by 2020 and to Mars thereafter.

No one has cared about any of these plans. Possibly because the first time a thing is done, it has a power no repetition can claim. But also perhaps because the first attempt succeeded. Which was not due only to its being first, of course, but to the factors that made its goal important to a great number of people for long enough that it succeeded.

Which brings us back to the Euthyphro-like dilemma of Hilbert’s Problems. Are they influential because Hilbert chose well, or did Hlbert’s choosing them make them influential? I suspect this is a problem that cannot be resolved.

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Thank you for reading. This and the other other A-to-Z topics for 2020 should be at this link. All my essays for this and past A-to-Z sequences are at this link. And I am taking nominations for J, K, and L topics. I’m grateful for anything you can offer me.

The End 2016 Mathematics A To Z: Zermelo-Fraenkel Axioms

gaurish gave me a choice for the Z-term to finish off the End 2016 A To Z. I appreciate it. I’m picking the more abstract thing because I’m not sure that I can explain zero briefly. The foundations of mathematics are a lot easier.

Zermelo-Fraenkel Axioms

I remember the look on my father’s face when I asked if he’d tell me what he knew about sets. He misheard what I was asking about. When we had that straightened out my father admitted that he didn’t know anything particular. I thanked him and went off disappointed. In hindsight, I kind of understand why everyone treated me like that in middle school.

My father’s always quick to dismiss how much mathematics he knows, or could understand. It’s a common habit. But in this case he was probably right. I knew a bit about set theory as a kid because I came to mathematics late in the “New Math” wave. Sets were seen as fundamental to why mathematics worked without being so exotic that kids couldn’t understand them. Perhaps so; both my love and I delighted in what we got of set theory as kids. But if you grew up before that stuff was popular you probably had a vague, intuitive, and imprecise idea of what sets were. Mathematicians had only a vague, intuitive, and imprecise idea of what sets were through to the late 19th century.

And then came what mathematics majors hear of as the Crisis of Foundations. (Or a similar name, like Foundational Crisis. I suspect there are dialect differences here.) It reflected mathematics taking seriously one of its ideals: that everything in it could be deduced from clearly stated axioms and definitions using logically rigorous arguments. As often happens, taking one’s ideals seriously produces great turmoil and strife.

Before about 1900 we could get away with saying that a set was a bunch of things which all satisfied some description. That’s how I would describe it to a new acquaintance if I didn’t want to be treated like I was in middle school. The definition is fine if we don’t look at it too hard. “The set of all roots of this polynomial”. “The set of all rectangles with area 2”. “The set of all animals with four-fingered front paws”. “The set of all houses in Central New Jersey that are yellow”. That’s all fine.

And then if we try to be logically rigorous we get problems. We always did, though. They’re embodied by ancient jokes like the person from Crete who declared that all Cretans always lie; is the statement true? Or the slightly less ancient joke about the barber who shaves only the men who do not shave themselves; does he shave himself? If not jokes these should at least be puzzles faced in fairy-tale quests. Logicians dressed this up some. Bertrand Russell gave us the quite respectable “The set consisting of all sets which are not members of themselves”, and asked us to stare hard into that set. To this we have only one logical response, which is to shout, “Look at that big, distracting thing!” and run away. This satisfies the problem only for a while.

The while ended in — well, that took a while too. But between 1908 and the early 1920s Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem paused from arguing whose name would also be the best indie rock band name long enough to put set theory right. Their structure is known as Zermelo-Fraenkel Set Theory, or ZF. It gives us a reliable base for set theory that avoids any contradictions or catastrophic pitfalls. Or does so far as we have found in a century of work.

It’s built on a set of axioms, of course. Most of them are uncontroversial, things like declaring two sets are equivalent if they have the same elements. Declaring that the union of sets is itself a set. Obvious, sure, but it’s the obvious things that we have to make axioms. Maybe you could start an argument about whether we should just assume there exists some infinitely large set. But if we’re aware sets probably have something to teach us about numbers, and that numbers can get infinitely large, then it seems fair to suppose that there must be some infinitely large set. The axioms that aren’t simple obvious things like that are too useful to do without. They assume stuff like that no set is an element of itself. Or that every set has a “power set”, a new set comprising all the subsets of the original set. Good stuff to know.

There is one axiom that’s controversial. Not controversial the way Euclid’s Parallel Postulate was. That’s the ugly one about lines crossing another line meeting on the same side they make angles smaller than something something or other. That axiom was controversial because it read so weird, so needlessly complicated. (It isn’t; it’s exactly as complicated as it must be. Or for a more instructive view, it’s as simple as it could be and still be useful.) The controversial axiom of Zermelo-Fraenkel Set Theory is known as the Axiom of Choice. It says if we have a collection of mutually disjoint sets, each with at least one thing in them, then it’s possible to pick exactly one item from each of the sets.

It’s impossible to dispute this is what we have axioms for. It’s about something that feels like it should be obvious: we can always pick something from a set. How could this not be true?

If it is true, though, we get some unsavory conclusions. For example, it becomes possible to take a ball the size of an orange and slice it up. We slice using mathematical blades. They’re not halted by something as petty as the desire not to slice atoms down the middle. We can reassemble the pieces. Into two balls. And worse, it doesn’t require we do something like cut the orange into infinitely many pieces. We expect crazy things to happen when we let infinities get involved. No, though, we can do this cut-and-duplicate thing by cutting the orange into five pieces. When you hear that it’s hard to know whether to point to the big, distracting thing and run away. If we dump the Axiom of Choice we don’t have that problem. But can we do anything useful without the ability to make a choice like that?

And we’ve learned that we can. If we want to use the Zermelo-Fraenkel Set Theory with the Axiom of Choice we say we were working in “ZFC”, Zermelo-Fraenkel-with-Choice. We don’t have to. If we don’t want to make any assumption about choices we say we’re working in “ZF”. Which to use depends on what one wants to use.

Either way Zermelo and Fraenkel and Skolem established set theory on the foundation we use to this day. We’re not required to use them, no; there’s a construction called von Neumann-Bernays-Gödel Set Theory that’s supposed to be more elegant. They didn’t mention it in my logic classes that I remember, though.

And still there’s important stuff we would like to know which even ZFC can’t answer. The most famous of these is the continuum hypothesis. Everyone knows — excuse me. That’s wrong. Everyone who would be reading a pop mathematics blog knows there are different-sized infinitely-large sets. And knows that the set of integers is smaller than the set of real numbers. The question is: is there a set bigger than the integers yet smaller than the real numbers? The Continuum Hypothesis says there is not.

Zermelo-Fraenkel Set Theory, even though it’s all about the properties of sets, can’t tell us if the Continuum Hypothesis is true. But that’s all right; it can’t tell us if it’s false, either. Whether the Continuum Hypothesis is true or false stands independent of the rest of the theory. We can assume whichever state is more useful for our work.

Back to the ideals of mathematics. One question that produced the Crisis of Foundations was consistency. How do we know our axioms don’t contain a contradiction? It’s hard to say. Typically a set of axioms we can prove consistent are also a set too boring to do anything useful in. Zermelo-Fraenkel Set Theory, with or without the Axiom of Choice, has a lot of interesting results. Do we know the axioms are consistent?

No, not yet. We know some of the axioms are mutually consistent, at least. And we have some results which, if true, would prove the axioms to be consistent. We don’t know if they’re true. Mathematicians are generally confident that these axioms are consistent. Mostly on the grounds that if there were a problem something would have turned up by now. It’s withstood all the obvious faults. But the universe is vaster than we imagine. We could be wrong.

It’s hard to live up to our ideals. After a generation of valiant struggling we settle into hoping we’re doing good enough. And waiting for some brilliant mind that can get us a bit closer to what we ought to be.