Four terrific posts by David Friedman, partly on psychic harm, partly on talking about psychic harm. I’d recommend these highly even if they hadn’t invoked my name.
Four terrific posts by David Friedman, partly on psychic harm, partly on talking about psychic harm. I’d recommend these highly even if they hadn’t invoked my name.
Brad DeLong appears to argue here that because pure reason once led him, Brad Delong, to an incorrect conclusion about which direction he was facing, it follows that pure reason can never be a source of knowledge.
(If that’s not his point, then the only alternative reading I can find is that Thomas Nagel is guilty of choosing a poor example to illustrate a point that DeLong would rather ridicule than refute.)
It would be too too easy to make a snarky comment about how we’ve known all along about Brad DeLong’s tenuous relationship with reason. Instead, here, for the record is a list of ten facts, of which I am willing to bet that DeLong is aware of at least 7 — none of them, as far as I can see, accessible to humans via anything but pure reason:
Over at Less Wrong, the estimable Eliezer Yudkowsky attempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
1. Yudkowsky poses, in essence, the following question:
Yudkowsky phrases the question a little differently. What he actually asks is:
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2″, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
2. Yudkowsky provisionally offers the following answer:
In The Big Questions, I argued that math is all there is: The Universe we live in is a mathematical object and is no more or less “real” than any other mathematical object. Thus, for example, the Godel universe, where time moves in circles, so that everything eventually returns to the time and place where it started, is as real as our own — though far, far, less complicated, because it contains, for example, no sentient beings). (Though on the other hand, it’s entirely plausible that there exists a Godel-like Universe that does contain sentient beings, and the existence of such a Universe can, in principle, be settled by purely mathematical inquiry.)
Obviously, I can’t prove this, but I’ve tried to explain why it strikes me as far more plausible than any of the alternatives. It all comes down to Ockham’s Razor. I know these mathematical Universes exist (pick up any issue of any theoretical physics journal and chances are you’ll find a couple described in detail), and it seems ontologically extravagant to suggest that some enjoy a different kind of existence than others. In other words, the notion of “physical reality” is exactly the sort of unnecessary baggage that Ockham’s razor wants to cut away.
People do seem to want to believe that the Universe we inhabit is somehow “special”, which is why I believe they’ve invented the unnecessary concept of “physical reality” to distinguish it from all the others. But the history of science has not been kind to viewpoints that cast human habitats as special. People used to think that the earth occupied a special place in the Universe; Copernicus (crying “Give up your Ptolemy! Rise up and follow me!”) rejected that notion in what can be seen as a slick application of Ockham’s Razor. Nowadays, people are tempted to think that the Universe we occupy has a special status in the zoo of mathematical Universes; but as good Ockhamized Copernicans, we should resist that temptation.
The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. Mochizuki’s work rests on so many radically new ideas that it will take the experts a long time to digest. I, who am not an expert, will surely die with only a vague sense of the argument. But based on my extremely limited (and possibly mistaken) understanding, it appears that Mochizuki’s breakthrough depends at least partly on his willingness to abandon the usual axioms for the foundations of mathematics and replace them with new axioms. (See, for example, the first page of these notes from one of Mochizuki’s lectures. You can find other related notes here.)
That’s interesting for a lot of reasons, but the one that’s most topical for The Big Questions is this: No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth.
There’s a widespread misconception that arithmetic is about “what can be derived from the axioms”, which is a lot like saying that astronomy is about “what can be discovered through telescopes”. Axiomatic systems, like telescopes, are investigative tools, which we are free to jettison when better tools come along. The blather of thoughtless imbeciles notwithstanding, what really matters is the fundamental object of study, whether it’s the system of natural numbers or the planet Jupiter.
Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.
Last week, we had some discussion of free will, which prompted some comments about determinism. I’m not convinced that determinism has all that much to do with free will one way or the other, but since the topic’s been raised, here are a few bullet points, jotted down late at night, which I hope will still make sense in the morning.
The subject of free will came up earlier this week, and I notice that Sam Harris has a new book on the subject, which I have not yet read. Some of you have asked for me to elaborate on my remarks on this subject in The Big Questions. Here are a few bullet points:
Here is a link to my piece in today’s issue of Time.com; and here’s the executive summary:
Deep inside a West Texas mountain, engineers are building a clock designed to click for 10,000 years. Amazon.com’s Jeff Bezos has sunk $43 million into this project and that’s just a start. What’s the purpose? “To foster long-term thinking and responsibility in the framework of the next 10,000 years”, say the organizers.
Now thinking and responsibility are two different things, but it’s hard to see how this project is likely to encourage either. Re thinking: The question of what we owe to future generations is subtle and difficult and requires close attention to difficult arguments; it’s hard to see how a ticking clock will do to foster that kind of effort. And as for responsiblity — if responsibility means making a better life for our distant descendants, then Bezos’s $43 million would almost surely be better spent on lobbying for lower capital taxes. Whether the goal is thinking or responsibility, a giant clock seems like a giant waste of time.
In the comments section of Bob Murphy’s blog, I was asked (in effect) why I insist on the objective reality of the natural numbers (that is, the counting numbers 0,1,2,3…) but not of, say, the real numbers (that is, the numbers we use to represent lengths — and that are themselves represented by possibly infinite decimal expansions).
There seem to be two kinds of people in the world: Those with enough techncal backgroud that they already know the answer, and those with less technical background, who have no hope — at least without a lot of work — of grasping the answer. I’m going to attempt to bridge that gap here. That means I’m going to throw a certain amount of precision to the winds, in hopes of being comprehensible to a wider audience.
Stanley Tennenbaum was an itinerant mathematician with, for much of his adult life, no fixed address and no permanent source of income. Sometimes he slept on park benches. He didn’t have a lot of teeth.
But if you were involved with mathematics in the second half of the twentieth century, sooner or later you were going to cross paths with Stanley, probably near the coffee machine in a math department. He’d proudly show you the little book he carried in his breast pocket, with the list of people to whom he owed money. Then he’d teach you something, or he’d tell you a good story.
Stanley had little tolerance for convention. His one permanent job, at the University of Rochester, came to an abrupt end during a faculty meeting where he spit on the shoes of the University president and walked out. Surely the same personality trait had something to do with his departure from the University of Chicago without a Ph.D., though the paper he wrote there (at age 22) has acquired fame and influence far beyond many of the doctoral theses of his more conventionally successful classmates. I’d like to tell you a little about that paper and what I think it means for the foundations of mathematics.
Yesterday’s nightmare scenarios triggered some good discussion, so let me throw out another one, which I think will help to isolate some of the issues that came up yesterday. Sometime next week I’ll try to summarize the best of the comments and ponder what we’ve learned.
In front of you are two childless married couples. For some reason, it’s imperative that you kill two of the four people. Your choices are:
A. Kill one randomly chosen member from each couple.
All four people agree that if they die, they want to be well remembered. Therefore all four ask you, please, to choose A so that anyone who dies will be remembered by a loving spouse.
If you care about the four people in front of you, what should you do?
Argument 1. For goodness’s sake, they’ve told you what to do. If you care about them, of course you should respect their wishes. Choose A.
Argument 2.Once the killings are over, Option A leaves two grieving spouses, whereas Option B leaves one relieved couple. Surely two dead plus two happy is better than two dead plus two sad. Choose B.
Here’s a question to ponder:
Question 1: If forced to choose, which of these nightmare scenarios would you prefer?
Scenario A: An evil alien flips a coin. If it comes up heads, he destroys all human life; otherwise he goes home.
Scenario B: The same evil alien flips 7 billion coins, one for each person on earth. He destroys anyone whose coin comes up heads.
I’ll tell you in a minute why I ask, but first let’s consider arguments in each direction:
Argument 1. In scenario A, I have a 50-50 chance of death, and a 50-50 chance of continuing my current life. In scenario B, I have a 50-50 chance of death, and a 50-50 chance of a life in which half my loved ones are gone. Surely I should take A.
Argument 2. In scenario A, there’s a 50-50 chance that all future generations will be destroyed-in-advance. In scenario B, even if the coin comes up heads, people will continue to be born, and in the very long run, the evil alien will be a forgotten memory. Surely I should take B.
Your answer to this question, I think, is likely to reveal a lot about how much you think we owe to future generations. If you think we owe them nothing, then Argument 1 is definitive. If you think we owe them the same respect we owe our contemporaries, then Argument 2 is definitive. If you think we owe them something in between, you might waver.
Now that sort of question might strike you as nothing more than Sunday-afternoon dorm room fare, but I don’t believe it can be dismissed so easily. It is pretty much impossible to take a coherent stand on issues ranging from Social Security reform to environmental conservation without first deciding how much we are obligated to care about future generations. A lot of people seem to think those issues are worth debating, which pretty much forces us to face up to the fundamental issues.
On the other hand, come to think of it, I suppose a person might prefer Scenario B to Scenario A for reasons that have nothing to do with future generations — namely the desire to be remembered. Is that something we care about? To focus on that issue, here’s another question:
Question 2: Suppose you’re happily married. If forced to choose, which of these nightmare scenarios would you prefer?
Scenario A: An evil alien flips a coin. If it comes up heads, he kills you and your spouse; otherwise he goes home.
Scenario B: The same evil alien flips a coin. If it comes up heads, he kills just you; if it comes up tails, he kills just your spouse.
Here again we have:
Argument 1. In scenario A, I have a 50-50 chance of death, and a 50-50 chance of continuing my current life. In scenario B, I have a 50-50 chance of death, and a 50-50 chance of a life in which my beloved spouse is gone. Surely I should take A.
Argument 2. In scenario A, there’s a 50-50 chance that my spouse and I will both be dead and unremembered (or at least unremembered by anyone who knows us as intimately as we know each other). In scenario B, however, we each get to live on, at least in the memory of a loved one. Surely I should take B.
Your answer to Question 2 will tell me something about how much being remembered matters to you, which will help me interpret your answer to Question 1.
The original version of this post included several more followup questions. But I suspect these two are grist enough for a day or two of discussions. I’ll post the followups after those responses start to peter out.
Related post here.
I don’t trust rocks. Rocks keep fooling me. They sit there looking all solid until you examine them more carefully and find out they’re mostly empty space, with a smattering of charged particles here and there. Then you look a little deeper and find out those charged particles are nothing like they first appeared. They don’t even have locations. Rocks, and their constituents, are nothing at all like they first present themselves. But at least they’re real. I think.
Now here’s what genuinely baffles me: Apparently there are people in this world (and even, occasionally, in the comments section of this blog) who haven’t the slightest doubt about the existence of rocks, galaxies, squirrels, and the rest of the physical universe, but who suddenly turn into hardcore skeptics re the existence of mathematical objects like the natural numbers. (Many of these people, I suspect, are in fact affecting skepticism because of a badly mistaken belief that it makes them look sophisticated. But that’s speculation on my part, so let’s put it aside and take their positions at face value.) I just don’t get this. Why on earth would, say, a scientist, commit to the belief that there’s a physical universe out there but not a mathematical one, when we know that our perceptions of the physical universe demand constant revision, whereas our perceptions of the mathematical universe are largely eternal. My conception of the natural numbers is very close to Euclid’s; my conception of an atom bears almost no resemblance to Demosthenes’s.
The apparently imminent discovery of the Higgs boson by scientists at CERN will have at least one quirky side effect that appears to have gone entirely unremarked until the appearance of this blog post — it threatens to inflict fatal collateral damage to the brilliant, eccentric and infuriating Omega Point Theory proposed by the physicist Frank Tipler.
Tipler, who is not a crackpot, once published a book called The Physics of Immortality, purporting, on the basis of orthodox physics plus some plausible auxiliary assumptions, to establish the existence of an omnipotent, omniscient, omnipresent and altruistic “being” who will one day resurrect everyone who has ever lived to eternal life.
The first step toward that startling conclusion is the assumption that our descendants will not allow all life to come to an end. This in turn will require them to control the evolution of the Universe so that it doesn’t collapse in anything that human beings perceive as a finite amount of time; Tipler argues that they’ll quite plausibly have the technology to do that. But all this future tinkering with the shape of the Universe has consequences that (in a very rough sense) radiate backward and forward through time. From this and some highly technical but more-or-less standard physics, Tipler manages to conclude the existence of an Omega Point — a place where (again speaking roughly) all the information in the Universe is stored. Writing in 1994, Tipler never considered the possibility that the Omega Pont might be located in Mountain View, California. Instead, he stressed that in its omniscience, it’s something very like God.
The ever-insightful philosopher Peter Smith has a number of interesting things to say about abortion, but I found one of those things particularly striking — partly because I don’t recall ever having thought of it before, and partly because, in retrospect, I don’t see how I could have failed to think of it.
Namely: The argument is made that zygotes/embryoes/fetuses, even at a very early stage, have the full moral status of human beings. Yet if that were true, surely we’d want to divert a substantial portion of the medical research budget away from relatively minor scourges like, say, cancer, to the spontaneous abortions that take the lives of something like 30% of these full-fledged humans. In a typical year, there are about 8 million cancer deaths worldwide; the number of early-stage spontaneous abortions must be at least twice that.
In Smith’s words:
very few of us are worried by the fact that a very high proportion of conceptions quite spontaneously abort. We don’t campaign for medical research to reduce that rate (nor do opponents of abortion campaign for all women to take drugs to suppress natural early abortion). Compare: we do think it is a matter for moral concern that there are high levels of infant mortality in some countries, and campaign and give money to help reduce that rate.
Smith is struck by the fact that this attitude is very widespread; I am more struck by the fact that it seems to be very widespread even among those who characterize themselves as pro-life.
Bertrand Russell, that most rational of men, was nonetheless plagued by intermittent depression and the occasional nightmare. Including this one, as reported by Russell’s confidante, the mathematician G.H. Hardy:
[Russell] was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated….
Principia Mathematica, to which Russell had devoted ten years of his life, was his (and co-author Alfred North Whitehead‘s) audacious and ultimately futile attempt to reduce all of mathematics to pure logic. It is a failure that enabled some of the great successes of 20th century mathematics. And — the first volume having been published in December, 1910 — this is its 100th birthday.
This is an extremely elementary post about numbers. (“Numbers” means the natural numbers 0,1,2 and so forth.) It is a sort of sequel to my three recent posts on basic arithmetic, which are here, here and here. But it can be read separately from those posts.
Today’s question is: Do numbers exist? The answer is: Of course, and I don’t believe there’s much in the way of serious doubt about this. You were familiar with numbers when you were five years old, and you’ve been discovering their properties ever since. Extreme skepticism on this point is almost unheard of among mathematicians or philosophers, though it seems to be fairly common among denizens of the Internet who have gotten it into their head that extreme skepticism makes them look sophisticated.
Let me be clear that I am not (yet) asking in what sense the natural numbers exist — whether they have existed since the beginning of time, or whether they exist outside of time, or whether they exist only in our minds. Those are questions that reasonable people disagree about (and that other reasonable people find more or less meaningless.) We can — and will — come back to those questions in future posts. For now, the only question: Do the natural numbers exist? And the answer is yes. Or better yet — if you believe the answer is no, then there’s obviously no point in thinking about them, so why are you reading this post?
Like everyone else I know, I am of course a longtime fan of the webcomic XKCD. But somehow it took me until last week to become aware of the frequently brilliant competitor Luke Surl, of which the above is a delectable example. What else out there am I missing?
Hat tip to Harry Brighouse of Crooked Timber.
Yesterday we started a conversation about whether mathematics is invented or discovered. Today I’ll give you my three best arguments for “discovered”. And to focus the discussion, I’ll talk not about mathematics generally but about the natural numbers (0,1,2, and so forth) in particular.
I believe the natural numbers exist, quite independently of whether anyone’s around to think of them. Here’s why: First, we perceive them directly. Second, we know non-trivial facts about them. Third, they can explain the Universe. In more detail:
Is mathematics invented or discovered? In my experience, applied scientists often think of mathematics as a human invention, while actual mathematicians (with a few notable exceptions) feel sure that mathematics was always there to be discovered. (Of course, it’s sometimes hard to tell how much of this is genuine disagreement and how much is a language barrier.)
I’ve just finished reading an excellent book by Mario Livio which is entirely about the invention/discovery question, though he’s chosen the (somewhat unfortunate) title Is God a Mathematician? Much of the book is a lively romp through mathematical history, with a well chosen mix of biography and exposition. Although he parts company with them in the last chapter, Livio gives a more than fair hearing to the many great mathematicians who have insisted that they are discoverers, from Pythagoras through Galileo, G.H. Hardy, Kurt Godel, and the contemporary Fields Medalist Alain Connes (among others). Here, for example is Connes:
The Intelligent Design folk tell you that complexity requires a designer.
The Richard Dawkins crowd tell you that complexity must evolve from simplicity.
I claim they’re both wrong, because the natural numbers, together with the operations of arithmetic, are fantastically complex, but were neither created nor evolved.
I’ve made this argument multiple times, in The Big Questions, on this blog, and elsewhere. Today, I aim to explain a little more deeply why I say that the natural numbers are fantastically complex.
As I mentioned the other day, I’ve recently (at the direction of my old friend Deirdre McCloskey) been reading some of the work of John Polkinghorne, the physicist-turned-theologian who seems to write about a book a week attempting to reconcile his twin faiths in orthodox science and orthodox Christianity.
Although Belief in God in an Age of Science is a very short book, it is too long to review in a single blog post. Fortunately, though, much of the non-lunatic content is concentrated in roughly the first ten pages, so I’ll comment here only on those.
Polkinghorne begins in awe. He is awestruck by the extent to which our Universe seems to have been fine-tuned to support life; this is the subject matter of the much-discussed anthropic cosmological principle. To take just one example (which Polkinghorne does not mention): The very existence of elements other than hydrogen and helium depends on the fact that it’s possible, in the interior of a star, to smoosh three helum atoms together and make a carbon atom; everything else is built from there. But it’s not enough to make that carbon atom; you’ve also got to make it stick together long enough for a series of other complicated reactions to occur. Ordinarily, that doesn’t happen, but now and then it does. And the reason it happens even occasionally is that the carbon atom happens to have an energy level of exactly 7.82 million electron volts. In fact, this energy level was predicted (by Fred Hoyle and Edwin Salpeter) before it was observed, precisely on the basis that without this energy level, there could be no stable carbon, no higher elements, and no you or me.
I saw with greater clarity than ever before in my life that when I say “Thank goodness!” this is not merely a euphemism for “Thank God!” (We atheists don’t believe that there is any God to thank.) I really do mean thank goodness! There is a lot of goodness in this world, and more goodness every day, and this fantastic human-made fabric of excellence is genuinely responsible for the fact that I am alive today. It is a worthy recipient of the gratitude I feel today, and I want to celebrate that fact here and now.
Logicomix is—I am not making this up—a graphic novel (that is, what we used to call a comic book) about Bertrand Russell and the writing of Principia Mathematica. Implausibly enough, it succeeds, making rather gripping drama out of the twentieth century crisis in the foundations of mathematics. The technical issues are portrayed clearly and accurately (a novice reader could learn a lot from this book) but never coldly; this is above all a saga about human obsession. I even like the device where the authors themselves appear as characters, trying to figure out how best to present this stuff. It works.
But there’s one part I find almost impossible to believe is accurate; maybe a reader can set me straight. The novel begins in 1939 and proceeds by flashback. In 1939 we see Russell, a lifelong pacifist confronted by the Nazi horror, being shaken to the core by the realization that his beloved Logic does not contain the answers to all of life’s problems. Can there be even a shred of truth to this? Surely the man who devoted his youth and over 300 printed pages to proving that 1+1=2 must always have been well aware that formal logic has its limitations as a practical guide to life.
Continue reading ‘Principia Mathematica: The Comic Book’