THE PLATONIST FILE: What makes 17 a prime?

FRIDAY, SEPTEMBER 14, 2018

When mathematicians wander:
Jim Holt got off easy.

Back in 2005, he wrote a slightly-disguised review of Rebecca Goldstein's new general interest book, Incompleteness: The Proof and Paradox of Kurt Godel. Holt's review appeared in The New Yorker, an upper-end general interest magazine.

In her general interest book, Goldstein had told the story of Godel's life. She'd also tried to explain his "incompletenesss theorems," on the basis of which he's often been called the greatest logician since Aristotle.

First, though, Goldstein tried to explain the doctrine which, she said, lay at the heart of Godel's intellectual life from the time of his first year in college. Because he was only writing a review, Holt described the doctrine very briefly, and then quickly moved on.

We'd have to say he got off easy! Here's his key passage again:
HOLT (page 8): Gödel entered the University of Vienna in 1924. He had intended to study physics, but he was soon seduced by the beauties of mathematics, and especially by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism, because it descends from Plato’s theory of ideas, has always been popular among mathematicians...
Platonism is built around the belief that "abstractions like numbers and circles have a perfect, timeless existence independent of the human mind." So said Holt, in The New Yorker, and then he quickly moved on.

Do numbers and circles have a perfect, timeless existence independent of the human mind? Friend, do you have even the slightest idea what that word sequence might possibly mean? We'll go ahead and answer for you:

No, you don't have the slightest idea. And neither does anyone else!

Because he was only writing a review, Holt got to leave things right there. Today, Holt's review, edited to remove most references to Goldstein, is the title essay of his own new book, When Einstein Walked with Godel: Excursions to the Edge of Thought.

Just this once, let's be honest! Holt's account of Godel's beloved "doctrine" is straight outta Jabberwocky. You don't know what the heck it means, and neither does anyone else.

But so what? When Holt's new book appeared, fronted by that title essay, major reviewers stood in line to exclaim, for the ten millionth, about how amazingly lucid and clear Holt's science/math writing is. Reviewers swore that Holt's writing was brutally lucid and clear.

In effect, reviewers swore that Holt had made Godel and Einstein easy. This has been standard journalistic practice dating at least to Einstein's own general interest book about relativity—to the brilliant physicist's failed attempt to make his own theories clear.

(More on that effort below.)

This is standard journalistic behavior—and Goldstein benefited from this practice when her book about Godel appeared. Her own account of Platonism is so murky that it seems to have come from the third planet beyond Jabberwocky. That said, the usual suspects stood in line to say how lucid her writing was.

Three major academic stars blurbed Goldstein's book on its jacket. You already know what they said:

In his dust jacket blurb, Alan Lightman praised Goldstein for her "penetrating, accessible, and beautifully written book."

Brian Greene went one step further. He said Goldstein's account of Godel was "remarkably accessible."

In a New York Times review, Polly Shulman said that Goldstein's writing was "surprisingly accessible." Meanwhile, back on the book's dust jacket, Stephen Pinker said this:
This book is a gem...Rebecca Goldstein, the gifted novelist and philosopher, offers us not just a lucid expression of Godel's brainchild but a satisfying and original narrative of the ideas and people it touched. Written with grace and passion, Incompletenesss is an unforgettable account of one of the great moments in the history of human thought.
"Lucid" was Pinker's word of choice. At Salon, Laura Miller stepped in to top him on this part of the color wheel, calling the book "eminently lucid."

This constitutes a familiar practice within several modern guilds. (Goldstein praises Holt's new book on that book's dust jacket!) Within the burgeoning publishing world of modern science-and-math-made-easy, the professors praise each other in these ways, as do the major reviewers.

Are these blurbs ever accurate? Back in 2005, Miller said that Goldstein's "masterful" book provided "an eminently lucid explanation of Gödel’s theorem and its implications.”

Does anyone think that Miller, a general interest reviewer, could string two coherent words together about Godel's highly abstruse theorems? We'd be very surprised if she could, but if she can, it's hard to believe that her ability stems from Goldstein's widely praised book.

At this point in our explorations, we haven't examined Goldstein and Holt's attempts to explain, elucidate, unpack or describe Godel's actual theorems. This week, we've been trying to see if either writer could explain, elucidate or describe the alleged "doctrine" called Platonism, which is said by Goldstein to lie at the heart of all Godel's ruminations.

What the heck is Platonism? We've already seen what Holt said. According to Goldstein's first bite at this apple, Platonism involves the belief that "the truths of mathematics are determined by the reality of mathematics"—and as we showed you yesterday, the project goes downhill from there, all the way to an impossibly strange rumination about the way the Platonist would evaluate the claim that Santa Claus exists.

Please understand—Goldstein isn't some second-rate shlub who got hauled in from the cold. As we explained in an earlier post, she lives a perfect timeless existence at or near the very top of modern academic elites.

She's a ranking philosophy professor, and a highly-regarded novelist. This helps explain why her book was blurbed so favorably by other elites—unless you think that writing like this really does deserve to be praised as transplendently lucid:
GOLDSTEIN (page 87): For a Platonist, mathematical truth is the same sort of truth as that prevailing in lesser realms. A proposition p is true if and only if p. "Santa Claus exists" is true if and only if Santa Claus exists.
"Santa Claus exists" is true if and only if Santa Claus exists? If that explains the doctrine with which Godel "fell in love" as a teen, then the whole world is a Platonist, including you and yours.

Casey Stengel is said to have said it when he managed the 1968 Mets: "Can't anybody here play this game?" the gent is said to have asked.

The later Wittgenstein said the same thing about a wide range of major "philosophers," not excluding himself in his own earlier phase. In time, we'll be perusing this major jailbreak which, according to Professor Horwich, is being strategically ignored.

That will come at a later date! For today, we're going to see what can happen when brilliant people decide to play out of position.

The greatest shortstop would probably make an extremely poor tight end. In 1968, Rod Laver was the world's top-rated male tennis player. There's little reason to think he could have helped Stengel's hapless Mets.

So too in the worlds of mathematics and physics! Consider what happened when Goldstein quoted G. H. Hardy, who was, by all accounts, a brilliant mathematician.

Who the heck was G. H. Hardy? The leading authority on his life answers your question here.

By all accounts, Hardy was a brilliant mathematician. For better or worse, he also crossed over to do some "philosophizing" in his iconic 1940 essay, A Mathematician's Apology.

According to Goldstein, Hardy, "an English mathematician of great distinction, expressed his own Platonist convictions" in this "classic" text. She seems to think that the passage she quotes in her book will help us understand this alleged doctrine.

Below, you see the passage Goldstein quotes, on page 46 of her book. We'll especially focus on the way Hardy puzzles over how we humans can know that 317 is a prime:
HARDY (1940): I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it...

[T]his realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more than what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy—I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
In our view, that passage is the work of a brilliant mathematician who's playing way out of position. In effect, Hardy—a brilliant mathematician—becomes your Uncle Charlie at Thanksgiving dinner, going on and on.

Step by step, a brilliant mathematician leads us away from clarity in that jumpy passage. This is the sort of thing which can happen when mathematicians wander far afield.

Hardy muddles his thinking in that passage at an array of points. At one point, he refers to the number 2 as a "mathematical object."

Do you have any idea why a person would want to do that?

That passage starts with Hardy saying that "mathematical reality" (whatever that is) "lies outside us." Consider:

You surely know what Homer meant when he said the battle between Achilles and Hector took place "outside the walls of Troy." But are you sure you understand what Hardy means when he says that "mathematical reality" (whatever that is) is somehow found "outside us?"

That's a rather unusual formulation. Are you sure you know what it means? Can you think of any conceivable way to disagree with that peculiar statement?

In this passage, Hardy plays with dueling "isms"—with "realism" and "idealism." This will almost surely work to further confuse the general reader. Indeed, we'd advise you to check your wallet even when full-fledged "logicians" start burying you in such jargon.

Eventually, the rubber meets the road. Hardy, a brilliant mathematician, chooses to tell us this:

"317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way."

What makes 317 a prime? According to Hardy, 317 is a prime because it is so! (Hardy's emphasis), not because our minds are shaped in some way.

Friend, would you have any idea how to disagree with that? Has it ever occurred to you to think that 317 (or, more simply, 7 or 17) is a prime because "your mind is shaped in some way," whatever that might mean?

Do you understand what you're being told when you're told that 317 (or, more simply, 17) is a prime because it is so? Are you really completely sure that you're being told anything at all? Does that sound a bit like Uncle Charlie arguing some favorite political point?

In this passage, you see a brilliant mathematician making little clear sense. Is the number 17 a prime just because we think it is? Do you have any idea why anyone would ever make such a claim? In the absence of any such idea, do you understand why Hardy seems to be aggressively "refuting" this claim?

Why is 317 a prime? Now that you've asked, we can explain it amazingly simply. The number 317 is a prime because it can't be divided evenly by the number 2, or by any other "natural number," as you will quickly be able to see if you just give it a try.

317 can't be divided evenly by any other number! Go ahead—you can try them all, though if you're arithmetically slick, you'll know that you only have to try these numbers: 2, 3, 5, 7, 11, 13, 17, 19.

After you try 19 and fail, you don't have to try any more. (Reason: 19 x 19 is larger than 317.) But go ahead—try them all! No other number will divide evenly into 317. That's the most straightforward, simple-minded answer to the (rather imprecise) question Hardy semi-poses in that peculiar passage.

Playing out of position, Hardy almost seems to fashion a tautology: 317 is a prime because it is so! Writing a general interest book, Goldstein presents this passage as if it will help us understand Hardy's "Platonist convictions" and the doctrine of Platonism as a whole.

We humans! If we weren't so inclined to defer to authority, we'd respond to this in the manner of the child who saw that the emperor forgot to put on his clothes. We'd marvel at the lunacy involved in Holt's lucid but ludicrous statement:
"[Gödel was] seduced by the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind. This doctrine, which is called Platonism...has always been popular among mathematicians."
If we weren't so strongly inclined to defer to academic (and journalistic) authority, we'd react as a sensible person might. We'd marvel at the peculiar claim that mathematicians are inclined to think that numbers and circles "have a perfect, timeless existence" of some undisclosed kind. We'd wonder why a man like Hardy was throwing various "isms" around as he heatedly seemed to explain how we can know that 317 is a prime.

What was Hardy trying to say in that passage? We have no idea.

That said, the burden of clarity falls on the person who's making the lofty claim. It isn't your task, as Hardy's reader, to pretend to make sense of something he's said. Nor should you ever simply assume that something makes sense just because it's being said by a ranking academic.

It was Hardy's job to make his statement lucid! If Goldstein is going to quote him, it's Goldstein's job to let us know why "317 is a prime because it is so" isn't simply the holiday raving of a type of Uncle Charlie.

This brings us back to Einstein's general interest book, even as it points us toward the work of the later Wittgenstein.

Einstein is widely viewed as the most brilliant physicist at least since Newton. After he fashioned his theories of relativity, a publisher asked him to write a general interest book to explain what he had done to the general reader.

The book appeared in German in 1916, in English in 1920. Manifestly, it didn't "make Einstein easy." In his recent biography of Einstein, Walter Isaacson told the comical story which explains how this happened.

Einstein, the world's most brilliant physicist, wasn't a general interest writer! In effect, he was the greatest athlete of all time, but not real good at cooking.

According to Isaacson, as Einstein tried to make Einstein easy, he selected his cousin Elsa's teen-aged daughter as his focus group. "He read every page" to her, Isaacson writes, "pausing frequently to ask whether she indeed got it."

She kept saying she understood, "even though (as she confided to others), she found the whole thing totally baffling." So it went when the planet's most brilliant physicist briefly played out of position.

Friend, do you have the slightest idea what it means to believe in the doctrine of Platonism? Was this really some deep philosophical view? Or was it possibly one of the first of Godel's crazy ideas?

We ask because Godel is described as the greatest logician since Aristotle. What might it mean if our greatest logicians was in thrall to crazy ideas? What does it our highest ranking professors can't make out this fact?

Could it mean that we in the west, like the Bushmen of the Kalahari, have been "seeing ourselves from afar?" Could it light the way toward the work of the later Wittgenstein, which we plan to discuss in coming weeks, if we get there before Mr. Trump decides to start his war.

Are Goldstein and Holt amazingly lucid? It certainly isn't a moral failing, but no, we don't think they are.

That said, Goldstein and her partners in blurbing are among our highest academic elites. When you see the way our top professors perform, are you surprised that our journalists perform even worse? Are you surprised that our cultural breakdown has reached the point where Donald J. Trump holds such power?

Next week: The incompleteness file

34 comments:

  1. "That said, the burden of clarity falls on the person who's making the lofty claim. It isn't your task, as Hardy's reader, to pretend to make sense of something he's said. "

    The classic complaint of the dilettante and the ignoramus.

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    1. Coming from an expert who can't explain themselves seems like a reasonable sacrifice to make?

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    2. " ... from an expert who can't explain themselves ...."

      Look who's talking.

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    3. If only Bob read his comments, this might help. The following statements true for different reasons:
      A. 6 = 2*3
      B. 6 = points for scoring a touchdown.

      B was an arbitrary choice. A is forced on us.

      Delete
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  2. Somerby doesn't realize the catch 22 that he has involved himself in. Hardy asks: why is 317 prime? Somerby says, because, dumbass, 317 has no whole number factors other than 1 and 317. Duh! Except, Bob has simply defined "prime number." In other words, Bob is simply saying "317 is prime because it is prime.", which is precisely what Hardy says. Except that Hardy is asking a deeper question. Is there a "mathematical reality" that lies "outside us?" To Somerby, that is laughable, crackpot, crazy. But what makes the term "physical reality" any more coherent? He is on the cusp of stating the problem, if he could open his mind to the possibility.

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  3. It was the 1962 Mets that Stengel managed.

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  4. If Somerby finds Goldstein's book too hard to understand, he should stop reading it and find something easier to grasp. Books come in a great variety so I am sure he will find something to his taste and level of comprehension.

    I do not agree that Goldstein is responsible for Trump and his lying ways, nor do I believe that the news is full of lies and fake news, as Somerby wants to claim (on the basis of some dustjacket blurbs).

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  5. Seriously? Steven Pinker wrote a blurb for Rebecca Goldstein? And anyone takes it seriously?

    https://en.wikipedia.org/wiki/Rebecca_Goldstein#Personal_life

    Anywho, let's talk about the notoriously prime number, 17. I betcha I can factor it. You wanna see my factorization? Do you, do you, huh? There, I thought you did! Well, here we go:

    17 = (4 + i)(4 - i)

    You like that? You want another factorization? OK, fasten your seatbelt, because here we go:

    17 = (1 + 4i)(1 - 4i)

    Math is great,
    Math is good.
    Not everyone likes math,
    But everyone should.

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  6. "You surely know what Homer meant when he said the battle between Achilles and Hector took place "outside the walls of Troy." But are you sure you understand what Hardy means when he says that "mathematical reality" (whatever that is) is somehow found "outside us?""

    I surely know that the world exists without my deceased friend in it. My friend passed away last year but the world went on. The world has an existence without my friend. If I generalize from my friend to all people, I can imagine a world that had no people in it at all, but continued to exist. If there were a massive plague for example. I would then say that the world had an existence outside of people.

    Why is that hard for Somerby to grasp? The moon has an existence even when people do not watch it.

    Why does Somerby keep saying that this is murky and hard to understand? Why is it hard for him? If he would explain his difficulties, we might understand his sticking point, but he appears to be incapable of stating his points of disagreement or confusion.

    Is that our fault or Somerby's? I think it is his fault.

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    1. Because "mathematical reality" is NOT a solid, physical thing. Although the earth is the 3rd Rock from the sun, there is no physical 3 circling the sun in the same way that there still is (at least for now until it smashes into a billion pieces when the comet Hale bops into it. (and yes, I know that Earth vs. comet is even more of a mismatch than freight train vs. smart car)) still a rock there.

      The world keeps turning, for some reason, but without the thinking of the human mind, does the number 3 exist any more than the rules of chess?

      We may think "of course it does" but can we prove it, or just assert and believe it? If man is no longer alive in the year 2525, and if a million years after that ants evolve to be even more sentient than they are now, will they also be talking about 317 and how prime it is? (in their own language, of course) Or will they have invented their own "math" which they will then use to build bridges, electric guitars and thermonuclear warheads in much the same way as they will invent their own board games and political divisions around the Missouri River?

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    2. The idea is that if those ants invent their own math, 317 must be prime in it.

      The relationship expressed by the words "above" and "below" are not solid physical things either. They are conceptual, abstract, designations of the relative positions of objects. But they exist objectively, independent of human beings.

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    3. I suppose the number "3", and other numbers, and math (aside from being interesting) are a handy way of understanding things, and getting things done. Contrary to TDH, I understand the concept that a 'circle" or numbers can be said to have existence outside the human mind - whether the concept is valid, or as TDH suggests, is stupid, probably can't be answered conclusively, though TDH seems to be approaching it at a shallow level. If TDH has a real valid point here, he seems to be failing much like he says Holt or Goldstein do - by failing to make anyone understand what in God's name his point is. He seems to have this idiosyncratic pet peeve, that certain authors attempt to popularize subjects like advanced physics or math, and critics praise them for their clarity, because TDH doesn't know what the heck they are talking about - possibly due to TDH's failures, not theirs. TDH reminds me of once I was listening to a talk radio show - there were 2 talk hosts, and they were talking about books, and one of them opined that "Shakespeare sucked." As one of TDH's sources of wisdom once wrote, 'don't criticize what you don't understand.'

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    4. I would also add, it seems screwy to tie this thing he is peeved about as having any relationship to the Trump phenomenon.

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    5. Somerby's point. First, you have to assume that Goldstein's writing is unclear. Then, the point is that several people wrote enthusiastic blurbs praising her lucidity (clarity) when she isn't clear at all. And if professors behave this badly when they write blurbs, journalists are even worse and that leads us to Trump.

      But this all starts with the assumption that Goldstein is unclear, and Somerby failed to prove that one, so the whole house of cards tumbles down. Do we then have to assume Trump is telling the truth? No, his lies are apparent no matter how badly the professors behave and no matter what journalists say about him. Maybe professors writing about Godel have nothing to do with political misbehavior after all!

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    6. As an undergraduate I worked summers in a cannery—long-ass hours (70-80 a week). And since he likes to city-name-drop Palo Alto and Cupertino, our resident math guy may remember these steamy, odorous plants near downtown Sunnyvale, long given way to massive blocks of condos. These were ancient mini-cities that churned non-stop from the first apricots until the fall rains rotted the last tomatoes.

      A lot of teachers worked there and a lot of college students. A group of us tended to hang out together and spent quite a bit of time during our long shifts discussing everything under the sun. Some were really smart dudes who went on to become doctors, professors, scientists and such (I became a chemist, meh, whatever). One year, like a contagion, reading and discussing literature spread through the lot of us. Camus, Sartre and Dostoyevsky were the rage. Every one of us would have a paperback in our back pocket which would be pulled out during breaks and lull periods. Books were traded back and forth. I managed to coax a few guys to read some of my favorites of that time: Steinbeck, Frank Norris, Hamlin Garland, Hardy and Faulkner. I was already into Camus but I did read The Idiot and The Brothers Karamazov at their behest.

      But here’s the point: the next year for some odd reason, many of these guys became obsessed with relativity. There were even two truck drivers that I had to deal with that got involved. I remember the three of us, outside at three or four in the morning, leaning on a metal railing, staring at the stars, discussing space-time and spitting chewing tobacco juice on the truck scale. I had no interest in understanding relativity then and I don’t now, but for me it was a way to kill time, so I joined in. There were four things I did to try understand this crazy crap, three of them helped and one was a waste. First, I had a couple of Physics texts that had relativity problems and I did every last one of them. As a science student, this was what worked for me the best. Second, I read The ABC of Relativity by Bertrand Russell and Einstein’s book on special and general relativity, the abridged version. These were readable books that could be followed pretty well and I would still recommend them to someone trying to tackle the subject. Third, on a rare day off, I went fly fishing with a buddy, in Pescadero Creek if I recall correctly, and sat for a time on a boulder with my legs dangling in the stream and really, really contemplated time and flux and the old “you can’t step in the same river twice, you can’t step in the same river once” concept as the water rushed by my waders. That helped too. Fourth, what was virtually worthless was reading several awful examples of the type of science-made-easy books that Bob critiques that were available at that time, loaned to me by my cannery buddies. Whatever the subject, my advice is: roll up your sleeves, learn the math, learn traditionally and don’t read that shit and expect to learn a complex subject by reading a single book.

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  7. A few days ago Somerby was asking what philosophy was good for. Most philosophy majors go on to become lawyers. Philosophy professors actively discourage their best students from going on to grad school in philosophy. They aim them toward cognitive science (which encompasses philosophy at the intersection of linguistics, psychology, computer science, and neuroscience). Another career path is bioethics, but there are very few jobs for academic philosophers. That means Goldstein is highly qualified and good at what she does. Somerby is making a fool of himself by criticizing her, but maybe he thinks women make easy targets?

    I was a philosophy major when I first entered UCLA as a Freshman. I quickly discovered that it was nothing but boys arguing with each other and not listening to anyone else. As a girl, I was talked over and interrupted and quickly lost interest because I couldn't get a word in edgewise. I switched majors to history.

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  8. I am reminded of one of Buchwald's jokes that stuck with me. He wrote that he was asked to give a blurb for an author. An author that I remember as being named Blinzjowski, although I am admittedly probably mis-remembering it.

    He then wrote something like - I didn't want to do it, because his recent book was crap, just like his last book was, BUT he also needed Blinzjowski to write a blurb for his future book, so it the offer to write a blurb was one that he could not refuse. Of course, he claimed, he didn't want to lie either.

    So he settled it by blurbing "Blinzjowski has done it again."

    Basically they were made men, supporting and being supported by other made me.

    Since what Hardy said seems so tautological they probably need to give more examples. Three examples of lesser truths that I would use are

    1) the rules of chess
    2) the capital city of South Dakota
    3) the pronunciation of said capital's name

    We might learn those things either in or out of school. However, unlike 317, those are truths that we humans made, and that can be changed. We can alter the rules of chess. Perhaps in the future we will be playing 3-D chess instead of the 2-D version we play today. Perhaps blitz chess will become more popular since it is clearly a lot more fun than chess.

    Before 1880, the question of SD's capital could not even be asked, because there was no such place. In the future there will be no such place. In other galaxies far far away, like Chicago, they do not CARE about that place, but their schoolchildren will still be forced to learn about 317.

    Also, the further away one gets from SD, the more likely they will say it wrong, pronounce it as Pee-air instead of the way the locals do, as Peer.

    We make, collectively we decide upon the rules of chess, the boundaries and capitals of states and the pronunciation thereof. They are temporarily true, because we make it so.

    OTOH, we don't get to decide if 317 is prime or not.

    The other day, though I happened to see a snip of a debate between Hitchens and a theologian. Hitches said that logic was man-made and the theologian scoffed at that idea. Would Hitchens also say that math is man made? That 317 is prime because of numbers and functions which are man-made?

    Truly, it may be the non-Platonist who has some explainin' to do.

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    1. Well put, Dr. T. It's frustrating that Bob doesn't read his comments, and so doesn't get the benefit of your lucid explanation.

      Another example to illustrate this principle is that various human societies created the natural numbers, and they're all the same. In every one of these systems, 2 + 2 = 4. This expression exists independent of which human society does arithmetic. That's why it's reasonable to say that the truth of that equation exists outside of human society.

      BTW Godel's proof depends on self-referential statements. Bob didn't realize it, but he made one such state in this post:

      The greatest shortstop would probably make an extremely poor tight end. In 1968, Rod Laver was the world's top-rated male tennis player. There's little reason to think he could have helped Stengel's hapless Mets.

      So too in the worlds of mathematics and physics!


      When Bob discusses Godel's theorem, he's the one playing out of position.

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    2. Aren't most of us though? I am just a semi-smart guy with a fair amount of education who, when an idea or conundrum is dangled before me, will chase after it, confident I can snare it with my powerful claws.

      Since tight end is mentioned in that example, I happened to recently see a YT video about the best undrafted players of all time. Antonio Gates spent four years playing college basketball, and then later became one of the best tight ends to date.

      Perhaps silly to pick at the analogy, but often size, strength and speed will make a person excel at multiple sports. In somewhat the same way, a person who is good at math and/or physics is probably gonna be good a logic.

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  9. What is wrong with one bushman being nice to another one?

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  10. "Has it ever occurred to you to think that 317 (or, more simply, 7 or 17) is a prime because 'your mind is shaped in some way,' whatever that might mean?"

    Well, yes, sort of. OK...but first, let me fully concede that the Platonism discussion is perhaps not terribly well-defined, and so quite wooly in places...but, with that caveat, here goes.

    First: bear in mind that many mathematicians for a long time were trying to put their field on a truly rigorous basis. They wanted to be explicit about a set of axioms, and they wanted all propositions to be derivable, via rules of logic, from those axioms and nothing more. That program got into difficulties (Russell, and Gödel!), but that didn't stop mathematicians from continuing to hope, wistfully, that something like a full axiomatic formulation might be possible. Now, if that's the case, isn't the choice of axioms just a free choice of the human mind? Doesn't that make mathematical statements, derivable from those axioms freely chosen by those humans, also pure, perhaps arbitrary, constructs of the human mind?

    Here's an analogy. "The cat sat on the mat" is a valid sentence in English. The rules of grammar of the English language, the words of the English language, even the letters of the alphabet are, it seems plausible to assert, pure constructs of the human mind. None of it "exists" outside the context of an animal endowed with language. And while we may believe that the statement of the cat sitting on the mat is a matter of objective truth or falsity, the proposition that "the cat sat on the mat" is a valid English sentence would seem to be much more a matter of tacit agreement among anglophones than a statement about objective reality. How would an alien who had never met an anglophone agree with the statement that "the cat sat on the mat" was a valid English sentence?

    Well, if the letters words and grammar of English are constructs of the human mind and NOT elements of objective reality, why would the choice of a set of mathematical axioms and the deductions from them not be similar human constructs?

    Note that Bob's argument, that of course 317 is a prime because we can check it, is analogous to the statement that of course "the cat sat on the mat" is an English sentence. Nobody doubts either claim. What's really being asked is whether the rules of arithmetic that we used to check the primality of 317 are arbitrary constructs of the human mind, or whether they are somehow more deeply encoded in the universe in which we live. Would the alien who had never met a human also be able to formulate in alienspeak the question of whether 317 is prime? More simply....would aliens have the same concept of number that we have? If he does, would that suggest that the choice of mathematical axioms is not arbitrary, but in some way intrinsic to existence?

    Perhaps the whole idea of reducing mathematics to axioms and logic is foolish, but it was an idea that certainly consumed a lot of mathematicians for many years. With that backdrop, it is not totally surprising that some folks asked the question as to whether, in woke-speak, mathematics is culturally determined.

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    Replies
    1. “Note that Bob's argument, that of course 317 is a prime because we can check it, is analogous to the statement that of course "the cat sat on the mat" is an English sentence.”

      317 is a prime only within the realm of mathematics. In the real world, the cat sat on the mat.

      I would posit this, perhaps I’m wrong: Whether or not a number is prime doesn’t mean Jack-Shit Squat. How does that advance human civilization?

      “With that backdrop, it is not totally surprising that some folks asked the question as to whether, in woke-speak, mathematics is culturally determined (sic).”

      Please expound on that idea.

      Leroy

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    2. The phrase "culturally determined" preceded woke speak.

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    3. Leroy -- Math fields like geometry, trigonometry, and calculus have obvious real world applications. Number Theory not so much, as you point out. However, Number Theory does play a big role in cryptography and cryptocurrencies, such as bit coins.

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    4. Leroy -
      1) whether "the cat sat on the mat" is a legitimate English sentence (is each collection of letters between spaces a "word", does it satisfy the rules of grammar...?) is a different question from whether "the cat sat on the mat" is a true statement. My point was that English is constructed from a rather arbitrary set of rules (the analog of axioms in mathematics), and so is, perhaps, a human construct. Dr. T. has a better example than mine: the rules of chess. Are they a construct of the human mind, or do they "exist" in some sense outside humans? (I still agree with Bob that it's a wooly question...but it's not prima facie absurd, as Bob seems to argue).

      Regarding woke-speak: My point was that some scholars argue that many of the intellectual edifices of Dead-White-Male civilization are in some sense culturally specific artifacts, so much so that they even lack objective truth. (Sometimes the more extreme among such scholars are very silly). The idea that mathematical axioms are free choices of their postulator would suggest that mathematics is, to some extent at least, a product of the culture of the postulator. (A Platonist, believing that mathematics has some existence outside that of any particular species, would presumably deny at least the more extreme versions of such a claim). In my view we should try to be alert to the cultural assumptions that underlie our thoughts (see my earlier comment on Tacitus and the bushmen), and the extent to which mathematics is such a cultural construct, and the extent to which it is universal, is worth thinking about. This alertness to the perspectives of members of other cultures or experiences not our own is now often characterized as "woke"...hence my slightly snarky use of the term.

      Anonymous - yes, "woke" (in this context) came into use after the expression "culturally determined". I would nonetheless characterize being alert to the possibility of some idea being "culturally determined" is one aspect of being what we would now describe as "woke". Wouldn't you?

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  11. Interesting stuff, thanks people. I don't know if the universe exist, or if consciousness is real. But I do like the factorization of 17 using imaginary numbers. In conclusion, upon waking I identified myself, and called me ONE. Then, inquired of the NOT ONE, also called ANOTHER ONE, and found two. Constructs of the human mind, but representing 'meta-' or the epiphenomenonal. Is it conceivable that a universe exists without the concept of a unitary?

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  12. Bill Maher thinks it makes liberals look bad to make sexual assault accusations against Kavanaugh when the accuser wishes to remain anonymous and the assault happened in high school.

    He is wrong about this for the following reasons: (1) the accuser is not anonymous to the FBI or Feinstein, only to the general public; (2) the sexual assault allegations are supported by any number of decisions and statements by Kavanaugh showing a fundamental disrespect toward women as autonomous agents and equals to men; (3) Kavanaugh's character witness and accomplice Mike Judge is even worse; (4) attempted assault affects the woman involved long after the event, it has no statute of limitations in the state in which it occurred; (5) the capacity to do that as a teen speaks to the character of a man who will make judgements affecting the lives of women; (6) there must be other candidates who have not attempted to assault women; (7) this matters to women and thus should not be dismissed as unimportant, simply because men can imagine themselves in Kavanaugh's situation; (8) the accuser has asked to remain anonymous because of the extreme threats received by anyone who has accused Trump or those he has supported of anything -- it is legitimate to wish to avoid death threats and harassment by Trump supporters; (9) the difficulty proving such allegations does not mean such crimes do not happen and should not be punished; (10) Kavanaugh's denial has not been accompanied by any sympathy toward women.

    Maher himself has issues with women. He isn't a fit arbiter of whether assault charges should or shouldn't be alleged in various situations.

    I fully expect that Somerby will defend Kavanaugh over the next week or two. Somerby and Maher have some things in common beyond being comedians. Louis CK might join them except he appears to be keeping his head down and his mouth shut about women's problems with men like them.

    "It looks bad..." Wouldn't want to get other men upset with you by siding with the wimmin. Especially not conservative men, he-men women haters, and men who have the guts to lock a woman in a room and jump her bones. Important to keep those kinds of guys happy with you, so say nothing and look embarrassed when the women start complaining about how they are treated, as HIGH SCHOOL GIRLS.

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  13. I agree with Somerby. The descriptions of "Platonism" are pathetically inadequate but treated as lucid and enlightening by reviewers. As always, the blog's focus is on media practices. Do they help us think we'll for ourselves by orienting themselves to the truth? As in political matters, no, they do not, to a systematically dangerous degree. Rather, they tell us what opinions to hold without helping us to figure out why.

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