THE PLATONIST FILE: Goldstein tries to explain what Godel believed!

THURSDAY, SEPTEMBER 13, 2018

Professor takes Platonist challenge:
Rebecca Goldstein is a ranking philosophy professor.

She's also a highly-regarded novelist. For our money, her inclination to tell the human story didn't serve her especially well when she wrote her 2005 general interest book, Incompleteness: The Proof and Paradox of Kurt Godel.

Goldstein goes into substantial internal detail about the way Godel "fell in love" when he was still a teen—about the "ecstatic transfiguration" produced by his love affair with Platonism.

She goes on and on, then on and on, about this "transfigurative intellectual love." At times, though, she also makes it sound like Godel had sworn allegiance, at this point in his life, to some revolutionary political group.

According to Goldstein, Godel "had become a Platonist in 1925," the year during which he turned 19. He "was already a committed Platonist in 1926," the year in which he began attending meetings of the super-elite discussion group called the Vienna Circle.

Godel was moving in lofty academic circles at a very young age. It's also true, according to Goldstein, that he didn't want the Circle to know about his commitment to Platonism. In effect, he was an Alger Hiss hiding within a circle of people all named Whittaker Chambers.

Back in 2005, Jim Holt reviewed Goldstein's book for The New Yorker. He told a shortened version of this slightly comical story in that review, which has now become the title essay of his own new book, When Einstein Walked with Godel:
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. In the philosophical world of nineteen-twenties Vienna, however, it was considered distinctly old-fashioned. Among the many intellectual movements that flourished in the city’s rich café culture, one of the most prominent was the Vienna Circle, a group of thinkers united in their belief that philosophy must be cleansed of metaphysics and made over in the image of science...

Gödel was introduced into the Vienna Circle by one of his professors, but he kept quiet about his Platonist views. Being both rigorous and averse to controversy, he did not like to argue his convictions unless he had an airtight way of demonstrating that they were valid...
Members of the Vienna Circle had no idea that a Platonist was lurking among them! For better or worse, Goldstein tells the story of the "clandestine Platonist" at much greater length, and with much more dramatic flair.

At any rate, Godel was "a committed Platonist" by the time he was 20. He was lurking among the "logical positivists," unwilling to let them know about his deepest beliefs.

By the time he was 24, Godel had formulated and publicized his "incompleteness theorems," the theorems which are said to identify him as the greatest logician since Aristotle. Essentially, Godel saw these theorems as the proof of his beloved Platonism. Or so Goldstein says, we would assume correctly.

As of 1930, this silliness was consuming one of the western world's highest intellectual elites. Along the way, we may puzzle a bit at the stakes involved in this war of the worlds.

As we saw yesterday,
Godel, the second greatest logician, was trying to figure out how we can know that 2 + 2 equals 4. He'd fallen in love with Platonism, and wanted to prove that the doctrine was true.

Can major battles of this type really revolve around matters like 2 + 2? As we proceed today and tomorrow, we'll see other apparently vapid puzzlements move to center stage in this remarkable tale.

Before proceeding, we ought to admit it—this story of the "committed Platonist" strikes us as amazingly silly. Today, though, we ask a more challenging question:

In the course of writing her book, was Goldstein able to explain the nature of this "doctrine?" To what beliefs had Godel committed when he committed to Platonism, if in clandestine fashion?

As we saw yesterday, Holt made virtually no attempt to describe this powerful doctrine in his review of Goldstein's book. Was Goldstein, a ranking philosophy professor, able to clarify matters further, writing at much greater length?

In her first attempt at taking the Platonist challenge, Goldstein had, rather unhelpfully, offered the formulation shown below. We've shown you this passage before. Try to ignore the technical language:
GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...

Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality.
There it stands. In this, her first pass at the Platonist challenge, Goldstein tells us this:

The Platonist believes in "the objective existence of mathematical reality," whatever that is supposed to mean. But wait—there's more.

According to the Platonist, the truths of mathematics are determined by the reality of mathematics! More specifically, the truths of mathematics are determined by the nature of the entities which make up that reality.

So this ranking professor has said. Presumably, everyone can see how unhelpful this first attempt at explication was. That leaves us asking an obvious question:

Does Goldstein go on, in her book-length text, to clarify, unpack, elucidate or explain the essence of this alleged doctrine? Does she ever do a better job helping us understand the nature of the doctrine to which the second greatest logician in history is said to have committed his life?

What the heck is Platonism? Does Goldstein ever provide an answer which is, quoting from the blurbs on her book, lucid, accessible, clear?

We're going to say that she doesn't. To give you a sense of what we mean, let's take ourselves to page 87 of her general interest book.

Friend, let's start with a basic admission. There's no way to present an excerpt from this book without exciting a possible objection. The reader may suspect that a tiny shard of Goldstein's brilliantly lucid exposition has been taken out of some larger context.

Go ahead—keep that possible objection in mind! Then proceed to read this passage, in which Goldstein tells us what a possible fool believes:
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. "Every even number greater than 2 is the sum of two primes" is is true if and only if there is no even number greater than 2 that is the sum pf two primes (even if we can never prove it).
What does a Platonist believe? With what sorts of beliefs did our second greatest logician fall in transfigurative love?

According to Goldstein, a Platonist believes the following. A Platonist believes that the proposition "Santa Claus exists" is true if and only if Santa Claus does exist!

No, really. That's what it says!

Friend, you must be a Platonist if that's what the doctrine is! Every person on your block is a committed Platonist too.

"Santa Claus exists" is true if and only if Santa Claus exists? Everyone believes that statement—and everyone believes the other two examples provided in that peculiar passage.

To marvel further at the type of work which routinely emerges from our academic elites, we turn to the footnote on that same page. In that footnote, Goldstein further discusses the third example from the passage we've already posted, the example involving the sum of two primes.

According to that footnote, "the Prussian mathematician Christian Goldbach (1690-1764) had conjectured that every even number greater than 2 is the sum of two prime numbers."

So far, so perfectly clear. But in her footnote, Goldstein further discusses Goldbach's conjecture. As she does, she tries again to let us know what a Platonist believes and asserts:
GOLDSTEIN (page 87): Goldbach's conjecture has been confirmed for every even number that has ever been checked; however, no proof has of yet been discovered for the universal conclusion that every even number greater than 2 is the sum of two primes. The fact that Goldbach's conjecture remains unproved means (at least according to the Platonist) that lurking out there beyond the point where mathematicians have checked there might be a counterexample: an even number that isn't the sum of two primes. Then again (according to the mathematical Platonist), there may not be a counterexample: every even number may be the sum of two primes, without there being a formal way of proving that this is so. A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other.
"A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other?"

Tell the truth! At least on its face, does that make any sense?

"A Platonist asserts that there either is or isn't a counterexample?" Who wouldn't make that "assertion?" And why would it take a Platonist to make the other assertions Goldstein lists in that passage about the conjecture?

Goldbach's conjecture remains unproved (and unrefuted)? Why would it take a Platonist to say that there might be a counterexample which hasn't yet been checked? Wouldn't anyone say the same thing? Ecstatic transfiguration to the side, what's love of Platonism got to do with it?

Goldstein's examples in this footnote bring us back to Santa Claus, who can only accurately be said to exist if he does exist. We flash on our second greatest logician trying to figure how we can know that 2 + 2 equals 4.

Misguided respect for academic authority will induce the trusting soul to assume that there must be some lofty explanation for these peculiar presentations. We'll strongly suggest that, in matters like these, such trust will be leading us wrong.

At substantial length, Goldstein tells a novelist's tale in her book—a tale of the powerful intellectual love which produced a committed Platonist. Respect for intellectual authority may incline us to believe that some actual great beliefs lie at the heart of this tale.

On the other hand, yesterday, in Holt's text, we saw our second greatest logician puzzling over 2 + 2 = 4. Today, we see a high-ranking professor offering what seems like perfect twaddle concerning the existence of Santa Claus.
.
In Goldstein's book, we also encounter the passage shown below. In this passage, Goldstein gives another example of what is, or was, at stake in this battle. Today, we'll highlight that passage:
GOLDSTEIN (page 44): Godel's commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...

Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist...and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity.
According to the Platonist, the claim that the number 4 is even is an "objective" claim. So is the claim that 25 is a perfect square. (That is, that it's the product of 5 x 5.)

These claims are "objective," the committed Platonist cries. But so does everyone you've ever known, along with all those you've never met. This account of what a Platonist believes doesn't seem to make any sense—and yet, this is the way a ranking academic authority explain this ancient "doctrine," with three academic stars blurbing how lucid she is.

According to Goldstein and Holt, Platonism is the doctrine which defined the world of our second greatest logician, a man who seemed to be mentally ill throughout his life and who was famous for believing a long list of crazy ideas. According to Goldstein, it was on this peculiar plain outside Troy that a giant battle was waged involving our second greatest logician and a lofty academic Circle.

Does any of this make any sense? Respect for authority tells us it must.

Experience tells us something different. Then too, there's the passage from G. H. Hardy, another alleged Platonist, which Goldstein quotes early on.

Tomorrow: When mathematicians stray, or how can we humans possibly know that 317 is a prime?

34 comments:

  1. What a loss for the schoolchildren of Baltimore when Somersby stopped teaching there !!

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  2. "Essentially, Godel saw these theorems as the proof of his beloved Platonism. Or so Goldstein says, we would assume correctly."

    Why does Somerby need to assume this? Why does he "assume" it at all? Can he not figure out what Gödel himself said about Platonism?

    He simultaneously ridicules Goldstein's presentation of Platonism, but then assumes she is right about what drove Gödel to his theorems?

    Next question: does Goldstein present a description of the non-Platonist view in her book, so that the reader may understand the difference? I have not read the book, and I do not trust Somerby to give it a fair hearing.

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  3. "Friend, you must be a Platonist if that's what the doctrine is! Every person on your block is a committed Platonist too."

    "Santa Claus exists" is true if and only if Santa Claus exists? Everyone believes that statement—and everyone believes the other two examples provided in that peculiar passage."

    Nice one, Bob. Be sure and try that one out at your next comedy gig. Except that this is a (we would say deliberately, giving the blogger the benefit of the doubt) boneheaded interpretation. Need it be said that the author is not saying that '"Santa Claus exists" is true if and only if Santa Claus exists" is a description of Platonism. Goldstein is talking about the way we human beings evaluate the truth of statements like the one about Santa Claus, and Platonism views mathematical assertions similarly....aw fuck it. Somerby is just bullshitting again. As he might ask: "Who cares?"

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  4. The Greeks regarded mathematics as inextricably entwined with religion. The study of geometric shapes and mathematical relationships was applied to discover the regularity and order of the natural world, the golden ratio as an example. Their philosophy had a purpose that Somerby doesn't seem to understand. It seems likely that Godel did understand the Greek inseparability of mathematics and nature in a spiritual way. Somerby is also refusing to see Godel in a historical context -- the application of empiricism and scientific method to all a priori beliefs that swept the scientific world. Somerby needs to be asking why Platonism was unpopular during that time period. Not implying that this is more of Godel's paranoia (which is described as part of his life in his 70s).

    When you ask what can be proven and what cannot, you are asking a very important question for science. Is Somerby so scientifically illiterate that he cannot see that?

    I am astonished that anyone who values learning would assert that any field of knowledge is irrelevant, unimportant, not worth studying. It is fine for Somerby to say that he himself doesn't want to spend his time studying something, but dismissing it entirely is just plain wrong. Next he'll be saying we should de-fund the space programs because who cares about what exists off-planet and we have too many practical problems close to home that can use the money. Oh, wait, that's what Obama said in 2008. Maybe that is what Bernie is saying too. Maybe we don't need to fund the arts either. Who needs made up stories when there are so many other non-fiction books to read? There are folks who believe that too.

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  5. Somerby keeps going back to the phrase "respect for authority." He loves to bash professors. In this case, those professors are also philosophers in their own right. A person doesn't become a professor like Goldstein or Holt without doing original work in their field. That is how you get your doctorate, how you are hired at a university and how you are promoted to full professor. These people are working philosophers, not journalists writing books about philosophy.

    The respect shown to these authors is not because they are authority figures. It is because they are working philosophers with training in philosophy, genuine accomplishments recognized by their peers (Somerby is not one of them), and the knowledge to comment in the ways they have done in their books.

    Somerby thinks we are blindly and mindlessly going along with whatever these folks say. He thinks all of journalism is doing that too. He thinks that the kind of reasoning he has applied to their work should lead others to be critical of these books, not rubber-stamp approval of them. It is incomprehensible to him that people might actually understand this topic well enough to respect the expertise of the authors, along with the contribution of Godel himself.

    Somerby dislikes professors because he (1) thinks he is smarter than they are, (2) thinks he knows more than they do, (3) thinks they haven't earned their position at universities, (4) thinks they are not doing their jobs properly, not protesting the right things, not writing the right op-eds, saying bad stuff when they do write, (5) resents the respect paid to them by reviewers.

    What happened to Somerby in college? I'll bet he found out that most of the other students were as smart or smarter than him and he couldn't deal with that fact. I'll bet he found out that he couldn't get good grades with minimal effort, that his professors expected more, and that made him angry, so he blamed them not himself for his laziness. I'll bet he learned very little while there because when you are God's gift to the world, you don't pay attention to what anyone else says or thinks. With his gentleman's C average, I'll bet he wasn't qualified for any real job, needed to avoid the draft, and became a teacher because, being lazy, it was the easiest opportunity open to him. Then he found out that teaching is hard too, and blamed the kids, their poverty, society's neglect, etc. And he's still doing that.

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  6. Bob mocks a statement that makes perfect sense:
    A Platonist asserts that there either is or isn't a counterexample, irrespective of our having a proof one way or the other.
    His lack of mathematics hampers him.

    Sad to see Bob mocking the concept of a number theory statement that could be true but unprovable. Even before Godel proved that such things exist, this was a natural question. Consider a simpler statement:
    Somewhere in the decimal expansion of pi, there are 1000 consecutive 3's.
    If the statement is true, it could be proved (in theory) by working out the expansion of pi until you reach the 1000 consecutive 3's. But, if it's false, you cannot disprove it by looking at the decimal expansion of pi, because that's infinitely long.

    Let's suppose that the statement is false: there is no sequence of 1000 3's in the expansion of pi. And, let's suppose this result it unprovable. A Platonist would nevertheless find it reasonable to say, "the statement is false." In fact, that was the first sentence of this paragraph, and, I think, most people would not find that sentence unreasonable.

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    1. In fairness to Bob, I would agree that Goldstein didn't explain this concept very well. Most readers probably would not understand it, just from that sentence in the footnote.

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    2. If you read something in a book that you don't understand, the problem is not with the author, but you just go look it up. Now we have the internet, so it is much easier to do this than it used to be. But that is the essence of being a literate, educated person. I am wondering why they didn't teach Somerby to do this at Harvard.

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    3. One person who is easy to mock is that fat coward Trump

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    4. Thank goodness! For many years we have mocked our Presidents. E.g. an uproarious phonograph record called "The First Family", that dropped out of sight after JFK was assassinated. https://www.youtube.com/watch?v=Xwu8S6Ekx9w

      We couldn't mock Obama, because he was black. I am glad to see the tradition returned.

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    5. Don't be a jerk. Lots of people mocked Obama, especially on the right. Do you think none of that tasteless racist trash reached the eyes of those who supported him?

      Those of us who work in offices got that stuff via email until a supervisor clamped down on it. It was all over the internet.

      There is no point in mocking Trump. Good people despise him. The rest are impervious, just like David.

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  7. One area that Plato got wrong was the idea that we are born with knowledge already in us, so that "all learning is recollection". https://www.thoughtco.com/slave-boy-experiment-in-platos-meno-2670668

    AFAIK there's no reason to believe that Godel's version of Platonism included this false belief.

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    1. All the Greeks got lots of things wrong.

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    2. We're born with the knowledge of how to see colors, how to see lines and triangles, how to walk, how to acquire a language, ... .

      We're born with lots of knowledge. If not, we couldn't begin to learn what we need to know.

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    3. Kids may be born with the capacity to walk, but they don't actually do it until they have experience with the physical world. Same with language. You cannot learn your native language without interacting with other speakers. People do not differentiate colors using words (as a concept) until they have the need to tell things apart that are the same in all respects except color. Color is a hard concept for preschoolers to learn. It would be easier if it were innate, but it must be taught using objects that are similar except for color. They pay more attention to shape and texture than color.

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    4. We see colors even if we don't have names for them. We're born with the knowledge of how to do that.

      We're born able to see contours and shapes, even if we don't have names for them. We're born with that knowledge.

      We're born knowing how to walk, even before we're strong enough to do it. If you hold a baby with its feet on the floor it knows how to alternate them, left and right.

      We're not born we a particular language, but we're born with the knowledge of how to learn one. We know to distinguish speech related sounds from background noise.

      When we're hungry we know that we need to eat.

      We know when somebody is trying to be nice to us or threaten us.

      If we itch, we know to scratch.

      We have a heck of a lot if innate knowledge.

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    5. We are not born with depth perception. That is acquired. We are not born forming those contours and shapes into objects and attaching names to them. That is taught. We are not born knowing that poison ivy should be avoided but regular ivy is OK.

      We have the capacity to learn a great deal but experience is also needed. If someone had to get by on only what is innate, they would not be able to function.

      That ability to walk by alternating foot movements is not "knowledge". It is a reflex that occurs at the level of the spinal cord and doesn't even involve the brain. It is why chickens can walk with their heads cut off. There is no point in glamorizing reflexes and instincts into "knowledge". No one is concerned about what is innate and what is learned any more because the answer is that innate knowledge and learning via experience interact to produce behavior.

      We know to distinguish speech from noise but we cannot recognize phonemes without hearing them spoken by others. We cannot segment speech sounds into words. We cannot attach meaning to those sounds.

      Not everyone sees colors and not everyone sees them the same way. Acquisition of language imposes order on our perceptions to the point where blind people and color blind people use color words the same way as sighted people do.

      Babies cannot judge the intentions of others. Even as adults, we frequently misjudge whether others are benign or hostile.

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    6. We couldn't see blue until recently. Not mentioned in the Bible or Koran etc..

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    7. Lots of cultures have a single term for a category that includes both blue and green (called grue by linguists). Aside from the middle east, also India and several Asian languages (Vietnamese for example). People can see blue and green differently but use that single term plus modifiers to distinguish them (grue like the sky, grue like a banana tree leaf, grue like the ocean). Russian not only divides blue from green but has two distinct terms for blue, dividing the colors into light and dark blue categories. English uses modifiers for this. Cross-cultural studies show that regardless of how people divide up the visual color space using words, they see colors the same (with variability due to individual differences, which can be extreme).

      So, we could see blue but not talk about it succinctly, Grunt. English uses more distinct terms for items in many kinds of categories while other languages have fewer words and use basic terms combined with modifiers to make the same distinctions.

      Obviously, I think this stuff is fascinating. For example, Spanish has more words for positive emotions whereas English has more words for negative emotions, comparing the two languages.

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  8. I’m enjoying this series, the comments not so much, given Elba’s lengthy intrusions, meant only to castigate rather than illuminate. Bob’s making good points, and he’s as acerbic as ever. No problemo. I’m looking forward to this “new Bob.”

    “We are not born with depth perception. That is acquired.”

    Assuming, that is, you are born with the gift of sight. And that you’re lucky enough to have two eyes.

    Many of the comments in this thread make that assumption about “innateness.” Several others, like walking, talking, perceiving, all rely on someone with all of the faculties needed for those tasks, which are acquired during gestation. Spina bifida, Zika, Down syndrome etc. make blanket statements on the subject of innateness seem out of place. Luckily (or not), they’re born with an involuntarily beating heart and breathing lungs. And as far as I know, the only innate or instinctual thing we inherit at birth is the ability to wail and suck a teat.

    After that, what is innate is only the human organism’s ability to evolve within its own lifetime, from embryo to death. And we seem to have equal parts compassion and brutality. It was necessary for our evolution, you see. But brutality seems to be center stage for our “leaders.”

    For Caesar, the best musical depiction of a human life cycle I’ve heard to date. It’s tearjerker, to me anyway. Maybe you have another example to share.

    https://www.youtube.com/watch?v=NpaKLbb1Kaw

    Leroy

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    1. Innateness doesn't refer to what an individual can do but to what the species can do.

      You seem to have some theories but there are actual studies of this stuff and scientific consensus.

      You should stop calling random people Elba. You aren't applying that term to the same person consistently -- it seems to refer to the posts you dislike. Why not just skip those comments instead?

      Babies who cannot see can still acquire depth perception and learn to walk using their sense of touch and hearing. They need to understand how far away things are from them, where they are located in space. Sighted babies do that by falling down occasionally, not just by looking at things (newborns do that) but by also reaching out and touching the faces of the people who are holding them.

      Look at the work of Joseph Campos at Berkeley.

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    2. After Napoleon Bonaparte retired, he adopted the hobby of composing English palindromes. His greatest achievement was this:

      Able was I ere I saw Elba.

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    3. 9:28

      “Innateness doesn't refer to what an individual can do but to what the species can do.”

      What an odd statement. Since the species is composed of individuals, and the species resides as a communal organism, your statement is both true and false. But I suppose that’s the way it is with all “either-or” questions. B.F. Skinner was right, at the same time he was wrong.

      https://en.wikipedia.org/wiki/B._F._Skinner

      All of the great questions concerning our existence end up being a synthesis of both, or many, ideas. There is no absolute truth, but rather a confluence of philosophies to explain the truth of our existence, with science as a guide. That’s why, as an atheist, I find some value in religious philosophy. All religions seem to contain a golden thread, which is compassion and understanding. It’s a very thin thread, indeed, but it’s there, and it’s the result of our “longing for life itself,” as Gibran put it.

      http://www.katsandogz.com/onchildren.html

      My “theories,” if they can be called that, whilst not scientific by any means, draw from the knowledge I’ve gained through science. I’m comfortable with what I know, despite the seeming paradox that the more I learn, the less I know. And I’m always willing to learn more. Gimmee a link sometime!

      You reminded me, somehow, of Helen Keller, an extraordinary case. It’s no wonder she was a socialist. She would have died without our innate understanding, as we mature, of the need for community for our very existence.

      As far as Elba, that person is identifiable through the great slabs of text, the point of which is to malign the proprietor. 3:01 is a perfect example. I don’t care if that is the one originally named, it carries the same imprimatur.

      Leroy

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    4. "Gimme a link sometime!"

      Okay: https://www.youtube.com/watch?v=CyDaacRLBTY

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  9. Watch James Spader's character in the original Stargate movie. How can anyone dislike academics like him? Like Godel, he has an esoteric passion.

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  10. Keto X Factor Sleep disorders such as insomnia and sleep apnea may contribute not only to pounds gain but likewise create tremendous problems in weight loss. Research have shown that getting only 5-6 hours of sleep or more than 9 hours of sleep will place you at risk to get weight.

    ReplyDelete
  11. Re the Goldbach conjecture the howler states:
    A Platonist asserts that there either is or isn't a counterexample?" Who wouldn't make that "assertion?"

    A person who isn't a Platonist wouldn't make that assertion.

    The howler has just proved that he doesn't understand what Platonism is all about. By assuming that there must either be a counterexample or not be a counterexample, then even if we could never actually prove it one way or the other, the assumption is that there must either exist a counterexample or there exists no such counterexample. That's assuming the independent real existence of numbers, even though one has no way of verifying that assumption one way or another - and that is Platonism.

    The Platonist can speculate that perhaps there might be a counterexample, but it might be so huge a number that it would be physically impossible to ever describe it, and so we might never prove its existence - or else there is no such number, and every number actually satisfies Goldbach, even though there is no finite way of proving it.

    On the one hand, a non-Platonist allows the possibility that we might never actually be able to prove Goldbach one way or the other, in which case the non-Platonist does not assume any independent reality of existence or non-existence of a counterexample. In the same way as an atheist does not assume the independent existence of a invisible god.

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  12. Who has an anti-science, anti-expertise bias? The right. Why is Somerby arguing the right's approach to topics these days, mocking Holt and Goldbach for being professors who wrote books, mocking Godel for advancing the field of logic and mathematics?

    Somerby has become the Trump of bloggers. Maybe he is decrying Trumpism by modeling it himself. If so, he is being too subtle. Too many people will take him at face value.

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  13. People who can see colors didn't learn to see colors. They have the innate capacity to see colors.

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    Replies
    1. Take a look at some of the experiences of people who were born with no vision, then had their vision restored. They had to learn to see.

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