THE PLATONIST FILE: Attempts to explain 2 + 2 = 4!

WEDNESDAY, SEPTEMBER 12, 2018

The perfect timeless world of our greatest logicians:
At this point, it's important to remember who we're talking about.

We're speaking here about Kurt Godel, "who has often been called the greatest logician since Aristotle." So says the widely praised science/math writer Jim Holt in the opening pages of his new book, When Einstein Walked With Godel: Excursions to the Edge of Thought.

Is it true? Was Godel really the greatest logician since Aristotle? Since Godel (1906-1978) lived in the twentieth century, that would make him the greatest logician in something like 2400 years!

You'd almost think that a person like that would have made some great, identifiable contribution to human thought. You'd almost think his name would be well known.

With Godel, the story is different. In Holt's telling, this second greatest logician seemed to struggle with mental illness since perhaps the age of 5—mental illness which became so extreme that he ended up dying of self-starvation.

Then too, there were the crazy ideas. Holt discusses them early on, contrasting Godel with his friend, Albert Einstein:
HOLT (page 4): Although Einstein’s private life was not without its complications, outwardly he was jolly and at home in the world. Gödel, by contrast, had a tendency toward paranoia. He believed in ghosts; he had a morbid dread of being poisoned by refrigerator gases; he refused to go out when certain distinguished mathematicians were in town, apparently out of concern that they might try to kill him. “Every chaos is a wrong appearance,” he insisted—the paranoiac’s first axiom.
Our greatest logician believed in ghosts. He had a morbid fear of gases from the fridge.

Holt essay first appeared in 2005 as a somewhat disguised review of Rebecca Goldstein's book, Incompleteness: The Proof and Paradox of Kurt Godel. As such, almost all Holt's material seems to be drawn from Goldstein's book, in which she presented an amusing list of Godel's apparently crazy ideas—crazy ideas which didn't seem to spring directly from paranoia or other such illness.

Does it seem strange to think that our greatest logician was perhaps best known, in his adult years, for his various crazy ideas? We'll let you wrestle with that one. For ourselves, we'd be inclined to see this syndrome as perhaps being instructive, illustrative of the vast intellectual dysfunction at the heart of the human experience.

Godel became known as "the greatest logician since" because of his incompleteness theorems, which Holt and Goldstein struggle to explain in their respective texts. We'll examine those struggles next week, marveling at the kind of journalistic/academic work which is reflexively praised for its clarity by long lines of scripted nimrods within the upper-end press.

This week, we're puzzling about something different; we're puzzling about Godel's alleged "Platonism." At great length and in flowery language, Goldstein describes the murky "doctrine" as the enduring, rapturous love of Godel's life—as a doctrine with which he "fell in love" when he was only 19.

Holt's essay condenses this portrait of Godel's affair—but what the heck is "Platonism" even supposed to be? As we've noted, the following passage includes Holt's first bite at that worm-infested apple. As he continues (see text below), some comical themes will emerge:
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...
It's just as we told you last week! According to Holt by way of Goldstein, the young Godel was "seduced by...the notion that abstractions like numbers and circles had a perfect, timeless existence independent of the human mind."

So says Holt, almost seeming to assume that this puzzling definition of Platonism makes some sort of earthly sense.

Does Holt go on to explicate this "doctrine?" We'll consider his fleeting effort below. First, let's consider the puzzling places to which we humans can be taken in explorations of modern "philosophy" at its highest levels.

Holt is halfway through a lengthy paragraph at the point where we've left off. From the rest of his graf, we can extract a minor attempt to flesh out the concept of "Platonism"—but we're also taken to a peculiar place, a place where we ponder 2 + 2 and the mysterious way in which 2 + 2 can be known to equal 4.

Are we humans the rational animal, or was that widely bruited assertion Aristotle's error? Is it possible that our highest intellectual elites have more often turned out ruminations which more closely resemble clown shows?

Have our highest academic elites sometimes resembled harlequins, clowns? We cut-and-paste, you decide:
HOLT (continuing directly): 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. Under the influence of Ludwig Wittgenstein, their reluctant guru, the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.
A significant name pops up in that passage—the name of Ludwig Wittgenstein. Before our post-philosophical explorations are done, we'll consider his admittedly murky, but conceptually simple, later work at some length.

Setting that matter aside for the present, let's look at Holt's attempt to explicate Platonism. Also, let's consider the fact that our greatest minds have struggled over the nature of the famous schoolboy proposition, 2 + 2 = 4.

In that extended passage, Holt seems to contrast young Godel's Platonism—his fervent belief that the number 3 lives a perfect timeless existence—with the hard-boiled views of a group of thinkers called the Vienna Circle.

Mercifully, Holt ascribes no particular "isms" to that particular group. But within that passage, a reader can discern a tiny attempt by Holt to explain Platonism in a bit more detail:

You might be a Platonist if! From that passage by Holt, it sounds like a Platonist believes that a proposition like 2 + 2 = 4 is true because "it correctly describes some abstract world of numbers."

Reader, tell the truth! Do you have even the slightest idea what that proposition might mean? Before you give a defensive answer in which you defer to academic authority, please remember that we've asked you to stick to the truth.

People, what makes a proposition like “2 + 2 = 4” true? According to Holt, our loftiest gangs of intellectuals were debating this chin-scratcher as recently as 1930, with Godel clinging to a belief which he was refusing to reveal:

According to Holt, Godel believed the proposition "2 + 2 = 4" is true because it correctly describes some abstract world of numbers. So believed our second greatest logician as he fell in love with the idea which would drive his work throughout his entire life.

Friend, do you have the slightest idea what any of that might mean? Do you feel sure that you can explain what it means to believe in the existence of "abstract worlds" at all? More specifically, do you know what it means to believe in the existence of the "abstract world of numbers"—which, presumably, is the place where the number 3 lives its perfect timeless existence, surrounded of course by the circles?

Do you have even the slightest idea what it means to believe in such things? We're going to guess that you do not—that you could never explicate, explain or unpack the essence of such alleged beliefs, which represent Holt's only attempts to explain what "Platonism" is.

Luckily, Holt briefly extends his explanation at one additional point. As it turns out, the truth of 2 + 2 = 4 was, for Godel, all about our ESP! For now, let's ignore the technical language and stick to the matter at hand:
HOLT (page 10): [Gödel] believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called “mathematical intuition.” It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives...
How can we know that 2 + 2 equals 4? According to Godel as told by Holt, we have "something like an extrasensory perception" which lets us discern such realities about the abstract world of numbers where the number 3, and all other numbers, live their perfect timeless existence in the company of circles.

Briefly, a word of praise. In her own book, Goldstein offered a perfect, simple explanation of how we can know that 2 + 2 equals 4.

Goldstein's perfect explanation doesn't involve ESP. You'll rarely see a philosophy professor make such a perfect statement. We'll get to that statement next week.

For today, let's leave matters at this. Back around 1930, the greatest logician in 2400 years was trying to determine how we can possibly know that 2 + 2 equals 4.

At Princeton, during his later years, he was known for his various crazy ideas. Starting around the age of 19, he developed the Platonistic idea that we can know about 2 + 2 because we have some faculty which resembles ESP.

This is the way our highest academic elites were playing during these not-so-distant years. According to Holt, even Lord Russell got dragged into this ridiculous mess. Again, ignore the technical language to focus on the sad absurdity of what is being said:
HOLT (page 9): Gödel’s incompleteness theorems did come as a surprise. In fact, when the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd. Even the once great logician Bertrand Russell was baffled; he seems to have been under the misapprehension that Gödel had detected an inconsistency in mathematics. “Are we to think that 2 + 2 is not 4, but 4.001?” Russell asked decades later in dismay, adding that he was “glad [he] was no longer working at mathematical logic.”
Lord Russell, "the once great logician," worried that he was being asked to believe that 2 + 2 might not equal 4 after all! This made him glad that we was no longer mired in the world of (mathematical) logic.

Respect for academic authority tells us that we must believe that these ruminations, by these great intellectual figures, simply must have made sense. By the end of the 1940s, Wittgenstein had torn such suppositions to shreds in a self-admittedly poorly-written book, Philosophical Investigations.

Professor Horwich has said that our current professors have stopped teaching Wittgenstein because his demonstrations mean that they would pretty much have to stop teaching everyone else. Down the road, we'll return to that claim, concerning which we'd occasionally joked on the world's greatest comedy stages before first encountering Horwich.

We'll return to Horwich's conjecture at some later date! Tomorrow, we'll return to Professor Goldstein's attempts to explicate "Platonism." We'll start with a comical story mentioned by Holt in passing:

A Platonist was hiding among them! Yes, it actually gets that silly, that puny, that tiny, that dumb.

Tomorrow: What a Platonist believes

28 comments:

  1. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it?

    Wrong, Bob. Godel said a mathematical proposition could be true although there was no possibility of proving it by means of an algorithm within a particular formal system. The truth of the proposition could be shown outside of that particular system.

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  2. "Our greatest logician believed in ghosts. He had a morbid fear of gases from the fridge."

    Our second-greatest logician, Aristotle, believed in the inferiority and baseness of women and that certain people deserved to be slaves.

    Somerby is an asshole.

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    1. Yes, Somerby is an asshole. But which great logician discovered that? And in which formal system can it be proved?

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    2. He was right in the former.

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  3. "Does it seem strange to think that our greatest logician was perhaps best known, in his adult years, for his various crazy ideas? "

    Our greatest logician (stipulating to Somerby's characterization of him), was not best known for his crazy ideas. He was best known for his incompleteness theorem.

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    1. You might as well say that his contribution to logic was undermined by the fact that he died at the end of his lifespan. His mental illness is irrelevant to his contribution to logic and mathematics. That stands on its own merits, not the personality or other characteristics of the man who formulated it. Does Beethoven's deafness mar his later musical compositions? Of course not. The rest of us are not deaf. The logicians who evaluated Godel's work were not themselves mentally ill.

      Somerby is being an ass again.

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    2. Gödel died at the end of his lifespan. Therefore he was not timeless.

      By the way, the German city of Königsberg is now the Russian city of Kaliningrad, so it's not timely either.

      I think Beethoven's late quartets would have been better if he could have heard what he was doing. That is, of course, a subjective opinion.

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    3. Here's another subjective opinion: Trump is a fat coward.

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    4. No such thing as objective. Even if it were a real thing, we would still have Trump voters.

      Leroy

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  4. "Professor Horwich has said that our current professors have stopped teaching Wittgenstein because his demonstrations mean that they would pretty much have to stop teaching everyone else. Down the road, we'll return to that claim, concerning which we'd occasionally joked on the world's greatest comedy stages before first encountering Horwich. "

    Here is the heart of Somerby's argument. Wittgenstein provides the excuse for ignoring and ridiculing all of the rest of philosophy and mathematics. And Somerby knew it before he encountered Horwich, so how great is Somerby? Wittgenstein says philosophy is crap and Somerby doesn't want to put any effort into understanding philosophy (or his effort doesn't yield much understanding) so it is far easier to dismiss all of it as bunk and ridicule the people who have contributed to the field he chose to major in (no one forced him to do it). Wittgenstein, in his poorly written book, somehow made the greatest logicians obsolete and Somerby knew it first!!! What a great man that must make Somerby, to have single-handedly brought down the pinnacle of logic.

    No wonder Somerby has no respect for humanity or any field of knowledge, and thinks everyone is crap.

    But this all hinges on whether Wittgenstein was right, doesn't it? What are the criticisms of Wittgenstein? Is it true that he isn't taught because the emperor would be shown to have no clothes? Why else might he not be taught? Is it possible that Wittgenstein is crap? Or irrelevant? Or not useful? How is Wittgenstein regarded by modern philosophers?

    Somerby will never tell. He is too busy deriding acquired knowledge to question the man who justified his hatred of academia. Too busy justifying his own academic failure (what were his grades, I find myself wondering).

    You cannot have any sense of integrity in your later years if your life's work has consisted of tearing down the work of others and showing that knowledge is bunk and humans are crap. So, if Somerby wants to look back on his life as well spent, he needs to reconsider whether the lives of others have been well spent too. Sucks to be him.

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    1. I did not know about this before making this criticism of Somerby, but here is what Bertrand Russell said about Wittgenstein:

      "The earlier Wittgenstein, whom I knew intimately, was a man addicted to passionately intense thinking, profoundly aware of difficult problems of which I, like him, felt the importance, and possessed (or at least so I thought) of true philosophical genius. The later Wittgenstein, on the contrary, seems to have grown tired of serious thinking and to have invented a doctrine which would make such an activity unnecessary. I do not for one moment believe that the doctrine which has these lazy consequences is true. I realize, however, that I have an overpoweringly strong bias against it, for, if it is true, philosophy is, at best, a slight help to lexicographers, and at worst, an idle tea-table amusement."

      Of course, we don't know whether Somerby is referring to the earlier or later Wittgenstein, but I'd be willing to bet money on which it turns out to be.

      Delete
    2. Maybe Somerby is referring to Ludwig Wittgenstein's brother Paul Wittgenstein, who lost an arm in World War I and commissioned piano works for one hand, such as this concerto by Maurice Ravel:

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

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    3. How could anyone honestly argue that philosophy has been anything other than worthless?

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    4. @1:27
      That must be why Somerby majored in it in college.

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    5. “This week, we're puzzling about something different; we're puzzling about Gödel’s alleged "Platonism."”

      Unfortunately, the commenters her often fail to notice that Bob is still being a media critic. He’s just moved on to different territory, which is cool with me. He doesn’t agree with the manner in which his targets, Holt and Goldstein, press their views. According to them, Gödel “has often been called the greatest logician since Aristotle."

      That does seem a long span of time between great proponents of logic. Me, I’m still trying to find the Moody Blues lyric in which they postulated that 1 + 1 is no longer true. At the time, I took it to mean that no matter the velocity of an object, the light reflected ahead of it would always be the same (velocity of object + speed of light still equals speed of light).

      Maybe it will come to me on the threshold of a dream. Meanwhile, can someone here define “philosophy?” My view is that it’s the exploration of how we humans should deal with our very existence in meaningful ways. Russell, despite his sometimes dim view of the subject, was one of its finest practitioners.

      Leroy

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  5. Here's my proof that 2 + 2 = 4. First, let's accept some definitions:

    2 = 1 + 1
    3 = 2 + 1
    4 = 3 + 1

    Then:

    2 + 2 = 2 + 1 + 1 = 3 + 1 = 4, QED.

    :)

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    1. Caesar -- that's exactly what's done using the Peano Postulates. The existence of the number zero is one of the postulated. Larger numbers are defined as "successors" of smaller numbers: 1=Szero; 2=SSzero; 3=SSSzero, etc. I have a vague memory of a math class where we started with the Peano Postulates and proved 2 + 2 = 4 by the means you laid out.

      But, ISTM that there's a remaining problem: how do you show that "SSSSzero" really is the number "4". Does "SSSSzero" have the essence of "4"? What is the essence of the number "4"? If you can determine the essence of "4", does that mean that the number "4" has its own timeless existence, independent of human thought? IMHO it's far from crazy to think that numbers do have their own timeless existence.

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  6. Holt and Goldstein create a portrait of Gödel, including many biographical elements such as his supposed mental illness. They also attempt to explain mathematical Platonism, representing their explanations as a satisfactory description of what Gödel actually believed. Somerby does not attempt to compare what they claim as Gödel's beliefs to Gödel's own thoughts on the subject.

    Somerby has framed his discussion of Gödel from the beginning by emphasizing his supposed mental illness, and he implies that Gödel isn't great because he isn't well-known and has made no "identifiable contribution to human thought." What Somerby means is that he, Somerby himself, can't identify any contribution, therefore it must not exist.

    A key passage in Somerby:
    "Does it seem strange to think that our greatest logician was perhaps best known, in his adult years, for his various crazy ideas? We'll let you wrestle with that one. For ourselves, we'd be inclined to see this syndrome as perhaps being instructive, illustrative of the vast intellectual dysfunction at the heart of the human experience."

    Gotta love the word "perhaps." ("perhaps best known, in his adult years, for his various crazy ideas"). Somerby sees an illustration of the "vast intellectual dysfunction at the heart of the human experience" based on something that is "perhaps" true, something which, even if true (which is debatable), does not imply any "vast intellectual dysfunction", except maybe Somerby's own.

    By the way, what is "this syndrome" Somerby identifies? Having crazy ideas, or being known for having crazy ideas? Those are quite different.

    At any rate, in what seems to be a post about the failures of people who write about Gödel, Somerby also accepts some of those writers' contentions (about mental illness) to imply that Gödel was a crackpot.

    He also betrays a lack of understanding of the activity of philosophers (or a desire to misrepresent it) when he says:

    "Respect for academic authority tells us that we must believe that these ruminations, by these great intellectual figures, simply must have made sense"

    That is *not* the way academics operate, not in the physical sciences, and not in philosophy. There is a healthy debate. In physics, it is continued experimentation to prove or disprove assertions. In philosophy, it is the attempt to provide a reasoned argument against certain philosophical pronouncements. Somerby's own example of Bertrand Russell shows that Gödel was not uncritically accepted.

    All Somerby has to say is that he doesn't think Gödel made sense. He doesn't need to vilify all academics in the process. He also doesn't need to feel resentful, indignant, or coerced because Holt and Goldstein think Gödel was great and Somerby doesn't.

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  7. 2+2=4

    A "true" statement.

    "2", "+", "=", and "4" are all symbols.
    But so are the words "matter", "proton", "electron", "atom."

    Physicists believe, at least for practical purposes, that their discoveries about "matter" are discoveries about things that exist apart from themselves. In other words, physicists do not assume they are merely discovering something about their own perceptions or the way the human mind works. Presumably, wherever matter exists, human physicists would observe that it consists of atoms consisting of protons, electrons, etc. Thus, despite not being able to test this hypothesis for all matter everywhere, physicists postulate a universality of this assumption.

    What about mathematics? The symbol "2" represents a concept of "2" that is independent of the symbol. There may be two coconuts and two bananas, but the idea that in some way both may be represented as "2" is the thought behind the notion that mathematics, in a practical sense, can be thought of as a discovery of truths that exist beyond our brains or our sense perceptions. It may not be true in either case, but it has important practical advantages.

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  8. Bob...you have really lost me on this one.

    "You'd almost think that a person like that would have made some great, identifiable contribution to human thought. You'd almost think his name would be well known."

    His great, identifiable contribution was his incompleteness theorem.

    His name actually is pretty well known for someone who labored in the very obscure and esoteric field of mathematical logic. (As I explained in an earlier post, he was well known to 18 year old me and my friends, who were probably a bit different from you and your friends at the same age).

    Also....academics, particularly scientists and mathematicians, are evaluated on their greatest work, not on some kind of average of everything they ever said or did....so Godel's oddness doesn't count any more than all that work that Newton did on Alchemy.

    Finally...you are welcome to discuss and have a view on the incompleteness theorem itself (although a little humility on the topic might be prudent, given the complexity of the subject). However, that does mean you should discuss the theorem itself---it's actually not totally inaccessible....probably more accessible than General Relativity, for example. Instead you seem to be throwing silly comments around about whether or not 2 + 2 = 4. That's on the same level as climate deniers observing that there was an unusually cold day last winter, or evolution deniers observing that their parents were not monkeys. That's not in contradiction to what either of those theories claim, and neither does Godel's work doubt that 2 + 2 = 4.

    Although some of us would disagree, you may even have the view that mathematical logic is a not-very-valuable endeavor. Sure. Tastes differ. But, if so, perhaps better to move on to another topic?

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  9. OK Bob, you're really barking up the wrong tree with this one. Let me start with where you're right, because that's the short bit, and then I'll explain why you need to stop banging this drum.

    First of all, you're right that journalists do a pretty poor job of explaining science and philosophy. However, in the case of science, there's a reason for that. Things like relativity require not just a serious study of the text, but the academic prerequisites. If you lack the mathematical background, then you're never going to really understand relativity. Layman's explanations are just doomed to failure because it's (a) very complicated, and (b) counter-intuitive.

    So I'm not sure if this is a great topic to complain about. Maybe journalists just shouldn't try. As someone who understands these things, it can be embarrassing to read their inevitably fumbling explanations. But don't expect it to get any better, because I don't think it really can get better unless everyone completes the requires courses in four semesters of calculus and two semesters of linear algebra.

    As for Gödel, the claims about his importance are not overstated. Computer technology would simply not be in the state it is today without his Incompleteness Theorem. It underpins the entire mathematical framework of computation. It's right at the enter of it all. Greatest logician in 2400 years? I don't know if that's true, but it isn't as absurd a statement as it sounds to you. Personally, I've always thought of him as a mathematician first, in which case I think he has a bit more competition (Gauss might better make that claim, for instance).

    The fact is that you're NEVER going to explain the incompleteness theorem to someone without the necessary mathematical background (mostly discreet math here, which is far more appropriate for the "logician" label). Plenty of computer science students stumble through their coursework on computability theory, because it's extremely abstract. You don't use it everyday (thank god), but it underpins the entire concept of a computation machine.

    It's one thing to mock the reporting of these discoveries; such projects are always doomed to fall short. But you're coming close to mocking Gödel, which exposes your vast ignorance on this topic. Don't assume that there's nothing there just because nobody can explain it to you. There's a reason you don't learn the Incompleteness Theorem in your first semester, or high school, or the newspaper.

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    1. Edgewise & Mark - very good posts. If TDH wants to criticize Godel, he should read Godel, not a journalist or popularizer trying to explain Godel. I have often defended TDH, but it seems he is becoming unhinged going on and on about this topic. Has his once great mind finally snapped? Why is he going on and on about this topic which he doesn't seem competent to be discussing,and not making much sense?

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    2. Well, Somerby seems to be focusing on Platonism, and it is a series (we keep hanging on).

      Is he steering us in the wrong direction? I'm a member of the illiterati, so I'll keep on for a bit.

      Leroy

      Delete
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    Thank Dr Alexzander for everything you did in my marriage.
    Thanks
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    ReplyDelete