THURSDAY, JULY 7, 2022
...in search of some comic relief: Provisionally, we issue a promise!
We promise that, on the morrow, we'll return to this week's supposed project.
On Tuesday, we said that we would be suggesting that our own tribe's very occasional, tiny/small errors might help explain the unhelpful way our political leaders are viewed.
In truth, we find that project depressing. For today, we go in search of the comic relief which lets our species fight on.
For today, we'll move to the terrain of our second ongoing series, the collection called "Took the Gödel Challenge." We'll discuss a possible tiny/small error of judgment committed by Bertrand Russell before Gödel came along.
Our material will be drawn from Stephen Budiansky's well-received biography, Journey to the Edge of Reason: The Life of Kurt Gödel (2021). Budiansky's book will be one of the four or five texts on which we' focus in our forthcoming work.
For today, we'll focus on the comic relief Budiansky provides at one point in the book, courtesy of Lord Russell. At issue is the mammoth task Russell undertook near the turn of the twentieth century—the effort he took, with Alfred North Whitehead, "to show," in Budiansky's words, "that all of mathematics could be derived from the propositions of logic."
Is it true? Could "all of mathematics be derived from the propositions of logic?"
We don't know it that could be done, nor do we have the slightest idea why anyone would want to do it. We're not even sure what this goal, as described, even means.
Budiansky never much tries to explain that point. Like many others, he doesn't seem to realize that he's dealing here with a type of technical language—with the type of technical language that doesn't appear to be technical language.
We'll discuss such matters in future reports. For today, let's restrict ourselves the comic relief brought to a grateful world in Budiansky's book.
Russell and Whitehead had assigned themselves the somewhat murky task described above. Eventually, Gödel would enter the fray, allegedly establishing himself as "the greatest logician since Aristotle."
We're going to set that aside for later. For today, we'll move to the comic relief.
Russell and Whitehead had displayed the way "mathematics can be derived from the propositions of logic" in a three-volume work, Principia Mathematica, which began appearing in 1910. Eventually, Budiansky starts to discuss some of the problems with the monumental work, drawing on Lord Russell's admirable propensity for humorous self-denigration:
BUDIANSKY (page 108): Gödel placed himself at the very center of the storm over mathematical foundations, which had broken with a deeply unnerving discovery Bertrand Russell had made at the turn of the century while working on Principia Mathematica. Russell's idea had been to establish the soundness of mathematics by showing how it all could be reduced to principles of logic so self-evident as to be beyond dispute. Defining even the simplest operations of arithmetic in terms of what Russell called such "primitive" notions, however, was far from an obvious task. Even the notion of what a number is raised immediate problems. The laboriousness of the methodology and notation was all too evident in the (often remarked) fact that it took more than seven hundred pages to reach the conclusion "1 + 1 = 2," a result which Russell and Whitehead described as "occasionally useful."
Say what? It took more than seven hundred pages to demonstrate the soundness of 1 + 1 = 2?
Arguably, a sensible person might start to wonder, at this point, about the soundness of the entire project. Soon, though, Budiansky was describing the "paradox" which brought Principia crashing down.
For today, we'll skip Budiansky's discussion of the wonderfully comical "set of all sets not members of themselves," the shoals upon which Russell came to believe that his ship had foundered. For today, we'll restrict ourselves to the most obvious points of comic relief.
Russell's handling of this troubling "set of all sets" would eventually bring Principia down. Along the way, his discovery of the troubling set had led to a conundrum known as "Russell's Paradox."
Budiansky's play-by-play reads as shown:
BUDIANSKY (page 109): "Russell's Paradox," as it came to be known, echoed paradoxes that had been around since antiquity. The prototype is The Liar's Paradox, attributed to Epimenides the Cretan, who asserted, "All Cretans are liars." Russell noted that this was akin to the conundrum posed by a piece of paper on which the sentence "The statement on the other side of this paper is false" is written on one side, and "The statement on the other side of this paper is true" on the other.
"It seemed unworthy of a grown man to waste his time on such trivialities," Russell later recalled, and "at first I supposed that I should be able to overcome the contradiction quite easily, and that there was probably some trivial error in the reasoning." The more he thought about it, the more he realized it was a flaw in his whole project too deep to be ignored.
Say what? A silly parlor trick of the type described brought a giant philosophical undertaking crashing to the ground?
We honor Russell for his self-deprecation; we compliment Budiansky for including this comic relief in his well-received book. But what can it mean when historical giants of the world of upper-end academic philosophy find their mammoth undertakings stymied by "such trivialities" in what seem to be such blatantly silly ways?
Could it suggest that something was wrong with the whole undertaking from the start—possibly with the whole cosseted upper-end academic culture from which this project emerged?
We'll set such questions aside for today. For today, Budiansky's flair for comic relief finally gives us this:
BUDIANSKY (pages 110-111): [T]here were other problems with the project which Russell himself recognized, and not long after finishing his herculean task he admitted feeling a deep sense of failure. The massive manuscript, with its complex notation which could only be written out laboriously by hand, had to be carted in a four-wheeler cab to the offices of Cambridge University Press when it was finally done. The press was so pessimistic about sale prospects that it demanded L600 to cover the anticipated loss; even with a grant from the Royal Society and other funding, Russell and his co-author Whitehead ended up earning "minus L50 apiece for ten years' work," Russell said. "This beats the record of Paradise Lost." A number of years later he told a friend, with his usual self-mocking humor, "I used to know only six people who had read the later parts of the book."
"One of those [six people] was Kurt Gödel," Budiansky notes at that point.
It took Russell 700 pages to establish that 1 + 1 = 2. The project was based on a ludicrous "triviality," and the reasoning was so laborious that the book, when finally done, had to be carted about in a four-wheeler cab—this monumental work of erudition which only six people had read all the way to the end.
Years later, Gödel turned his attention to this peculiar forcefield. Out of it came a famous theorem no one can explain to the general reader—the famous "incompleteness theorem[s]" which established its creator as "the greatest logician since Aristotle."
What does this say about one major part of the intellectual and cultural world within which we all live? In part, it says that Lord Russell had a good sense of humor, and we'll leave it right there for right now.
Russell came to see that there were problems with his mammoth undertaking! Errors, including those which may go unremarked, can form a very large part of the world, even within our own blue tribe, a social and cultural aggregation which is often perhaps somewhat landlocked and is famously quite self-impressed.
Much of the red tribe has lost its mind. But what's going on within ours?
Tomorrow: What we've provisionally promised
"Could it suggest that something was wrong with the whole undertaking from the start..."ReplyDelete
One question; only one: how is it any skin off your nose, dear Bob? What's your problem?
You're the one with the problem, professional troll.Delete
Having studied Russell's paradox under Alfred Tarski, I'm in a better position to defend it than most. What Bob is ignoring is the desire to created an incontrovertible basis for mathematics. Russell's work and Godel's work showed that this project was more difficult than it seemed.ReplyDelete
As a philosophy major, Bob ought to appreciate the desire to create an incontrovertible basis for beliefs. That's a part of many aspects of philosophy. E.g., is there a single basis for ethical decisions that all should agree with? Or, is there simply a choice of ethical bases?
I found all this interesting. The study of knowledge, epistemology, is it also knowledge? Will it perhaps have real-world applications as AI get more sophisticated? Or is it just worthless navel gazing? All valid questions IMO.Delete
Myself I feel there's no such thing as absolute incontrovertible truth, everything has a premise that must be accepted to build upon. I also think assuming 1+1 is correct is a premise that can be accepted as wholly true, for convenience. If we can't agree on the nature of counting things, we're going to spin our wheels and I tend to think the efforts Bob were describing were *largely* a waste of time but may perhaps inspire some other research or work at some point which is applicable to AI.
It has already done so.Delete
Both Turing and Einstein were enthusiastic about Godel's work. I would take that as a measure of its worth before I took Somerby's word that it is useless.Delete
We don't have AI yet, we have expert systems. It's a point of argument among experts, but no I don't agree at all that we have Artificial Intelligence yet.Delete
You're second post is propaganda and not worth responding to.
Somerby does not assert that the work by Godel or Russell is useless. As I read him, he challenges someone to assert that it is useful and explain how, without resorting to appeals to authority or empty cliches.Delete
Another voice of reason entered the chat. Thanks David.Delete
@2:23 We also have machine learning. Quibbling over what is true AI is a waste of time. Godel's work is important to computational modeling, which we do have now.Delete
@David Stein: regarding Russell’s paradox, I have tried over and over to explain in comments what Russell was doing. It relates to set theory. Somerby has mocked Russel’s paradox, because he doesn’t understand it. Go back and read his posts. He has suggested that “Lord” Russell was an elitist.Delete
@mh: I will try to go back and review. Thanks,Delete
"We don't know it that could be done, nor do we have the slightest idea why anyone would want to do it. We're not even sure what this goal, as described, even means. "ReplyDelete
This should have been taught in Somerby's philosophy major at Harvard. Why would someone choose to be a philosophy major if they weren't interested in such things?
This is Somerby's way of casting aspersions on knowledge. Every field begins by teaching students why their field of study is important, introducing them to the basic terminology of the field, organizing the subfields, and explaining how knowledge is generated in that field. Without such an introduction, any field can seem useless. Somerby deliberately tries to portray fields such as mathematics and logic without that basic introduction, as fields that contribute nothing to the human condition. That is far from true and it rests on the ignorant person's expectation that understanding must be handed to them without effort on their part.
Somerby claims to have been an educator. That makes it especially hard to understand his current attitude about Godel, relativity, and other areas of knowledge. I don't understand much about electrical enginnering or musical recording or forensic accounting. Shall we toss out those fields too? If the layperson cannot immediately grasp what such things are "good for", shouldn't we disparage them as much as Somerby disparages Godel? And once we have tossed all knowledge on the trash heap, what will our society look like? Maybe Medieval times? But I don't know about that, since history has been tossed on the trash heap too.
I think Somerby is simply challenging us to question whether we know what we think we know, and to show our work when asserting what we believe. With obvious political implications.Delete
Wouldn't it be nice if logic had a place in politics. But one might argue it is illogical by nature.Delete
Politics is more statistical in nature.Delete
Logic has a place in speech, which is center stage in a democracy.Delete
"...we said that we would be suggesting that our own tribe's very occasional, tiny/small errors might help explain the unhelpful way our political leaders are viewed."ReplyDelete
Can tiny small errors produce large effects? Maybe, but more likely, when it comes to the way our political leaders are viewed, it takes larger influences to produce big effects. Tiny/small errors tend not to be noticed.
Why has Somerby adopted this new usage? Is he implying that liberals are defensive, so our errors need to be minimized with mocking speech because we are too sensitive to listen otherwise? Or is he being sarcastic about the size of the errors he thinks he finds? It is hard to know what Somerby means by this, because he has given up actually talking about any errors, actually providing evidence of what such errors might be.
Today the NY Times jumped all over Biden because he is helping soccer player Brittney Griner deal with her trial in Russia but has allowed some other Americans incarcerated in Russia to languish in jail. It preferred to publish an article about that other man and his family, instead of giving Biden any favorable press for helping Griner. Also without telling us what the other guy did. The complaint was that Biden apparently only cares about famous athletes. There was no mention that when you commit a crime in a foreign country, you are tried by that other country's laws and you do your time if you are convicted. Griner is different because she was arrested in the context of the Ukraine war and became a political pawn. Also, it is questionable whether she did anything wrong.
But the obvious negative tone of the NY Times coverage of positive activities by Biden should attract Somerby's attention. Did Biden do anything small/tiny wrong or is he damned if he does and damned if he does nothing in the eyes of the NY Times, who jump on any excuse to print something critical of Biden? And why does Somerby never point such articles out here, where his focus is supposedly the media? Why does he assume that liberals are making mistakes, instead of a biased press pointing out non-errors in an obvious attempt to make Biden look bad?
"In new comprehensive polls of battleground states Future Majority found the Congressional Generic shifting from 43-45 (-2) in March to 44-42 (+2) now, a 4 point shift. With Democrats now consistently leading in the generic ballot it is a new election, a competitive not a wave election and all talk of a Republican wave should end. "ReplyDelete
It is time to stop talking about a Republican wave and Democrats struggling. It is now a close election in the House and even more positive for Democrats in the Senate. This is largely due to Roe v Wade and a Republican reaction to both the Supreme Court decision and the 1/6 hearings.
Somerby needs to acknowledge that Democrats are no longer in trouble and start focusing on how to help the red tribe deal with the aftermath of their own stupidity.
You need to acknowledge that your comments are asinine and the only people reading them at this point are newbies or long term readers that are possessed of a morbid curiosity.Delete
And you, of course. You seem to be reading them...Delete