Comments on: G.H. Hardy reviews Principia Mathematica
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica/
Comments on MetaFilter post G.H. Hardy reviews Principia MathematicaMon, 12 Sep 2011 14:41:35 -0800Mon, 12 Sep 2011 14:41:35 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60G.H. Hardy reviews Principia Mathematica
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica
<a href="http://www.cl.cam.ac.uk/~ns441/files/hardy-principia.pdf">"Perhaps twenty or thirty people in England may be expected to read this book."</a> G.H. Hardy's review of Whitehead and Russell's <em>Principia Mathematica</em>, published in the Times Literary Supplement 100 years ago last week. "The time has passed when a philosopher can afford to be ignorant of mathematics, and a little perseverance will be well rewarded. It will be something to learn how many of the spectres that have haunted philosophers modern mathematics has finally laid to rest."post:www.metafilter.com,2011:site.107360Mon, 12 Sep 2011 14:37:37 -0800escabechemathmathematicsphilosophyhardyghhardyrussellbertrandrussellprincipiamathematicalogicwhiteheadBy: GuyZero
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917419
"... some of the jokes are very good."
Way to focus in there guys.comment:www.metafilter.com,2011:site.107360-3917419Mon, 12 Sep 2011 14:41:35 -0800GuyZeroBy: blue_beetle
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917424
<small>warning: pdf</small>comment:www.metafilter.com,2011:site.107360-3917424Mon, 12 Sep 2011 14:43:56 -0800blue_beetleBy: It's Raining Florence Henderson
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917434
Something about this doesn't add up.comment:www.metafilter.com,2011:site.107360-3917434Mon, 12 Sep 2011 14:51:38 -0800It's Raining Florence HendersonBy: oonh
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917436
This is the book in which they take ~362 pages to prove that 1+1=2.comment:www.metafilter.com,2011:site.107360-3917436Mon, 12 Sep 2011 14:52:17 -0800oonhBy: yeolcoatl
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917440
<i>This is the book in which they take ~362 pages to prove that 1+1=2...</i>
... and don't actually fully succeed.comment:www.metafilter.com,2011:site.107360-3917440Mon, 12 Sep 2011 14:55:59 -0800yeolcoatlBy: baf
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917443
Also, it's the book whose chief goal, of completing mathematics, was proved futile by Kurt Gödel a few years later. He even calls it out by name: the paper in which he presented his famous Incompleteness Theorem is titled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems".comment:www.metafilter.com,2011:site.107360-3917443Mon, 12 Sep 2011 14:58:46 -0800bafBy: zachlipton
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917445
<em>"The time has passed when a philosopher can afford to be ignorant of mathematics, and a little perseverance will be well rewarded. It will be something to learn how many of the spectres that have haunted philosophers modern mathematics has finally laid to rest."</em>
And conversely, how much philosophy haunted mathematics after <em>Principia Mathematica</em> led to <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">Gödel's Incompletenes Theorem</a> and related developments. Russell and Whitehead went off on this massive effort to prove that all mathematics can be rigorously defined through logical operations on a few basic axioms, and the universe came along and taught us that it can't truly ever be done.comment:www.metafilter.com,2011:site.107360-3917445Mon, 12 Sep 2011 14:59:05 -0800zachliptonBy: Zed
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917468
<a href="http://www.metafilter.com/81265/I-See-Your-The-Watchmen-and-I-Raise-You-Logicomix">Logicomix</a>, a comic book that's mostly a bio of Russell, attributes to him (IIRC) the statement that Gödel's the only person he was sure had actually read the whole thing.comment:www.metafilter.com,2011:site.107360-3917468Mon, 12 Sep 2011 15:10:02 -0800ZedBy: ennui.bz
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917498
<i>And conversely, how much philosophy haunted mathematics...</i>
Yes, haunted by the ghost of logical positivism, long after the body of it had been extinguished...
Goedel's results goes back to "<a href="http://en.wikipedia.org/wiki/Hilbert's_second_problem">Hilbert's Second Problem</a>":
<blockquote>
"When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. ... But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. ... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.</blockquote>
The point was whether all questions of mathematics could be phrased in terms of one formal language with a defined syntax and then resolved (in principle) algorithmically.
The Principia of R&W really starts with <a href="http://en.wikipedia.org/wiki/Carnap">Carnap's</a> "Logicist Foundations of Mathematics" i.e. logical positivism and suffers from the 'garbage in, garbage out' problem.
The philsophy was always there.comment:www.metafilter.com,2011:site.107360-3917498Mon, 12 Sep 2011 15:33:30 -0800ennui.bzBy: whuppy
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917505
<a href="http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917468">Zed</a> beat me to the punch, but I'll go ahead and recommend Logicomix as an excellent overview of the whole kerfuffle. Frankly, I wish I had had a chance to read it before tackling Hofstadter.comment:www.metafilter.com,2011:site.107360-3917505Mon, 12 Sep 2011 15:40:21 -0800whuppyBy: jamjam
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917506
Surprised me to (re)discover that Hardy was the <a href="http://books.google.com/books?id=beImvXUGD-MC&q=principia#v=snippet&q=principia&f=false">ultimate source</a> of a story of Russell dreaming about the long-tail worth of <em>Principia Mathematica</em>:
<em>I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated....</em>
In overthrowing the system of the <em>Principia</em> (modest title, that), Gödel essentially immortalized it, and ensured that Russell's nightmare could not come true.comment:www.metafilter.com,2011:site.107360-3917506Mon, 12 Sep 2011 15:40:48 -0800jamjamBy: madcaptenor
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917518
I prefer less obscure math jokes, such as "a comathematician is a device for turning cotheorems into ffee".comment:www.metafilter.com,2011:site.107360-3917518Mon, 12 Sep 2011 15:58:31 -0800madcaptenorBy: madcaptenor
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917525
Apparently the joke in my previous comment dates back <a href="http://mathforum.org/kb/thread.jspa?messageID=48418&tstart=0">at least to 1996</a>. I am disappointed because that means I was certainly not the first person to invent it, as that joke would have made no sense to me in 1996.comment:www.metafilter.com,2011:site.107360-3917525Mon, 12 Sep 2011 16:11:31 -0800madcaptenorBy: Zed
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917531
Of related interest: <a href="http://blog.plover.com/math/Gdl-Smullyan.html">World's Shortest Explanation of Gödel's Theorem</a> from <a href="http://www.metafilter.com/104970/What-is-the-title-of-this-post">Smullyan's</a> <a href="http://www.amazon.com/exec/obidos/ASIN/0312295170/metafilter-20/ref=nosim/"><i>5000 BC and Other Philosophical Fantasies.</i></a>comment:www.metafilter.com,2011:site.107360-3917531Mon, 12 Sep 2011 16:18:16 -0800ZedBy: nebulawindphone
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917533
Yields falsehood when preceded by its quotation! Yields falsehood when preceded by its quotation!comment:www.metafilter.com,2011:site.107360-3917533Mon, 12 Sep 2011 16:21:10 -0800nebulawindphoneBy: oonh
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917557
madcaptenor: see Jeremy Martin's <a href="http://www.math.ku.edu/~jmartin/morepubs.html">Additional Publications</a>, specifically "Cocomputing cocohomological coobstructions to cocombing the cohairs on a cococonut using coCoCoA"comment:www.metafilter.com,2011:site.107360-3917557Mon, 12 Sep 2011 16:36:56 -0800oonhBy: Revvy
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917573
<i>The time has passed when a philosopher can afford to be ignorant of mathematics, and a little perseverance will be well rewarded.</i>
As someone who was taking Philosophy, Computer Science, Math, and Logic courses every semester in college, I can attest to the failure of many philosophy students to understand this idea.comment:www.metafilter.com,2011:site.107360-3917573Mon, 12 Sep 2011 16:46:07 -0800RevvyBy: Vibrissae
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917578
Wittgenstein figured out some of this on his own, from a different direction than Gödel. I think that Gödel and Wittgenstein, along with some of the quantum and relativity theorists, THE intellectual giants of the last century (excepting a few poets). They freed us from the myth that pure logic (reason) could solve all, thank god! Neuroscience is discovering that they were right.comment:www.metafilter.com,2011:site.107360-3917578Mon, 12 Sep 2011 16:49:13 -0800VibrissaeBy: wittgenstein
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917620
My ears are burning for some reason...comment:www.metafilter.com,2011:site.107360-3917620Mon, 12 Sep 2011 17:22:39 -0800wittgensteinBy: twoleftfeet
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917683
<a href="http://quod.lib.umich.edu/g/genpub/aat3201.0001.001">The Principia is online</a> (the site seems to be loading slowly though.) It has 666 pages.comment:www.metafilter.com,2011:site.107360-3917683Mon, 12 Sep 2011 18:17:56 -0800twoleftfeetBy: twirlip
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917703
<i>Of related interest: World's Shortest Explanation of Gödel's Theorem</i>
For a slightly longer explanation, check out <i>Gödel's Proof</i> by Ernest Nagel and James R. Newman. It does an excellent job of taking you through the proof step-by-step, so that you understand <i>why</i> the theorem is true -- even if you're a <s>moron</s> non-mathematician like me.comment:www.metafilter.com,2011:site.107360-3917703Mon, 12 Sep 2011 18:48:55 -0800twirlipBy: GentleReader
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917709
It is not true that the Principia project was made futile because of Gödel's incompleteness theorems. Principia was the effort to reduce arithmetical theory to logic (plus set theory). There was no attempt in Principia to demonstrate the completeness or consistency of an axiomatic theory of arithmetic. (What was made futile was Hilbert's second problem.)
The value of Principia has to do with the representation of arithmetic, and proofs in arithmetic, in terms drawn solely from formal logic and set theory. The mathematical consequences of this were significant in terms of the systematization of science; but practically as well, it is safe to say that unless and until this reduction had been accomplished, there certainly could never have been a digital computer
Also, the Principia of R&W could not start with Carnap's "Logicist Foundations of Mathematics," because that was written only 20 years later.
Bertrand Russell was the greatest philosopher writing in English in the 20th century, and this was far from his greatest achievement.comment:www.metafilter.com,2011:site.107360-3917709Mon, 12 Sep 2011 18:53:07 -0800GentleReaderBy: ovvl
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917713
Russell: "I wouldn't be willing to die for my beliefs. I could be wrong."comment:www.metafilter.com,2011:site.107360-3917713Mon, 12 Sep 2011 18:55:39 -0800ovvlBy: Crabby Appleton
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917733
<i>Bertrand Russell was the greatest philosopher writing <b>in English</b> in the 20th century, and this was far from his greatest achievement.</i>
Way to narrow the field. Wittgenstein? Sartre? Not writing in English.
What about Quine?comment:www.metafilter.com,2011:site.107360-3917733Mon, 12 Sep 2011 19:18:22 -0800Crabby AppletonBy: meese
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917736
<em>Bertrand Russell was the greatest philosopher writing in English in the 20th century, and this was far from his greatest achievement.</em>
A bunch of philosophers (and whoever else was there) <a href="http://leiterreports.typepad.com/blog/2009/03/lets-settle-this-once-and-for-all-who-really-was-the-greatest-philosopher-of-the-20thcentury.html">disagree.</a>comment:www.metafilter.com,2011:site.107360-3917736Mon, 12 Sep 2011 19:22:22 -0800meeseBy: No-sword
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917840
David K. Lewis? How many divisions does he have?
<small>Wait, I mean autobiographies. How many autobiographies.</small>comment:www.metafilter.com,2011:site.107360-3917840Mon, 12 Sep 2011 21:09:28 -0800No-swordBy: Joe in Australia
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3917850
Twoleftfeet wrote:<em> The Principia is online (the site seems to be loading slowly though.) It has 666 pages.</em>
But is it complete?
<small>If so, I bet it's not consistent.</small>comment:www.metafilter.com,2011:site.107360-3917850Mon, 12 Sep 2011 21:19:25 -0800Joe in AustraliaBy: Decani
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3918411
<em>Logicomix, a comic book that's mostly a bio of Russell, attributes to him (IIRC) the statement that Gödel's the only person he was sure had actually read the whole thing.
posted by Zed at 11:10 PM on September 12</em>
Yet another example of why Russell was immensely cool.comment:www.metafilter.com,2011:site.107360-3918411Tue, 13 Sep 2011 09:14:25 -0800DecaniBy: GentleReader
http://www.metafilter.com/107360/GH-Hardy-reviews-Principia-Mathematica#3918995
In promoting Russell, I am presenting my own opinion. I am certain that he was far more influential than Wittgenstein--the revolution in the 1900s set the stage for the entire 20th century in Anglophone philosophy, including Wittgenstein's own work (he was Russell's student and "designated heir"). It's hard to appreciate, but consider that Russell is the founder of analytic philosophy, and that transformed every area of Anglophone philosophy, even aesthetics and the history of philosophy. One could argue that the German-language immigrants of the 1930s and 40s were as influential, but they wouldn't have had a receptive audience in the USA if not for Russell. Thus, Russell's methods and style are as important as his philosophical conclusions, but if he had not overturned British Idealism then the course of Anglophone philosophy would have been completely different.
I am happy to note that in the informal poll on the Leiter Reports website cited by meese, Wittgenstein led with 17% while Russell was second with 16%--so my opinion is pretty mainstream. But in a poll of historians of the period, Russell would dominate. Wittgenstein is more popular partly because he's such an interesting person and partly because his later work is taken to undo certain analytic extravagances.
Quine was certainly a major figure. GE Moore was enormously influential and does not get the credit he is due; likewise Frank Ramsey.
For myself, I would give second place to Rawls, and he is a significant challenger to Russell. But they're so different, and removed in time, that it's hard to compare.
I think the most influential non-English-writing philosopher of the century was Heidegger, though I don't know how to compare him to Cassirer or, for that matter, Lenin. For quality I would take Carnap. (Carnap did publish in English, but I consider him a German philosopher; Wittgenstein wrote in German but I consider him English because he had no association with his Austrian or German contemporaries.)comment:www.metafilter.com,2011:site.107360-3918995Tue, 13 Sep 2011 13:05:43 -0800GentleReader