G.H. Hardy reviews Principia Mathematica

September 12, 2011 2:37 PM

"Perhaps twenty or thirty people in England may be expected to read this book." G.H. Hardy's review of Whitehead and Russell's Principia Mathematica, 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."
posted by escabeche (29 comments total)

"... some of the jokes are very good."

Way to focus in there guys.
posted by GuyZero at 2:41 PM on September 12, 2011

warning: pdf
posted by blue_beetle at 2:43 PM on September 12, 2011

Something about this doesn't add up.
posted by It's Raining Florence Henderson at 2:51 PM on September 12, 2011 [2 favorites]

This is the book in which they take ~362 pages to prove that 1+1=2.
posted by oonh at 2:52 PM on September 12, 2011

This is the book in which they take ~362 pages to prove that 1+1=2...

... and don't actually fully succeed.
posted by yeolcoatl at 2:55 PM on September 12, 2011 [6 favorites]

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".
posted by baf at 2:58 PM on September 12, 2011 [6 favorites]

"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."

And conversely, how much philosophy haunted mathematics after Principia Mathematica led to Gödel's Incompletenes Theorem 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.
posted by zachlipton at 2:59 PM on September 12, 2011 [1 favorite]

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 3:10 PM on September 12, 2011 [26 favorites]

And conversely, how much philosophy haunted mathematics...

Yes, haunted by the ghost of logical positivism, long after the body of it had been extinguished...

Goedel's results goes back to "Hilbert's Second Problem":

"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.

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 Carnap's "Logicist Foundations of Mathematics" i.e. logical positivism and suffers from the 'garbage in, garbage out' problem.

The philsophy was always there.
posted by ennui.bz at 3:33 PM on September 12, 2011 [2 favorites]

Zed 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.
posted by whuppy at 3:40 PM on September 12, 2011 [1 favorite]

Surprised me to (re)discover that Hardy was the ultimate source of a story of Russell dreaming about the long-tail worth of Principia Mathematica:

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....

In overthrowing the system of the Principia (modest title, that), Gödel essentially immortalized it, and ensured that Russell's nightmare could not come true.
posted by jamjam at 3:40 PM on September 12, 2011 [5 favorites]

I prefer less obscure math jokes, such as "a comathematician is a device for turning cotheorems into ffee".
posted by madcaptenor at 3:58 PM on September 12, 2011 [12 favorites]

Apparently the joke in my previous comment dates back at least to 1996. 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.
posted by madcaptenor at 4:11 PM on September 12, 2011

Of related interest: World's Shortest Explanation of Gödel's Theorem from Smullyan's 5000 BC and Other Philosophical Fantasies.
posted by Zed at 4:18 PM on September 12, 2011 [4 favorites]

Yields falsehood when preceded by its quotation! Yields falsehood when preceded by its quotation!
posted by nebulawindphone at 4:21 PM on September 12, 2011 [2 favorites]

madcaptenor: see Jeremy Martin's Additional Publications, specifically "Cocomputing cocohomological coobstructions to cocombing the cohairs on a cococonut using coCoCoA"
posted by oonh at 4:36 PM on September 12, 2011 [1 favorite]

The time has passed when a philosopher can afford to be ignorant of mathematics, and a little perseverance will be well rewarded.

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.
posted by Revvy at 4:46 PM on September 12, 2011 [1 favorite]

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.
posted by Vibrissae at 4:49 PM on September 12, 2011

My ears are burning for some reason...
posted by wittgenstein at 5:22 PM on September 12, 2011 [2 favorites]

The Principia is online (the site seems to be loading slowly though.) It has 666 pages.
posted by twoleftfeet at 6:17 PM on September 12, 2011 [2 favorites]

Of related interest: World's Shortest Explanation of Gödel's Theorem

For a slightly longer explanation, check out Gödel's Proof 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 why the theorem is true -- even if you're a moron non-mathematician like me.
posted by twirlip at 6:48 PM on September 12, 2011 [2 favorites]

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.
posted by GentleReader at 6:53 PM on September 12, 2011 [5 favorites]

Russell: "I wouldn't be willing to die for my beliefs. I could be wrong."
posted by ovvl at 6:55 PM on September 12, 2011

Bertrand Russell was the greatest philosopher writing in English in the 20th century, and this was far from his greatest achievement.

Way to narrow the field. Wittgenstein? Sartre? Not writing in English.

What about Quine?
posted by Crabby Appleton at 7:18 PM on September 12, 2011

Bertrand Russell was the greatest philosopher writing in English in the 20th century, and this was far from his greatest achievement.

A bunch of philosophers (and whoever else was there) disagree.
posted by meese at 7:22 PM on September 12, 2011 [1 favorite]

David K. Lewis? How many divisions does he have?

Wait, I mean autobiographies. How many autobiographies.
posted by No-sword at 9:09 PM on September 12, 2011 [1 favorite]

Twoleftfeet wrote: The Principia is online (the site seems to be loading slowly though.) It has 666 pages.

But is it complete?

If so, I bet it's not consistent.
posted by Joe in Australia at 9:19 PM on September 12, 2011 [1 favorite]

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

Yet another example of why Russell was immensely cool.
posted by Decani at 9:14 AM on September 13, 2011

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.)
posted by GentleReader at 1:05 PM on September 13, 2011 [1 favorite]