Hardy and Godel gone wild: Just for today, let's be fair.
Professor Livio didn't invent the cultural phenomenon which is, in fairness, on full display at the very start of his well-received 2009 book, Is God A Mathematician?
As we noted yesterday, the cultural practice is on full display by page 3 of Livio's book. At that point, in just the fifth paragraph of his book, we're confronted by the spectacle of a ranking astrophysicist citing a renowned Oxford mathematical physicist, with the latter said to have made incoherent remarks about the constitution of this big crazy cosmos of ours.
For the fuller text, see yesterday's report. Meanwhile, this is a taste of the oddness:
LIVIO (page 3): The Platonic world of mathematical forms, which to [the mathematical physicist] has an actual reality comparable to that of the physical and the mental worlds, is the motherland of mathematics. This is where you will find the natural numbers 1, 2, 3, 4,..., all the shapes and theorems of Euclidean geometry, Newton's laws of motion, string theory, catastrophe theory, and mathematical models of stock market behavior. And now, [the mathematical physicist] observes, come the three mysteries. First, the world of physical reality seems to obey laws that actually reside in the world of mathematical forms...According to the astrophysicist's account of the views of the mathematical physicist, the Platonic world of mathematical forms doesn't just "have a reality," whatever that means. It has an actual reality!
Through a process which doesn't get explained, that "motherland of mathematics" is said to be "where you will find the natural numbers 1, 2, 3, 4,...," along with Newton's laws of motion. Later in the passage, we're told that's where they "reside."
In what way will you "find" the natural numbers there? Readers, please don't ask! But that's the world where the natural numbers "reside." Rather, that's where the number 3 "actually resides," or so we're crazily told.
A peculiar culture is on quick display in Livio's well-received book. In fairness, though, he didn't invent it. This puzzling culture of apparent incoherence has been around for an extremely long time.
Within this puzzling culture, peculiar metaphysical statements are hatched by physicists, astrophysicists and mathematicians gone wild. One such earlier adept was the brilliant British mathematician G. H. Hardy (1877-1947), whose famous treatise, A Mathematician's Apology, was reverentially quoted by Professor Goldstein in her well-received 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel:
HARDY (1940): I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it...Warning! In this passage, the terms "realism" and "idealism" are technical terms taken from a long, less-than-coherent "philosophical" tradition. Having allowed for that, let's consider what Hardy does in this passage.
[T]his realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more than what they seem. A chair or a star is not in the least like what it seems to be; the more we think of it, the fuzzier its outlines become in the haze of sensation which surrounds it; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best into some framework of idealistic philosophy—I do not believe it, but there are eminent physicists who say so. Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way.
In this hoary, much-quoted passage, Hardy invents a construct called "mathematical reality." He then starts debating where it "lies" (i.e., where it "resides") and the way it is "built."
For reasons which go unexplained, he refers to numbers like 2 and 317 as "mathematical objects." In this way, he continues venturing down as unfortunate road, in which, to quote the later Wittgenstein, "language goes on holiday," generally with bad results.
The numbers 2 and 317 have now been classified, somewhat oddly, as "mathematical objects." Having introduced that strange formulation, Hardy tells us that such objects "are much more than what they seem."
More specifically, he tells us that the chair on which you're currently sitting "is not in the least what it seems to be," while the properties of the number 2 "stand out the more clearly the more closely we scrutinize it."
From there, we're invited to wonder about what makes 317 a prime. It isn't a prime "because we think so," this brilliant mathematician somewhat peculiarly says. Instead, he says that 317 is a prime "because mathematical reality," whatever that is, "is built that way," whatever that might mean.
Why is 317 a prime? Within the context of Hardy's rumination, the question starts to seem odd, but we'll offer an extremely simple answer:
Once you know what it means to say that some number is a prime, it's very easy to determine that 317 qualifies. First you try to divide it (evenly) by 2. Then you try to divide it evenly by 3, and then by 5, and then 7.
By the time you've been unable to divide it evenly by 17, your search is over, though people like Hardy—mathematicians who have wandered outside their field of expertise—will try to engage you in a debate about a range of fuzzy concepts they couldn't explain or clarify if they were competently asked to.
Alas! Within our academic world, such challenges are rare. In that famous passage from Hardy's famous essay, we're looking at the ruminations of a brilliant mathematician gone wild.
Hardy's ruminations are barely coherent. But Professor Goldstein presented the passage with full respect, and she tends to treat Godel's crazy ideas about "Platonism" in a similar way.
Livio behaves the same way in his opening chapter. He quotes a string of award-winning mathematicians as they make fatuous remarks about the structure of the cosmos—after they have strayed beyond the bounds of their expertise.
In effect, Livio treats these mathematicians' fuzzy statements as "things that make us go hmmm." In his years on late-night TV, Arsenio Hall performed that trademark bit as a rollicking entertainment. But at the higher ends of the academy, astrophysicists, mathematicians and "philosophers" gone wild still play the game for real.
In the first two paragraphs of his book, Livio starts taking us down the amusing road of "things that make us go hmmm." Tomorrow, we'll review those opening paragraphs to see what can happen when the mind of a ranking astrophysicists is allowed to stray.
For today, we'll only say this. Once we start down that road—the road where language has gone on holiday—we can reach some very strange destinations. By page 9 in his opening chapter, Livio is offering the following passage as he tries to fight his way through a long-standing semantic muddle:
LIVIO (page 9): As I noted briefly at the beginning of this chapter, the unreasonable effectiveness of mathematics creates many intriguing puzzles: Does mathematics have an existence that is entirely independent of the human mind? In other words, are we merely discovering mathematical verities, just as astronomers discover previously unknown galaxies? Or, is mathematics nothing but a human invention? If mathematics indeed exists in some abstract fairyland, what is the relation between this mystical world and physical reality? How does the human brain, with its known limitations, gain access to such an immutable world, outside of space and time?...We're now being asked to consider the possibility that mathematics "exists in some abstract fairyland"—in some sort of "mystical world," which lies "outside of space and time." This is where we can get by page 9 when, on pages 1-3, language has been allowed to go on a bit of a journey.
The later Wittgenstein warned about this ubiquitous practice. According to Professor Horwich, his work has been thrown under the bus because our "professional philosophers" still want to gambol and play.
According to Horwich, they want the right to retain their long-standing "linguistic illusions and muddled thinking. This helps explain why so little aid has come from the academy as our journalistic discourse has descended into inanity and grinding technical incompetence—and by the way:
This is the type of work which is done by our highest-order thinkers! By our astrophysicists, our mathematicians, our winners of all those medals!
Man [sic] is the rational animal? As we await the start of Mister Trump's War, is it possibly time to thrown that pleasing old story away?
Tomorrow: Things that make us go hmmm
From the original text: "Philosophical problems arise when language goes on holiday."
So said the later Wittgenstein, in Philosophical Investigations.
To review the original text, click here, then move to page 19—though, as Wittgenstein essentially acknowledged, the writing is quite obscure.