THURSDAY, MAY 20, 2021
The anthropologist's question: As we've acknowledged, it's easy to make fun of this apparently silly, "1 + 1" academic stuff.
As the poet so thoughtfully said, Easy to be hard!
That said, the anthropologist's question remains—and it remains unanswered. Before we quote this credentialed expert, let's get ourselves moving right along:
Let's move to 2 + 2.
Those twins appear, though in disguised form, in Rebecca Goldstein's 2005 general interest book, Incompleteness: The Proof and Paradox of Kurt Godel.
The book was intended for general readers. 2 + 2 shows up in the early passage where Professor Goldstein tries to explain what modern-day "Platonism" is.
For the record, Goldstein is a ranking philosophy professor and a well-regarded novelist. In terms of the academic / high intellectual firmament, she's a significant player.
Incompleteness was well-regarded, and was extremely well-blurbed. That said, we regard the book as fascinating because so much of it doesn't seem to make sense. The passage to which we refer goes exactly like this:
GOLDSTEIN (page 44): [S]ince Godel's field is mathematics, the "out yonder" in which he is interested is the domain of abstract reality. His commitment to the objective existence of mathematical reality is the view known as conceptual, or mathematical realism. It is also known as mathematical Platonism, in honor of the ancient Greek philosopher...
Platonism is the view that the truths of mathematics are independent of any human activities, such as the construction of formal systems—with their axioms, definitions, rules of inference and proofs. The truths of mathematics are determined, according to Platonism, by the reality of mathematics, by the nature of the real, though abstract entities (numbers, sets, etc.) that make up that reality. The structure of, say, the natural numbers (which are the regular old counting numbers: 1, 2, 3, etc.) exists independent of us, according to the mathematical realist...and the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity.
According to that puzzling passage, Godel was committed to "the objective existence of mathematical reality," whatever that might mean. This view, whatever it may turn out to be, is known as "mathematical Platonism," or as "Platonism" for short.
What does Platonism turn out to be? Before she gives a few examples, Goldstein offers this puzzling account:
"The truths of mathematics are determined, according to Platonism, by the reality of mathematics," Goldstein unhelpfully says.
We almost sense the ghost of Meinongianism in that puzzling opening statement. But then, the examples appear:
According to Platonism, we're now told, "the properties of the numbers 4 and 25—that, for example, one is even, the other is odd and both are perfect squares—are as objective as are, according to the physical realist, the physical properties of light and gravity."
Is the number 4 even? We would agree that it is! In fact, anyone would agree that the number 4 is even, just so long as the person in question knows what the meaning of "even number" is.
An "even" number is any number which can be divided into two equal, whole-number parts. And since 4 can be split into 2 + 2, every schoolchild will understand that the number 4 is "even!"
What could any of that have to with someone's being a "Platonist?" What could any of that have to do with a "commitment to the objective existence of mathematical reality," whatever that formulation might turn out to mean?
Anyone who knows the meaning of the term would agree that the number 4 is even; it's as simple as 2 + 2! What could any of this have to do with some vast philosophical "commitment," of the type Goldstein ascribes to the Platonist?
The ghost of Meinong linger in that prose—the ghost of the Austrian philosopher who engaged in the conceptual madness of saying, as part of a "theory of objects," that even though unicorns don't exist, they must have some type of being. (See Monday's report.)
By way of difference, Meinong wrote in the late 1800s. Goldstein's book appeared in 2005.
Goldstein's initial stab at defining Platonism seemed to make no sense at all. But then, a mere two pages later, she extended her exploration, quoting a very well-known figure from the 1940s. That passage started like this:
GOLDSTEIN (page 45-46): Godel's mathematical Platonism was not in itself unusual. Many mathematicians have been mathematical realists...G.H. Hardy (1877-1947), an English mathematician of great distinction, expressed his own Platonist convictions in his classic A Mathematician's Apology, with no apology at all.
Hardy was a Platonist too, as are many mathematicians!
Goldstein then presented a lengthy excerpt from Hardy's famous text. Hardy was a highly-regarded mathematician, but the excerpt presented in Goldstein's book makes no obvious sense. This is the excerpt she presented, exactly as it was presented:
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...
[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.
Hardy said he believed "that mathematical reality lies outside us," whatever that might mean. Trafficking in some monumentally fuzzy philosophical terms (idealism v. realism), he went on to offer an example of what he means:
In that passage, he said that 317 is a prime, "not because we [merely] think so," but because "mathematical reality is built that way." This would seem to make Hardy a "Platonist," according to Goldstein's book.
That said, everyone on the face of the planet agrees that 317 is a prime. You simply have to know what the term "prime number" means, then be willing to execute a very small amount of long division.
(You can stop after you've tried dividing by 19.)
As you will quickly see, 317 can be divided evenly by no number except 1 and itself—and that's the definition of a prime! It's hard to know how any of this is supposed to lead us on to some complex philosophical theory, some theory dating all the way back to Plato in his famous, bewildering cave.
What was Goldstein talking about in her well-blurbed book? It seems to be true that many mathematicians regard themselves as "Platonists." But it's also true that Hardy, who actually was a great mathematician, was a complete and total bust when it came to defining this alleged belief.
Sadly, the same problem afflicted Goldstein when she offered her own examples of Platonism, including one example as simple as 2 + 2. But even as late as 2005, this was the world into which a youngster would be drawn when he or she decided to sign on as a philosophy major!
Does any of this make any sense? To appearances, no, it doesn't. Nor is this the part of Goldstein's book which struck us as most remarkably odd—which we we found most strikingly incoherent.
This leads us to the anthropologist's question! Sadly, though, time is fleeting. We'll save that for tomorrow.
Tomorrow: The anthropologist's tale