ANTHROPOLOGY UNBOUND: Defining the meaning of 1 + 1!

TUESDAY, MAY 18, 2021

Anthropologists float theory: Very frankly, we almost felt triggered. 

We'd turned to the leading authority on Bertrand Russell's 1903 classic text, The Principles of Mathematics. 

"The book presents a view of the foundations of mathematics and Meinongianism and has become a classic reference," we were quickly told.

We'll admit that we knew nothing of this Meinongianism. But when we clicked the kink we'd been provided, we were confronted with this:

[Meinong's] theory of objects, now known as "Meinongian object theory," is based around the purported empirical observation that it is possible to think about something, such as a golden mountain, even though that object does not exist. Since we can refer to such things, they must have some sort of being. Meinong thus distinguishes the "being" of a thing, in virtue of which it may be an object of thought, from a thing's "existence", which is the substantive ontological status ascribed to—for example—horses but not to unicorns. 

Since we can refer to some entity we have imagined, it must have some sort of being! 

Already, our youthful analysts felt triggered. They felt triggered even before they reached the important distinction between the "being" and the "existence" of some such imagined entity.

We understood the youngsters' deep discomfort. For ourselves, we'll have to admit, we felt divinely amused!

For now let's return to Meinong's theory of object:

Since we can discuss some entity we have imagined, it must have some sort of being! Stating the obvious, that passage traffics in a kind of conceptual madness.

In the face of such attacks on sanity, we humans will often defer to academic authority. Given the high academic source, we'll assume that there must be something of substance to the puzzling claim at hand.

In making such charitable assumptions, might we humans be wrong? 

Here at this site, we'd dug out our antique copy of Russell's text as an act of renunciation. Our Town's public discourse had reached a point of such vapidity that we found it hard to watch "cable news" or to bother with the foolishness found in our major newspapers.

In a flight from this banality, we had returned to loftier concerns, and to one favorite book—Rebecca Goldstein's Incompleteness: The Proof and Paradox of Kurt Gödel. That favorite book had sent us back to Russell's concerns with the "foundations of mathematics."

Soon, we were reading about Meinong's theory of objects. While the analysts were triggered by the passage we've quoted, we were filled with a type of delight.

Sometimes, you just have to laugh, a bit like the god on Olympus. As we were first told in undergraduate days, the tortured ruminations of the western world's "philosophers" may, or may not, make sense.

That said, these ruminations will often provide a type of amusement. On page 4 of Russell's classic text, we found him saying this about the state of philosophy as it had existed:

RUSSELL (page 4): The Philosophy of Mathematics has been hitherto as controversial, obscure and unprogressive as the other branches of philosophy. Although it was generally agreed that mathematics is in some sense true, philosophers disputed as to what mathematical propositions really meant: although something was true, no two people were agreed as to what it was that was true, and if something was known, no one knew what it was that was known. So long, however, as this was doubtful, it could hardly be said that any certain and exact knowledge was to be obtained in mathematics. We find, accordingly, that idealists have tended more and more to regard all mathematics as dealing with mere appearance, while empiricists have held everything mathematical to be approximation to some exact truth about which they had nothing to tell us. This state of things, it must be confessed, was thoroughly unsatisfactory. 

At the start of that passage, Russell seems to speak poorly of the various branches of philosophy circa 1903. The amusement begins when he describes the disputes among philosophers as to what mathematical statements mean.

"Although something was true, no two people were agreed as to what it was that was true, and if something was known, no one knew what it was that was known," Russell lamented. 

Given this state of affairs, "it could hardly be said that any certain and exact knowledge was to be obtained in mathematics," Russell  then said. "This state of things, it must be confessed, was thoroughly unsatisfactory.". 

One who is inclined to defer may assume that Russell is speaking here of some highly abstruse region of higher mathematics. 

Apparently not! Two pages later, his classic text offers this:

RUSSELL (page 6): Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain variables. The notion of the variable is one of the most difficult with which Logic has to deal, and in the present work a satisfactory theory as to its nature, in spite of much discussion, will hardly be found. For the present, I only wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. Elementary Arithmetic might be thought to form an exception: 1 + 1 = 2 appears neither to contain variables nor to assert an implication. But as a matter of fact, as will be shown in Part II, the true meaning of this proposition is: “If x is one and y is one, and x differs from y, then x and y are two.” 

In that passage, Russell seems to explain what 1 + 1 = 2 truly means. This simplest of all arithmetical statements actually means this:

“If x is one and y is one, and x differs from y, then x and y are two.” 

Finally, someone had said it! 

Russell said he was going to prove it in Part II of his book. Meanwhile, at the end of that paragraph, Russell takes things even further: 

RUSSELL (continuing directly): And this proposition both contains variables and asserts an implication. We shall find always, in all mathematical propositions, that the words any or some occur; and these words are the marks of a variable and a formal implication. Thus the above proposition may be expressed in the form: “Any unit and any other unit are two units.”

Any unit and any other unit are two units! 1 + 1 = 2 may be expressed in that form.

Those who are inclined to defer will assume that we're on our way to something truly significant. For ourselves, we decided to turn to the leading authority for a quick overview of Russell's classic text. 

Instantly, we were linked to Meinong's theory of objects, which—let's be perfectly clear—comes from from the realm of conceptual madness, at least as it is described.

(According to the leading authority, Russell agreed with that "theory of objects," until such time as he didn't)

Does any of this make any sense? We who defer will be inclined to assume that it simply must.

Russell occupied the western world's highest level of academic authority. It's hard to believe that the work of such intellectual giants could really have been the work of a type of madness.

That said, anthropologists have whispered to us that the gods on Olympus continue to laugh, right to the present day. According to these credentialed experts, the later Wittgenstein sits by their side, obsequiously assuring them that they had it right all along.

Within the journalistic realm, Our Town's daily discourse has reached a point of almost complete banality. The dumbness is on full display, even over here in Our Town..

Might it be, anthropologists ask, that the jumbled wiring of our human brains has produced similar manifestations on much higher levels, possibly extending all the way back to the dawn of the west? Could our journalistic incompetence be part of a larger mess?

Were the Olympians right when they laughed? In our experience, leading experts—anthropologists—glumly continue to ask.

Tomorrow: Whatever may seem to come next