I really enjoyed talking to our old friend John Perry about the philosopher Gottlob Frege, whose 19th-Century about logic and mathematics are still influential today.  Our listener Daniel sent in a great question for John, which we didn’t time to answer during the show.  I wanted to say a little about Daniel’s question here—especially since today is Frege’s birthday!

Daniel’s is curious about Frege’s attempt to establish math as an extension of logic.  Frege believed that math is analytic, meaning that the definitions of mathematical terms like “2” and “4” guarantee the truth of sentences like “2+2=4”.  Frege’s theory explains how we know about math; as long as we can understand what we mean by mathematical terms, and can reason logically, our mathematical knowledge is guaranteed.  But in order to work, the theory has to rely on Frege’s definition of a number.

Frege explained numbers by appealing to functions, like the addition function labeled by the + sign.  The name “Ray” picks out a person… but what does the plus sign pick out?  It doesn’t seem like it picks out a number.  At best, you can use it to pick out a number by writing it next to the names of other numbers: “2+3” picks out the number 5.  But “+” by itself doesn’t name an object; it stands for a function.

Frege thought predicates like “philosopher” stood for functions too.  The word “philosopher” doesn’t stand for any philosopher in particular—it refers to Ray, Josh, Mary Wollstonecraft, Frege, Franz Fanon… and so on.  Instead, it picks out a function.  If you fill in a name and a few connecting words, you can use it to build a sentence like “Ray is a philosopher,” and that sentence will pick out something—maybe a state of affairs where Ray is a philosopher, or maybe some mysterious Boolean object called True.

Frege thought that instead of being the name of a particular kind of thing (a number) “3” is actually more like a word for a function.  Consider the property of being a Brontë sister.  This property has properties of its own, like the property of applying to at least one person.  (You might say that while a regular property belongs to an ordinary object, a higher-level property applies to a property.)  One property of being a Brontë sister is the property of applying to three things.  We can think of the numeral “3” as picking out a higher-level property of all the properties that apply to exactly three things (for instance, being a Brontë sister, being a star in Orion’s belt, or being a novel in the Lord of the Rings trilogy).

There’s a lot more to Frege’s theory of numbers, but Daniel’s question comes in here.  Daniel writes:

As there are first level functions, representing incomplete objects, there are also second level functions, representing other functions.  And the same holds for concepts:  First level concepts refer to complete objects and second level concepts to other references.  It is the distinction between the levels of functions however which Frege finds to be non-conventional and belong to nature itself, laying potential ground for the exhaustive reduction of mathematical reasoning to logical reasoning, thus an explanatory ground for the efficacy of the former in the study of nature as deriving from a conceived intrinsic unity with nature of the latter.
 
He then asks: do second-level functions (and properties) really exist, or can we somehow reduce them to first-level functions (and properties)?

I don’t know what John would say, but I think second-level properties must really exist.  The property of applying to three things isn’t really a property of ordinary objects.  (I guess I could apply to three jobs, but that’s not anything like applying to three Brontë sisters, three stars, or three Lord of the Rings novels.)  Logicians know I can express by appealing to second-level properties that I can’t possibly express in any other way.  (Here’s a simple example from George Boolos: there are some critics who admire only each other.  Boolos’s sentence says that there’s some property, such that every critic admires every other critic with that property and no one else.)

Second-level functions are really useful in mathematics.  If I want to study a branch of the mathematics of voting called social choice theory, I might start with a function that assigns preferences to each voter.  I can then ask questions about what society’s preferences should be, based on the individual voter preferences.  Different possible answers can be expressed as second-level functions, which take voter preference functions as their inputs, and return society’s preference ordering as their outputs.  

Second-level functions also matter in measurement theory, which studies scales—which you can think of functions that take objects as their inputs and return numbers as their outputs (representing features like mass in kilograms, temperature in Kelvins, or price in dollars).  We can learn interesting things about scales by considering transformation functions that take one scale, like the temperature-in-Kelvins scale, to another scale, like the temperature-in-degrees-Fahrenheit scale.  Since a scale is already a function, a transformation function is a second-level function that maps one function to another.

There are some reasons to be cautious about which second-level properties functions to believe in.  In semantic paradoxes of self-reference, some phrases that seem to pick out perfectly good second-level properties give rise to strange and potentially contradictory consequences.  (Consider the property of being heterological, that is, being a word that isn’t true of itself.  The word “long” is heterological, since it’s short; the word “monosyllabic” is heterological, since it has multiple syllables.  Is the word “heterological” heterological?  Oh no…)

Maybe, despite appearances, second-level properties and functions are completely imaginary.  I can’t rule that possibility out.  Some nominalists think that all properties and functions are completely imaginary, including the first-level ones, and nobody has definitively proven that they’re wrong.  But since I usually think things are roughly as they appear, I believe that there are second-level properties and functions.  I think Frege might have gotten some other things wrong, though!  I certainly enjoyed arguing about it with Josh and John.