Update: Fixed an apparent contradiction on whether critical truths are isomorphic or not.
What does it mean when we say something is true? Most people are fairly confident they know what truth is when they see it, but can we actually define it? I think when pressed, many people answer something along the lines of "A statement that contains no falsehoods," but is that adequate?
Consider the statement "All meebles are foob." There are no falsehoods in that statement, but it contains no truth, either; it is meaningless.
Ah, but maybe we can get away with, "A meaningful statement that contains no falsehoods."
Alas, no. Consider "Chocolate is better than brussel spouts." As I've discussed before on this blog, normative statements are meaningful, and yet cannot be meaningfully assigned a truth-value. The statement is neither true nor false.
So, can we exclude normative statements, and define truth as "A meaningful positive claim that contains no falsehoods?"
A few weeks ago, I might have said yes, but I have come to reconsider that position. The problem is that there are certain statements which are not positive--they are not statements about the empirically measurable properties of physical entities--and yet also appear to be true. For example, "Frodo Baggins is a hobbit," or "Two plus two equals four."
And while previously I would have regarded them as a subtle type of positive statement (the first can be viewed as really being a statement about the arrangement of ink on the pages of a particular set of books, the second as a statement about how the human brain works), they actually differ from positive truths in quite important ways.
I am thus forced to conclude that there are multiple kinds of truth, and it is to those I now turn.
Positive truths are the easiest category to discuss. We appear to live in a universe that is consistent--it obeys rules. We may not be entirely certain what those rules are, but so far the universe and everything in it has recognizably behaved very consistently.
We can therefore conclude that the set of positive truths is consistent: No positive truth contradicts any positive truth, including itself.
More importantly, we have defined positive statements as being about the empirically observable properties of physical entities. That means that there is an external standard against which to test positive truths; if a positive statement is true, that it means it is isomorphic to (exists in one-to-one correspondence with) some feature of physical reality. Put another way, the set of all positive truths is isomorphic to the physical universe.
Finally, as there is only one physical universe about which there can be positive truths,* and we already have the requirement of consistency, it follows that a positive truth is universally true.
Hence we have a definition for positive truth: Positive truths are consistent, universal, and isomorphic to the physical universe.
Mathematical statements, like "Two plus two is four," alas, are not isomorphic to the physical universe. There is no physical "two" out there, and no physical entity has the property of "two-ness." We also cannot argue that it's a description of the behavior or interactions of physical entities, because there are no physical entities in the statement.
"Two elephants plus two elephants is four elephants," might be true in some circumstances, and we could argue that "Two plus two is four" is merely an abstraction and generalization of this specific case, but unfortunately it is an abstraction that does not always hold. If the elephants are violent, two plus two may equal one. If, alternatively, they are amorous, it may equal six.
Let's try a statement that seems more concrete: "The shortest distance between two points is a straight line." Simple enough, and it seems like it conforms to measurable properties of real objects. Only problem is, when we're talking about physical entities, it isn't true.
The shortest distance between two points is a straight line only on a perfectly flat surface. If you are constrained to follow a curved surface, the shortest distance is a curve whose properties depend on the curvature of the surface. If you dig a tunnel straight through the Earth from New York to Paris, and compare that to the shortest path an airplane can take between them, the plane curves notably to the north of the tunnel.
Fun fact: Space is measurably curved. "The shortest distance between two points is a straight line," is not strictly true anywhere in the physical universe (though for distances short enough relative to the local curvature it is close enough for practical purposes).
So shall we discard that statement as untrue? Certainly not! There are many contexts within mathematics in which the statement must be treated as true; specifically, in Euclidean geometry.
So, can we say that mathematical truths are isomorphic to particular contexts? Unfortunately, no. Consider my favorite chair. It is a dark-brown leather recliner with some cosmetic damage, but structurally sound.
The previous sentence is isomorphic to certain properties of my chair. It would be absurd, however, to say that my chair is isomorphic to my chair. My chair is my chair; to exist in one-to-one correspondence, two things must be different. In other words, identity precludes isomorphism.
Alas, when we do geometry, we are examining and thinking about a set of statements. The set of statements to which "The shortest distance between two points is a straight line," belongs is Euclidean geometry, and thus cannot be isomorphic to it.
Euclidean geometry, to be specific, is a set of axioms--statements which do not contradict each other and are treated as self-evident--and everything which can logically be derived from those axioms. If you change an axiom (for example, the definition of "parallel"), you get a different geometry (such as elliptical or hyperbolic geometry).
Math is, in effect, a number of different sets of statements. Each set has the property of being consistent within itself, but is not necessarily consistent with other sets.
And thus a definition for mathematical truths: mathematical truths are consistent, contextual, and identical, rather than isomorphic, to their contexts.
Are we done? Have we covered all true statements? We have yet to discuss normative statements; are they a kind of mathematical truth? Perhaps the context is a particular person's opinion and beliefs, and certainly it seems that the identity property is present--the set of true statements about my beliefs and opinions would be identical to my beliefs and opinions.
However, if you honestly believe that a person's beliefs and opinions are necessarily, or even often, consistent? I have a bridge to sell you.
It seems we need a new category, but what could it be? Consistency seems like it ought to be the most fundamental of all requirements for any definition of truth--how can truth possibly contradict truth?
Consider the following true statements:
"The sole author of the original Sherlock Holmes stories is Sir Arthur Conan Doyle."
"The Sherlock Holmes stories are accounts by Watson of his adventures with his friend, Sherlock Holmes."
These two statements are both true statements about the Sherlock Holmes stories, and yet they contradict--if they are accounts by Watson, Doyle cannot be the sole author, and vice versa.
Ah, says the savvy reader, but they don't contradict because they represent different perspectives on literature: In literary theory, the Doylist perspective is that from which our world is real and the story is fiction; the Watsonian perspective is that from which the story is true, and our world is irrelevent. Claims made from one perspective need not be consistent with claims made from the other--they are independent contexts.
That's all well and good, but what if I write a story which intentionally contradicts itself? Consider the Invisible Pink Unicorn. For real, physical entities, it is a contradiction to be both invisible and pink, and descriptions of the IPU make it clear it is just as contradictory for this imaginary entity. Thus, if I write a story about the IPU, true statements about that story include both "The IPU is invisible" and "The IPU is pink."
This is even more the case when we get into analysis and criticism. Art is a symbolic activity, and a given symbol may have more than one meaning in one person's mind, let alone many people's. Even the cursory analysis involved in looking at a painting, recognizing its subject, and having an immediate emotional response involves extensive interpretation of symbols(1), and these symbols may have multiple meanings. As analysis goes deeper, the number of available meanings multiplies, and likelihood that they will all be consistent drops.
However, all these meanings are true. Consider the syllable pronounced "hi." It is true that it is a greeting in English, and also true that it means the opposite of "low" in English. Both meanings are true. While we can usually identify from context which meaning is intended, the fact that the other meaning is not intended in this instance doesn't make it less true that that is a meaning of the symbol.
If we are dealing with more ambiguous symbols, and as the number and complexity of symbols increases, the number of interpretations which each correspond to the work (and yet may contradict each other) increases exponentially.
Thus it is that Buffy the Vampire Slayer is both a feminist critique of the horror-movie staple of the helpless young woman cornered in an alley, and a misogynistic celebration of woman being punched in the face. Both are true because both can be demonstrated as meaningful interpretations of the images and sounds that comprise the show, and yet the two statements contradict each other--and depending on which interpretations occur more immediately and naturally to me, I might have a very strong emotional response to one or both!
Like mathematical truths, critical truths are statements about human mental constructs(2). However, an individual critical truth is not identical to an element in these constructs(3). They are also contextual--there is a set of true statements about Buffy the Vampire Slayer, and it is obviously not the same as the set of true statements about Mozart's Eine Kleine Nachtmusic.
Are critical truths isomorphic? No; the elimination of consistency means that critical truths can exist in a two-to-one (or seventeen billion-to-one) relationship to the works they describe. Some critical truths may be identical as well: if a story begins "This is a story about hope," the statement that the story is about hope would be identical to that element of the story. The common element of critical truths is that they are derived from a mental construct--they draw on elements of it, as filtered through a unique human consciousness, to create a new (possibly inconsistent) set of statements about that work which may well be larger than the original work(4).
We can therefore define critical truths as contextual and derivative, with no requirement of consistency or isomorphism.
At last we have the tools to begin to re-examine the truth value of normative statements, which I will do in a future post.
1: Ceci n'est pas une pipe.
2: One might object to calling art a mental construct, as a work of art must take some physical form in order to be shared. However, that physical form is a representation of a mental construct of the artist, and encountering the symbol creates a mental construct for the viewer. This is perhaps more obvious with storytelling forms such as literature and film, where the author creates mental settings and characters represented within the work, but still just as true of any work of art which requires the artist to first imagine or envision something (which is to say, all art).
3: The signifier-signified distinction, in other words.
4: As a college freshman, I had an assignment to write ten pages about any two lines in Hamlet that were not from the "to be, or not to be" speech.