"Can be interpreted as" vs "Is"

Question

Consider the following pairs of statements:

• "I see what I interpret as a chair" vs. "I see a chair."

• "This chair can be interpreted as a set of atoms" vs. "This chair is a set of atoms."

• "This set of atoms can be interpreted as a chair" vs. "This set of atoms is a chair."

• "The universe can be interpreted as a mathematical structure" vs "The universe is a mathematical structure."

• "The mind can be interpreted as a pattern of neural activity" vs "The mind is a pattern of neural activity."

• "The group of rotations of a cube is isomorphic to S_4" vs "The group of rotations of a cube is S_4."

Let us consider in particular the last of these sentences, about the group of rotations of a cube. It's quite common in mathematics to speak this way. When we have shown two structures are isomorphic, this means that theorems (of a certain kind) that we deduce about one of the structures can be directly translated into theorems about the other structure, and vice versa. A mathematician thus will casually refer to the two structures as "the same," even though technically there is a difference in how the structures are defined. They are the same for his purposes.

An isomorphism in mathematics is a kind of rigorous interpretation; we look at one structure and interpret it as a second one. And from then on we can treat them interchangeably, and comfortably refer to them as the same structure, as long as we restrict ourselves to propositions of a certain kind that can be "translated across" the interpretation in either direction.

The other statement pairs listed above are similar to the statement pair about S_4. In each case, there is an object, and an interpretation of the object, so that statements of a certain kind made about the object translate into statements of a certain kind about the interpretation, and statements of that kind made about the interpretation translate back into statements about the object.

So, is it reasonable to say that the statements in each pair are equivalent? In other words, to use the word "is" in a manner that does not denote exact equality (whatever that is), but that denotes equivalence of structure with respect to a certain class of statements?

Answer 1

No. The two statements in each case but the last have different meanings. And different truth values. Discussing each:

• "I see what I interpret as a chair" vs. "I see a chair."

We perceive chairs. Most people consider their perceptions to be basic. Both perceptual psychology, and perceptual neuroscience, have shown with strong evidence that we are interpreting, not directly perceiving chairs. The difference is between fundamental phenomenology, vs the view of scientific realism. I consider scientific realism correct here, but most people disagree, and the difference between the two statements is significant. The issue here is basically between expert opinion, and popular opinion.

• "This chair can be interpreted as a set of atoms" vs. "This chair is a set of atoms."

Both assume atomic reductionism. The first is just a claim that an atomic reductionist version of scientific realism is a valid way to view chairs, the second is the stronger claim that it is the only valid way to view chairs. Additionally, unlike perception being interpretive, this claim that reduction of chairs to atoms is valid is NOT true. See the SEP article on scientific reduction -- the reductionist project has stalled out with only about 90% of physics reduced, half of chemistry, and only a subset of biochemistry. Science has now adopted pluralism, which is assumed to be arrived at through a TBD and as yet uncharacterized process of emergence. Chairs do not reduce to atoms, they provide functions which are independent of their reduced components. For an explanation of the reason that functions cannot be reduced, explore the literature on multiple realizability.

Note that chair reductionism is CONTRARY to expert opinion, so this example is not equivalent to the first.

• "The universe can be interpreted as a mathematical structure" vs "The universe is a mathematical structure."

This one has the same two issues as above: "one of many possible views" vs "this IS how it is" issues that the statement above had. And has the same problem that not even the first has been shown to be valid, much less the unsupported and almost certainly untrue second claim that the universe fully reduces to mathematics.

And once more, expert opinion is also against the second claim, and a strong interpretation of the first. I put the caveat "strong interpretation" in because the utility of an inaccurate but interesting mathematical representation may provide insights, and most experts would endorse mathematical representation as an approximation to the universe.

• "The mind can be interpreted as a pattern of neural activity" vs "The mind is a pattern of neural activity."

Repeat objections to the above two. The two statements make different claims, and neither is actually supported by evidence. The SEP article actually cites the long term FAILURE of neural reduction of consciousness, as one of the stronger evidences for scientific pluralism.

The same points apply as above to this statement -- experts disagree with both as "true" but the first is a potentially fruitful line of investigation as an approximation.

• "The group of rotations of a cube is isomorphic to S_4" vs "The group of rotations of a cube is S_4."

Despite this statement having a somewhat similar structure to the others, it is significantly different in that the understanding of what mathematics IS, does not exclude functional equivalence as part of what constitutes a math feature. Experts, and the public, mostly agree with both statements.

Answer 2

Those statements are not analogous to each other, and many of them are not proper, meaningful English.

"I see what I interpret as a chair" vs. "I see a chair."

The first statement is naturally taken as a comment about the nature of perception, and not about chairs at all. It seems to be based on a view that takes sensations as basic and discusses how the mind interprets sensations.

"This chair can be interpreted as a set of atoms" vs. "This chair is a set of atoms."

This statement has no use in normal English because a chair is not the sort of things that can be interpreted. Only things with meanings can be interpreted, and chairs typically don't have meanings. Now, you could attribute a meaning to a chair (or anything else), but the most natural meaning to attribute to a chair is its expected usage, or why it was placed where it was or something like that. It is hard to come up with a meaning in which a chair can be "interpreted as" a set of atoms. Now, a chair can be viewed as a set of atoms, but that's not the same thing as being interpreted as a set of atoms.

"This set of atoms can be interpreted as a chair" vs. "This set of atoms is a chair."

Again, a set of atoms is not something that normally has a meaning. However, I can imaging the following context in which this sentence might be used: suppose you are examining an image that was produced by an instrument that detects individual atoms. Then you might use that sentence to describe how to interpret the image.

"The universe can be interpreted as a mathematical structure" vs "The universe is a mathematical structure."

Again, the universe isn't the sort of thing that can be "interpreted" because it doesn't have a meaning (except possibly in a religious context, but then it wouldn't be interpreted as a mathematical structure). Saying that the universe is a mathematical structure is equally odd. The universe is a physical object. No physical object is a mathematical structure. Probably what you mean is "the universe can be modeled or represented as a mathematical structure".

"The mind can be interpreted as a pattern of neural activity" vs "The mind is a pattern of neural activity."

Again, this doesn't make sense. The mind doesn't have a meaning that can be interpreted. You could say "the mind can be viewed as a pattern of neural activity" or "the mind is constituted by a pattern of neural activity".

"The group of rotations of a cube is isomorphic to S_4" vs "The group of rotations of a cube is S_4."

Saying "A is isomorphic to B" is nothing at all like saying that "A can be interpreted as B". Interpretation is about meaning; isomorphism has nothing to do with meaning.