# "Can be interpreted as" vs "Is"

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?

• Isn't the group of rotations of a cube a subset of the rotations of the rotations of a general object? The symmetry properties of the rotations of a sphere are different from those of a cube. Apr 14, 2022 at 19:20
• @Felicia Yes, the rotation group of a cube is S_4 while the general 3d rotation group is SO(3) Apr 14, 2022 at 19:26
• 'is' - 'to be' - has different meanings and connotations in different languages... Apr 16, 2022 at 5:23
• In all these cases when asserting P instead of Q, though P implies Q, there is a Gricean implicature that Not Q, since the Principle of Cooperation is violated. Apr 17, 2022 at 21:30
• @jlawler I don't understand. What is P and Q? Apr 18, 2022 at 1:20

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.

• Note, while I am disagreeing with the assumptions behind the question, I think it is a good question, and serves as helpful to reveal hidden/suspect assumptions. Apr 14, 2022 at 20:16
• Let me ask you: would you say that in Conway's game of life, patterns such as a glider or a blinker reduce to the arrangement of black-and-white cells that make them up? I ask to determine whether your argument against reductionism is independent of the specific laws of physics (and thus also applicable to Conway's game of life), or dependent on the specific laws of physics (and thus only applicable in our own universe). Apr 14, 2022 at 22:20
• @Causative -- The behaviors of macro structures in Life, was surprising to the creator. That there ARE macro structures, and that the HAVE behavior, was surprising. These are emergent phenomenon. LIFE is studied empirically, an an emergent/macro scale rather than reductively. I cannot show that LIFE macro structures CANNOT be reduced, I instead suspect they can. One also cannot show that reduction simply doesn't apply in principle, such as with atomic reduction and chairs. Loss of pixels usually breaks up a structure, while loss of innumerable atoms does not. Apr 15, 2022 at 3:11
• Emergence comes in degrees, and LIFE's emergence seems to be a lower degree than many in our world. Note however, I am not an expert on LIFE, and am partially guessing about it here. Apr 15, 2022 at 3:14
• One significant difference, objects in our world do not satisfy A = A, and are therefore not valid subjects of logic. But objects in LIFE have definite boundaries and characteristics, and what they are IS what they consist of, definitively, so ARE valid subjects of logic. Apr 15, 2022 at 3:20

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.