## Monday, March 4, 2019

### Isomorphism of inputs

For simplicity, I’ll stick to deterministic systems in this post. Functionalists think that if A is a conscious system, and B is functionally isomorphic to B, then when B receives valid inputs that correspond under the isomorphism to A’s valid inputs, B has exactly the same conscious states as A does.

Crucial to this is the notion of a functional isomorphism. A paradigmatic example would be a computer built of electronics and a hydraulic computer, with the same software. The electronic computer has electrical buttons as inputs and the hydraulic computer uses valves. Perhaps a pressed state of a button has as its isomorph an open valve.

But I think the notion of a functional isomorphic is a dubious one. Start with two electronic systems.

• System A: Has 16 toggle switches, in two rows of 8, a momentary button, and 9 LEDs. When the button is pressed, the LEDs indicate the sum of the binary numbers encoded in the obvious way by the two rows of toggle switches.

• System B: Has 25 toggle switches, in three rows, of 8, 8 and 9, respectively, a momentary button, and 9 LEDs. When the momentary button is pressed, the LEDs indicate the positions of the toggle switches in the third row. The toggle switches in the first two rows are not electrically connected to anything.

These two systems seem to be clearly non-isomorphic. The first seems to be an 8-bit adder and the second is just nine directly controlled lights.

But now imagine that the systems come with these instructions:

• A: 8-bit adder. To use, move the toggle switches in the two rows to correspond to the bits in the two input numbers (down=1, up=0), and press the momentary button. The input state is only validly defined when the momentary button is pressed.

• B: 8-bit adder. To use, move the toggle switches in the two rows to correspond to the bits in the two input numbers (down=1, up=0), move the toggle switches in the third row to correspond to the bits in the sum of the two input numbers, and press the momentary button. The input state is only validly defined when the momentary button is pressed and the third row of switches contains the sum of the numbers in the first two rows.

There is now an isomorphism between valid inputs of A and B. Thus, the valid input of A:

• 00000001,00000001,momentary pressed

corresponds to the valid input of B:

• 00000001,00000001,000000010,momentary pressed.

Moreover, the outputs given the isomorphically corresponding valid inputs match: given the above inputs, both devices show (left to right) seven LEDs off, one LED on, and one LED off.

So it seems that whether A and B count as functionally isomorphic depends on what the instruction manuals specify as valid inputs. If the only valid inputs of B are ones where the third row of inputs corresponds to the sum of the first two, then B is an 8-bit adder. If that restriction is removed, then B is no longer an adder, but something much less interesting.

This point generalizes. Any computational system can be made isomorphic to a much simpler system with a more complex instruction manual.

This is all well and good if we are dealing with computers and software that come with specifications and manuals. But it is disastrous for the functionalist project. For the functionalist project is supposed to be a contemporarynaturalistic naturalistic account of our minds, and given naturalism, our brains do not come with specifications or manuals if contemporary naturalism is true. (If we have Aristotelian naturalism instead, we might get something akin to specifications or manuals embedded in our teleology.)

Objection 1: We need only allow those systems where the specification of valid inputs is relatively simple in a language whose linguistic structure corresponds to what is perfectly natural (Lewis) or structural (Sider), or only count as an isomorphism something that can be described in relatively simple ways in such a language.

Response: First, where is the line of the “relatively simple” to be drawn. Precise specification of the position of a toggle switch or water valve in the language of fundamental physics will be very complicated.

Second, System A is a bona fide electronic 8-bit adder. Imagine System A* is a very similar bona fide hydraulic 8-bit adder. It is very likely that a specification of what counts as a depressed toggle switch or an open valve in the language of microphysics is quite complex (just describing electricity or the flow of water in microphysics is really hard). It is also quite likely that the specification of one of these inputs is quite a bit more complex than the specification of the other. Let’s suppose, for simplicity, that A* is the system where the microphysical specification of how valid inputs work is quite a bit more complicated. Intuitively, fluid dynamics is further from the microphysics than electricity. Then the specification of the valid input states of System B may welll turn out to be closer in complexity to the specification of the valid input states of System A than that of the hydraulic A*. If so, then counting A* as isomorphic to A would force one to likewise count B as isomorphic to A.

Objection 2: The trick in the argument above was to use the notion of a valid input. But perhaps functional isomorphism needs a correspondence between all inputs, not just valid ones.

Response: This is implausible. Amongst invalid inputs to a human brain is a bullet, which produces a variety of outputs, namely death or a wide variety of forms of damage (and corresponding mutations of other behaviors), depending on the bullet trajectory. It is too stringent a requirement on an isomorph of the human brain that it should have the possibility of being damaged in precisely the ways that a bullet would damage a human brain, with exactly isomorphic mutations of behaviors.

More generally, the variety of invalid inputs is just too great to insist on isomorphism. Think of our electronic and hydraulic case. The kind of output you get when you press a toggle switch too hard, or too lightly, is unlikely to correspond to the kind of output you get when you open a valve too much, or too little, and such correspondence should not be required for isomorphism.

Conclusions: We need a manual or other source of specifications to talk of functional isomorphism. Functionalism, thus, requires a robust notion of function that is incompatible with contemporary naturalism.