This objection, more subtle and powerful than its predecessors, begins by bringing before us the argument as it stands given the dialectic to this point:
Now, the new objection runs as follows: ``The inference at ...But Theorem 1 implies that... is invalid, because Theorem 1 applies to abstract mathematical objects (of a sort studied in theoretical computer science), whereas is now, given the rebuttal to Objection 3, a physical object. To link the two as the argument now does, to ascribe to properties possessed by mathematical machines, is to commit a fallacy."
The first problem plaguing this objection is that Strong AI and Cog Sci are founded upon the notion that there is a level of analysis, in the study of the mind, which corresponds to computation. As Dennett  recently puts it:
The central doctrine of cognitive science is that there is a level of analysis, the information-processing level, intermediate between the phenomenological level (the personal level, or the level of consciousness) and the neurophysiological level (p. 195).
It may well be, for example, that at the neuromolecular level, computation is a dispensable notion. It certainly seems that at the phenomenological level computation is at least unwieldy, to say the least. [We routinely give exclusively ``folk-psychological" (= phenomenological) explanations of human behavior, as when, e.g., we say such things as that ``Bob left the room because he wanted to."] But the information-processing level, the level at which AI and Cog Sci is canonically done, is a different story: it's a level surely governed by results in computability theory; and one such result is none other than our Theorem 1.
There will doubtless be those inclined to sustain the fight for the position that an unbridgeable chasm separates computability theory from (and the like), so let's explore, in a bit more detail, the relation between such results and the ``real" world.
Some well-known theorems in computability theory, in particular those concerning uncomputability, clearly enjoy direct connections with (or at least have immediate implications concerning) the physical world. Consider, for example, the halting problem.
In order to briefly review this problem, we begin by writing
to indicate that TM M goes from input u through a computation that never halts. We write
when machine M goes from input u through a computation that does halt. And we write
when machine M goes from input u through a computation that prints out v and then halts.
Let the traditional property of decidability be handled by way of the symbols Y (``yes") and N (``no"). (For example, the problem of whether a given object a is a member of a set A is decidable iff some machine M' exists which is such that
Finally, let represent an encoding, in the form of a natural number, of machine M.
The halting problem now amounts to the theorem (call it `HP') that it's not the case that there is a TM M such that for every input u, and every :
Now that we have HP firmly on the table, let's return to our dialectic. The question before us is whether there is an unbridgeable chasm separating HP from the physical world. Put another way: Does HP apply to the physical world, as well as the mathematical?
The answer is an unwavering ``Yes." In fact, HP can be ``physicalized," in the sense that it can be reworded so as to make a most concrete assertion. One possibility is what we might call HP , which says that it's not physically possible that one build a TM M (or, if you like, a computer) such that for every input u (in the ``real world" sense of `input' attached to the word in, say, business and engineering applications), and every physical :
Since HP applies to the corporeal world in the form of HP the objection in question looks to be evaporating.
But perhaps it will be said, ``You've just been lucky with HP. This theorem does link the `Platonic' realm with the corporeal one; with this I agree. But this is a coincidence. Other theorems, and in fact perhaps the bulk of computability theory, stands inseparably apart from the physical realm. If I had to guess, I would say that HP turned out to apply because it's a negative result. Notice that you yourself turned first to uncomputability. Things will turn out differently for theorems that are not negative. And when things do turn out this way, we can return to scrutinize Theorem 1 and its relatives, which are themselves not negative."
Well, let's see if this is right; let's pick a theorem which isn't negative. Specifically, let's consider a fact mentioned above, viz.,
``Theorem" 2. Turing Machines with multiple tapes are no more powerful than standard one-tape TMs.
For any fixed natural number k, a k-tape Turing Machine has k two-way infinite tapes, each of which has its own read/write head. We assume that a k-tape TM can sense in one step the symbols scanned by all heads and, depending upon what those heads find, can proceed with standard actions (erase, write, move right or left).
Though a fully worked out proof of Theorem 2 is perhaps a bit daunting in its detail, the key idea behind the proof is actually rather simple. (Specifying the proof, once one grasps this key idea, is downright tedious.) It is that k tapes can be converted into one tape with k ``tracks." An example will make the idea clear: Suppose that we have a 3-tape TM whose configuration, at some moment, is captured by
Then the algorithm at the heart of Theorem 2 would simply convert this into one tape with three tracks (where the single-tape TM's alphabet is suitably composed), as in
Now, the question we face is whether
``Theorem" 2 . Physical Turing Machines with multiple tapes are no more powerful than standard one-tape physical TMs.
is true. And the correct answer would seem to be an affirmative one -- because it seems physically possible to follow the algorithm at the heart of the proof of Theorem 2 in order to convert a physical k-tape TM into a single tape physical TM.
Objection 4, then, is beginning to look like a dead end. There does appear to be a link between computability theory and the physical world sufficiently strong to undergird the Argument from Irreversibility. And this is just what we would expect, given that computability theory provides much of the mathematical framework for Computationalism.
At this point some may be inclined to sustain Objection 4 in the following manner: ``The two of you say that computation is reversible, and with this I heartily agree. But you also go on to point out that no physical process is reversible; and you then capitalize on this fact. But what you fail to appreciate is that at the information-processing level to which Dennett has drawn our attention consciousness is reversible. The problem is that you have surreptitiously moved at the same time to a level beneath this level, the level of Max and entropy and thermodynamical equilibrium, which is indeed a level where reversibility fails to apply. Your argument is nothing more than sleight-of-hand."
The problem with this objection is that it conveniently ignores the fact that Computationalism is wed not only to information processing, but also to agent materialism, the view that cognizers are physical things, and that therefore cognition is a physical process. In light of this, introducing at least elementary considerations from physics, as we have done, is not only natural, it's unavoidable. And, as we have shown, once these considerations are introduced, the Argument From Irreversibility is off and running.
The situation can be specified by returning to the adumbration of the argument we offered at the outset of the paper. There, as you may recall, we said that the argument would involve a quartet of propositions numbered (i) through (iv), now unpacked as
It's important to note that our conclusion needn't worry those who seek to pursue only ``Weak" AI, essentially the program satisfied with engineering intelligent systems -- systems not intended to replicate those properties (e.g., self-consciousness, command of a language, autonomy, etc.) traditionally thought to be constitutive of personhood. Weak AIniks are free to reject Proposition 1''. We return to this point at the end of the paper.