Copyright © 1996, Allan Randall

Plato's theory of forms is developed and compared to the modern theory of recursion. I show how Plato's theory, as it applies to mathematical objects, is essentially a primitve version of modern recursion theory, which has all the essential elements of the ancient theory. However, Plato himself thought there was more than mathematics to his forms. He believed that form had a noncomposite, unanalyzable component. So, while recursion theory provides an adequate formalization of Plato's theory, it cannot be considered identical to it. I argue--drawing from material in the "Meno," "Phaedo," and "Republic"--that Plato's arguments for noncomposite form are largely fallacious. Plato would, I believe, have taken the computational version developed here seriously, since mathematics was his primary source for clear examples of forms (one could argue that it was his only source short of dipping into mysticism). In fact, there is a long-standing oral tradition that Plato developed a more formal version of his theory in lecture notes for courses he taught at the Academy (the university he founded in Athens). Any such notes, if they existed at all, have been lost. But if such a formal version did exist, it is tantalizing to wonder to what extent Plato may have anticipated modern theoretical computer science and metamathematics.

Mathematical entities are often used as examples of such immateriality. Would "1+1" not equal "2", whether a material universe existed or not? Using modern concepts of computability, I will sketch what a purely mathematical version of Plato's Theory of Forms might look like--I will call it the Theory of Computable Forms. I will then ask whether this new theory is the same as Plato's theory (and if not, how it differs). I will argue that it is in Plato's discussion of the immortality of the soul that the biggest differences arise between the two theories. Although Plato felt mathematics was important to his theory, he seems to have held out, at least, for a nonmathematical soul. Such arguments have their modern parallel in the position against artificial intelligence. I will argue that what philosophy and science today need, if the remaining riddles of the universe are to be solved, is something very much like a computational version of the Platonic Theory of Forms.

- (((())(()()()())(()()))((())((())()((()()()()()()()()))()(()))(())(())))

- ((

One could imagine creating all kinds of complex mathematical structures
out of this simple method of bracketing atoms. Yes (we again say to ourselves)
perhaps any structure at all, including all that we see in the world around
us, can be expressed as some such set. But on further reflection, we discover
that this Theory of Sets is not sufficient. Platonic forms explain "sameness
in difference"--how different things can be said to be somehow the same.
The forms are abstractions, "one on many", as they explain how it is that
different objects can be similar, mapping one object to many forms, and
one form to many objects. But with this Set Theory, we still have these
same puzzles, just in the realm of mathematics rather than physical objects.
Certainly, these objects *seem* more abstract, like Platonic Forms,
than physical objects. They are mathematical and as such are immaterial
and not tied to the sensible world. But they are still just mathematical
*objects*. They are *particular *things, not abstract things.
They do not account for sameness in difference.We are still just as baffled
as ever as to why two *different* idealized Set Theory "cats" are
the *same* with respect to Catness, even though they may be different
with respect to Color.

To Plato's mind, this robs such "concrete" objects of reality. Only
abstract things are truly real. To see the sense in this, note the two
different representations above of the same set. It is easy to see that,
on their own terms, these two structures are completely equivalent. The
use of (()) instead of **a** is purely a notational difference, not
a difference in what the symbols *refer *to*. *On their own terms,
then, all atoms *as things in themselves *are identical. It is only
by putting these things together in a certain way that we can have two
different things *in themselves. *But there are convincing arguments
why even these complex sets are not things-in-themselves. Most such arguments
(for example, those of F.H. Bradley [3]) are in one
way or another a variation on Plato's point that structure ultimately has
to do with "sameness in difference", with abstraction. Indeed, Plato's
arguments are themselves derived from those of his predecessor, Parmenides
of Elea [4].

There is an important sense in which any two sets cannot even be distinguished
as *things-in-themselves. *Sets need an interpretation external to
the set itself to make it a *something*. Even complex sets that seem
to have structure ultimately suffer the same "dissolution into unreality"
that befell the two apparently "different" atoms above. Take the set (**a**
(**a** **b**)), for example. What *is *this thing? Well, somehow,
it is connecting together two separate things, **a** and (**a** **b**).
One might interpret it as a *relation*: **a** ---related-to-->
(**a** **b**). Or we might want to say instead that the set is simply
collecting the two things together, in a grab-bag of symbols, so to speak.
But in terms of the set *itself*, these are just different ways of
describing the same thing. The problem in treating a set as a thing-in-itself
is in defining what makes these collected things *different* things
at all. If all we really have is a collection consisting of **a** and
(**a** **b**), and there is nothing more to it than that, then there
really is *no* intrinsic difference between these two things. They
are just different atoms, in themselves equivalent to the empty set (sometimes
called "void"). But, one might argue, (**a** **b**) seems to have
the internal structure that **a** alone lacks. But this falls away when
we try to examine what (**a** **b**) is *in itself. *We see
that it is *also* just a collection of two things, **a** and **b**,
which have no distinction in themselves. This argument can be used to dissolve
any set (or relational structure, which is the same thing) away to nothing.
And if all a set is is a collection of nothings repeated over and over
again, what right have we to say it is anything at all? How can a collection
of indistinguishable unconnected entities be said to be any different than
just one such entity?

The key to getting out of this mess is contained in that last line:
"how can they be distinguished"? Two sets *cannot *be distinguished
without a process or procedure that acts on the structures in some way
in order to distinguish them, or to treat them differently. But this is
an *interpretation *of the set external to the set itself. One interpretation
might distinguish two sets as different, while another will see them as
identical. Only when such an interpretational process is brought in, so
that we can actually *test *two sets *for *a difference, can
we say they constitute things-in-themselves.

For instance, you might want to argue that a set of 2000 "voids" could
still be said to represent the number 2000, and so *is* distinguishable
from a single void, or 100 voids. But this cannot make sense unless the
2000 voids are being *counted. *Only under the "counting" interpretation,
which is not in the definition of the set itself, can the sets be distinguished.
Indeed, we know from Computability Theory [5] that sets
are insufficient to account for everything we would normally consider computable,
or thinkable. There is something going on in the physical world, and in
our thinking, that simply is not reflected in this set theory model. It
has to do with process. And we have seen that process, like Platonic abstraction,
can create sameness in difference--it can generalize. Indeed, we will see
that process and abstraction are actually equivalent.

The logician Alonzo Church came up with a theory that *does* account
for anything computable. He called it the "lambda-Calculus." In many ways,
it can be considered an extension to the basic set theory just developed.
To set theory, Church added the missing piece of the puzzle: "lambda-abstraction"
[6]. He defined a special kind of ordered set, or list,
called a "lambda-expression", introducing two new symbols: a dot `**.**'
and the Greek letter **lambda**. The sets that can be constructed out
of this raw material are both of the traditional concrete kind, and the
new abstract "lambda" kind. Taken together, these sets are sometimes called
S-expressions (S is for "symbolic", but you could also think "set").

A lambda-expression has the following form:

One way to think of this is literally as a "form", as in an application form you might fill out, that asks for your name, address and personal information. This is a very common and straightforward example of abstraction. A single health insurance form can be filled in with many different combinations of names and addresses. When John Smith and Jane Doe fill in the same application form, they end up with two different particular instantiations of the same general form. To instantiate the general form into a particular filled-in form, they fill in the blanks. Each blank is a "variable". A lambda-expression is just a set with blanks, or variables, to be filled in. In the above form,

wherexandPstand for S-expressions [7].

Lambda-forms get much more interesting when we allow Jane to fill in
the "Name" blank with yet another form! Perhaps she wants the application
form to apply to the entire Doe family, so in the "Name" blank, she enters
"**____ **Doe".

If a lambda-form is complicated enough, its evaluation can require many
more than just two steps before halting like this. In fact, it might *never*
halt. Note that "never halting" does *not *mean "requires an infinite
number of steps to halt". The latter is a nonsensical statement--there
*are *an infinite number of steps in the evaluation, so the lambda-form
*never *halts. There is no final result that we could see if only
we could evaluate the form for an infinite amount of time. An infinite
evaluation simply *has no *final result [8].

It is easy to see that the process of substitution at each evaluation
step is analogous to physical time, and thus we could call it "lambda-time".
The result of each step depends on the previous result, so evaluation can
be thought of, not just as abstraction, but also as a model of causation
or mechanism. So we see that "abstraction", "causation" and "mechanism"
are just different words for the same thing. All will allow us to perform
a *test-for-sameness *on two different set-objects.

You may have guessed by now that the lambda-calculus can also be viewed as a simple and elegant model of computing:

In fact, the popular computer programming language LISP is a version of the lambda-calculus [9]. It is generally believed that this simple model can account for anything computable (and perhaps anything at all). The "Church-Turing Thesis" expresses this idea, and it is commonly taken as a Truth in information science, although it has never been proven in any of its incarnations. Following are three versions of the Thesis, each with an increasingly liberal notion of what is covered by the lambda-forms. The original intent of Church and Turing probably did not go beyond version #1 or 2.Pis the "program".Iis the "input", or "environment" in which the program is run.Ois the "output" or "value" of the computation.

1. AllThere is general agreement in information science that anything computable (i.e. mechanical) falls under the lambda-calculus [10]. If you believe the physical universe operates on mechanism, then the lambda-calculus can also describe physical reality. If you believe that your mind is mechanistic as well, then the lambda-calculus can account for anything thinkable. If you believe there is nothing coherent, or meaningful, that cannot in principle be thought of by the human mind,computablethingscorrespond to a lambda-form.(Mathematical Version)

2. Allphysicalthingscorrespond to a lambda-form.(Physical Version)

3. Allthinkablethingscorrespond to a lambda-form.(Cognitive Version)

4. Allthingscorrespond to a lambda-form.(Ontological Version)

One important question arises: even given that the Church-Turing Thesis
is true, and the lambda-calculus is "comprehensive" (i.e. it can describe
anything describable, within one of the above domains), could it not also
"dissolve" into void, as our simple sets did? This dissolution amounted
to a realization that we really had no idea what we meant by "grouping"
things together into "sets" in the first place. Do we have any better idea
of what it means to "substitute" one thing for another thing, or for one
thing to "cause" another, or be "transformed" into another? These are just
different ways of describing lambda-substitution, and it really is ultimately
as unknowable as "grouping". The difference that allows us to "fudge" this
and call lambda-forms "things-in-themselves" is Church-Turing comprehensiveness,
or "Turing Equivalency"--the idea that any two languages that are comprehensive
can be somehow mapped onto each other, or made to mimic each other. Set
grouping by itself dissolves into void because one can produce different
Church-Turing interpretations of different sets that yield completely different
results, and sets *need *some kind of Church-Turing interpretation,
since they cannot by themselves express all possibilities. So we see that
the dissolution into void--the *incoherence*--of static sets, is intimately
tied in with the lack of comprehensiveness. If we could show Church-Turing
to be wrong, that there *are *things beyond the lambda-forms, then
they would suffer the same dissolution. F.H. Bradley called this single
criterion of comprehensiveness/coherence the criterion of "system". What
has more "system" is closer to being real. This is in its essence what
Platonism is all about.

But, that means of course that lambda-forms, if they are truly things-in-themselves,
must contain process as *part of* their definition, not as an external
counting or other process. S-expression #12398387, then, is not real in
itself, but the lambda-form under which it is so defined *is*. Only
in that sense does this number refer to something external to our own minds.
But then this particular S-expression is *only real to the extent that
it is considered part of the entire enumeration, or counting, process.*

One of the results of computability theory is that there exist lambda-expressions
which we can refer to *exactly* as precisely as we referred to S-expression
#12398387, but for which we cannot ever compute the full expression, even
in principle [12]. If we accept this as a mathematical
truth, it is difficult to see how *something *of this S-expression
is not truly self-existent. Results like this give many mathematicians
the feeling that mathematical objects have Platonic, objective existence.
But as we have seen, it is not the S-expression itself as a concrete object
that is real, but its embedding in a larger process. This is true of C-expressions
and lambda-expressions! The lambda-expression as a *particular* form
is in itself not truly real unless embedded in a larger process. By "larger
process", of course, I mean "more comprehensive lambda-form". Even individual
lambda-forms, then, are only "closer" to truly being real than the C-forms.
Only the full comprehensiveness of a Turing Equivalent lambda-form is truly
real. And even then we only say this because we cannot conceive of anything
more complete. We will see later, that while this whole structure fits
well with Plato's Forms, Plato himself thought there *was *a complete
Form, something that went beyond mathematics.

But once again, the "translation" between these languages is itself arbitrary; we only accept it as somehow "objective" and "absolute" because of its comprehensiveness. In fact, in an objective sense, there can be no need to "translate"--what is real is what all such languages have in common. Nothing else is truly real.

So why do we accept this translatability, this absolute nature of computability,
or abstraction, or mechanism, or structure or whatever you want to call
it? Even though comprehensive languages are not absolutely knowable on
their own terms, we accept their reality because we can directly experience
this in our minds. We "just know" that mathematics and logic somehow *do
*make sense, even if we cannot justify them in terms of themselves.
This direct "innate" knowledge of forms was an important, almost mystical,
aspect to Platonism.

Our lambda-forms certainly do account for this kind of similarity in
difference, at least within their domain of computable things. The lambda-form,
being a computer program, can be "programmed" to test for any kind of difference
we want. A single lambda-form can evaluate to the same thing for many completely
different input expressions. Assume that a value, or virtue, like "beauty"
can somehow be mathematized. Label the result **BEAUTIFUL(x,y)**, which
is actually some frighteningly complicated lambda-expression. Now if **BEAUTIFUL(x,y)**
is taken to mean "**x** is more beautiful than **y**", and the lambda-expression
was written with some arbitrary S-expression taken to be **TRUE** and
another taken to mean **FALSE**, then an evaluation of the beauty of
two women might go something like this:

This distinction fits well with Plato's analogy of the divided line
[15]. Plato divides the world of *epistêmê*
(Knowledge) into two sections: *dianoia* (Thought) and *noêsis*
(Understanding). *Dianoia *is closer to the world of sensibles, including
all objects of thought, such as concrete mathematical objects, which do
not account for similarity and unity. However, these entities are still
above the crucial sensible/nonsensible (i.e. material/immaterial) dividing
line. It is implied there is a continuity up the line from objects of *dianoia
*to objects of* noêsis. *Bradley would say objects higher
up have more "system". As forms become more lambda-abstract, they refer
to forms of forms of objects of *dianoia* and forms of forms of forms
of objects of *dianoia*, until you get to the Superform, the Form
of the Good, from which all the forms derive.

*Epistêmê *is knowledge of S-forms, *dianoia *is
knowledge of C-forms and *noêsis *is knowledge of lambda-forms.
lambda-forms that are closer to the Good are more abstract lambda-forms,
whose lambda-expressions contain many layers of lambda-expressions nested
within lambda-expressions nested within lambda-expressions. C-forms are
at the bottom of this hierarchy. All of the lambda-forms are ultimately
grounded in C-forms, in that it is the C-forms that are manipulated by
the lambda-forms, even though the C-forms are literally nothing without
the lambda-forms.

We could label the top half of the divided line, then, in two different ways: the modern computational way, and the ancient Platonic way:

It is tempting to be Platonic in our approach and try to construct an absolutely ultimate form. But modern metalogic has shown conclusively, in results like Gödel's incompleteness thereom, that such a form is nonsensical. Any attempt to construct one will imply there is something outside of it. This fundamental incompleteness of the forms stems from the ever-present thorn in our side: our inability to fully justify our translation between comprehensive languages, and thus our very notion of comprehensiveness itself. This arbitrary translation, while clear and distinct in our minds, is still an interpretation, which implies an external perspective.

In a sense, we really can talk about Turing Equivalency, or Universal
Language, since we can build within one comprehensive language, a model
of any other. So the logical lambda-interpretation of the Form of the Good
would be a lambda-form that itself is able to represent and interpret "lambda-expressions"
of some kind. In fact, it is easy to build such an interpreter. Why call
this Plato's Form of the Good? Because *epistêmê *of this
would mean *epistêmê *of all possible lambda-forms--total
and complete knowledge of everything there is (assuming for now that there
are no incomputable things). But this assumes the language we are speaking
in is modelled within itself. And this is, in absolute terms, nonsensical.
Any such translation between a language and something expressed within
that language will assume a further perspective outside the language. It
is an interpretation of the language. Gödel showed that this means
any such form-of-all-forms will be incomplete. The Absolute, then, is not
itself a form. Forms cannot be ultimately, metaphysically explained in
terms of forms. Thus, most Platonists accept some kind of Cartesian justification:
"I think therefore I am". Mathematical truths just *are *true to me;
they are clear and distinct in the mind.

However, if there are immaterial things, of which we can have *epistêmê,
*and which are *not* computable (i.e. we reject Church-Turing Thesis
#3), then the Theory of Computable Forms is just one aspect--an incomplete
picture--of the full Theory of Forms (a kind of mini-version that excludes
immaterial incomputables).

It is in his arguments concerning the immortality of the soul that Plato seems to take stock of these issues. We will look at two of the more relevant arguments. But first, the following diagram may help sort out the difference between materials, immaterials, computables and incomputables.

Note first that there are computable materials (physical things)--the
normal physical objects we see in the world. These are below the divided
line--there can be no *epistêmê *of these objects, since
all our minds can really contemplate are objects of thought (there is always
some chance our connection to the external world is faulty). Indeed, any
attempt to define physical things as things-in-themselves apart from the
forms suffers dissolution into nothingness, as we saw for static sets.
Even if we try to say a physical thing is a physical instance of a form,
recognizing that it cannot be static and concrete in itself, we are still
left helpless to say what it is about the thing that makes it physical
and not *just *formal. Any attempt to do so inevitably leads to void.
Since all we can ever really talk about or think about are computable things,
which are not in themselves physical, we can never actually define what
makes a physical thing physical. This is Parmenides' Principle, that what
is real is only what can be spoken or thought of (i.e. what is computable).
However, as we will see, Plato did not subscribe to this principle.

Then there are computable immaterials. These are mechanical entities, just like physical objects, but they are immaterial mathematical versions. These correspond to all that is above the divided line, in the Theory of Computable Forms. Since they are computable, and do not have any invisible "physical" tags on them, they are closer to being real than physical things. But even so, the concrete mathematical objects, near the bottom of the top part of the divided line, are less real than Universal Language, or the Form of the Good, which is the ultimate abstraction, containing all possibility within it. Other forms are more real the closer they come to this Universality. This idea that the more possibilities that are included, the more real a thing is, is called the Principle of Plenitude, another concept which Plato inherited from Parmenides.

Next, there are immaterial incomputables. If you have rejected Church-Turing Thesis #3, then you would consider the mind to be an example of one of these. If you accept Thesis #4, then you do not believe this area of the diagram has any meaning at all. The Theory of Computable Forms as I have developed it takes this latter stance. Only the intersection of the two circles above--the computable immaterials--are things which can be talked about or spoken of, so they only are real. Even they are only real when considered as an aspect of Universal Language, but the other areas in the diagram do not even qualify for this ability to "participate" in the forms, since they cannot be grounded in the forms at all. Physicality and incomputability are nonsensical [17].

Finally, there is the unlabelled region outside the circles. This would correspond to material incomputables, but we have already decided to assume there is no such thing. We decided by definition that anything incomputable about the world would be considered nonphysical. However, if instead we decided to call some of these incomputables "physical", then this new category, "incomputable materials" would be another nonsensical category.

I should stress that there is a real difference between the four kinds
of incoherence we have talked about: *(1)* the nonsensical nature
of incomputable and material things,* (2)* the nonexistence of void
concrete objects *in themselves*, *(3) *the lesser reality of
abstract, nonvoid forms that fall short of full comprehensiveness and *(4)
*the absolute unjustifiability of even full comprehensiveness.

In *(1), *we are talking about "things" that are just not things
at all, *even relative to our own private language. *Even as relative
things, they are incoherent. This is really the only kind of incoherence
that justifies an accusation of inconsistency. In *(2)*, however,
we are talking about things which *can *be precisely defined *relative
to *a form. It is only when we try to imagine that they are coherent
on their own that our idea of them becomes completely incoherent as in
*(1)*. In *(3), *however, we are talking about things which,
under the assumption of the Ontological version of Church-Turing, are truly
objectively self-existent. But we are recognizing that they, in a sense,
fall short of the full reality of comprehensiveness, the Absolute. However,
they have the Absolute built into their nature and are defined in terms
of it; whereas concrete objects can be defined without including abstraction
in their definition at all. *(2)* removes language from the thing
talked about, while *(3)* retains the full comprehensiveness of language
in the thing itself. Nonetheless, we recognize that a particular form,
in itself, cannot be thought of as independent of the Universal Language,
in which it is expressed, without losing self-existence. In *(4) *we
are being total sceptics and recognizing that the Ontological Church-Turing
Thesis cannot be ultimately defended rationally, and thus nothing is really
coherent on its own terms. If we take this view, all three previous kinds
of incoherence reduce to an equal incoherence: the incoherence of Form
itself. This sceptical attitude can be tempting, but it seems to deny the
one thing we can all directly experience: our own thinking.

Plato himself did not take the modern view that the Forms are intrinsically incomplete. However, he did not see them as limited to mathematics or Turing Computability. He thought there was more to structure than what can be thought of or spoken of, and so went against his predecessor Parmenides. Recall the Jane/Helen/Aphrodite example used earlier. If we had used 2/3/4 and the forms "Greater than" and "Less than", we could actually have written the relevant lambda-programs in short order [18]. The forms involved here are obviously mathematical. However, it is less than clear whether "Beauty" really could be mathematized like this. The same goes for the other virtues: justice, piety, et cetera. To many people, it seems a tall order to expect this from mere symbols, when we are dealing with such uniquely human traits. But do not dismiss the power of the lambda-forms too lightly. Recall that they can be as complex as you want. For instance, if you accept Church-Turing Thesis #2, that the physical universe is mechanical, then you could feasibly simulate the entire physical universe in a lambda-program. Out of all the myriad complex lambda-forms, could there not be some wildly strange looking, complex and abstract lambda-expression for the Form of Virtue? Plato thought not. He generally liked mathematical interpretations of the Forms, and he used mathematical examples all the time. But he seemed to draw the line at the mind, the human soul. To him there was something ultimately "noncomposite", or nonmechanical, about the soul. So Plato in a sense came to the same conclusion concerning the ultimate unknowability of the Forms as modern Gödelians, except that the Platonic tendency was to see these "noncomposite" aspects less as a limitation or incompleteness, and more as a real knowable thing, but one that just happens to be nonmechanical. Our minds, after all, said Plato, are themselves noncomposite, so there is no reason they cannot know something noncomposite [19]. Personally, I think it makes more sense to view the mind as strictly mathematical, and hence mechanical, but to view mathematics itself as ultimately unjustifiable.

Much of Plato's argument for the immortality of the soul in the Phaedo seems to amount to a claim that the mind cannot be mechanized, although it is not entirely clear to me that this has to follow from his arguments. I will look at two of these arguments, which are especially relevant today: the affinity argument and the harmony argument.

This argument, it seems to me, is confused. If the soul is an emergent
property of the body, then this material body the soul emerges from *includes*
the brain. The emergent property emerges from the *entire* body, including
the cognitive part, which is what is opposing the other part. In no way
does the emergent computable soul oppose the material stuff from which
it emerges.* *It only opposes the noncognitive part of the body. It
would be nonsensical to claim that it opposed the entire body-brain system.
Plato is assuming the very division between body and soul that he is trying
to prove. It is not the body minus the brain that the soul emerges from--it
is body *and* brain. That is, the soul only opposes *part* of
the whole body, and it certainly does not oppose the part from which it
emerges.

So Plato's arguments that the Forms need to be incomputable, or "noncomposite", are unconvincing, and we should not feel compelled to drop the idea of a computable immaterial soul [22]. His arguments against materiality, however, are more compelling. It seems likely that Plato's own conception of the soul was of an incomputable immaterial (something which I claim to be incoherent). So the Theory of Computable Forms is not identical to the Platonic Theory.

There is so much about the two theories that is in harmony, that it
may not be all that much of a stretch to imagine that Plato would have
taken the Computable version seriously, although he would probably have
considered it an incomplete picture. The essential feature of sameness
in difference is there in both theories. There is a long-standing oral
tradition that Plato developed a more formal version of his theory in lecture
notes for courses he taught at the Academy (the university he founded in
Athens). Any such notes, if they existed at all, have been lost. But if
such a formal version did exist, it is tantalizing to wonder to what extent
Plato may have anticipated modern theoretical computer science and metamathematics.

[2] See: Plato, "Timaeus". In: *Timaeus and
Critias*, D. Lee (Trans.), Penguin Books, London, 1977.

[3] F.H. Bradley. *Writings on Logic and Metaphysics*.
Clarendon Press, Oxford, 1883, 1893, 1897, 1914, 1922, 1935, 1994.

[4] Parmenides. On Nature, Allan F. Randall (Trans.). http://www.allanrandall.ca/Parmenides/, Toronto, c. 475 BC, 1996.

[5] For a readable introduction to the computability concepts used in this paper, see Roger Penrose, "The Emperor's New Mind," Oxford U. Press, New York, 1989. For a more technical treatment, see G.J. Chaitin, "Algorithmic Information Theory," Cambridge U. Press, Cambridge, 1987.

[6] Throughout this paper, I will refer to lambda-expressions as being "lambda-abstract" or just "abstract." This is a purely mathematical abstraction, not a mental abstraction. The meaning of "abstract" here is the mathematician's meaning: one mathematical object which covers, or subsumes, many others. The psychologist's sense of a "mental abstraction" or "concept" is not intended.

[7] The bold-face characters are place-holders for S-expressions, but are not themselves S-expressions. They are a way of talking about S-expressions, and are called, along with the S-expressions containing them, "meta-expressions," or "M-expressions." Most so-called S-expressions you see in print are actually M-expressions. An M-expression that stands for a lambda-expression is called a "function."

[8] Although it is possible the expression could
approach a particular S-expression in the limit. However, it takes *another
*lambda-expression referring to the nonhalting one to establish this.
*In itself *the lambda-expression just produces an infinite sequence
of new expressions.

[9] In fact, I have used the terminology of LISP in this paper. The actual terms used vary somewhat from one version of the lambda-calculus to another.

[10] The exceptions are those who believe there should be literal infinities allowed for. Although the evaluation process can go on forever, an S-expression itself can never be infinite in size (although there is no limit to its size). Although an evaluation can go on forever, there is no S-expression that is the result of an infinite evaluation--the evaluation simply never halts. Discussing this issue further would be beyond the scope of this paper, so we will just assume for now that the lack of this type of literal infinity does not hamper the expressiveness of the lambda-calculus.

[11] I do not myself make any such distinction, but it is one that many people do make, so I will leave it an open question for now.

[12] This result is known as Gödel's Incompleteness Thereom, and it applies to any formal system with the expressive power of the lambda-calculus.

[13] Another result from Gödel is that number theory is in this class of languages.

[14] One problem that some might see in this
is the arbitrary declaration of what is TRUE and FALSE. Here, we define
them as atomic, and thus *in themselves *they are void...nothing.
Bradley explains this by making the extremely important point that Truth,
*as *something opposed to False, is ultimately incoherent, void. Absolute
truth is just the totality of all lambda-forms, which in themselves just
evaluate. As self-contained things, there is no room for Falsehood. Any
new expression produced from a previous expression just *is *true.
Falsehood requires the evaluation of one form to set up within it a model
of another form and to simulate *its *evaluation, and draw conclusions
on the results, *based on an interpretation.*

[16] For example, the open interval **(1,2]**
has a closed boundary of 1, and an an open boundary of 2. There are an
infinite number of points on the interval, of which 1 is included, but
2 is not. From the point of view of a line segment living strictly within
this interval (if we agree to be fanciful), 1 exists, but 2 literally is
void. However, Mr. Line Segment will be *very *tempted to talk *about
*2 as if it were real, since he can imagine approaching it in the limit.

[17] This may seem to contradict my earlier claim that there are forms which we can label, yet can never compute. Thus, these are "real things" that are incomputable. But note that the form is still a computable. It is not the form itself that cannot be expressed as a computation, it is a computation that produces the form. It is this second level of computation that is incoherent, not the self-existence of the form itself. To talk of a procedure to compute such a thing is talk of void.

[18] Writing the necessary arithmetical operations with a version of the lambda-calculus as pristine as ours would be a bit tedious (especially if we restricted ourselves to a 1-atom alphabet), but it is well established that it is doable. (You might want to try it as an exercise.) Any Turing-equivalent language can do arithmetic.

[22] This is not to say that an argument for
immortality could not be made based on the lambda-soul. If all possible
lambda-forms exist, then unless your continued survival in "Form-space"
is a logical inconsistency (like a square circle, for instance), there
are bound to be *many* lambda-programs that contain whatever it is
that makes you *you*. So even if your physical instance dies, there
could be many multiple "you"s surviving out there in the mathematical space
of S-forms. But Plato did not, to my knowledge, address his theory in this
manner.