<HTML> <HEAD> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <META NAME="resource-type" CONTENT="document"> <META NAME="description" CONTENT="Some basic principles of mathematical Platonism are developed using the lambda-calculus. The Platonic forms are associated with Turing Computability, and the modern computational Platonism that results is contrasted with the views of Plato himself."> <META NAME="keywords" CONTENT="Plato, Plato of Athens, Platonic, Platonism, Parmenides, F.H. Bradley, Church, Turing, forms, ideas, metaphysics, Absolute, ontology, existence, idealism, objective idealism, absolute idealism, modal realism, Greek philosophy, mathematical platonism, computational platonism, formal system, lambda-calculus"> <META NAME="GENERATOR" CONTENT="Mozilla/4.03 [en] (Win95; U) [Netscape]"> <TITLE>Computational Platonism</TITLE> </HEAD> <BODY BGCOLOR="#FFFFFF"> <FONT FACE="Times New Roman,Times"><FONT SIZE=-1><B>&nbsp;</B>http://www.allanrandall.ca/Plato.html</FONT></FONT> <BR><FONT FACE="Times New Roman,Times"><FONT SIZE=-1><A HREF="copyright1996.html">Copyright</A> &copy; 1996, Allan Randall</FONT></FONT> <P><FONT FACE="Times New Roman,Times"><FONT SIZE=-1>&nbsp;</FONT></FONT> <CENTER> <H2> <FONT FACE="Times New Roman,Times"><FONT SIZE=-1>&nbsp;</FONT> Computational Platonism</FONT></H2></CENTER> <CENTER><B><FONT FACE="Times New Roman,Times">Allan F. Randall</FONT></B></CENTER> <CENTER><B><FONT FACE="Times New Roman,Times">Toronto, Ontario, Canada</FONT></B></CENTER> <CENTER><B><FONT FACE="Times New Roman,Times"><A HREF="mailto:research@allanrandall.ca">research@allanrandall.ca</A>, <A HREF="">http://www.allanrandall.ca/</A></FONT></B></CENTER> <CENTER><FONT FACE="Times New Roman,Times">&nbsp;</FONT></CENTER> <CENTER> <H3> <B><FONT FACE="Times New Roman,Times">Abstract</FONT></B></H3></CENTER> <BLOCKQUOTE><B><FONT SIZE=-1><FONT FACE="Times New Roman,Times">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, t</FONT>here 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.</FONT></B> <CENTER>&nbsp;</CENTER> </BLOCKQUOTE> <CENTER> <H3> <B>I. Introduction</B></H3></CENTER> Twenty-four hundred years ago, Plato proposed his Theory of Forms <A HREF="Plato/#fn0">[1]</A>. In it, he envisioned a world apart from our physical world of sensation--a world of "forms." These Platonic forms are not material objects, nor are they concepts in our heads--they exist on their own terms, apart from the physical universe, eternal and immutable. Physical objects are what they are by virtue of their participation in certain forms--reflecting these forms imperfectly in the material world. <P>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. <CENTER> <H3> II. The Platonic Theory of Forms</H3></CENTER> What distinguishes our physical world from complete chaos? What makes it intelligible (what the Greeks would call a "Cosmos")? Some intuitive answers might be "pattern," "structure," and "order." Plato felt the Forms were necessary to explain the structure he saw in the world around him <A HREF="Plato/#fn1">[2]</A>. The only reason the physical universe is intelligible at all is that different things take on the same form. Yet we need to explain where we get such concepts that declare completely different objects somehow "the same" with respect to this or that (height, color, catness, et cetera). No one would doubt that two cats are somehow instances of the same thing: <I>Cat</I>. But how can this be? How can two things made of completely different physical stuff, two things that are not at all identical, be the same with respect to Catness, but different with respect to, say, Color? It is easy to just shrug this off and say that the existence of the similarity is obvious, but that doesn't <I>explain</I> it. Where do <I>you</I> get the ability to tell that things are the same? Plato believes you have mental access to the Forms, which are, to put it simply, that which makes different things similar. <CENTER> <H3> <B>III. The Theory of Computable Forms</B></H3></CENTER> We will begin our attempt to mathematize the Theory of Forms by starting with something much like Set Theory, since this is an intuitive way of looking at the structure of things that most people are familiar with. Surely (we might say to ourselves) this will account for much of the variety and order we see around us. Let us start with three symbols: two "delimiters," <B>(</B>, <B>)</B> and one "atom", <B>a</B>. A larger alphabet of atoms can be used, but it is not necessary. For instance, instead of introducing a second atom <B>b</B>, we could just use (<B>aa</B>). In what follows, I will go ahead and use whatever characters I want for atoms, with the understanding that they could be reduced to single-atom expressions. In fact, we really don't need any special atom characters at all! If we want, we can use the empty set as our only true atom, building our alphabet up from that, so (()) is <B>a </B>and (()()) is <B>b</B>, et cetera. By bracketing these atoms into larger sets, which can contain atoms or further sets, we can create complicated structures. For example: <UL>(((())(()()()())(()()))((())((())()((()()()()()()()()))()(()))(())(())))</UL> Or, equivalently: <UL>((<B>adb</B>)(<B>a</B>(<B>a</B>()(<B>h</B>)()<B>a</B>)<B>aa</B>))</UL> It is important to remember that there is literally <I>no </I>difference between the above two expressions. The difference between (()) and <B>a</B> is nil--they are simply two alternate notations for the same thing. The above set could be interpreted either as a regular unordered set, where the order in which the elements are listed is irrelevant, or as an ordered set, where the order matters. Another name for an ordered set is a "list". <P>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 <I>seem</I> 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 <I>objects</I>. They are <I>particular </I>things, not abstract things. They do not account for sameness in difference.We are still just as baffled as ever as to why two <I>different</I> idealized Set Theory "cats" are the <I>same</I> with respect to Catness, even though they may be different with respect to Color. <P>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 <B>a</B> is purely a notational difference, not a difference in what the symbols <I>refer </I>to<I>. </I>On their own terms, then, all atoms <I>as things in themselves </I>are identical. It is only by putting these things together in a certain way that we can have two different things <I>in themselves. </I>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 <A HREF="Plato/#fn2">[3]</A>) 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 <A HREF="Plato/#fn3">[4]</A>. <P>There is an important sense in which any two sets cannot even be distinguished as <I>things-in-themselves. </I>Sets need an interpretation external to the set itself to make it a <I>something</I>. 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 (<B>a</B> (<B>a</B> <B>b</B>)), for example. What <I>is </I>this thing? Well, somehow, it is connecting together two separate things, <B>a</B> and (<B>a</B> <B>b</B>). One might interpret it as a <I>relation</I>: <B>a</B> ---related-to--> (<B>a</B> <B>b</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 <I>itself</I>, 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 <I>different</I> things at all. If all we really have is a collection consisting of <B>a</B> and (<B>a</B> <B>b</B>), and there is nothing more to it than that, then there really is <I>no</I> 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, (<B>a</B> <B>b</B>) seems to have the internal structure that <B>a</B> alone lacks. But this falls away when we try to examine what (<B>a</B> <B>b</B>) is <I>in itself. </I>We see that it is <I>also</I> just a collection of two things, <B>a</B> and <B>b</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? <P>The key to getting out of this mess is contained in that last line: "how can they be distinguished"? Two sets <I>cannot </I>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 <I>interpretation </I>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 <I>test </I>two sets <I>for </I>a difference, can we say they constitute things-in-themselves. <P>For instance, you might want to argue that a set of 2000 "voids" could still be said to represent the number 2000, and so <I>is</I> distinguishable from a single void, or 100 voids. But this cannot make sense unless the 2000 voids are being <I>counted. </I>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 <A HREF="Plato/#fn4">[5]</A> 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. <P>The logician Alonzo Church came up with a theory that <I>does</I> 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" <A HREF="Plato/#fn5">[6]</A>. He defined a special kind of ordered set, or list, called a "lambda-expression", introducing two new symbols: a dot `<B>.</B>' and the Greek letter <B>lambda</B>. 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"). <P>A lambda-expression has the following form: <BLOCKQUOTE><IMG SRC="fig1.gif" ALTERNATIVE="(<b>lambda</b> <b>x</b>.<b>P</b>)" HEIGHT=24 WIDTH=56> <BR>where <B>x</B> and <B>P</B> stand for S-expressions <A HREF="Plato/#fn6">[7]</A>.</BLOCKQUOTE> 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, <B>P</B> is an S-expression, which may (or may not) contain instances of the variable <B>x</B>. The <B>x</B>'s are "filled-in" when this form is given another S-expression, call it <B>I</B>, to "fill in the blank" wherever an <B>x</B> appears in <B>P</B>, producing a new filled-in S-expression, call it <B>O</B>. This filling-in is called "evaluating" the form, or sometimes "applying", "simplifying" or "reducing": <BLOCKQUOTE><IMG SRC="fig2.gif" ALTERNATIVE="((<b>lambda</b> <b>x</b>.<b>P</b>)<b>I</b>) --evaluate-> <b>O</b>" HEIGHT=19 WIDTH=164></BLOCKQUOTE> For example, if Jane Doe were filling in an application form, the part that asks for her name could be formalized as follows: <BLOCKQUOTE><IMG SRC="fig3.gif" ALTERNATIVE="((<b>lambda</b><b>____ .</b> &qt;<b>Name: ____</b>&qt;) (<b>Jane Doe</b>)) <b>--</b>evaluate<b>-></b> (&qt;<b>Name: </b>(<b>Jane Doe</b>)&qt;)" HEIGHT=20 WIDTH=474></BLOCKQUOTE> So the "lambda" part of the form, to the left of the period, simply defines the "blank" to be filled in. To the right of the period is the expression containing the "blanks" that actually get filled in with whatever further expression the form is "applied to", appearing to the right of the lambda-expression. The lambda-expression is sometimes just called a "form", but we will call it a "lambda-form" to distinguish it from Platonic forms (the two may or may not be equivalent). <P>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 "<B>____ </B>Doe". <BLOCKQUOTE><IMG SRC="fig4.gif" ALTERNATIVE="((<b>lambda</b><b>____.&qt;Name: ____&qt;</b>) (<b>lambda</b><b>____.</b>(<b>____ Doe</b>))) <br> <b>--</b>evaluate<b>-></b> (<b>&qt;Name: </b>(<b>lambda</b><b>____.</b>(<b>____ Doe</b>))<b>&qt;</b>)" HEIGHT=39 WIDTH=300></BLOCKQUOTE> Notice that although the form evaluates to an expression that contains yet another form, no further evaluation is possible, since the internal form is not being applied to anything--nothing is given to fill in the blank. Furthermore, this final expression cannot itself be applied to anything, since it is just a regular concrete S-expression <I>not </I>a lambda-expression. However, it does <I>contain </I>a lambda-expression. This form within a form could be applied internally, within the S-expression, to either "Jane", or any of the first names in Jane's family, such as that of her husband John. Evaluation would then require two steps before giving a result that cannot be further evaluated: <BLOCKQUOTE><IMG SRC="fig5.gif" ALTERNATIVE="( (<b>lambda</b><b>____.&qt;Name: ____&qt;</b>) ((<b>lambda</b><b>____.</b>(<b>____ Doe</b>)) <b>Billy</b>) ) <br> <b>--</b>evaluate<b>-> </b>( (<b>lambda</b><b>____.&qt;Name: ____&qt;</b>) (<b>Billy Doe</b>) ) <br> <b>--</b>evaluate<b>-> </b>( <b>&qt;Name: </b>(<b>Billy Doe</b>)<b>&qt;</b> )" HEIGHT=61 WIDTH=352></BLOCKQUOTE> Again, we end up with a final S-expression that is not a lambda-expression: not abstract, but concrete. From a Platonic perspective, this concrete object lacks objective reality; it is not anything in itself. So it is the "filling-in" of the blanks, the evaluation, that is "real", not the end result, which has no reality without the lambda-substitution. In this case, the final result is completely concrete; it does not even contain a lambda-expression within it. But even if it <I>did </I>contain a lambda-form deep within its structure, it would only be <I>that </I>form that is a thing-in-itself. The rest of the concrete object is void. Such void S-expressions that are <I>not</I> lambda-expressions I will call "C-expressions". Think "C" for "concrete" (as opposed to abstract) mathematical object. Just as "lambda-form" refers to the thing-in-itself, rather than our particular notation for it, "C-form" could be used to refer to the thing-in-itself represented by a concrete C-expression. We would then have two kinds of "S-forms": "C-forms" and "lambda-forms". But remember my claim that only lambda-forms are things-in-themselves. C-forms are non-entities; they do not exist in themselves, apart from our interpretation of them. I am claiming that a C-expression cannot coherently be separated from the entire context of the scratches and markings on paper, my seeing of those markings and my understanding them in a certain way. Lambda-expressions, however, <I>can</I> be so separated (although even this position has its problems, which we will see later). <P>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 <I>never</I> halt. Note that "never halting" does <I>not </I>mean "requires an infinite number of steps to halt". The latter is a nonsensical statement--there <I>are </I>an infinite number of steps in the evaluation, so the lambda-form <I>never </I>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 <I>has no </I>final result <A HREF="Plato/#fn7">[8]</A>. <P>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 <I>test-for-sameness </I>on two different set-objects. <P>You may have guessed by now that the lambda-calculus can also be viewed as a simple and elegant model of computing: <BLOCKQUOTE><IMG SRC="fig2.gif" ALTERNATIVE="((<b>lambda</b> <b>x</b>.<b>P</b>)<b>I</b>) -> <b>O</b>" HEIGHT=19 WIDTH=164> <BR><B>P</B> is the "program". <BR><B>I</B> is the "input", or "environment" in which the program is run. <BR><B>O</B> is the "output" or "value" of the computation.</BLOCKQUOTE> In fact, the popular computer programming language LISP is a version of the lambda-calculus <A HREF="Plato/#fn8">[9]</A>. 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. <BR>&nbsp; <CENTER><B>Some Church-Turing Theses</B></CENTER> <BLOCKQUOTE>1. All <B>computable</B> <B>things</B> correspond to a lambda-form. <I>(Mathematical Version)</I> <BR>2. All <B>physical</B> <B>things</B> correspond to a lambda-form. <I>(Physical Version)</I> <BR>3. All <B>thinkable</B> <B>things</B> correspond to a lambda-form. <I>(Cognitive Version)</I> <BR>4. All <B>things</B> correspond to a lambda-form. <I>(Ontological Version)</I></BLOCKQUOTE> There is general agreement in information science that anything computable (i.e. mechanical) falls under the lambda-calculus <A HREF="Plato/#fn9">[10]</A>. 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, <I>and</I> you believe that the mind is a mechanism, then any <I>thing</I> at all falls under the lambda-calculus. Later, we will ask which, if any, of these Church-Turing Theses Plato himself would have adopted. Provided we remain uncommitted on the issue of whether the human mind is "physical," thesis #2 seems uncontroversial enough that we will accept it as a premise for the remainder of the paper. It seems to capture what most people mean when they talk about something being "physical" as opposed to perhaps "supernatural" or "mental" <A HREF="Plato/#fn10">[11]</A>. <P>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 <I>need </I>some kind of Church-Turing interpretation, since they cannot by themselves express all possibilities. So we see that the dissolution into void--the <I>incoherence</I>--of static sets, is intimately tied in with the lack of comprehensiveness. If we could show Church-Turing to be wrong, that there <I>are </I>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. <CENTER> <H3> IV. Are Platonic Forms lambda-Forms?</H3></CENTER> We are now in a position to consider the lambda-calculus as an alternative to Platonic Forms. If system is what is essential to objective reality, it is imperative that we decide what version of the Church-Turing Thesis to accept--this will greatly influence our whole metaphysics. I will tackle this question by asking what Plato himself would have thought of this Theory of Computable Forms. Is this theory consistent with Plato's Forms, or did Plato have something else entirely in mind? Is it consistent, but only within the domain of a weak version of Church-Turing? I will start by giving a run-down of the most important features of Platonic Forms. <H4> The Independent Existence of Forms</H4> The lambda-forms are purely mathematical and hence immaterial. Like the integers, they exist mathematically whether we are around in the real world to contemplate them or not. "2+2=4" would be a mathematical truth, Plato tells us, even if there were no humans to think about it. The same is true of statements about the lambda-forms. In fact, if we impose an alphabetical ordering on our alphabet, the S-expressions can be enumerated--counted out just like the integers. We could even write a lambda-program to do this (obviously a non-halting one). Let us assume, then, the following (arbitrary) ordering: <BLOCKQUOTE><IMG SRC="fig6.gif" ALTERNATIVE="<b>( )</b> <b>a</b> <b>.</b> <b>lambda</b>" HEIGHT=17 WIDTH=79></BLOCKQUOTE> The notion of mathematical existence (and hence mathematical truth) is very close to Plato's notion of the existence of forms, whether he saw them as mathematical or not (as we will see, he certainly thought at least <I>some</I> of them were). There <I>exists</I>, given the above alphabetical ordering, an S-expression #12398387. Whether or not anyone ever bothers to calculate what that expression is does not change the fact that it exists, and has a precise and consistent definition. But, one might argue, concrete S-expression also exist in this fashion. That is true, but <I>only </I>under a counting process or procedure, a program to enumerate them. To make the concrete thing real, we see again, a process, an abstraction, an interpretation, a <I>form, </I>must be brought in. <P>But, that means of course that lambda-forms, if they are truly things-in-themselves, must contain process as <I>part of</I> 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 <I>is</I>. Only in that sense does this number refer to something external to our own minds. But then this particular S-expression is <I>only real to the extent that it is considered part of the entire enumeration, or counting, process.</I> <P>One of the results of computability theory is that there exist lambda-expressions which we can refer to <I>exactly</I> as precisely as we referred to S-expression #12398387, but for which we cannot ever compute the full expression, even in principle <A HREF="Plato/#fn11">[12]</A>. If we accept this as a mathematical truth, it is difficult to see how <I>something </I>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 <I>particular</I> 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 <I>was </I>a complete Form, something that went beyond mathematics. <H4> Innate Knowledge of Forms</H4> You may wonder how the "eternal" and "immutable" Platonic forms could ever correspond to something like the lambda-calculus, which we just chose arbitrarily. After all, we did not have to choose a language as simple and elegant as the lambda-calculus. We could have chosen anything from a traditional computer language like FORTRAN to English to number theory itself <A HREF="Plato/#fn12">[13]</A>. These languages are all comprehensive, or "universal". There really is nothing special about the lambda-calculus in particular. The only way the lambda-forms can be equivalent to the Platonic forms is if we view the lambda-calculus as a language for talking about the forms, but recognize that <I>the details of the language are completely irrelevant to the nature of the forms themselves</I>, provided the language is at least as expressive as the lambda-calculus. Any such comprehensive languages, having this expressive power, qualifies as "Turing-equivalent". A one-to-one correspondence, or "isomorphism", can be drawn between the programs of any two Turing-equivalent languages. This is what establishes the mathematical independence of lambda-forms from the language in which they are described. <P>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. <P>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 <I>do </I>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. <H4> Unity in difference</H4> One of the most important features of Plato's forms is that they account for similarity in difference--they are abstract. The question Plato is really asking here is no less than "Why is there order in the Cosmos instead of random chaos?" Order requires forms because it requires unlike things to be like. The intelligibility of the universe demands forms. <P>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 <B>BEAUTIFUL(x,y)</B>, which is actually some frighteningly complicated lambda-expression. Now if <B>BEAUTIFUL(x,y)</B> is taken to mean "<B>x</B> is more beautiful than <B>y</B>", and the lambda-expression was written with some arbitrary S-expression taken to be <B>TRUE</B> and another taken to mean <B>FALSE</B>, then an evaluation of the beauty of two women might go something like this: <BLOCKQUOTE><IMG SRC="fig7.gif" ALTERNATIVE="(<b>lambda</b> (xy).(<b>BEAUTIFUL(</b>x,y<b>)</b>)) (<b>Helen</b> <b>Jane</b>) -> <b>TRUE</b><br>Or:<br> (<b>lambda</b> (xy).(<b>BEAUTIFUL(</b>x,y<b>)</b>)) (<b>Helen</b> <b>Aphrodite</b>) -> <b>FALSE</b>" HEIGHT=36 WIDTH=438></BLOCKQUOTE> Thus, Helen takes part in the form "Beautiful" <I>and</I> the form "not Beautiful" (or "Ugly," if we want to be less kind to Helen): sameness in difference. Similar lambda-programs could be written for "Greater than" and "Smaller than," but they would be much simpler <A HREF="Plato/#fn13">[14]</A>. <H4> The Divided Line</H4> It is <I>only</I> the lambda-forms that can account for similarity. The C-forms are just concrete, nonabstract mathematical objects. They just sit there. Without the process involved in evaluation, there is no way to compare things to determine similarity. Even if two huge S-expressions differ by only a single symbol, without a lambda-form to operate on them, where is the justification for saying their S-forms are "similar"? In some other language, they might even differ by many more characters. <I>Looking at the similarity in the S-expressions doesn't tell you anything about similarity between the forms they represent</I>, since such a procedure is not language-independent (unless we really <I>do </I>want to talk about the S-expressions in terms of the particular notation we are using, but then we are not talking about their objective existence). There is simply no way to call one similar to the other unless you make use of a lambda-form. You cannot use a C-form for this purpose, since there is no way to apply one C-form to another. Only lambda-forms can map the Many onto the One. <P>This distinction fits well with Plato's analogy of the divided line <A HREF="Plato/#fn13">[15]</A>. Plato divides the world of <I>epist&ecirc;m&ecirc;</I> (Knowledge) into two sections: <I>dianoia</I> (Thought) and <I>no&ecirc;sis</I> (Understanding). <I>Dianoia </I>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 <I>dianoia </I>to objects of<I> no&ecirc;sis. </I>Bradley would say objects higher up have more "system". As forms become more lambda-abstract, they refer to forms of forms of objects of <I>dianoia</I> and forms of forms of forms of objects of <I>dianoia</I>, until you get to the Superform, the Form of the Good, from which all the forms derive. <P><I>Epist&ecirc;m&ecirc; </I>is knowledge of S-forms, <I>dianoia </I>is knowledge of C-forms and <I>no&ecirc;sis </I>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. <P>We could label the top half of the divided line, then, in two different ways: the modern computational way, and the ancient Platonic way: <BLOCKQUOTE><IMG SRC="line.gif" ALTERNATIVE="C-forms -> lambda-forms -> The Universal Language<br>Concrete Mathematical Objects -> Forms -> The Good" HEIGHT=33 WIDTH=499></BLOCKQUOTE> <H4> The Form of the Good--the Absolute</H4> One difference between our computable forms and Plato's forms is that Plato believes in an ultimately absolute form with "maximum system". My formulation follows Bradley in rejecting that idea. The so-called Form of the Good does not actually exist, except perhaps as an unreachable limit. A mathematician would say that the "Good" or the "Absolute" was an "open boundary" <A HREF="Plato/#fn14">[16]</A>. <P>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 G&ouml;del'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. <P>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 <I>epist&ecirc;m&ecirc; </I>of this would mean <I>epist&ecirc;m&ecirc; </I>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. G&ouml;del 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 <I>are </I>true to me; they are clear and distinct in the mind. <H4> Immateriality, Noncomposite Forms and The Immortality of the Soul</H4> The correspondence we have established so far between the Theory of Forms and the Theory of Computable Forms relies on the notion that all immaterials (things we can have <I>epist&ecirc;m&ecirc; </I>of) <I>are</I> computables. There is no aspect of them that could not be simulated in a computer program. Our preceding arguments seem to indicate that, if we accept this (which requires that we accept Church-Turing Theses #'s 1-3), then the two "formal" theories are essentially identical. <P>However, if there are immaterial things, of which we can have <I>epist&ecirc;m&ecirc;, </I>and which are <I>not</I> 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). <P>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. <P>&nbsp; <CENTER><IMG SRC="materials.gif" ></CENTER> <P>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 <I>epist&ecirc;m&ecirc; </I>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 <I>just </I>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. <P>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. <P>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 <A HREF="Plato/#fn15">[17]</A>. <P>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. <P>I should stress that there is a real difference between the four kinds of incoherence we have talked about: <I>(1)</I> the nonsensical nature of incomputable and material things,<I> (2)</I> the nonexistence of void concrete objects <I>in themselves</I>, <I>(3) </I>the lesser reality of abstract, nonvoid forms that fall short of full comprehensiveness and <I>(4) </I>the absolute unjustifiability of even full comprehensiveness. <P>In <I>(1), </I>we are talking about "things" that are just not things at all, <I>even relative to our own private language. </I>Even as relative things, they are incoherent. This is really the only kind of incoherence that justifies an accusation of inconsistency. In <I>(2)</I>, however, we are talking about things which <I>can </I>be precisely defined <I>relative to </I>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 <I>(1)</I>. In <I>(3), </I>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. <I>(2)</I> removes language from the thing talked about, while <I>(3)</I> 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 <I>(4) </I>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. <P>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 <A HREF="Plato/#fn16">[18]</A>. 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 G&ouml;delians, 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 <A HREF="Plato/#fn17">[19]</A>. 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. <P>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. <H5> <I>The Affinity Argument</I></H5> In the affinity argument <A HREF="Plato/#fn18">[20]</A>, Plato is essentially saying that "it takes one to know one." Only an immaterial can know an immaterial. Objects of thought are immaterial. Therefore, the mind must be immaterial if it is to know these objects of thought. This sounds like an argument against a computable soul, but the real meat of his argument really does not go that far. He talks about "visibles" (materials) and "invisibles" (immaterials). But a lambda-mind <I>is</I> immaterial and invisible, yet nonetheless computable. This kind of mind is not some kind of inscrutable non-mechanical object without any parts, yet it <I>still</I> is immaterial. If you doubt that a computable mind would be immaterial, ask yourself what happens after all the atoms in your body have eventually been replaced with new ones. You are no longer the same physical stuff, but you are still <I>you</I>. It is your pattern--your participation in the forms--that makes you <I>you</I>. Or imagine that your brain state is read by a sophisticated machine that can download your mind into a robot. This would be a completely different physical entity from you <I>now</I>, but it would have your memories, thoughts and desires. Surely this would still be <I>you</I>. Some people are convinced by this argument, but others feel they are more than just a computational process--more than a lambda-form--and require a nonmaterial entity that is more than just an "emergent property" of the way material things are arranged. <H5> <I>The Harmony Argument</I></H5> The harmony argument <A HREF="Plato/#fn19">[21]</A> makes this implication more explicit. Here, Plato argues directly that the soul cannot be an immaterial emergent property of material things. The analogy is of a musical instrument, which produces a harmony. The harmony is an immaterial emergent property, yet it dies when the instrument is smashed. The soul is a nonmechanical, noncomposite entity without parts, and does not die. Plato argues that a soul consisting of emergent properties cannot oppose the materials that it emerges from, since it is no more than a mathematization of the arrangement of those material things. This is unlike the real soul, which can actively oppose the body's tendencies. <P>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 <I>includes</I> the brain. The emergent property emerges from the <I>entire</I> 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.<I> </I>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 <I>and</I> brain. That is, the soul only opposes <I>part</I> of the whole body, and it certainly does not oppose the part from which it emerges. <P>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 <A HREF="Plato/#fn20">[22]</A>. 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. <CENTER> <H3> V. Conclusion</H3></CENTER> The Theory of Computable Forms relies on the assumption of the computational nature of human cognition. This is a highly controversial debate, and one on which a lot of research dollars in Artificial Intelligence are spent. I do not have room to explore its intricacies, but Plato seems to be on the side of the non-computationalists. Parmenides, who motivated much of Plato's work, seems more in line with modern computationalism. We could say that the Computational Theory of Forms is a Parmenidean version of Plato's theory. So in a sense, some of its implications actually predate Plato. But to Plato, our theory is necessarily incomplete, even though it may well be part of what is going on in the Platonic Theory. The other difference in the two theories is Plato's insistence on the completeness of Form, and a Form of all Forms. However, Plato's whole system would probably not suffer much if we replaced the Good with an open boundary Absolute, and called it "comprehensiveness" as discussed earlier. <P>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. <BR>&nbsp; <CENTER> <H2> Notes</H2></CENTER> <A NAME="fn0"></A>[1] The Platonic "Theory of Forms" referred to in this paper is gleaned mostly from the "Meno," "Phaedo," and "Republic"--particularly the "Phaedo." These are early to middle works in Plato's career. The translations used are: Plato, <I>Five Dialogues</I>, G.M.A. Grube (Trans.), Hackett, Indianapolis, 1982 and Plato, "Republic," G.M.A Grube, C.D.C. Reeve (trans.), Hackett, Indianapolis, 1992. <P>&nbsp;<A NAME="fn1"></A>[2] See: Plato, "Timaeus". In: <I>Timaeus and Critias</I>, D. Lee (Trans.), Penguin Books, London, 1977. <P>&nbsp;<A NAME="fn2"></A>[3] F.H. Bradley. <I>Writings on Logic and Metaphysics</I>. Clarendon Press, Oxford, 1883, 1893, 1897, 1914, 1922, 1935, 1994. <P>&nbsp;<A NAME="fn3"></A>[4] Parmenides. On Nature, Allan F. Randall (Trans.). http://www.allanrandall.ca/Parmenides/, Toronto, c. 475 BC, 1996. <P>&nbsp;<A NAME="fn4"></A>[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. <P>&nbsp;<A NAME="fn5"></A>[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. <P><A NAME="fn6"></A>[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." <P>&nbsp;<A NAME="fn7"></A>[8] Although it is possible the expression could approach a particular S-expression in the limit. However, it takes <I>another </I>lambda-expression referring to the nonhalting one to establish this. <I>In itself </I>the lambda-expression just produces an infinite sequence of new expressions. <P>&nbsp;<A NAME="fn8"></A>[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. <P>&nbsp;<A NAME="fn9"></A>[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. <P>&nbsp;<A NAME="fn10"></A>[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. <P>&nbsp;<A NAME="fn11"></A>[12] This result is known as G&ouml;del's Incompleteness Thereom, and it applies to any formal system with the expressive power of the lambda-calculus. <P>&nbsp;<A NAME="fn12"></A>[13] Another result from G&ouml;del is that number theory is in this class of languages. <P>&nbsp;<A NAME="fn12"></A>[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 <I>in themselves </I>they are void...nothing. Bradley explains this by making the extremely important point that Truth, <I>as </I>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 <I>is </I>true. Falsehood requires the evaluation of one form to set up within it a model of another form and to simulate <I>its </I>evaluation, and draw conclusions on the results, <I>based on an interpretation.</I> <P>&nbsp;<A NAME="fn13"></A>[15] Republic 509e-511e. <P>&nbsp;<A NAME="fn14"></A>[16] For example, the open interval <B>(1,2]</B> 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 <I>very </I>tempted to talk <I>about </I>2 as if it were real, since he can imagine approaching it in the limit. <P>&nbsp;<A NAME="fn15"></A>[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. <P>&nbsp;<A NAME="fn16"></A>[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. <P>&nbsp;<A NAME="fn17"></A>[19] Phaedo 78b-c. <P>&nbsp;<A NAME="fn18"></A>[20] Phaedo 78b-84b. <P>&nbsp;<A NAME="fn19"></A>[21] Phaedo 86a-86d, 91d-95a <P>&nbsp;<A NAME="fn20"></A>[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 <I>many</I> lambda-programs that contain whatever it is that makes you <I>you</I>. 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. <P> <HR WIDTH="80%"> <CENTER><I>Go Back to <A HREF="">Allan Randall's Home Page</A>.</I></CENTER> <CENTER>&nbsp;</CENTER> </BODY> </HTML> <!-- FILE ARCHIVED ON 12:40:51 Jun 9, 2007 AND RETRIEVED FROM THE INTERNET ARCHIVE ON 15:59:16 Nov 21, 2012. JAVASCRIPT APPENDED BY WAYBACK MACHINE, COPYRIGHT INTERNET ARCHIVE. ALL OTHER CONTENT MAY ALSO BE PROTECTED BY COPYRIGHT (17 U.S.C. SECTION 108(a)(3)). -->