Mirror Notation: Symbol Manipulation without Inscription ... · Mirror notation illustrates...

Mirror Notation: Symbol Manipulation without Inscription ManipulationAuthor(s): Roy A. SorensenSource: Journal of Philosophical Logic, Vol. 28, No. 2 (Apr., 1999), pp. 141-164Published by: SpringerStable URL: http://www.jstor.org/stable/30226667 .

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ROY A. SORENSEN

MIRROR NOTATION: SYMBOL MANIPULATION WITHOUT INSCRIPTION MANIPULATION

ABSTRACT. Stereotypically, computation involves intrinsic changes to the medium of representation: writing new symbols, erasing old symbols, turning gears, flipping switches, sliding abacus beads. Perspectival computation leaves the original inscriptions untouched. The problem solver obtains the output by merely alters his orientation toward the input. There is no rewriting or copying of the input inscriptions; the output inscriptions are numerically identical to the input inscriptions. This suggests a loophole through some of the computational limits apparently imposed by physics. There can be symbol manipulation without inscription manipulation because symbols are complex objects that have manipulatable elements besides their inscriptions. Since a written symbol is an ordered pair of consisting of a shape and the reader's orientation to that inscription, the symbol can be changed by changing the orientation rather than inscription. Although there are the usual physical limits associated with reading the answer, the computation is itself instantaneous. This is true even when the sub-calculations are algorithmically complex, exponentially increasing or even infinite.

KEY WORDS: algorithmic complexity, computation, Cambridge event, duals, mirror, NP-completeness, symbol manipulation, Turing machine

Instructions on how to perform some logical operations with a mirror are

presented below. The basis of the technique is a notation that ensures that the mirror image of any well formed formula is itself a well formed formula. These reflected formulas are the correct answers to questions posed with the original formulas.

Mirror notation illustrates perspectival computation. The stereotypical form of computation involves intrinsic changes to the medium of representation: writing new symbols, erasing old symbols, turning gears, flipping switches, sliding abacus beads. Perspectival computation leaves the original inscriptions untouched. The problem solver merely alters his orientation toward the input. There is no rewriting or copying of the input inscriptions; the output inscriptions are numerically identical to the input inscriptions.

Perspectival computation is symbol manipulation without inscription manipulation. Symbols are complex objects that have manipulatable elements besides their inscriptions. Since a written symbol is an ordered pair of consisting of a shape and the reader's orientation to that inscription, the

Journal of Philosophical Logic 28: 141-164, 1999. @ 1999 Kluwer Academic Publishers. Printed in the Netherlands.



142 ROY A. SORENSEN

symbol can be changed by changing the orientation rather than inscription. For instance, a parking lot attendant rotates his -+ sign to direct traffic left

+- or right --, is using a single inscription token to form opposite symbol types.

The motive for manipulating a symbol's orientation rather than its shape is that the shape is harder to alter. Writing a line through 1111, replacing 51 by 63, or even just appending a 0 to 4, requires physical energy, not a mere decision. Put more dramatically, the perspectival calculator side-steps the physical constraints imposed by the manipulation of inscriptions (as in digital computing) or the manipulation of physical models (as in analog computing). For instance, since the input symbols need not be causally altered, the speed of the computation is not constrained by Einstein's principle that no signal can travel faster than the speed of light.

Although there are the usual physical limits associated with reading the answer, the perspectival computation itself has no duration because nothing substantive is done to the data. The transformation of -+ from part of a right turn symbol to part of a left turn symbol is a 'Cambridge event' (Geach, 1969, 71). Given this non-causal aspect of perspectival computation, 'faster than light' performance is achieved even when the sub-computations are algorithmically complex, exponentially increasing or even infinite. This raises the hope of a loophole through looming com-

putational limits (Garey and Jonson, 1979).

1. REFLECTIVE TRUTH TABLES

The dual of a logical connective is computed by exchanging true for false and vice versa, for every line in a truth table - including the initial columns. For instance, the dual of conjunction is disjunction:

Original Dual

A H Aand H A H Aor H

T T T TF TF TF

T F F TF FT FT

F T F FT TF FT

F F FT FT FT



SYMBOL MANIPULATION WITHOUT INSCRIPTION MANIPULATION 143

The computation is faster if we change notation. Let conjunction be / and disjunction be \. And let true be > and false be <. The table for

conjunction is then

ARH A/H > > >

> < <

< > <

< < <

If you hold this new table up to the mirror, you obtain the dual of conjunction. So instead of changing twelve truth values by hand, you obtain the dual with a single operation of mirror reversal.

The mirror makes some conceptual points observational. A look in the mirror reveals that the dual of negation is negation. (Let's signify negation by the' mark. I realize that a bar above the letter would avoid the distracting right side to left side mirror switch. But I am working with a pre-mirror- notation font.)

A A'

>I<

< >

General principles of duality also take on a welcome directness. We can 'see' that the dual of any valid schema must be inconsistent because mirror reversal will change a pure > column into a pure < column.

The mirror corrects common misconceptions about duals. Students who are introduced to duality via a mirror are less likely to confuse duality with negation. For they realize that a single plane mirror reverses the whole field, not selected columns.

2. MULTIPLE DIMENSION MIRROR NOTATION

Negation can be performed by mirror if we increase our stock of symbols with true A and false v, while still retaining our old symbols > for true and < for false



144 ROY A. SORENSEN

o x O/X A A>

A V <

V A <

V V <

Our new symbols A and v, are vertically symmetric i.e. they look the same when turned 180 degrees about their vertical axis. Therefore, the mirror does not change the appearance of the first two columns. The mirror only switches the orientation of the column to be negated. The dual of O/X can be calculated by turning the truth table upside down. For A and v are horizontally asymmetric, i.e. each looks different when turned 180 degrees about its horizontal axis. When an observer views > and < upside down, his change of perspective switches what counts as being to the left and right of him. Hence the upside down perspective changes left and right along with up and down. Since one negates by mirror reversal and duals by inverting up/down, one can calculate the negation of the dual of

conjunction by viewing the mirror image of the table upside down. Notice that the introduction of a second sign for truth does not commit

us to a double-truth theory. Both A and > mean the same thing even though they have distinct syntactic roles. Also notice that we can dispense with the mirror in favor of side-views. The vertical line I only changes when viewed from a side. By letting I be true and the horizontal line - be false, we get an equivalent table:

o x O/X A AI A V

VA

V V

We start from a southern perspective with conjunction (AND). The view from the west yields the dual of conjunction (OR) while the view from the north yields the negation of this dual (NOR). The view from the east is the negation of the dual (NAND). Hence, this two-dimensional truth table

expresses four distinct truth functions. The expressive power of a flat truth table can be doubled by reintro-

ducing the mirror. Let I indicate truth exactly when it is vertical (north) or tilted northeast /. With A, the table now uses three symbols for truth:




ox O/X A^A

Av\

VA

v v\

This two-dimensional truth table expresses eight distinct truth functions. Prior to the introduction of the mirror, we could only operate along two axes.

3. COMPLEX SYSTEMS OF REFLECTION

Reflecting the mirror image of the truth table into a second mirror shows that the dual of a dual is equivalent to the original formula. Images can be doubled with a reflection from a pair of mirrors set at a slight angle. Positioning the mirrors opposite each other creates an infinite regress of smaller and smaller images.

The mirrors themselves can be sophisticated. An infinite variety of curved mirrors can be constructed from six basic surfaces: plane, convex, concave, convex cylinder, concave cylinder, and the saddle shaped (Thomas, 1980). Some of these mirrors turn the image upside down, others rotate the image 90 degrees, still others multiply or annihilate images. And, usefully, some of the mirrors do nothing at all. Topologists accurately predict these effects. This has enabled them to construct matrix mirrors that achieve complex image effects.

By sophisticating the mirrors and increasing their numbers, complex reflection systems miniaturize, multiply, divide, blend, distort, and can- cel images. One can envisage perspectival computing devices that would involve the subtle optical geometry that governs telescope design. Mir- rors seemed unlikely scientific instruments even to early developers of reflective telescopes. These 'parlor trick' devices were first sketched but left unbuilt. Then built but used half-heartedly. Isaac Newton constructed the first serious reflective telescope out of desperation. He (mistakenly) believed that chromatic aberration of lenses would never be overcome (Gregory, 1997, 160). Soon reflective telescopes totally supplanted direct view telescopes.

The range of current telescope designs includes flexible membrane mirrors and liquid mirrors formed from flat pools of mercury or pools curved by rotation. There is also a futuristic proposal to create a giant gaseous mirror to be used for a spaceborne radio telescope:



146 ROY A. SORENSEN

Two lasers would be used to produce a standing wave field of light within a disk-shaped volume of hundred of meters or perhaps several kilometers across. Gas molecules trapped in the field by photon pressure would act as a reflection hologram to radio waves and thus focus them onto a conventional radio telescope receiver. Such a structure may seem utterly fantastic to us but can you imagine trying to explain a rather conventional radio telescope array or a far UV telescope in space to, perhaps, Isaac Newton? (Manly, 1991, 19)

However, we should walk before we blast off. Sticking with a simple tool confers a methodological advantage. Recall how geometers restricted themselves to figures that could be drawn with a straight edge and compass. Accordingly, I confine myself to reflections performable with a single, plane mirror. In Euclidean space, this is not much of a limit because any change you make by rotating or re-locating ('translating') a figure can also be done through a finite number of successive reflections. (Bas van Fraassen has an accessible proof of this theorem on pages 262-263 of Laws and Symmetry.) In addition to restricting myself to one mirror, I shall work with a single symbol for truth and concentrate on duality.

4. MIRROR NOTATION FOR SCHEMAS

Most laws of duality concern direct relationships between schemas. For instance, the first law of duality concerns any schema Si that is built up of sentence letters solely by the following connectives: negation, conjunction, and disjunction. Perspectival computation requires that invari- ance be achieved through reflection symmetry. So we restrict ourselves to reflectively symmetrical sentence letters such as A and H.

Now consider the schema S2 that results from substituting conjunction / with disjunction \ and vice versa. The first law states that S1 and S2 are duals. The law confirms our truth table result that (A \ H) and (A / H) are duals.

Well, strictly speaking, the mirror says (H / A) is a dual of (A \ H). Disjunction and conjunction are commutative operations, so anything that is a dual of 'H or A' is a dual of 'A or H'. This distracting commutation effect could be eliminated by writing vertically rather than horizontally. For instance, the vertical variant of the reflective truth table is

A > > < <

H> < > <

A

\ > > > < H




Suppressing commutation preserves customary order. But I shall forego this advantage of familiarity in favor of the greater typographical conve- nience offered by horizontal mirror notation. Happily, there is no logical difference between vertical and horizontal mirror writing. Schemas are duals of each other by virtue of a relationship between their truth-functions. The syntactic properties of two schemas only make them duals by virtue of a connection with this more fundamental property. Since infinitely many schemas have the same truth function, 'Which schema is the dual of S?' falsely presupposes uniqueness.

The first law excludes schemas that contain symbols for material implication and the biconditional. We dispense with these symbols in favor of their definitions in terms of the accepted symbols:

A materially implies H = A' \ H

A if and only if H = (A / H)\(A' / H')

Thus the first law of duality lets us test whether S2 is a dual of S 1 by formulating S 1 in accordance with the above method and checking whether it is equivalent to S2. Mirror notation is suited to detect those rare formulas that are self-dual, such as (A / H) \ (A / M) \ (H / M).

The second law of duality (which holds for schemas regardless of their connectives) says that you get a dual if you negate each of the letters and also the whole schema. This law follows from the definition of duality. After all, to get a dual of a schema, you swap all the truth-values for the sentence letters and for the schema as a whole. Applying the second law to a conjunction A / H / ... / M delivers the distributively negated dual: (A' / H' / ... / M')'. That is, the mirror image of A / H/ / ... / M is logi- cally equivalent to (A' / H' / ... / M')'. Or to put the point in more familiar notation,

(A v H v v M) +-> (^A& - ,H &...

& & M).

This constitutes a proof of DeMorgan's first law i.e. a disjunction holds if and only if it is not the case that each disjunct is false. Parallel argument proves DeMorgan's second law: a conjunction is true if and only if it is not the case that one of its conjuncts is false. Thus the mirror seconds the common observation that DeMorgan's laws are duality principles.

The second law could be made more vivid by using a notation to effect distributive negation. This requires us to address each sub-sentence explicitly by either affirming it with f or denying with 4. Hence, the old X' / O would read t (4 X / t O). The upside down perspective negates 4 X into X t (remember t appears indifferently to the right or left of the sentence letter just like the ' in X' and 'X). Hence, inverting t (4. X / " 0) distributively negates each component sentence along with the compound



148 ROY A. SORENSEN

sentence itself. So by the second law of duality, the upside down inversion of t (4 X / t O) yields a dual that is equivalent to the mirror image inversion of the same formula. Now we have two independent ways to obtain a dual of a formula.

Our 'dual duals' can help us establish interesting equivalences. Con- sider an enriched language where (/x) is used for the universal quantifier and (\x) for the existential quantifier. One way of obtaining duals of the following four formulas,

t (/x) t Hx t (/x) Hx 4 (/x) t Hx 4 (/x) - Hx,

is by turning them upside down. This done, we can apply our second method of deriving duals, mirror imagery, to obtain the dual of these duals. Since the dual of a formula's dual is truth functionally equivalent to the

original formula, it follows that the above formulas are truth-functionally equivalent to mirror images of their upside-down inversions. (This pair of operations can be coalesced into the single operation of holding the mirror

perpendicular to the page.) Since their mirror image formulas feature existential quantifiers rather than the original universal quantifiers, we have just established the standard quantifier negation equivalences familiar from

predicate logic. The above point can be extended to modal logic. Since necessity be-

haves like a universal quantifier with respect to possible worlds, we can mimic the above results via the symbols // (for necessity) and \\ (for possibility). For instance, 4 // f H looks like t \\ H when upside down in a mirror. This shows 'It is false that it is necessary that H is true' is

equivalent to 'It is possible that H is false'. The third law of duality says that a schema is valid if and only its dual is

inconsistent. As mentioned earlier, this must hold because a valid schema has a pure column of >'s. Reversing that column yields a pure column of

<5s.

According to the fourth law, S1 entails S2 if and only if the dual of S2 entails the dual of S 1. The duals of S 1 and S2 behave exactly like S1 and S2

except that their truth values are interchanged. So if there is no assignment of truth values that makes S1 true and S2 false, there is no assignment that makes the dual of S1 false and the dual of S2 true. Hence, the dual of S2 must entail the dual of S 1. Thus a proof in mirror notation can be turned into a distinct proof by reading its mirror image from bottom to top.

(H / I) / (X \O) (H / I) H

H\X




Usually the fourth law of duality is presented as a way of getting a free theorem without the trouble of going through a proof. But if the original theorem was proved in mirror notation, the dual proof is automatically encoded.

The mirror proofs based on the fourth law require that each new step follow solely from its predecessor. Proofs that have steps in which a new line is drawn with the help of non-immediate predecessors will not validly invert: For instance, there is an apparent invalid transition from H \ X to X for the bottom-up inversion of the following proof.

(H / I) / (X / O) (H / I) H

(X / 0)

H/X The solution is to reconstruct the proof in tree form. The premises for each inference occur immediately above the conclusion and no formula is a premise for more than one inference. For example, the above proof converts into a tree via the introduction of conjunctive and disjunctive branches:

(H / I) / (X / O)

H/I X/O

H X

H/X

Forks diverge when a connective is eliminated and converge when a connective is introduced. Conjunctive forks branch from a / node. Disjunctive forks branch from a \ node. The mirror reverses these nodes, thereby preserving the validity of the dual argument.

The fifth law of duality follows from the fourth law: schemata are equivalent exactly when their duals are equivalent. Thus a proof that uses only



150 ROY A. SORENSEN

rules of replacement will yield an equally valid proof for the dual when held to a mirror. Notice that this proof does not need to be read bottom-up. An equivalence rule proof yields two dual counterparts. The reasoning in the mirror will be valid top down (by virtue of the equivalences and the fifth law) and bottom up (by virtue of the implication relations and fourth law).

The last three laws are labor savers. They let logicians infer new results (validities, inconsistencies, implications, equivalences) from established results without new analysis. For instance, dual counterparts of all the results obtained from computations using normal schema hold once conjunction is interchanged with disjunction.

Interest in the economies offered by duals emerged early in the history of computing. Ralph Slutz (1976), one of the pioneers of American computing, knew that OR-gates required fewer diodes (or equivalently, vacuum tubes) than AND-gates. Inspired by a study of Boolean algebras, Slutz tried to simplify the design of the Standard Electronic Automatic Com- puter, SEAC. (SEAC began running in 1950 and was the first American electronic stored program machine.) Slutz proposed that every AND-gate be replaced by a OR-gate, and vice versa, and then re-interpreted pulse as false and no pulse as true. Although Slutz calculated that this achieved a 30% saving, he had trouble explaining his 'dual machine' to potential users and operators. Consequently, his clever re-design was not implemented for SEAC.

5. TRUTHDIALS AND MANY-VALUED LOGIC

In many-valued logic, propositions may receive intermediate truth values. For instance, in a three valued logic, a statement can receive a value intermediate between 1 (true) and 0 (false), namely .5 (usually taken to express indefiniteness). This requires a new rule for computing negation. The usual rule is to let the negation of p equal the absolute value of 1 minus the value of p. So if p has a truth value of .5, its negation has a value of .5. We might accommodate this possibility by designating the value of .5 with a downward arrow, v. Our table for negation would be

A A

> <

< >

V V




The standard rule for computing conjunction is to assign it the value of its lowest conjunct. Disjunctions get the value of their highest disjunct. Hence, the mirror image of this matrix style truth table for conjunction yields the table for its dual:

A/H

H > < v

A

> > < V

< < < <

V V < <

Some many-valued logics have four or five truth values. But it is more

popular to postulate an infinite number of intermediate truth-values. In-

deed, the most popular alternative to three values is to use uncountably many values; the intermediate truth values correspond to the real numbers between 0 and 1. We can represent this possibility with the help of a

symbol that looks like a sundial.

Truthdial

The gnomon of a sundial casts a shadow that indicates the time of day. The gnomon of a truthdial casts a shadow that indicates a proposition's degree of truth. The positions of the truthdial can be conveyed by picturing it as a clock. At one o'clock is 0 for full falsehood. At six o'clock is the intermediate truth value of .5 and at eleven o'clock, is 1 representing full truth. The midnight position is left undefined to prevent the truth values 0 and 1 from coinciding.

Holding a truthdial up to a mirror reveals the complementary truth- value. This means that the magnitude of the shadow's shift varies with the magnitude of the initial truth value. In particular, the size of the shift from a truth-value of n is In - .51 degrees of truth. Although stereotypical relational changes, such as becoming an uncle, are qualitative, there are quantitative relational changes that vary in scale - just like 'real' causal

changes.



152 ROY A. SORENSEN

6. RACES WITH TURING MACHINES

When the number of truth values is countable, a complete truth table of truthdials is possible. The table might be impractically large. Indeed, a countable table could contain the same number of rows as the natural numbers. Although a Turing machine can write out any truth table of finite

length, it cannot compute a denumerable truth table for, say, conjunction. The machine's memory is unbounded but still finite. All Turing machines

proceed by scanning each input symbol. So when the number of inputs is infinite, a standard Turing machine cannot complete the task. True, the machine eventually gets to any given row but there is never a time at which it has read all the rows.

This obstacle cannot be overcome by relaxing the constraint that a Tur-

ing machine have an upper bound on the speed to which it can compute. An 'accelerated Turing machine', that manipulates symbols at an ever faster rate, can solve many uncomputable problems (Copeland, 1998a). But to evaluate an infinite truth table, even this souped up machine would need the further supplement of an accelerated read-write head that would permit it to read all the initial truth-values. This doubly accelerated machine reads the first row in the first minute, two more rows in the next half minute, four rows in the next quarter minute, and so on. At the end of two minutes the accelerated Turing machine has completed the truth table for conjunction.

If there is only finite mass in the universe, the doubly accelerated Turing machine must economize by also writing smaller and smaller. Indeed, I

prefer that the machine write this truth table for conjunction on a single transparency (normally used for overhead projectors). That way I could

compute the truth table for disjunction by simply turning the transparency over. I would not have time to read over all my work row by row. But I would have been infinitely productive.

Although a robotic arm can also perform the operation of turning over a transparency, this action is not in the repertoire of a standard Turing machine. Its stock of primitive actions is limited to moving forward or back- ward along its input tape, reading symbols, erasing, and writing symbols in blank squares. This basic stock does not include semantic operations such as 'Change the meaning of '/' to that of '\'. Of course, it can perform the purely syntactic operation of rewriting each input '/' as a '\' and each output '\' as a '/' finitely many times. But these local, piecemeal symbol manipulations are scale sensitive. The more changes that need to be made, the longer the run time of the program. And in the case where the input is infinite, the task is not computable.




Jack Copeland (personal communication, 1998) suggests that a Turing machine can perform global mirror reversals by using transparent tape. The data to be reversed are prefixed with 'FRONT'. Mirror reversal is achieved by deleting 'FRONT' and writing 'BACK' in its place. Copeland's method makes essential use of the transparency of the tape. We can appreciate its role by comparing Copeland's idea with a less appealing proposal. This involves opaque tape and the indicator 'NOT MIRROR REVERSED'. To mirror reverse the tape, the machine deletes the 'NOT'. This second proposal is unsatisfying because the Turing machine is relying on the reader to finish the computation by doing the mirror reversal. We would not be satisfied with an alphabetizing program that works by prefacing the data to be alphabetized with 'READ IN ALPHABETICAL ORDER'. The output must be in a form that does not require ingenuity to read.

Copeland's exploitation of a transparent medium satisfies this mechanical criterion. You just read off the results by changing your perspective in accordance with the reading key. But a Turing machine is no more permitted to exploit the transparency of the tape than it is permitted to exploit the opacity of tape. For example, Copeland's FRONT/BACK technique can just as effectively be used to 'erase' symbols in one swoop by flipping over opaque tape.

These remarks are made in the same spirit as the geometrical criticism that a certain proof uses more than a straightedge and compass. I am not advising engineers to restrict themselves to the resources available to a Turing machine. Turing's own doctoral dissertation considers a range of more powerful machines, 'Oracle machines', with enriched sets of prim- itives (Copeland, 1998b). The transparency proposal is fully in the spirit of Alan Turing's requirement that a computable problem is solved by rote procedure. But the proposal violates the letter of the Turing's requirement, i.e. Turing's particular regimentation of this problem solving ideal. This regimentation, whether in the vivid form of the Turing machine or one of its equivalents, is an entrenched standard in the theory of computation.

A machine that can perform global mirror reflections can do some things that a Turing machine cannot. This thesis is comparable to the claim that a computer with access to a Geiger counter and a pile of uranium can do something a Turing machine cannot: genuinely randomize. Just as a Turing machine can approximate randomness, it can also approximate global mirror reversal. However, genuinely global mirror reversal has an insensitivity to scale and complexity that Turing machines cannot match.

This insensitivity creates a qualitative difference in method, not a mere ergonomic novelty. The DVORAK keyboard is quantitatively superior to the QWERTY keyboard because one can type on a DVORAK keyboard



154 ROY A. SORENSEN

faster and more accurately. QWERTY, named after the first six letters of its

keyboard layout, was designed to minimize collisions of typewriter keys. But it survives in an age where typewriter keys are obsolete. QWERTY's status as an industrial standard ensures that established users will have to pay transition costs to switch to DVORAK and that new learners will encounter few DVORAK keyboards. The analogy is depressingly com-

pelling. Even if perspectival computation offers some practical advantages, it may not be able to surmount the QWERTY entrenchment effect. But the theoretical interest of perspectival computation would remain because it offers a qualitatively different alternative. Perspectival computation does not expedite inscription manipulation, it avoids inscription manipulation altogether. Moreover, the dodge highlights the computational relevance of the metaphysician's distinction between intrinsic and extrinsic properties.

The perspectival ideal of computing simply in virtue of extrinsic properties should also be distinguished from the semiotic project of designing more suggestive signs. It is convenient to have a symbol suggest what it

signifies. Nicer still is to have combinations of symbols that systematically suggest their relationships. For instance, the mirror opposites > and < reflect the fact that the relations of GREATER THAN and LESS THAN are opposites. Charles Peirce was more ambitious. He coordinated all of the sentence connectives into a unified system of iconography. Shea Zell-

wanger (1997) has continued this project of cognitive ergonomics with

special attention to mirror reversals. He has emphasized that selected mirror reversals achieve a calculative effect. However, these piecemeal reversals do not display insensitivity to complexity and scale. To achieve a

purely perspectival computation that avoids the physical limits of inscription manipulation, one must treat the input holistically, reversing all the

inscriptions. 'Wide' computationalists view calculative aids as extensions of the mind

- on a par with parts of the brain (Clark and Chalmers, 1998). They might be inclined to group mirrors with blackboards, slide rules, and mechanical calculators. My view is that the mirror is more like a pair spectacles; the mirror is a visual aid that helps us read the result of the computation.

Mirrors are dispensable in mirror notation if the individual can mentally mirror reverse figures. Lewis Carroll amazed his friends by reading and

writing mirror reversed sentences. However, mirror reading is a widely learnable skill, never a serious obstacle in the professions that require it. Take relief printing. To set type-face by hand, one must load mirror reversed letters into a composing stick. The metallic sentence (when turned upside down) will read TUO but will print as OUT. Beginners use mirrors to check their work. But an experienced type-setter reads mirror image




writing. If he also knows mirror notation and logic, he can compute the duals simply in virtue of his decision to read them from another orientation.

I am using 'orientation' in the same abstract way as artists and pro- jective geometers use 'perspective'. When I say that mirror notation is reader relative, I mean the reader's orientation rather than the reader himself. Polite lecturers adopt their audience's orientation when using 'left'. After all, it is more efficient to have one lecturer make the adjustment in

describing objects from an alien perspective rather than have the many people in the audience go through the effort. A lecturer who wishes to face his audience as he writes symbols could use a glass board and write in mirror image form. This is difficult in ordinary notation. However, if he were using mirror notation, then the lecturer could just write what would be the duals from his natural orientation. Since the intended orientation is his audience's, the lecturer would be asserting the sentences as they would be read from an orientation other than his own natural orientation.

If the lecturer went on to inquire about the duals of the sentences he had

just written, he would not need to turn the board around or walk around to the opposite side. He could just adopt his natural orientation and read off the answers.

Given that the switch in orientation is merely a matter of decision, no 'real' information processing takes place. Hence the computation is instantaneous. Once the agent decides to switch perspectives, the computation is complete. Of course, it takes a little time to form a decision. But that is not relevant to the computation time. When timing how fast someone can convert a Fahrenheit temperature reading to a Celsius reading, one does not include the time it takes him to decide to make the conversion.

The perspectival computation of a dual does not require that input values be read. A digital computer that computes mirror images must read each coordinate of the image it has been instructed to reverse. Since no

signal can accelerate past the speed of light, there is a limit on how fast a symbol is read. Therefore, a computer cannot deductively reverse an image that has infinitely many coordinates. The best the computer can do is to inductively infer an image from a sample of coordinates.

These are only limitations on what a computer can compute. A computer can solve the problem at the level of reading. Mirror notation can be inscribed on computer punch cards. To mirror reverse the encoded pro- grams one need only turn the stack of cards around. Or one could make the card-reading end portable and just move it to the rear of the stack. This portable reader would be even more useful if it could be turned on its side and turned upside-down. For then it could do all three perspective shifts. The cards sit untouched in the hopper as the card-reader re-orients. This



156 ROY A. SORENSEN

advantage increases with the size of the stack and becomes crucial when the stack is infinite.

7. RE-ORIENTATIONS COMPARED WITH SHIFTS

The stereotypical difference between symbols lies in their contrasting intrinsic properties. An object's intrinsic properties are those it has on its own - ones that are not due to relationships with other things (Langton and Lewis, 1998). Roundness is an intrinsic feature of the letter O but it being rounder than D is an extrinsic property of O.

An object's relationships to its parts count as intrinsic properties. For instance, the letter X has the intrinsic property of being composed of two intersecting strokes. Internal relations are crucial for understanding com-

plex symbols. The Roman numeral IV is composed of the same simple symbols as VI. However, the numerals designate different numbers because they are part of a system in which symbols to the left of the main

symbol are subtracted and symbols on the right are added. 18 differs from 81 in their relation to a positional framework. This

framework was explicit in the days of counting boards and the abacus. But thanks to the introduction of zero as a placeholder, the framework is left implicit. This invisible relatum becomes explicit in discussions of the Hindu-Arabic number system:

5TH 4TH 3RD 2ND 1ST

Base x Base x Base x Base Units

Base x Base x Base

Base x Base

Base

18 is a distinct numeral from 81 by virtue of how 1 and 8 relate to the columns in the above grid. We can discern these column positions by noting whether 1 is to the left or to the right of 8. But unlike the Roman numerals, this is just an heuristic for determining the digits's relative position to another relatum - the invisible grid in which the digits are being located.

The existence of an invisible third relatum is more readily seen with chess. Suppose two chessplayers lose their chessboard but not their pieces. They set the pieces on a table and commence to play. The boardless players do well in the opening because they can use the relative positions of pieces to figure out each piece's relationship with an invisible chessboard. The




game becomes more difficult as more and more pieces are eliminated from the board.

Actually, some chessplayers deliberately forego both the chessboard and the pieces. In 'mind chess', the players just call out the moves to each other in chess notation. In effect, this is two-sided blindfold chess. In blindfold chess, one player foregoes the heuristic advantage of looking at the board. In mind chess, both players accept this handicap.

The chess example suggests that ancient abacuses and counting boards would become merely tacit as people became more proficient with arithmetic. The tendency to overlook the invisible background relatum ob- scures the profound difference between the 18/81 distinction and the IV/VI distinction.

In positional notation 4 x 100 is computed by simply shifting the relative position of 4. Writing two zeros after the 4 makes it an indicator of hundreds rather than units: 400. The same numeral token can change its role in positional notation. This point is salient when many digits are involved: 495433 x 100. Appending two zeros to 495433 systematically changes the roles of all the digits. Sometimes the change is achieved by tacitly shifting the unstated positional grid. When a real estate agent says that the price of a house is 288, he means that the price 288,000. Symbols can be manipulated without inscription manipulation by manipulating the unstated background.

Simon Stevin introduced floating point notation to extend decimal computation to all rational numbers. Stevin's point extends the scope of shifts because one can multiply by the n-th base simply by shifting the 'decimal

point' n places to the right. As Nicholas Chuquet noted around 1500 in the algebraic part of his Triparty en la sciences des nombres, counting the shifts reduces multiplication to addition. If I multiply by the n-th base and then multiply by the m-th base, I have merely moved the decimal place m + n positions.

Shifts work by virtue of positional notation rather than the fact that the base happens to be 10. Any base will do. For instance, the multiplication of 2 by base x base proceeds by adding two zeros after 10 (which is the binary representation of 2). This suggests that meta-addition can handle any re- peated multiplication. We need only have a notation that states the number to be multiplied (the base) and how many times it is to be multiplied (the exponent). Hence the familiar 23 means 2 times 2 times 2, or equivalently, the third base in base 2 arithmetic. More generally, the product of n x'es is x". So the meta-addition trick yields the law xmxn = xm+n. Ordinary addition governs fractions as well as natural numbers. What about meta-



158 ROY A. SORENSEN

addition? Well, x1/2X1/2 = x works if x1/2 is taken to denote yfT. These fractional exponents are the basic idea behind logarithmic computation.

The benefits of positional notation compound because some notation governs other notation. By replacing direct computations with computations about computations, we exploit a multiplier effect in representational power. Gains in calculative efficiency are self-magnifying.

Shifts reduce the amount of inscription manipulation needed to yield an answer. Inscription manipulation is a form of labor. The more manipulation, the greater the opportunity cost in time, the more wear and tear on the calculator, and the more chance for error. Speed limits on the performance of each step will render some problems infeasible. Techniques that reduce the amount of inscription manipulation increase speed, accu- racy, and feasibility. Perspectival computation pushes the minimization of symbol manipulation to the zero point. Thus it qualitatively contrasts with inscription manipulation.

The counterpart of the shift in mirror notation is the re-orientation. Un- like the shift, the re-orientation does not require a place-holder inscription. The relatum for a shift is the positional number grid. With mirror notation, the relatum is the calculator himself. The re-orientation alters the identity of the inscription. This is clear when / is used to signify one in binary notation while \ is used to signify zero. Horizontal reflection changes which numeral is constituted by the inscription.

Familiar computation involves a mixture of intrinsic changes to the

symbols. Some are absolute (crossing out a digit when subtracting), some alter internal relations between constitutive symbols, and some alter internal relations with a (usually unstated) background relatum. Mirror notation extends this relationality to the calculator himself.

8. THE SYNTACTIC NATURE OF RE-ORIENTATION

The reader's orientation has a syntactic role in mirror notation. After all, the marks < and > have the same shape. They only differ in which direction they point. Thus the inscription token > qualifies only as a symbol fragment, like the dot that is part of i. The whole symbol is an ordered

pair consisting of the inscription and an orientation. Since orientation af- fects whether / is conjunction or disjunction, orientation has an indirect semantic impact. But all symbol defining syntactic features have this triv- ial semantic influence. Distinguish between defining a symbol (say what a solidus is, namely, the oblique stroke /) and defining the meaning of a symbol (specifying what solidus signifies, namely, that / means conjunction).




Frank Ramsey (1927, pp. 161-162) once suggested that the negation sign could be eliminated by adopting the convention that false sentences be written upside-down. Here is a sample sentence followed by its negation:

I WON HOW MOM WON

NOM WOW MOH NOM I

Ramsey's proposal is intriguing but sketchy. The elimination of 'not' might seem to make the liar paradox inexpressible. But now our talk about the orientation of sentences is unexpectedly pregnant with semantic implications. In particular, the innocent looking

THIS STATEMENT IS UPSIDE-DOWN

appears to harbor the liar. How are type-setters to talk about the orientation of sentences without becoming embroiled in semantic paradox?

Ramsey's proposal also needs to be refined to avoid the inconsistency that precipitates from palindromes that are up-down invertible. I can create one by first stipulating 'XO' means 'stink'. 'Ox' has two plurals 'oxen' and 'ox'. So my neologism makes OX XO a sentence of English. But

Ramsey's upside-down rule makes OX XO its own negation. Presumably Ramsey would add an ancillary rule outlawing upside-down palindromes. But what counts as an upside-down palindrome is sensitive to writing style. Scott Kim (1989) has made an art form of turning words into am-

bigrams. ('Ambigram' is Douglas Hofstadter's term for a word that look the same when turned upside down). David Holst and his collaborators at Word Net have computerized the inversion process. Their WEB site is

http://ambigram.matic.com/ambigram/matic/graphics/index.htm. You can automatically invert words by typing them into the 'ambigrammatic'.

Ramsey happily noted that double negation would involve two inversions and so would return the original sentence. An intuitionist would object that this notation prevents him from even expressing doubt about the inference rule of double-negation elimination. A paraconsistent logician would complain that the prohibition of upside down palindromes would

prevent him from expressing the thesis that a proposition and its negation can be both true. For a single statement cannot be both right side up and upside down. The classicist replies that these are virtues of Ramsey's proposal. Misconceived objections to valid inference rules should be pre- vented by rendering them ungrammatical. Bertrand Russell was pleased that St. Anselm's ontological argument for God's existence cannot be ex- pressed in the notation of Principia Mathematica.

By 'upside-down' Ramsey means 'upside-down relative to the page'. If Ramsey were relativizing to the reader's orientation, then turning a page



160 ROY A. SORENSEN

upside down would negate all the sentences. Ramsey attaches neither semantic nor syntactic significance to the reader's orientation. An upside- down sentence is a well formed formula on its own. Only the meaning of the formula is affected by its orientation to the page.

In Ramsey notation, the reader's orientation is relevant to the episte- mology of meaning. If Ramsey's convention for negation were adopted, there would be more ambiguity. We normally tell which side of a page is up by looking at the other sentences. Hence, a page containing only a single sentence would always be ambiguous. Adding a second sentence 'This way f up' would not definitely settle the issue because the sentence could be upside down.

The epistemological relevance of the page's orientation has been dra- matized with standard logical notation. Roberto Casati and Achille Varzi (1998) tell a tale in which Tictac and Tactic find a message in a bottle. All it says is: p v d. Since the message writer failed to indicate which way is up, Tictac and Tactic cannot tell whether the message is the disjunction p v d or the conjunction p A d.

Although uncertainties about the page's orientation create ambiguities for the message receivers, the sentence has a definite meaning independent of the reader's orientation. The message sender knows what the meaning is and could decisively resolve the ambiguity that bedevils Tictac and Tactic.

This determinacy of meaning also distinguishes extrinsic notation from

uninterpreted symbols. Formalisms can sometimes do double-duty by be-

ing interpreted and then re-interpreted. In modal logic, O is sometimes interpreted as necessity, sometimes as knowledge, and sometimes as deter-

minacy. These re-interpretations do not require that one touch the original symbols. However, the uninterpreted formalism contrasts with interpreted mirror notation in two ways. First, in the case of mirror notation, the

change takes place within a system, not between systems. Hence, there is no equivocation. Second, perspectival computation changes the syntactic identity of the symbol. The inscription component for conjunction symbol / becomes an inscription component for disjunction symbol. Symbols are ordered pairs consisting of an inscription and an orientation. Hence, a change of orientation is a change of symbol type.

9. LIBERALIZING THE ALPHABETS OF FORMAL LANGUAGES

Formal languages use a finite stock of primitive symbols that can be com- bined into more complex parts of sentences. Orientation plays an implicit role because many of these letters cannot be distinguished solely by shape.




These belong to a set of what type setters call the 'demon characters':

69 pdbq nu Ill OOo >< ][ }{ \/

Logicians have felt no need to state the orientation of their inscriptions because the orientation is a constant. The only permissible orientation to the page is the one customary to European languages.

The innovation in mirror notation is that it exploits orientation as a variable instead of a constant. In ordinary arithmetical notation, > ex-

presses greater than and < expresses less than. Since these inscriptions have the same shape, we must be tacitly using their orientation to dis-

tinguish them. Most European alphabets are orientational in this sense.

(That's why dyslexia is a serious learning disorder for Westerns but not for users of non-orientational languages such as Japanese.) The feeling that mere marks on the page are the whole symbol, is an illusion encouraged by the unvarying nature of the hidden relatum of orientation. A similar illusion arises for movement. Since we habitually relativize to the earth, we fail to realize that we are relativizing. (Constants become invisible.) Hence, movement seems absolute rather than relative.

There are natural languages that alternate orientations. Egyptian Hiero-

glyphics, Babylonian Cuneiform, and Ancient Greek writing follow the

rebounding boustrophedon pattern. Alternate lines are written left to right and then right to left just as an ox plows a field. Boustrophedonic computer printers emulate this pattern to save time on carriage returns. The printers must be designed to avoid writing every other lines in mirror-reversed letters. However, boustrophedonic writing systems accept the alternate mirror reversals.

In mirror notation, the mirror image of any well formed sentence is itself well-formed. Boustrophedonic sentences only satisfy a conditional version of this condition. The mirror image of a boustrophedonic sentence is guaranteed to be grammatical only if it belongs to a line with a distinct parity. One sentence must be on an odd numbered line while the other is on an even numbered line. For instance, if English were boustrophedonic, the occurrence of MAX on line 10 would not guarantee the grammatically of XAM on line 12.

Mirror notation also differs in that each inscription token does double-

duty. The inscription is part of one symbol from one orientation and part of a distinct symbol from another orientation. This is what allows the perspectival calculator to manipulate symbols without manipulating inscriptions.

There are a few forms of notation that naturally allow for orientational variation. If you turn a player-piano roll around, the high and low notes will be reversed in a way that some find stimulating. Bach intentionally designed the 12th and 13th fugues of Die Kunst der Fuge to be reversed.



162 ROY A. SORENSEN

However, musical notation was not designed for reversal and it is a feat to construct musical pieces that have merit when mirror reflected. A calculative notation must be artificially constructed for the purpose.

The theoretical motive for the construction is that perspectival computation has illuminating contrasts with inscriptive computation. According to the inscriptive paradigm, the formal systems that developed under the leadership of David Hilbert reveal how the semantic properties of symbols can be mimicked by their syntactic properties. These syntactic properties reduce to differences in the shape and location of the inscriptions. Since these differences make causal differences, engineers were able to automate formal systems. To some extent, the automation had already taken place in the form of computing devices such as the abacus, Napier's rods, and more mechanically sophisticated adding machines. But with the development of Charles Babbage's programmable operations and conditional branches, there was a potential for versatile, autonomous implementation of formal systems by machines. Later electronic technology made such machines a practical reality. It then became attractive to picture the brain as just another medium for the coherent manipulation of inscriptions. Valid reasoning is just a matter of one string of symbols appropriately causing the formation of another string of symbols. Jerry Fodor summarizes:

You connect the causal properties of a symbol with its semantic properties via its syntax. The syntax of a symbol is one of its higher-order physical properties. To a metaphorical first approximation, we can think of the syntactic structure of a symbol as an abstract feature of its shape. Because, to all intents and purposes, syntax reduces to shape, and because the shape of a symbol is a potential determinant of its causal role, it is fairly easy to see how there could be environments in which the causal role of a symbol correlates with its syntax. It's easy, that is to say, to imagine symbols tokens interacting causally in virtue of their syntactic structures. The syntax of a symbol might determine the causes and effects of its tokening in much the way that the geometry of a key determines which locks it will open. (Fodor, 1987, pp. 18-19)

Fodor needs to include the orientation of the symbols in addition to their shapes. A cork-screw shaped key that threads clockwise can open a lock that cannot be opened by an isomorphic counterpart that threads counter- clockwise. Fodor is not wedded to shape:

Any nomic property of symbol token, however - any property in virtue of the possession of which they satisfy causal laws - would, in principle, do just as well. (So, for example, syntactic structure could be realized by relations among electromagnetic states rather than relations among shapes; as, indeed, it is in real computers.) This is the point of the Func- tionalist doctrine that, in principle, you can make a mind out of almost anything. (Fodor, 1987, 156fn)

However, electromagnetic states also have orientations. As Hans Christian Oersted discovered in 1820, the direction of a magnetic compass needle




shifts when electric current is passed beneath it. Reversing the current 'mirror reverses' the needle from an east to a west reading.

The letters d and b have the same shape but their distinct orientations relative to other symbols give them different causal relations with these symbols. But in the case of perspectival computations, there are no orientation differences relative to the other symbols. If I write a truth table on a glass blackboard, all the relations between the inscriptions must be preserved when viewed from the opposite side of the board. They only change their orientation with respect to the reader of the symbols. The input symbols coherently determine what the output symbols will be but not via a causal relation (or any other relation) between distinct strings of inscriptions.

Perspectival computation is a counterexample to causal theories of inference that represent all inferences as inscription alterations. A broader causal theory would recognize the possibility of an oriented inscription causing a different orientation to the same inscription. Orientation does play a role in causal explanations. Few lightbulbs are stolen from New York subways. Why? Because those lightbulbs are threaded counter- clockwise. They cannot be screwed into normal light sockets.

A simple perspectival computation yields its fruit in a single global step. Unlike recursive symbol manipulation, it cannot work its way to a solution in small, adaptive increments. Hence, simple perspectival computation could only be of practical value if it figured as one element of a mixed methodology. Possibly, perspectival computation already performs this kind of eclectic teamwork in those natural processes that seem to solve computationally intractible problems.

After all, nature is no stranger to duals. The four base elements of the DNA molecule form a mirror system because the base guanine will only bond with cytosine, while the base adenine will only bond with thymine. Abbreviate these four nucleotides as G, C, A, and T. Thus the 7-molecule long strand CTAATGT uniquely determines a complement strand GAT- TACA. Now represent guanine as / and cytosine as \, and let adenine be < and thymine >. The CTAATGT molecule is then \ > < < > / >. Mirror reversal yields its complement. Since the mirror is indifferent to the length of the molecule, we can see how a bacterium can afford to have a DNA molecule that is a million nucleotides long.

I have no detailed evidence that nature has furnished any precedents for collaboration between perspectival computation and inscriptive computation. I hope recognition of the possibility of perspectival computation will lead to recognition of instances in real life.



164 ROY A. SORENSEN

ACKNOWLEDGMENT

This paper benefitted from audience reactions at City College. I also wish to thank Roberto Casati, Jack Copeland, Chris Landesman, Kit Fine and Roy Mash for their reflections.

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Department of Philosophy, New York University, U.S.A. (e-mail: rs3 @is2.nyu.edu)



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