Let's Say: A Child's First Calculus

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Topic 2: Revealing an Operator

2.1 Are the two values exact opposites?

2.1.0 We don't know yet

We cannot know yet whether our two values, the simple expressions, are exact opposites, because we don't yet do anything for which the question even makes sense.

True, we can evaluate expressions and get different results depending on the particular expression, but that is only enough to say "different values" or "distinct values." We are probably starting to think of them informally as opposites, but there is no operation we do on them yet that would give the concept opposite practical meaning.

Moreover, the pancake arithmetic alone is not practical at all, we are finding. It's a fun game for a short time, handling bigger and bigger expressions and having them come out the same as someone else finds by a different sequence of wand-waving steps, but we are not yet seeing a way to apply this knowledge and skill directly to a practical problem.

2.1.1 "What -- is your Quest?"

What we are after, ultimately, is not the arithmetic at all, but rather the algebra. That is where the main fun and use will be, and opposite or reversal will be a central idea there.

But we need not wait for the pancake algebra to give us a reversing operator! We don't even need to define a reversing property for the pancake, either. We have already established enough to get it from the pancake arithmetic directly.

Let's do it now.

2.2 Seeing the Pancake as an Operator

2.2.1 Operator

In mathematics generally, an operator definitively has input, action and output. Its action is to produce some particular output value as a result of having particular values (perhaps just one) presented as its inputs, or arguments. For example, a null operator would always give out the same value it receives as its argument, unchanged, as its output. (Above some level of analysis we consider the action to take place without delay, whenever a new input value comes along.)

2.2.2 The Pancake as an Operator

A pancake, we said, has an inside and an outside. The important thing about its outside is where it sits -- on what plaza. The important thing about its inside is, that is where inputs are presented when we start looking at pancakes as operators. The pancake will always have exactly one action (or property) as an operator.

We soon show that this reversing action follows from the pancake arithmetic itself. The pancake's operator property will figure heavily in the algebra and all the uses we put it to (applications). As we prepare to establish the operator property of the pancake, start picturing the inside of the pancake as the input plaza.

2.3 Is the pancake a reversing operator?

This question is closely related to the earlier question: are the two values exact opposites?

2.3.1 "What -- is your Test?"

We should answer "yes" to the "opposites" question if and only if we find some operator which, when we give it one of the values as its input argument, gives back the other value as the output result, and gives back the first value as output when we give it that earlier result as the input.

2.3.2 Exact Opposites test

Our test for the question about exact opposites, then, is 2.3.1 It will also be our test for the question of whether any operator that operates on these values is a reversing operator. Such an operation could then be called opposite, complementation, reversal, negation, or something similar. (It is sometimes called "oops" by at least one writer in this field.)

2.4 The Pancake is a Reversing Operator

2.4.1 "Why -- should anyone care?"

The pancake calculus can be seen as a formalism for using logic gates of either NOR or NAND flavor exclusively to build computer-like machines. Indeed, that was its origin. It worked reliably for G. Spencer-Brown and his brother as they worked on engineering an elevator control system, before ever G. sat down to develop its foundations with mathematical rigor.

This echoes the much longer time from when Newton and Leibnitz published the infinitesimal calculus until Cauchy, Dedekind and others established its foundations with mathematical rigor. Nevertheless the infinitesimal calculus was exploited right away anyway in science and engineering just because it worked. It solved problems which cannot be done by numerical algebra alone.

The N of NAND and NOR indicates a reversal of the result from letting the nominal Boolean function (AND, or OR) work on the inputs first.

We could not build computers, digital memories or other logic devices of any usefulness whatever if this reversal were not available in their logic-level structure. AND gates and OR gates would be just about useless if we couldn't build NOT gates. This is one reason why it is interesting to know about value reversal in pancake calculus.

Another reason a reversing operator is interesting is that negation -- truth versus falsity, validity versus fallacy -- is another practical interpretation (application) of reversal and is the central issue in the other logics we are preparing to play with.

2.4.2 "What -- is your Plan?"

We intend to prove that the pancake can be seen and used as an operator --in particular, a reversing operator. We get the plan for the proof right from our Test for Both Questions in 2.3.1 The Test has two parts, specifying two experiments involving input and output (i/o experiments).

First we are to see what output results when we present one of the values (simple expressions) as input. Then we are to take that first output value, whatever it might be, and feed it to the same operator as the input value and again observe what output value comes out. The test succeeds if and only if that second output value is the same as the original input value.

Did I say that right? is that what the Test in 2.3.1 requires? That agreed, we can start in on the proof under this two-phase plan.

2.4.3 Reversing Operator Proof, Phase 1: the Invisible Symbol as the Input Value

Consider the formal identity Fid.1

( ) = ( ) Fid.1

First we need agreement on its truth. We might say, This statement says that something is interchangeable with itself, so we agree it is true. We'll say this is "identically true."

( ) = ( ) Fid.1

Now we read Fid.1 another way: we need to see it as a statement of the pancake's action on an input value. That is, in the Rhs see just a simple expression, the empty-looking pancake, as the output value in our Test. In the Lhs, see the pancake as an operator acting on an input.

What is the input, anyway? and where is it? Here is where we exploit the everywhereness of the invisible symbol. What is on the input plaza (topside, inside) of our candidate operator, the empty-looking pancake in Fid.1? I seen it, I seen it! You seen it? I seen it! One of the simple expressions, the Everywhere Invisible Symbol. Thus we can now read identity Fid.1 this way: The output of the pancake operating on the invisible symbol as its argument is a different value, the empty-looking pancake.

2.4.4 Reversing Operator Proof, Phase 2:
empty-looking Pancake as the Input Value

Our proof is proceeding according to plan. We have a First Output value: the empty-looking pancake. According to our plan, this now becomes the input value for the next part of the Test. So let’s see what value the pancake operator gives as output when we give it the empty-looking pancake as input.

Where is this information to be found? Let's first express the question formally as a help in looking for the answer. Writing down carefully --formally-- what we want to know, then:

(( )) = what?

As before, we are formally presenting the input value (an empty-looking pancake this time) at the inside (the input plaza) of an outer pancake (the operator). Now we are ready to root around in the cellar of our calculus for something similar. I suggest looking among the primitives of the pancake arithmetic, the Good Luck arithmetic primitive axioms. We do indeed come across something similar -- so similar it is actually the direct answer:

(rummage rummage rummage, rummage rummage rummage, ---
AHA!)

Short Stack Good Luck!

(( )) = , I seen it! I seen it!

The Lhs of the question is none other than the Lhs of Short Stack Good Luck, so we can re-read SSglax this way: "The result of the pancake operating on an empty-looking pancake as input is the invisible symbol."

2.4.5

We have now almost finished carrying out the plan of our proof. It only remains to compare the output just seen in Phase 2 to the input presented in Phase 1. We observe that the original input was the invisible symbol, so this completes the proof that the pancake is a reversing operator with respect to the two values which are the simple expressions. []

2.5 vizzo, invizzo Are Exact Opposites

In the course of showing that the pancake is a reversing operator we also showed the two values -- the simple expressions -- to be indeed exact opposites now that there is something we can do with them beyond simply discovering them by evaluating expressions.

2.6 True, False

In the logics we are planning to examine, values are typically True, False. As we explore practical applications of the pancake algebra we shall be translating between these 2-valued logics and pancake by substituting values.

For example, in Boolean B2 algebra the values are 0, 1, interpreted as False, True. We can translate between B2 and pancake either by letting 0 and invizzo correspond (leaving 1 to correspond to vizzo) or the other way 'round, and it doesn't seem to make any difference, so long as, in any particular investigation, we do both directions of translation in the same flavor.

Later on, when we use pancake calculus to model predicate calculus, translation flavor will make a difference.

2.7 Coming Attraction: First Algebraic Conjecture

That's all there is to let's-say about the pancake arithmetic. We are on the verge of algebra.

You who are new to the game might want more practice in evaluation before proceding to the pancake algebra, so the next Topic begins with plenty of worked-out examples.

These examples will serve another purpose as well. The pancake expressions I have cooked up for you to eat will all have something in common which should prod you to suspect and invent your first algebraic conjecture -- your "Algebraic Conjecture Number One."

Your reward for inventing the conjecture will be, as in any study worth your time and effort, a further task: state the conjecture (formally) and prove it.

2.20 Spencer-Brown's "Cross" Notation

G.Spencer-Brown's original work presents a notation that has been called "two-dimensional." It is easy to deal with visually. It was not designed for convenient handling in software tools.

S-B's name for the pancake is cross. His cross operator is an overbar with a downward stroke at one end. The inside (input side) of a cross is its underside. Overarching crosses contain (or nest) what is under them.

Visually, all deepest-nested elements are seen along the bottom of the arrangement. Depth of nesting is reflected in the count of overarching crosses above an element.

2.21 Boundary Mathematics Notation

Boundary Mathematics (William Bricken) seems, at least basically, to be the same as the math we're developing here. Based alike on Spencer-Brown's Laws of Form , Boundary Mathematics notation uses a closed circle or oval for its region-delimiting symbol. Boundaries may never intersect (these are not Venn diagrams); they may only contain or not contain other Boundary arrangements or variable-names representing them affording precisely the same nesting possibilities as pancake's paren-pairs.

I have not yet seen a complete presentation of Boundary Mathematics.

2.30 Side-View Pancake Notation

Among alternative notations for this mathematics is a non-serial notation similar to that of Laws of Form -- side-view pancake notation. In side-view, pancakes are indicated by underbars. A side-view pancake contains what is above it, so that an element more deeply nested (in layers of containing pancakes) is seen in a higher position.

Here is an expression in side-view pancake notation, with its equivalent written just below it in standard pancake notation (paren-pairs):

__ __
________ __
______________ Side-view pancake notation

( ( ( ) ( ) ) ( ) ) Standard pancake notation

2.31 Transcribing between Pancake Notations

You may now be ready to try writing down a procedure for transcribing any side-view expression into standard pancake notation. I have found that this is easier than the reverse: writing down a procedure for transcribing from standard pancake into side-view pancake.

The two views are easy to relate visually, whatever you thought of the difficulty writing transcription procedures. We are now free to use these two notations interchangeably.

I find side-view somewhat easier to analyze in work on paper when searching for patterns to exploit in algebra.

Of the notations available for this mathematics, the only one which lends itself easily to entry into software tools is the paired delimiters of standard pancake.

The only notation which clearly and intuitively shows that serial order on a plaza is inconsequential is Boundary Mathematics. Being able to place things anywhere within or outside a circle without lining them up makes it obvious that it's not necessary to arrange things serially, hence all apparent serializing is arbitrary.

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