the fixed point of Asa Zook/paul pietsch -- copyright 1996 by Paul Pietsch

chapter 21 Fixed Axes

They'd limit their inquiry, Asa opened, to what gives and sustains life and would not attempt to encompass all Nature. Much of what he would say, at first, might seem very strange. For the internal parts of the theory confound common sense and clash with "academic sophistry."

Joyce interrupted. "Can't we use something less abrasive than academic sophistry dear?"

He'd yield to her in matters of usage, he said. "However," and now spreading over his face was the expression she'd first seen at Annabelle's lecture (And to her astonishment he looked just like Poppa.) "while we must avoid pejoratives, we cannot permit euphemisms to erode truth. We cannot flirt with perfidy and expect Nature to be revealing to us."

And, Joyce added an unspoken phrase to her thoughts, 'If Sophistry's head pops up my Gadfly will gladly lop it off with his dialectically swift sword.'

Asa continued. "Life consists of finite individuals. That is obvious. But the living state would quickly vanish from the universe without true continuity between one generation and the next."

"True continuity?"

"The continuity exhibited, for example, by waves; a logical continuity, knowable only in the ideal."

"But Asa, couldn't we account for biological continuity with the genes?"

"Only to a degree, Joyce. Only to the extent that we do not need theory, to what we can directly experience and intuitively grasp.

"DNA is not a theory. It is a molecule. But our theory, Joyce, will encompass genes, and go far beyond the ken of the geneticist or the biochemist of our day."

"For instance?"

"The mind."

Joyce considered his answer. "Suppose, Asa that a biochemist asserts, 'Mind is really brain; and the brain is merely the sum of what its molecules do. And we can relate all that biological systems do back to what the genes are -- including the mind.' How could you...we counter an argument like that, dear?"

"Your hypothetical biochemist does not truly believe the Second Law of Thermodynamics. Only a few years ago, Norbert Wiener called attention to the fundamental difference between information and matter. It is true that we employ matter and energy as storage and transport media for information. But we can make monumental changes in a parcel of information with tiny changes in mass- energy."

"Aren't you being harsh with my hypothetical savant, Asa? I mean, if you say he doesn't respect the laws of physics, then you imply that he's a vitalist. And isn't that an insult to most modern scientists?"

"Biology, en masse, ignored Wiener's advice ."

Joyce drew her lower lip between her incisors. "I dislike the tone, Asa. You seem to be indicting all of science."

"Indict is the wrong word, Joyce. 'Describe' is what I intend. The experimentalist must operate within a narrow context, Joyce, a context too narrow to appreciate the implications of his results in terms of remote bodies of facts.

"Our theory is general -- or will be after we construct it. But let me return to the notion of continuity. Let me summon Riemann's words to state how we can know when one entity is truly continuous with another. 'When the one is part of the other."

Joyce looked up from her notes and smiled. 'When the one is part of the other'; that was one of her favorite phrases. No. It was her favorite phrase of all. And she was glad, finally, to know where it originated.

Asa continued: "To go further with Riemann, the one can become truly continuous with the other when the paths to the point of union describe a curve, and not a Euclidean straight line."

"Dear," she held up her pen. "You've used 'true' several times here about continuity. Is there such a thing as false continuity?" "Yes. Contiguous aggregations often mimic continuity, often trick the intuition into thinking they form a continuum."

"Such as?"

"Any situation where the whole equals the sum of discrete parts or sets."

"Toy blocks? Would a stack of blocks exemplify contiguity?"

"Excellent, Joyce! Excellent!"

"For eventually they always topple?"

"Exactly. When the contiguous grouping becomes very large or exceedingly complex, the contiguum disperses or crumbles or falls over."

"And a wave is a specific example of a true continuum?"

"Or a circle. Any system with attributes Jan Christiaan Smuts has called holistic -- where no matter how much we dissect it, some fraction of the whole is left over."


"The numerical value of pi." "Three point one four and on and on into infinity?"

He now had her draw a circle and within it mark three points: "A, B and C.

"Assume for the sake of illustration, Joyce, that A, B and C are the only points in our system. Assume also that as long as we stay both on and within the circle, we can make any transformations of the system we please. Suppose, now, we move through the path of least curvature from A to B, that we merge the two points to form a new point, AB. Now suppose we are moving AB along the least curvature to C but as we're doing so C keeps shifting before we arrive at it. What would occur?"

Joyce pondered the problem. "Wait. If C shifts, then we can't move through the path of least curvature to point C from point AB. We can't..." she paused and bit the nail off her index finger (shit!). "If C shifts, we can't make any continuous transformation we please. Because we're denied the use of the path of least curvature!" She was frowning. But Asa was smiling, as though genuinely pleased.

Her observation, Asa went on, illustrated the proof of Brouwer's fixed- point theorem. Fixed- point theorems, he said, guarantee (no less!) that one point in a truly continuous system must remain "unchanged!" after any continuous transformations. The fixed point, he went, was a necessary but also sufficient condition for a true continuum.

"Is the fixed point what contiguums lack, Asa?"

Again, he seemed delighted. "With an exception I'll take up shortly, that is correct. Contiguums in general lack fixed points and are, as a direct consequence thereof, discontinuous."

What he said next at first puzzled but then pleased her. "Suppose, Joyce, we must make all our measurements within -- and not from outside -- our system. Suppose that to measure, we must place --or superpose -- one point atop another. Now what if we've just folded AB onto C to make ABC. When we have point ABC, how do we measure it?"

Joyce pondered. "We can't!" she realized. "There's nothing left to measure it with or by!" And when her own words registered, "Asa, there's uncertainty in the continuous system. Indeterminism! Free will. Whee! Poetry has a chance, after all."

"More than just a chance, Joyce. Free will is a guarantee of a true continuum."

Asa went on to talk about logic. He reminded her that logical statements were those that can be unequivocally proved true or false, that "illogical is not the same as false"; that a false statement can be perfectly logical. Provability was the key idea here, she noted.

He had her get a pencil and two sheets of graph paper from the shelf above his desk. Then he asked her to divide each sheet into four sectors: "Quadrants," he said. "Please mark True in two quadrants and False in the others." She did as he asked.

"Assume, Joyce that each quadrant represents a statement. Assume that we can only test one statement by folding another statement onto it. One folding is one proof. Now, Joyce, take the paper and fold it to measure -- prove or disprove -- the statements."

Joyce immediately saw the outcome. "Asa. One square remains unprovable!"

"You can prove the underlying theorem for yourself by imagining that you fold and fold until you have only two remaining statements, each the size of a point. After you make the last possible fold, nothing remains to serve as a reference for the last statement."

"It's the same as the fixed point, isn't it?"

"In principle, exactly. In a logical continuum, one statement is unprovable."

"Just suppose, Asa, that by some quirk of things we really can prove every statement in a system? Then what?"


Joyce saw it herself: "We'd have a contiguum, not a continuum. Is that right?"

"It is."

But before she had a chance to gloat, he asked her to take the second graph and cut it into four quadrants. Now, testing involved simply placing one quadrant on another. Unlike before, now she could lay any square on any other. "Like shuffling cards," she observed. "I can scramble them around and prove whatever I want, right down to the last statement in the pile." The reason, he said as she shuffled and reshuffled, was that there were now no rules inherent in the system to restrict how she went from one test to another, "Whereas with each folding in the continuous system," he observed, "we limit our options for the next round of testing."

Then something occurred to her. "Asa, isn't the contiguum the more complete system?" Asa roared when she said that. Although she laughed, too, she hadn't the foggiest idea of what was so funny. And she pressed on with her argument. "I mean...with the contiguum, first of all, we don't have to worry about that unprovable statement. Second, we don't have the restrictions imposed by folding. And third we can make the truth of the whole equal to the sum of the parts. We know more when we're done then, don't we dear?"

"But how do we know all statements of the contiguum necessarily belong together? How can we guarantee we are dealing with a whole system and not merely an arbitrary pile of your blocks, Joyce?" He digressed to give credit to Gödel and to emphasize the incompleteness theorem and its proofs were not Asa Zook's inventions.

Incompleteness! So that's why he laughed. "Wait Asa! Completeness takes away our guarantee of continuity. Is that right?" He was nodding yes. "Because contiguums lack fixed points, is that right dear?"

"Yes, Joyce. But now is the appropriate juncture to introduce the exception. Contiguous entities may share a fixed point only if the entities are without content, only when we contrive in advance, by means of our definitions, to have a completely arbitrary relationship between the two entities -- which we can do in logic if the statements are either empty or meaningless."

"Such as?"

"The many famous paradoxes in conventional logic. One such employs 'grue' and 'breen' instead of blue and green. But there's a young metaphysician at Yale who is showing rather convincingly that necessary relationships grow out of the nature of entities."

"Isn't that obvious, Asa?"

Asa roared again. "To all but the faithful followers of David Hume and of conventional logic."

"Oh, you mean the lack of connection between cause and effect? Golly, I never put the two thoughts together in my mind before, Asa. I mean, I always accepted Hume's arguments that no necessary relationship exists between a cause and an effect. And yet I've spent my entire career in politics searching the nature of causes to explain effects."

Another thought struck her. "Asa. Doesn't what you say make logic sort of illogical?"

He chuckled. "No, dear." Content-free statements could be logical.

"But the moment we put stuff in them..." What she was about to write seemed nasty. "Let me start that over, Asa. When our statements have meaning, they can only be consistent when they are part of a continuum. Is that correct, Asa?"

"We can only guarantee their consistency if the statements enjoy a fixed point and are thus part of a continuum. The key word is guarantee."

"And you mean that in the rational sense, of course."

"Yes, dear."

Joyce simply had to take a break. She took the Journal along with her to the bathroom and pondered what he'd been saying. This free will business really appealed to her. She'd try to paraphrase it in her own words. I need a 'for instance' for this fixed point thing, she thought. Upon return to the room, she went to the dresser, took out a diaphragm and slipped it over her fingers.

"In other words, Asa" she flexed the rubber dome, "when this is on me, one point between it and my cervix remains unchanged following continuous transformations, right? And the fixed point is an existential necessity of the union between the diaphragm and me. And while we can't measure the fixed point -- because we don't have a reference -- we can establish its existence in a system continuity. By demonstrating true continuity. Does that capture what you've been saying?" He nodded, yes. She slipped the diaphragm into the pocket of her jeans and gave it a pat.

Millie brought tollhouse cookies and hot chocolate at midmorning. Time seemed to be flying, Joyce observed after Millie left. How were they going to get the continuum into living systems?

"With memory, Joyce. Through phase."

"The phaseogram?"

"Yes. Do you remember how Gabor constructed the hologram?"

"By making the object and the reference waves collide --interfere!"

"Do you recall the condition necessary for waves to produce interference patterns?"

What was that damn word. Yes. "Coherence! Being in step. Mathematically in step! "

"Ordinarily two light beams are very incoherent and don't form interference patterns. But Gabor -- and Young and Fresnel long ago -- created interference patterns with ordinary light."

"I remember you saying that, Asa. And it's always troubled me. But I didn't want to ask dumb questions...until now."

"The question is not dumb. Not at all. Gabor, like Young and Fresnel, produced object and reference beams from the same source. Before the light reached the target, the future object waves and reference waves formed a continuum."

"Had to have a fixed point?"

"Guaranteed. And the distortions that objects make on the object waves -- what will make for their images -- the distortions do not change the fixed point."

Joyce jumped to her feet. "I see it. I see it. Oh Asa, I see it. For the first time, really. The object and reference waves share the fixed point. By definition! That's what coherence comes down to, doesn't it -- potentially sharing in the fixed point."


He paused as though waiting for her say it. And by jumping- g- sassafras, she was going to say it. "That's what the phaseogram, in principle, is. A memory of fixed points! Reminiscences or a regenerated organs are...are...can you help me out with an appropriate phrase here, Asa."

"Reconstructions when the readout and the phaseogram recreate the original entity via their mutual fixed- point relationships.

"Regeneration and remembering are utilizations of fixed points to allow the original entity to live again -- to exist once more in physical reality."

"Oh, Asa, I..." But before she could think of something appropriately poetic, he went on.

"I'd like now to show you a fundamental feature of a memory- bearing continuum, Joyce."


"Illustrate our Hegelian logic."

"By Hegelian, Asa, you mean thesis and antithesis combining for synthesis. Opposites attract!"

"Yes, except that, as we shall eventually see, the least number of primitive components is five, not three."

"That sounds like Karl Marx, Asa." He nodded yes. (Her political enemies would just love to have heard her say, Karl Marx!)

He asked her to get several strips of paper tape from the adding machine, and "a handful of bar magnets from the drawer of my typing desk, please, while you're there. Also a roll of Scotch tape. And a scissors, too, dear.

"We can use opposite poles of the magnet as analogs of thesis and antithesis," he said when she returned.

"A magnet itself would be a synthesis then?"

"Good observation."

"And its fixed point will be at some uncertain locus approximately half way between north and south pole?" He smiled.

"Imagine, Joyce, that we're at the beginning of time for our bar magnet universe. Imagine that we cleave a magnet lengthwise into two sister magnets. You can simulate creation by placing two bars parallel to each other with their north poles oriented in the same direction. What occurs?"

"They-- whoops!..repel each other." He'd painted one end of each magnet red and the other black. The magnet she was holding in her right hand jumped loose and fused with the one in her left hand, red to black. "I had an Hegelian synthesis happen to me here." She pulled the two magnets apart and then forced them to repel each other.

"Now, Joyce, please take Scotch tape, and fasten a magnet crosswise to either end of a long strip of paper. Orient the red poles toward one edge and the blacks toward the other. Next, print THESIS at the red edge and ANTITHESIS on the black. All right, now turn over the paper strip, and write THESIS and ANTITHESIS at the same locations as THESIS and ANTITHESIS on the other surface."


"Now Joyce, bring the two magnets together, in parallel. But this time let them interact naturally -- let red and black fuse."

When she did as he asked, the strip of paper formed a loop. "As a marker, put an X on the THESIS of one side, Joyce. Ready? Now beginning at the X, draw a line along the strip... around ...around...until...that's it... until you arrive back at the X where you started from."

She carried out his instructions, drawing and drawing...until: "Asa! The line jumped from the THESIS to the ANTITHESIS surface of the tape. But...Wait."

"There is only one surface, Joyce."

"Oh! This is a Möbius strip. Let me write it down in the Journal." Timothy was going to love this one. For he'd first playfully made her aware of Möbius strips, the little rascal. And she suddenly realized that she was writing for him. When she finished, she read aloud Asa what she'd written: "'If we create a thesis and an antithesis from the same source and then let thesis and antithesis rejoin, the resulting synthesis is a Möbius strip.'" Is that reasonably put, Asa?"

"For the simplest condition, yes. Add, though, that the newly synthesized Hegelian universe has five fundamental components."


"Two theses, two antitheses and an indeterminate domain of one or more fixed points -- a fixed axis."

"I get six domains, Asa. Two theses, two antitheses and two fixed axes. Wait! I see, I see, I seeeee!" She jumped up, kissed him, and he flinched. "Oh, I'm sorry, dear. But, I see it, I see it I seee it!. The domains share the fixed axis. And"; she stopped to reexamine the Möbius strip; "even though we know it is there, and roughly where it is, we can't precisely localize it -- the fixed axis, I mean." She suddenly realized something else. She had to go -- no run! -- to the ... she didn't quite make it this time.

After she returned, Asa had her find a reprint of an article he'd written many years before. "The diagrams will be useful, Joyce." The first, "Step 1," the captions said, was simply the division of a graph into four quadrants. In step 2, he'd labeled REAL the quadrants to the right of the vertical meridian and IDEAL those on the left. In step three, he'd used the equator; the upper quadrants he identified as OBJECTIVE, and the two below he'd called SUBJECTIVE. In the fourth step, he gave names to each quadrant: the upper left, OBJECTIVE-IDEAL, he called Math (presumably short for mathematics); the upper right -- OBJECTIVE-REAL, was Science; SUBJECTIVE-IDEAL was religion; SUBJECTIVE-REAL, he called Art. In the fifth diagram, which he entitled "Function" he had drawn what looked like a 4- leaf clover, one leaf in each quadrant, and the connecting stem at the point where the equator crossed the vertical meridian.

The sixth and last diagram was sort of weird, Joyce thought; the caption said, "Local Constants; Asa had added a heavily stippled zone just outside each clover and in the central area of the figure. Bounded with a heavy line, the stippled zone and the cloverleaf function within it, took on the form of a solid upper case H. The upper leaves were marked Thesis, and the lower two, Antithesis. In the cross piece of H, the word overlapping the four domains, he'd printed PHILOSOPHY.

Joyce looked quizzically at Asa. What is this all about, sweet? "

"It's a unified Hegelian epistemology. We can use the H as a model of our general theory, dear. Also the analytical proof of the theorem is at the end of the article. Do you think you could redraw the essence of what is there in Figure 6?"

Joyce took a fresh piece of graph paper and, speaking as she worked, made a rough likeness of the H. "I find the H fascinating, Asa. I mean, what you seem to be saying is that you can go from every branch of knowledge to any branch of knowledge."

"Via philosophy."

"Via philosophy! But dear, why the stippled zone?"

"The cloverleaf is our universal, our function -- that with which we can transform from one domain to any other. The stippling stands for our local constants -- quirks peculiar to a given neighborhood. But, if you please Joyce, take the scissors and cut the H free."

When she finished cutting, she held up the H.

"Now Joyce, please bring each thesis and antithesis across the diagonal and, with Scotch tape, join them into a Möbius strip"

After she finished cutting and pasting, Joyce worked her fingers along the now folded- over H and convinced herself that it, indeed, had a single surface. "Through philosophy! Is that where the fixed point is, Asa?"


"But... Maybe I'm too stupid or something...but..."

"You're not stupid my darling. I know precisely your concern. And it is your gift that allows you to see it. How can we know precisely where the fixed point is? Is that your concern, Joyce."

"Yes it is, Asa. I mean, you made such a strong point of that before...."

"Examine the crossbar of the H.

She looked for a moment. Then she realized the stippling obscured the point of contact between the four leaves. "Asaaaaaa!

I see it. I really seeeeeeee it." She jumped up and clapped her hands. "The quirks make it impossible to say just where the fixed point lies. Is that it, dear.

Doc Clever arrived just as Joyce was about to ask Asa about the analytical proof he'd mentioned. Both she and Asa had declined Millie's offer of lunch. While Doc Clever was examining Asa, Joyce tried to use the forced break to fix toasted cheese sandwiches. After she permanently welded cheddar to the skillet (one of Millie's new set, no less) and the bread caught fire, and she wasn't sure just how much, or precisely where, the coffee went in a percolator (and, naturally, there was no such thing as a jar of instant in a Zook house), Joyce fortuitously peeked into the cookie jar. It was full of an assortment of fresh oatmeal- tollhouse hybrids and raisinized coconut macaroons -- Asa's favorites. And a note, "Kisses and loves for Joyce and Asa. X.X. Millie, et al. PS. I'll bring supper up to you."

Doc Clever said Asa looked fine, "considering." He wanted that wrist kept taped up. He declined the glass of milk but did help himself to a handful of cookies to munch between house calls. Asa emphatically rejected any medication. Doc Clever did not press the issue. He picked up his bag and Joyce walked him to his car.

She should make sure Asa kept that wrist bandaged up. "And," he paused and took a deep breath. "I don't like his blood pressure, at all, Missus Zook." If it didn't start coming down, Asa was going to have to go on medication. "And stay on it. I'm sure he knows that as well as I do." He'd be by again in the morning, if that was convenient for them. It was, Joyce said.

Back in the room, she started to open a dialog on health. "Dear- - "

Asa raised his hand: "Please, Joyce."

She took a deep breath, walked to the bed and gave him a light kiss on the top of the head. Then she plunged back into philosophy.


Asa had her read the proof of the theorem in his article. But it was a hieroglyph of mathematics. "I can't come to grips with it Asa. I'll have to have a visual metaphor or something, dear."

"I shall have to work on that, Joyce. I do not want to lose momentum and at this time must defer the task of illustrating the proof of the theorem. But I believe you can readily see how the continuity of the system depends on the crossbar of the H."


"Continuity, the cardinal attribute of our Möbius H universe, vanishes without PHILOSOPHY." Her words gave her a cold chill. And almost as though his ears had heard her thoughts, he spoke:

"It may comfort you to know Joyce, continuity is about us, everywhere."


"When the sun rises above the stillness between waves of sea. Whenever or wherever an Hegelian thesis and antitheses remember; when the one becomes part of the other. Nature chooses life, Joyce. And when we elect beauty or goodness or love or life, we reenact Nature's most essential principle: When the one becomes part of the other." Joyce set the Journal on the floor, rose and held out her hands, one palm up and the other down. Asa offered his hands. They touched middle fingers, and then complemented the others. "And as we are this moment," she said. "As you and I freely choose to be, Asa."

She was immortally part of him, Joyce knew. And for as long as either lived, immutably so. In spite of his sore and battered body, despite every promise she'd made to herself, she let him take her into his arms. Their love had remembered itself.

Copyright, 1996 by Paul Pietsch, all rights reserved. May be copied for personal, educational or other non-commercial "fair-use" purposes, as defined by U.S. copyright law.

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