Toward Footnote GA156

Footnote GA156 (added Jan. 18, 2023):

Consider this passage from The Interpretation of Dreams by Sigmund Freud (A.A. Brill's 1914 translation):

If we read, e.g., three of Hildebrandt’s “Alarm Clock Dreams,” we will then have to inquire why the same stimulus evoked so many different results, and why just these results and no others.

 “I am taking a walk on a beautiful spring morning. I saunter through the green fields to a neighbouring village, where I see the natives going to church in great numbers, wearing their holiday attire and carrying their hymn-books under their arms. I remember that it is Sunday, and that the morning service will soon begin. I decide to attend it, but as I am somewhat overheated I also decide to cool off in the cemetery surrounding the church. While reading the various epitaphs, I hear the sexton ascend the tower and see the small village bell in the cupola which is about to give signal for the beginning of the devotions. For another short while it hangs motionless, then it begins to swing, and suddenly its notes resound so clearly and penetratingly that my sleep comes to an end. But the sound of bells comes from the alarm clock.”

“A second combination. It is a clear day, the streets are covered with deep snow. I have promised to take part in a sleigh-ride, but have had to wait for some time before it was announced that the sleigh is in front of my house. The preparations for getting into the sleigh are now made. I put on my furs and adjust my muff, and at last I am in my place. But the departure is still delayed, until the reins give the impatient horses the perceptible sign. They start, and the sleigh bells, now forcibly shaken, begin their familiar janizary music with a force that instantly tears the gossamer of my dream. Again it is only the shrill sound of my alarm clock.”

Still a third example. “I see the kitchen-maid walk along the corridor to the dining-room with several dozen plates piled up. The porcelain column in her arms seems to me to be in danger of losing its equilibrium. ‘Take care,’ I exclaim, ‘you will drop the whole pile.’ The usual retort is naturally not wanting—that she is used to such things. Meanwhile I continue to follow her with my worried glance, and behold! at the door-step the fragile dishes fall, tumble, and roll across the floor in hundreds of pieces. But I soon notice that the noise continuing endlessly is not really a rattling but a true ringing, and with this ringing the dreamer now becomes aware that the alarm clock has done its duty.”

Here we note that the dream construction appears to extend time backward. That is, the stimulus of the external sound is seemingly "forgotten" long enough for the formation of a passage of dream time.

These dream reports demonstrate the principle of dream reality construction that includes a sense of presence and an "unfolding of history" or passage of time. That is, the dream self invents its own history and dwells within it as if it is within a flow of objective time. The "now" of the dream self is out of synchrony with the "now" of the waking self.

Of course, we must concede another possibility: that the dreamer's body is precisely clocking the passage of linear time (as with a computer clock) and is somehow anticipating the alarm and so signals the self to begin preparing to wake up. In that case, a part of the person's mind-body system is setting up a defense against too much of a wake-up shock. Further, that would appear to be a motive for the first case. By "forgetting" the noise and inventing a dream reality, the mind-body system avoids what it regards as too forceful a transition into the unforgiving world of "cold reality," where it is no longer free to easily express its impulse life.

In fact, there is reason to suspect that both forms of mentation occur, tho not usually simultaneously.

Freud goes on:

The following dream of Maury[48] has become celebrated. 21He was sick, and remained in bed; his mother sat beside him. He then dreamed of the reign of terror at the time of the Revolution. He took part in terrible scenes of murder, and finally he himself was summoned before the Tribunal. There he saw Robespierre, Marat, Fouquier-Tinville, and all the sorry heroes of that cruel epoch; he had to give an account of himself, and, after all sort of incidents which did not fix themselves in his memory, he was sentenced to death. Accompanied by an enormous crowd, he was led to the place of execution. He mounted the scaffold, the executioner tied him to the board, it tipped, and the knife of the guillotine fell. He felt his head severed from the trunk, and awakened in terrible anxiety, only to find that the top piece of the bed had fallen down, and had actually struck his cervical vertebra in the same manner as the knife of a guillotine.

This dream gave rise to an interesting discussion introduced by Le Lorrain and Egger in the Revue Philosophique. The question was whether, and how, it was possible for the dreamer to crowd together an amount of dream content apparently so large in the short space of time elapsing between the perception of the waking stimulus and the awakening.

From the perspective of the awake state, the "internal clock" explanation will not do. The "crowding" surely implies that the dreamer invented a reality that projected "back in time" from the rude interruption from the external world. The dream self proceeded down an invented time stream. The dream's self's "now" was encased in a flow of time that was invented by the organism's psyche. The Maury dream's "crowding" idea, Freud adds, has been countered by "many arguments." We may suspect that the objection refers to the back formation of a time flow.

Yet, we must concede the possibility of a false memory of a dream sequence that was concocted after the intrusion of the external stimulus. But such an explanation is incomplete without accounting for the purported experiencing of the subjective passage of time.

Are the reals actually well-ordered?

The well-ordering theorem, also known as Zermelo's theorem, says that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering.

A strict total order on a set X is a strict partial order on X in which any two elements are comparable. That is, a total order is a binary relation < on some set X, which satisfies the following for all a,b and c in X:

Not a < a (irreflexive).
If b < c and a < b, then a < c (transitive).
If a =/= b, then a < b or b < a (connected).

Zermelo's approach was to argue that each element of a set could be withdrawn from it sequentially and then, if necessary, put into correspondence with a set controlled by the < relation. That is, the members of X are ranked in accordance with Y, a set of numbers ordered by <.

In the point set (0,1] it is obvious that no smallest number can exist after 0.

That is, lim_n-->inf. rⁿ = 0. Otherwise, rⁿ --> rⁿ⁺¹. That is a "small number" always implies a yet smaller number. Or see the epsilon-delta proofs.

But then, the same applies to any real, not only 0, that appears in the point set [0,1].

This then means that if there is a smallest set after 0, that set must be empty -- which would violate the well ordering concept. Only if we permit well-ordering to mean that any two members of a set imply a strict ordering, are we out of the woods.

Consider:

x₁ < x₂ -->
x₂ < x₃ -->
.
.
.
x_n-1 < x_n

That is, redefining well-ordering and using ordinary mathematical induction should help.

Of course, it has been argued that the axiom of choice proves well-ordering (in first order logic). I suppose that would mean if we have a set of ever smaller elements, we should be able to pick out one by some metaphysical method and call that the smallest. Rather than go to that length, I prefer a modified definition of well-ordering. In fact, all we really need to say is that any two members of a set of numbers obey the relation <, as in a < b or b < a. That is, ALLx e R(x_a < x_b or the converse).

Where are the non-computablesin the binary tree picture of the reals?

 

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NOTE. Occasionally, over the years, I have published discussions of the binary tree picture of the reals. This post continues in that vein.

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We accept the idea that the set of decimal extensions is bijective with the reals. For convenience, we convert to the base-2 number system, such that the reals are comprhended by the set of binary extensions.

To help us in our intuition, we employ a binary tree with a denumerable infinity of stages/levels. That is, at each level is a set of 2ⁿ nodes, or branching points. At level 1, there are 2 nodes...at level 10, 1024 nodes... Every left branch is designated 1 and every right branch 0.

Example: At level 3, we have,

  
  L R L R L R L R
  
  which is equivalent to
  
  1 0 1 0 1 0 1 0
  
  

The complete set of stages corresponds to the set of natural numbers N. The number of nodes in the "final" stage is taken to be 2^ℵ_o.

Hence, we see that every path is well-ordered. (But AC is needed to tell us those distinct paths reach the "bottom" stage.) Now we know from the Goedel and Turing theorems that the cardinality of the computables is ℵ_o, which is that of N.

It is also evident that every rational is represented as a finite digit string that recurs infinitely often. That is a set of all 0's means 0; a set of all 1's means 1; a finite string of 0's and 1's that is either followed by all 0's or that recurs infinitely often means some rational.

In addition, we have the set of computable irrationals. This means there exists at least one procedure that converts a numerical input into some 1's and-or 0's without ever doing so periodically and that this procedure is infinitely (or indefinitely) recursive, as in f(x) = x_next.

Now each path, up to any finite level, we must construe as a proto-number. All one can prove about any such path one follows is that if we know the algorithm, we can say whether the number is rational or irrational. And, if one does not know the algorithm, all one can say is that -- thus far -- the number is computable, as it can be defined stage by stage. In fact, at any finite level, the path has thus far been computed. So there is really no way to discern a noncomputable from the ordinary perspective.

But if we accept that the power set of N exists, in that case it must be that the "bottom line" point set of 2^ℵ_o "originates" the non-computables.

That could only mean, I suggest, that there must be many digit strings (from the "top" or "our" perspective) followed by an infinity of 0's that are then followed by a "finite" (from "bottom" perspective) string of 0's and 1's.

Such numbers are said to be undefined because their paths vanish as they climb skyward. Their paths all converge to the number 2. But they have vanished "before" coming into our finite range.

I concede such talk is inexact. Our language is unsuited for such conceptualization. Yet, I suggest, that intuitively we can discern that the "bottom up" paths must contain the non-computables, but that we cannot, from "top down" obtain them.

This picture does not suggest much beyond what is already known: that the continuum hypothesis is independent of the axioms of standard set theory.

A way to think about that is to note that the set of the computables K and its complement K^c comprise the reals. And R\K is equinumerous to R.

But we must use AC to "pick out" an arbitrary x ∈ K^c because there is no method in our posession for specifying any x. That is, if we define K^c by its elements, we say K^c is defined such that x is not definable. This is almost an abuse of the usual set notation {x|x ... }, which means that "any element x is defined..." So for K^c we have {x|x is undefinable}. That expression, to me, makes clear why CH is such an elusive beast.

A proof sketch of the Jordan curve theoremthat covers all wild loops

I can't remember many of the details of the following post from 2007. But I can visualize what I meant, which is that if the reals are mapped onto any finite closed loop, this set of reals has 2^N permutations, whereby each point is 0 distance from two "adjacent" points.

So that implies that there must be "wild" curves where x < y < z does not necessarily hold. Since the set of permutations covers all possibilities, we have the subset of all possible continuous loops, including an infinitude of unvisualizable ones.

As there is no way for a curve or line to pierce the continuous curve, there is a set of points which that segment does pierce and another set which cannot be intersected. That latter set is called the "inside" of the loop and the other set is outside the loop. (The case where outer points come arbitrarily close to the boundary is covered.)

The article below was intended to cover not only single loops, but pretzels, but from this vantage point I don't find it all that clear.

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Kryptograff contains Paul Conant's thoughts on scientific and mathematical matters. Conant is a journalist who holds no scientific degrees. This blog was set up after problems at the previous address: http://kryptograff.blogspot.com. Please check there for previous posts.

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The Jordan curve theorem holds for wild curves

Draft 3

Assertion
The Jordan curve theorem holds for a set of non-visualizable, "wild" curves that can be posited in accord with the Well-Ordering Principle.

Prefatory remarks
An intuitive idea of the well-ordering of the reals can be had by considering the following:

We have that any infinite digit string of 0s and 1s may represent a real. Now a string of length n digits may be ordered from least to greatest in 2ⁿ ways, with the awareness that any digit after n is a 0.

This denumerable ordering holds for any finite n. However, denumerable ordering does not hold for the entire set 2^N of course. But the Well-Ordering Principle can be interpreted to mean that the set of string permutations is precisely ordered at 2^N.

We can obtain the set of all curves, including wild non-differentiable and non-fractal-like curves thus:

Orienting ourselves on the x-y grid, using Quad I for convenience, we can arbitrarily assign any y height to any x value in the interval, say, [1,2]. By this, two neighboring x points can have wildly different heights, though there would still exist a slope for them. But, by the Well-Ordering Principle, there must exist two points that are precisely ordered and that have 0 distance between them. These two points will have no slope and constitute a wild, non-visualizable curve "section." That is, we have a situation where there is a y height for every real but no slope for the curve anywhere, even though y does not equal 0.

Though the area under such a curve must be less than 1*y_max, we may find it difficult to evaluate this integral or even give a good approximation by numerical methods.

To complete the set of all planar curves, we mention fractal and fractal-like curves, which are discussed briefly below.

Proof
We form what I call a molecule, or bubble, with the understanding that such an entity may not be visualizable with a drawing.

We may define an origin-based molecule as r = cosx where, with a well-ordering of the set of radians, r is any finite length. Additionally, r(x) = cosx is a relation with 2n-1 values, accounting for "fingers" -- any of which may be infinitely short -- and "interior molecules" beyond the neighborhood of origin. An interior bubble is defined in the same way as a basis bubble, except that its relation r' = cosx requires that for every value of x, r'(x) is less than r(x) and falls between origin and r(x). (Note: an interior bubble is defined by the relation and is not considered an a priori figure here.)

This will suffice to describe any n-loop; i.e., simple loop or pretzel, though we must shortly consider some sets.

[To help one's mental picture, we can proceed thus after forming a basis molecule whereby there are no fingers or holes. We form a second molecule, which may or may not be congruent, and map it onto the first by arranging a translation of coordinates, the effect of which is to intersect the two bubbles such that they share at least two points. All points other than the join points are then erased. The join points are those which do not intersect only a bubble's points.

[We can orient a bubble any way we like and add bubbles to bubbles to our heart's content. We may get a simple loop or an n-loop pretzel.

[A pretzel may appear when two or more bubbles intersect and the intersection set is construed to be "empty" or "background." If a pretzel hole appears, then the intersection obeys the relation requirements of a molecule. A single-hole pretzel is defined by the relation s(1) and s(2) each have one value and there is a subset for which there are exactly four distinct values of s(x).]

Now the interval [r(x)_low,r(x)_high] is (ignoring fractals), a finite line segment. So we regard those two values as the end points of the line segment. We then require a Dedekind cut between such an end point and the end point of the corresponding, coinciding half-line. In the case of fingers and pretzel holes, we have rather than a half-line, another line segment, of course.

Now the set of such Dedekind cuts maps as a continuous curve about a finite area with no end points. Suppose the boundary curve had two end points. In that case the relation r would have 0 or 2n values, a contradiction.

So, with respect to a molecule, we have that a point on the plane is either an element of the figure's Dedekind boundary set, an element of a line segment r = cos(x) such that there are 2n-1 values of r(x) (not including origin), or neither. So then, the Jordan curve theorem is proved for molecules -- n-loops -- with continuous or wild curves. (We have not bothered with the matter of nested sets of interior loops, which clearly follows from the preceding.)

In the matter of fractal, or fractal-like, curves, it is plain that a fractal construction at any finite step n is composed of a set of self-similar bubbles of diminishing size. Clearly our proof holds for any such step. By the way, we can see immediately that we can apply, at least notionally, a wild curve to a fractal, giving us a whole new set of fractals.

In the case of a fractal, the non-trivial fractal "slopes," though non-differentiable, take on infinitesimal form in the infinite limit, but what form does a wild fractal curve take? The wild part has 0 slope. So whether the curve can be said to exist in the algorithmic limit must be determined from consideration of axioms. At this point, I am content to point out such a situation.

The Jordan curve theorem also applies to any curve of infinite length that is found at, above or below the x axis. We have the relation r(x) which may have 2n-1 distinct values. We then follow the arguments above and have that a point is found in a Dedekind boundary, between a Dedekind boundary point and r(a)_2n-1 or not in the Dedekind boundary but above r(a)_max or below r(a)_min.

Footnote LN31

LN31. Two useful accounts of Leibniz's philosophy are found in Bertrand Russell's A Critical Exposition of the Philosophy of Leibniz (Cambridge 1900) and H. Wildon Carr's Liebniz (Little, Brown 1929). In his inimitable style, Russell throws a spotlight on many issues typically glossed over in standard texts. Carr's account of Leibniz's Monadology is succinct and perceptive.

I don't think Carr would agree with Russell that Leibniz was a closet atheist, though I can see why Russell thought so. Once the universe is off and running, in Leibnizian thinking, it appears that there is very little for God to do.

Though I am no Leibniz enthusiast, I draw attention to the brilliance of Leibniz's philosophical thinking, as understood by these two 20th Century British philosophers.

But, as Carr observes, Monadology is more a work of art than an instrument of truth.

Leibnez's monads were inspired by the new discoveries of the microscope: microbiotic life forms, or animalcules. He decided that it was "animalcules all the way down" in an infinite series, requiring the existence of infinitesimal monads, analogous to the infinitesimal quantities of calculus, which he helped to develop. Each monad is some sort of primeval soul.

As for evolutionary development of life forms, that could not be, according to Liebniz. The hierarchy of animalcules meant to him that all things were preformed, that each form of life tracked to a monad that was built in to it.

Yet one can pick up a whiff of the theory of evolution with his thought that life forms don't really die, but change form. The monadic souls march on. That notion certainly is evocative of the modern idea that DNA constitutes a "monad" of sorts, as we have from the book The Selfish Gene by Richard Dawkins (Oxford 1976).

Footnote Vhu53

Vhu53. In a jibe at empiricists, Leibniz compared them to unreasoning animals.

Common souls are ruled like empirics, purely by sense examples; but rational souls examine by reason (wherever possible) how far past examples are applicable to their present case. The brute souls, therefore, cannot apprehend necessary and general truths, just as an empiric can never be sure that what has often succeeded with him, without his knowing why, will again succeed with him in the future.

Further, said Leibniz,

It is probable there are rational souls more perfect than we are, we think of them as Genii and hope to be one day of their number. The order of the universe seems to require it.

As quoted in Leibniz by H. Wildon Carr (Little Brown 1929).

Footnote wyz23

wyz23. In Leibniz (Little Brown 1929), H. Wildon Carr writes:

The word monad originally was used to denote the unit of arithmetic, the monad, the dyad, the triad, etc. Yet even in ancient philosophy it was sometimes used to mean simply the individual, something which like the atom of Democritus was by definition indivisible. Leibniz meant by it a living being, using it to denote the individual which is really indivisible, as distinguished from a mathematical unit or atom, which is only indivisible by definition and cannot be indicated in any real existent. In modern philosophy the word had been used by Giordano Bruno in the identical meaning which Leibniz afterwards gave it, and Bruno had developed from it a doctrine in all essentials resembling Leibniz's conception. So striking is the resemblance that it seems as though Leibniz must have derived his doctrine from him. This cannot be the case, however, for Leibniz had worked out his system long before he adopted the name, and he had thought it out independently of any previously existing doctrine and of the name he afterwards gave it. The name monad is, in fact, employed for a new definition of substance, a definition intended to express the distinctive meaning of a new concept.