Alexander’s Sphere and Impossible Objects
David Pierce has written an article on the Alexander sphere at Knottedness, where he makes a reference to my page Alexander’s Horned Sphere, stating that I “show no sign of understanding what the horned sphere is for.” We’ll leave the discussion regarding what it might mean to talk about what a mathematical construct is “for ” - that is, what its purpose might be till later; first we will talk about some aspects of Pierce’s notions regarding the Alexander sphere.
As I point out on other pages, such as Understanding Limits and Infinity, definitions involving infinities can require the inclusion of a limit condition to avoid contradictions - and it is also important to realize that, depending on what aspect of the original definition that one wants to precisely delineate, different limiting conditions can be applied. It is important to recognize that such different limiting conditions can result in different mathematical objects, and this can simply mean that the original definition did not in fact define a well-defined single mathematical object.
In his article, Pierce starts off by using a physical analogy of an elastic snake with a string loosely tied around it. (Footnote: Pierce’s analogy is: If you roll out a lump of clay into a snake, then tie a string loosely around it, can you contort the ends of the snake, without actually pressing them together, so that you cannot get the string off? You can stretch the clay into a Medusa’s head of snakes, and tangle them as you like, again without letting them touch. If you are allowed to rest the string on the surface of the clay, then you can get it off: you just slide it around and over what was an end of the original snake. If however no point of the string may touch the snake, then you can trap the string so that it cannot be removed.) I’ll talk about that first simply because it ties in quite well with his later statements. As he says, you can pull and twist the snake any way you like and as long as you don’t make the snake touch itself to make an enclosed loop, then you can always remove the loop of string. He then says, if you also include a rule that the string cannot touch the snake, then you can prevent the string being removed even if no points of the snake actually touch to make a loop.
Clearly, there are two obvious cases here, one is where the snake does touch itself into a loop and the other is the case where it does not touch itself into a loop, and there is a distance between opposing points on the surface of the snake (where this distance is exterior to the snake).
What Pierce is trying to claim in his article, in a rather roundabout way, is that one can have a variety of mathematics where one can define a mathematical equivalent of the snake and the string, where besides the two cases just mentioned, there can be a third case where the snake is “infinitesimally close” to touching itself into a loop - where the “infinitesimally close” implies a distance that is smaller than any other distance but nevertheless, at the same time it is greater than zero. This provides a convenient basis for Pierce’s claim that the string cannot be removed from the supposedly transformed snake, since the string around the snake cannot have a smaller thickness than this “infinitesimally close”. (Footnote:
J. W. Alexander originally stated the claim as:
“An example of a simply connected surface bounding a region which is not simply connected.”
In other words the claim is although every protruding horn of the surface of the transformed sphere is still separated from its opposing protruding horn of that surface, they are in effect described by the oxymoronic “infinitesimally close but not coincident”, so that there are parts of the volume that are exterior to that surface that are not simply connected to each other - since no part of this volume could be in any region that is this “infinitesimally close but not coincident”.)
The notion that Pierce and others have regarding the Alexander sphere is that since the iterations of creating new horns never terminates, then the opposing surface of the horns can never coincide - and at the same time, the distance between them must become “infinitesimally close” - but where there is also the concomitant claim that it cannot be zero since the iterations never terminate. This notion simply ignores the fact that the notion of “infinitesimally close but not zero” is an illogical concept that fails to ascribe any definitive numerical term to it - in common parlance it is a fudge. The result that inherently accompanies the notion is the illogical result that two opposing surfaces can be closer than any other distance, but at the same time they cannot be coincident.
This claim is closely related to the notion in one dimension that a line is actually composed of infinitely many points that simultaneously exist, a notion that began with Cantor’s ideas that elements of infinite sets can somehow all “exist” simultaneously. (Footnote: See Cantor’s Grundlagen, Section 1 where he talks about the “actual-infinite”. David Hilbert eulogized Cantor’s notion of two different types of infinity, a “potential infinity” and an “actual infinity”, see “On the Infinite”.) This assumption means that, given any defined point on a line, that necessarily implies that there “exists” a specific point that is immediately adjacent to one side of that point, and another specific point that is immediately adjacent to the other side of that point. But that is an impossibility - two “adjacent” points cannot exist - since given any two points, there are always infinitely many points between them (since any two different points have different numerical values, there is an interval between them in which there are limitlessly many points). In other words, there is an assumption that such points “exist” in the mathematical system even though that entails an illogical impossibility. Furthermore, the assumption necessarily ensures that any such postulated impossible points cannot be defined in that mathematical system. (Footnote: See also Indefinable Numbers.)
In higher dimensions, the corresponding assumption is that a surface is actually composed of infinitely many points that simultaneously “exist”, and that a volume is composed of infinitely many points that simultaneously “exist”. So that given any defined point, the assumption necessarily implies that there “exist” limitlessly many specific points that are all immediately simultaneously adjacent to that point, even though, as for the one-dimensional case, that is illogical and impossible - as for the one-dimensional case, two “adjacent” points cannot exist.
For a one dimensional line, the notion of two closed intervals that might be supposed to be “infinitesimally close” but not coincident cannot correspond to any actual interval between the two intervals, since by definition for any such interval there is another smaller interval - that leaves the only other possibilities are that this “infinitesimally close” might correspond to the notion of two open intervals touching each other, or else there is a single point separating two opposing endpoints of the two open intervals. But both of these necessarily imply that there “exist” at least two “adjacent” points, which, as we have just noted, is illogical and impossible. See also the Appendix below.
At this juncture I would note that I am not claiming that mathematicians actually claim that they define any specific adjacent points - but they create scenarios which necessarily imply the existence of such impossible things, where the inherent absurdity is ignored on the basis that the positions of such adjacent points are never specifically defined.
The alternative to such illogical notions is that, given a specific point on a line one can state that there is no limit to the quantity of points that can be defined that are closer to that point than any other given point. There is no need for any Platonist assumption of the simultaneous existence of limitlessly many points. (Footnote: See also Platonism and Platonism and the Infinite.)
Similarly, for three-dimensional surfaces, the notion of “infinitesimally close” but not coincident cannot correspond to any actual volume between two planar surfaces, since by definition for any such volume there would be another smaller volume - and, as for the one-dimensional case, that leaves the only other possibilities are that this “infinitesimally close” might correspond to the notion of two surfaces touching each other, or else there is a single plane separating any two opposing points of the two planar surfaces. Similar considerations apply for non-planar surfaces. And as for the one-dimensional case, both of these necessarily imply that there “exist” at least two “adjacent” points, which, as we have just noted, is illogical and impossible, and see also the Appendix below.
And as for the one-dimensional case, the logical alternative is to simply state that limitlessly many points can be defined that are closer to a given point than any other given point. The key point here (no pun intended) is that if a definition results in an implication of impossible adjacent points, then that indicates that there is a problem with that definition and/or the system that the definition resides in. (Footnote: This is not to assume that defining mathematical systems that ensure that such scenarios cannot arise is a simple matter, see below What is the Alexander sphere for?)
Pierce’s case involves a definition that includes infinite iterations, but Pierce avoids any mention of a limiting state of the repeated iterations, which is the key to generating a logically valid result. We can use a specific case to illustrate the fallacy involved in Pierce’s argument. We take an initial solid torus with part of it removed, with the cut ends parallel and separated (this object is homeomorphic to a sphere - it can be “stretched” to transform it into a sphere). Can these cut ends be separated by a mid-plane, that is, by a flat planar surface - a surface that is precisely one point "thick" everywhere?
No, as explained above, since that would mean that there would be three planes, the mid-plane, and planar surfaces immediately “adjacent” to it on either side, where these adjacent planar “surfaces” are part of the transformed surface of the original cut solid torus. That scenario would ask you to believe that the notion of such adjacent planes is logically coherent, which is no different to asking you to believe that the impossible notion of “adjacent points” on the real number line is a logically coherent notion. And for the same reason, the cut end surfaces also cannot be immediately adjacent to each other with no intervening points.
We can consider the limiting state of iterations where the distance between the cut ends of the solid torus decreases by some ratio at each iteration. Consider the simple case where we start with a cut solid torus with the parallel cut ends by separated by a specific distance. The mid-plane between the two cut ends can be precisely defined. We define iterations where the cut ends stay the same area but the separation distance is reduced at each iteration to 1⁄3 of the previous separation distance. At each iteration we still have a surface that is the boundary of its interior and the boundary of its exterior, and at each iteration the new ends are part of that surface. The description of infinitely many iterations without any specified limit condition implies that the cut ends never meet. But the crucial point is that without the application of a limit condition, it is also the case that after every iteration, there always exists a specific well-defined non-zero distance between the cut ends, where that distance is defined in terms of the nth iteration - and at no iteration in the limitlessly many such iterations does the integer n that is the number of an iteration somehow magically transform into a non-integer “infinity”. This fact simply indicates that a logically valid limit condition must be added to provide a valid definition of the transformation of the original sphere, and which gives a logical limiting state for the limitlessly many iterations.
The analysis of the one dimensional case of an interval on a straight line extends directly to cases of more than one dimension. For the case of an open interval on a one-dimensional line, where the endpoints are not included in the interval, the limiting state of the repeated decrementation of the open interval is an open interval of zero width, which by definition does not include any points - that is, the interval disappears completely. This is explained in detail in Understanding Limits and Infinity and Understanding sets of decreasing intervals.
Consider the three-dimensional case where the cut end surfaces of the cut solid torus are not defined as part of the volume included between the cut ends, but are defined as a part of the cut solid torus. This corresponds to the case above of an open interval on a one-dimensional line. In the same way, the repeated decrementation of the space between the cut ends of the solid torus is where that intervening space disappears completely, and does not include any points at all - and hence the limiting state of the cut solid torus is a complete solid torus, and where there is no surface with no local exterior. Hence the original cut solid torus, which is homeomorphic to a sphere, has been transformed into a solid torus which is not homeomorphic to a sphere.
Another three-dimensional case is where the cut surfaces are included as part of the volume included between the cut ends, and in that case we get a somewhat different result, which is that the surface is transformed into something where part of it is embedded into the overall resultant transformed volume, and has no local exterior: the diagram below is a cross-section of the result.
Cross-section of transformed solid
Hence for both this case and the previous case, the original cut solid torus has been transformed into a solid which is not homeomorphic to a sphere.
The case where the surface area of the ends also decreases at each iteration makes no difference to the above principle; a limiting state is possible where the space between the ends limit to meet at a single point, and where this single point is a point of the surface - and is also a point of the transformed solid. For reasons already explained above, there cannot be a single point where the ends meet and where the point is not a point of the surface, since that would entail that there are specific points of the surface that would be adjacent to that point, which is impossible - the diagram below is an attempt in two dimensions to illustrate that impossibility:
An impossible case: A point flanked by two surfaces
The case where at each iteration each end splits into two also makes no difference to the principle involved here, and hence the limiting state of the transformed sphere must have connected loops, and it will have infinitely many such loops. On the other hand, if one defines that there is a definitive limiting distance that the distance between the protrusions approaches then there cannot be any loops in the limiting state.
What is the Alexander sphere for?
I am not claiming that the definition of continuity of surfaces and the definition of transformations of a surface are simple matters. But if we consider the question that Pierce suggests, which is: “What is the Alexander sphere for? ”, that is, “What is its purpose? ” we can note that he doesn’t offer any answer to the question.
- One answer might be that it is useful as it indicates that there is a problem in the foundations of the mathematics, and hence the purpose of such a case is to stimulate a search for improvements to the foundations of mathematics.
- Another answer might be that it does not have any purpose at all, it is simply an amusing triviality that is part and parcel of mathematics.
The Alexander sphere is just one of a multitude of strange mathematical results that arise in conventional mathematics. But instead of reflecting on why such results can be generated, it is perplexing to observe that the overall mindset seems to be one of “Well, that’s just the way things are.” rather than questioning “Why is this happening? ” or “How can we improve on this? ”. It is a mindset that is epitomized by Von Neumann’s aphorism: “…in mathematics you don’t understand things. You just get used to them.” (Footnote: As quoted in The Dancing Wu Li Masters: An Overview of the New Physics by Gary Zukav, Bantam Books, 1979, footnote at p.208.)
For some unfathomable reason, conventional mathematics of the 21st century proceeds according to the dogma that its current foundations (Footnote: Zermelo-Fraenkel set theory is the favored foundational theory, although it comes in different versions; ZFC includes what is called the The Axiom of Choice. Another version includes the assertion that the Continuum Hypothesis is true, another asserts that it is false. Several other set theories have been proposed but their differences from Zermelo-Fraenkel theory are insignificant.) nearly all of which were set down over 100 years ago and have remained essentially unchanged since, are the best possible foundation of mathematics for all time, and that it is not possible that there could ever be any better foundational system. (Footnote: The similarity to religious dogma that is based on an unchangeable set text is striking.)
This is astonishing - in no other field of study would such an attitude be accepted. In no other field of study is there an unwritten rule that no journal shall publish an article that questions foundations of its subject that result in problematic results. In other fields of study, strange results, contradictions and conundrums are taken as indicators that something is not quite right in the overall understanding of their subject, and as a result, efforts are made to try and uncover deficiencies in that understanding. Today’s mathematicians, on the other hand, sit back, and faced with any challenge, simply repeat ad nauseam foundational ideas that were almost all established between one hundred to one hundred and fifty years ago and which in many cases were idea borne out of religious beliefs rather than any logical considerations, see for example Cantor’s religious beliefs and his transfinite numbers.
Impossible adjacent points
In two dimensions one can define a curve consisting of a circle and a radius as in the diagram below, so that we have a curve where for part of that curve there is no local outside (in the vicinity of the radius in the diagram) although the curve as a whole has a well defined inside and outside:
Now, suppose that one pretends that one can transform the curve so the radius is not itself part of the curve but the curve itself lies immediately adjacent to either side of the radius - so that along either side of this radius the curve is separated from itself by a line that is a single point “wide”; the diagram below tries to illustrate the impossible case, viz:
But, of course this is impossible, since there cannot be a line that is immediately adjacent to the radius, because then we would have impossible adjacent points, consisting of lines either side of the straight line that is the radius. And now suppose that one can take single points on the radius and pretend that those points are not part of the curve itself but the curve itself lies immediately adjacent to either side of these points; the diagram below tries to illustrate the impossible case, viz:
And again, this is impossible - one cannot have a single isolated point that is surrounded by the continuous curve, since that would be a circle of zero radius enclosing a single point, and we would then have adjacent points - the single isolated point and a “circle” of zero radius that consists of a “circle” of points all immediately adjacent to that single point, which is impossible.
In three dimensions the same considerations apply - for a well-defined continuous surface such as a sphere - if one defines modifications to that surface such that the new definition still defines every point on that surface, then it follows in the same manner that it is impossible for there to be another surface where the transformed surface is immediately adjacent to it, and also impossible for the transformed surface to be immediately adjacent to it on either side of that surface. Figure 2 can be viewed as an attempt at a cross-sectional illustration of the impossibility of such a three-dimensional case.
And in the same way as for the two-dimensional case, it is impossible for there to be a point where the transformed surface is immediately adjacent to that point on either side of that point, and Figure 3 can be viewed as an attempt to illustrate the impossibility of such a three-dimensional case by a cross-sectional illustration.
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