C-II Analysis

Under the metaphysical hypothesis C-II (defined in my 2005), the objective geometrical line (i.e. the structure instantiated by such linear physical continua as time and spectra; if indeed they are, and are continuous) would be full of points, # of them (defined in my 2005), even though the natural numbers would form a relatively potential infinitude (in some realistic way; see below) and the question arises, what would a mathematical theory of such a line have to be like?

To begin with, calling an arbitrary point ‘0’ gives us our origin, and calling any other point ‘1’ defines a unit of length and a positive direction. Any point, say p, between the points 0 and 1 lies in one of the tenths of that interval (e.g. between the points called ‘0·0’ and ‘0·1’), and in one of the tenths of that tenth (e.g. between 0·00 and 0·01), and so forth, and is therefore associated with an endless decimal expansion, say £(p) (e.g. 0·000…), where £ is going to be a function from the points of the line to such numerical labels. Each point in the line generates (in the endlessly hierarchical, quasi-temporal way suggested by that description) a particular w-sequence of digits, which (being potentially infinite; see below) will generally depend upon its generating point for its definition, being identifiable as that expansion via no other method.

For some points (e.g. the point exactly midway between 0 and 1) the associated w-sequence of digits (e.g. 0·5000…) will also be given by a finite formula; and for such expansions (those that can be independently specified by finite laws) we can talk about the line via those expansions (which is of course useful because the points themselves are relatively intangible). An analysis of the line in case C-II should therefore be founded upon such expansions (the legal reals, as I call them), and they would presumably resemble (the standard and/or the constructive) recursive or computable reals.

But of course, prima facie it would not even be possible for a line to be full of points, were the natural numbers a potential infinitude; and more generally, were there objective continua, whether full of points or not, the natural numbers would, it seems (as below), have to form their usual transfinitude. So perhaps I ought to say more about how C-II might be possible (about that “endlessly hierarchical, quasi-temporal way” mentioned above). First, for my problem, consider a line of points. For each natural number n, the line contains more than n points. But therefore there seem to be À₀ points by definition, since there is (at least) one distinct point per natural number.

One way of picturing that problem is to imagine bisecting a line segment, in order to pick out the midpoint (which was already in the line of points), and then endlessly repeating the bisection, of one of the two resulting lines, in order to pick out an endless sequence of points, which were there already. Note that this problem (for C-II) would appear to arise for any continua, even those in which points (or other parts) have to be constructed, since presumably something (that was not a conceptually indivisible individual) would have to be extended across the continuum—cf. how the substance of the two bits of a twig that has been broken into two is just the wood of the original twig; how, in general, any extension (other than a point), just by being extended (across more than one place), may be thought of as already containing (however fuzzily) at least two things.

It seems, then, that the existence of objective continua would force the infinitude of the natural numbers to be actual rather than potential. If so, then the arguments of Metaphysica 6, 91-109 (see my 2005) could only be valid in case C-I, while my arguments for a potential infinitude (see my 2003 and unpublished) would be simultaneously arguments against objective continua (or at least against them being full of points). Nonetheless, our experience with the standard foundations of mathematics may well have taught us to be wary of intuitive arguments about infinity (note that the argument above concerns the totality of the natural numbers) so, is it really unreasonable to consider the epistemic possibility of C-II?

After all, various attempts have been made (since the days of Aristotle) to coordinate the natural numbers’ potential infinitude with the existence of continua that are not full of points, whence, since my problem seems to arise for such continua too, some variant of such an attempt might apply to C-II. The Intuitionists, for example, maintain their (not uncommon) intuition about that potential infinity by regarding all such mathematical objects as our own creations. Now, I don’t want to go as far in that direction as they have gone, because I want to be primarily hypothesizing about the abstract structures that objective reality instantiates, but there is of course an element of truth to the idea that mathematical objects are our own creations—similarly, all the things that we see around us (e.g. cars, trees, dogs, that dog), which clearly exist relatively independently of us, are hypotheses (i.e. our own, if usually unconscious, creations) that seem to explain our experiences.

So, even granted that each natural number is (clearly) an objective mathematical (or abstract structural) object, it may be that our innate concept of the natural numbers does not precisely accord with what pluralities actually are.

It is not prima facie implausible that a degree of discord could arise at the level of their totality (and its relations with other kinds of totality) because people do have diverging intuitions about that infinitude (which lead them to conclusions that they are often less sure of, including philosophical foundations and justifications of those basic intuitions). And lest you imagine that our idea of a natural number could not possibly be imperfect, note that even our most elementary intuitions about what is and what is not objective are not always correct—e.g. we name shades of colour (e.g. Lincoln green) as though they were objective things, because the green of the leaf clearly seems to belong to the leaf, and yet we find nonetheless that colours are surprisingly subjective (and similarly, consider how our basic intuitions about the objective space that we actually occupy may be surprisingly unrealistic; or our classical intuitions about physical objects, etc.).

Perhaps, for example, our innate concept is only perfectly realistic when it comes to the individual products of the endless process, not its entirety—perhaps the extension of our concept (the totality of the products of such an endless process) is not an actual (if abstract structural) part of objective reality.

How could that be possible? Unfortunately, it will not be possible to find any very good analogies, since anything more clear would have to be finite; but we might begin with Vopenka’s paradox (cf. Lavine’s “Understanding the Infinite” 1994: 248). Consider how sure we are, that you and I are individual people, who are wholly human. Nonetheless, if some plausible theory of evolution is granted (if only for the sake of this analogy) then our intuitions about that are challenged. Intuitively, the parents of people are also people, just as the children of people are always people, and yet somehow we descended from non-human ancestors. While we could conclude that humanity is a vague or fuzzy concept, it certainly does not seem to be, not when we consider each other as we now are, our essential humanity. Note that I am not suggesting that the natural numbers are really vague or fuzzy (despite appearances), because that analogy concerns a (presumably) vague or fuzzy sub-collection of a finite set (e.g. the set of all multicellular organisms), whereas the natural numbers form an infinite collection. But that is why it is, I suggest, possible that, despite how well defined the definitive process of repeatedly adding 1 is, the extension of that process could nonetheless be only potentially infinite (the conceptual fuzziness would be infinitesimal, so to speak, affecting only the relationship between that extension and different kinds of infinite totality, such as lines of points).

The basic idea behind C-II needs some such intuitive backing (a not too inappropriate analogy, a not too implausible story) before it would be reasonable to take it seriously, as an epistemic possibility, but there are common enough intuitions (in various forms) both for and against this relatively realistic potential infinity—and such fundamental possibilities as types of continua (or types of collections) may need no more than that to begin with, because (just as with our standard mathematical foundation, standard set theory, which is a partial formalization of an intuitive, but apparently impossible, naïve set theory; or, indeed, as within much of modern physics) their justifications would come, if at all, from the coherence and utility of the resulting theories. We just need some motivation for developing such hypotheses into theories.

For a very short story, consider having an infinitely big lump of stuff (an actually infinite amount) from which we might remove unit-sized lumps, one after another, endlessly. The static nature of the original lump is to be contrasted with the endless process of the removals—each removal being followed by many more, they are all part of a process that is intrinsically dynamic, is essentially incomplete.

So, although the original lump contained enough stuff to allow us to continue to remove unit-sized lumps endlessly, would it have had to contain À₀ units? Not if the definite process of endlessly adding units, which defines the natural numbers, produces not so much an ultimate totality of all the natural numbers (an actually infinite collection), but only larger and larger numbers without end (a potentially infinite collection). In that case, the non-existence of a biggest natural number and the infinite size of the original lump merely allow the removals to go on and on endlessly—to say that the original lump already contained one unit of stuff for each natural number (i.e. my problem) is just to say that the process could continue endlessly, that the original lump was not a finite lump. Admittedly that possibility is hard to see, it being an unusual one (the usual being, of course, finite), but the alternative—that it is to say that it did contain À₀ units—is also counter-intuitive. E.g., whilst the endlessly bisected line segment (the first half, the next quarter, the next eighth, etc.) clearly goes all the way up to the endpoint, at least on the standard view, it is hard to see how the natural numbers could go all the way up to infinity, because none of them get anywhere near infinitely large.

Now, that story involves a spatial original lump, and a temporal process of removing lumps, but the idea is that those spatio-temporal aspects (for all that they may form a natural part of any informal picture of the potential infinity) are not part of what is being described, which is why I have preferred to speak of non-hierarchical (or quasi-spatial) and hierarchical (or quasi-temporal) structures, in my previous papers. And note that although the story may well seem most plausible when each of the removals takes the same time (since it would then take forever to remove À₀ unit-sized lumps), it should not be less plausible if they were done quicker and quicker (so that it would take only a finite time), or even instantaneously, because in any case were there not À₀ units (and assuming the actual infinity) the removals could not be continued endlessly (since we would run out of unit-sized lumps).

That analogy and that story were partially describing the epistemic possibility, of a relatively realistic potential infinity, they were not arguments for it; and another way to see that possibility is to see intrinsic problems with the epistemic necessity of the actual infinity. Suppose, for example, that our intuition is essentially that, because the natural numbers are all actual mathematical objects (i.e. possible numbers of actual things), therefore so is À₀ (since that is just their number). Then we have our first transfinite cardinal, and Cantor’s diagonal argument can yield more and more of them, endlessly (given a realistic view of mathematical objects, such as would justify the full power-set)—but therefore we face a dilemma, as follows. If the transfinite cardinals are all as realistic (as actual) as the natural numbers, then our intuition (for the necessity of the actual infinity) seems to give us an actually infinite totality of cardinals (which is impossible, via Cantor’s paradox). But if they are not, then some of them are less real (more potential) than their predecessors; and not only does that run counter to the intuition in question, it is hard to see why the rot would have to set in after À₀ (e.g. at À_w).

Still, stories are only stories, and our first impression may have been correct—maybe continua must contain À₀ parts, after all. If so, then it ought to be possible to construct a more precise argument against C-II. E.g. (as suggested by Peter Fletcher in 2002), given an endless line of points (two of which are 0 and 1), consider the following properties of collections of points: (Plus) Given any point in the collection, the point one unit along the line from that point in the positive direction is also in the collection; and (First) the point 1 is in the collection.

There are many collections of points in the line that satisfy both Plus and First, and such collections will be called PFs. E.g., there is a PF that contains only the points 1, 2, 3, ..., m, for some natural number m, together with all of the points in the line from m + y onwards, where y is some point between 0 and 1, and such a PF will be called a PF_my.

By definition, let the intersection of the PFs contain all, and only, those points that are in each PF. Clearly, each point named by a natural number is part of every PF; and no other points are common to every PF because, for each point between m and m + 1, there is one closer to, but still smaller than, m + 1, which is the y of some PF_my. So, the argument is that, since that intersection contains exactly À₀ points, hence any endless line of points must contain À₀ points.

But that conclusion only follows if the infinite intersection of the PFs exists, and while we would certainly expect it to exist (as a part of the line—as it would have to be if it exists, since it would be a part of each of the parts of the line that are the PFs), that expectation may well be based (perhaps subconsciously) upon an extrapolation from the finite case, and such extrapolations are notoriously unreliable. In fact, since each PF is just a collection that satisfies Plus and First, the nature of their intersection is determined by the total effect of those defining properties, in the context of the totality of the PFs—which, as a collection of collections of points, is not necessarily part of the line.

So if lines were like C-II, the intersection of the PFs would go against the grain of such actually infinite totalities of points, so to speak, because the intersection of the PFs would be a potentially infinite w-sequence (no less so than the union of the points 1, 2, 3, …). The picture is of quasi-spatial lines having a quasi-temporal intersection because of the quasi-temporal variables in the definitions of those lines. In other words, if the cardinality of a potentially infinite w-sequence is represented by ‘¥’ (and note that there is a one-to-one correspondence between natural numbers and their squares, whence we might perhaps speak of such a cardinality, although the legitimacy of such a symbol would of course depend upon how the mathematical analysis of C-II develops), then the cardinality of the totality of the PFs would involve a factor of ¥·# (via such as the m and the y of the PF_mys respectively), whence that totality would have the same sort of irreducibly hierarchical properties (via the factor ¥) as the intersection.

Assuming for now, then, that there are reasonable doubts about the impossibility of C-II, I shall now return to C-II’s analysis. Incidentally I now (2007) regard my diagonal argument (Metaphysica 6, 91-109: p. 103) for a bigger cardinal than # as a little misleading. It seems relatively clear (from the above) that the collection of all subcollections of points in some line would include the union of all the PFs, and hence involve a factor of ¥·#, whence it would not be bigger so much as incomparable (even if open-ended lines are possible, which I'm a little unsure of), although it is in some sense bigger (as the potentially infinite natural number sequence is bigger than any of those natural numbers). Furthermore, maybe the diagonal argument could not get started anyway, because I'm unsure whether open-ended intervals are more than the simultaneous consideration of closed intervals and their end-points; that is, whether the properties of the points of an instantiated line of # points would be so independent of each other, or not.

But anyway, since the size of a point is 0, while our w-sequences are only potentially infinite, a point’s location in the line (relative to 0 and 1) will contain far more information than any w-sequence could express; which is why there are infinitesimals in lines of # points (as mentioned in Metaphysica 6, 91-109: pp. 105-7), as follows. In any infinitely extended line of points, there are points that are n unit lengths from 0 for every n, but there is no sequence of all such lengths in our line (since there are not À₀ points). Now, any point lying in the positive direction from 0 would, if not in any of those lines from 0 to n, have to be infinitely distant from 0. Is there such a point (infinitely distant from 0)? Well, if there was no such point then each point (in that direction) would belong to some line from 0 to n; so in the absence of a sequence of all such lengths the line would then have to be finite (whereas it is infinite), whence there are indeed points infinitely distant from 0 (relative to any unit of length).

The uniformity of the line means that such a point (infinitely far from 0) might have been chosen as the point called ‘1’ (following our choice of origin), and so the point that was actually called ‘1’ shows (under the alternative labelling) that there is a point, say i, such that although i is not the point 0, £(i) = 0. Such infinitesimals (legal irreal numbers, as I call them) will have whatever properties the demands of logic (to be consistent) and metaphysics (the assumption of C-II) dictate—they will be consistent (unless C-II is impossible) and applicable (unless C-II is not true).

A point infinitesimally distant from p will clearly generate the same expansion, i.e. £(p + i) = £(p), where ‘+’ denotes the geometrical addition of line segments (a vector addition), so each legal real is associated with many points; but what is the precise nature of the relationship between the points that can be individually specified (via the points 0 and 1) by finite formulae, and the legal reals? That is, are there finite formulae, for particular expansions, which do not pick out particular points (given the two that we started with)?

Also, since the line contains n points for any n, and # points, but no intermediate amounts (so far as we know), another question that immediately arises concerns whether or not ‘#’ and ‘–#’ would make good labels (within £) for points that are infinitely distant from 0. Prima facie they would make good labels for the gradients of vertical lines (which are usually written ‘±¥’). However, ‘a gradient of #’ would not mean that if we went a distance of 1 in the x-direction then we would go a distance of # in the y-direction. And although # is a cardinal number, to use ‘#’ to denote such points (that are infinitely far from 0 in the positive direction) could not mean that such points are # units from 0 (since the initial w-sequence of units does not exist). So, we might have too much information (for this labelling job) with #. However, using ‘0’ to refer to i (within £) does not imply that i is not distinct from 0—and we might well use ‘0’ to refer to infinitesimals, e.g. to refer to such as i² (since that is one way of making infinitesimals useful) or to a point that is nearer to the origin than is iⁿ for any natural number n (if such points exist). So, that may not be a particularly good reason for not using # to label such points after all.

Nonetheless, it might be better to associate such points with Q_L (see my 2005), since that object encapsulates only an algebraic part of the information contained within the pair # and –#. Also, since Q_L is (by definition) the object that turns the legal real number field into the legal real number pitch, it is more closely related to the number systems that are usually used in the various mathematical analyses. And the prima facie adequacy of Q_L for such a label, within £, was shown in Metaphysica 6, 91-109: p. 106. Of course, it may turn out to be better to use some other symbol entirely, depending upon how the analysis of C-II develops, so perhaps I should first address such questions as, by how much would both the standard and the constructive recursive analyses of computable reals need to be tweaked, for them to accommodate the geometrically motivated infinitesimals and potential infinity of C-II?

Furthermore, since my motivation is the apparent realism of the hypothesis that the natural numbers form some sort of potential infinitude (see my 2003 and unpublished), together with the mathematical elegance of the hypothesis that geometrical lines contain # = 1/0 points (see my 2005), presumably I ought to have begun my analysis with a better-defined object than the straight line of our intuition (e.g., something satisfying Hilbert’s axioms of incidence, order and congruence, and so forth). And of course, C-II may not be the best way to analyse continuity anyway, so before any of that I shall first consider more fundamental questions about continua, which connect with the fundamental questions about pluralities.