When someone’s walking speed is two miles per hour, there are not two things, “one mile per hour walkings”, that are present.
When we say that a sculpture has three dimensions, we are not saying there are exactly three things—dimensions?—that are present in it. But are there not width, height and depth? In a way. But rotate the sculpture by 45 degrees, and “width”, “height” and “depth” refer to measurement along three other axes. There are, it seems, infinitely many axes along which the sculpture can be non-trivially measured.
These are examples of what one might call “fake counting”. We speak as if there were n of something, but the following argument is invalid:
There are n Fs.
n ≥ 1.
So, there are some Fs.
And, similarly, this is invalid:
There are exactly two Fs.
So, ∃x∃y(F(x)&F(y)&∀z(F(z)→(z = x ∨ z = y))).
In fake counting of Fs, there is counting involved, but it is not counting of Fs. For instance, when we say that the sculpture has three dimensions, we mean something like this:
- there are three mutually perpendicular axes such that the sculpture has non-zero extent along each of them, but there are no four such axes.
So, there is a counting of axes, but it is not a counting of dimensions. If we were counting dimensions, we would have to have say what the first one is, what the second one is and what the third one is, and as the rotation thought experiment shows, that doesn’t work. And the counting of axes doesn’t involve counting axes overall, but rather axes in a particular set of them.
We need to beware of fake counting when making metaphysical arguments for the existence of entities of some sort. For instance, topologists have ways of “counting holes”. But topological properties are invariant under deformations. Now, imagine a pancake with, as we would say, “one hole in the middle”. Well, however we distort the pancake, it has one topological hole. But if we ask where that hole is, there is no topological answer to it (in the animation below, is the hole outlined in red or in blue?). So, topological hole counting is fake counting.
22 comments:
Pruss
I realise that the ontological status of an hole is vital to you.
The theological privation concept hinges on it.
There are, it seems, infinitely many axes along which the sculpture can be non-trivially measured.
However, the number of symmetries is delimited in 3-dimensional space. We live in a very special type of space. It is only possible to have 2, 3, 4, 5, and 6 fold symmetries in either natural or man-made objects.
Simply examine a piece of pyrite; its symmetry is imposed on it by the type of space we live in... one could even accept the natural state of pyrite as being a modern man-made sculpture.
Long ago I attended a really good philosophy of science lecture by William Wallace in which he discussed what he called 'the invention of per', that is, the shift from measurements like "one mile in an hour" to "one mile per hour". There was a time in the early modern period in which a lot of physicists were strongly inclined to reject outright any mathematical account of physical systems that used the latter, and it only slowly began to be used, at first as nothing more than a convenient calculating fiction.
There's an interesting question of how one would distinguish counting vs. fake counting in practice; perhaps most of what we would usually think of as counting is arguably fake counting.
Brandon:
That's interesting. I wonder if the resistance was due to qualms about the status of dispositions. The property of moving at one mile per hour sounds dispositional.
One test for fake counting is whether negative and fractional values of the count make sense. If so, probably this is fake counting. (Counterexample: there are 2.5 cookies on the table. But maybe that is not exactly right. A half-cookie either is a defective cookie or not a cookie at all. If it is a defective cookie, then there are, strictly, 3 cookies on the table. If it is not, then there are, strictly, 2 cookies on the table.) But what interested me most about the case of dimensions and holes is that there negative and fractional values don't make sense (in the cruder sense of "dimension" that I was using; fractal dimension is something else), but the counting is fake.
An interesting possibly related phenomenon is how mathematicians sometimes count objects "with multiplicities", so they can say things like: "Every nth degree polynomial has n roots, counting multiplicities."
It wouldn't surprise me if counting particles of a given type is fake counting. Thus, there is such a quantity as the "number of photons", but one cannot identify: here's photon 1 with its distinct identity, here's photon 2, ..., here's photon n.
The hypothesis about dispositions is interesting and would make a lot of sense; the early modern period certainly was a hotbed of skepticism about dispositions. If I recall the lecture correctly, what eventually gave 'per' its place is the combination of the overwhelming usefulness of the idea of instantaneous velocity and mathematicians finally getting the calculus into order, and I think one can argue that at least some skepticism about the calculus was also tied to skepticism about dispositions (ghosts of departed quantities, and all that).
The negative test makes sense, and negative numbers were also resisted well into the nineteenth century. (I wonder if this is disposition-related, too. In physical systems, a lot of times negative numbers come about when you are measuring with respect to some kind of equilibrium.) A worry I have about the fractional test, related to your cookie example, is that it can't be true that all counting of parts of a whole is fake counting, and counting parts of a whole can always be represented as a fraction. But perhaps it's a matter of direction, comparing identifiable parts to the whole as opposed to starting with the whole and constructing the parts from it? That might be related to the topological case; counting bulges on a rubber sheet would be fake counting for topological reasons similar to those for holes (a hole as an empty possible bulge, so to speak).
But if we ask where that hole is, there is no topological answer to it (in the animation below, is the hole outlined in red or in blue?).
The single hole in your animation is the evolving inner parametric surface (red, blue, yellow) of the pancake.
Pruss
Do the same animation with your idea of annihilating space within the hole at initialisation.
The boundary condition for the annihilation of space would be Dirichlet at t=0; u=v=0 for inner surface points during the entire simulation.
Then run the animation...doing this will demonstrate to you that holes are not fake counting.
Brandon
A worry I have about the fractional test, related to your cookie example, is that it can't be true that all counting of parts of a whole is fake counting, and counting parts of a whole can always be represented as a fraction.
You are correct to worry. Pruss is conflating fake-count and count-noun.
Hi Alex
I don't understand much of this debate, but it strikes me, as a mathematician, that the numbers involved in speed and dimensionality are not cardinal numbers because there can be fractional speeds and fractals. So I wonder, are there fractional holes?
The picture, entertaining though it is, just shows that it makes no topological sense to ask where a hole is. That is not a topological concept, and so the absurdity of asking about it hardly tells against the idea that topological holes can be counted.
Martin Cooke
In topology an hole would be defined as a mathematical object/structure which prevents the object from being continuously shrunk to a point.
The animation fully conveys this principle.
Martin Cooke
Further, the animation clearly defines the location and scantling of the hole region within the pancake.
This is the reason I chose the word parameter to describe the hole region rather than the word prismatic in my earlier comment above.
A Pruss
The single hole in your animation is located at position(0,0,0); it's size is min(diameter) of the hole; here the inner prismatic surface of the pancake is prevented from further shrinkage.
The hole is countable.
Pruss: I've often wondered if -- at the very least -- a good heuristic in the case of what you're calling "fake-counting" would be whether we can re-word the sentence without the putative "object" without losing any meaning. I don't think you can do that with real objects (e.g. "The apple was delicious" could be reworded in such a way that you just describe the apple without using the word, but you could not remove reference to the object altogether without losing meaning). But, maybe you can do it with any grammatical objects of sentences that are not real objects in the world (e.g. "I fell into a 6ft hole" could be "I fell in an area where there was no ground for 6ft" or something like that).
In other words, saying "there's a hole in this cheese" is really just saying "there isn't any cheese right here" and pointing at the missing spot. Saying "there is a piece of cheese here" has no such rewording. You'd have to say the "non-cheeseness of the world fails to obtain in this area" or something ridiculous like that.
I dunno. It may not be a great rule, per se; but maybe it hints at a heuristic. It would at least get rid of the silly reifying of "nothing" that people like Lawrence Krauss and Peter Atkins have attempted in recent years, since I can reword the sentence "nothing was unstable" to "there wasn't anything that was unstable", without changing its English meaning at all; but it reveals that the first version of the sentence doesn't mean what they want it to at all.
Martin:
There are, of course, many kinds of dimensionality. I was talking of something more like vector space dimension (my gloss was in terms of mutually perpendicular axes). That differs from topological dimension, Hausdorff dimension, etc.
If topological holes can be *really* counted, then there are such *things* as topological holes. But if there really are such things, then it should be possible to say things like: this is the first hole, this is the second, etc. But we can't. Facts about "how many" n-dimensional holes there are are just facts about the structure of the nth homology group, rather than counting facts. There is no such thing as "the set of holes".
Michael:
That is a useful sufficient condition, but it is rarely met. One not only has to paraphrase existential statements about holes, but statements about their persistence, shape, size, etc. And it is very difficult to find such a paraphrase.
Martin:
Here's another way to think about fake counting. Here is a general kind of thing: numerical characterization, the characterization of a situation with a number. Pretty much everything done in the hard sciences is an example of this, and there is a lot of it in daily life, too. For instance, I will soon get to assigning numbers to student papers.
Now, counting is a special case of numerical characterization. In the mathematical arena, counting works as follows: We count the Fs by forming the set { x : F(x) } and taking its cardinality. Counting Fs is, however, only one of many kinds of numerical characterizations.
When, for instance, we characterize the dimensionality of a vector space V, we do not form the set { x : x is a dimension of V } and take its cardinality, because "x is a dimension of V" is nonsense. (We could try to say "x is a dimension of V" means that x is a member of a basis of V. But then every non-zero member of V will count as a dimension, and every non-trivial vector space will have dimension infinity.)
When we characterize the dimensionality of a vector space V, we do engage in counting. But we don't count dimensions. Rather, we look at the minimal cardinality of set that linearly spans V. The process of characterizing a vector space with a dimensionality involves counting: but it is not a counting of "dimensions", and it is a counting within the scope of an infimum operator.
I think a number of cases of fake counting are cases of numerical characterization where counting is involved, but it is not the counting of the objects that on its face it is advertised to be. When we characterize spacetime with four dimensions, it seems like we are counting dimensions. But we're counting the minimal number of factors of R such that the space is locally homeomorphic to a product of that many factors of R. For holes, we count the number of factors appearing in a homology group (but these factors are not "the holes").
Thanks for that, Alex, and apologies for my not replying earlier.
You say, in your shorter reply to me, that if there are such things as holes then we should be able to say "this is the first hole," but I do not see why we cannot.
I now see, from your longer reply, why holes can be an example of what you call "fake counting". That is how topology is done, within set theory. But that does not mean that holes cannot be counted.
There is clearly one hole in your picture. When the substance at the edge of the hole rotates around the body, the hole remains where it clearly is.
And here is a picture of a body with two holes: 8. They can be counted in the following sense: I associate the top one with the counting number 1, the bottom one with the counting number 2.
Could the notion of fake counting make the Kalam argument clearer? WLC is making the distinction between actual infinity and potential infinity, and many people can't grasp why e.g. the number of points in time is not an actual infinity. "Fake counting" seems a lot easier to understand.
If it works, this would be like the Raft algorithm vs the Paxos algorithm in distributed computing. Paxos was revolutionary, but notoriously hard to understand and implement. Then Raft solved both those issues.
(Or maybe I'm completely wrong and "fake counting" doesn't apply to the Kalam argument at all)
Martin:
Mid-way through the distortion, the substance looks like the outside of a cylinder with two openings. Where is the hole then? At the top or the bottom?
WM:
That's very interesting, but it wouldn't be synonymous with the actual/potential infinite distinction for two reasons, and I think it might not help Craig.
Let's assume, with Craig, that the future isn't real (i.e., presentism or growing block is true). Then any counting of future things is fake counting. And this is true regardless of whether we are counting a finite number (e.g., the number of people who will be born next year) or an infinite number (e.g., the number of future days). Moreover, if presentism is true (as Craig himself thinks), then any counting of past objects is fake counting, too.
So, the real/fake counting distinction helps Craig only if he is a growing blocker, but not if he is a presentist, since on presentism there is an ontological symmetry between the past and future. And I think he's a presentist.
I may not be seeing the picture properly, Alex, but to me it looks like the hole is always in the middle of the object, with the object changing its shape around it. When the object looks like a cylinder, the hole therefore appears to be the inside of the cylinder. To me, it looks like the hole is at the top and the bottom, and all the way in between.
Martin:
So in my example, you'd say that the hole in the shape S is ConvexHull(S)\S?
But what if we make the shape S thinner than in the picture, thin like a sheet of paper? (That's how I was thinking about it, but in my code I thickened it for visual clarity.) Now imagine that we're at a point where the shape S is almost flat, but not quite: it's a paper-thin truncated cone with an angle very close to 180 degrees. In that case, ConvexHull(S)\S is the inside of that truncated cone. But it doesn't seem right to say that ConvexHull(S)\S is the hole: it's too big.
Imagine that I have a flat round rubber sheet, 100 mm in diameter and 1 mm thick, and I punch a 5 mm hole in the middle. I put the sheet on a flat table. The sheet now has two circular edges: one 100 mm in diameter and the other 5 mm in diameter. And it has a hole that is intuitively 5 mm in diameter. I now hold down the outer edge while lifting the inner edge by 2 mm off the table (I can use some sort of tools to do that).
Intuitively, the following did NOT happen: the hole grew from being 5 mm in diameter to being 100 mm in diameter just by the inner edge getting lifted.
But suppose I keep on lifting that inner edge, stretching the sheet, until now the inner edge is 100 mm in the air. I now have a 60 degree cone. By your intuition on holes, my hole is now also conical in shape, and its flat bottom side has a diameter of 100 mm.
So, now, the question is: When the did the bottom of the cone jump from 5 mm to 100 mm as I was lifting the inner edge of the punctured rubber sheet?
An elegant answer, suggested by the ConvexHull(S)\S suggestion (which of course isn't meant to work for every shape, but only for shapes like the ones in my story), is that the slightest lift causes the diameter to jump to 100 mm. But this seems counterintuitive.
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