Wednesday, February 27, 2008

The smedlium case

There's a short piece by me here on systems of measurement - for the philosophers among you. It's on something I call the smedlium case:

Imagine a world quite similar to our own that contains large quantities of a metal-like material – let's call it smedlium – which gradually and unpredictably alters in size. All smedlium objects expand and contract in unison. At one o'clock on one particular day all the smedlium objects are 5% larger than they were at mid-day; at two o'clock they are all 10% smaller, and so on. Despite this peculiarity, smedlium remains a useful material. In fact, it is the strongest and most durable material available. One of the inhabitants of this world builds machinery made wholly out of smedlium. The machines are used in situations where their size relative to non-smedlium objects doesn't matter. The smedlium engineer constructs and calibrates a measuring rule made out of smedlium to use when manufacturing such machines. She measures dimensions in ‘S’s, one S being measured against the length of her smedlium measure. Of course, so far as manufacturing smedlium machines is concerned, a smedlium measure is far more useful than is a rule made out of some more stable material, for it allows the smedlium engineer to ignore the changes in size of the object upon which she is working. For example, she knows that, say, if the hole for the grunge lever measured 0.5 S in diameter at one o'clock, then a grunge lever which measures 0.5 S in diameter at two o'clock will just fit into that hole, despite the fact that the hole is now noticeably smaller than it was at one o'clock.


NAL said...

That's because the grunge lever is made out of smedlium.

scott gray said...

solid items would be measurable. interfacing moving parts might be a bit of a problem.

what applications would such a material be good for? frying pans? soccer balls? not watch pieces.

Stephen Law said...

I don't see why interfacing moving parts would be a problem.

Except for one reason: if weight stays constant, but mechanisms shrink, watches will run fast or slow, etc. (I talk about this in the paper, in fact)

But in terms of the relative geometry of the moving parts, that stays constant. Gears will continue to mesh just as before, etc.

scott gray said...

when you talk about expanding, an s-piece expands along three dimensions, into space, but it has to expand from somewhere as well. my default is center mass. say there's a ball bearing inside a hollow shell made of woven threads, both made of smedlium, with infinitesimal separation. if the ball expands from its center mass, and the threads expand from 'thread center' that lies insode each thread, the pieces will lock together. like metals heated. unless the smedlium thread woven piece expands outward from a point not inside its mass.

the whole argument seems to depend on what point it expands from.


Anonymous said...

I would expect gears to shrink away from each other and becoming unmeshed, or to expand into each other and effectively fused together.

Unless Smedlium either reorientates itself to keep the relative geometry the same, or as Scott suggests the contraction/expansion happens relative to some common external reference point.

More interesting to me is the type of device that could be made from such a material.

Say a medical device for de-plaquing arteries. It could be "easily" put together at a larger size, all the while shrinking. Upon reaching the correct size it is injected into your blood stream, performs it's task while shrinking into noting.

could use it for frying pans for people on a diet... Each time they come to cook a meal it gets smaller...

NAL said...

Gears won't be a problem if their mountings were also smedlium. As the teeth shrank, the gears would move closer together.

Stephen Law said...

Nal is right. All dimensions of all smedliuum objects increase/decrease by same percentage. So as long as the machine is 100% smedlium, there's no problem.

Take a blue print of a machine. Scan it and reduce by 10%. Every dimension is reduced by 10%. Yet it all still fits together perfectly. That's why 1/10th scale working models work!

scott gray said...

when you reduce the blueprints, all the atoms reduce from the same point of view (upper right hand corner, center, etc). how can we be sure smedlium acts the same?

Anonymous said...

Although the substance in question is hypothetical it is not quite as fanciful as some engineers would like. Apparently the standard 1Kg mass has lost a little recently


(Both seem to be from AP)

I think physicists much prefer units that an be defined in more abstract numerical terms. For example using light emitted from a particular type of atom under specific conditions. If you know the definition you can build your own instrument.

Anyway enough of the mechanics, back to the K vs W debate...

The Celtic Chimp said...

Hi Stephen,

I read the post on blogging the question and I don't really see why the issue of contingency cause a problem in the K-type system. Taking the example of industrial espionage:

It would be obvious to the smedlium engineer that the measuring rod had been altered. Every other smedlium object would now be 10% larger as measured by the rod. If the rod is also made of smedlium, other smedlium objects should remain the same relative size. The only two possibilities would be that:
A) Every smedlium object had indeed grown in size, which smedlium has a tendancy to do, but the measuring rod somehow was immune on this occasion.
B) Someone chopped 10% off the end of the measureing rod. The rod is now 0.9S. one S, does indeed rigidly designate a (smedlium) length. The lenght 1s is now effectively independent of the rod. 1s is the lenght that the rod was in relation to smedlium objects at the time it was designated to be 1s. To put it another way, smedlium objects only have variable length when compared to non-smedlium objects. Smedlium to smedlium measurements would be as rigid as non-semdlium measurement in general. All the same contingencies apply. If smedlium somehow became the most common material in the universe, then all non-smedlium objects would seem to inherit the odd tendancy to change size, whilst smedlium would be the standard against which this stange phenomenon was measured.

Although most of these comments seem a little off topic, they are interesting in their own right. I think Scott makes an excellent point about the point of inflation/deflation. It would indeed seem that in order for smedlium objects to remain intact, all atoms (except the one at the focal point of the expansion/contraction) would have to actually move. The futher from the focal point, the futher they would have to move. If the focal point was in the centre of the object, they would not even be moving in the same direction. The movements would in fact have to be extremely orcastrated in order for the atoms to maintain their positions relative to one another and prevent the object from shattering.

The Barefoot Bum said...

The Smedlium case is interesting, but I don't think it does the job you expect of it in the full article, which is to distinguish between W and K measurement concepts.

The original article suffers from a fatal flaw: We cannot say that smedlium changes length: We must say that the relative lengths vary between smedlium and non-smedlium materials. To pick out one material as invariant implicitly endorses Kripke's view. We might just as well say that smedlium possesses the unique property of measuring absolute length, and it is non-smedlium material that changes length. But it is precisely the fact that smedlium changes absolute length that argues against Kripke's view.

Personally, I think the smedlium case argues that the supposed conflict between W and K concepts is a a pseudo-problem. The fact that smedlium does indeed match our intuitions about measurement shows that consistency, not "truth", is the fundamental basis of our intuitions.

The Barefoot Bum said...

Also, your article uses outdated science. A metre is now defined to be 1⁄299,792,458 of a light-second (and delineated (not defined) today in labs as 1,579,800.298728(39) wavelengths of helium-neon laser light in a vacuum). [Wikipedia]; it's hard to imagine someone sneaking in and chopping 10% off the end off of c or laser light.

Anonymous said...

Sorry to go back a few posts but:


This is only true if the operation is dilation - basically changing the scale of the object (altering size not shape) - as with photocopying.

Expansion/Contraction is a different process (to my understanding):
Imagine a doughnut (torus) of Smedlium, if it expanded the hole in the middle would eventually disappear - draw a torus then expand it by drawing another torus around the first. (The same would happen with gear teeth - the gap between them getting bigger or smaller respectively).

Dilation would however allow the shape to remain the same; to achieve that kind of change a complex object would have to translate itself while expanding/contracting
(this would not affect a cube for example, but gears are most definitely affected).

I think it's just my understanding of expansion/contraction is different from what is meant here.

Of course if the space around the Smeldium also expanded/contracted the shape would be maintained; but so would the size - relatively speaking that is.

Stephen Law said...

Yes there are different kinds of expansion. There's what you might call photoscanner expansion, on which all dimensions increase by the same percentage, which is what I have in mind, and then there's what we might call balloon expansion, which is what you'd get if the object were a balloon that was then inflated (resulting in straight lines being distorted, holes actually closing, etc. That's not what I have in mind.

P.S. We don't have to suppose smedlium is made out of atoms, of course. It's a thought experiment, remember...

In the full paper I sort of take Celtic Chimp's line. But it requires we give up on the idea of the kind of absolute length that K seems to have in mind.

B.B. In the full paper I of do discuss modern definitions of one metre. The same intuitions of contingency apply e.g. while we might fix the reference of "one metre" by saying it'. e.g. the "wavelength of X", it's still true that X might have had a longer or shorter wavelength - i.e. a wavelength other than one metre.

Anonymous said...

I agree it's a thought experiment, I'm just focusing on different aspects of the thought experiment.

For example in the main article:

"The saboteur shaves 10% off the end off the smedlium measuring rod..."

This interests me because if the rod expands uniformly not only the length of the rod changes, its diameter also changes. (or if it is a rectangular rod it's height and width change).

So our Smeldium Engineer - probably a practical chap - as part of his process would record the ratio of height to diameter. If that ratio ever changed he would know he was being messed with or that the properties of Smeldium had changed.

A Smeldium Philosopher on the other hand may find the problem more intriguing :)


So as long as the Smeldium Engineer uses a constant reference point for a design (it could be a different reference point for every design) he is happy. (With any scalar quantity 4X - 4 is the magnitude , X is the reference, be this 4 Kg or 4 Metres or 4 miles or 4 yards - it's always a magnitude and an arbitrary reference).

I don't think I even see a problem with the sabotage (I've not read the full paper yet). If I have a smedlium widget and part of it was designed to be 0.5 S long, I would always expect that part to be 0.5 S. I believe the rod has been defined as S. So what I am really saying is that part of my widget is 0.5 RODS long (not just any rod but this rod). If it is ever not 0.5 RODS long then either Smedlium does not posses the characteristics I expect or the rod has changed.

To my mind S does not have to be fixed or a constant; what matters is the magnitude and that Smeldium behaves uniformly.

But then I've never understood the fuss about metric or imperial. It's just a number with an arbitrary scale.

I believe the problem is the assumption that the reference point has to be constant. For keeping life simple we tend to expect this.

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Things defintely won't be an issue in the event that their own mountings had been additionally smedlium. Since the the teeth shrank, the actual things might proceed nearer collectively.

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