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Re: [Phys-l] data ± model ± trust



Having only 3 legs I hypothesize that the tables will fall down.

Vern

On Jan 21, 2011, at 6:28 PM, John Denker wrote:

jOn 01/21/2011 02:45 PM, William Robertson wrote:
You can't completely ignore the legs, because they provide a 3D
perspective. Without them, what the figure shows could just as easily
be a top view as a perspective view.

That completely fails to answer the question that was asked
... but it is also the correct answer.

That is to say, the original question was unanswerable.

The thing that makes this tricky and interesting is that
to most people, the question /looks/ answerable. That is,
if you've seen enough perspective drawings, you "know"
how to interpret them.

This has direct relevance for teaching how science is done.
(Some people talk about "the scientific method" but I prefer
to avoid that term.)

In this case, there is an obvious hypothesis, namely that
the red tabletop is more-or-less square (like a card table)
while the blue tabletop is longer and narrower (like a
banquet table). This model fits all the presently-available
data.

My point is that having a model that fits the available
data means "something" ... but it does *not* mean you can
trust the model. The problem is, there could be other
models out there that fit the presently-available data
just as well, yet lead to wildly different predictions
about what happens next.

The rule in science is simple:
Consder *all* the plausible hypotheses.

It makes my hair stand on end when I walk through the
science fair and see project after project where
only one hypothesis was considered, and the goal was
to show that "the" hypothesis was "true". This is a
travesty of science and a mockery of common sense.

Again: the point is that having a model that fits the
data does *not* mean you can trust the model. As we
saw the other day, if you have a model that says Monty
always opens a goat-door, and that model fits the
available data points (all one of them), you do *not*
know that your model is a winning strategy. In the
very next instance of the problem, Monty might decide
not to open any doors ... and the distribution of his
actions might be /correlated/ with the distribution of
prize-locations. You might have a model that says
Monty is your ally, but the real Monty might not know
about -- let alone adhere to -- your model.

Returning to the two tables: Kid #4 was telling the
truth: If we look at the two-dimensional diagram,
the parallelogram representing the red tabletop is
the same as the parallelogram representing the blue
tabletop. A lot of people refuse to believe this.
Sometimes even after measuring the parallelograms
they refuse to believe it, because the two things
"look" so different.

You might think this is just an optical illusion, such
that two things that are really the same appear not
the same. Maybe that's true, but whether or not it's
true there is a deeper explanation which is worth
pursuing.

Even though I warned everyone to disregard the table
legs, many people find this difficult or impossible
to do. They know the rules for interpreting diagrams
of this sort, and they subconsciously (or otherwise)
insist on visualizing the tables in 3D. They build
a 3D /model/ of the situation. And in 99% of the
cases this is the right thing to do. This is an idea
that goes back to Plato's allegory of the cave. In
99% of the cases you *should* try to understand the
situation in terms of the 3D model, rather than directly
in terms of the 2D projection.

I reckon that sometimes it helps to have a certain
childlike innocence (and ignorance of Plato). In
particular, this makes it possible to notice that
in the projective plane, the red parallelogram is
the same as the blue parallelogram. Most grown-ups
are so entranced by the 3D model that they can't
see the 2D projection for what it is.

The hallmark of real mastery is being able to see
things in more than one way. You want to see the
2D parallelograms for what they are *and also*
build the 3D model.

Meanwhile ... That's still not the end of the story.
There is even bigger game afoot here. The problem
with model-building is that sometimes undocumented
assumptions get built into the model.

In this case, I can tell you what one fatal assumption
is: Normally, the legs of a table are perpendicular
to the tabletop, or at least symmetrically disposed
around the table. (In fact that's where the word
"normal" comes from; the original core meaning was
"upright".)

Now it is a mathematical fact that there is no way kid
#2 can be telling the truth /unless/ one of the tables
has out-of-kilter legs. They don't need to be very
far out-of-kilter; here's another model that fits
the data, using reasonably plausible leg-angles, and
two identical square table-tops:
http://www.av8n.com/physics/img48/tables-more-perspective.png
According to this alternative model, both tabletops are
square. The red table is standing flat on the floor,
while the blue table is walking down the stairs, with
no two feet at the same level. So now we have identified
a second fatal assumption, namely the assumption that
both tables were standing on level ground.

To prepare this diagram, all I did was make a copy of
the blue table and rotate it in the plane of the diagram
so that it lined up with the red table, top-to-top.
Any rotation in 2D is also a rotation in 3D, so this
is certainly an allowed physical operation.

Of course there are other models i.e. other hypotheses
to be considered, with various degrees of plausibility,
including the one that W.R. pointed out, namely that
both tabletops are parallelograms and the diagram is
a top view.

I'm not saying don't build the model. I'm saying you
need to consider *all* the plausible models, and test
them against plenty of data before you trust them.

On 01/21/2011 03:57 PM, Bernard Cleyet wrote:
bc used a ruler,

Good move.

and suspects this will lead to special relativity

Good call.

In my book, special relativity is just Plato's cave all
over again, just pushed up one dimension. Plato's cave-
dwellers could only see the two-dimensional projections
of three-dimensional objects. It would be the height
of foolishness to think that the 2D projections were
"reality".

Similarly, in special relativity, in some circles it is
fashionable to concentrate on what things "look like"
in 3D ... but IMHO that is the height of foolishness.
Anybody with any sense knows that 3D is not reality.

In case anybody missed the metaphor: The "long narrow
banquet table-top" is the famous pole, and the "square card
table-top" is the famous barn. In the higher-dimensional
reality they are the same size. They might "look"
different when projected onto a lower dimension, but
that's not where the real physics takes place.

Special relativity is not nearly as weird as it is cracked
up to be. The /projection/ of spacetime onto 3D space is
pretty weird, but that's not where the real physics takes
place.

Again: You pretty much have to build the model. So build
it carefully. Try to keep dubious assumptions out of the
model. Test the model with /lots/ of data before trusting
it.
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