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Conterintuitive facts in mathematics, CS, and physics

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Re: Conterintuitive facts in mathematics, CS, and physics

#171

Post by stan_rogers »

version_five wrote: post #53
lotsofpulp wrote: post #35

Earlier quoted context omitted.

Oh wow, learning a lot today. I was taught in the US that pounds are a measure of force, and that is how it was always used in physics problems.

I think there is some weird dual usage that makes them either mass or weight depending on the context. For example, torque is in ft•lbs or N•m so the pounds there are lbs force. Though checking wikipedia again, it actually specifies that torque is measured in as lbf•ft. I take that to mean that 1 ft•lb is the torque of 1/2 oz (1/32 of an lb) at 1 foot. I expect that's a test question almost everyone would get wrong,…

Think like an ordinary person. You know, an ordinary person who would say that they weigh about 80 kilograms. Only science nerds would say that they mass about 80 kilograms, or that they weigh about 785 Newtons. Similarly, anyone who's used to living with US customary (or Imperial) units understands that a pound of force is what a pound of mass weighs and would see no reason, under any circumstances, why anyone would want to divide the gravitational acceleration out of a pound-foot to arrive at a "real" torque value. When the pound value is expressing a weight-equivalent force, that force is the force of a pound under normal gravitational acceleration at or near the surface of the Earth.

Re: Conterintuitive facts in mathematics, CS, and physics

#172

Post by twic »

caf wrote: post #124
robocat wrote: post #121

Earlier quoted context omitted.

- a chair with four even legs placed on any undulating continuous surface can always be rotated such that all four legs touch the ground at once.

Do the legs have to be even?

Not even, but the feet have to be coplanar.

Re: Conterintuitive facts in mathematics, CS, and physics

#173

Post by cycomanic »

MatteoFrigo wrote: post #19

The list is missing one of the most astonishing discoveries of all time: if you reflect the universe in a mirror, you can tell whether you are in our universe or in the mirror because the laws of physics are different in the mirror. See https://en.wikipedia.org/wiki/Wu_experiment

It is a great discover I agree, it might not quite fit in the list because it's not so counterintuitive for someone not into particle physics though.

There are actually many examples of things working different for the mirror image, e.g. many medical/chemical compounds work differently based on chirality.

The topic of chirality is fascinating, one of the big puzzles in nature is for example why it so strongly prefers right-handedness.

Re: Conterintuitive facts in mathematics, CS, and physics

#174

Post by amelius »

MatteoFrigo wrote: post #19

The list is missing one of the most astonishing discoveries of all time: if you reflect the universe in a mirror, you can tell whether you are in our universe or in the mirror because the laws of physics are different in the mirror. See https://en.wikipedia.org/wiki/Wu_experiment

How can someone rigorously prove that? E.g. perhaps there is something inside quarks that has chirality which we haven't discovered yet.

Re: Conterintuitive facts in mathematics, CS, and physics

#175

Post by jean_tta »

User23 wrote: post #74
mint2 wrote: post #22

Earlier quoted context omitted.

I really don’t like that example because it makes no sense. In no logical circumstance could the potatoes dehydrate so quickly when left out over a single night.

You probably won’t be thrilled by the spherical cows in the nearby pasture then.

Do spherical cows dream of mathematical potatoes?

Re: Conterintuitive facts in mathematics, CS, and physics

#176

Post by BalinKing »

0xmarcin wrote: post #154
ncmncm wrote: post #150

Number 19 is a clinker. Banach-Tarski applies only to objects in real-number space, but there are no such objects. For that to work, objects have to be infinitely divisible, but all of our objects are made out of atoms. Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.

Also this only works when the highly contested Axiom of Choice is used.

But isn't Banach-Tarski one of the main reasons why AoC is highly contested in the first place? (I don't know much about higher math, so this only my amateur impression.)

Re: Conterintuitive facts in mathematics, CS, and physics

#177

Post by kristjansson »

londgine wrote: post #86

> There are as many whole positive numbers as all fractions According to a specific non-colloquial definition of "as many".

There’s an infinite number of each, so we’re already stretching colloquial definitions by comparing them

Re: Conterintuitive facts in mathematics, CS, and physics

#178

Post by joshuamorton »

jiggawatts wrote: post #157
d_burfoot wrote: post #114

Here is one of my favorites. Take a light source and shine it at a filter that is polarized up/down that is in front of a filter that is polarized left/right. None of the light will get through: the first filter removes all of the L/R components of the light, and the second filter removes all the remaining light. Now add a third filter, between the two, which is polarized at 45 degrees. Now some of the light goes thr…

It makes more sense if you think of the polariser sheets as fences with vertical bars, and light like a string going through between two bars and being wiggled. If the direction of the waves in the string are at right-angles to the slit, the wave can't propagate through the bars. If the wave is at an angle, then some of the wave gets through, with a maximum of 100% when the wave wiggles up and down in the same direct…

IIRC, you can grok this with linear transforms:

given X = [[1,0],[0,0]] and Y = [[0,0],[0,1]], aXY = 0 for all a (because XY == [[0,0],[0,0]]).

But, given V = [[1,1],[1,1]] (or, in fact, a lot of other matrices/linear transforms), XVT = [[0,1],[0,0]], which blocks all vertical components and rotates the horizontal component 90 degrees. So if you imagine light as a wave with a horizontal and vertical component, and the polarizing filter applies a linear transform to the light.

Re: Conterintuitive facts in mathematics, CS, and physics

#179

Post by mjburgess »

ncmncm wrote: post #150

Number 19 is a clinker. Banach-Tarski applies only to objects in real-number space, but there are no such objects. For that to work, objects have to be infinitely divisible, but all of our objects are made out of atoms. Real-numbered space is a good enough approximation to our experience that we hardly ever encounter a model failure like this one.

I'd deny that our objects do not live in a real space. Spacetime is real-valued.

The issue isn't the reals, but that "solid object" isn't defined properly, ie., the sets under question don't have well-defined volumes.

As soon as you fix that problem, via measure theory, the paradox resolves. You dont need to ditch real numbers.

Re: Conterintuitive facts in mathematics, CS, and physics

#180

Post by jacobolus »

p1necone wrote: post #101
p1necone wrote: post #91

Earlier quoted context omitted.

You're being overly pedantic (and I would argue actually incorrect). Kilograms and pounds are both referred to as "weight" in general conversation and nobody is going to be confused by this. Go to any supermarket in a country that uses the metric system, potatoes will be sold by the kilogram - it's the natural way to phrase this outside of America. In a physics context the definition of kilogram might be specifically…

Although the more I think about this the more I think the difference between technical and colloquial is actually that "weight" in colloquial use refers to mass, because force is not commonly relevant.

"Weight" historically referred to mass, in common speech dating back forever. It’s the Germanic word which has been used throughout the history of English, whereas "mass" comes from Latin via French, like 5–6 centuries ago. The two words are almost exact synonyms, in historical/colloquial use.

Both historically and today, a "pound" (Roman libra) is a unit of mass. People use a pound-force as a unit of force only in somewhat specialized contexts.

At some point in the relatively recent past, someone (not sure who) decided that we needed to have 2 separate words for mass vs. force, and we should keep the Latin word for mass and use the Germanic word to mean force.

Now pedantic people are constantly insisting that using the standard English word weight to mean mass is "wrong".

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