Which is simpler and easier to imagine?

[marcus]

For some people, an infinite sheet of paper is simpler (because it doesn't have the extra detail of edges) and easier to imagine than a regular standard-size sheet of computer paper, while for others it is evidently simpler and easier to picture the paper with edges. I'm curious to know which people find which simpler to think about.

I believe this actually has something to do with science, because of Occam's razor. The timehonored Razor principle that you don't put more in the picture than you need to fit the data. If both models fit, use the one with fewer elements. (William of Ockham said it in Latin.) IOW avoid unnecessary detail.

The Razor may have something to do with mathematical cosmology in particular, and how different people seem to approach it. I think there are four main possibilities.

1. You find the infinite sheet simpler and easier to imagine.

2. You find the infinite sheet simpler, but it's harder for you to imagine.

3. You find the sheet with edges simpler, but harder to imagine.

4. You find the sheet with edges both simpler and easier to imagine.

I have no idea what people will say, but my guess is that in any case it won't be unanimous one way or the other.

I could have put the same question about a line. Which do you find simpler to think about, an infinite line or a line segment with endpoints? But it seemed more relevant framed in terms of something 2D. I was inspired to ask because of King Ordo's thread about the universe.

 

[nicky nichols]

My findings are

When I think about the infinite sheet, it is not made of paper, it is something definitely abstract.

Thinking that it has a finite size, does not immediately put the edges into view.

Thinking that the finitely sized sheet is made of paper, puts the edges into view as a thin black frame with the white paper inside it and all upon an indefinitely colored background.

Forcing the background to become blacker makes the sheet hazier, as I lose the edge to background contrast.

Reconsidering the finitely sized sheet as an abstract entity forces it to snap back to infinitely sized.

I understand that this is what is called a perception catastrophe.

 

[cristo]

I go for option number 2. An infinite sheet is simpler, as you say, since it does not have the added detail of a boundary, whereas I think a finite sheet is easier to imagine. I say this because I know exactly what a finite piece of paper looks like; I can pick one up, feel it, and thus my brain ties the concept of a finite piece of paper with the thing that I have everyday experience with. For an infinite sheet, it is not possible to do this.

Just my thoughts, anyway!

 

[marcus]

Nicky, Christo excellent! Your subjective experience is very clear and different from mine so I'm glad you tossed them into the pot. Thanks for replying.

Nicky, the poll options don't specify whether paper or abstract (although in my post I sometimes said paper for definiteness) so you can really imagine it however. So as to record your take on it, would you kindly indulge me and pick one of the four and click the poll? I'd like to get a bunch of responses.

 

[KingOrdo]

The sheet with edges is obviously easier to imagine. We can all bring up a mental image of a piece of computer paper with ease and clarity. But there is something inconceivable about an infinite plane; though we may understand what is being talked about, none of us can actually imagine it. At best, we can imagine a local region of the infinite plane and can correspond that to a certain propositional attitude (viz. the belief that the plane in question is infinite).

I would go further and argue that the finite case is simpler. Nowhere in our common experience do we encounter infinite things. Nothing goes infinitely fast. Nothing is infinitely big or infinitely small. Indeed, all a posteriori knowledge is finite: Nothing that comes through the senses is infinite. Now, it's true that we "encounter" the infinite through certain sorts of abstraction (e.g. the set of integers is infinite: Got it). But those are unscientific claims that reside in philosophy of mathematics. Science is, and should be, strongly--if not fully--empirical. Finite is simpler; finite is better.

 

[marcus]

...But those are unscientific claims that reside in philosophy of mathematics. Science is, and should be, strongly--if not fully--empirical. Finite is simpler; finite is better.

King, this is a clear thoughtful statement. It does two things I like very much:
1. it is consistent with your point of view that the burden of proof is on the other guys to show you evidence that there is NOT an edge.
(You can invoke Occam's razor of simplicity on YOUR side. If you see space-with-edges as the simplest picture then you can insist that the other guys prove it's wrong before you give it up.)

I think that's fair.

2. You put your finger on the fundamental essence of mainstream cosmology (how it differs from verbal talk and commonsense intuition about the shape of the world at large.) You say that it is in the MATHEMATICS realm where edgeless pictures are simpler!

In fact for someone using math models, the real line is simpler than a finite interval. You just say "R, the real line" and you don't have to specify endpoints. You use that simple concept almost every hour of the day, as a working mathematician. Or the complex plane, or R^2 or R^3. It is much much easier to say R^3 than to specify some finite solid shape like a cube or prism or rectangle. And from long habit also easier to THINK R^3. Some people say E^3 for Euclidean 3-space, just trivially different notation.

So the essential thing about mainstream cosmology, which is basically what we discuss here at Forum, is that it's fitting the observational data with MATHEMATICAL models of spacetime, not with verbal models or with commonsense intuitive models.

This means that your take on things----that may be native to you---that the other guy has to prove NO edges, is always going to be just slightly out of line from other people for whom mathematical universe modeling is the name of the game. Because they automatically think no edges UNLESS someone can prove edges with observational evidence.

And in your case I do not perceive any difference in intelligence or moral virtue or individual merit between the two viewpoints. What you are seeing is a cultural difference. So I find that interesting and I hope you do as well. Not to worry about. just be conscious of. Awareness.

 

[marcus]

We have 6 responses! So far the viewpoint that is leading is that
edgeless is simpler but harder to imagine.
Hope more people weigh in on this issue (very basic but still influencing how one thinks)

 

[wolram]

Why is edgeless hard to imagine? take the matter out of our universe and what do you have, you have some thing that is unobservable, why should a measly .5% or whatever it is matter
make some thing real, add to that the matter HAD to come from some where, it existed for all time, then why not an infinite space that has existed for all time, this may be philosophy
but it comes down to can some thing be created from nothing, is there an edge to nothing?

 

[Contrapositive]

I picked "the sheet with edges is both simpler and easier to imagine." Obviously, something finite is easier to imagine than something infinite, and something with edges is easier to imagine for a human mind because we never experience anything edgeless in daily life.

However, because it is simpler and easier to imagine, does not make it true.

 

[wolram]

I would not want to shave with ockham's razor.

 

[marcus]

...
However, because it is simpler and easier to imagine, does not make it true.

I agree! the simplicity idea has more to do with judging who has the burden of proof (who has to do the work of arguing and demonstrating to prove something isn't so)

I think you know this perfectly already but I'll just repeat it anyway...Occam says pick the model with fewest elements that fits the data so far. Then it's up to the others to show evidence that it doesn't work. Simplicity is not about what is true, it is about whose job it is to do the falsifying. I guess. I'm not a philosophy of science expert by a long shot!

Imaginary example: if I can match spacetime and matter dynamics with a model with 4D spacetime, and you do it with 10D, then mine is simpler and you have to do the work.
I don't have to show there AREN'T the extra dimensions, I just wait calmly for you to offer evidence that my 4D model is inadequate (new data that it doesn't match).

Thing is, Occam's razor really does have a subjective aspect! We can differ as to what we think is simple! Science is very much a human community activity depending on vague ideas like judgement and goodfaith participation. I don't see how you could ever automate the business of arriving at consensus, so you could turn a crank on some logic engine.

 

[marcus]

I would not want to shave with ockham's razor.

LOL!
Dull as a butter-knife, wolram. But it's what we've got and we make do.

 

[nicky nichols]

Regge.

 

[Wallace]

Infinite is always easier to imagine, since as long as you grasp how a small piece works you've got the whole lot sorted. Edges introduce more difficult conceptual problems.

Applied to Cosmology, I think if the Universe is finite, but much bigger than the observable universe (so $$\Omega$$ just slightly greater than unity) then assuming the Universe is infinite is easier conceptually and mathematically than not.

Infinities are often used in maths and physics in place of things that are known to be finite, since it often drastically reduces the complexity of a calculation, without affecting the result by any noticeable amount.

That's my view anyway, and it looks like option 1 is making a comeback :)

 

[marcus]

Regge.

http://en.wikipedia.org/wiki/Regge_calculus

 

[Cantari]

I have a hard time hearing people saying that infinite is simpler. If one just thinks about what infinite means for a few moments, you can realize it is beyond our comprehension. Which I think means is not simpler than a printer sized sheet we deal with everyday.

It may sound simpler to say infinite compared to 8.5 inches by 11 inches, although thinking about it a bit deeper it is certainly not simpler.

 

[marcus]

I probably should not have said a sheet of PAPER because that introduces other issues like atoms and molecules and cellulose fibers and mass and gravity etc. And the paper getting in the way of stars and so on. Let's revise and make the plane and rectangle more abstract than actual paper.

I have a hard time hearing people saying that infinite is simpler. If one just thinks about what infinite means ...

Cantari thanks for making your viewpoint known. You're encouraged to state your personal view whatever it is, but please don't tell me you "have a hard time hearing" me state mine.

I don't think there is a right answer here and I am not trying to SELL you on my point of view. I'm interested to know who thinks what and what the range of perceptions is. What I'm looking for here is some cross cultural understanding.
I don't care if you think similarly or differently, and I'm fine with being in a minority. But I'd like for you to understand my point of view as a possible and even natural one.

If INFINITY bothers you, or you think we cannot comprehend it, then you should already be allergic to bounded intervals. There is the same number of points in the interval (0,1) between zero and one as there is in the interval (-oo, +oo) between minus infty and plus infty. You can easily map one interval on the other by simple algebra.

For much of my life I have used the line, the plane, ... things like that on an almost daily basis: R, R^2, R^3.
I just naturally think of those things as simple and comparatively familiar and easy to imagine. If it is just the line, then I'll grant there is not much difference between the line and an interval. But if we are talking about R^3 versus a cube or cone or triangular prism or tetrahedron, then I just instinctively feel that the whole R^3 is simpler than some bounded solid figure.
With a solid figure there are all those details you have to answer---what's the HEIGHT of the tetrahedron, what's the VOLUME of the cone, what's the DIAGONAL of the cube. None of these questions arise when one is dealing purely with the entire space R^3.
So I feel relaxed about R^3, compared with any bounded solid figure subset of it. And some of that spills over into how I think about the 2D case as well.

I'm willing to believe that you think an 8 by 11 rectangle is simpler than R^2 (by the way what length is its diagonal? ) So how about you granting that I think of R^2 as simpler than an 8 by 11 rectangle?

 

[Cantari]

My background is in math so it probably affects the way I think. There is the same number of points in the interval (0,1) between zero and one as there is in the interval (-oo, +oo) between minus infty and plus infty. You can easily map one interval on the other by simple algebra.

So if INFINITY bothers you, or you think we cannot comprehend it, then you should already be allergic to bounded intervals.

Yes... I also have done my fair share of math. Allergic to bounded intervals? ... I don't see how this relates to the discussion of thinking an infinite sized plane is "simpler" than a finite plane. I didn't think we were talking about an infinite amount of numbers between any 2 numbers, which is fairly easy to say.

A finite plane has boundaries and a size. We have all seen a finite plane, such as your example of computer paper. Have any of us seen an infinite plane, let alone visualize it? I don't think it is possible to visualize an infinite plane. So I would think that simpler is going to be directly related to imagining in this question.

So how about you granting that I think of R^2 as simpler than an 8 by 11 rectangle?

Maybe your definition of "simpler" is different than what I am thinking about for this problem. Sure your way of expressing r^2 instead of a given dimension is more convenient and easier than having to measure a plane and list the dimensions. Though saying it is more "simple" in the actual comprehension of the two planes, I don't understand.

 

[hurk4]

I have difficulty with the example i.e. "the sheet of paper". Was this example taken because (IMO the universe) would have been the good but controversial example? I voted for 1) because I replaced the sheet of paper by the universe. I don't want to introduce "the nothing" which realistically doesn't exist. If my replacement is not acceptepted then forget my vote.
Can anyone define an ideal Xtal which is not infinite?
regards
Hurk4

 

[marcus]

... I voted for 1) because I replaced the sheet of paper by the universe...

The universe, and the abstract models of space used to describe it, are just what I had in mind.

We can't argue about perceptions of simplicity---it is just how different people feel about it and you couldn't ever convince anybody or settle the issue. the most we can do is just explain our viewpoint and try to understand the other guy's.

I agree with your perception, hurk. Edges and boundaries are extra complication that introduce more to worry about. What happens there? What is the surface area of the boundary etc.? The attitude I think we both share is, to put it concisely:

Infinity is not a complication. Edges and boundaries are.

Maybe I've misunderstood and you think differently, But just speaking for myself, when I picture the universe about half the time I do think of it as spatially R^3. That is such a good approximation. But I also balance that by picturing it a spatially S^3.

KingOrdo mentioned S^3, and the fact that (although it is finite volume) it is boundaryless. It has the simplicity of something without edges but also the interesting feature of finite volume which means (since we assume matter more or less uniformly distributed throughout space) it has a finite amount of matter.

In S^3 the total amount of matter and energy in the universe is defined. Whereas in R^3 it's infinite. Both are edgeless. The data fit either one equally well at present.

There are other pictures usually with more complex topology which i don't bother to think about. So that's my mental picture----half R^3 and half S^3----anyone is welcome to it.

But the mental pictures are only provisional for the time being---data is not good enough yet to pick from among the front-runners---so no right answer and you go with the mathematical model that works for you.

 

[nicky nichols]

My infinite abstract sheet turns out to be triangular. There is some sort of structure is the far distance background vertex. I thought it might be a sierpinski fractal, but it is too fine to make out. Looking harder, I could see convex curvature downhill away from my viewpoint.

Earlier, I had been considering how my perception might oscillate between finite and infinite sheets.

I have also found a secondary finite sheet far over to the side can appear simultaneously with the infinite sheet in primary imagination space.

This is what I wondered about.

Apparently, there is some sort of theory to do with the way in which high dimensional space might be built up by oscillation of a low dimensional system.

I wondered if Regge might have thought about this idea first of all, but considered it much too difficult to formulate rigourously, and instead settled upon his triangulation calculus.

 

[staf9]

I went with option one, as simple as possible (opinion). This is indeed a debate when you take universal expansion into account. Infinity is infinity, it can't get bigger, yet could the universe be getting bigger? And if it is, does that necessarily make it finite?

 

[marcus]

I went with option one, as simple as possible (opinion). This is indeed a debate when you take universal expansion into account. Infinity is infinity, it can't get bigger, yet could the universe be getting bigger? And if it is, does that necessarily make it finite?

when cosmologists talk about expansion, they mean something that can happen just as well in the infinite case.
Expansion means that the distance between stationary points increases.
There is an idea of points being at rest with respect to the microwave background (or equivalently with respect to the expansion flow)expansion is defined without reference to any overall total volume---which is good because the total volume might not be defined due to the universe being spatial infinite.

==================
the idea of being at rest is basically very simple
Hubble caught on to it back in the 1930s even before the microwave background was observed

if you are not at rest, then there is a DISTINGUISHED DIRECTION in which the microwave background looks hotter because of Doppler shortening of the wavelength

the solar system is moving approx 380 km/s in the direction of the constellation Leo.

Hubble could tell this even without observing background because he could see there was a direction in which the distant galaxies were NOT RECEDING AS FAST as they should be because our motion was causing us to slightly catch up with them.

so being at rest is essentially just not having a bias in anyone particular direction as to what redshifts one measures----that is, the observed distribution of redshifts (or CMB temp) is not skewed somehow along an axis.
(operationally it is a simple pragmatic idea, and astronomers use it all the time to make necessary small corrections)

Corrections for individual motion are small because speeds like this 380 km/second are generally small compared with the recession speeds of distant objects. Nobody in the universe is moving very fast compared with the rate that large distances expand. All galaxies are APPROXIMATELY at rest with respect to the expansion flow. And the distances between them grow on average one percent every 140 million years.

this is what expansion really means. but since it is too technical to tell the whole story people just say "space expands" and they get misled into thinking that it must have some finite volume in order that the volume could expand and give meaning to the idea. It is a language problem with the words "space expands".

several people have written articles (like in SciAm) trying to fight against this confusion but it is deep-rooted.

 

[Kurdt]

I voted yesterday and didn't post my reasoning. I think an infinite sheet is simpler mathematically but a lot harder for the majority of us to understand. With the proper training of course it can become simpler to imagine.