Do curves, circles and spheres really exist?

[Cody Richeson]

Obviously, they exist as mathematical concepts, and those concepts are real, but in physical reality, everything is made up of subatomic particles and, if the theory is ever verified, strings. So if you try to construct a curve, circle or sphere, you are necessarily stacking a bunch of subatomic particles in an arrangement that, on a macro scale, appears curved, but is really pixelated. Does this mean that a true curve cannot really exist?

 

[DaveC426913]

Not all real things are made of particles.

 

[Cody Richeson]

Not all real things are made of particles.

So are we talking about forces and fields here? Or what?

 

[muscaria]

If we walked around a glass while looking at the top, we would say that the closed curve is in fact an ellipse and would abstract the union of the appearances of the ellipses as deriving from an essence which is a circle. Then you go, well actually the circle is made up of atoms, which are mostly empty space, and the atoms themselves are merely appearances of yet more fundamental particles. Then we find out these particles are certain excitations of a field such that the fields are now the essence and the particles correspond to an appearance of the field. It seems that the essence keeps receding in some fashion and it seems very possible that the fields will be considered at some point to be some appearance of yet a deeper order. A feature that is common to both relativity and quantum theory is that the abstracted physical properties of systems in these theories enter in the form of relationships between systems (more fundamentally from quantum theory though). If you're interested in this, there's a rather compelling appendix about perception in David Bohm's book on special relativity which tries to bring out these points.

 

[DaveC426913]

So are we talking about forces and fields here? Or what?

Sure.

But the point is, your assertion that circles must be made of discrete elements is specious, therefore there's no basis for stating real circles are impossible. You can't - until you've eliminated all possibilities. i.e.the onus is on you to demonstrate that you have.

 

[ZapperZ]

Obviously, they exist as mathematical concepts, and those concepts are real, but in physical reality, everything is made up of subatomic particles and, if the theory is ever verified, strings. So if you try to construct a curve, circle or sphere, you are necessarily stacking a bunch of subatomic particles in an arrangement that, on a macro scale, appears curved, but is really pixelated. Does this mean that a true curve cannot really exist?

This is puzzling, and I don't think you realize the actual depth of the question you just asked. This is because I can generalize your question even more, and see if you agreed that this is what you are asking:

"Is the mathematics that we use to describe nature is an accurate representation of nature itself?"

Don't you think this is really what you are asking?

If it is, then you need to first consider what you think is "accurate" enough, and whether such accuracy implies that the description is representing something "real".

Zz.

 

[DaveC426913]

The S-orbital electron shell around a free hydrogen atom forms a perfect sphere. The probability that an electron will be observed at radius r at point x,y,z is precisely the same probability that it will be observed at radius r at point x₁,y₁,z₁.

Whether our instruments can measure it to some arbitrary degree of precision is beside the point.

 

[muscaria]

"Is the mathematics that we use to describe nature is an accurate representation of nature itself?"

I'd agree with you and think that is really the essence of the question.. any kind of curve or geometry is arbitrary and we're asking about the relationship of our abstractions of reality to reality itself. This has been one of the oldest and deepest questions which has fascinated philosophers and physicists for thousands of years, thinking back of Plato's allegory of the cave etc.. I feel that the belief that mathematical and physical theories can be taken as a direct reflection of reality leads to many inconsistencies. I find it makes more sense to consider these as abstractions that have relevance (if any at all) within a certain domain of experience and become limited when one asks more subtle questions and this process keeps receding. It is when the limits become apparent that new orders are called for in Physics and I feel this is really the driving force behind scientific theories.

 

[ZapperZ]

I'd agree with you and think that is really the essence of the question.. any kind of curve or geometry is arbitrary and we're asking about the relationship of our abstractions of reality to reality itself. This has been one of the oldest and deepest questions which has fascinated philosophers and physicists for thousands of years, thinking back of Plato's allegory of the cave etc.. I feel that the belief that mathematical and physical theories can be taken as a direct reflection of reality leads to many inconsistencies. I find it makes more sense to consider these as abstractions that have relevance (if any at all) within a certain domain of experience and become limited when one asks more subtle questions and this process keeps receding. It is when the limits become apparent that new orders are called for in Physics and I feel this is really the driving force behind scientific theories.

But you really haven't explained your question at all, then. What do you consider to be "accurate enough" to be "real"? Do you consider your spouse/children/parents/friends/etc. to be "real"? Yet, they are less predictable than our ability to predict the behavior of electrons in an atom, i.e. we know more about those electrons than what your friend might do tomorrow. So who is less real here?

A very common component that is often neglected by many when question like this arises is the experimental aspect of science. People who are not in science are often enamored by strange, exotic theory and the metaphysical implications of them. Yet, the biggest part of science, and what differentiates it from many other studies, is the crucial experimental aspect, and the reproducibility of observations. I think the general public tend to not have a greater appreciation of this part of science. In fact, there was even an article written on why the public tends to gravitate more towards theorists rather than experimentalists.

So when something like this pops up, I always wonder why the experimental aspect of science is usually neglected. Why is there very little significance placed on the reproducibility of our experimental results that are consistent with theoretical descriptions. This is MORE and STRONGER verification than anything you do elsewhere in real life. So why is this more in doubt than those?

Zz.

 

[muscaria]

But you really haven't explained your question at all, then.

we're asking about the relationship of our abstractions of reality to reality itself.

What do you consider to be "accurate enough" to be "real"? Do you consider your spouse/children/parents/friends/etc. to be "real"?

I consider them to be real :p, but not my abstraction of them, that remains abstract and has no direct correspondence with reality. I don't consider theories to be real in the sense of a direct correspondence with reality because it seems like any form of theory is limited. Theories are abstractions of reality which have predictive "power" over the expectation or outcome of a physical measurement - if they are relevant to reality in some domain of experience, if not they're just cerebral masturbation. When the limits of a theory become apparent, such as Newtonian mechanics in view of special relativity, it's not that Newtonian mechanics suddenly became wrong, but just that its limits became apparent. I feel that because of these limits, theory has no $\textbf{direct}$ correspondence with reality, in the sense of the theory being the reality or something along those lines, which I find is a somewhat crazy point of view.

 

[fresh_42]

Obviously, they exist as mathematical concepts, and those concepts are real, but in physical reality, everything is made up of subatomic particles and, if the theory is ever verified, strings. So if you try to construct a curve, circle or sphere, you are necessarily stacking a bunch of subatomic particles in an arrangement that, on a macro scale, appears curved, but is really pixelated. Does this mean that a true curve cannot really exist?

Platonist? Yep, the "real" world is a very discrete thing. Concepts like smoothness, differentiability and so on cannot be found in reality. And I'm looking forward to the answer of whether time is quantized.

 

[fresh_42]

The S-orbital electron shell around a free hydrogen atom forms a perfect sphere. The probability that an electron will be observed at radius r at point x,y,z is precisely the same probability that it will be observed at radius r at point x₁,y₁,z₁.

Whether our instruments can measure it to some arbitrary degree of precision is beside the point.

But all you said is already a hypothetic construction. The fact that you can handle a sphere doesn't make it existent! The elctron is still at a place in time and space, not on the whole shell. And even fields are built by their carrying particles. It's simply math and not "existence".

 

[muscaria]

And I'm looking forward to the answer of whether time is quantized.

Ditto. I would like to study Bohm and Hiley's implicate order which seems to investigate this question.

 

[Cody Richeson]

Very interesting responses. I'd like to add the following:

Science typically relies on approximations in order to describe an isolated aspect of reality. Huge clusters of atoms and molecules become averages because it is impossible to take into account every variable involved. I would imagine fluid dynamics would be an example of this, because the math describing it is not taking into account every single elementary particle in the fluid.

What's really weird about curves, circles and spheres is that, as concepts, they are infinitely smooth. I'm not sure how this is, since a circle is said to be made of a finite number of degrees, and yet each of these degrees has an infinite number of points between it. In real life, there are a countable number of points making up the curve. So it's almost like a reversal. Usually, science has to dumb down things due to the immense complexity of what is being described, and yet here, the thing being described (a curve) seems to contain more information (an infinite amount) than the physical thing being described (which is obviously countable).

 

[ZapperZ]

I consider them to be real :p,

Why do you consider them to be "real"?

And where is the clear and unambiguous definition of "real"? Why is the BCS theory less real than your friends and family? Where does it say that a theory that has a range of validity is not real?

Isn't this idea of yours itself a "theory"? What makes it valid or real? I would even say that this idea of yours has less degree of validity than the BCS theory. It makes it really absurd that a "less real" idea is claiming that something that has been so thoroughly verified to be not real.

Please note that previous threads on this very same topic have suffered untimely death.

Zz.

 

[Mark44]

Obviously, they (Mark44: curves, circles, and spheres) exist as mathematical concepts, and those concepts are real, but in physical reality, everything is made up of subatomic particles and, if the theory is ever verified, strings. So if you try to construct a curve, circle or sphere, you are necessarily stacking a bunch of subatomic particles in an arrangement that, on a macro scale, appears curved, but is really pixelated. Does this mean that a true curve cannot really exist?

Curves, circles, and spheres are abstract geometric concepts that I would not call "real." If you construct a curve or circle, your construction will be made using paper and pen or pencil, hence the marks you make will have some finite width, unlike that of the ideal you are approximating. Even if you could construct the curve using atoms laid out in order, the curve would have some finite width, again unlike that of the mathematical idea.

What's really weird about curves, circles and spheres is that, as concepts, they are infinitely smooth. I'm not sure how this is, since a circle is said to be made of a finite number of degrees, and yet each of these degrees has an infinite number of points between it.

A better way to state this is: A circle has a central angle of 360°. One degree of angle subtends an arc that is an interval on the circumference of the circle, and this interval contains an uncountably infinite number of points.

In real life, there are a countable number of points making up the curve.

I don't see how this matters. If we draw a circle "in real life" the best we can hope for is an approximation to the geometric ideal, in which there are an uncountably infinite number of points making up the curve.

So it's almost like a reversal. Usually, science has to dumb down things due to the immense complexity of what is being described, and yet here, the thing being described (a curve) seems to contain more information (an infinite amount) than the physical thing being described (which is obviously countable).

The question has been asked and answered, so I am closing this thread.