Undergrad Can mathematics disagree with the real world?

[fresh_42]

As I said before, the question of whether a mathematical object exists is only a specific question when we define particular ways of mapping mathematical concepts to things in the real world.

So there is a way to map a circle onto something existing without losing any properties of the circle? I could only imagine an orbit doing so, but then you're left with the fact, that there is no point in reality.

Since you're not allowed to change the mathematical concept, your statement claims that there is always a reality that covers this concept. Thus you reduce reality to make it fit. But this implies that you are only allowed to consider certain aspects of reality contradicting the set of properties of the existing object that come along with it. In any case you will have to make adjustments which contradicts the existence on one side of the equation.

 

[micromass]

I don't get what "space as a continuum" has to do with matter being quantized. If matter is quantized, how do you make a physical circle out of matter?

You can't. So what's the point. If spacetime is a continuum, then circles do exist.

 

[zinq]

It's hard to answer the question "Does math describe things that can't exist in the real world?" because math is one thing; the real world is another. Our understanding of the real world is mediated by our consciousness and the physics that we know so far. Physics, as humans comprehend it and model, has undergone tremendous expansion in the past 500 years and there is no reason to believe this will not continue. This means that what we mean by "the real world" is a moving target. So, we don't know what it would mean to say something in math does not exist in the real world.

One example suggested above is the mathematical concept of a perfect circle. Certainly when we draw a circle one paper with a pencil in a compass, that is not the mathematical ideal — it's a bunch of graphite molecules on an imperfectly flat piece of paper. But the force field of an electron seems to be spherically symmetric, and a sphere certainly contains circles. (Maybe! This has to be adjusted for gravity and other things. But there might be a perfect circle in there somewhere, anyhow.) So we don't really know if a perfect circle exists in the real world.

Someone said there are no things that mathematics claims do not exist. This is not true. There are plenty of things that math shows cannot exist. For instance, there does not exist any pair of even integers whose product is an odd integer. There does not exist any positive integer with two distinct prime factorizations (i.e., where the two sets of prime factors are unequal).

Someone said our real world is only 4-dimensional but math can define spaces of any number of dimensions, even infinitely many. But how can anyone say such things cannot exist in the real world? In fact quantum mechanics makes use of infinite-dimensional Hilbert space, and any finite-dimensional space can be found inside such a Hilbert space. But then again we don't know if the Hilbert space model misses some fine points of physics that we don't know about yet.

Someone said that the math of a black hole predicts an impossible thing at its center: infinite mass density. But nobody has really visited the center of a black hole, and probably never can. So we don't really know if infinite density is impossible; we have just never experienced that and are unsure exactly what that would mean in terms of how to mathematically understand infinite mass density. But ruling this out assumes facts not in evidence.

These kinds of issues mean the question is not really possible for humans to answer.

 

[fresh_42]

You can't. So what's the point. If spacetime is a continuum, then circles do exist.

But spacetime isn't flat from the mathematical point of view.

 

[micromass]

But spacetime isn't flat from the mathematical point of view.

Can't we define circles in a nonflat space? I don't see a reason why we can't. They just won't satisfy the usual properties we're used to.

 

[phinds]

That only shows a circle doesn't exist when a "point" fails to represent matter. It doesn't deal with situations when a "point" represents some other aspect of the real world. As I said before, the question of whether a mathematical object exists is only a specific question when we define particular ways of mapping mathematical concepts to things in the real world. You are selecting one particular way of associating mathematical concepts with something in the real world. That creates an example where a mathematical object doesn't exist in the context of using that association. It doesn't eliminate the possibility that there are other ways of doing associations where the mathematical object does exist.

Fair enough. I'm an engineer and think in simple concrete terms so perhaps this is beyond me.

 

[fresh_42]

Can't we define circles in a nonflat space? I don't see a reason why we can't. They just won't satisfy the usual properties we're used to.

Of course. But that leaves us alone with a flat circle that cannot be found elsewhere. But even in a perfect flat space there is still a difference between the mathematical imagination of a circle and some imaginary curve in space. I think there can't even be a precise radius. But that's probably another discussion about the Planck length. And whether there is actually such a border or not, I believe there is a border beyond which we cannot measure or define anything.
But I confess guilty for being a Platonist.

 

[ehild]

Do we know what real world is? We can feel, see, smell, hear and touch different things and we call these real world. We have tools to extend our observations, and we say these non-direct observations also belong to the real world, but do they really?
On the other hand, I exist, I am real, so I belong to the real world. I do not know if my observations are real or not. Maybe all of them are shadows on the wall of the cave I live in, like in Plato's cave.
On the other hand, if I am real, my brain is real, and my thoughts really exist for me. I imagine a circle with my real brain. A circle is real for me, as set of points on a plane all at equal distance from the centre. If somebody tries to make that circle, it will not be perfect. It will be only a model of my circle.
And I can imagine a sphere in 3-dimensional space. And spheres in 4, 5, ...N dimensions. I even can derive their volume ( I can not now, I forgot what I had learned about it) . I can use my N-dimensional spheres to model gases, and derive their properties. And my observations about gases support these derivations. So are N-dimensional spheres not real? What we call real world at all?

 

[rootone]

I think if you were to drop a 10kg weight on your foot, you would come to the conclusion that kilograms are real.

 

[chiro]

You can quantize continuous objects in language just like discrete objects.

The only difference is what the information maps to and the basis for the language.

One can represent objects that have continuity and analyticity (analytic being based on complex variable calculus) with a finite number of symbols.

Just because information is quantized does not mean it has to correspond to some fixed discrete system representation.

 

[jack476]

Are there "things" mathematics describes that can't exist in the real world; things mathematics says can exist but are actually impossible to ever exist in the world we live in?

Also, are there things that do exist in our world that mathematics says can't exist? The obvious answer to me is no, but I am far from informed on any of this.

Your insights are greatly valued, to me.

Well, does math "exist" in the universe at all?

And yes, of course you can mathematically construct things that do not correspond directly to the real world. See for instance the physics engine of any video game.

And as for physics describing things that can't be mathematically explained, at least with current tools, that's pretty much the summary of the entire history of theoretical physics.

 

[lavinia]

It's hard to answer the question "Does math describe things that can't exist in the real world?" because math is one thing; the real world is another.

Strip the mathematical form completely out of the "real world" including our perceptions which are intrinsically geometrical - what is left over to be called the "real world"?