# Can mathematics disagree with the real world?

Tap Banister
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.

Last edited:
• jedishrfu

## Answers and Replies

Staff Emeritus
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?

How about a tesseract?

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.

I'd say no as well, but I'm not math expert.

• Tap Banister and jedishrfu
Mentor
I think you are confusing some things: mathematics and models. When I was in college a very long time ago, one the faculty members there said, "Current aerodynamics models say that bumblebees cannot possibly fly." Well, bumblebees do fly. So where was the problem? The bumblebee model used was constructed wrong.

Nothing to do with math. It's like blaming the messenger for the contents of the message.

The math in the model simply followed the "directions" given. Which were wrong.

http://www.snopes.com/science/bumblebees.asp
cites a minor excerpt from a book, based on a faulty model as the reason for this bit of nonsense.

• olgerm, billy_joule, Tap Banister and 2 others
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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?
As far as is known, the "infinity density" area at the center of a black hole, which we call a "singularity" is CALLED a "singularity" because although the math says it exists, it is believed that such a thing cannot exist physically. The assumption is that this, like the example Jim gave, is because our math model is wrong but it does point out a definite case where it appears that the math describes something that is physically impossible. The fact that this particular model does not properly describe physical reality does not mean that it's invalid math, just that it is applied incorrectly in this case.

It was all summed up many decades ago by Korzibsky: "The map is not the territory"

• Tap Banister, QuantumQuest and jedishrfu
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Personally I'm convinced that something as easy as a circle does not "exist" in "the real world". Whenever you try to show me one I will refer to the next electron microscope. Did you know that mathematics counted as a Geisteswissenschaft (science of spirit) in former times rather than a science of nature? And although mathematics nowadays is widely used as the language of science of nature it is still closer related to philosophy. It is a framework of logical conclusions within a given frame of conditions. "The real world" has nothing to do with it.

• ehild and jedishrfu
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Personally I'm convinced that something as easy as a circle does not "exist" in "the real world".
Good point, and another great example of the map not being the territory, the model not being the reality. A model (math) is ideal, not real. The confusion only arises at all because math does SUCH a great job of describing reality (particularly when you don't probe down to the 20th decimal place)

• Tap Banister and jedishrfu
JonnyG
There are solutions to the Einstein field equations that give a geometry of the universe, which as far as we know, does not describe the actual universe. Mathematical predictions about the real world must be backed up by actual observation.

• Tap Banister and jedishrfu
Mentor
• Tap Banister
• DrewD and Tap Banister
Mentor
Lots of things exist in mathematics, but not in the "real" world, the world that we perceive, which is four-dimensional, with time being one of the dimensions. In mathematics we can work with vectors of three, four, five, ten, or more dimensions (including an infinite number of dimensions). The fact that something doesn't exist in the real world is of no consequence to mathematicians.

• Tap Banister
Tap Banister
Thank you all so much for your answers. They are much appreciated.

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?

That's an interesting question. How do we identify the mathematical existence of something in the real world? We are allowed a lot of arbitrary choices. For example if a person says "That gate is rectangle", he needs to identify something about the gate as its vertices and something about the gate as its edges. Another person might look at the same gate and say "That gate is a random process with two states, 'open' and 'closed'. So a given real object can be regarded as different mathematical structures - and, of course, as given mathematical structure can be applied to more than one real world object. Given that there is so much freedom in matching up mathematical concepts with actual things, it would be surprising to me if a mathematical object exists that can't be matched up to something in the real world if we are given enough freedom in doing the matching-up. If you put restrictions on how the matching-up is done then that's a different question. For example if you insist that "point" and "edge" are defined as places were two planar pieces of sheet metal meet then this greatly restricts what real world objects can be identified as mathematical objects involving points and edges. By contrast, if we are given complete freedom in defining points and edges, we could say that a "point" will be a person, or a color, or a voltage value.

• Drakkith
"Mathematics can never prove anything. No mathematics has any content. All any mathematics can do is -- sometimes -- turn out to be useful in describing some aspects of our so-called 'physical universe'. That is a bonus; most forms of mathematics are as meaning-free as chess."

- Robert A. Heinlein

Mathematics largely looks at consistency in a variety of contexts.

The "pure" mathematicians focus most of their energies on such consistency and they look to create and resolve paradoxes with the existing body of mathematical knowledge.

This consistency need not reflect all forms of consistency found in nature (you will find that developments - "major" developments in mathematics actually consider something thought to be impossible or inconsistent and then find a way to actually make it consistent and eventually make some sense of it).

You have to question whether nature is consistent before you can answer your original question.

VinayS
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.
There are no things that Mathematics claims cannot exist. In fact, Mathematics can formulate some things to claim its existence that do not exist in reality.
Each process may have infinite results that chain to infinite more results. So the probablity of anything is never 0. Anything existing can always be proved for its possibility of existence.

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There are no things that Mathematics claims cannot exist.
He did not ask if there is anything that math CLAIMS cannot exist, he asked if there are things that math DESCRIBES that cannot exist in the real world, and has been shown over and over in this thead the answer to that is yes.

In fact, Mathematics can formulate some things to claim its existence that do not exist in reality.
I have no idea what this means.
Each process may have infinite results that chain to infinite more results.
I have even less idea what that means.

So the probablity of anything is never 0
That is demonstrably incorrect to the point of being ridiculous.

Anything existing can always be proved for its possibility of existence.
That seems like a tautology. "Things that exist, exist". Yep, they sure do.

he asked if there are things that math DESCRIBES that cannot exist in the real world, and has been shown over and over in this thead the answer to that is yes.

That result hasn't been shown.

What has been argued is that if you match up certain simple mathematical concepts to things in the real world (like "point" and "surface") then more complicated mathematical objects that are defined in terms of those simple concepts do not exist. The failure of mathematics to describe a real world object is only a failure within a certain context - namely a context where you have stipulated a particular way in which mathematics succeeds in describing certain real world things.

rootone
Math can be used to describe negative quantity, such as in the case where my friend gives me an apple on the condition that I will give him an apple tomorrow.
Mathematically until I give the apple to him, I am the owner of -1 apple, but -1 apple does not actually exist.

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he asked if there are things that math DESCRIBES that cannot exist in the real world, and has been shown over and over in this thead the answer to that is yes.
That result hasn't been shown.
Math describes a ball which cannot exist in the real world. That has been said more or less.

What has been argued is that if you match up certain simple mathematical concepts to things in the real world (like "point" and "surface") then more complicated mathematical objects that are defined in terms of those simple concepts do not exist. The failure of mathematics to describe a real world object is only a failure within a certain context - namely a context where you have stipulated a particular way in which mathematics succeeds in describing certain real world things.
This argumentation is in the opposite direction. It wasn't about whether a model can't be set up to describe mathematically a real world object which I agree would be a matter of context.
Nevertheless, a mathematical ball cannot exist in reality. There is no smooth object that can be made out of a finite number of molecules. In any context.

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That result hasn't been shown.

What has been argued is that if you match up certain simple mathematical concepts to things in the real world (like "point" and "surface") then more complicated mathematical objects that are defined in terms of those simple concepts do not exist. The failure of mathematics to describe a real world object is only a failure within a certain context - namely a context where you have stipulated a particular way in which mathematics succeeds in describing certain real world things.
I am dumbfounded that you would say so. I give the following argument (basically, what has already been said in this thread)
1. Mathematics can describe a perfect circle consisting of an infinite number of points in an extremely precise geometrical relationship
2. Such an object cannot exist in the real world.
So do you reject this argument? If so why and if not, how can this not be called an object that math describes that does not exist in the real world?

Homework Helper
Gold Member
"Can Mathematics disagree with the Real World"?

Yes! Further, Mathematics can also be misapplied.

I am dumbfounded that you would say so. I give the following argument (basically, what has already been said in this thread)
1. Mathematics can describe a perfect circle consisting of an infinite number of points in an extremely precise geometrical relationship
2. Such an object cannot exist in the real world.

So do you reject this argument?

You haven't made an argument. Are you saying that a "point" cannot exist as a real world object ? Or are you saying that particular real world things exist that are "points" and the particular arrangement of those material things that forms a circle cannot exist ? Are you saying that such an arrangement cannot exist because of some physical law ? Or are you merely saying such arrangements can't be created by a particular manufacturing process ?

In modern mathematics there is no requirement that a "point" represents something in physical space.

• lavinia
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You haven't made an argument. Are you saying that a "point" cannot exist as a real world object ? Or are you saying that particular real world things exist that are "points" and the particular arrangement of those material things that forms a circle cannot exist ? Are you saying that such an arrangement cannot exist because of some physical law ? Or are you merely saying such arrangements can't be created by a particular manufacturing process ?

In modern mathematics there is no requirement that a "point" represents something in physical space.
I'm saying that matter is quantized and ideal circles are not so a mathematical circle cannot be formed in the real world.

Gold Member
I'm saying that matter is quantized and ideal circles are not so a mathematical circle cannot be formed in the real world.
In many theories in physics e.g. classical mechanics, the theories of relativity, string theory, quantum mechanics .... space is modeled as a smooth manifold. So why can't a perfect circle exist in a smooth manifold?

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In many theories in physics e.g. classical mechanics, the theories of relativity, string theory, quantum mechanics .... space is modeled as a smooth manifold. So why can't a perfect circle exist in a smooth manifold?
I don't get how a smooth manifold gets around the fact that mass is quantized. Look, I'm not arguing that math doesn't do an amazingly good job of describing the real world, but I continue to believe that mathematical ideal are exactly that ... ideals ... and cannot be realized perfectly in the real world. I don't see that as a problem, just a fact.

Gold Member
I don't get how a smooth manifold gets around the fact that mass is quantized. Look, I'm not arguing that math doesn't do an amazingly good job of describing the real world, but I continue to believe that mathematical ideal are exactly that ... ideals ... and cannot be realized perfectly in the real world. I don't see that as a problem, just a fact.
i dont think you answered the question. So what if some theory says that mass is quantized?

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i dont think you answered the question. So what if some theory says that mass is quantized?
Do you think the quanization of mass is just a theory? Do you think mass is continuous?

Gold Member
Do you think the quanization of mass is just a theory? Do you think mass is continuous?
That was not my point. Suppose some theory says that mass is quantized. How does that change the fact that physical theories model space as a continuum?

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That was not my point. Suppose some theory says that mass is quantized. How does that change the fact that physical theories model space as a continuum?
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?

• fresh_42
I'm saying that matter is quantized and ideal circles are not so a mathematical circle cannot be formed in the real world.

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.

• zinq
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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.

Staff Emeritus
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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.

• lavinia
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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.

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