Problem with Maths and Nature

[CarbonCopy]

If Mathematics is supposed to describe nature perfectly then why do we sometimes get infinities as solutions. What does it mean ? Is it fine to just discard them or do we need to look deeper ?

[ajb]

If Mathematics is supposed to describe nature perfectly then why do we sometimes get infinities as solutions. What does it mean ? Is it fine to just discard them or do we need to look deeper ?

 

It is usually understood as a signal that the equations do not hold in that particular situation. This is a signal that new physics is needed.

[CarbonCopy]

What about for eg, when we put r=0 in newton's law of gravity we get infinity. how do we rationalize that. and what about singularities,

Edited October 26, 2012 by CarbonCopy

[ajb]

What about for eg, when we put r=0 in newton's law of gravity we get infinity. how do we rationalize that. and what about singularities,

 

So that would represent masses concentrated to a point, that is classically not allowed. You are trying to apply physical laws to rather unphysical situations.

 

The singularities in general relativity are more subtle than this and much more interesting.

[Dekan]

If Mathematics is supposed to describe nature perfectly then why do we sometimes get infinities as solutions. What does it mean ? Is it fine to just discard them or do we need to look deeper ?

 

A good question. It strikes at the heart of Mathematics as applied to Nature.

 

Let's consider: When we look at Nature, we find that all natural objects are made of round things. Thus the Earth is round, and it orbits round the Sun, which is also round.

And the Earth and Sun are supposed to be made of smaller particles, like protons, electrons and neutrons - which are also round.

 

"Roundness" seems a basic property of Nature.

 

However - what happens when we try to investigate this Natural "roundness" by applying our Mathematics to it. Even if we use the simplest example of "roundness" - a perfect circle - we find notoriously, that Maths cannot provide an exact solution to such a basic question as - what's the ratio between the circle's radius and its circumference.

 

All the Maths comes up with, is the dispiriting and fundamentally unsatisfactory series : 3.14159265358979323...... and so on to infinity.

 

Doesn't this point to some kind of disconnect between Maths and Nature?

[CarbonCopy]

1351363811[/url]' post='710707']

A good question. It strikes at the heart of Mathematics as applied to Nature.

 

Let's consider: When we look at Nature, we find that all natural objects are made of round things. Thus the Earth is round, and it orbits round the Sun, which is also round.

And the Earth and Sun are supposed to be made of smaller particles, like protons, electrons and neutrons - which are also round.

 

"Roundness" seems a basic property of Nature.

 

However - what happens when we try to investigate this Natural "roundness" by applying our Mathematics to it. Even if we use the simplest example of "roundness" - a perfect circle - we find notoriously, that Maths cannot provide an exact solution to such a basic question as - what's the ratio between the circle's radius and its circumference.

 

All the Maths comes up with, is the dispiriting and fundamentally unsatisfactory series : 3.14159265358979323...... and so on to infinity.

 

Doesn't this point to some kind of disconnect between Maths and Nature?

So your saying that to describe all this we need to describe a circle. But we cannot get a solution for even this.

 

 

 

[CarbonCopy]

Doesn't this point to some kind of disconnect between Maths and Nature?

so Math does not really describe nature to the fullest ?

Edited October 28, 2012 by CarbonCopy

[ajb]

so Math does not really describe nature to the fullest ?

 

It seems that nature does make use of mathematical laws, everything does look rather "clockwork", but not in the sense that Newton would have meant. It is an amazing fact that mathematics has been so useful in the physical sciences.

 

Us humans use mathematics to describe nature, but does that truly mean that nature is really mathematical?

 

I have no idea and doubt we could ever find an answer to that.

[Villain]

It seems that nature does make use of mathematical laws, everything does look rather "clockwork", but not in the sense that Newton would have meant. It is an amazing fact that mathematics has been so useful in the physical sciences.

 

Us humans use mathematics to describe nature, but does that truly mean that nature is really mathematical?

 

I have no idea and doubt we could ever find an answer to that.

 

I would say no.

 

Take a simple equation of 1+1, 1 is the ideal in the Plato sense, the perfect example of 1. In the world that we exist in however there is no such thing as a perfect 1, only something that resembles 1 in certain ways, futhermore this object resembling 1 is never the same as that object resembling 1 but yet we impose that this (object) 1 + that (object) 1 = 2 (ideals of 1). In mathematics 1+1=2, but it has no real value in our reality since two 1's don't exist.

Edited November 2, 2012 by Villain

[swansont]

A good question. It strikes at the heart of Mathematics as applied to Nature.

 

Let's consider: When we look at Nature, we find that all natural objects are made of round things. Thus the Earth is round, and it orbits round the Sun, which is also round.

And the Earth and Sun are supposed to be made of smaller particles, like protons, electrons and neutrons - which are also round.

 

"Roundness" seems a basic property of Nature.

Some of those examples are because "round" minimizes some feature that is (or is related to) a quantity that should be minimized. Round minimizes surface/volume (or area, in 2-D), and distance to the center.

 

However - what happens when we try to investigate this Natural "roundness" by applying our Mathematics to it. Even if we use the simplest example of "roundness" - a perfect circle - we find notoriously, that Maths cannot provide an exact solution to such a basic question as - what's the ratio between the circle's radius and its circumference.

 

All the Maths comes up with, is the dispiriting and fundamentally unsatisfactory series : 3.14159265358979323...... and so on to infinity.

 

Doesn't this point to some kind of disconnect between Maths and Nature?

The ratio is exactly pi, and always pi. That pi is irrational rather than being an integer is a separate question, and nature is not obligated to conform to your notion of what is satisfying.

[Dekan]

Some of those examples are because "round" minimizes some feature that is (or is related to) a quantity that should be minimized. Round minimizes surface/volume (or area, in 2-D), and distance to the center.

 

 

The ratio is exactly pi, and always pi. That pi is irrational rather than being an integer is a separate question, and nature is not obligated to conform to your notion of what is satisfying.

 

That's true - Nature is satisfying - all her circles and spheres have a exact ratio of circumference to radius. Otherwise - there'd be gaps.

 

Problems only arise when we try to measure the ratio. Then our Math fails to give an exact answer.

 

This failure shows that our present-day Math doesn't provide a complete explanation of Nature. Of course we can prevaricate, and try to explain it away. By inventing terms like "pi", and "irrational numbers". But that doesn't sound very convincing. Surely the fault lies in our Math, not in Nature.

 

Perhaps we'll advance to better Maths in the future!

[swansont]

That's true - Nature is satisfying - all her circles and spheres have a exact ratio of circumference to radius. Otherwise - there'd be gaps.

 

Problems only arise when we try to measure the ratio. Then our Math fails to give an exact answer.

 

This failure shows that our present-day Math doesn't provide a complete explanation of Nature. Of course we can prevaricate, and try to explain it away. By inventing terms like "pi", and "irrational numbers". But that doesn't sound very convincing. Surely the fault lies in our Math, not in Nature.

 

Perhaps we'll advance to better Maths in the future!

But that's based on a premise that nature behaves in a certain way. A premise that hasn't been shown to be true.

[Villain]

But that's based on a premise that nature behaves in a certain way. A premise that hasn't been shown to be true.

 

Sadly such an honest statement would probably get negative rep in the religious section.

[CarbonCopy]

But that's based on a premise that nature behaves in a certain way. A premise that hasn't been shown to be true.

 

so your saying nature does not follow certain rules ?

I thought Science is based on that very premise.

Edited November 4, 2012 by CarbonCopy

[timo]

I think it's rather obvious that there are rules you can formulate mathematically that nature does not follow. For example the inverse-cube law, by which the gravitational attraction of an object to a spherical mass is proportional to the distance from the object cubed (F ~ 1/r³), or the inverse-charge law, by which the electrostatic force between two objects is proprtional to the inverse product of their charges (F ~ 1/(q1*q2)).

[swansont]

so your saying nature does not follow certain rules ?

I thought Science is based on that very premise.

Not at all. I'm saying it does not have to follow an artificial constraint we place upon it, in this case that the ratio d/C be a "nice" number.

[ydoaPs]

A good question. It strikes at the heart of Mathematics as applied to Nature.

 

Let's consider: When we look at Nature, we find that all natural objects are made of round things. Thus the Earth is round, and it orbits round the Sun, which is also round.

And the Earth and Sun are supposed to be made of smaller particles, like protons, electrons and neutrons - which are also round.

 

"Roundness" seems a basic property of Nature.

 

However - what happens when we try to investigate this Natural "roundness" by applying our Mathematics to it. Even if we use the simplest example of "roundness" - a perfect circle - we find notoriously, that Maths cannot provide an exact solution to such a basic question as - what's the ratio between the circle's radius and its circumference.

 

All the Maths comes up with, is the dispiriting and fundamentally unsatisfactory series : 3.14159265358979323...... and so on to infinity.

 

Doesn't this point to some kind of disconnect between Maths and Nature?

 

Pi is an exact number. As is e as is Phi. Irrational numbers aren't fuzzy wuzzy things; they're just numbers that cannot be represented as ratios of whole numbers. And since pi is defined as a ratio, that means for every perfect circle in a Euclidean (the ratio that gives us pi varies by the curvature of the space in which you're doing the geometry) geometry, the circumference, the diameter, or both are themselves irrational!

 

On a side note, I like the "and so on" there; it implies a pattern where there is none. If it were a regular pattern, it would be rational.

 

It seems that nature does make use of mathematical laws, everything does look rather "clockwork", but not in the sense that Newton would have meant. It is an amazing fact that mathematics has been so useful in the physical sciences.

 

 

Us humans use mathematics to describe nature, but does that truly mean that nature is really mathematical?

 

One must take care to distinguish between the description and the thing being described. Mathematics is an abstract description of reality, so it shouldn't be terribly surprising that it accurately reflects how the universe behaves; that is what it was designed to do.

 

It seems that the issue here is when people ask why mathematics describes reality so well, what they really want to know is why reality behaves so regularly that it allows for description via formal languages and systems.

[imatfaal]

...

 

On a side note, I like the "and so on" there; it implies a pattern where there is none. If it were a regular pattern, it would be rational.

...

 

This line - which I am not arguing with - made me think of something, it caused me to think of 2d patterns and then of tilings. Could I equate the lack of a repeating pattern to the lack of translational symmetry? Because we can make tilings with an absence of translational symmetry - but which is made of identical tiles, and which demonstrate self-similarity over different scales. Do we know that this doesn't exist in the irrationals and transcendentals. Or am I just seeing links where there are none?

 

Sorry if a bit fey and wishy-washy.

[Pugdaddy]

Can someone explain the following? If an observer is watching a point on a rotating plane, does not the circumference decrease due to length contraction and yet the radius stays the same. Would this not change the pi ratio of radius to circumference?

[ydoaPs]

Can someone explain the following? If an observer is watching a point on a rotating plane, does not the circumference decrease due to length contraction and yet the radius stays the same. Would this not change the pi ratio of radius to circumference?

 

Yes, because you're no longer dealing with a flat space. Geometry is different depending on the curvature of the space.

[LaurieAG]

Pi is fascinating because, when derived from distances or time, it becomes a dimensionless constant.

 

If I started photographing the sparkler around 6 and a bit feet away, the circle was 2 feet in diameter and I captured the light from the spinning sparkler in one complete circle the ratio ( A ) of the time between the rotating source and the observer over the diameter of rotation would be roughly equal to Pi.

In this case the ratio ( B ) of the actual distance between source and observer over the distance travelled by light in a year would be very small and the ratio ( C ) of the observation period over the time it takes for the sparkler to be rotated once will equal one. All observations should have a width of field that covers the complete diameter of rotation of the source being observed.

 

If I halve the exposure period I get half a circle and capture half as much light and when I double the exposure period I get 2 circles over each other and twice as much light in my photograph. If the sparkler is rotated twice as fast I would expect something that looked similar to when I doubled the exposure period but I would also expect to capture the same amount of light from only one rotation despite the doubling of the speed of rotation. If I taped two sparklers together I could halve the exposure time and double the speed of rotation to capture a similar amount of light from 1 sparkler doing 1 complete rotation. If the sparkler was moved at an angle to me I would observe an oval instead of a circle but the amount of light captured would remain the same as for a complete circle.

 

In this simplest base context A = Pi, B = tiny, C = 1 and the observer will capture one complete cycle. On any scale where C >= 1 the observer will capture at least one complete cycle despite the size of B.

 

On any scale where A = Pi * x, B >= 1 and C < 1 the observer will only capture the light from B * C = x of one rotation during any observation regardless of the speed of rotation of the same object.

 

On a galactic year scale where A = Pi * x, B = 230 million and C = 1/230 million you would expect to capture the light from B * C = x rotations or roughly one rotation regardless of the speed of rotation.

 

On a galactic year scale where A = Pi * x, B = 4.2 billion and C = 1/4.2 billion you would expect to capture the light from B * C = x rotations or roughly one rotation.

 

This should be what we see when we observe light from an undistorted source rotating around a galactic centre that is stationary.

Edited November 22, 2012 by LaurieAG

[PeterJ]

Is it possible that pi is irrational because in real life the 'point' at which a radius meets a circumference is not in fact a point, and cannot be defined accurately until the two lines are infinitely thin and therefore not there anymore? Iow, Pi is an approximation to an impossible perfection.

[imatfaal]

But geometry works with lines with only one extension, points with no area, and impossible precision. irrationals are not the product of inaccuracy - and there are proofs (many and from many years ago) of the irrationality and transcendental nature of pi and e. The proofs do not use measurement - they are geometric and algebraic and introduce no inaccuracy. Some of the ideals of mathematics are never ending transcendentals - in a way that makes sense; how could an ideal be representable by a combination of something less complex?

 

Although by using names like irrational and transcendental mathematicians are opening themselves up for all sorts of follow up questions that really have nothing to do with the maths. Almost as bad an idea of naming as dark matter and dark energy.