Pythagoras' Theorem. Evidence of elsewhere or our ingenuity.

[Sean Hayden]

But that is exactly my point. If it is restricted by what it describes, then it's not purely an invention.

Purely a map is what I'm thinking.

[Svartalf]

None of the straight lines listed above are actually straight. Even light is bent by gravity.

Nope it isn't, at least not if Einstein is right. The universe itself is bent by gravity. Light still travels in a straight line through that universe.

No, quanta of light (photons) take every possible path between one point and another! It is only when they interact with another particle that their position becomes fixed - for all of the space between their creation and their destruction, they are literally everywhere at once (with varying probabilities.)

And, to take the general relativity stance instead of the quantum mechanical one, light travels in geodesics, not straight lines. It takes the shortest route, not the straightest, that route being defined by the relative positions of the matter in the universe and its associated gravitational field.

That's why I started by saying "it's a darned good approximation"

But isn't the shortest route also the straightest? I thought that was how we define straight.

Look at plane trajectories when going around the earth... They are not taking straight lines.

[Xamonas Chegwé]

If it's purely an invention, what would restrict it?

My first thought is usefulness, what it describes.

But that is exactly my point. If it is restricted by what it describes, then it's not purely an invention.

It is not purely an invention. It is a model of reality. But it is limited in its accuracy by its ideal nature.

Pythagoras' Theorem applies to ideal, two-dimensional spaces and three-dimensional spaces constructed from them. It translates well to the surface of the Earth because, at human scales, that surface approximates very well to a two-dimensional plane. But start measuring large enough lengths and it breaks down. Draw a triangle on the Pacific with sides of 3,000, 4,000 & 5,000 kilometers and you will not find a right angle in it!

All mathematics (indeed all science) is based on such idealised models. Their usefulness depends on their accuracy at the scale under examination. There is no need to allow for relativity in a car's speedometer. There is no reason to consider quantum effects when striking a pool ball. It is only when the model is extended to a scale beyond its original purpose that its flaws become obvious.

[MiM]

But that is exactly my point. If it is restricted by what it describes, then it's not purely an invention.

Purely a map is what I'm thinking.

I can agree on the concept of map. The important issue here is that it is linked to something real, that is outside human thought.

[MiM]

None of the straight lines listed above are actually straight. Even light is bent by gravity.

Nope it isn't, at least not if Einstein is right. The universe itself is bent by gravity. Light still travels in a straight line through that universe.

No, quanta of light (photons) take every possible path between one point and another! It is only when they interact with another particle that their position becomes fixed - for all of the space between their creation and their destruction, they are literally everywhere at once (with varying probabilities.)

And, to take the general relativity stance instead of the quantum mechanical one, light travels in geodesics, not straight lines. It takes the shortest route, not the straightest, that route being defined by the relative positions of the matter in the universe and its associated gravitational field.

That's why I started by saying "it's a darned good approximation"

But isn't the shortest route also the straightest? I thought that was how we define straight.

Look at plane trajectories when going around the earth... They are not taking straight lines.

No. The straight lines goes through the earth, so?

[MiM]

None of the straight lines listed above are actually straight. Even light is bent by gravity.

Nope it isn't, at least not if Einstein is right. The universe itself is bent by gravity. Light still travels in a straight line through that universe.

No, quanta of light (photons) take every possible path between one point and another! It is only when they interact with another particle that their position becomes fixed - for all of the space between their creation and their destruction, they are literally everywhere at once (with varying probabilities.)

And, to take the general relativity stance instead of the quantum mechanical one, light travels in geodesics, not straight lines. It takes the shortest route, not the straightest, that route being defined by the relative positions of the matter in the universe and its associated gravitational field.

That's why I started by saying "it's a darned good approximation"

But isn't the shortest route also the straightest? I thought that was how we define straight.

Straightness is an ideal. As has been claimed by others, it does not exist in nature. It is a very good approximation indeed when it comes to the human scale on Earth - but it is still an approximation.

The shortest route through curved space well is not straight - and all space is curved to some degree - in 4 dimensions.

All animals are gay

[MiM]

If it's purely an invention, what would restrict it?

My first thought is usefulness, what it describes.

But that is exactly my point. If it is restricted by what it describes, then it's not purely an invention.

It is not purely an invention. It is a model of reality. But it is limited in its accuracy by its ideal nature.

Pythagoras' Theorem applies to ideal, two-dimensional spaces and three-dimensional spaces constructed from them. It translates well to the surface of the Earth because, at human scales, that surface approximates very well to a two-dimensional plane. But start measuring large enough lengths and it breaks down. Draw a triangle on the Pacific with sides of 3,000, 4,000 & 5,000 kilometers and you will not find a right angle in it!

All mathematics (indeed all science) is based on such idealised models. Their usefulness depends on their accuracy at the scale under examination. There is no need to allow for relativity in a car's speedometer. There is no reason to consider quantum effects when striking a pool ball. It is only when the model is extended to a scale beyond its original purpose that its flaws become obvious.

Obviously it's a model. And obviously all our models are approximative and prone to errors. No arguing here. But in my view the fundamentals of mathematics are experimental. If experiment would show that adding two apples to three would yield a different result from adding three apples to two, our mathematical rules would be different from what they are now. So there is something fundamental out there that guides the rules we can apply in mathematics, while still keeping it a useful tool to describe reality.

[Svartalf]

That's why I started by saying "it's a darned good approximation"

But isn't the shortest route also the straightest? I thought that was how we define straight.

Look at plane trajectories when going around the earth... They are not taking straight lines.

No. The straight lines goes through the earth, so?

Even clinging to the surface, they still follow curvy trajectories rather than ones that are as straight as possible, or you wouldn't pass above the pole when crossing the Atlantic.

[MiM]

That's why I started by saying "it's a darned good approximation"

But isn't the shortest route also the straightest? I thought that was how we define straight.

Look at plane trajectories when going around the earth... They are not taking straight lines.

No. The straight lines goes through the earth, so?

Even clinging to the surface, they still follow curvy trajectories rather than ones that are as straight as possible, or you wouldn't pass above the pole when crossing the Atlantic.

They are following the same paths as a string pulled tight on the surface of a globe. Are you mixing map projections with three dimensional geometry?

[Robert_S]

But then again, there aren't even lines in nature. We use the word "line" to describe patterns, trajectories, and so on. But there are no actual lines when you look close enough.

I think this is a kind of human hubris of which solipsism is the extreme. Obviously thoughts does not exist if nobody is thinking them, but just as obviously the cosmos doesn't give shit about us and our thoughts. If all life on earth will be annihilated tomorrow, the earth will still revolve around the sun obeying the same physical laws as always, and so will the other planets, and Pluto. And the number of bodies heavier than Pluto doing so will still be the same, even when nobody is counting, or defining "number" and "heavier".

I meant in the commonsense way that there are not many object shaped even very much like a line in nature if you look closely enough. What objects are shaped like lines lose their linearity when you look closely enough. They usually get all bumpy and craggy at some point. I'm not saying that things don't exist without humans, but that human observation tends to smooth out and simplify. Things are never exactly as they appear. They're more or less approximately as they appear a good deal of the time.

I'm also saying I think we might overestimate the difference between our sensory systems interpreting the "walking around" world and Pythagoras' Theorem. Just another way to abstract and simplify things enough to work with them. We see "lawn" and we miss the uniqueness of every blade of grass, but you kida have to do that in order to work the lawnmower, or even walk to the front door.. We see "sand" and we miss the incredible diversity of shapes and composition: each grain unique and probably never repeated anywhere else in the universe.

But is we get hung up on that, we have a real hard time making concrete. Or in my case, I had a hard time keeping focused on road work while being aware of all the cool little stones I was reburying, especially crinoid, gastropod and coral fossils.

I had to shut that out and think "pea gravel"

[Xamonas Chegwé]

I hate peeing gravel. It hurts.

[Svartalf]

That's why I started by saying "it's a darned good approximation"

But isn't the shortest route also the straightest? I thought that was how we define straight.

Look at plane trajectories when going around the earth... They are not taking straight lines.

No. The straight lines goes through the earth, so?

Even clinging to the surface, they still follow curvy trajectories rather than ones that are as straight as possible, or you wouldn't pass above the pole when crossing the Atlantic.

They are following the same paths as a string pulled tight on the surface of a globe. Are you mixing map projections with three dimensional geometry?

Just mentioning that on non Euclidean surfaces, the straight line is not the shortest way.

[Xamonas Chegwé]

All surfaces in our non-Euclidian space are non-Euclidian. Fortunately for us, there are plenty of very nearly Euclidian ones to play with.

[pErvinalia]

If it's purely an invention, what would restrict it?

My first thought is usefulness, what it describes.

But that is exactly my point. If it is restricted by what it describes, then it's not purely an invention.

It is not purely an invention. It is a model of reality. But it is limited in its accuracy by its ideal nature.

Pythagoras' Theorem applies to ideal, two-dimensional spaces and three-dimensional spaces constructed from them. It translates well to the surface of the Earth because, at human scales, that surface approximates very well to a two-dimensional plane. But start measuring large enough lengths and it breaks down. Draw a triangle on the Pacific with sides of 3,000, 4,000 & 5,000 kilometers and you will not find a right angle in it!

All mathematics (indeed all science) is based on such idealised models. Their usefulness depends on their accuracy at the scale under examination. There is no need to allow for relativity in a car's speedometer. There is no reason to consider quantum effects when striking a pool ball. It is only when the model is extended to a scale beyond its original purpose that its flaws become obvious.

Obviously it's a model. And obviously all our models are approximative and prone to errors. No arguing here. But in my view the fundamentals of mathematics are experimental. If experiment would show that adding two apples to three would yield a different result from adding three apples to two, our mathematical rules would be different from what they are now. So there is something fundamental out there that guides the rules we can apply in mathematics, while still keeping it a useful tool to describe reality.

All you are really saying there is that numbers are real. But as I said earlier, maths is about much more than just numbers.

[Xamonas Chegwé]

Addition is commutative over all number systems yet devised; ie. a + b = b + a. Good luck finding a counter-example - just one would bring the whole house of cards tumbling and have mathematicians running around for years trying to revise their entire science.