Pythagoras' Theorem. Evidence of elsewhere or our ingenuity.

[JimC]

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.

Obviously you can create any kind of mathematic system system by choosing arbitrary axioms, but asfaik there has been no basically different systems devised that describes our world in a useful way. That is exactly why I believe there is an element of discovery in mathematics.

A good example exists in Euclidean geometry. Most of it was a logically consistent set of related axioms, that described a consistent pattern within an abstracted 3-dimensional space.

Mathematicians tried to show that parallel lines never meeting, and triangles always having total angles of 180 degrees were automatic corollaries of that set of axioms, but couldn't. It was realised that such things were actually derived from the nature of space in this universe; it could be so, but only if our universe had one particular type of space (often called "flat"), rather than an infinite number of other, potential curved geometries a physical universe could have.

Mathematics describes a series of nested, self-consistent logical models; typically they contain statements that cannot be proved without making use of a larger model, with a broader set of axioms (Godel...). We check parts of this mathematical world for potential correspondences with the observed universe; surprising often, they occur...

[Audley Strange]

Mathematics does not deal with nature? Hmmmm. Fibonacci sequence in sunflowers, fractal geometry of coastlines. Mathematics is natures operating system.

Well that's one view, the other is that some of the mathematical models we make up allow us to see the patterns we made up to depict those models in other things that we percieve as external. That we've drawn the map first and then added the territory that fits.

You seem to think the former.

Also consider that nature may little more than an entropy bug in the OS that's crashing the system.

[Clinton Huxley]

I think that if mathematics was "just made up", life would be impossible. I think humans discovered it, rather than invented it. There may be universes where the inital conditions were such that maths works differently but I bet they are horrid, worse than Hatfield or Stevenage.

[Audley Strange]

Since we're talking countability, is discreteness a fundamental feature of anything in existence, or is it projection of how the brain/mind evolved to effectively analyze experience?

I was getting to it. Honest.

Remember a while ago there was some conversation in which I kept persisting that "universe" by definition was a totality, that no matter what you describe, parallel dimensions, the past, phone books, pornographic drawings of Linus from Peanuts, the letter P, all of it is part of one single thing?

Well there is your discreteness. However there is a problem in that the concept of universe as a unit or an object, is that A) we are part of that object
and B) every experience we have of it is dim and internalised. So it definitely is a projection. However is the projection correct, like a torch in a dark forest, or is it just us looking at hallucinations on the cave walls?

[Audley Strange]

Since we're talking countability, is discreteness a fundamental feature of anything in existence, or is it projection of how the brain/mind evolved to effectively analyze experience?

I was getting to it. Honest.

Remember a while ago there was some conversation in which I kept persisting that "universe" by definition was a totality, that no matter what you describe, parallel dimensions, the past, phone books, pornographic drawings of Linus from Peanuts, the letter P, all of it is part of one single thing?

Well there is your discreteness. However there is a problem in that the concept of universe as a unit or an object, is that A) we are part of that object
and B) every experience we have of it is dim and internalised. So it definitely is a projection. However is the projection correct, like a torch in a dark forest, or is it just us looking at hallucinations on the cave walls?

[JimC]

I think that if mathematics was "just made up", life would be impossible. I think humans discovered it, rather than invented it. There may be universes where the inital conditions were such that maths works differently but I bet they are horrid, worse than Hatfield or Stevenage.

I think both invention and discovery can be valid descriptions. There are mathematical systems that have been created that have nothing whatsoever to do with the current universe, or maybe even any possible physical universe. Pure mathematicians can invent a bewildering variety of self-consistent logical systems. In one sense, the mathematical universe is a much vaster thing than our own. However, patterns within physical universe can typically be matched with patterns in the mathematical universe - perhaps what we discover is the correspondences between the two...

[Audley Strange]

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.

Obviously you can create any kind of mathematic system system by choosing arbitrary axioms, but asfaik there has been no basically different systems devised that describes our world in a useful way. That is exactly why I believe there is an element of discovery in mathematics.

A good example exists in Euclidean geometry. Most of it was a logically consistent set of related axioms, that described a consistent pattern within an abstracted 3-dimensional space.

Mathematicians tried to show that parallel lines never meeting, and triangles always having total angles of 180 degrees were automatic corollaries of that set of axioms, but couldn't. It was realised that such things were actually derived from the nature of space in this universe; it could be so, but only if our universe had one particular type of space (often called "flat"), rather than an infinite number of other, potential curved geometries a physical universe could have.

Mathematics describes a series of nested, self-consistent logical models; typically they contain statements that cannot be proved without making use of a larger model, with a broader set of axioms (Godel...). We check parts of this mathematical world for potential correspondences with the observed universe; surprising often, they occur...

Awesome.

One question, is logical consistency a linguistic/symbol game rule or a universally objective application. That is to say is it etiquette that ousts non conformists or is it an actual information/noise filter?

[Audley Strange]

I think that if mathematics was "just made up", life would be impossible. I think humans discovered it, rather than invented it. There may be universes where the inital conditions were such that maths works differently but I bet they are horrid, worse than Hatfield or Stevenage.

I think both invention and discovery can be valid descriptions. There are mathematical systems that have been created that have nothing whatsoever to do with the current universe, or maybe even any possible physical universe. Pure mathematicians can invent a bewildering variety of self-consistent logical systems. In one sense, the mathematical universe is a much vaster thing than our own. However, patterns within physical universe can typically be matched with patterns in the mathematical universe - perhaps what we discover is the correspondences between the two...

Are you talking about configuration spaces?

[JimC]

It may be described as a linguistic/symbol game rule, I suppose, but the term "game rule" allows for some very arbitrary decisions; arbitrary rules are greatly distrusted in mathematics, because they can be used to prove anything. A logically consistent set of mathematical axioms must have a bridge, a connection to the broader mathematical universe, or it is pure solipsism. Godel's proof that sufficiently complicated sets of axioms as to contain arithmetic cannot be proved within their own frame of reference was once though to be a blow to the edifice of mathematics. It was in fact the opposite - it showed that we can make a vital connection to a broader, more over-arching set of axioms; mathematics has its own ecological web...

[JimC]

I think that if mathematics was "just made up", life would be impossible. I think humans discovered it, rather than invented it. There may be universes where the inital conditions were such that maths works differently but I bet they are horrid, worse than Hatfield or Stevenage.

I think both invention and discovery can be valid descriptions. There are mathematical systems that have been created that have nothing whatsoever to do with the current universe, or maybe even any possible physical universe. Pure mathematicians can invent a bewildering variety of self-consistent logical systems. In one sense, the mathematical universe is a much vaster thing than our own. However, patterns within physical universe can typically be matched with patterns in the mathematical universe - perhaps what we discover is the correspondences between the two...

Are you talking about configuration spaces?

Potential confusion exists in this terminology. Physics makes use of the term configuration space to describe the vector space of an actual physical system, deriving from a set of parameters. It is an idealised description, sure, but it is tied to an actual physical system.

In mathematics, this broadens considerably, being able to describe a set of potential topological spaces. A sub-set of these may correspond to actual pysical systems in the space we occupy.

[Audley Strange]

Ah, thank you Jim. I go forage now.

[pErvinalia]

Audley, what about the discreteness-countability issue I mentioned?

[cough]I mentioned it on the first page[/cough]

[FBM]

Since we're talking countability, is discreteness a fundamental feature of anything in existence, or is it projection of how the brain/mind evolved to effectively analyze experience?

I was getting to it. Honest.

Remember a while ago there was some conversation in which I kept persisting that "universe" by definition was a totality, that no matter what you describe, parallel dimensions, the past, phone books, pornographic drawings of Linus from Peanuts, the letter P, all of it is part of one single thing?

Well there is your discreteness. However there is a problem in that the concept of universe as a unit or an object, is that A) we are part of that object
and B) every experience we have of it is dim and internalised. So it definitely is a projection. However is the projection correct, like a torch in a dark forest, or is it just us looking at hallucinations on the cave walls?

So if the totality is a singular unity, where is the discreteness? What's it discrete from?

[Audley Strange]

Since we're talking countability, is discreteness a fundamental feature of anything in existence, or is it projection of how the brain/mind evolved to effectively analyze experience?

I was getting to it. Honest.

Remember a while ago there was some conversation in which I kept persisting that "universe" by definition was a totality, that no matter what you describe, parallel dimensions, the past, phone books, pornographic drawings of Linus from Peanuts, the letter P, all of it is part of one single thing?

Well there is your discreteness. However there is a problem in that the concept of universe as a unit or an object, is that A) we are part of that object
and B) every experience we have of it is dim and internalised. So it definitely is a projection. However is the projection correct, like a torch in a dark forest, or is it just us looking at hallucinations on the cave walls?

So if the totality is a singular unity, where is the discreteness? What's it discrete from?

The void. The Abyss.

[MiM]

I think that if mathematics was "just made up", life would be impossible. I think humans discovered it, rather than invented it. There may be universes where the inital conditions were such that maths works differently but I bet they are horrid, worse than Hatfield or Stevenage.

I think both invention and discovery can be valid descriptions. There are mathematical systems that have been created that have nothing whatsoever to do with the current universe, or maybe even any possible physical universe. Pure mathematicians can invent a bewildering variety of self-consistent logical systems. In one sense, the mathematical universe is a much vaster thing than our own. However, patterns within physical universe can typically be matched with patterns in the mathematical universe - perhaps what we discover is the correspondences between the two...

.
Thank you Jim, shows you're the teacher