Are Numbers Real?

[Aliza]

We create maths. We discover ways in which particular maths has applications when applied to reality.

Remember that maths need not have any bearing on reality at all. To say that you "discover" a new mathematical system would be metaphorical only; it would be like exploring the platonic plane of abstract concepts.

Maths makes statements which are either true or false within their own framework. They are true, essentially, because we say they are true. They are a logical result of applying the rules which we say are true. Verifying they are true is a matter of making sure the rules have been correctly followed.

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

We may at some point in time in the future discover a new way of categorizing values, but even if we find a new way, it doesn't take away from the truth of our current system. Such advancements wouldn't nullify our current understanding of numbers... it would only add a new dynamic to what we know.

[robvalue]

We create maths. We discover ways in which particular maths has applications when applied to reality.

Remember that maths need not have any bearing on reality at all. To say that you "discover" a new mathematical system would be metaphorical only; it would be like exploring the platonic plane of abstract concepts.

Maths makes statements which are either true or false within their own framework. They are true, essentially, because we say they are true. They are a logical result of applying the rules which we say are true. Verifying they are true is a matter of making sure the rules have been correctly followed.

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

It’s very much a matter of nuanced language here. Remember that there are any number of mathematical systems. In some systems, this would not be true. So essentially, the result follows, and is true, based on the assumed truth of the axioms we choose. It’s a series of tautologies. If we find a new result, then we’ve discovered it insofar as we didn’t know it before, but it would always have been true (or not) because of the axioms. Most people would call me a pedantic cunt for having said what I just said, and I accept that totally. It’s meant to be highly technical, because of the discussion at hand.

We pick a mathematical system, and we could be metaphorically described as exploring the truths it contains. This is the case because we don’t simply have immediate access to all these truths.

Remember to differentiate between pure maths and applied maths; even when talking about a right-angled triangle, it is still pure maths. When we go to apply our maths to reality, this is a different matter. The difference here is so subtle that I think many people don’t even realise it. As a way of illustrating the difference, I could have a "pure maths" triangle with sides of length 3, 4 and 5. I can’t have that in a practical setting. I must pick some units, and I’m really only projecting an abstract idea onto reality for convenience. I could also have a triangle in a different maths system that doesn’t transfer at all into "real" triangles, regardless of units. So it’s about picking the right tool.

I can only prove that all modelled, theoretical perfect triangles behave a certain way within my maths system. I can’t prove that reality itself behaves a certain way universally. I essentially simplify reality through filters, so that my model can be applied exactly.

[vulcanlogician]

It’s very much a matter of nuanced language here. Remember that there are any number of mathematical systems. In some systems, this would not be true. So essentially, the result follows, and is true, based on the assumed truth of the axioms we choose. It’s a series of tautologies. If we find a new result, then we’ve discovered it insofar as we didn’t know it before, but it would always have been true (or not) because of the axioms. Most people would call me a pedantic cunt for having said what I just said, and I accept that totally. It’s meant to be highly technical, because of the discussion at hand.

We pick a mathematical system, and we could be metaphorically described as exploring the truths it contains. This is the case because we don’t simply have immediate access to all these truths.

Remember to differentiate between pure maths and applied maths; even when talking about a right-angled triangle, it is still pure maths. When we go to apply our maths to reality, this is a different matter. The difference here is so subtle that I think many people don’t even realise it. As a way of illustrating the difference, I could have a "pure maths" triangle with sides of length 3, 4 and 5. I can’t have that in a practical setting. I must pick some units, and I’m really only projecting an abstract idea onto reality for convenience. I could also have a triangle in a different maths system that doesn’t transfer at all into "real" triangles, regardless of units. So it’s about picking the right tool.

I can only prove that all modelled, theoretical perfect triangles behave a certain way within my maths system. I can’t prove that reality itself behaves a certain way universally. I essentially simplify reality through filters, so that my model can be applied exactly.

I must admit that you have lost me here, Rob. Math isn't my strong suit. I have a vague inkling of what you are talking about, however. Things like non-euclidean geometry wherein the things like circumference of a circle=pi times the diameter are not necessarily true. I am out of my element here, but doesn't each "system" of mathematics work on assumptions that are axiomatic? I mean, without axioms you cannot have facts of any kind.

But, like I said... out of my element, so I can't really stand by what I'm putting forward here.

[fromdownunder]

01001111 01101110 01100101 00100000 01101001 01110011 00100000 01110100 01101000 01100101 00100000 01101100 01101111 01101110 01100101 01101100 01101001 01100101 01110011 01110100 00100000 01101110 01110101 01101101 01100010 01100101 01110010 00100000 01110100 01101000 01100001 01110100 00100000 01111001 01101111 01110101 00100111 01101100 01101100 00100000 01100101 01110110 01100101 01110010 00100000 01100100 01101111 00001010 01010100 01110111 01101111 00100000 01100011 01100001 01101110 00100000 01100010 01100101 00100000 01100001 01110011 00100000 01100010 01100001 01100100 00100000 01100001 01110011 00100000 01101111 01101110 01100101 00001010 01001001 01110100 00100111 01110011 00100000 01110100 01101000 01100101 00100000 01101100 01101111 01101110 01100101 01101100 01101001 01100101 01110011 01110100 00100000 01101110 01110101 01101101 01100010 01100101 01110010 00100000 01110011 01101001 01101110 01100011 01100101 00100000 01110100 01101000 01100101 00100000 01101110 01110101 01101101 01100010 01100101 01110010 00100000 01101111 01101110 01100101 00001010 

01001110 01101111 01110010 01101101

[robvalue]

It’s very much a matter of nuanced language here. Remember that there are any number of mathematical systems. In some systems, this would not be true. So essentially, the result follows, and is true, based on the assumed truth of the axioms we choose. It’s a series of tautologies. If we find a new result, then we’ve discovered it insofar as we didn’t know it before, but it would always have been true (or not) because of the axioms. Most people would call me a pedantic cunt for having said what I just said, and I accept that totally. It’s meant to be highly technical, because of the discussion at hand.

We pick a mathematical system, and we could be metaphorically described as exploring the truths it contains. This is the case because we don’t simply have immediate access to all these truths.

Remember to differentiate between pure maths and applied maths; even when talking about a right-angled triangle, it is still pure maths. When we go to apply our maths to reality, this is a different matter. The difference here is so subtle that I think many people don’t even realise it. As a way of illustrating the difference, I could have a "pure maths" triangle with sides of length 3, 4 and 5. I can’t have that in a practical setting. I must pick some units, and I’m really only projecting an abstract idea onto reality for convenience. I could also have a triangle in a different maths system that doesn’t transfer at all into "real" triangles, regardless of units. So it’s about picking the right tool.

I can only prove that all modelled, theoretical perfect triangles behave a certain way within my maths system. I can’t prove that reality itself behaves a certain way universally. I essentially simplify reality through filters, so that my model can be applied exactly.

I must admit that you have lost me here, Rob. Math isn't my strong suit. I have a vague inkling of what you are talking about, however. Things like non-euclidean geometry wherein the things like circumference of a circle=pi times the diameter are not necessarily true. I am out of my element here, but doesn't each "system" of mathematics work on assumptions that are axiomatic? I mean, without axioms you cannot have facts of any kind.

But, like I said... out of my element, so I can't really stand by what I'm putting forward here.

Yes you are right, each system has axiomatic assumptions. You then "discover" what those assumptions necessarily lead to, if you like. My point is that if we change the assumptions, then we get different truths. The axioms are true because we say they are true, and by extension any result they produce. These are truths that only apply to the system itself as we choose it to be. This is why they cannot be wrong, unlike scientific facts.

[I_am_not_mafia]

What say you? Are numbers real? If so, in what way are they real?

They are real only in that we have created a system which consists of numbers. But wipe out the human race and they wouldn't be. Like all mathematics.

---

Some of them are.

... but some of them are imaginary.

Ah here you go, making everything complex!

[robvalue]

Putting it another way...

I prove something about right angled triangles. Someone asks why it is true.

I show them the workings. They ask why a particular facet of my workings is true.

I refer to another principle. They ask why that is true. And so on. Eventually, I must refer to axioms. It’s all true by definition, so cannot be false. But that is totally unrelated to any practical applications a system may or may not have. Scientific facts attempt to model truths about reality, using maths as a tool. But maths is not a monolith, just like morality isn’t.

[I_am_not_mafia]

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

Or to use the example of PI, you could ask if that was discovered. But there are no perfect circles in nature. Presumably true of right angle triangles. Our mathematical descriptions are just approximations of them.

Take numbers. We think in terms of single objects. Like one or two oranges. But actually you don't have to think in this way. It's useful for every day situations but maybe an alien race would think entirely differently depending on how they evolved. Like part of a hive mind for example. Because we are making an arbitrary distinction between what is and is not an orange. That sounds like a daft statement but take humans and the debate about when life begins, is it at conception? When an egg is fertilised? When it is born etc? What about a bee? Is it a single object or part of a super organism? Why can't we say the same thing about a human? After all, there is never only one thing in existence, everything is part of a larger environment. The orange is part of a tree and part of its life cycle. But for convenience we talk about multiples of a single orange. Sometimes though it is not convenient to think in these terms.

[vulcanlogician]

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

Or to use the example of PI, you could ask if that was discovered. But there are no perfect circles in nature. Presumably true of right angle triangles. Our mathematical descriptions are just approximations of them.

Take numbers. We think in terms of single objects. Like one or two oranges. But actually you don't have to think in this way. It's useful for every day situations but maybe an alien race would think entirely differently depending on how they evolved. Like part of a hive mind for example. Because we are making an arbitrary distinction between what is and is not an orange. That sounds like a daft statement but take humans and the debate about when life begins, is it at conception? When an egg is fertilised? When it is born etc? What about a bee? Is it a single object or part of a super organism? Why can't we say the same thing about a human? After all, there is never only one thing in existence, everything is part of a larger environment. The orange is part of a tree and part of its life cycle. But for convenience we talk about multiples of a single orange. Sometimes though it is not convenient to think in these terms.

But now I feel like we are getting away from the significance of the question in the first place. To follow your and Rob's skepticism any further, we may as well say "there are no such things as facts, because for there to be facts in the first place relies on a basic set of assumptions."

The hive mind aliens may not be able/willing to understand math the way we do, but this has no bearing on the fact that math is a real, objective thing. The hive mind aliens may have facets of their reality that are beyond our comprehension, but this does not necessarily indicate that those things are not objectively real. There may be a state of deep communication which all members of the hive mind enter into that we humans could never possibly fathom, but that does not mean that this state is devoid of objective and/or actual qualities.

[ignoramus]

Mat, I wouldn't imagine that to be the case with aliens.

The periodic table has beautifully simple structures ... eg: Hydrogen = 1 proton, Helium - 2 protons, Lithium = 3 protons., etc

I believe any advanced race will discover all these correlations via maths.