Assuming Pythagoras discovered irrational numbers, what are we to say about irrational numbers before Pythagoras discovered them?

To my way of thinking, there are things and there are ideas. Numbers are ideas. As a working hypothesis, subject to correction, I'll even say that if it's a something, or if it just is, and it is not a thing, then it is an idea. Ideas, of course, can and do evolve - change.

The idea that ideas must somehow be things of some kind is simply an idea that leads to confusion and useless (i.e., without a use) contention.

Do ideas exist? Of course they do! Just not in the same sense that things exist. To ask if ideas exist (as things) is to confuse the two senses of "exist" - to confuse content with substance.

Assuming Pythagoras discovered irrational numbers, what are we to say about irrational numbers before Pythagoras discovered them? — tim wood

I guess the circumference of a circle was just as many times longer than the diameter before the discovery of irrational numbers as it was after the discovery.

I guess the circumference of a circle was just as many times longer than the diameter before the discovery of irrational numbers as it was after the discovery. — litewave

Yes but there is no such thing as a circle in the world. The circle whose circumference divided by its diameter is exactly pi is not any object that can exist in this mortal world of ours. The circle of mathematics is an ideal circle, a pure mental abstraction.

It's the set of points, whatever they are, in the plane, whatever that is, that are all exactly the same distance from some other point.

Before there were humans, there were round-ish things like planets and stars. But there were no circles. There were no circles until human beings came along and conceived them as an abstraction.

Pi already has abstract mathematical existence. But you can't argue that pi is any realer than that by invoking idealized circles. Circles have exactly the same mode of existence as pi: as idealized mathematical abstractions. Circles and pi are contingent on the evolution of abstract reasoning in humans.

tl;dr: Were there circles before humans?

If our space or at least some part of it is continuous and flat then it contains perfect circles. A perfect circle is simply the set of all points in a plane that are the same distance from some point. But even if our space is not continuous and flat, it doesn't mean that a perfect circle is just a mental object. There seems to be no reason why a continuous and flat space couldn't exist; it just would not be the space we live in.

If our space or at least some part of it is continuous — litewave

That's a wild assumption with very little evidence for it, and considerable physical evidence against it.

and flat — litewave

Ditto.

But surely you are not making the claim that mathematical Euclidean space is actually the literal truth about the physical world? If not, please explain what you are claiming. And if so ... well, frankly the burden's on you to provide evidence.

Let me give you some thought questions. If the universe the same as Euclidean space, and points in physical space are like points in Euclidean n-space where I don't care what n is, then is the Continuum hypothesis true or false? That is, how many points are in the unit n-cube?

If anyone seriously believed that physical space was Euclidean space, then set theory would become an experimental science. When the physics postdocs start getting grants to determine the truth value of the large cardinal axioms then maybe you'll have a remote hope of my believing this absurd claim.

Is this the line of argument you are putting forward? If not, then what are you saying exactly?

ps -- I apologize if I sound too emphatic. The physical world and mathematical Euclidean space are very different things. The idea of mathematical continuity is an abstraction.

I will also note for the record that even a discrete space can have a notion of continuity. In fact imagine that the universe consists of a lattice of bowling balls with nothing between them. So there is no continuity as you might think of it.

Nevertheless we can put a metric on a discrete space, and say that the distance is 0 from a point to itself, and 1 between any two distinct points. With this metric, every function whose domain is this discrete space is continuous, by the formal definition of continuity. In topological terms this is the discrete topology, in which every subset is an open set. Math gives us the tools to create many universes; but math can't tell us which model is our universe. My guess would be none of them. Or all of them.

But the major point here is that there are no circles in the real world. Do you believe in the physical reality of dimensionless points?

Is this the line of argument you are putting forward? If not, then what are you saying exactly? — fishfry

I think that our space is probably quantized, as suggested by contemporary physics. In that case there seem to be no measurable perfect circles in our space because no measurement instrument can penetrate into the minimum length interval (Planck length). Still, we might take the Planck length as further divisible but not "physically", that is, beyond possibility of measurement / physical interaction.

And of course, as I said, there may actually exist continuous Euclidean spaces, even though we don't live in one.

There are no integers in reality since everything is unique. Whatever we chose to nominate as a thing, there is no other thing that is the same as that thing.
Two oranges are not the same. Numbering the oranges asserts that those oranges are the same and equal to one another. One orange plus another orange is two oranges is not exact but an approximation.
The universe is analogue, numbers are digital. PI is irresolvable because of this contradiction.
There are no straight lines in nature; maths imposes them.
I think the thread is a no brainer.

Numbers are ideas. They could have instances in reality.

Ideas do not have instances in reality. Ideas can only attempt to represent reality, or try to describe it.

And of course, as I said, there may actually exist continuous Euclidean spaces, even though we don't live in one. — litewave

Do you suppose that the axiom of choice is true in such a space? Then the Banach-Tarski paradox is true as well. Then matter could be created, contrary to the laws of physics.

Is it possible that you (like me, like Kant, like everybody) have a strong intuition of Euclidean space, yet that intuition is simply misleading? And that in fact mathematical Euclidean space is inconsistent with physical reality?

Do you suppose that the axiom of choice is true in such a space? — fishfry

I don't know. If the axiom of choice is consistent with Euclid's axioms then there can be a Euclidean space with axiom of choice. If the negation of the axiom of choice is consistent with Euclid's axioms then there can be a Euclidean space without axiom of choice.

Then the Banach-Tarski paradox is true as well. Then matter could be created, contrary to the laws of physics. — fishfry

I don't claim that our laws of physics apply in such a space. I don't even claim that such a space is bound up with a time dimension into a spacetime.

Is it possible that you (like me, like Kant, like everybody) have a strong intuition of Euclidean space, yet that intuition is simply misleading? And that in fact mathematical Euclidean space is inconsistent with physical reality? — fishfry

Our space is generally not Euclidean but in everyday life the curvature is usually negligible.

I don't know. If the axiom of choice is consistent with Euclid's axioms — litewave

Ah. Interesting point, sort of a category mismatch. When you say Euclidean you mean Euclid's axioms of geometry. When I say Euclidean I mean modern Euclidean space $\mathbb R^n$. The axiom of choice of course applies to the latter (unless one chooses to accept its negation) but not to the former.

Our space is generally not Euclidean but in everyday life the curvature is usually negligible. — litewave

It's not the curvature that's the problem, it's the idea of dimensionless points. There's no such thing in physics except as conceptual abstractions.

Dimensionless points are common to both classical and modern definitions of Euclidean space.

How do you justify the idea of dimensionless points as physical entities? Even in an alternate universe?

Is number something realizable in external world that we experience and we try to discuss? 1 apple +1 apple=2 apple. Something just appears in our mind which we can individuate it. It shows a property of the world.

How do you justify the idea of dimensionless points as physical entities? Even in an alternate universe? — fishfry

It seems to be no problem in mathematics. What do you mean by "physical"?

It seems to be no problem in mathematics. What do you mean by "physical"? — litewave

The physical world. The object of study of physicists.

Well, according to physicist Max Tegmark, there's no difference between physical and mathematical structures.

Well, according to physicist Max Tegmark, there's no difference between physical and mathematical structures. — litewave

That's a speculative idea, not physics. And besides, he's also suggested a stricter idea of the computable universe. If the universe is a computation, then it's not continuous! Because most real numbers are not computable, so the real number line is full of holes. These are interesting ideas but they are not physics.

If you don't understand the distinction between abstract mathematics and the actual, physical world that we live in, that's something you should try to understand. You've fatally weakened your own argument by admitting you don't know the difference between the two.

If you don't understand the distinction between abstract mathematics and the actual, physical world that we live in, that's something you should try to understand. You've fatally weakened your own argument by admitting you don't know the difference between the two. — fishfry

And do you know the difference? You didn't explain it.

If you regard as "physical" only the world we live in, then I already said that Euclidean space is not the space we live in.

I think numbers are some form of code in the sense of a computer program AND in the sense of language. Numbers can carry information that is both passive (like a newsreport) and active (like a software program).

Does that mean numbers are real? May be. I'm not completely sure but there's something odd about how the laws of nature are mathematical. It can't be all coincidence me thinks. Godel's incompleteness theorems, from what I understand, pokes a hole in formal mathematics but I think completeness of math (I don't know how mathematicians call it) is not necessary to understand our world. I could be wrong.

Numbers can carry information that is both passive (like a newsreport) and active (like a software program). — TheMadFool

I would really like to hear more on what you mean by numbers being passive and active. I'm currently struggling to get my thoughts together on this, but I think we might have very similar views.