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. — litewave
If our space or at least some part of it is continuous — litewave
and flat — litewave
Is this the line of argument you are putting forward? If not, then what are you saying exactly? — fishfry
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? — fishfry
Then the Banach-Tarski paradox is true as well. Then matter could be created, contrary to the laws of physics. — fishfry
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
I don't know. If the axiom of choice is consistent with Euclid's axioms — litewave
Our space is generally not Euclidean but in everyday life the curvature is usually negligible. — litewave
Well, according to physicist Max Tegmark, there's no difference between physical and mathematical structures. — litewave
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
Numbers can carry information that is both passive (like a newsreport) and active (like a software program). — TheMadFool
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. — fishfry
So circles and numbers are the idealised limit of physical reality? — apokrisis
They represent perfect symmetry — apokrisis
and to "physically exists" means always to be individuated - a "materially" broken symmetry. — apokrisis
Therefore mathematical forms are not real. — apokrisis
There is only imperfect matter and its approximations of these forms - always inevitably marred by "accidents". — apokrisis
Every physical circle is a bit bent. — apokrisis
Any collection of things may be given a number, but no two things are actually alike. — apokrisis
This is certainly a familiar ontological view. — apokrisis
A substance ontology is what we experience, and any mathematical notions about form seem so clearly an abstraction produced by the creative human mind. — apokrisis
I said no such thing. Circles and numbers are abstractions. Limits have a technical definition and I would never use that word imprecisely in a mathematical discussion. This is not the first time you've quoted me as saying something I never said. — fishfry
Surely I don't have to explain to you the difference between abstract and physical objects. You're just being disingenuous. — fishfry
Mathematical forms are real, they're just not physical. — fishfry
But of course electrons are right on the border between the physical and the abstract. I do understand your point that saying that physical things are "really there" is a stretch once we get into the higher realms of physics. Still, one can distinguish between a number and a rock, one being abstract and the other physical. Even you would agree to this distinction, yes? — fishfry
Surely you can understand that my response was to someone claiming that the number pi proves that numbers are physical or have material existence. I'm not on any soapbox about the ontology of physics. I understand the traps therein. — fishfry
That your point?
Ok. I don't disagree.
But the number pi is a lot different from a rock. — fishfry
Wait!! It seems you agree with me after all. — fishfry
a human invention, a mere accidental notion — apokrisis
But that is just an ontology endorsing a sharply divided dualism. — apokrisis
This is a deep mystery. Our abstractions are telling us something about the world. We're not sure what.
I don't think you and I disagree all that much. — fishfry
My understanding is that we can accommodate abstract mental constructs quite easily within physicalism. Abstractions are thoughts, biochemical processes in my brain.
But thoughts are still different from rocks. Thoughts and rocks are both physical processes, but they have a different character. One doesn't need dualism. — fishfry
Our brains go quite comfortably back and forth between the real and the unreal. Yet sane people alway know the difference. — fishfry
There's some mathematical constant pi "out there." — fishfry
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