But it raises the question of what, in modern parlance, is “canon”.
As far as I can tell, mathematics becomes canonical if enough mathematicians think it’s beautiful. There is plenty of seriously presented mathematics that never achieves canon status. The practical effect of being canon is that other mathematicians care to do proofs with it, or about it.
Riemann’s hypothesis is canon even though it hasn’t been proven, because so many proofs use it as an axiom.
AFAIK, Riemann's hypothesis is not quite canon but many mathematicians posit that it is likely to be true due to a variety of "evidence" they feel leans in that direction.
I'm also not equipped to understand that paper, but my find tool is equipped to say the words "decidable", "undecidable", and "decidability" are not present.
I tried to think of any mathematics that used to be canon but isn't anymore. All that comes to mind is the Principia Mathematica, demolished by Gödel. Is there mathematics that has just become unfashionable, not because anybody found anything wrong with it, but just because it turned out not to lead anywhere interesting?
It would be hard to claim that such mathematics existed coeval with (non-avian) dinosaurs.
For one thing we learned to understand "axioms" as something less than self-evidently necessary truths. For another we learned that the original axioms didn't really prove what they were thought to prove -- at least up to modern notions of rigour.
We would now understand the original Euclidean proofs to have been "sneaking in" other, implicit, axioms disguised as obvious deductions, and so modern formulations of the geometry need more elaborate axioms.
>What does it mean to exist independent of time and space? Nothing, as far as I can see. I was once at a UVIC philosophy seminar where this came up. I asked if, in the time of the dinosaurs, bounded linear operators already existed. Yes, I was told. I have no idea what this meant.
I think a more interesting question is if an intelligent species that lived a billion light years away from us would eventually come up with bounded linear operators (or something homomorphic to it).
This is of the same flavor as wondering if said species would invent money, a collective fiction of usefulness so obvious it has been created many times. Or perhaps it’s closer to wondering if they would develop futures on a stock market.
Dinosaurs didn't have numbers as they hadn't been invented yet. Merely being surround by quantities of stuff doesn't make human abstractions spring into existence.
The natural world doesn't perform arithmetic beyond summation and subtraction or symbolic manipulation of anything. We may use invented symbols to model abstract concepts aspects of the natural world but that doesn't make such things exist outside of collective human knowledge.
I suppose it works well because it’s designed that way. We abstract away the non essential and do our math. The results measure out in the non-abstract world well enough to be useful. So we keep on with it. This process is recurring, so we do math on our math.
I think it’s fun to ask if this abstract world is real. I think it is. It’s a wonderful place in the mind to visit if you’re inclined to wander around without trying too hard to get anywhere. That’s my hobbyist point of view anyway.
> the Cantor set or the Borel Hierarchy is like the USS Enterprise or the Iron Throne. In fact Star Trek fans talk of the Star Trek universe and GOT fans of Westeros and they talk like these places really exist.
You're right, it feels like quite a loose analogy, and only holds when discussing a "canonical" fictional universe. Still, I'm sympathetic to the idea that when we're talking about a circle we're talking about a shared abstract understanding rather than something which objectively exists.
The pupil in the eye. The sun through the clouds. The craters after an impact. The waves when you drop a rock in water. The section of a bamboo stick... And it gets more evident when you go to microscopic world. Some eggs are perfect spheres, as small bubbles in water.
I think circles are a real natural shape. Not an invention.
Nature brings us all kind of shapes, some of them are discovered studying living creatures, like the scutoid.
It's not a very heavy argument anyway. Circles are in swirls, tornadoes and even the Sun looks like a bright circle looking from the Earth.
Water waves draw circles when you throw a rock in it. Circles are everywhere.
Some people says "nature hates perfect shapes" but it's not true, nature creates perfect cubes in some minerals, some fish eggs are perfect spheres... And so on.
Water waves are perfect circles as waves move at same speed in every direction, so its points are always equidistant.
Let a soap bubble settle in the table and take some measurements with a ruler. They are almost always perfect circles.
You can deny they're circles, but you know it's not true.
Even at atomic scales there are always minor distortions due to external influences. For example, although the shape of an electron s-orbital is said to be a sphere, the only way this could theoretically be a perfect sphere is if the atom is and always has been completely isolated from all external fields and forces.
I'm sorry for being dismissive but you threw a barrage of contentious examples at me without really trying to engage with my point.
The way I see it, the circle is an abstract idea which resides within the language that we use to share our understanding about the patterns we observe in the external world. Those patterns may conform to the abstraction to varying degrees but they are fundamentally not the same thing. Mathematics is a very precise and formal way to communicate this type of knowledge, but that doesn't mean the ideas therein exist objectively. This contrasts with the views of someone like Max Tegmark, who has the impression that the universe is inherently mathematical.
So my understanding is he believes math is invented, and exists in a "neo-fictional" space where things can be both true and false. Because if it were pure fiction like Star Trek then Math's statements would be false.
To me this seems a bit off, but I'll assume that it's my understanding of what he is trying to say that is flawed.
36 comments
[ 3.1 ms ] story [ 90.2 ms ] threadI believe the actual term is τ (FWIW: Verdana's Greek glyphs are absolutely atrocious)
"Bleen" is as good a name for it is "tau".
https://en.wikipedia.org/wiki/List_of_humorous_units_of_meas...
But it raises the question of what, in modern parlance, is “canon”.
As far as I can tell, mathematics becomes canonical if enough mathematicians think it’s beautiful. There is plenty of seriously presented mathematics that never achieves canon status. The practical effect of being canon is that other mathematicians care to do proofs with it, or about it.
Riemann’s hypothesis is canon even though it hasn’t been proven, because so many proofs use it as an axiom.
The P=NP question would be an example of a canonical unsolved problem in CS.
http://phys.lsu.edu/~fmoxley/bbm.pdf
http://phys.lsu.edu/~fmoxley/r1.pdf
It would be hard to claim that such mathematics existed coeval with (non-avian) dinosaurs.
For one thing we learned to understand "axioms" as something less than self-evidently necessary truths. For another we learned that the original axioms didn't really prove what they were thought to prove -- at least up to modern notions of rigour.
We would now understand the original Euclidean proofs to have been "sneaking in" other, implicit, axioms disguised as obvious deductions, and so modern formulations of the geometry need more elaborate axioms.
I think a more interesting question is if an intelligent species that lived a billion light years away from us would eventually come up with bounded linear operators (or something homomorphic to it).
The natural world doesn't perform arithmetic beyond summation and subtraction or symbolic manipulation of anything. We may use invented symbols to model abstract concepts aspects of the natural world but that doesn't make such things exist outside of collective human knowledge.
I think it’s fun to ask if this abstract world is real. I think it is. It’s a wonderful place in the mind to visit if you’re inclined to wander around without trying too hard to get anywhere. That’s my hobbyist point of view anyway.
> According to intuitionism, mathematical objects are products of our mind, like characters in a novel. I agree, as far as it goes...
> Fictionalism holds that the mathematical universe is a collective fiction, like Star Trek or Game of Thrones.
You're right, it feels like quite a loose analogy, and only holds when discussing a "canonical" fictional universe. Still, I'm sympathetic to the idea that when we're talking about a circle we're talking about a shared abstract understanding rather than something which objectively exists.
Now go with your atomic imperfections and external influences and take a nap, because it must be exhausting to be so repellent.
The way I see it, the circle is an abstract idea which resides within the language that we use to share our understanding about the patterns we observe in the external world. Those patterns may conform to the abstraction to varying degrees but they are fundamentally not the same thing. Mathematics is a very precise and formal way to communicate this type of knowledge, but that doesn't mean the ideas therein exist objectively. This contrasts with the views of someone like Max Tegmark, who has the impression that the universe is inherently mathematical.
To me this seems a bit off, but I'll assume that it's my understanding of what he is trying to say that is flawed.
Also, didn't we all use the unit circle in Calc?