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I assume you're diving things into "from logic" or "from nature". However that's a highly misleading (if not false) dichotomy. "Logic" refers to the most basic

I assume you're diving things into "from logic" or "from nature". However that's a highly misleading (if not false) dichotomy. "Logic" refers to the most basic set of logical operations (AND, OR, etc.) and their use. Pi does not come from this. It comes from Mathematics. If we define World to be "a set of true claims" then we can define worlds without any mathematical claims in; however the process of defining the world likely requires logic. (And so Mathematics != Logic). In any case, as the article shows, the definition of Pi is relative to a distance metric. "Pi = Circ/Diam" which is ill-defined. One can construct a World (as defined above) which admits only claims relative to a given distance metric ("A Taxi-Cab World"). At this world, "Pi" has a different value. It isn't also clear that the value of "Pi" here is given purely by mathematics. As its value is relative to a distance metric, and we choose that metric for empirical reasons ("it applies to our world").

pdpi 5y ago
The question "Is pi the same in all universes?" is poorly defined. You're talking about pi as "the circle constant for whatever universe I'm in", whereas the GP is talking about pi as "the circle constant for a very specific geometry (that we once accepted as representative of the universe we're in)"
lordnacho 5y ago
None of the other metrics come from nature either, that's the point. "Logic" is more of a catchall for "not from nature" than a specific axiomatic system.
mypalmike 5y ago
I'm no mathematician but it's my understanding that axiomatic set theory is considered foundational for both what you call mathematics and also for discrete mathematics (aka logic).
- foldr 5y ago
  Axiomatic set theory can't very well be considered foundational for logic, as its axioms are formulas of first-order predicate logic.
  - mypalmike 5y ago
    Yeah that's fair. I was more thinking about the natural mapping of set operations to boolean logic operations (e.g. OR equivalence to union for given sets, de Morgan's laws applied to sets and boolean operations, etc.).
ganafagol 5y ago
> "Logic" refers to the most basic set of logical operations (AND, OR, etc.) and their use. That's not what mathematicians mean with "logic". It's a much broader field than just boolean operators. Those are what computer architects and perhaps electrical engineers mean with "logic".
- mjburgess 5y ago
  I meant the whole field of logic; I was keeping it simple. Pi is not "from logic" however broadly you construe it. Unless you endorse logicism, but I think that's both highly implausible and somewhat incidental.
mekoka 5y ago
Logic refers to its philosophical branch of study. The dichotomy is apt. Math is pure logic, meaning that you can be locked in a bunker for generations and keep discovering new laws. You can't do the same with physics. A mathematical or logical circle is nothing but a thought, an idea, it's perfect and has no physical equivalent. Pi is bound to that idea in any universe. The metric system is irrelevant (it mathematically cancels out).
- foldr 5y ago
  Mjburgess is referring to the failure of the logiscist program in research in the foundations of mathematics: https://plato.stanford.edu/entries/logicism/ https://plato.stanford.edu/entries/logicism/ The article doesn't make it entirely clear that the program is widely considered to be a failure, but it is. A better way to express the claim that you're making about mathematics would be to say that mathematical truths can be known a priori.
LeifCarrotson 5y ago
Logic refers to the most basic set of logical operations, and the axioms which we can build on top of those fundamental operations. Mathematics builds on these fundamentals with non-logical axioms, postulates which are used as building blocks for a more complex system. Euclidean geometry is a mathematical system which starts from a set of five basic postulates: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. Through any given point not on a line there passes exactly one line parallel to that line in the same plane. These, with corresponding idealized definitions of 'point' and 'straight' and 'indefinite' and 'circle', rather than physical constructions, concretely define pi as a ratio between the circumference of the circle and the length of the line segment. Any universe where a mathematician can imagine these five postulates and use them to calculate pi will result in the same value. If you discard or modify one of these postulates, you can come up with a system like a hyperbolic or elliptical geometry where pi could have a different value. Euclidean geometry closely corresponds to a our macroscopic physical world, one can form something fairly close with a straightedge and compass. If you're imagining other universes, you might as well imagine one where the universe itself has a different geometry and the basic axioms are different. In that universe, children drawing a line in the dirt with their fingers would fundamentally understand that there are infinitely many lines parallel to the first in the same plane and would be taught so in primary school, but if they grew up to be a mathematician and replaced that postulate with ours, independently inventing Euclidean geometry, he could discover our definition of pi. Conveniently, typical algebraic postulates like those resulting in Euler's formula, differential equations, integration, power series, and other mathematical constructs also result in the same definition of pi as that derived from Euclidean geometry. This seems to indicate to me that it's not just a constant in our preferred flavor of geometry but perhaps more fundamental.

pdpi 5y ago

The question "Is pi the same in all universes?" is poorly defined. You're talking about pi as "the circle constant for whatever universe I'm in", whereas the GP is talking about pi as "the circle constant for a very specific geometry (that we once accepted as representative of the universe we're in)"

lordnacho 5y ago

None of the other metrics come from nature either, that's the point. "Logic" is more of a catchall for "not from nature" than a specific axiomatic system.

mypalmike 5y ago

I'm no mathematician but it's my understanding that axiomatic set theory is considered foundational for both what you call mathematics and also for discrete mathematics (aka logic).

foldr 5y ago
Axiomatic set theory can't very well be considered foundational for logic, as its axioms are formulas of first-order predicate logic.
- mypalmike 5y ago
  Yeah that's fair. I was more thinking about the natural mapping of set operations to boolean logic operations (e.g. OR equivalence to union for given sets, de Morgan's laws applied to sets and boolean operations, etc.).

ganafagol 5y ago

> "Logic" refers to the most basic set of logical operations (AND, OR, etc.) and their use. That's not what mathematicians mean with "logic". It's a much broader field than just boolean operators. Those are what computer architects and perhaps electrical engineers mean with "logic".

mjburgess 5y ago
I meant the whole field of logic; I was keeping it simple. Pi is not "from logic" however broadly you construe it. Unless you endorse logicism, but I think that's both highly implausible and somewhat incidental.

mekoka 5y ago

Logic refers to its philosophical branch of study. The dichotomy is apt. Math is pure logic, meaning that you can be locked in a bunker for generations and keep discovering new laws. You can't do the same with physics. A mathematical or logical circle is nothing but a thought, an idea, it's perfect and has no physical equivalent. Pi is bound to that idea in any universe. The metric system is irrelevant (it mathematically cancels out).

foldr 5y ago
Mjburgess is referring to the failure of the logiscist program in research in the foundations of mathematics: https://plato.stanford.edu/entries/logicism/ https://plato.stanford.edu/entries/logicism/ The article doesn't make it entirely clear that the program is widely considered to be a failure, but it is. A better way to express the claim that you're making about mathematics would be to say that mathematical truths can be known a priori.

LeifCarrotson 5y ago

Logic refers to the most basic set of logical operations, and the axioms which we can build on top of those fundamental operations. Mathematics builds on these fundamentals with non-logical axioms, postulates which are used as building blocks for a more complex system. Euclidean geometry is a mathematical system which starts from a set of five basic postulates: 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. Through any given point not on a line there passes exactly one line parallel to that line in the same plane. These, with corresponding idealized definitions of 'point' and 'straight' and 'indefinite' and 'circle', rather than physical constructions, concretely define pi as a ratio between the circumference of the circle and the length of the line segment. Any universe where a mathematician can imagine these five postulates and use them to calculate pi will result in the same value. If you discard or modify one of these postulates, you can come up with a system like a hyperbolic or elliptical geometry where pi could have a different value. Euclidean geometry closely corresponds to a our macroscopic physical world, one can form something fairly close with a straightedge and compass. If you're imagining other universes, you might as well imagine one where the universe itself has a different geometry and the basic axioms are different. In that universe, children drawing a line in the dirt with their fingers would fundamentally understand that there are infinitely many lines parallel to the first in the same plane and would be taught so in primary school, but if they grew up to be a mathematician and replaced that postulate with ours, independently inventing Euclidean geometry, he could discover our definition of pi. Conveniently, typical algebraic postulates like those resulting in Euler's formula, differential equations, integration, power series, and other mathematical constructs also result in the same definition of pi as that derived from Euclidean geometry. This seems to indicate to me that it's not just a constant in our preferred flavor of geometry but perhaps more fundamental.