Why?
Finding out that Pi is irrational and then transcendental are discoveries. I won't be able to make anything work if I fork mathematics to make it rational.
Same thing that finding out that if you tie a string to a stake and then go around keeping it tight, you have to walk a distance that is two times pi times the length of the string to get where you started. I don't need a language for that, just more pieces of string.
It is this complete logical decoupling from any physical, observable reality, that I cannot accept mathematics as anything other than an, ideally, internally consistent, logic used to define and discuss symbolic logical objects. And I am of the opinion that logics are merely invented by conscious beings as means to systematize areas of discourse and to derive internally consistent inferences and implications built there on.
I would agree that if irrational numbers were prohibited from being inferred in the logical system of math then some things would not work within the logic. Because a change in the axiomatic or inferential rules of a logic are changed the logic as a whole changes. But, if, as you describe it, math was forked to be limited to rationals, and the precision of those radicals was calibrated to be on the order of 1/10 of a Planck length, I am not convinced that any equation attempting to describe 3 dimensional space would be inaccurate. However, I could wrong.
But it is uncommon in my experience to actually define Pi as this ratio, if you build math axiomatically; Rather, the "shortest way" to get to pi which is well defined is by first defining the exponential function ( \exp x = \sum_i=0^\inf \frac{x^i}{\fact i} ) with all the pre-requisite for that (numbers with order, addition and multiplication; limits and convergence; then imaginary numbers). Then you define pi to be the smallest positive number such that exp(2pii) is 1, and e to be exp(1). All the properties of pi follow "easily", including it being the ratio of am euclidean circle's circumference to a diameter.
The thing is, in math, all of these things end up the same regardless of where you start; Whether you start with a geometric definition of a circle and work hard to discover said ratio, or you start with exp(), you'll end up with pi=3.14159.... and it having the other properties. It is in that sense, not arbitrary.
You could (and would) take a step back, and say that being euclidean is arbitrary - which is true; but any description compatible with euclidean axioms will get the same value of pi as the ratio of circumference to diameter; and that value will be the same of the exp() value that has no concept whatsoever of euclidean space.
It is in that sense that math is "discovered" - there is no euclidean construction in which pi is different. There is no peano-compatible (set, games, p-adic, or other) construction of natural numbers in which primes do not exist. The peano axioms themselves are, indeed, logically arbitrary. But the "discovery" is that "peano -> existance of prime numbers" -- and that does not depend on language.
It’s similar in my perspective to a nuance that exists pervasively today within scientific models. Just because the model allows a prediction doesn’t mean that the model reflects “base reality.”
Our models, our tools, are the results of our observations. If it works to provide a prediction, then it becomes useful. However I feel there’s a disconnect as the nuance of a model vs what is, has been lost among many.
The map is not the territory.
It’s very interesting to see science use mathematical models as objective reality and orthodoxy. I’d say that isn’t science, but words change in meaning with the zeitgeist. Unfortunate in many ways.
I like the way you framed the argument.
Any recent things you’ve read in this realm of subjects that were interesting to you? Any favorite past time papers/books you’d recommend?
For some topic adjacent past time papers, a lot of my comments concerning logic come from the research for my programming language, so I have been immersed in logic and proof theory work for the last six months pretty hard. I think any of the course note PDFs from Frank Pfenning (CS prof at Carnegie Mellon Univ.) are great read in general (and are easily found by googling Pfenning logic course notes and just looking around). If you like video lectures, or just listening to them, any of Pfenning’s lecture sets from the OPLSS session, which are all on YouTube are wonderful, particularly the 5 lectures from 2017 on Substructural Type Systems and Concurrent Programming [2] Also, Noam Zeilberger’s OPLSS lectures on Refinement Types [3] are great and his dissertation was a great read [4]. Finally, Neel Krishnaswami and Dunfield‘s 2 papers on Higher Rank Bidirectional Type Checking [5] was really good (and like all the above the cite lists are a trove of good stuff).
I literally could keep going for hours on great reads in logics and type theory, so if you want more in some area I’ll provide.
Cites: 1 - https://plato.stanford.edu/entries/philosophy-mathematics/
2 - https://m.youtube.com/watch?v=ZW-VWWa1o9g
3 - https://www.youtube.com/playlist?list=PLiHLLF-foEeyS8V4lwN1q... 3a (course notes) - http://noamz.org/oplss16/refinements-notes.pdf
4 - https://www.cs.cmu.edu/~rwh/theses/zeilberger.pdf
5a - https://arxiv.org/abs/1306.6032 5b - https://arxiv.org/abs/1601.05106 5(2019 Version) - https://www.cl.cam.ac.uk/~nk480/gadt.pdf
I think it’s important to expand further in this direction. Confirmation bias or not, I’m compelled to dig further under these ideas. Thanks.
I notice you’re on a throwaway. If you’re on any other platform I’d enjoy staying in touch. My profile has sigs.
Not very divorced. I can glance at my coffee cup and notice that there is a distance around the circular edge and another if you were to go straight across. Maybe they are just having a trial separation?
Pi is not calculated by running around in a circle with a string (nor by any other real-world measurement), it's calculated by setting up a rule-based system (math) and following the rules to derive its value. You can get pretty far in the real world by using a rational value for pi - ten decimal places should suffice for atomic-level precision, I believe.
One of the interesting consequences of the recent redefinition of the SI units is that now h, k, e and a few others are considered exact constants, just like c. A caveat: physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself, whatever that might be it's something dealt with epoché.
So now the Stefan-Boltzmann constant, which involves powers of c, k and h is exact. It also involves the fifth power of Pi (through the value of the Riemann zeta fuction for argument 4). As you can measure the Stefan-Boltzmann constant being creative with a radiometer, you could determine increasingly accurate values of Pi experimentally. In the sense of magnitudes as used in physics, which is a fruitful approach to being quantitative in this confusing physical world, this "thing" we call "Pi" behaves like a dimensionless physical constant.
Before you'd object, if we came up with more precise theories, experience shows that that would mean corrections to the laws implied above. It also shows that when this happens instances of "Pi" don't deviate from the Platonic value of Pi.
This is entirely oxymoronic and double thinking and exactly the type of mental gymnastics that manipulates what truth is. Physics is based on the physical because it was built upon observation of reality using our senses. Yet modern physics is based on the imaginative and has nothing to do with the physical.
Any measured unit is imaginary. The only thing that exists is the measure.
Exists is based on physical. To be is to exist. To exist must be within physical reality. Without this distinction then we might as well give into the esoteric notion that everything is entirely a mental construct and nothing actually “is.”
How far we’ve come from the physical to the mental and then lack the nuance to distinguish as such and furthermore blur the lines is a disgrace to critical thinking and cognition.
This is only a general comment because I don't understand anything you've written so I can't address your points, which undoubtedly must exist.
If following the logical inference chain that leads to determining the geometric concept of a circle and then inferring the ratio between the circumference and radius is a discovered thing, then any and all logical inferences which can be proved are then discoveries, but I don’t think that argument can be taken seriously.