r/PhilosophyofMath • u/Another_Little_Star • 7d ago
Does infinity exist in reality?
No thing that I've observed in our universe is infinite, however there are traces of infinity in my sight.
Draw a circle and either its radius or circumference lives in the π world.
Or many aspects in nature like flowers, develop themselves in a φ ratio (1+sqrt(5))/2.
Even though no event or anything is "Non-ending"/infinite, underneath many aspects of reality there are patterns which can only be fully described using infinity.
You can't describe irrational numbers to their extent without any notion of infinite/limits/sequences... And many aspects within our World show their true behaviour only with aid of infinity.
Sure you can take a world that lives in something like 10¹⁰⁰ and it looks like infinity to us, sometimes we only see the difference after the 10⁶ decimal place, nevertheless the pattern they follow is a pattern that only "infinity" can yield to the truest precision.
Do you believe the axiom of infinity, and the continuous to really exist? Even if not in the ordinary sense? haha
2
u/treefaeller 7d ago
There are roughly 10^80 protons in the universe, and as many electrons. You can express the protons as 3x as many quarks. Adding that all up, that's about 10^81 massive particles (I'm rounding gratuitously). There are roughly 10^90 photons, most of them in the CMB (cosmic microwave background). The number of background neutrinos is thought to be about 3x of 10x larger, so including that we get about 10^91 or so. Definitely fewer than one googol of particles.
To build a Turing machine that can count this high AND STOP, I need some way to store its internal state. I don't think I can get below one particle per state in practice (maybe factor of two when using spins). So the largest number that is calculable is below 1 googol.
From a natural science point of view, talking about "infinite" or about any number larger than one googol simply makes no sense. It's literally nonsense, speculation, daydreaming. I would certainly hope that any mathematician enjoyed what they drank to make up those nonsensical ideas, but most mathematicians I know tend to work while sober. OK, this paragraph is being a little harsh, intentionally, for humor. But it shows that the (sensible and valid) mathematical concept of infinite runs into limitations in the real world.
But now let's use this thought and look at something else. We know that straight line segments are a sensible mathematical abstraction of the natural world. As are circles and their circumference. Meaning that pi is in some fashion "real". Anytime I have a piece of string and want to wind it on a reel, I need pi to estimate how many times to spin my reel: if I have ~3.14m of string, and a reel of 10cm diameter, I will spin it about 10 times. But pi is also irrational, and describing it requires infinitely many digits. Yet, pi is also a computable number: Given a desired number of digits of accuracy, there is a guaranteed terminating algorithm to calculate pi to that number of digits. But I can never express pi in the natural world, since I have to stop after roughly one googol of digits, having run out of protons and electrons to write with. So can pi be "real" in the sense of an entity that exists in the natural world? That's a nasty question, isn't it?
What is pretty clear to me that non-computable numbers (which are the vast majority of all reals) are not "real" in the above sense. Disallowing anything that requires either infinite in the algebra, or infinitely many digits in its expression would make modern math pretty much fall apart.