I. Math Is Not Reality
Math is an abstract logical system that does not necessarily have anything to do with reality. Although you can use math as a very accurate model of reality, and use it to predict things like the path of a model rocket as is launches into the sky accurately, it is still a model. It can have principles that makes its logical system internally consistent, but doesn’t necessarily have to correspond with reality at all. It also means there is nothing necessarily scientifically empirical about mathematics, and it is self contained within its own thought space, without any empirical realities butting into how it works.
Because math does not need to be connected to reality, it can use principles that are physically impossible for reality itself. Such as take the limit of something to infinity, or even the concept of infinity itself. Eventually reality decomposes down to plank lengths and constants and is finite1. No physical object can be infinite anything, be it mass, length, energy and so on. Even a black hole is finite in some form. Mathematics, since it’s an abstract thought experiment, can state something that can keep on going on beyond those plank constants, and since it’s a mostly internally consistent2 logical system be perfectly valid onto itself.
When you enter college, that is traditionally the point in North America when you start encountering mathematics that engages with concepts that are hard to translate to finite reality, such as Calculus or proof by induction.
You can get along very well until about logarithms by believing that math is fully physically correspond-able to reality itself, and all of math is expressible as a decomposable finite equivalent and all of it’s operations are easy to invert and apply. Most of K-12 school math lets you gloss over the ‘infinity’ of things like trigonometry and irrational numbers. Logarithms is when you start running into operations that are difficult invert or solve mechanistically, and it’s the reason why things like logarithm tables existed before calculators and computers, but exponentiation tables not so much.
College math also starts a shift from teaching you techniques and concepts like it was a timed puzzle sport with concrete numbers and operations, and into a different kind of puzzle sport that involves logically proving things. These two hurdles, along with the generable horribleness of most college math education, can make someone who was decent in high school math start to struggle in college.
So when you study math that starts to detach from reality on some fundamental level, such as infinity, detach from the need for mathematics to actually correspond to reality itself. Treat math as its own game that has no need to actually correspond with physical reality at its own limits. Treat it like the internally self consistent game that it is and only use logic as your rules.
With this freedom, things like proof from induction, logarithms, irrational numbers, limits or what not become far more believable and understandable, because there are no unknown edge cases with large numbers from physical reality that could butt into what you are trying to learn and there is no need to actually express what a non-expressible thing is completely, like the number pi. Just describing the process of generating or approximating the physically inexpressible is usually enough.
II. A Way to Think about Infinity
Since in Math, describing how something works formally in a finite way is fairly equivalent to the thing itself, you can use a finite description of infinity to define infinity.
For me, mathematical infinity is something that you could think of as adding 1 to the a number, taking that new number, adding one to it and that process never ending in a loop. Infinity as a result is more of a process and not an actual expressible, explicit number.
With something that is describing a process vs. an actual physical number, it allows you to state things like the infinity of the integer number line is 2 times bigger than the infinity of the natural number line. Since the integer line goes infinitely in two directions, while the natural number line goes infinitely in one direction. Or you can say that for every natural number, there is two integer numbers, and if you go on forever, the ‘length’ or cardinality of the integer number line is twice as big as the infinity of the natural number line. Similarly you can say that since there are an infinite number of irrational numbers between two fractional numbers 3.
Similarly, the concept of 0 is the inverse of infinity and it took longer than you think for human mathematics to discover it and accept it.
III. Many Math Textbooks Are Bad
Most math is very badly explained and taught at the college level. Calculus text books use concepts to explain & prove things that are not even taught to you until 3 math classes later (proofs, set theory, etc), which can make reading their chapters an exercise in confusion if you are a North American high school graduate. The text books are written to professors as a form of sales, since the real customers of the text books are the professors themselves, kind of like enterprise software sales where the buyers are executives who don’t use it, and the real users are employees. Just treat most textbooks as a source for problem sets, not as something to gain understanding from. If your math textbook is actually readable and understandable, then congratulations on your luck.
When I was learning math, khan academy didn’t exist quite yet. I remember scouring the internet and library for better calculus resources, and not being very satisfied. Nowadays, there is khan academy and many other good youtube videos and other such resources. Use them if your textbook is not good for you.
IV. How Infinity Translates to Finite Reality
Since math does not need to actually correspond to reality, you may think, then why use math? Because it’s a very precise approximation or model of reality that has a very good history of effectiveness. So even if your math model might say ‘take the limit to infinity’ when applied to our finite reality it reduces down to planck limits or expands to the entire universe and is plenty good enough as a tool.
Similarly, you can apply the same logic with something like proof by induction. Since math is an abstract thought system and not reality, there is no ‘how can you actually prove something with induction, since you don’t actually know if the proof would change with some large iteration or edge case’. There is no unknown empirical rules of reality which can butt into your proof, just a finite set of rules in a thought experiment in a very simple math model world. ‘Unknown rules of reality butting in’ is something you think of with a finite empirical scientific or engineering / software mindset, but you don’t necessarily have to worry about that in an abstract math world.
So with a situation that says a dot will move to the right one spot with every clock click, you could prove by induction that the dot will move to the right for an infinite distance once you can prove the base case and & n+1 is correct, since empirical or physical reality will never stop this thought experiment. In physical reality, you think the dot flying through space will eventually reach the edge of the universe or hit something along it’s way. In abstract math land, there is none of that stuff to stop your dot.
Therefore, you don’t need the crutch of physical finite reality to learn mathematics, think of it more of a puzzle game where the rules don’t have to correspond to reality, but have to be logically consistent. But at the same time, you can use this puzzle game to model something in reality and make predictions about future actions via logic and calculation. You verify your mathematical model is correct by doing tests against actual reality against it. If they don’t match, either your math model is wrong, or your measurement of reality is wrong and your model just pointed out something you might of been wrong about.
But realize math is still a model, and models are tools to understand reality and are not actual reality and can be inaccurate in some details compared to actual reality itself. Therefore, a model that says physically impossible things can still be useful for modeling and understanding reality itself. This is where the warning ‘the map is not the territory’ comes from.
Reality might one day be found to be infinite in some manner, but that reality is so far removed from what you can experience and understand as a physical human being that it might as well be a mathematical model or religion to you. ↩︎
This is my own conceptualization of it from skimming articles and thinking about it myself a bit, I wouldn’t rely on this on being the math community accepted definition. I initially pondered about this in elementary school when we first learned about irrational numbers but back then I thought infinity was this crazy sauce number because reality is finite, so infinite irrational numbers exist between two fractions seemed a bit nuts and I just threw up my hands about it, in confusion, in my head. It wasn’t until many years after doing badly in calculus classes I came to this realization about infinity. ↩︎