[dmfdmf]

>Infinity invites resistance. Aristotle rejected the existence of the infinite entirely; to him, infinity was simply a limit that could never be reached, not a true mathematical entity.

Off to a bad start. Aristotle was not making a mathematical point but a metaphysical one. Infinities do not exist and is not a number. For example, Pi is not a number but a symbol that stands for an open-ended (infinite) process to calculate a rational number and is a perfectly valid mathematical concept that, I am sure, Aristotle would agree. On any computer, despite protestations by the mathematical platonists, Pi is ultimately a rational number in all use cases involving actual measurement or calculations.

The error is illustrated in the first image in the article.

https://www.quantamagazine.org/wp-content/themes/quanta2024/...

The third set in this example is an invalid and undefined set by including Pi since Pi is indeterminent and thus cannot be an object to be counted. All of Cantor's nonsense rests on this type of error, i.e. treating a mathematical process as a number. All of these errors are implicit in Newton's calculus and Berkeley's Ghost of Departed Quantities critique still needs to be answered. Hint; there is no such thing as infinite precision and epsilon/delta needs to be defined in a consistent way, not arbitrarily as it is now.

[sixeyes]

> Pi is indeterminent and thus cannot be an object to be counted

this seems weird to me. doesn't pi (the symbol) point to one specific concept, whether or not we can determine its exact shape?

[dmfdmf]

Pi is not a number, it stands for a method (i.e. infinite series) to calculate a number. Conceptually it is the ratio of circumferance to the diameter of a circle which are incommensurate quanties, i.e. can't be represented as a rational number.

It is a subtle distinction but important. We define the exact value based on the context. If I am tiling my circular patio then 3.14 is fine to calculate how many tiles I need. If I going to the moon or mars then I need more decimals or I will miss the target.

[pavel_lishin]

By that logic, the square root of two and i are also not numbers, right? In fact, any non-rational "number" isn't a number by that definition.

[dmfdmf]

Setting aside i, math symbols that stand for irrational numbers can be treated as numbers in theorems or derivations or proofs but one needs to be careful that a symbol such as Pi or SQRT[2] actually designates an infinite series to calculate a rational number. This distinction is important when math equations are actually used to calculate or measure something specific which is the whole point of math. All valid measurements can only result in a rational number. It is the distinction between doing math and doing physics or engineering (i.e. applied math) which has to be integrated, they are not separate fields with regards to measurement.

The imaginary number i=SQRT[-1] is the base solution to the polynomial equation -y= x^2. If you read the history it was invented to solve certain types of cubic equations, i.e. as a heuristic of method. So not only is it not a number it is a bare contradiction. While i was useful to solve some subclass of all cubics it did not lead to a general solution to cubic equations. Nevertheless the mathematicians ran with it and added complex numbers to the definition of number so that the number system could solve all possible polynomial equations.

In my opinion imaginary numbers are a kludge and deadend and it is masking the real issues in math, i.e. the ghosts of departed quantities. In math some symbols stand for an infinite series but you can't just choose any arbitrary series for Pi or e or SQRT[2], they all have to be defined as part of a system of measurement with clearly defined and globally defined epsilon/delta (i.e. precision) of measurement to get valid results.

[kccqzy]

Found a finitist in the wild. You would probably also think most real numbers don’t exist.

[dmfdmf]

Yes, you can call it "finitist" if you like but that is an error too. Define "exist".

Not "most" but all real numbers are similar to Pi, i.e. they are symbols that stand for an infinite process to calculate a rational number. Both irrational numbers like Pi, e, etc. and reals exists and are legit and useful math concepts but infinite precision does not exist "in the wild" only in your mind as an abstraction. In any actual calculation or measurement your infinite series must stop and the dedekind cut must be made.

The ghost of departed quantities still haunts math. Is Pi 3.14 or 3.1416? Mathematically, it is neither and both because math intentionally abstracts from the precision of the ratio of the circumference to the diameter of a circle. These open-ended (infinite) processes are ultimately used to define a rational number, a ratio of integers.The finitist -vs- infinitist is a false binary which ignore that actual measurement must use rational numbers.

[112233]

Most natural numbers don't exist, for any useful definition of exist.

When you prove, say, by induction, that p(n) holds for any natural number n, and hear you teacher say that p(n) holds for all natural numbers, you start forming the idea that "all natural numbers" is a thing that exists. The set N, you think, surely by writing it, all natural numbers are called into existance.

And then, much later, you come upon problem, where actual existence of the number becomes better defined. Say, like finding a large prime. And suddenly "all numbers" becomes a confusing mental burden.