11–20 of 39 posts
The hypotenuse of a triangle with unit length sides is real enough.
I assume you mean in the physical sense? What if you get down to the quantum level? You don't have a straight line in the Euclidean sense any longer.
There are the useful ones we know about, like pi or e, which can be described to arbitrary precision, but you cannot "write them out" bc that would require infinite paper and a lot of pencils. But they're useful, they have meaning to us, so we give them nicknames and use those. But what about all those other ones? Is it possible that there are certain ones, of future usefulness, that we haven't yet recognized? Are th…
The subset of numbers for which you can compute successively better approximations, such as pi and e, is countable. It’s called the set of computable numbers and contains everything that we could even in theory ever describe or “use” somehow. The actually uncountable subset of reals is by definition forever out of our reach, which is what makes uncountable sets… rather ontologically suspicious.
I do not find anything ontologically suspicious about uncountable sets.
What I would really found ontologically suspicious would be the existence of convergent sequences of numbers without a limit or the inexistence of continuity, e.g. the impossibility of knowing whether moving from one side of a curve to the other side of the curve will intersect the curve or not.
Even if most programmers are among the people who use the least the mathematics based on real numbers, for most of the mankind the mathematical models that would not be possible without the concepts of limit and continuity have become essential for survival, regardless if they are able to understand them or not.
I strongly dislike the meaningless term "real numbers", which was created as an opposite to the equally meaningless term "imaginary numbers". However what matters are not the numbers or whether they can be counted or not. What matters are the concepts of limits and continuity.
Already Aristotle classified the quantities into discrete quantities and continuous quantities, using a definition of continuity not too different from the modern, even if it was non-rigorous. In Antiquity, what are called now "real numbers" were called "measures", i.e. results of measurement operations, in contrast with the "numbers" obtained by counting.
There are the useful ones we know about, like pi or e, which can be described to arbitrary precision, but you cannot "write them out" bc that would require infinite paper and a lot of pencils. But they're useful, they have meaning to us, so we give them nicknames and use those. But what about all those other ones? Is it possible that there are certain ones, of future usefulness, that we haven't yet recognized? Are th…
"Real" numbers are really the "realizable numbers", on a potential sense, like how every sperm is a potential mammal -- anything you construct on the "real" line is real, but the "real line" is mostly space for numbers that will never be realized.
Whether real numbers are real is a metamathetical questions. a choice of models/axioms is either the rules of a fun game or a map for the territory of something that exists in the real Universe. If you are looking for a good map, you choose between non-equivalent models based on how accurate the map is, per experiment, not philosophical preference. But then you are doing physics, not pure pure math, so then the metamathetical debate is irrelevant.
I.e., reals are number-like entities that are actually described via a mathematical expressions, but can successfully be used in expressions as if they were numbers.
In this account, while a real can be used in the place of numbers in mathematical expressions, it's really more of a placeholder for a function that provides the operation (or for an expression nested inside the larger one; take your pick).
As a loose programming analogy, think of how in some languages you can use a generator in the exact same way as an array (at least for iteration), even though the array is a static value and the generator is actually backed by a function with dynamic behavior. Also think of Haskell with its lazy data structures that are represented the same as "actual" values (e.g., infinite arrays)
The author sort of covers this here:
> In particular, it seems intuitive that there are countably many descriptions of numbers (assuming that descriptions are finite-length strings of some finite language). And the real numbers are uncountable, so accepting the existence of the reals means that there are numbers that exist but can never be described. And this feels deeply unsettling to me: if we can never interact with these numbers even conceptually, it would be impossible to distinguish between two universes where in one they exist and in the other they don’t. Which to me makes their existence feel tenuous at best.
So that's a good objection. But what if we weaken the claim and say that only the "describable" reals actually exist, and for the rest, we can talk about them in principle but they actually don't have corresponding objects? This would divide reals into two categories: function-esque ones that we can describe, and "hypothetical" reals that can't be successfully represented using any formal syntax. Just as how in a programming language with generators, you have a countably infinite number of valid generators and an uncountably infinite number of generators you could dream up that fail to produce values in some way (e.g. bad syntax, perform illegal operations like div-by-0, or even just a generator someone came up with in a fever dream that isn't representable in symbols at all).
Numbers aren’t real.