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.<p>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 there, perhaps, different classes of such special numbers?<p>Maybe it's just a trick of our notational system; writing infinite strings of digits does let us describe some numbers in a vast uncountable sea of possibilities, but just bc the notation makes it possible, doesn't mandate that for any arbitrary string there is ultimately some succinct description, that elaborates to that string. In effect, the names <i>are</i> the numbers.<p>Yeah, these are fun things to think about.
I only "believe" in the computable numbers - that can be defined by an algorithm at arbitrary precision.<p>That includes all rationals along with irrational that we have a way to approximate by algorithms - like sqrt(2) and PI. It does not include the incalculable numbers like Omega.<p>Being linked to algorithms, such numbers are only countable many, so they cannot contain "most" of irrational and therefore they lack most of "Real numbers".<p>There are mathematicians that take this view seriously, like "Constructivists" and "finitists". For example: <a href="https://www.youtube.com/watch?v=REeaT2mWj6Y" rel="nofollow">https://www.youtube.com/watch?v=REeaT2mWj6Y</a>
Constructivism isn't really a "camp". It's a set of axioms. (Probably more than one, since mathematicians are clever.)<p>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.
Would anyone or anything do anything differently if they did or did not exist? The same people would still write insisting they do or don't, regardless, because nobody could tell.<p>If there is no need for them to exist, Ockham's Razor says we should not assume them.<p>Suppose they don't exist, then what are we talking about? We exist, and we do stuff (such as writing blog posts). So, numbers -- all numbers -- are stuff we do. And write about.<p>They don't need to be more than that. <i>We</i> don't need them to be more than that. If we did, we would be out of luck, anyway.
A lot of this is way over my head both mathematically and philosophically, but could we get away with saying that the reals exist in some slightly weaker sense, more like how a function exists?<p>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.<p>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).<p>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)<p>The author sort of covers this here:<p>> 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.<p>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).
If I understand correctly, the problem is not so much real numbers, but real number that are not otherwise defined(integers, rational numbers, floating point numbers, etc) call them pure real numbers.<p>Because while it is easy to imagine there are numbers that are not defined, the minute you try define what they are... they are now defined, thus the problem/paradox, can something exist that is not defined?
Do integers exist? Do sequences of integers exist? If they do, so do the real numbers.<p>This is because any positive real number has a decimal expansion which determines it. For example pi = 3.1415926535... or 1/3 = 0.3333333...
Real numbers are abstractions. And abstractions are embedded in Physics. So if you believe that Physics is <i>real</i> then Real numbers are also <i>real</i>.