Sure, otherwise you're just solipsistic, we have to assume things exist, it's how you make progress. If you think quantum is "the real stuff that exists" then you should assume, for now, that the model actually models something, and our best guess is that it's merely probability theory.
Absolutely. There's no reason to believe space is Euclidean, in fact, relativity says it is curved. Quantum mechanics says that it is not infinitely divisible.
Algebraic reals are the easy part, as are the few familiar classes of transcendentals that include numbers like pi and e. There’s only countably many of those. The set of all reals that can be described is countable. It’s the literally undescribable ones that are the problem, ontologically speaking
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 there, perhaps, different classes of such special numbers?
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 are the numbers.
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.
Limits work just fine without dealing with infinity. When you do an epsilon delta proof, you only ever deal in the finite. You can also have continuity while only dealing in finite constructions.
The real numbers are a very convenient abstraction, but they are also a leaky abstraction (edit: when paired with axiom of choice), which is anathema to mathematics.
We can still study the real numbers, they are still valid in some abstract sense, and they are convenient, but an element of the set of reals minus computables can never be constructed, so you'll never actually need to deal with them.
It's pretty easy to improve that almost every "real" number is impossible to name.
"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.
Agree. It's difficult to guess the definition of "exist" while reading it. The phrase
> ...accepting the existence of the reals means that there are numbers that exist but can never be described ...we can never interact with these numbers even conceptually
is also full of vagueness: what is "be described"/"interact"?
Constructivism isn't really a "camp". It's a set of axioms. (Probably more than one, since mathematicians are clever.)
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.
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?
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).
The “laws of physics” are mathematical models (abstractions) created by humans. And those abstractions are embedded in the physical world (our brains, written text, magnetic disks etc.)
Right, that's also roughly what I default to (because Platonism seems to have more unnecessary complexity/hypotheses). The actual nature/ontology of abstractions isn't entirely (or at all) solved by that (e.g. https://plato.stanford.edu/entries/abstract-objects/).
I'm pretty sure (although definitely can't prove) that all solutions that come up in physics are computable numbers which is a countable subset of the reals.
Do infinite sequences of integers exist? Especially all the ones that can't be compressed into a finite description? I don't think that's clear 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.
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?
Real numbers are defined by properties, usually, and it’s just a “I say these things exist” then.
For example, axioms of real numbers state that increasing bounded sequences have a limit, and it is a real number. It could be that this leads to some contradiction somewhere which we don’t know about.
For example, if you erroneously thought that rational numbers in fact themselves satisfy these axioms, you could construct an increasing sequence whose squares converge to 2, and yet the sequence itself doesn’t converge to a rational (because sqrt(2) is not rational). Perhaps some such contradiction exist for real numbers, and in that respect they could “not exist”. That would mean the axioms are somehow inconsistent.
As the article states, we can construct real numbers with Dedekind cuts that only rely on rationals; I didn’t realise there is any nuance to that definition.
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.
If there is no need for them to exist, Ockham's Razor says we should not assume them.
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.
They don't need to be more than that. We don't need them to be more than that. If we did, we would be out of luck, anyway.
I only "believe" in the computable numbers - that can be defined by an algorithm at arbitrary precision.
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.
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".
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[ 3.1 ms ] story [ 87.1 ms ] threadWhat if you get down to the quantum level? You don't have a straight line in the Euclidean sense any longer.
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?
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 are the numbers.
Yeah, these are fun things to think about.
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.
The real numbers are a very convenient abstraction, but they are also a leaky abstraction (edit: when paired with axiom of choice), which is anathema to mathematics.
We can still study the real numbers, they are still valid in some abstract sense, and they are convenient, but an element of the set of reals minus computables can never be constructed, so you'll never actually need to deal with them.
"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.
> ...accepting the existence of the reals means that there are numbers that exist but can never be described ...we can never interact with these numbers even conceptually
is also full of vagueness: what is "be described"/"interact"?
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).
This is because any positive real number has a decimal expansion which determines it. For example pi = 3.1415926535... or 1/3 = 0.3333333...
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?
For example, axioms of real numbers state that increasing bounded sequences have a limit, and it is a real number. It could be that this leads to some contradiction somewhere which we don’t know about.
For example, if you erroneously thought that rational numbers in fact themselves satisfy these axioms, you could construct an increasing sequence whose squares converge to 2, and yet the sequence itself doesn’t converge to a rational (because sqrt(2) is not rational). Perhaps some such contradiction exist for real numbers, and in that respect they could “not exist”. That would mean the axioms are somehow inconsistent.
As the article states, we can construct real numbers with Dedekind cuts that only rely on rationals; I didn’t realise there is any nuance to that definition.
If there is no need for them to exist, Ockham's Razor says we should not assume them.
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.
They don't need to be more than that. We don't need them to be more than that. If we did, we would be out of luck, anyway.
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.
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".
There are mathematicians that take this view seriously, like "Constructivists" and "finitists". For example: https://www.youtube.com/watch?v=REeaT2mWj6Y