God created the real numbers
ethanheilman.com
ethanheilman.com
> H.C. Agrippa's work is still considered authoritative in its fields: the numbers represent the ideas, which were both created in the first moments by God.
That does sound like the computability idea of numbers being programs.
``` In the beginning was the Word, and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. ```
The "Word" in Greek is translated from logos which also means logic, rational, metric.
``` In the beginning was the Number, and the Number was with God, and the Number was God. ```
Looking around on the internet, there is a lot here: https://en.wikipedia.org/wiki/Rationes_seminales
The article's claim seems to be about the mathematical formalisms humans have invented for integers and real numbers. And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
Everything you can express in integers you can express in reals, but there are many things expressable in reals not possible in integers. It would be surprising if the formalism for a thing that completely supersets another thing had an equally simple formalism
Are you sure there is anything we can express in the Reals that isn't an integer in disguise?
The first answer might be the sqrt(2) or pi, but we can write a finite program to spit out digits of those forever (assuming a Turing machine with integer positions on a countably infinite length tape). The binary encoding of the program represents the number, and it only needs to be finite, not even an integer at infinity.
Then you might say Chaitin's constant, but that's just a name for one value we don't know and can't figure out. You can approximate it to some number of digits, but that doesn't seem good enough to express it. You can prove a program can't emit all the digits indefinitely. And even if you could, is giving one Real number a name enough? Names are countable, and again arguably finite.
It seems to me we can prove there are more Reals than there are Integer or Computable numbers, but we can't "express" more than a finite number of those which aren't computable. Integers in disguise.
No, not any more. :-)
I liked the idea a year or two ago, but I've come to believe that even the Integers are too bizarre to worry about. For now, I'm content just playing with fixed and floating point, maybe with arbitrary precision. Stuff I can reason about.
I just think people are a little too casual thinking they are really using the Reals. It might be like Feynman's quote about saying you understand QM.
Does tuna casserole exist independently of humans? If not, how is the idea of the number 7 different from the idea of a tuna casserole? Or what about the concept of decision by majority, which isn't as basic as 7, but doesn't have the physicality of a casserole?
No of course not, but that's not the question. The question is whether the concepts were created by the beings.
In a system with 7 stars the number of stars doesn't change when all humans die or humans never even developed.
Others would say they all exist independently of humans.
Some think abstract mathematical concepts are more privileged than physical categories like stars.
That is the question. I can't even tell if you're arguing one side or the other because you say "of course not" followed by "doesn't change" when those appear to be contradictory positions. But of course there are a lot of subtleties here. There's no "of course" to any of it. These things take entire books to argue one side and try to refute the other.
If only this was the stuff humans wasted their energy on not shooting explosives to improve their ego. :-)
> If humans never even developed, many philosophers would say the concepts of "number" and "7" and "star" would not exist.
Yes, I'm in the camp that thinks this is just plain wrong.
> abstract mathematical concepts are more privileged than physical categories like stars.
I agree to that.
> I can't even tell if you're arguing one side or the other
Sorry, if I was unclear. You wrote:
> Do or can concepts and categories exist without the beings that create them?
This takes it a priori, that the categories are created by the beings. If this is true, then I think by definition this categories can't precede the beings. That's why I wrote "of course not". When things are created by beings, they don't precede them, when they are not, they do.
What I already disagree with is what you already implied as a given. Honestly I don't even know how to argue for that, because this is what I think is part of the definition of the concepts and categories. When I coin some term it always happens in reaction to things I perceived with my senses. So given my senses don't lie to me, of course this is outside of me.
There are also things that are in fact invented by humans, for example color names. This doesn't mean that colors don't exist independently, but the exact boundary is arbitrary.
There are also concepts where I think they can be both. For example beauty. When applied to skin or shape, these are just invented, but the beauty of complexity or completeness exists outside of us. And beauty in general only exists as it points as to a thing that exists independently of us.
It is arguing that the integers separate from the reals is the formalism and that the abstract entity is the reals.
We also have a formalism of the reals, but it is closer to the abstract entity.
> And I agree that our formalism of integers is simpler and more elegant than our formalism of real numbers. But that could just be because we've done a worse job formalizing real numbers!
As we create better more useful formalisms, we interact more with the formalism than the thing itself. It is like putting on oven mitts to pick up a hot tray from the oven. This has already happened with the reals!
Consider the question of if the reals are well-ordered or not. The question applies to the actual entity of the reals itself, but the absoluteness theorem shows that the question has no arithmetic consequences. You can simple ignore the question. Thus, those seeking the mental comfort and utility of the formalism do not have to concern themselves with the true nature of the real number line.
Why believe in abstract entities at all, as something distinct from the formalism? We have various formal or abstract concepts, that are useful in science and its applications, but they make contact with reality only through these uses: the natural numbers are used in counting, which is useful because there are many physical objects or events that are similar enough to each other that their differences can be neglected, then it's meaningful to count them and come up with a number for the collection, e.g. apples in a basket, planets in the solar system, or whatever.
Real numbers developed from our perception of continuous physical magnitudes, and from the usefulness of applying the concept of number to these magnitudes. Then the formalism of the real numbers was developed based on mathematical constructions from the natural numbers/integers.
We don't have to posit some abstract entity that the concept of real numbers refers to: it's a symbolic or mathematical construction that helps us in reasoning about (what we perceive as) continuous magnitudes, which aren't abstract, but concrete aspects of our experience of the world
Because they seem to have some actual reality independent of human ideas. Not that everyone agrees with this, but there are good arguments for it (see the last few thousand years of debates on the subject of platonic idealism).
The fact that something has been discussed for thousands of years also has nothing to do with whether there are good reasons for believing it, e.g. the Earth being flat or stationary at the center of the universe. People can be wrong for thousands of years
I agree people can be wrong for thousands of years, but a flat Earth was conclusively disproved early on. I'm not talking about random people debating this issue, I'm talking a continuous intellectual process over thousands of years to understand the nature of mathematics.
It may turn out that various theories put forward will be shown to be wrong. This has already happened, Godel's incompleteness was a major blow to people that argued that mathematics did not have an independent reality because it was just a logical game defined by axiomatic rules. Godel showed that axiomatic systems beyond a certain level of complexity have a reality beyond that logic can investigate.
Godel's theorems are notoriously liable to misinterpretation, and to be honest this sounds like one.
What Godel actually proved is that given a consistent formal system capable of representing arithmetic, there are statements in the language of that system such that neither the statement nor its negation can be proved in the system.
The thing is these statements will be different for different formal systems. The theorem doesn't say that there are statements that are in general unprovable in any formal system, which is a common misconception. Maybe that's not what you're saying, but I find it hard to relate the claims your making about Godel's theorem to the statement of the theorem itself.
Godel's proof itself can be formalized, so I don't see how it places a limitation on formalism in general
It goes what step further, there are statements that are true but can not be proved.
> Godel's theorems are notoriously liable to misinterpretation, and to be honest this sounds like one.
What do you think was Godel's philosophical motivation for investigating incompleteness?
"Many philosophers and logicians have explored the significance of these theorems. However, they would often find a reason to distance themselves from the interpretation their author, Kurt Gödel, ascribed them. Gödel believed that his results provide a strong argument for the objective existence of a rationally organized world of concepts, which can be to some extent described by a deductive system but cannot be changed or manipulated (cf. [12], p. 320)." [0]
You can argue that Godel misinterpreted incompleteness and There is a case to be made for that. Godel probably did not misinterpret his own philosophy about incompleteness.
> Godel's proof itself can be formalized, so I don't see how it places a limitation on formalism in general
It uses formalism to show that formalism is limited and that the positivist notion of truth does not include things which are true but can not shown to be true (within a axiomatic system capable of expressing arithmetic).
[0]: Kurt Gödel and the Logic of Concepts (2024) https://arxiv.org/html/2406.05442v2#bib.bib12
No, this is the misconception. I know it's often stated this way, but it's wrong. Godel showed that given any formal system, you can construct a statement that the system can neither prove nor disprove. But Godel actually proves this statement in his proof, so it's certainly not unprovable in general.
You can construct a statement in an axiomatic system complex enough to express arithmetic:
1. Which is true,
2. AND which can be proved to be true under the assumption the system is consistent,
3. and which can not be proven true within that axiomatic system because you can't prove consistency of the system itself within the axiomatic system itself.
The crass positivist position would be that such undecidable statements are meaningless, i.e. neither true nor false. The incompleteness theorems provide a strong counter-example: a statement which can not be proved within that axiomatic system, but is true (if the system is consistent). The positivists must then amend their position to include things which can't be proved within the axiomatic system but can be proved true. This resulted in the watered down late-positivism.
In fact, what it means for a mathematical statement to be true is that it's provable. Godel showed that the concept of provability is much more subtle than following from a single axiom system, but it doesn't show that there's some kind of transcendent form of mathematical truth that doesn't depend on proof, in my view.
A statement is provable if it's possible to write a proof for it.
But you're unlikely to find that satisfactory...
Spelling out the bounds of what can possibly constitute a proof is a big question in mathematical logic, model theory, and proof theory, and I'm far from an expert in these subjects.
I think any proof has to be based on some set of axioms, but Godel showed that not all provable statements can be formally derived from a single set of axioms.
Another way to put it might be that the space of all possible proofs can't be defined by a single axiom system with its rules of inference, but I'm not sure about this, because as I understand it we should always be able to tell mechanically, given a putative proof, whether it really proves the theorem.
In any case, I believe all these subtleties and difficulties also have to be faced if we try to spell out precisely what we mean by mathematical truth. In my view, this is the same question: mathematically true = it's possible to prove it.
It is from that era that they developed systems of rigorous debate, formal logic, and things like peered reviewed papers that we call "the scientific method".
As far as the history of these sorts of mathematical discussions the concept of negative numbers didn't exist until the 15 century. I am sure that each new concept was faced with some resistance and debate on its true nature before it became widely accepted.
So I am sure that somebody looking through the historical record could find all sorts of wild quotes from different theologians trying to grasp new concepts and reconcile them with existing mathmatical standards.
No, I don't think so. It seems much more based on ancient Greek geometry and logic, the Indian numeral system and Arabic algebra. Modern science really took off after Galileo, at the time when the ancient Greek works were recovered in Europe and could be synthesized with the arithmetic and algebra of these other cultures. Galileo himself credits the "divus" Archimedes as his main inspiration.
What aspects of Christian scholasticism do you think developed into modern science?
There is no fundamental unit of "star". Maybe we can talk about electrons or protons or something, but what is and is not a star is a model, not a reality.
Concretely, a bundle of pre-stellar gasses at some point transitions to being a star, but when in that time spectrum does it make that transition? When in the process of stellar exhaustion does it stop being a star?
Which would mean integers are baked in, rationals too, but non-constructible reals (essentially all reals, given any degree of approximation) are a useful abstraction but don't actually exist in any way.
Reals are not real.
(Roughly) Equivalently: There may be no perfect circles in nature.
Constructible reals are also a subclass of reals, but you can't claim anything about the class of reals, which are vastly dominated by un-constructible reals, because constructible reals have a property.
There are many reasons to doubt un-constructible numbers exist in nature.
Just for starters, you can never actually define a specific un-constructible real. If you did, you would have defined it, making it constructible.
An un-constructible real requires infinite information to define. Not infinite digits (pi is constructible, e is constructible), but an infinite list of uncompressible digits, or some other expression with infinite numbers of symbols!
The name "reals" is highly deceptive/unfortunate. (What could be more reasonable than a "real" number?)
We need a pithier name for constructible numbers, and that is what should be introduced along with algebra, calculus, trig, diff eq, etc.
None of those subjects, or any practical math, ever needed the class of real numbers. The early misleading unnecessary and half-assed introduction of "reals" is an historical educational terminological aberration.
(It would be nice to rename "real" numbers, to mean actually real numbers that we could actually use. But given the generations of confusion that would incur, I propose "actual numbers", to be all constructible numbers. Nobody but mathematicians, who play abstract games with higher order infinities, need "real" numbers.)
Well no, you will never have it.
You can't start out with finite things, and built an infinite thing, even if you have infinite components to put together, and infinite time to do it.
That is what countably infinite means. It is a very practical kind of infinity.
And the concept comes directly, and inevitably, from the integers.
Just like integers, and all the theorems/patterns we discover in them, reality may be countably infinite. Filled with an infinite number of structures of unbounded sizes, and infinitely large structures parameterized with finite constraints.
The class of real numbers is not what our familiarity with its name makes us think it is. It is a mathematician's mind game, regarding properties of abstract made up things that got called numbers by fiat. Not by induction or other mathematical inevitability. Nowhere in the chain of numbers built up from the integers. With no hope of ever encountering a single concrete instance that isn't already in a smaller better defined subclass, that doesn't require the concept of reals.
Mathematicians get to have their games. At a minimum they are useful as ways of stretching mathematical skill. Concepts that don't correlate with things that exist, can still be interesting, challenging, and spin off insights.
If you accept countably infinite rationals then you also accept Cauchy Sequences, no? Then we see that the reals arise naturally from rationals. I'm a noob at this btw so I would appreciate guidance.
Absolutely.
When most people think of reals, they are thinking constructible reals. Which are countably infinite in number.
If there is an equation (known or in principle), they are constructible.
If there is an algorithm (known or in principle), they are constructible.
Limits, continuous fractions, all those are constructible.
From a mathematics perspective, I think it's a loss that the distinction between countable/constructible numbers vs. uncountable/un-constructible is completely blurred under one name "reals" early in everyone's math education. Even though the difference is significant when reasoning about information, the relationship between math and physics, math and computation, etc.
And about infinites. Most of us famously know that there are infinitely more reals than integers.
But how many people know that all well defined reals, constructible reals, calculable reals, and their equations, probably everything they imagine when they think of "real" numbers in practice, remain countably infinite. Exactly like the integers.
The set of constructible reals is the same size as the set of integers.
Brownian motion?
Similar to how we model pressure, temperature and volume relationships of gases, irrespective of the individual particles they represent.
But in principle, with enough computing power, quantum field equations can model the same phenomena at the particle level.
Basically all of calculus needs all the reals because if you don't you get some really pathological results.
If you do calculus over the constructible reals, almost all continuous functions will have infinitely many jumps (and these jumps will be infinitely dense too). This means that if a function is continuous, that doesn't mean the integral exists. The only way to avoid these pathological functions is to apply some restriction on how they would behave over the reals, but then you're back to square one.
This is how deceptive the name and early introduction is.
No, algebraic numbers, calculus, diff eq, topology [ , ... ] only need constructible numbers, structures, relationships and other forms and concepts.
Pi, e, infinitesimals, converging series, limits, smooth connected continuity, are all constructible. They all have finite symbolic expressions (with finitely describable algorithms for arbitrarily accurate decimal expansion).
There is never a situation in calculus where an un-constructible real does anything useful.
There is never a situation in calculus where the concept of un-constructible reals is necessary.
Bonus challenge: Name a single un-constructible real constant that you have ever encountered. What is the 0, -1, pi, i or e of un-constructible numbers?
Now think of the function f(x)=0 if x<r and 1 if x>r.
Edit: I originally said this is a function over the un-constructible reals. That was wrong!
This is a function over the constructible reals. What I mean with that is that it only has a value for constructible real numbers, because r isn't constructible, f doesn't have a value at r. That's why I haven't defined what f(r) is.
While this function exists (or something like it does, I can prove this fact if you like), this is not a construction because of course r is un-constructible
This function is continuous.
Let's prove this with the definition of continuity. We say that a function is continuous at a point x if lim a->x f(x) = f(x). We say that a function is continuous if it is continuous at all points.
The only point where this function could hypothetically be discontinuous is at the boundary point r, but actually this boundary point doesn't exist for this function (because f(r) isn't defined), so this function is continuous at every point, and is thus a continuous function.
Edit: forgot to add, but based on this concept you can then show that almost all continuous functions over the constructible reals have these jumps at infinitely many places, and that means that you can't hope to define a way to integrate continuous functions.
I appreciate that this is a pretty complex topic, so that I probably haven't been that clear (or made a mistake), so I value any and all comments.
This argument is somewhat adapted from an argument about why we do calculus with the reals and not the rationals. See also this video: https://www.youtube.com/watch?v=vV7ZuouUSfs
But it doesn't hold up. (He says with an overconfident flourish!)
So we have: f(x) = 0, over constructible reals
And we can prove it is continuous over constructible reals, because in that case,
The limit of f(x), as x —> c, trends to f(c) = 0
(1) Exist. This requires postulating some kind of infinite information oracle for each independent un-constructible number.
I have questions, but ... for now ok.
(2) That these un-constructible numbers are somehow "in" the constructible real line, even though they cannot be "on" the constructible real line, in a way that is coherent. Not all numbers are, i.e. imaginary numbers are not.
I have questions, but .. ok for now.
(3) That by defining f(r) = 1, for unconstructible r, we can create a case where f(x), as x —> r, does not trend to f(r)
I will concede this third postulate whole heartedly!
But that can't and doesn't invalidate our original constructible continuity proof.
We had to not only generalize "number", but redefine "f".
And I don't think it says much or anything about un-constructible numbers either.
For instance: Let's call regular numbers "blue reals", and define "red reals" such that for every blue real x, there is a red real red(x), that is exactly to the right (i.e. positive) of x. In such as way that they are ordered, but no number can come between them.
So (red(x)-x) = red(0), for all x, and there can never be a "blue-red" number smaller than red(0) (other than zero).
Then we can take our original "blue" proof, define f(red(1)) = 1, and declare we have broken continuity over blue-red reals.
So this breaking of continuity is a trivial trick, and it has no dependence on un-constructibility.
What we have really done, is define a new class of number and redefine "f" to get a motivated result.
We could just as easily have simply redefined f, to be the same but include f(1) = 1. If we get to redefine f, we get to redefine f.
So un-constructibility isn't needed to prove continuity (our original practical proof holds).
It can't "break" continuity either. Given redefining "f" was both a necessary and sufficient condition to do that. Nor does doing so shed any special light on unconstructibility or continuity.
Thoughts? :)
You make some very good points! I have to commend your mathematical reasoning.
Because I've seen some more formal math, I have some pretty good answers to your points. But I want to emphasize again that you bring up some excellent concerns.
> We could then postulate un-constructible numbers are something that:
> (1) Exist
Very unfortunately, there isn't really a rigorous way to define a function that tells you if a real number is constructible, you can get pretty close! However, we don't need something like this.
There are only countably many constructible numbers. This is because that for every constructible number, there is at least one finite description. However, (because of Cantor's diagonal argument) we know there infinitely more real numbers than constructible real numbers. So there must be a large amount of un-constructible real numbers.
> (2) That these un-constructible numbers are somehow "in" the constructible number line
There's a pretty rigorous way to assert this. Lets say that r is an un-constructible number. Like all real numbers, it has a decimal expansion. Lets say that it starts:
0.1637289458946...
Now I can compare constructible numbers and see if they are larger or smaller. Lets consider x = Pi-3 = 0.1415...
We'll look at it digit by digit. 0 = 0, so we don't know yet which one is larger. 1 = 1, so we still don't know. 4 < 6, so now we know that x=Pi-3 actually has to be smaller than r. This process finishes in finite time for any constructible number, because if there is no decimal place where they differ, they are the same number (which is impossible because x is constructible and r isn't).
Because I can compare the size of un-constructible numbers, the un-constructible numbers are "in between" the constructible numbers on the constructible number line. This is similar to how the irrational numbers (or if you prefer, the irrational constructible numbers) are "in between" the rational numbers on the rational number line.
> (3) That by defining f(1) = 1 (for instance)
I'm not sure exactly what your concern is, but I get the feeling that it will (hopefully, at least somewhat) be addressed by the next bit about "red" and "blue" numbers.
> Then I can take our original "blue" proof, define f(red(1)) = 1, and declare I have broken continuity, but again, I am just making up an arbitrary new class of numbers, and using it break continuity over the new class.
For a very subtle reason, this construction doesn't work. To see why, we'll have to take a look at the definition of the limit of a function.
When we say: lim a->x f(a) = L, we mean "if for every number ε > 0 , there exists a number δ > 0 such that whenever 0 < |a-x| < ε, we have that L - ε < a < L + ε.
(I'm happy to elaborate on this definition)
Now we can see why piecewise function I gave is continuous. let's consider x and f(x). (This is the f from my previous comment.) We know that x!=r, because x is constructible and r isn't), so we know that |x-r| is some positive real number. Now we can take δ=0.5|x-r| (or if you prefer, a constructible real number smaller than 0.5|x-r|).
Diagram of f:
y=1 ___________________________
y=0 _______________________
^ ^ ^
| | |
x x+δ r
This works because for any x, we can "zoom in" close enough to x that r is out of view and f is just a horizontal line. This is also the key difference between this and your color construction (at least what I think you mean).
My understanding is you define g (lets use a different function) as: g(x) = 0 if x < red(1) and 1 if x > red(1) Because red(1) and blue(1) are right next to each other, the function is discontinuous at g(blue(1)). Because for any ε, there exist some a>red(1), because blue(1+ε)>red(1) (by definition). So you see g(x)=0 and g(x)=1 no matter how much you zoom into the function at x=1. Your function is discontinuous, and continuity hasn't been broken!
Because the discontinuity occurs at an un-constructible r, the constructible limit "process" of x approaching r, never includes r. So f(x->r) encounters a limit of 0. And 1 from the other side.
That does leaves the validity of defining continuity of constructible functions over constructible reals intact.
It is only a problem if un-constructible numbers are "brought" in.
Previously you stated:
> Basically all of calculus needs all the reals because if you don't you get some really pathological results.
This is then what I don't understand.
Why would un-constructible reals be needed?
I would agree that all constructible reals are needed (after all, constructible functions as a class are by definition, sensitive to constructible reals as a class).
But why would un-constructible reals be necessary for anything?
At best I can see them as a kind of contrivance that shortens some proofs, that could be made without them with more careful reasoning.
But ... ?
--
> However, (because of Cantor's diagonal argument) we know there infinitely more real numbers than constructible real numbers.
Cantor uses sleight of hand out of the gate. From out of nowhere, he postulates that there can be infinite digit numbers, without finite description.
That is really interesting thing to suggest.
Because until then, numbers were built up in steps, from the concepts of increment, repetition, and orthogonal units (to post-formalize the actual progression of informal to formal).
Constructibility (in practice and theory) was inherent in what it meant to "have" a number.
A number is a relationship. A relationship isn't a relationship because I can pick a name "r" and say it is a relationship.
But suddenly Cantor talks about infinite digit numbers with no expression. No defined relation. And then an (attempted) list of all of them.
So constructibility of numbers is abandoned as a precondition for his arguments. Before any point of diagonal incompleteness is made.
It gets worse. Even if we had access to magic oracles that will generate any number of un-constructible reals we desire, with any number of digits supplied, so we can add, subtract and compare them: The process would remain indistinguishable from the same process in which we are actually being given constructible numbers.
My conviction (very open to being shot down of course!) is that un-constructible reals are an interesting concept, and interesting to reason about as mathematical puzzles, perhaps good exercises for inspiring new proof tactics, but in direct relation to anything we do with "actual" real numbers, they are unnecessary.
Also, any actual reasoning about infinite information structures and higher order infinities is going to itself be isomorphic to a finite (or countably infinite) system with 1-to-1 behaviors not interpreted as about un-constructible things. Because anything we do, even reasoning about the un-constructible, remains a constructible domain.
If I am wrong, I would be very interested to understand why.
What I think you're getting at is that even though my function f breaks what we think of as continuity of functions over the constructible reals, f is clearly un-constructible. So if we only do analysis using constructible functions, all is well.
I was thinking about how this works and trying to think of an example proof that doesn't work with only constructible reals, but actually the same proof basically works, so I'll just share that instead:
Intermediate value theorem: if f is continuous, f(a) is negative and f(b) is positive, then there is some c such that f(c)=0.
Proof for real functions: Define S = { x ∈ [a,b] : f(x) ≤ 0 }.
S is nonempty because a ∈ S (f(a) < 0).
S is bounded above by b, so S has a least upper bound c = sup S with c ∈ [a,b].
We claim f(c) = 0.
Suppose first that f(c) > 0. By continuity at c there is δ > 0 such that for all x with |x−c| < δ we have |f(x)−f(c)| < f(c). In particular for such x we get f(x) > 0 (since f(c) − |f(x)−f(c)| > 0). But then every x in (c−δ, c+δ)∩[a,b] is not in S, so there is no point of S greater than or equal to c−δ/2. That contradicts c being the least upper bound of S because then c−δ/2 would be an upper bound smaller than c. Hence f(c) ≤ 0.
Now suppose f(c) < 0. By continuity at c there is ε > 0 such that for all x with |x−c| < ε we have |f(x)−f(c)| < −f(c) (note −f(c) > 0). Then for such x we get f(x) < 0, so every x in (c, c+ε)∩[a,b] also satisfies f(x) ≤ 0 and hence belongs to S. But that gives points of S strictly greater than c, contradicting that c is an upper bound of S. Thus f(c) < 0 is impossible.
If we instead talk about constructible functions, note that f is constructible, so S is constructible, so c = sup S is constructible. We know that c is in the domain of f, and using the proof above we can show f(c)=0.
So maybe if we limit ourselves to constructible functions analysis works out. There are still two reasons why you might not want to do this. Adding a line at the end of every proof explaining why all the numbers you're talking about are constructible feels unnecessary when you can just talk about the reals. Secondly, (as far as I understand) its impossible to actually formalize our idea of constructible.
As far as not wanting to pedantically refer to "constructible reals", I agree that doesn't sound fun.
The better solution would be having a clear common pithy term for "unconstructible reals", for:
1. Teaching math related to numbers, until unconstructible reals or other structures had any relevance. I.e. most people never, ever.
2. For talking about algorithms, physics and other constructible structures, where the term reals is pervasivably used to mean constructible reals.
3. Most students get introduced to the fact that the cardinality of reals is greater than the cardinality of integers. But would be surprised, and get more use, out of knowing that the cardinality of instantiatable numbers (the ones they could define, calculate, measure or apply in virtually every situation but highly abstract math games), is EXACTLY the same as the integers.
Un-constructible structures are an interesting but exotic concept that shouldn't be riding around sereptitiously in common vocabulary.
A fair number of responses here involved confusion about what "reals" covers. And this is HN.
For one thing, they're speculative: the current theories that give the most precise and accurate predictions within their respective domains of applicability are general relativity and quantum field theory. These theories are based on continuous space and time, and no attempt to base them on discretized space and time has been successful (AFAIU both QFT and GR rely on Lorentz invariance, which means there's no absolute rest frame, hence no absolute unit of time and space, but a discretized spacetime would require absolute unit values for space and time, hence an absolute rest frame).
Should we conclude then that the reals are real, because they're components of our best current physical theories? Maybe, maybe not: these are features of our current best models, but we don't know, and possibly will never know, the ultimate nature of physical space and time.
For another thing, even if space and time are fundamentally discrete, there's still no doubt that the mathematical theories based on real numbers are effective in making predictions, and we would still like to use them. That's means they should have some logical foundation which can guarantee that reasoning using them is correct.
But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).
"It seems wrong to me to base mathematical ontology on speculative theories about the nature of physical space and time."
Given the theme and title of the article.
Given one of the primary (but often not emphasized) properties of the class of reals, is that it contains and is actually dominated by the un-constructible reals, an argument that the total class of reals isn't represented in nature seems unremarkable to me.
Un-constructible reals are a highly exotic abstract concept. You can never actually identify or operate on one.
I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality.
It is a mishap of terminology, that the larger class, which is unnecessary for middleschool math, or any math most people will ever encounter, has a pithy name "real".
While the practical concept, that directly correlates with arithmetic, algebra, calculus, diff eq, etc., has the unwieldily title of "unconstructible reals".
So in order to avoid having to talk about constructible vs. unconstructible numbers to kids who shouldn't need to care, we use the pithy term and actually throw unconstructible reals into the mix, where it wasn't necessary at all.
We tell kinds how the reals have a higher-order uncountable infinite cardinality, relative to integers. Which is true, but gives the impression that cardinality is somehow a concept necessarily linked to algebraic and other practical numbers that we introduce at the same time, which it is not.
The set of constructible reals has the same countably infinite cardinality as the integers.
I'll use this definition: a constructible real is a real that you can describe uniquely in a purely mathematical way (and prove that your description identifies a unique real). IE the positive solution to x^2=2 identifies a unique real. Also, the first positive solution to sin x.
Now if you accept that the universe is infinite (and has infinite matter in every direction), you could get representations of un-constructible reals in your universe. One pretty contrived way to get an un-constructible real from this infinite universe is this: start at earth with some velocity, lets say 0.01c in some direction. Start with r = "0." Every second, find the closest particle. Take the amount of meters (rounded down) that this particle is away from you and append it to r.
So if after one second the closest particle is 145m away, r becomes 0.145. If after another second the closest particle is 0.14m away, r becomes 0.1450
The value this process converges to could be un-constructible.
I would argue that is still a constructible real. Only practical issues make calculating that value difficult.
Since we are instances of physical constraints ourselves, just because we can't do a particular measurement, directly or indirectly, doesn't make a value un-constructible in the mathematical sense.
(Also side noting, that we handle superposition/quantum collapse explicitly, by actually generating many alternate counts, or an expression that covers all the counts.)
Note that your "algorithm" was finitely statable, and that its "data", consists of a finite number of particles (in any given superposition).
But if I were going to argue for an un-constructible number with a physical counterpart, your thought experiment is a good starting point!
Well if the universe is actually infinite, the amount of data in the number this process approaches is infinite.
> I would argue that is still a constructible real.
That is what I was going for. I was trying to think up a construction that leads to uncountably many reals, but the construction I gave doesn't really work.
Consider a different situation:
Start with r = 0. (a number in binary)
Look for an unstable radioactive isotope. Wait for its half time. If it decays within its half time, concatenate 1 to r. Else concatenate 0. Look for another radioactive isotope and repeat.
The number this process approaches could be real number between 0 and 1 (including both bounds). Is the resulting number constructible?
That's a good one.
I am going to say it absolutely is. Then acknowledge why others may feel very strongly that it isn't.
So that's quantum mechanics, which from a field theory standpoint is completely deterministic. It just appears non-deterministic to us, because we are also superpositions. We are quantum structures too. And our field would keep splitting in two, at each measurement/decision point, so our total quantum field would remain completely predictable.
But, it is true that each of our superpositions would have the experience of a completely random set of digits, going off to infinity.
But, despite it adding additional physics and not explaining any more, some physicists seem to still think that there is a real collapse, not just an already explainable experience of collapse, of quantum fields.
So, I think it is fair to say that if that was true, then truly unconstructible events would be happening. There would be no way to form an expression or algorithm to ever predict the flow of digits, even in principle.
So you nailed the best possibility for it that I can think of.
And this is a little circular, but between collapses adding a new phenomenon with no additional explanatory power (Occam's Razor be damned!), and the magic event decisions, are why I don't believe collapses happen.
Collapses don't just imply that a magical event decision is made whenever we set up some careful experiment with one particle, but that all possible event situations in space-time, even in us, are constantly being magically decided as we are exposed to information about them, all the time.
Given virtual particles are constantly frothing around even in empty space, this means that all of space-time is constantly flooding us with an unimaginable amount of magically created information. The magic bandwidth would be insane.
One magical fundamental physical constant seems implausible to me. But 10^(very very big number) of magical decisions animating all of our universe and us every pico-second? Well, that would just be ... unconstructible!
This interpretation of reality, at least how I stand it, seems like the correct one to me. (At least that's how it feels to me.)
But this combined with:
> I am going to say it absolutely is.
Means that all reals in [0, 1] are constructible, and as a result of that, all reals by modifying the starting value of r to ie 1. instead of 0., or 10. (2. in decimal).
Constructible reals are continuous over [0, 1] in that there are no gaps between any interval between constructible reals [r1 r2] that are not filled by more constructible reals (and in fact, the cardinality of the constructible reals within any interval is the same, i.e. countably infinite, in a fractal way).
So there is no obvious motivation for anything but constructible reals from that standpoint.
Unconstructible reals were invented (or at least used) by the mathematician Cantor to explore ideas about different infinities. A "real" number with infinite decimal digits but not any finite description lets him create a number space larger than the constructible reals, a larger infinite cardinality.
So there is nothing missing in a [0 1] constructible interval. Or to put it another way, constructible numbers are closed. There is no sqrt(-1) situation requiring unconstrucible numbers to fill, like the square root of -1 required imaginary numbers (or geometric algebra dimensions) to fill.
But [0 1] contains the higher infinity of unconstructible reals in it, if you want. But I am unaware of any claim that they solve any problems by being included, other than exploring interesting puzzles related to unconstructible numbers as interesting ideas in themselves.
Numbers and other mathematical concepts are used to describe and reason about physical systems. More or less everyone agrees on that much. Why make the further claim that some of these mathematical concepts or objects are "real"?
There isn't any one-to-one mapping between numerical concepts and physical systems. Even for a finite collection of physical objects, we could associate with it a number, which is the number of items in the collection, but we could also associate with it another number, which is the number of possible combinations of items of the collection. It depends on our interests and what we want to do. Even grouping items into particluar collections is dependent on the goals we might have in some situation. We might choose to group items differently, or just measure their total mass. More generally, any physical magni9tude can be associated with an arbitrary number, just by a choice of unit.
We use numbers, and mathematical concepts more generally, in many different ways to reason about physical systems and in our technical constructions. I don't see why we need any more than that, and to say that some mathematical concepts are "real" while some are not.
> I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality.
Can you explain what it would mean to "find a constructible real in nature"? Maybe we just have different ideas about how this would be spelt out
That there may be structures in nature that are 1-to-1 with any given (constructible) mathematical concept.
Anywhere there is conservation of quantity we get addition and subtraction. Anywhere quantity can be looked at from two directions we get reversibility, i.e. positive and negative perspectives of the quantity. Multiplication happens anywhere two scalar values operate on each other, or orthogonal quantities create a commutative space between each other.
We find reversibility, associativity, commutativity, and many more basic algebraic structures appearing with corresponding structures in physics. And more complex algebra where simpler structures interact.
Wherever there is a dependency between constraints applying, we have logical relationships.
So that is what I mean about mathematical structures appearing in nature. Numbers/quantities just being subset of those structures.
> Can you explain what it would mean to "find a constructible real in nature"? Maybe we just have different ideas about how this would be spelt out
My emphasis is really that we don't/won't find un-constructible reals.
I WEAKLY claim (given that reality increasingly looks likely to extend beyond our universe, and more conjecturally, may be infinite), that any given constructible math structure has a possibility of appearing somewhere. Perhaps all constructible math appears somewhere.
However, that is the weaker claim I would make.
I more STRONGLY claim that un-constructible mathematical structures are highly unlikely to have counterparts. Which includes un-constructible reals.
The un-constructible real invented by Cantor, was a value r, which has infinite decimal digits, but with no finite description. No algorithm to even generate.
It is an interesting concept, but an un-instantiatable (even in theory) one. Infinite information structures immediately present difficult problems just for abstract reasoning. How corresponding structures might exist and relate in a physical analogue isn't something anyone has even attempted, as far as I know.
--
I don't think I am saying something controversial.
If there is anything surprising in what I am saying, it is that I am addressing the fact that reals got defined in a way that includes un-constructible reals.
The only implication most people know about that, is that the cardinality of reals is greater than the cardinality of integers.
But what might be very surprising for many, is that the cardinality of constructible reals, every possible scalar number we could ever calculate, measure or apply, is in fact the same as the integers. A distinction/insight that seems highly relevant and useful when dealing with instantiatable math, physics and computation.
There may be, or there may not be. I don't think we can make a definitive argument either way without perfect knowledge of the structure of the physical world, or at least some part of it. What we have are mathematical models of physical systems that are valid to within some error margin within some range of parameters. The ultimate structure of the physical world is so far unknown, and may be forever unknown. Any actual physical situation is too complex for us to fully analyze, and we can only make our mathematical models work (to within some error) when we can simplify a physical system sufficiently.
I think I understand better though what your main point is: that whatever physical theories or models we might have, the unconstructible reals won't be an essential part of it, i.e. even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?
That might be true for all I know. I'm not sure how to evaluate it (IANA mathematical physicist). It seems plausible though. There's the example of synthetic differential geometry, which has a different conceptual basis to the standard formulation of differential geometry, and at least suggests that you can't a priori rule out the possibility of alternate formalizations of any mathematical structure. I don't know enough about it to say whether or not it postulates something equivalent to unconstructible reals, it's just something that came to mind as maybe being along the lines of your point of view
https://ncatlab.org/nlab/show/synthetic+differential+geometr...
Yes, that is a good way to put it.
Unconstructible reals are a major and interesting "what if". What if there were numbers that had no finite relation to other numbers?
It is a great idea, from a mathematical boundary pushing way. So abstract we can never do anything not abstract with it!
> even if we have some physical theory or model whose standard formulation might be committed to unconstructable reals, we could always reformulate it into a predictively equivalent model which doesn't have this commitment. Is that fair?
But unconstructible structures can never be reformulated as constructible by definition. That would mean they were constructible.
We can never define a specific unconstructible real.
But anyone who manages to create an interesting systems theory that uses them, with dynamics that constructible math can't match, would have created a major work of mathematical art!
God certainly had a fondness for the real subset. Measurements are real scalars -- so much so that it really does look like God created the reals. That's what's important to us. But the fundamental laws seem to require the complex numbers (or their equivalent, like matrices), and closure under arithmetic operations really does feel like it should be a requirement for the reality of the universe.
0.59999999999999999999999...
+ 0.00000000000000000000000...
------------------------------
0.?
I think you can make pathological cases for every arithmetic operation, so maybe (I'm not sure) none of the operations are computable. (Need to be careful with the definitions though, and I'm being pretty sloppy)
It doesn't mean that you can compute a complete representation in decimal (or any other positional numeral system) of the number using an algorithm which halts. This is of course impossible with computable numbers like pi or 1/3.
But you can compute the value of pi or 1/3 within error bound n, for any rational number n. Thus we say pi and 1/3 are both computable numbers. This isn't quite the same as saying you can always generate the first n digits of the decimal representation, because as you point out, any decimal digit can be sensitive to arbitrarily small changes in value.
But given these definitions, we can see that adding two computable numbers is indeed computable. In your example, the decimal representation of the output could begin with 0.5 or 0.6, depending on the precision you chose and the values of the two inputs at that chosen precision. Regardless, the output will be within the chosen precision.
Your example also comes close to illustrating that testing the equality of two computable numbers is not computable. There is no finite algorithm which can tell you if any two computable numbers are equal (or tell you which one is larger). Again, you can compute whether they are within any chosen bound of each other, but not whether they are equal.
If the 9’s repeat forever then there is no possible number between that and 0.6, so it must be the same…
Put another way, any single digit “n” repeating is just n/9.
Your problem is equivalent to (4/9 + 5/9 - 0.4) + 0/9
(Equivalent to 0.6)
which would be consistent with their interest in the question of the "divine" and human reasoning at all, especially as argued about by theologically inclined philosophers much admired by Judeochristians.
That subtext being, discovering that our models or knowledge are incomplete somehow increases the territorty of what he's calling mysterious. By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
It's a relief that such people have participated in construction of a society within which such beliefs are considered personal, as it saves a lot of embarassment for people such as myself, who find such notions wince-inducing, and, both their origins and utility quite transparent.
Author here. I'd describe myself as atheist/agnostic.
I just dislike the God of the gaps argument. I understand its utility in debating the dishonest moving goal post arguments of young earth creationists, but taken outside of that debate, I don't think it holds water.
> One of the great and persistent bemusements of my adulthood is discovering that other adults take their religiosity not just seriously but central to their understanding of themselves, and their context generally.
I take religion and religious questions seriously. If I took religion less seriously I'd be religious because I enjoy religion. We owe each other a certain level of honesty on truly serious matters even if it is uncomfortable.
> By which I take it he means, knowable to and to not beat around the burning bush, attributable to the divine. By which I take it for him that he means a Judeochristian god.
That is certainly how a 16th Century religious Italian would understand it and I find that an interesting perspective to contrast with my own somewhat blander late-modernist beliefs. One of the reasons I enjoy reading books from prior ages is seeing how much that was taken for granted as a universal truth of that culture has changed.
For example, in fault tolerant quantum computing, rotations are synthesized using sequences of 45 degree rotations around the X, Y, and Z axes. The matrices that describe those 45 degree rotations contain rational and irrational numbers (in particular: sqrt(2)). If those irrational numbers are actually truncated, this would have observable consequences. You'd need a sufficiently large quantum computer running for sufficiently long to do sufficiently accurate tomography of sequences of those rotations in order to resolve the truncation, and to be frank some of those "sufficientlies" would be very impractical to achieve especially if the truncation was small (and woe unto you if adding more qubits somehow reduces the amount of truncation!), but in principle it'd be possible.
> All abstract concepts brought about solely by humans.
That is a defendable philosophical position to hold but many thinkers disagree with it or at least admit there is more nuance [1]. People have been debating the question of the independent reality of mathematical concepts for thousands of years.
If abstract concepts are brought about solely by humans, does that mean the first human to invent a proof of some property actually decides the reality of that property? If Godel hadn't proved incompleteness, could another mathematician created a completeness proof instead? On the hand, if that isn't the case and the truth or falsity of a math statement exists prior a human thinking it, doesn't it have some reality beyond human thought? What about mathematical statements which are true which can't be proven?
[0]: Platonism in the Philosophy of Mathematics https://plato.stanford.edu/entries/platonism-mathematics/
[1]: Mathematics: Discovery or Invention? https://royalinstitutephilosophy.org/article/mathematics-dis...
That said I don't think there's much to be gained from dragging God into it all. Even He would probably have a headache changing pi to 3.2.