How real are real numbers? (2004)
arxiv.org
arxiv.org
I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adoption of new axioms (such as for example the axiom of choice) is justified by their resulting in new, interesting mathematics.
But here, it seems his goal is to arrive at a somewhat Platonic conclusion, that either the reals are valid numbers or (seemingly he prefers) not.
My lay and naive take would be: if you adopt these rules (this Formal Axiomatic System) then you can have Big Fun in the playground of ever more esoteric and complex infinite cardinals, or if you adopt this other FAS you get to discover which results can and can't be obtained under a strict constructivist regime, and whichever FAS you choose it's just the same process of choosing axioms and applying valid deductive steps to arrive at a result you find interesting, with no more "existence" implied than the thoroughly non-Platonic existence of a solution to a problem, which can be demonstrated by solving it: if you take such-and-such steps then such-and-such result will follow.
I suppose the point being made is that avoiding axioms which imply the "existence" of the reals is more useful for doing physics, but that seems non-obvious in a field which for the last 200 years has seemingly sought to progressively make more and more phenomena intelligible by means of differential equations from infinitesimal calculus!
0: see, for example: https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical
> Why should we believe in real numbers, if most of them, it turns out,[^15] are maximally unknowable like Ω? [^16]
The footnotes:
> [^15]: See the chapter entitled The Labyrinth of the Continuum in [Chaitin, 2005]
> [^16]: In spite of the fact that most individual real numbers will forever escape us, the notion of an arbitrary real has beautiful mathematical properties and is a concept that helps us to organize and understand the real world. Individual concepts in a theory do not need to have concrete meaning on their own; it is enough if the theory as a whole can be compared with the results of experiments.
---
The reference [Chaitin, 2005] in footnote 15 links to..
Meta Math! The Quest for Omega - http://arxiv.org/abs/math/0404335
> This book presents a personal account of the mathematics and metamathematics of the 20th century leading up to the discovery of the halting probability Omega. The emphasis is on history of ideas and philosophical implications.
Irreducible Complexity in Pure Mathematics - http://arxiv.org/abs/math/0411091
> By using ideas on complexity and randomness originally suggested by the mathematician-philosopher Gottfried Leibniz in 1686, the modern theory of algorithmic information is able to show that there can never be a "theory of everything" for all of mathematics.
¯\_(ツ)_/¯ For some reason I get the same vibe from this as people referring to LLM inference using gendered pronouns instead of "it".
Can this be true? DNA is the information system of living creatures and as far as I know, it is not coded with 0s and 1s. So, how can we justify that "the world is built entirely out of digital information"?
Every age has their technology which they will project onto the world. A century ago one may have started to talks about everything being electrical wires and switches.
All that matters is whether it facilitates reasoning towards the goal.
I view "real" numbers, in the context of uncomputable, unnameable numbers, to be as unfortunately named as "imaginary" numbers.
They're worse. Just having another dimension is significantly more relevant to reality.
And the reals also ruin the word "normal".
It's complicated in a completely different way from how the real world is complicated. I don't think that gives you useful intuition.
Likewise for the square root of 2 and for Euler's Number. These numbers are ever so real and yet they sure fucking look normal to me. If you assure me they are not normal, but you can't prove it, I shall not believe you.
My point isn't about digits, it's about precision. You can do math by hand with more precision than actually exists in the real world. And once you add any slack at all, even one part per googol, your numbers stop being normal and they can all be computed and represented in rational form.
This claim can't mean anything because it disappears up its own backside. If you, here in the real world, can "do math by hand" that math is in the real world.
But Pi shows up in other places and I'm not at all sure you can show there are no such places which distinguish some arbitrary approximation from the actual ratio.
Waves are pervasive and described by relationships involving those numbers. They show up in other relationships. With important properties such as conservation of energy that any partial precision wouldn't be able to achieve.
Numbers are not just evident by single value measurement, but even more powerfully when they govern a system, where any problem with the definition would result in an easily recognizable failure of an entire theory.
I think "precision" is the wrong way to look at what can mean something or not.
I think the boundary between numbers that "make sense", relative those that don't is better found by looking at the progression of numbers.
From naturals, to integers, to rationals, to algebraic (both non-rational roots, and roots of negatives), all the way to limits and series. (Note that the infinite computation associated with expanding digits is not a definition problem. Even 1/3 requires infinite digits in decimal, but the relationship between 1 and 3 is clear.)
What is true about all these numbers is not precision, but that they emerge from a finite number of relationships.
They can be written exactly, defined perfectly, with finite numbers of symbols. (Meaning, abstracting away notation, with a finite number of relationships.)
And all those types of numbers do show up exactly (for all appearances), in waves, and other relationships. The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.
So pi really exists. All kinds of physics would fail if it didn't. That doesn't mean we can make a perfect pi circle with plan length, since that would be an arbitrary test, and if the medium is discrete units, one chosen to a priori fail.
Contrast with: The uncomputable, undefinable numbers, which we can't define, can't measure, etc., and are introduced via shaky (relative to the general body of mathematics) means. They require infinite information to define exactly. Not just measure, but even to define. Which is a remarkable postulation, and is not needed to solve any problems they don't themselves introduce.
If your measurement of energy is 150 +/- 2, you only need a handful of digits to do calculations involving that value that preserve it just fine. Insisting that that billionth digit and more still match is no longer working with the real world.
> The relationships themselves make predictions more powerful than the practical precisions we might have in measuring single values.
You can make a prediction to infinite precision but it's not falsifiable. The infinite precision isn't real any more than aether theory is real.
Any powerful results that impact the real world don't need all that precision.
> if the medium is discrete units, one chosen to a priori fail
I choose discrete units because that's what the universe is, as far as we can measure.
If there's something more subtle than Planck, we can't measure it.
It's possible the universe does round at some point. We can't tell.
Can you describe any theoretical experiment that could tell the difference between perfect pi and thousand digit pi?
> Contrast with: The uncomputable, undefinable numbers,
I agree that there's a stark difference there. But I don't think computability is the specific point where it detaches from reality, it's just where the disconnect gets the most obvious.
You just repeated the misunderstanding.
Numbers like pi are not just magnitudes, but form critical relationships. And relationship tests offer (unimaginable) orders of magnitude more stringent testing.
"Weak" relationship test: The 3-body problem. There are stable modes, but even small discrepancies results in an unstable system falling apart. Accuracy rapidly compounds over observation or reconstructible time.
Strong example: If wave equations were not exact to pi, the discrepancy would be obvious in a nanosecond, much less thousands, millions or 14 billion years.
Pi isn't just a magnitude, it is a very special magnitude, where any offset completely destroys its properties. Properties that have held for billions of years of plank time intervals, themselves distributed over non-linear space time and all the other disturbances of the universe's complexities.
Try to come up with a non-pi number that does not radically alter quantum mechanics and chemistry. The maximum discrepancy you can come up with would be an unimaginable infinitesimal, shrinking faster and faster every Plank unit of time since the Big Bang. And also shrinking relative to the increasing volume, in Plank lengths, of observable space ever since the Big Bang.
There is no direct magnitude measurement that begins to compare with that.
It is impossible to create a circle made up of discrete lengths (Plank or not) in flat space, due to basic geometry. So using that as a test, when no theory predicts or depends on a "perfect" spacial circle, is a red herring. We already know it does not exist.
(If this does not make sense to you, point out the problem.)
Yes, relationships. But the only way to test relationships is to eventually get to measurements of magnitudes.
> And relationship tests offer (unimaginable) orders of magnitude more stringent testing.
How?
> "Weak" relationship test: The 3-body problem.
You can't prove the 3-body problem isn't rounding to the nearest planck unit. You can't measure it precisely enough.
> Strong example: If wave equations were not exact to pi, the discrepancy would be obvious in a nanosecond, much less thousands, millions or 14 billion years.
I don't think you're conceptualizing "a thousand digits" properly.
> where any offset completely destroys its properties
What's an experiment we could do that verifies pi doesn't have an offset of 1e-1000?
Also keep in mind that just the slightest bit of gravity or cosmic expansion has a much much bigger warping effect then 1e-1000 and yet physics keeps working the way we expect, and we can't tell the difference for small enough amounts of those things.
> The maximum discrepancy you can come up with would be an unimaginable infinitesimal, shrinking faster and faster every Plank unit of time since the Big Bang. And also shrinking relative to the increasing volume, in Plank lengths, of observable space ever since the Big Bang.
Why would it have to keep shrinking?
I think you're saying that for it to have no difference at all it would have to be that small. But my challenge is for an experiment that measures the difference. If some physical effect shifted over by 5 Planck units would you be able to tell? What if reality has just a tiny itty bit of jitter to it that ruins perfect numbers like Pi?
> It is impossible to create a circle made up of discrete lengths (Plank or not) in flat space, due to basic geometry. So using that as a test, when no theory predicts or depends on a "perfect" spacial circle, is a red herring. We already know it does not exist.
It's not a red herring when I'm suggesting that no test is even possible.
Can you come up with a test?
Quantum mechanics, the wave equation. Electron shells, photochemistry, general chemistry, just about everything if we are talking about pi, or e, or i.
1 part off in trillions ^ trillions would impact the fusion of stars, the rates of chemical reactions, require adjustments to basic laws, violate conservation of energy as we know it, ... really obvious impacts. As in: "we would not be here" impacts.
Cosmology has run experiments for us that ran billions of years.
Contrast: Not everything can be tested with virtually unlimited precision, but basic mathematical constants in physics often can be. The gravitational constant is not testable like that. We don't have a mathematic derivation that we can leverage to test for violations like we do with pi, e, i, and other basic mathematical relationships that show up in physics.
But often, even a tiny difference becomes obvious. We exist because the production of matter and anti-matter at the beginning of the universe was off by a tiny amount. Despite the small discrepancy, that there was a discrepancy is very clear. Another "we would not be here" test.
Pointing at entire fields is not helpful. Can you give me one specific measurement and an estimate of how far off it would be?
> really obvious impacts. As in: "we would not be here" impacts.
That sounds pretty nonsense to me. The range of possible values for life isn't that narrow. And relativity is already in there ruining any straightforward conservation of energy and mass by constantly shifting the weight of things as their state changes. But it still works just fine! And we don't know exactly how strong that effect is, which could hide all sorts of imprecision in the real world. It would not be obvious.
> Not everything can be tested with virtually unlimited precision, but basic mathematical constants in physics often can be.
I'm begging you, name a specific test. One that could tease out 1e-1000.
> But often, even a tiny difference becomes obvious. We exist because the production of matter and anti-matter at the beginning of the universe was off by a tiny amount. Despite the small discrepancy, that there was a discrepancy is very clear. Another "we would not be here" test.
And if the matter-antimatter imbalance was 1e-1000 it would be imperceptible. It would be less than one atom in the entire visible universe, by an unimaginable factor. It was somewhere around 1e-9, probably, sort of. Not that small at all.
You are dismissing my point, then asking me to make it.
You can think of tests/measurements of values as falling into different classes. The strongest tests of all are for the critical values of systems. Because any discrepancy would result in entirely different system behaviors.
Single highly accurate measurements are much lower on the rung. Complementing those are many tests with known statistical inaccuracy. Etc.
Single tests? Every experiment involving quantum mechanics tests pi's role in those equations to a much lesser extent. Similarly, any test involving gravity tests the gravitational constant. But we have much stronger tests for pi in quantum mechanics that we do for g in gravitation.
There isn't just one kind of measurement/test, there are many. And we want the strongest test we can make in any given situation.
But of course, we can always perform weaker tests.
Validating pi in quantum mechanics can be done with extreme robustness, because the entire theory depends on that value critically. Even the tiniest discrepancy would result in different physics compounding over all Plank space and time units, over billions of years and universe expansion, and we wouldn't be here.
Of course, we can't rule out any discrepancy. But in this case, we can rule out discrepancies down to unimaginable infinitesimals. I doubt anyone even knows how to characterize how much of a discrepancy from pi would still be consistent with what we know. That tiny.
The criticality is what gives us this far stronger test. Non-critical values cannot be tested this way. Pi can. Particular tiny ranges of stable constants in a stable 3-body system can (to a lesser extent, given the smaller system and higher bounds on criticality).
Another way to view the systemic criticality of pi, is to recognize that pi is not just a representation for a particular magnitude, but a representation of conserved cyclic behavior. Any deviation from pi breaks cyclic behavior. Thus, the implications of pi in a theory, and our ability to test pi, are profoundly greater than for most other constants. Because the difference between cyclic vs. non-cyclic behaviors, is profound. Not just slightly different behavior, but entirely different behavior.
And if you had two separate copies of the universe, you could measure this compounding.
But we only have one universe. How do we know which one we're in?
> Any deviation from pi breaks cyclic behavior. Thus, the implications of pi in a theory, and our ability to test pi, are profoundly greater than for most other constants. Because the difference between cyclic vs. non-cyclic behaviors, is profound. Not just slightly different behavior, but entirely different behavior.
Or it just knocks off the frequency by an absurdly small amount.
But even if it did ruin cyclic behavior, how long are your cycles? The universe is only 1e61 planck times old. A discrepancy of 1e-100 would have no effect yet, let alone 1e-1000.
> And if you had two separate copies of the universe, you could measure this compounding.
> But even if it did ruin cyclic behavior, how long are your cycles? The universe is only 1e61 planck times old. A discrepancy of 1e-100 would have no effect yet, let alone 1e-1000.
When this is true:
When there are no threshold conditions, then discrepancies accumulate just as you have described. And for any given t, a small enough discrepancy can be chosen so that it does not impact measurements of a given accuracy.
When it is not true:
For any system with thresholding conditions, any discrepancy can have profound immediate effects. A collision can result in a particle heading off in an entirely different direction after a collision, cascading into an entirely different system state.
How critical constraints change structure:
When pi, i and e are used structurally, i.e. "pi" represents traversal of a cycle, "i" a quarter turn traversal, "e" a positive feedback traversal, each of them represent something invariant: for "pi" some sum of two squared units is conserved (the squared radius on two units), for "i" a position may be conserved while an orientation rotates, for "e" some feedback value maintains an invariant relation between its position, its rate of change, and its accumulation. These relations define the system itself, not just some proportions.
So for those cases, where constants define structure, any change to those constants changes the structure. Suddenly, there are differences where they did not exist before, non-unity proportions where they did not exist before. The system has new state values, new interactions. The system itself has changed, not just proportions. The system likely has more states.
Structural change is threshold change:
Changing the system's structure is the most significant threshold-type change one might imagine. The entire system is different starting at time = zero.
Can a structural constant be changed in a way that leaves it only proportionally changed? So that discrepancies simply accumulate, until they are measurable? Sometimes, yes.
But will that hold in general? No. In general, the difference between interactions that exist, vs. interactions that do not exist, states that exist vs. states that do not, includes systems that can behave qualitatively differently from the very first time step.
Does that make sense? (I rewrote this several times!)
Sometimes numbers define structure. They define what interacts with what. And what does not interact. Not just proportions.
Changing structure is a threshold-type change: 0 to something, equal to unequal. A state that didn't exist, to one that exists. No interaction, to interaction. That might result in a system that simply accumulates discrepancy. But it may also result in entirely different behavior from the first step onward.
I understand that there might be a threshold in some cases. But given how much of the universe is already effectively random when you get down into the itty bitty details, and that you can only make an experiment so big to tame the statistics, there could easily be untold rough edges that are invisible to us. Some particle went a different way than it "would have", but you can't measure "would have", it still looks like a normal interaction with a normal distribution of outcomes.
Even if you had access to a parallel universe and you knew either you or them had 'correct' physics and the other didn't, I don't see how you'd be able to figure out which is which.
I'm sorry, what did you mean by "how complicated the world we inhabit is" because I thought you were talking about physical interactions of matter and energy.
If you're including all math as "real world", then I think the claim that math teaches you something useful about the complexity of the real world is what actually disappears up its own backside.
There are no numbers in the real world, that require and infinitely long definition.
1/3 has infinitely many digits in decimal, but has a finite definition. So its good.
I would write down the opposite, a definition of an uncomputable, unnameable real, but there are not enough atoms in the universe (multi-verse, ...) to being to do that.
The mean value theorem isn't true for the computable reals, differentiation of computable function isn't always computable, sequences tend to behave poorly, etc. There's still a lot you can say: https://en.wikipedia.org/wiki/Computable_analysis but in general calculus doesn't care about computability, that's a human problem.
As in, not even computable in theory. It is isn't about normal "computable" concerns. It is "proven to never be characterizable".
Which is a class of numbers whose "existence", if that can term can even be applied coherently for undefinable things, is contested, in theory. In practice they certainly do not exist.
1/3 has infinite decimal digits, but is definable with a finite number of symbols, so it is a computable real. Even if we had no algorithm yet to compute those digits.
Try and define a specific number, that requires infinite symbols to define. As far as I am aware of, no part of calculus involves specific values that have no finite definition, except when the need for uncomputable/undefineable reals are asserted on a circular basis (i.e. they are needed to resolve problems with assumptions that already assume them.)
(Note that a number defined by interpreting the infinite digits of pi as mathematical relations, would still be considered a definable number, assuming some form of convergence could be proven. Because pi is finitely defined.)
If any are a "randomly" chosen real number, the answer would almost certainly be no. But a test of sufficient precision would of course be impossible.
This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.
You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.
dhosek is saying that in a quantized space, there are reals that cannot be demonstrated this way, and he is right, but the same thing is untrue of rationals.
("In a quantized space", by the way, just means that all measured quantities are necessarily integers. That causes all kinds of problems, but "lacking examples of arbitrary rational numbers" isn't one of them.)
You seem to have your units confused. Half an inch is a distance, while ratios, or comparisons of ratios, are all dimensionless scalars.
Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which is why it was absent from Western mathematics for so long). Constructing arbitrary ratios is possible through similar triangles and square roots through right triangles, but some basic algebraic numbers like cube roots cannot be constructed with a ruler and straight edge (which raises the question of whether, in a quantized universe, whether irrational cube roots actually exist). Of course there’s no guarantee that the quantization is going to be uniform and we also have the ɣ factor of special relativity (1/sqrt(1-v²/c²)) which gives us a non-Euclidean space to complicate things, but it’s not clear that if you can find a value for 1 that allows you to get a measurement for every irrational number.
It's a pretty linear increase in complexity between the math and the apples.
And I'm just explaining how to use the same methods as the comment I replied to.
But don't ask me about that. That part wasn't my idea at all. Ask dhosek about that way to do fractions. My contribution was the square root and the subtraction.
Any rational number has some meaning if you just add the right unit, even if the units become increasingly ridiculous. But for reals that trick does not work
But it would get complicated, for any given allowable velocity and allowable length, you’d get more lengths from Lorentzian contraction.
There are really a couple of different ideas being combined: are there an infinite number of quantum states for the universe, are space and time continuous, is the forward direction of time resolved by computable processes.
And even bigger ones lurk: are space and time emergent properties from quantum waveforms that lack an inherent idea of space and time (but things that are highly correlated give rise to a notion of being near each other in “space time”)?
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=24029791 - Aug 2020 (112 comments)
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=14080024 - April 2017 (265 comments)
One of his mantras is Time is real; Real numbers are not. He conceives of real numbers resulting from processes (approximations, relaxations, computable calculations) that unfold over time. So there is a Heisenbergish uncertainty principle of observable precision and elapsed time.
Selected papers:
Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?
https://arxiv.org/pdf/1803.06824
Real Numbers are the Hidden Variables of Classical Mechanics
https://philarchive.org/rec/GISRNA
Time Really Passes, Science Can’t Deny That
https://arxiv.org/pdf/1602.01497v1
Popular articles:
Real numbers don’t cut it in the real world, this physicist argues
https://www.sciencenews.org/article/real-numbers-physics-fre...
But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.
In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. And any useful number can be approximated arbitrarily closely by rationals.
And for computable numbers there's even less of a gap. With rationals you can only approximate. But you can have a computable number that is exactly equal to the square root of 2, because a computable number is the algorithm by which you form arbitrarily close approximations. The square of that computable number is itself computable and is exactly equal to 2.
What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
And if you're worried that sticking to the rationals and the computable numbers is too much of a concession to "physical reality", rest assured -- the rationals are just as unphysical as the real numbers because they are continuous already, and physics does not give us the power to measure the difference between two sufficiently precise rational numbers just as it barfs when you throw "real" numbers at it.
It buys you the rigor of doing calculus, which buys you a lot of results that, while could be computed without calculus, would also be very difficult without it.
Doing calculus with computable numbers is totally possible and you get all the continuity you need. You need to drop the Lebesgue formulation of the integeral and either use a Reimann integral or the gauge integral (Henstock–Kurzweil) if you need a well-behaved integral in the face of very poorly-behaved functions, but in physical reality these don't exist and in abstract mathematics they are rarely of interest and the gauge integral is as robust as Lebesgue without all the measure theory nonsense.
Intuitionalist analysis and calculus are very well established; the only thing you can't do with them is nonsense like showing that integrating over the characteristic function of the rationals is zero (who cares) or showing that you can break a three dimensional sphere up into three pieces are reassemble them after translations and rotations into a larger sphere (obviously not true).
But can you do stats without measure theory? Normal distribution in the limit and all that?
There is no linearly additive measure on rationals, and therefore no way to grab a rational uniformly from (0,1). Has to be skewed to some level of complexity in the denominator.
Approximate relative to what? All actual measurement is implicitly or explicitly approximate such as L = x meters +/- epsilon. There is no infinite precision by which to discount rational measures as "approximate" and thus "invalid" in any way.
>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? Basically nothing.
You "buy" all of mathematics which operates on the assumption of "infinite" precision. It is an abstraction necessary to prove theorems and relationships of math. Abstracting from precision isn't a denial that it exists, it is the assumption that I can ignore it or leave it undefined. This is the assumption that distinguishes math from physics/engineering. Mathematicians deal with abstract e or pi but in the real world pi=3.14 if you are tiling your patio and 3.14159265... or whatever is necessary to get to the moon.
pi and e and sqrt(2) are real numbers and not rational, to be sure. But they are computable! Computable just means that they are arbitrarily approximable. "approximate relative to what" is that whatever criteria defines the number. You can't represent the "true" value of a non-rational number in the rationals, but you can prove that the error of an approximation is (rationally) bounded above and below, and you can have another approximation with a tighter bound.
Rational numbers are already infinitely precise relative to other representations -- finite decimals are another representation that is functionally equivalent to the rationals, but even a simple rational like 1/3 does not have a finite decimal value.
You can prove all the interesting theorems with computable numbers and rational/decimal numbers. You don't need the real numbers because you can't name a real number that exists and is not computable, BY DEFINITION! No mathematical construction can define a real number that is not constructible. These numbers are useless and there's no reason to continue even in abstract mathematics to pretend that they are useful because we have the formalisms to ignore them.
However, I thought things like Chaitin's Constants (you could make one per programming language) are real numbers you can name but not compute. I think you could do this from any undecidable problem.
Of course there only a countable number of those Reals. And they still don't seem useful for much more than naval gazing.
> But when we say things like "the rationals are discrete"
In the usual topology they are not?
> In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers.
This characterization captures neither the intuitive nor the formal definition of continuity. You are effectively saying that Q is dense in R, but this is insufficient to prove, for example, the intermediate value theorem.
> measure theory, a theory which yields almost nothing of value except endless paradoxes
Come on now. The usual definition of concepts as basic as areas is tethered to measure theory. We say it's "obvious" that the integral is the area under the curve (and it is: e.g. the Riemann integral is trivially the Peano-Jordan measure) but this only works because we're appealing to it. You can route around it, but let's not pretend we're doing it for no reason.
I can see the elegance of a purely intuitionistic construction, but the "usual" real numbers are much closer to how we intuitively (no pun intended) work with numbers.
The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techniques (Dedekind cuts or Cauchy sequences) effectively only yield the computable numbers, the real numbers outside of the computables are inherited from the diagonal argument rather than being foundational to the construction. I mean, this is trivially true because constructions are constructive.
I disagree that area is tethered to measure theory; I certainly learned about areas in geometry long before I ever heard of anything with measure theory. Measure theory exists to tie up some of the horrifying poorly behaved functions that increasingly wily mathematicians invented to break our notions of area and continuity. But we have better tools now for dealing with those that don't involve measure theory so there's no reason to ever hear the phrase "almost everywhere" or "subadditive" ever again.
To back it up to your closing and my main point -- the constructive numbers are way closer to the way we work with numbers because all numbers we ever deal with, even abstractly, fit this definition much better.
Again, I don't think your position is indefensible, but this doesn't strike me as particularly convincing. The usual definition of R is that there exists a unique ordered complete Archimedean field up to isomorphism. We get the kitchen sink from the least upper bound property. As a constructivist you're gonna say that I don't get to define R like that, but you can't pretend it's done for no reason or that it buys nothing.
> I certainly learned about areas in geometry
And how were they defined? In elementary geometry we just sweep the question under the rug, usually...
If you get to say that being able to articulate why the measure of Q is 0 is unimportant and uninteresting, then I get to claim that the supposed problems with the usual definitions are also unimportant!
Saying that the non-constructive world leads to worse problems is a respectable position. Pretending the usual way of doing things is completely arbitrary isn't very honest.
They're well-known and have a simpler implementation, and we are familiar with their quirks. There is a giant body of useful knowledge built up around standard real analysis. That doesn't really exist if you insist on using only computable numbers.
The computables are also more fiddly in many ways. Because equality is undecidable, you can't have discontinuous functions, you need to carry around error epsilons all over the place, and we lose useful tools like the Heine-Borel theorem, I think.
Try proving some results in PDE theory, and I think you might change your mind.
In general, I find clarity in thinking of numbers as the system that implements them, rather than as platonic objects with individual reality. What does using Old Boring tech buy you over using Shiny New Thing?
equality is always undecidable until you see the light of intuition. consider the rational number whose numerator is 0 if $theorem is true, and 1 if it is false, and whose denominator is 1.
But there are numbers in constructivism for which it is unknown whether they are zero. Some of which must remain unknown, if mathematics is consistent. This is a rather important and weird edge case.
We define computable numbers to be Turing machines, lambda reduction processes, or whatever your favorite model of computation happens to be. If you don't like this kind of definition, then we need to talk philosophy of computation.
To decide equality, we let your machines clunk along until they both produce a result, which we then compare (using another machine). Hello Mr. Halting Problem. Specific programs are fine, but comparing against arbitrary classes of program is the bugger. This is why discontinuous functions cannot exist in a hardline computable analysis theory.
I guess the naive answer is completeness. Every Cauchy sequence converges to a member of the space. For example, quantum mechanics relies on the formalism of Hilbert space, defined as a complete inner product space. This gives us nice things like the spectral theorem for unbounded operators, without which we wouldn't be able to define probability (the Born rule) or time evolution (the operator exponential e^-iHt).
Can you formalize quantum mechanics using computable numbers? I don't actually know, but let's say yes. I assume it's more work with more edge cases, so I would ask the same question: what do you get for the trouble of building a formalism around computable numbers?
What you get for the formalism around computable numbers is this. Every mathematical object in the theory is something that can be, at least in principle, actually written down. When we say that it exists, this existence is of the most tangible form that any mathematical thing could have.
I don't see the benefit of being able to write something down "in principle." A number can only ever be computed to a finite number of digits in practice. If we're talking about finite approximations, then the standard approach using numerical solutions to the Schrödinger equation handles this just fine, no alternative mathematics needed. If we're talking about theories, then we should choose whatever abstraction is most convenient for expressing the theory.
Personally, I don't believe numbers "exist." The physical universe exists, and numbers are abstractions that we invent to describe it. In that sense, uncomputable numbers are just as "real" as computable ones.
But you get there by a path of increasing fuzziness, and no clear boundaries, from numbers that we can both write down, and work with.
Then there is a jump to numbers that cannot be written down. Not even in principle.
Some people feel that that jump matters. Others don't. I feel it matters. But I accept that most mathematicians, don't.
They're a powerful abstraction - the base concept of a smooth continuous complete domain which encodes non-trivial relationships, and is a prototype for other analytic abstractions.
The reals are the philosophical base class for some very useful mathematical objects. Computability and physicality are both side issues.
Measure theory is used for lots of practical things, for example probability theory.
You may enjoy a recent update on that story [0] that maybe avoids a few paradoxes and looks at things other than navels.
No numbers are real. They are all an entirely abstract construction, like lines and planes and open sets and closed balls and metric spaces and everything else.
If I construct a number by saying it’s the limit of the sequence sqrt(2)-1/n as n->infinity that’s just as real as the number 1 or 1/2.
It just seems really arbitrary to privilege one kind of construction over another. We want a complete ordered field, so we built the reals. Saying they don’t exist or they aren’t real or whatever seems just to be completely beside the point. They are real enough to do the thing we want them to do.
They are just as real as anything else in maths.
Does it exist if it is impossible to show an example? Only if "exist" is interpreted to mean "you cannot deduce a contradiction from assuming them", which is a logically consistent position. But if you mean "they are in some sense actually there", in some describable way, then it's muddier. And you can build a logically consistent position from declaring "they are not there" as well. Precisely because no counterexample can be produced.
And then there's ultrafinitists, and yeah, they are a bit bonkers.
More importantly, a function is its graph so if I want my functions to be continuous I need them to be there.
:-)
The historical context about the constructivist movement is that it was a religiously-inspired objection to the work of Cantor, who some random bishop said was challenging God with his work on transfinite numbers because God owned infinity. I just find it weird that now people try to pretend that it's somehow more rigorous when really it's just an alternative axiomatic perspective that started in this shonky way and has grown to a point where it's just about respectable.
The problem to me is that the Reals which aren't computable are absurd. We've never used any of them in all history. We can only put names on a very special few, those are countable, and we don't even know their value very well.
And the main way we prove that the Reals are uncountable is to use a proof by contradiction. It would take too long to spell it out, but they aren't really just contradicting "Reals are Countable". It's "All that other stuff we think is true AND Reals are Countable" that gets contradicted.
Once you accept the Reals, the Axiom of Choice is not simply obvious any more. And if you go down that path you get things like the Banach-Tarski paradox. To me THAT ought to be a proof by contradiction that we've made a mistake somewhere.
More interesting than that though: If the universe we live in requires non-computable Reals to describe it carefully, then it says something very weird about determinism. In order to compute a future state of a system, we need to use numbers we can't compute?!?
You just used them yourself in your previous post to make your argument that computable numbers are dense in incomputable numbers.[1] So presumably that makes you the first person in all of maths history to use them. Congratulations I guess? The other possibility is they get used a lot and we just don’t make a fuss about it because most of the time it’s exactly like you used them - to express an argument where it doesn’t matter whether they are computable or not.
I have no idea why you think they are absurd but as I say it’s an alternative axiomatic perspective. It just makes a huge amount of maths very inconvenient without really yielding (as far as I can see) much of anything.
I don’t have a perspective on your questions about the universe. Non-computable numbers don’t really trouble my world view. We invented the calculus and the language of continuous functions which gave rise to the rigorous construction of the real numbers precisely to better describe the universe, so it seems that they are pretty useful in that context.
[1] Which is interesting right, because this is a trivial argument in the sense that in standard first year analysis you learn that the rationals are dense in the irrationals and rationals are obviously computable so it must be that rationals are dense in the non-computable numbers. But this is a very basic argument that we can only make because we permit the Reals to be complete. It’s not possible to express the argument in the restriction to computable numbers because there is no ambient space outside the computable numbers for them to be dense into.
Btw that's not true. Cantor's diagonal argument isn't a proof by contradiction, it's a purely constructive argument. The way he made it in his original paper is a bit more technical than this but this is the "modern" version that's a bit easier to put into layman's terms.
Say you say you can construct a (countably finite) list containing all the real numbers. I say I don't care how you made your list I can give you a number that's not on it, and in fact cut your list down to just numbers between 0 and 1. If you have all the real numbers you must have all the numbers between 0 and 1, but even if you make a countably infinite list of numbers between 0 and 1 I'll give you a procedure that will construct a number that's not on your list no matter how you made it.
1) Read the first number on your list. If it has 1 in the first decimal place, make the first decimal place of my number a two otherwise make it a 1. 2) Read the second number on your list. If it has 1 in the second decimal place, make the second decimal place of my number a two otherwise make it a 1. ....
Proceed in that manner.
At the n-th step I read the n-th number on your list. If it has 1 in the n-th decimal place make the n-th decimal place of my number a 2 otherwise make it a 1.
Now: My number is clearly nowhere on your list as it differs at in least one decimal place from every number on your list.
Therefore it is not possible to construct a countably infinite list of real numbers.
Here's a great discussion on Curt Jaimungal's podcast:
https://www.youtube.com/watch?v=l7LvgvunVCM
And a good debate on the topic with Daniel Rubin, who takes the more orthodox position:
https://www.youtube.com/watch?v=edh5bbgSKqo
Wildberger has tons more on this topic on his own channel. His arguments are thought-provoking, even if you don't agree with them.
Nicolas Gisin: Time, Superdeterminism, & Quantum Gravity
Interesting, I think everyone else calls this Kolmogorov complexity.
Or sometimes as real as you can fathom :)
Since Lean has become more popular as a proving system I've stumbled upon one very annoying feature of reals: they are not computably comparable. The system says you can never know whether two arbitrary real numbers are the same because you don't have enough time to compare them.