God Created the Real Numbers
Original: God created the real numbers
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The debate rages on: did God create the real numbers, or are they a human construct? As commenters dive into Ethan Heilman's provocative article, constructivists and skeptics of Cantor's infinite sets duke it out, with some arguing that the real numbers are "unphysical" yet remarkably useful. While some, like andrewla, agree with the article's thesis, others, such as nh23423fefe, dismiss key concepts like bijections between infinite sets as "stupid" and "not useful." The discussion reveals a surprising consensus that our minds can, in fact, work with abstract concepts like real numbers, even if they're unnameable, as SabrinaJewson and shkkmo passionately argue opposing views on this very point.
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Nature and the universe is all about continuous quantities; integral quantities and whole numbers represent an abstraction. At a micro level this is less true -- elementary particles specifically are a (mostly) discrete phenomenon, but representing the state even of a very simple system involves continuous quantities.
But the Cantor vision of the real numbers is just wrong and completely unphysical. The idea of arbitrary precision is intrinsically broken in physical reality. Instead I am off the opinion that computation is the relevant process in the physical universe, so approximations to continuous quantities are where the "Eternal Nature" line lies, and the abstraction of the continuum is just that -- an abstraction of the idea of having perfect knowledge of the state of anything in the universe.
They're unphysical, and yet the very physical human mind can work with them just fine. They're a perfectly logical construction from perfectly reasonable axioms. There are lots of objects in math which aren't physically realizable. Plato would have said that those sorts of objects are more real than anything which actually exists in "reality".
On the one hand, this article is talking about the hierarchy of "physicality" of various mathematical concepts, and they put Cantor's real numbers at the floor. I disagree with that specifically; two quantities are interestingly "unequal" only at the precision where an underlying process can distinguish them. Turing tells us that any underlying process must represent a computation, and that the power of computation is a law of the underlying reality of the universe (this is my view of the Universal Church-Turing Thesis, not necessarily the generally accepted variant).
The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no. It leads to rabbit holes that are just uninteresting; trying to distinguish inifinities (continuum hypothesis) and leading us to counterintuitive and useless results. Fun to play with, like writing programs that can invoke a HaltingFunction oracle, but does not tell us anything that we can map back to reality. For example, the idea that there are the same number of integers as even integers is a stupid one that in the end does not lead anywhere useful.
it leads to the idea that measuring 2 sets via a bijection is a better idea than measuring via containment
I am not sure what you are arguing here. We’ve been teaching this to all undergraduate mathematicians for the last century; are you trying to make the point that this part of the curriculum is unnecessary, or that mathematics has not contributed to the wellbeing of society in the last hundred years? Both of these seem like rather difficult positions to defend.
Otherwise it's pretty much a dead end unless you're in the weeds. You just mutter "almost everywhere" as a caveat once in a while and move on with your life. Nobody really cares about the immensely large group of numbers that by definition we cannot calculate or define or name except to kowtow to what is in retrospect a pretty bad theoretical underpinning for formal analysis.
I didn't mean to suggest that the reals are the floor of reality, rather that they are more floorlike than the integers.
> The other question is whether Cantor's conception of infinity is a useful one in mathematics. Here I think the answer is no.
Tools are created by transforming nature into something useful to humans. Is Cantor's conception of infinity more natural? I can't really say, but the uselessness and confusion seems more like nature than technology.
Well, there are the same number. So, uh, sorry?
Can it? We can only work with things we can name and the real numbers we can name are an infinitesimal fraction of the real numbers. (The nameable reals and sets of reals have the same cardinality as integers while the rest are a higher cardinality.)
In fact, if you are to argue that we cannot know a “raw” real number, I would point out that we can’t know a natural number either! Take 2: you can picture two apples, you can imagine second place, you can visualize its decimal representation in Arabic numerals, you can tell me all its arithmetical properties, you can write down its construction as a set in ZFC set theory… but can you really know the number – not a representation of the number, not its properties, but the number itself? Of course not: mathematical objects are their properties and nothing more. It doesn’t even make sense to consider the idea of a “raw” object.
Personally, I’d go with the sideline cut definition.
You might say, I can imagine 2 apples, but I can't imagine pi apples, but you could just as easily imagine unrolling a circle with a diameter of 1, and you have visualized "pi" just as well as you can visualize 2 apples.
Nah, you're likely thinking of the rationals, which are basically just two integers in a halloween costume. Ooh a third, big deal. The overwhelming majority of the reals are completely batshit and you're not working with them "just fine" except in some very hand wavy sense.
the first 2 naturals form an integer.
that integer and a 3rd natural constitute a real (but this 3rd natural best be bigger than zero, else we're in trouble)
what I choose to focus after observing the "unphysical" nature of numbers. is the sense of natural opposition (bordering on alternation) between "mathematical true" and "physical true". both are claiming to be really real Reality.
in the mathematical realm, finite things are "impossible", they become "zero", negible in the presence of infinities. it's impossible for the primes to be finite (by contradiction). it's impossible for things (numbers or functions of mathematical objects) to be finite.
but in the physical reality, it's the "infinite things" which become impossible.
the "decimal point" (i.e. scientific notation i.e. positional numeral systems) is truly THE wonder of the world. for some reason I want something better than such a system... so I'm still learning about categories
A skeptic in what way? He said a lot.
Math is math, if you start with ZFC axioms you get uncountable infinites.
Maybe you don't start with those axioms. But that has nothing to do with truth, it's just a different mathematical setting.
So yes, generally not starting with ZFC.
I can't speak to "truth" in that sense. The skepticism here is skepticism of the utility of the ideas stemming from Cantor's Paradise. It ends up in a very naval-gazing place where you prove obviously false things (like Banach-Tarski) from the axioms but have no way to map these wildly non-constructive ideas back into the real world. Or where you construct a version of the reals where the reals that we can produce via any computation is a set of measure 0 in the reals.
Well you can be skeptical of anything and everything, and I would argue should be.
Addressing your issue directly, the Axiom of Choice is actively debated: https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_...
I understand the construction and the argument, but personally I find the argument of diagonalization should be criticized for using finities to prove statements about infinities.
You must first accept that an infinity can have any enumeration before proving its enumerations lack the specified enumeration you have constructed.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
> Math is math, if you start with ZFC axioms
This always bothers me. "Math is math" speaks little to the "truth" of a statement. Math is less objective as much as it rigorously defines its subjectivities.
https://news.ycombinator.com/item?id=44739315
The axiom of choice is not required to prove Cantor’s theorem, that any set has strictly smaller cardinality than its powerset.
Actually, I can recount the proof here: Suppose there is an injection f: Powerset(A) ↪ A from the powerset of a set A to the set A. Now consider the set S = {x ∈ A | ∃ s ⊆ A, f(s) = x and x ∉ s}, i.e. the subset of A that is both mapped to by f and not included in the set that maps to it. We know that f(S) ∉ S: suppose f(S) ∈ S, then we would have existence of an s ⊆ A such that f(s) = f(S) and f(S) ∉ s; by injectivity, of course s = S and therefore f(S) ∉ S, which contradicts our premise. However, we can now easily prove that there exists an s ⊆ A satisfying f(s) = f(S) and f(S) ∉ s (of course, by setting s = S), thereby showing that f(S) ∈ S, a contradiction.
I don't think it's debated on the ground of if it's true or not.
And I was imprecise with language, but by saying "math is math" I meant that there are things that logically follow from the ZFC axioms. That is hard to debate or be skeptical of. The point I was driving was that it's strange to be skeptical of an axiom. You either accept it or not. Same as the parallel postulate in geometry, where you get flat geometry if you take it, and you get other geometries if you don't, like spherical or hyperbolic ones...
To give what I would consider to be a good counterargument, if one could produce an actual inconsistency with ZFC set theory that would be strong evidence that it is "wrong" to accept it.
you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.
Any rotating/spinning black hole will no longer be a perfect sphere.
> > The idea of arbitrary precision is intrinsically broken in physical reality.
There is no contradiction here.
But even then, the biggest black hole we think is possible measured down to the planck length gives you a number with 50 digits. And the entire observable universe measured in planck lengths is about 60 digits.
So how are you going to get a physical pi of even a hundred digits on the path toward arbitrary precision?
The accuracy of their volume and radius did not reach the level of a 64-bit float, but it was several orders of magnitude better that of 32-bit FP numbers.
While you cannot build a thing made of molecules with an accuracy better than that of a FP64 number, you can have a standing wave in a resonator, which stays in a cryostat, where the accuracy of its wavelength is 4 orders of magnitude better than the accuracy of a FP64 number, and where the resonator is actively tuned, typically with piezoelectric actuators, so that its length stays at a precise multiple of the wavelength, i.e. with the same accuracy. Only the average length of the resonator has that accuracy, the thermal movements of the atoms cause variations of length superposed over the average length, which are big in comparison with the desired precision, which is why the resonator must be cooled for the best results.
However, it does not really matter whether we can build a perfect sphere or circle. What it matters that modelling everything while using a geometry that supposes the existence of perfect circles we have never seen errors that could be explained by the falseness of this supposition.
The alternative of supposing that there are no perfect circles is not simpler, but much more complicated, so why bother with it?
When talking about whether arbitrarily precise numbers are real in the universe, it extremely matters.
Sadly, atoms exist. In some ways that makes things more complicated, but it's the truth. Anything made of discrete chunks in a grid can't have arbitrarily precise dimensions.
Is there a circle in physical reality? Or only approximate circles, or things we model as circles?
In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
What does it mean to "exist in physical reality"?
If you mean there are objects that have physical characteristics that involve pi to infinite precision I think the truth is we have not a darn clue. Take a circle, that would have to be a perfect circle. Even our most accurate and precise physical theories only measure and predict things to 10s of decimal places. We do not possess the technology to verify that it's a real true circle to infinite precision, and many reason to think that such a measurement would be impossible.
One could argue that nature always deals in discrete quantities and we have models that accurately predict these quantities. Then we use math that humans clearly created (limits) to produce similar models, except they imagine continuous inputs.
There have been attempts to create discrete models of time and space, but nothing useful has resulted from those attempts.
Most quantities encountered in nature include some dependency on work/energy, time or space, so nature deals mostly in continuous quantities, or more precisely the models that we can use to predict what happens in nature are still based mostly on continuous quantities, despite the fact that about a century and a half have passed since the discreteness of matter and electricity has been confirmed.
I'm under the impression that all our theories of time and space (and thus work) break down at the scale of 1 plank unit and smaller. Which isn't proof that they aren't continuous, but I don't see how you could assert that they are either.
It’s fairly easy to go from integers to many subsets of the reals (rationals are straightforward, constructible numbers not too hard, algebraic numbers more of a challenge), but the idea that the reals are, well real, depends on a continuity of spacetime that we can’t prove exists.
Because energy is action per time, it inherits the continuity of time. Action is also continuous, though its nature is much less well understood. (Many people make confusions between action and angular momentum, speaking about a "quantum of action". There is no such thing as a quantum of action, because action is a quantity that increases monotonically in time for any physical system, so it cannot have constant values, much less quantized values. Angular momentum, which is the ratio of action per phase in a rotation motion, is frequently a constant quantity and a quantized quantity. In more than 99% of the cases when people write Planck's constant, they mean an angular momentum, but there are also a few cases when people write Planck's constant meaning an action, typically in relation with some magnetic fluxes, e.g. in the formula of the magnetic flux quantum.)
Perhaps when you said that energy is discrete you thought about light being discrete, but light is not energy. Energy is a property of light, like also momentum, frequency, wavenumber and others.
Moreover, the nature of the photon is still debated. Some people are not convinced yet that light travels in discrete packets, instead of the alternative where only the exchange of energy and momentum between light and electrons or other leptons and quarks is quantized.
There are certain stationary systems, like isolated atoms or molecules, which may have a discrete set of states, where each state has a certain energy.
Unlike for a discrete quantity like the electric charge, such sets of energy values can contain arbitrary values of energy and between the sets of different systems there are no rational relationships between the energy values. Moreover, all such systems have not only discrete energy values but also continuous intervals of possible energies, usually towards higher energies, e.g. corresponding to high temperatures or to the ionization of atoms or molecules.
Perhaps our theories of time and space would break down at some extremely small scale, but for now there is no evidence about this and nobody has any idea which that scale may be.
In the 19th century, both George Johnstone Stoney and Max Planck have made the same mistake. Each of them has computed for the first time some universal constants, Stoney has computed the elementary electric charge in 1874 and Planck has computed the 2 constants that are now named "Boltzmann's constant" and "Planck's constant", in several variants, in 1899, 1900 and 1901. (Ludwig Boltzmann had predicted the existence of the constant that bears his name, but he never used it for anything and he did not compute its value.)
Both of them have realized that new universal constants allow the use of additional natural units in the system of fundamental units of measurement and they have attempted to exploit their findings for this purpose.
However both have bet on the wrong horse. Before them, James Clerk Maxwell had proposed two alternatives for choosing a good unit of mass. The first was to choose as the unit of mass the mass of some molecule. The second was to give an exact value to the Newtonian constant of gravity. The first Maxwell proposal was good and when analyzed at the revision of SI from 2018 it was only very slightly worse than the final choice (which preferred to use two properties of the photons, instead of choosing an arbitrary molecule besides using one property of the photons).
The second Maxwell proposal was extremely bad, though to be fair it was difficult for Maxwell to predict that during the next century the precision of measuring many quantities will increase by many orders of magnitude, while the precision of measuring the Newtonian constant of gravity will be improved only barely, in comparison with the others.
Both Stoney and Planck have chosen to base their proposals for systems of fundamental units on the second Maxwell variant, and this mistake made their systems completely impractical. The value of Newton's constant has a huge uncertainty in comparison with the other universal constants. Declaring its value as exact does not make that uncertainty disappear, but it moves the uncertainty into the values of almost all other physical quantities.
The consequence is that if using the systems of fundamental units of George Johnstone Stoney or of Max Planck, almost no absolute value of any quantity can be known accurately. Only the ratios between two quantities of the same kind and the velocities can be known accurately.
Thus the Max Planck system of units is a historical curiosity that is irrelevant for practice. The right way to use Planck's constant in a system of units has become possible only 60 years later, when the Josephson effect was predicted in 1962, and SI has been modified to use it only after other 60 years, in 2019.
The units of measurement that are chosen to be fundamental do not matter in any way upon the validity of physical laws at different scales. Even if the Planck units were practical, that would give no information about the structure of space and time. The definition of the Planck units is based on continuous models for time, space and forces.
Every now and then there are texts in the popular literature that mention the Planck units as they would have some special meaning. All such texts are based on hearsay, repeating affirmations from sources who have no idea about how the Planck units have been defined in 1899 and about how systems of fundamental units of measurement are defined and what they mean. Apparently the only reason why the Planck units have been picked for this purpose is that in this system the unit of length happens to be much smaller than an atom or than its nucleus, so people imagine that if the current model of space breaks at some scale, that scale might be this small.
Therefore using the Planck length for any purpose is meaningless.
For now, nobody can say anything about the value of a Schwartzschild radius in this range, because until now nobody succeeded to create a theory of gravity that is valid at these scales.
We are not even certain whether Einstein's theory of gravity is correct at galaxy scales (due to the discrepancies non-explained by "dark" things), much less about whether it applies at elementary particle scales.
The Heisenberg uncertainty relations must always be applied with extreme caution, because they are valid in only in limited circumstances. As we do not know any physical system that could have dimensions comparable with the Planck length, we cannot say whether it might have any stationary states that could be characterized by the momentum-position Heisenberg uncertainty, or by any kind of momentum. (My personal opinion is that the so-called elementary particles, i.e. the leptons and the quarks, are not point-like, but they have a spatial extension that explains their spin and the generations of particles with different masses, and their size is likely to be greater than the Planck length.)
So attempting to say anything about what happens at the Planck length or at much greater or much smaller scales, but still much below of what can be tested experimentally, is not productive, because it cannot reach any conclusion.
In any case, using "Planck length" is definitely wrong, because it gives the impression that there are things that can be said about a specific length value, while everything that has ever been said about the Planck length could be said about any length smaller than we can reach by experiments.
So, like, I’m saying that if Einstein’s model of gravity is applicable at very tiny scales, and if the [p,x] relation continues to hold at those scales, then stuff gets weird (either by “measurement of any position to within that amount of precision results in black-hole-ish stuff”, OR “the models we have don’t correctly predict what would happen”)
Now, it might be that our current models stop being approximately accurate at scales much larger than the Planck scale (so, much before reaching it), but either they stop being accurate at or before (perhaps much before) that scale, or things get weird at around that scale.
Edit: the spins of fermions don’t make sense to attribute to something with extent spinning. The values of angular momentum that you get for an actual spinning thing, and what you get for the spin angular momentum for fermions, are offset by like, hbar/2.
So Einstein's theory depends in an essential way on matter being continuous. This is fine at human and astronomic scales, but it is not applicable at molecular or elementary particle scales, where you cannot approximate well the particles by an averaged density of their energy and momentum.
Any attempt to compute a gravitational escape velocity at scales many orders of magnitude smaller than the radius of a nucleus are for now invalid and purposeless.
The contradiction between the continuity of matter supposed by Einstein's gravity model and the discreteness of matter used in quantum physics is great enough that during more than a century of attempts they have not been reconciled in an acceptable way.
The offset of the spin is likely to be caused by the fact that for particles of non-null spin their movement is not a simple spinning, but one affected by some kind of precession, and the "spin" is actually the ratio between the frequencies of the 2 rotation movements, which is why it is quantized.
The "action" is likely to be the phase of the intrinsic rotation that affects even the particles with null spin (and whose frequency is proportional with their energy), while those with non-null spin have also some kind of precession superposed on the other rotation.
I don’t expect this to work. For one thing, we already know the conditions under which the spin precesses. That’s how they measure g-2 .
Also, orbital angular momentum is already quantized. So, I don’t know why you say that the “precession” is responsible for the quantized values for the spin.
the representations of SU(2) for composite particles, combine in understood ways, where for a combination of an even number of fermions, the possible total spin values match up with the possible values for orbital angular momentum.
Could you give an explanation for how you think precession could cause this difference? Because without a mathematical explanation showing otherwise, or at least suggesting otherwise, my expectation is going to be that that doesn’t work.
Obviously this does not exclude the possibility that in the future some experiments where much higher energies per particle are used, allowing the testing of what happens at much smaller distances, might show evidence that there exists a discrete structure of time and space, like we know for matter.
However, that has not happened yet and there are no reasons to believe that it will happen soon. The theory about the existence of atoms is more than 2 millennia old, then it has been abandoned for lack of evidence, then it was revived at the beginning of the 19th century, due to accumulated evidence from chemistry, and it was eventually confirmed beyond doubt in 1865, when Johann Josef Loschmidt became the first who could count atoms and molecules, after determining their masses.
So the discreteness of matter had a very long history of accumulating evidence in favor of it.
Nothing similar applies to the discreteness of time and space, for which there has never been any kind of evidence. The only reason of the speculations about this is the analogy made with the fact that matter and electricity had been believed to be continuous, but eventually it has been discovered that they are discrete.
Such an analogy must make us keep an open mind about the possibility of work, time and space being discrete, but we should not waste time speculating about this when there are huge problems in physics that do not have a solution yet. In modern physics there are a huge amount of quantities that should be computable by theory, but in fact they cannot be computed and they must be measured experimentally. Therefore the existing theories are clearly not good enough.
https://youtu.be/GL77oOnrPzY?si=nllkY_E8WotARwUM
Also Bells Therom implies no locality or non realism which to me furthers the nail on the coffin of spacetime
There are already several decades of such discussions, but no usable results.
Time and space are primitive quantities in any current theory of physics, i.e. quantities that are assumed to exist and have certain properties, and which are used to define derived quantities.
Any alternative theory must start by enumerating exactly which are its primitive quantities and which are their properties. Anything else is just gibberish, not better than Star Trek talk.
However, the units of measurement for time and length are not fundamental units a.k.a. base units, because it is impossible to make any physical system characterized by values of time or length that are stable enough and reproducible enough.
Because of that, the units of time and length are derived from fundamental units that are units of some derived quantities, currently from the units of work and velocity (i.e. the unit of work is the work required to transition a certain atom, currently cesium 133, from a certain state to a certain other state, i.e. which is equal to the difference between the energies of the 2 states, while the unit of velocity is the velocity of light in vacuum).
The standard construction for the real numbers is to start with the rationals and "fill in all the holes". So why even bother with filling in the holes and instead just declare God created the rationals?
Citation needed.
Especially since there are well-established math proofs of irrational numbers.
Let me add that we have no clue how to do a measurement that doesn't involve a photon somewhere, which means that it's pure science fiction to think of infinite precision for anything small enough to be disturbed by a low-energy photon.
Even for elementary particles, we can't be sure that all electrons, say, are exactly alike. They appear to be, and so we have no reason yet to treat them differently, but because of the imprecision of our measurements it could be that they have minutely different masses or charges. I'm not saying that's plausible, only that we don't know with certainty
The logic is circular, simply because mathematicians are the ones who invented irrationals. Of course they have proofs on them. They also have proofs on lots of things that don't exist in this universe.
And as I pointed out elsewhere, many analysis textbooks define a real number to be "a (converging) sequence of rationals". The notion of convergence is defined before reals even enter into the picture, and a real number is merely the identifier for a given converging sequence of rationals.
My point was that it is possible that all values in our universe are rational, and it wouldn't be possible for us to tell the difference between this and a universe that has irrational numbers. This fact feels pretty cursed, so I wanted to point it out.
I think the conceit is supposed to be that analysis—and therefore the reals—is the “language of nature” more so than that we can actually find the reals using scientific instruments.
To illustrate the point, using the rationals is just one way of constructing the reals. Try arguing that numbers with a finite decimal representation are the divine language of nature, for example.
Plus, maybe a hot take, but really I think there’s nothing natural about the rationals. Try using them for anything practical. If we used more base-60 instead of base-10 we could probably forget about them entirely.
So every subset that allows you to do your daily calculations contains the rationals.
I think teachers lie to children and say that decimals are just another way of representing rationals, rather than the approximation of real numbers that they are (and introduce somewhat silly things like repeating decimals to do it), which makes rationals feel central and natural. That’s certainly how it was for me until I started wondering why no programming languages come with rational number packages.
Is sqrt(2) computable?
Is BB(777) computable?
Is [the integer that happens to be equal to BB(777), not that I can prove it, written out in normal decimal notation] computable?
So yes sqrt(2) is computable.
Every BB(n) is computable since every every natutal number can be computed. It's the BB function itself that is not computable in general, not the specific output of that function for a given input.
> A busy beaver hunter who goes by Racheline has shown that the question of whether Antihydra halts is closely related to a famous unsolved problem in mathematics called the Collatz conjecture. Since then, the team has discovered many other six-rule machines with similar characteristics. Slaying the Antihydra and its brethren will require conceptual breakthroughs in pure mathematics.
https://www.quantamagazine.org/busy-beaver-hunters-reach-num...
But given the answer, I suppose you could write a program that just returns it. This seems to hinge on the definition of “computable.” It’s an integer, so that fits the definition of a computable number.
My mistake.
So a specific BB(n) is just a number and is computable.
Then HH the function itself is not computable, but the numbers 0 and 1, which are the only two outputs of HH are computable.
Integers themselves are always computable, even if they are the output of functions that are themselves uncomputable.
So as you noticed, it only makes sense to talk about whether a function is computable, we can't meaningfully talk of computable numbers.
I've solved multiple continuous value problems by discretizing, applying combinatorics to the techniques, and then taking the limit of the result - you of course get the same result if you had simply used regular integration/differentiation, and it's a lot easier to use calculus than combinatorics.
But the point is the "rational", discretized approach will get you arbitrarily close to the answer.
It's why many analysis textbooks define a (given) real number as "a sequence of converging rational numbers" (before even defining what a limit is).
But taking the limit of a sequence of rationals isn’t guaranteed to remain in the rationals (classic example: https://en.wikipedia.org/wiki/Basel_problem. Each partial sum is rational, but the limit of the partial sums is not)
So, how does that statement rebut “You can't do rigorous calculus (i.e. real analysis) on rationals alone.”?
I'm not saying it does. What I'm saying is that you can make a correspondence with the reals by using only rationals.
You can define convergence without invoking the reals (Cauchy convergence). If you take any such sequence, you give that sequence a name. That name is the equivalent of a real number. You can then define addition, multiplication - any operation on the reals - with respect to those sequences (again, invoking only rational numbers).
So far, we have two distinct entities: The rationals, and the converging sequences.
Then, if you want, you can show that if you take the rationals and those entities we're calling "converging sequences" together, you can make operations involving the two (e.g. adding a rational to that converging sequence) and eventually build up what we know to be the number line.
Computation can only use rationals, and of course can get arbitrarily close to an answer because they are dense in the reals.
However, the entire edifice of analysis rests on the completeness axiom of the reals. The extreme value theorem, for example, is equivalent to the completeness axiom; the useful properties of continuous functions break down without it; the fundamental theorem of calculus doesn't work without it; Etc. So if the maths used in your physics (the structure of the theory, not just the calculations you perform with it) relies on these things at all, you're relying on the reals for confidence that the maths is sound.
Now you could argue that we don't need mathematical rigour for physics, that real analysis is a preoccupation of mathematicians, while physicists should be fine with informal calculus. I'm not going to argue that point. I'm just pointing out what the real numbers bring to the table.
Here's Tim Gowers on the subject: https://www.dpmms.cam.ac.uk/~wtg10/reals.html
To me at least, if you can write down a finite procedure that can produce a number to arbitrary precision, I think it is fair to say the number at that limit exists.
This made me think of a possible numerical library where rather than storing numbers as arbitrary precision rationals, you could store them as the combination of inputs and functions that generate that number, and compute values to arbitrary precision.
The continuum is the reality that we have to hold to. Not the continuum in the Cantor sense, but in the intuitionalist or constructivist sense, which is continuously varying numbers that can be approximated as necessary.
Calculo, ergo sum?
Pretty much:
https://en.wikipedia.org/wiki/Church_encoding
I’m not convinced that we could have our current universe without irrationals - wouldn’t things like electromagnetism and gravity work differently if forced to be quantized between rationals? Saying ‘meh it would be close enough’ might be correct but wouldn’t be enough to convince me a priori.
The idea that a quantity like 1/3 is meaningfully different than 333/1000 or 3333333/10000000 is not really that interesting on its own; only in the course of a physical process (a computation) would these quantities be interestingly different, and then only in the sense of the degree of approximation that is required for the computation.
The real numbers in the intuitionalist sense are the ground truth here in my opinion; the Cantorian real numbers are busted, and the rationals are too abstract.
Hard disagree. This is the problem with math disconnected from physics. The real world is composed of quanta and spectra, i.e. reality is NOT continuous!
But that's tatamount to the belief that the minutest particle of the universe requires the equivalent of an infinite number of bits of state.
Integers come into existence long before god - as the only presumption required is a difference between one thing and another (or nothing). The integers also create infinite gaps. The primes.
So no - I do not think reals are closer to the divine. They require we import infinity twice to be defined, and I'm undecided on whether our reality has unbounded 'precision' like that - or if 'just' an infinite number of discrete units.
Caveat: former Catholic; 50+ years of fervent atheism.
ps. Various numerology phenomena have a similar vibe, and no wonder so many people who go off the deep end tend to get trapped by them. Maybe I will be one of them as I become old and senile :-D
yes, I also enjoy trying to answer this question.
what is such an structure even mean? how could it be that simply defining numbers, obersving addition, and generalizing it away into multiplication would yield this natural structure?
It all begins with zero. the predecessor of One, the best known number.
zero can be assumed by anyone. the surprise is how all zeros are the same zero. (by uniqueness of emptyset; but as I hope you can see, I'm a crank. a nutjob. I'll stop
The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
This is a problem of modeling optimization. The models based on uncountable "real" numbers are logically consistent and simple to use, so they are adequate for predicting what happens in natural or artificial systems.
All attempts to avoid the uncountable infinities produce models that are both more complicated and also incomplete, as they do not cover all the applications of traditional infinitesimal calculus, topology and geometry.
Unless someone will succeed to present a theory that avoids uncountable infinities while being as simple as the classic theory and being applicable to all the former uses, I see such attempts as interesting, but totally impractical.
The real numbers are a useful mathematical trick that make it possible to prove results in calculus. What you surrender in return for being able to prove statements is to give up the ability to compute expressions. This may be a worthwhile trade-off for physicists but for the universe (which does many computations and zero proofs) it's quite a burden.
The real numbers exist and are approximable, either by rationals or by decimal expansion. The idea of approximability and computability are the critical things, not the specific representation.
Egg cartons might sound contrived but the reals don't necessarily make sense without reference to rulers, scales, etc. And in fact the defining completeness / Dedekind cut conditions for the reals are necessary for doing calculus but any physical interpretation is both pretty abstract and probably false in reality.
A better way to dispute the unit square diagonal argument for the existence of sqrt(2) would be to argue that squares themselves are unphysical, since all measurements are imprecise and so we can't be sure that any two physical lengths or angles are exactly the same.
But actually, this argument can also be applied to 1 and other discrete quantities. Sure, if I choose the length of some specific ruler as my unit length, then I can be sure that ruler has length 1. But if I look at any other object in the world, I can never say that other object has length exactly 1, due to the imprecision of measurements. Which makes this concept of "length exactly 1" rather limited in usefulness---in that sense, it would be fair to say the exact value of 1 doesn't exist.
Overall I think 1, and the other integers, and even rational numbers via the argument of AIPendant about egg cartons, are straightforwardly physically real as measurements of discrete quantities, but for measurements of continuous quantities I think the argument about the unit square diagonal works to show that rational numbers are no more and no less physically real than sqrt(2).
However, at each finite n we are still dealing with discrete quantities, i.e. integers and rationals. Even algebraic irrationals like sqrt(2) are ultimately a limit, and in my view the physicality of this limit doesn’t follow from the physicality of each individual element in the sequence. (Worse, quantum mechanics strongly suggests the sequence itself is unphysical below the Planck scale. But that’s not actually relevant - the physicality of sqrt(2) ultimately assumes a stronger view about reality than the physicality of 2 or 1/2.)
> They were both put in a room and at the other end was a $100 and a free A on a test. The experimenter said that every 30 seconds they could travel half the distance between themselves and the prize. The mathematician stormed off, calling it pointless. The engineer was still in. The mathematician said “Don’t you see? You’ll never get close enough to actually reach her.” The engineer replied, “So? I’ll be close enough for all practical purposes.”
While you nod your head OR wag your finger, you continuously pass by that arbitrary epsilon you set around your self-disappointment regarding the ineffability of the limit; yet, the square root of two is both well defined and exists in the universe despite our limits to our ability to measure it.
Thankfully, it exists in nature anyhow -- just find a right angle!
One could simply define it as the ratio of the average distance between neighboring fluoride atoms and the average distance of fluoride to xenon in xenon tetrafluoride.
Some eggs are smaller than others; some are more dense, etc. Yes, the "count" is maybe sort of interesting in some very specific contexts, but certainly not in any reductive physical context. It only works in an economic context because we have standards like what constitutes a "chicken egg large white grade AAA".
Doesn't it though?
What happens when three bodies in a gravitationally bound system orbit each other? Our computers can't precisely compute their interaction because our computers have limited precision and discrete timesteps. Even when we discard such complicated things as relativity, what with its Lorentz factors and whatnot.
Nature can perfectly compute their interactions because it has smooth time and infinite precision.
That doesn't follow. Nature can perfectly compute them because they are nature. Nowhere is it required to have infinite precision, spatial or temporal.
The set of real numbers is almost all extraneous junk that the universe definitely doesn't care about but is very important to mathematicians.
This is a Jewish and Christian conception of God. How can this be true when so many things that give us comfort in the natural world: fresh fruit, shade trees, sunshine and warm sand between our toes, etc., were not created by man?
Even in mathematics itself: how improbable, how ludicrous, how miraculous is it that the 3rd, 4th, and 5th natural numbers -- numbers you could discover by looking at your own hands -- have the amazing property of demonstrating the Pythagorean theorem?
The Islamic ideal of God (Allah) is so much more balanced. God created both the integers AND the reals. He created everything, some things for our comfort and rest, some things to drive us close to madness, and a lot of stuff in between. Peel back enough layers of causality and all of creation has the stamp of the divine.
...
> The Jewish [Christian] ideal of God (YHVH) is so much more balanced.
There's enough bigotry out there. Let's not make assumptions about people's beliefs.
[0] Jesus being human changes the calculus quite a lot, of course, as elaborated in e.g. Hebrews 4:14–16. God, who was fully transcendent, became human, hence why Jesus is also called Immanuel/Emmanuel (lit. “God with us”) in the Bible.
I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are.
Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But this just creates another mystery: what is the relationship between the material and the immaterial realm? What governs that? (Calling one or more of the immaterial entities "God" doesn't really make it any less mysterious.)
If we add entities to our model of reality to answer questions and all it does is create more and more esoteric questions, we should take some advice from Occam's Shovel: when you're in a hole, stop digging.
then maths is really THE absolute best description available of language and nature.
but non-mathematical minds will simply wonder and be amazed at how "maths explains the world", a clear indication that somebody is not thinking like a mathematician.
> Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But this just creates another mystery: what is the relationship between the material and the immaterial realm?
the relationship between the material and the immaterial pattern beholden by some mind can only be governed by the brain (hardware) wherein said mind stores its knowledge. is that conscious agency "God"? the answer depends on your personally held theological beliefs. I call that agent "me" and understand that "me" is variable, replaceable by "you" or "them" or whomever...
oh, and I love (this kind of figurative) digging. but I use my hands no shovels.
(And the discrepancy might not be in the physical continuum being simpler than the mathematical reals, as some here postulate, but rather in the continuum being far stranger than the reals, in ways we may never observe nor comprehend.)
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