But is computation enough? The computable reals are a subset of the reals.
That is: Real numbers describe real, concrete relations. For e.g., saying that Jones weighs 180.255 pounds means there's a real, physical relationship -- a ratio -- between Jones' weight and the standard pound. Because both weights exist physically, their ratio also exists physically. Thus, from this viewpoint, real numbers can be viewed as ratios.
In contrast, the common philosophical stance on numbers is that they are abstract concepts, detached from the actual physical process of measurement. Numbers are seen as external representations tied to real-world features through human conventions. This "representational" approach, influenced by the idea that numbers are abstract entities, became dominant in the 20th century.
But the 20th century viewpoint is really just one interpretation (you could call it "Platonic"), and, just as it's impossible to measure ratios to infinite precision in the real world, absolutely nothing requires an incomputable continuum of reals.
Physics least of all. In 20th and 21st century physics, things are discrete (quantized) and are very rarely measured to over 50 significant digits. Infinite precision is never allowed, and precision to 2000 significant digits is likewise impossible. The latter not only because quantum mechanics makes it impossible to attain great precision on very small scales. For e.g., imagine measuring the orbits of the planets and moons in the solar system: By the time you get to 50 significant digits, you will need to take into account the gravitational effects of the stars nearest to the sun; before you get to 100 significant digits, you'll need to model the entire Milky Way galaxy; the further you go in search of precision, the exponentially larger your mathematical canvas will need to grow, and at arbitrarily high sub-infinite precision you’d be required to model the whole of the observable universe -- which might itself be futile, as objects and phenomena outside observable space could affect your measurements, etc. So though everything is in principle simulatable, and precision has a set limit in a granular universe that can be described mathematically, measuring anything to arbitrarily high precision is beyond finite human efforts.
Rationals > irrationals on computing them. You can always approximate irrationals with rationals, even Scheme (Lisp, do'h) has a function to convert a rational to decimal and the reverse. decimal to rational.
This entire discussion is about mathematical concepts, not physical ones!
Sure, yes, in physics you never "need" to go past a certain number of digits, but that has nothing to do with mathematical abstractions such as the types numbers. They're very specifically and strictly defined, starting from certain axioms. Quantum mechanics and the measurability of particles has nothing to do with it!
It's also an open question how much precision the Universe actually has, such as whether things occur at a higher precision than can be practically measured, or whether the ultimate limit of measurement capability is the precision that the Universe "keeps" in its microscopic states.
For example, let's assume that physics occurs with some finite precision -- so not the infinite precision reals -- and that this precision is exactly the maximum possible measurable precision for any conceivable experiment. That is: Information is matter. Okay... which number space is this? Booleans? Integers? Rationals? In what space? A 3D grid? Waves in some phase space? Subdivided... how?
Figure that out, and your Nobel prize awaits!
There are plenty of ratios that are ratios of other things than integers, so they are not rational numbers.
Ratios of collinear vectors are scalars a.k.a. "real" numbers, ratios of other kinds of vectors are matrices, ratios of 2D-vectors are "complex" numbers, ratios of 2 voltages are scalars a.k.a. "real" numbers, and so on.
In general, for both multiplication and division operations, the 3 sets corresponding to the 2 operands and to the result are not the same.
Only for a few kinds of multiplications and of divisions the 3 sets are the same. This strongly differs from addition operations, which are normally defined on a single set to which both the operands and the result belong.
In practice, multiplications and divisions where at least one operand or the result belong to another set than the remaining operands or result are extremely frequent. Any problem of physics contains such multiplications and divisions.
Wat? Vectors do not form a division algebra. You can't "divide" vectors! Perhaps you're thinking of the dot product, which returns a scalar.
Also, you're conflating physical/engineering concepts (such as units of measure) with mathematical abstractions such as number spaces, which don't have units.
Physical measurements exist in the real world, with its limitations, units, and practicalities.
Mathematical numbers exist in a pure theory space that is completely and totally independent of any physical reality. They're just axioms and rules. Definitions and logical conclusions.
Don't mix up the two!
On the contrary, there exists a huge number of kinds of division operations, but only very few division algebras.
For any kind of multiplication operation you can define an associated division operation (or 2 associated division operations, for non-commutative multiplications), which accepts as input the product as the dividend and one factor as the divisor, and which returns the other factor as the quotient.
In general, the division operation can never be defined for all pairs of product with factor for which the multiplication is defined, not even in a division algebra, where division by zero remains undefined. Nevertheless, in almost all cases a division operation can be defined for many of the possible product-factor pairs.
While there are some vector pairs for which the division operation is undefined, for most vector pairs the division operation is perfectly defined. Given such a pair of vectors there exists a unique matrix that will produce the dividend vector when multiplied with the divisor vector. Every kind of division is the inverse of some kind of multiplication, and this kind of division is the inverse of the matrix-vector multiplication.
The division of vectors is quite important in practice, for example for finding the rotation that will transform one object into another object, which is identical, but rotated.
Any physical measurement is a division that is not performed in a division algebra, because the operands belong to the set of values of the measured physical quantity, while the result belongs to a different set, i.e. it is a rational number that approximates the scalar a.k.a. "real" number that corresponds to the measured value.
Saying that a measurement is a division operation is not some kind of metaphor, but it is a concrete description of how to implement it. For example, there are a very large number of kinds of analog-digital converters, which are used for voltage measurement. Because they must compute the division between the measured voltage and a reference voltage, each of the many possible structures (e.g. flash converters, successive-approximation converters, pipelined converters, cyclic converters, ramp converters, sigma-delta converters, etc.) corresponds to one of the algorithms that can be used for computing a division, even with pen and paper.
All the useful parts of mathematics are not isolated of the physical world, but they abstract the properties of the physical world, to enable the description of the physical world by models about which one can reason formally.
What are called now "real numbers" were called "measures" by the ancients (who called "numbers" only the values that can be obtained by counting). The theory of "measures", i.e. "real numbers", has not been conceived in a dream, separated from the physical world, but it was conceived as an axiomatic model that attempted to match the perceived behavior of the physical quantities, such as lengths.
Similarly all the other valuable mathematical concepts have been conceived for using them in mathematical models of the physical world.
Once a useful axiomatic framework existed, there have also been many mathematical researches that were motivated only by intellectual pleasure disconnected from immediate practical necessities, like in the case of many results in number theory, or in the study of systems where only less axioms are used or some axioms are replaced, in comparison with any system applicable in practical mathematical models.
However, completely "pure" mathematical research, where one would study the consequences of some random axioms completely unrelated to the physical world (i.e. which would not use any of the mathematical concepts abstracted from the physical world, e.g. numbers, topologies etc.) has been very rare and it has not produced interesting results.
This requirement is necessary to allow the definition of a division operation, which has as operands a pair of values of that physical quantity, and as result a scalar a.k.a. "real" number. This division operation, as you have noticed, is called "measurement" of that physical quantity. A value of some physical quantity, i.e. the dividend in the measurement operation, is specified by writing the quotient and the divisor of the measurement, e.g. in "6 inches", "6" is the quotient and "inch" is the divisor.
In principle, this kind of division operation, like any division, could have its digit-generating steps executed infinitely, producing an approximation as close as desired for the value of the measured quantity, which is supposed to be an arbitrary scalar, a.k.a. "real" number. Halting the division after a finite number of steps will produce a rational number.
In practice, as you have described, the desire to execute the division in a finite time is not the only thing that limits the precision of the measured values, but there are many more constraints, caused by the noise that could need longer and longer times to be filtered, by external influences that become harder and harder to be suppressed or accounted for, by ever greater cost of the components of the measurement apparatus, by the growing energy required to perform the measurement, and so on.
Nevertheless, despite the fact that the results of all practical measurements are rational numbers of low precision, normally representable as FP32, with only measurements done in a few laboratories around the world, which use extremely expensive equipment, requiring an FP64 or an extended precision representation, it is still preferable to model the set of scalars using the traditional axioms of the continuous straight line, i.e. of the "real" numbers.
The reason is that this mathematical model of a continuous set is actually much simpler than attempting to model the sets of values of physical quantities as discrete sets. An obvious reason why the continuous model is simpler is that you cannot find discretization steps that are good both for the side and for the diagonal of a square, which has stopped the attempts of the Ancient Greeks to describe all quantities as discrete. Already Aristotle was making a clear distinction between discrete quantities and continuous quantities. Working around the Ancient Greek paradox requires lack of isotropy of the space, i.e. discretization also of the angles, which brings a lot of complications, e.g. things like rigid squares or circles cannot exist.
The base continuous dynamical quantities are the space and time, together with a third quantity, which today is really the electric voltage (because of the convenient existence of the Josephson voltage-frequency converters), even if the documents of the International System of Units are written in an obfuscated way that hides this, in an attempt to preserve the illusion that the mass might be a base quantity, like in the older systems of units.
In any theory where some physical quantities that are now modeled as continuous, were modeled as discrete instead, the space and time would also be discrete. There have been many attempts to model the space-time as a discrete lattice, but none of them has produced any useful result. Unless something revolutionary will be discovered, all such attempts appear to be just a big waste of time.
The Heisenberg uncertainty principle is just a trivial consequence of the properties of the Fourier transform. It just states that there are certain pairs of physical quantities which are not independent (because their probability densities are connected by a Fourier transform relationship), so measuring both simultaneously with an arbitrary precision is not possible.
The Heisenberg uncertainty principle says absolutely nothing about the measurement of a single physical quantity or about the simultaneous measurement of a pair of independent physical quantities.
There is no such thing as a quantity that cannot be measured, i.e. the set of its values does not have the required Archimedean group algebraic structure. If it cannot be measured, it is not a quantity (there are also qualities, which can only be compared, but not measured, so the sets of their values are only ordered sets, not Archimedean groups; an example of a physical quality, which is not a physical quantity, is the Mohs hardness, whose numeric values are just labels attached to certain values, a Mohs hardness of "3" could have been as well labeled as "Ktcwy" or with any other arbitrary string, the numeric labels have been chosen only to remember easy the order between them).
Various proposals exist in which the fundamental physical structure of space form a kind of "quantum foam". And so, somewhere near the Planck length, it doesn't even make sense to talk about distance.
What you have in mind about "ranges of values" has nothing to do with Heisenberg, but with the formalism of quantum mechanics as developed by Schroedinger, Dirac and others (not counting the matrix mechanics of Heisenberg, which was an inferior mathematical method, soon forgotten after superior alternatives were developed, and which is something completely else than the Heisenberg principle of uncertainty).
In the various variants of the formalism of quantum mechanics, the correspondent of a single value in classic dynamics is a function, the so-called wave function, but even those functions cannot be called "ranges of values". There are several interpretations of what the "wave" functions mean, but the most common is that they are probability densities for the values of the corresponding physical quantity.
Depending on the context, in quantum mechanics the value of a physical quantity may be unknown, when only the probability of it having various values is known, but the value may also be known with absolute certainty, usually in some stationary states.
In both cases the value of the physical quantity can be measured, but in the former only an average value can be measured, which nonetheless may have unlimited precision, while in the latter case the exact value can be measured, exactly like in classical dynamics.
There are pairs of physical properties, connected by a Fourier transform, that cannot both be known at once. Therefore each property is, in general, only known to be in a range. And therefore we cannot always compare these properties between different particles.
As for measurement, in the Everett interpretation the values are not known after measurement either. All that is knowable are correlations due to entanglement.
But now we are far afield. You made an a priori statement about physics. Physics is not required to oblige you. My understanding of physics says that it does not. But even if I am wrong, and it currently does oblige you, it is still not required to.
A priori assertions do not have a good history. For an example, look at Kant's assertion that reality must be described by Euclidean geometry. He was not in a position to understand the ways in which he would prove to be wrong.
Granularity is implied by some, but not all, post standard model physics. It's a very open question.
> You can measure any value to arbitrary precision
Quantum mechanics lets you refine one observable indefinitely if you are willing to sacrifice conjugate observables.
You can measure that one observable to an arbitrarily (not infinitely!) high precision -- if you have an correspondingly arbitrary duration of time and arbitrarily powerful computational resources. That measurement of yours, if exceedingly precise, might require a block of computronium which utilizes the entire energy resources of the universe. As a practical matter, that's not permitted. I'm certainly unaware of any measurement in physics to more than 100 significant digits, let alone "arbitrary precision".
In fact, as it turns out, no experiment has ever resolved structure down to the Planck length. A fundamental spatial resolution is likely to be quite a lot smaller than the Planck length; even the diameter of the electron has been estimated at anywhere from 10^-22m to 10^-81m.
The question I was responding to asked whether reals are "necessary" -- plainly, insofar as reality is concerned, they are not.
My concern with the reals goes the other direction. The reals are not closed under common operations. A lot of the work is done in complex numbers, which are. The rationals are not closed under radicals, and radicals seem pretty fundamental.
You can define a finitist model despite this. But it's ugly, and while ugliness is not physically meaningful, it tends to make progress difficult. A usable solution may yet arise; we shall (perhaps) see.
Now, are the reals necessary?
During the last decades, there have been published many research papers exploring the use of discreteness instead of the traditional continuity, but they cannot be considered as anything else but failures. The resulting mathematical models are much more complicated than the classic models, without offering any extra predictions that could be verified.
The reason for the complications is that in physics it would be pointless to try to model a single physical quantity as discrete instead of continuous. You have a system of many interrelated quantities and trying to model all of them as discrete reaches quickly contradictions for the simpler models, due to the irrational or transcendent functions that relate some quantities, functions that appear even when modeling something as simple as a rotation or oscillation. If, in order to avoid incommensurability, you replace a classic continuous uniform rotation with a sequence of unequal jumps in angular orientation, to match the possible directions of nodes in a discrete lattice, then the resulting model becomes extremely more complicated than the classic continuous model.
Everything that is known about numerical analysis is based on discreteness, instead of continuity. Every useful predictive model that we have about the world, for example for forecasting weather, depends on numerical analysis.
I consider this a success!
Forecasting weather, designing semiconductor devices, or any other such activities are all based on continuous mathematical models, which use e.g. systems of equations with partial derivatives or systems of integral equations.
Only in the last stage of solving such a problem, in order to use a digital computer the continuous mathematical model is approximated by a discrete mathematical model, which is constructed using a standard method, e.g. finite elements, boundary elements, finite differences, most of which are based on approximating an unknown function with an infinity of degrees of freedom with a function determined by a finite number of parameters, then by approximating the operations required to compute those parameters by operations with floating-point numbers.
Such an approximating method is something extremely different from formulating a discrete mathematical model of physics that is considered the exact model.
Even the graphics for a game are based on continuous models of space, not on discrete models, where it would be very difficult to implement something like the rotation of an object or perspective views.
The failures to which I have referred are the attempts to create such discrete models of physics, e.g. where the space and time are discrete not continuous.
These attempts have nothing in common with the approximating techniques, which are indeed the base for most successes in using digital computers.
Time-scale calculus is a pretty niche theoretical field that looks at blending the analysis of difference and differential equations, but I'm not aware of any algorithmic advances based on it.
In a sense this is wrong. Some say the indispensability of modern math, which includes the incomputable reals, shows that such abstract objects are required for science.
The idea that uncountable means more comes from a bad metaphor. See https://news.ycombinator.com/item?id=44271589 for my explanation of that.
Accepting that uncountable means more forces us to debatable notions of existence. See https://news.ycombinator.com/item?id=44270383 for a debate over it.
But, finally, there is this. Every chain of reasoning that we can ever come up with, can be represented on a computer. So even if you wish to believe in some extension of ZFC with extremely large sets, PA is capable of proving every possible conclusion from your chosen set of axioms. So yes, PA is enough.
If you're not convinced, I recommend reading https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden....
It all gets mind-bendingly mind-bending really fast, especially with people like Harvey Friedman ( https://u.osu.edu/friedman.8/foundational-adventures/fom-ema... ) or the author of this post trying to break the math by constructing an untractably large values with the simplest means possible (thus showing that you can encounter problems that do not fit in the universe even when working in a "simple" theory)
(i saw the username and went to check audiomulch website, but it did not resolve :( )
Of course not, but that would invalidate the entire project of some/many here to turn reality into something clockwork they can believe they understand. Reality is much more interesting than that.