I'm still reading it, but I don't agree with the following:
>In Plato and the Nerd, I review Claude Shannon's channel capacity theorem [Shannon, 1948], which states that any noisy observation of anything conveys only a finite number of bits of information. Borel's know-it-all number cannot be encoded with finite number of bits unless the list all possible yes-no questions is finite, which it is not. It is easy to construct an infinite sequence of valid yes-no questions. For example, let the first question be "Is one a whole number?" Let the second question be "Is the answer to the first question YES?" Let the third question be "Is the answer to the second question YES?" And so on. As a consequence, Borel's number cannot reveal itself to us unless we invent some noiseless way of observing a thing in itself.
It is true that he showed an example of an infinite list of valid yes/no questions, which would lead to a number with an infinite amount of digits. But despite it being infinite, by the way it was constructed you already know all the digits (they will be all 1s), meaning that that number's entropy is remarkably low and that number can be expressed with a finite number of bits of information. So Shannon's theorem isn't a problem here.
The presentation in the quote is awkward and doesn't work, as you pointed out, but Lee could instead have just mentioned that there's such a construction with questions of the form "is the answer to the previous question YES [NO]?" which is not constant (not all 1s), and does have the property he wants, i.e. the are such sequences that cannot be specified finitely (corresponding to binary sequences with that property).
I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem.
Tangentially, I'm interested to see a formal statement/proof of the form of Shannon's theorem add used in this article, because I can sort of intuitively see why it makes sense but I'm not familiar enough with the technical details to guarantee that this is even an appropriate use of the theorem, assuming the yes/no question construction has all the properties Lee wants it to have.
> I don't think you've shown that Shannon's theorem isn't a problem here so much as the argument presented did not successfully convey that it's a problem.
Yes, I was referring to the specific example he made, I wasn't generalizing my conclusion to all possible cases. At the moment of writing my comment I wasn't sure if there was an example that would have worked or not, but the comment about Collatz numbers is a good example.
Edit: actually it could be a good example, because wheter that number can be represented in a finite way or not depends on whether Collatz's conjecture is true or false.
> Despite these difficulties, the cognitive notion of a continuum is not at all difficult to grasp. The difficulties arise only when trying to communicate, for example by naming or describing all the real numbers. But one can understand without communicating.
I feel it's important to point out that humans are very good at convincing themselves that they have done a great job of understanding things when they have not. Including brilliant minds.
Since this is in "trust our intuitions" territory about, essentially, philosophy of mind, it's worth bringing along this awareness.
>> [...] cognitive notion of a continuum is not at all difficult to grasp. The difficulties arise only when trying to communicate, for example by naming or describing all the real numbers. But one can understand [...]
Notice that one can do a very similar analysis after replacing "continuum" with "rules of the game of chess" and "naming or describing all real numbers" with "name or describing all positions which could occur in a chess game". You don't need uncountable infinities, topological oddities of R, subtle but deep differences from how the real world works, etc. to get out to where humans aren't all that good at understanding things.
I also think it's notable that somehow DNA "encoding" our minds for mind-inheritance is ruled out using the channel capacity theorem.
Shouldn't this same kind of argument rule out the transferral of minds from one moment to the next, unless minds do not actually arise from our bodies? In other words, does this criticism by Lee commit to mind-body dualism?
In fact, the formalization of calculus (and in particular, continuity and differentiability via limits) came about because people had different intuition about “continuous” functions.
I would say the implications of a continuum versus a dense cloud of rationals is incredibly unintuitive.
> If in fact the mind relies on a continuum for its cognitive functions, this would explain why cognitive functions cannot be inherited and why our minds can deal with real numbers, despite the paradoxes.
I disagree with the assertion that "our minds can deal with real numbers". AFAIK there's no way to distinguish an uncomputable number from a computable one, using a finite set of computations (i.e. there's always some computable model which gives the same answers as our uncomputable number, even if it's highly convoluted); so we're still only learning about the countable, computable sub-set.
"Deal with" in detail, absolutely not, just as you say. Pardon my rephrasing you point (I think):
You can show that some functions that give real numbers are computable. (Let's say, computing that number digit by digit. Binary digit or base-ten, you choose.) You can't show that all Turing machines halt, so you can't use the mere existence of a Turing machine as proof of a computable function. Therefore, we can't know or compute the set of all real numbers because we can't compute the set of irrational numbers; there will always be some Turing machine that, unbenownst to us, calculates a (likely irrational) function that spits out a real number. If we can't describe even a way to sort functions into two piles: "computes a real number (digit by digit)" and "doesn't compute a real"; then we can't be said to "deal with" the set of real numbers at all well.
This rather presupposes the (now pretty much accepted) concept that numbers "exist" if they can be computed (whether they have been computed, or conceived of, or not.) "Intuitionism" in other words, IIRC.
What is it good for if there's no mechanism to reach it?
"What they’ve found is that some games have Nash equilibria that are impossible for a computer to determine and even more equilibria are computable only if one of the largest open problems in computer science, P = NP, is solved."
All of physics stands as an objection to "digital physics" which uses real numbers indispensably. Many things fall apart with discrete numbers (classical physics becomes non-deterministic, quantum mechanics becomes non-linear, etc.).
The loons on the "digital bandwagon" have yet to produce a single plausible work showing the geometrical spaces hiterto essential to physics can be replaced with anything discrete.
This is just another pipe-dream of idealist-Platonist-AI fanatics for whom the world is an abstract discrete number only because their favourite toy is so-described.
> In 1900, Max Planck derived the correct form for the intensity spectral distribution function by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can be emitted or absorbed only in discrete packets, called quanta, of energy
Of course reality has discrete aspects. That really has nothing to do with whether it /only/ has discrete aspects, and indeed, whether they are abstract (ie., lack spatiotemporal properies).
People who believe that today are mostly loons -- with the eyes, fervour and askew chalantanism of loons
> I thought one of the central ideas of quantum mechanics is that space is discrete? Or am I misunderstanding what the Planck length is?
Yes, that is a misunderstanding. The fundamental idea of quantum mechanics is that certain properties of matter are quantized, but space is not one of them - they are spin, charge, electroweak charge, color charge, and perhaps a few others.
Space and time though are continuous. Planck's length comes in when we discuss the maximum precision that can be measured, as it's related to the Heisenberg uncertainty principle (and to other fundamental concepts, such as c or even black holes). So, the Planck length is the smallest unit of distance that can be meaningfully discussed, but that doesn't mean that distance between particles is an integer multiple of the Planck length.
Wow I had no idea that was the case I had the same misunderstanding as the other posted. Thank you for that, I will be revisiting my understanding of time and space
The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks.
Once Algebra is invented, we learn that not only is "zero" a useful concept (additive identity), but so is "one" (multiplicative identity). More importantly, the inverse of multiplication can be defined as "blah times number == 1". IIRC, the Greeks somehow understood this concept before the invention of Algebra, but its easier to see once you have Algebra.
Once we have those, its not too difficult to extend into irrational numbers, such as sqrt(2). The Greeks almost discovered it, but were too stubborn. There's a lot of discussion in Greek Mathematics over sqrt(2)... they're so so close but they were too much of a fanboy of integers / rational numbers to actually reach the concept.
Once we understand sqrt(2), its not too difficult to extend the principle to sqrt(-1) or sqrt(-2), reaching the concept of "i".
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sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering (https://en.wikipedia.org/wiki/Phasor). Its as "real" as any other number. IE: We can truly measure real-world objects (ie: the voltage going through your power-lines) and assign them a value of "i", and use that to predict what happens to the electricity as it flows through your equipment and/or house.
So all of these things are "real" numbers, that describe truly physical real-world objects. Its just an unfortunate naming convention of "Real" vs "Imaginary" vs "Rational" vs "Irrational", which comes down to history / order of discoveries, rather than usefulness.
The difference is that the probability of any given "real" number being used for anything, ever, or even being nameable, or even being potential covered by a naming scheme, is 0.
This is also true for countably infinite collections of real numbers.
Real numbers are unreal, at a scale incomprehensibly beyond the merely slightly hidden numbers like complex algebraics and the known transcendentals. Real numbers aren't numbers. R is a background continuum in which numbers exist. It is almost entirely void, yet contains an infinite collection of actual numbers.
You in fact did write down "Graham's number" and "googleplex". Most real numbers cannot even be written down as a finite sequence of symbols in any way. However for algebraic numbers ("Plenty of irrational numbers"), every single one has a finite description.
There are uncountably many reals and only a countable number of "names" we can give them. This is a very real problem that makes people say that the real numbers are not real.
Oh jeez, its like the space of rational numbers is well described by Arabic Numerals or something. And that anything that's a non-rational number cannot be described by Arabic Numerals + decimal points.
:-p
Don't confuse our current way of writing down numbers with alternative writing methods. That's my point. "Rational numbers" are simply one-and-the-same as "The space described by finite-length Arabic Numerals + decimal points", no more, no less.
EDIT: To put it another way: you're describing a limitation of Arabic Numerals. Not a limitation to math in general. Much like how Roman Numerals are a terrible representation of modern math, we need other representations (ie: Algebraic notation) to represent other concepts in modern math.
I did not mention Arabic numerals at all in my comment; I think you have missed my point entirely. I was actually thinking of sequences of symbols like "lim(inf, (1 + 1/x)^x)" being a name for e. And all algebraic numbers can be expressed by a polynomial equation and a description of which root you are referring to.
Given that I literally wasn't even thinking about Arabic numerals, I was not describing a limitation of them. The limitation is with math. When talking about numbers you need a way to write down a specific number (Needs a finite representation). If it can be put into a computer, that finite representation comes from some countable set. There are uncountably many real numbers. That means we cannot given a computer understandable name to every real number.
You can give a name to every algebraic number or rational number. That is a key difference that is not tied in any way to Arabic numerals.
> When talking about numbers you need a way to write down a specific number (Needs a finite representation)
There are plenty of rational numbers that cannot be written down. Indeed, an infinite number of them. All the numbers larger than say... a googleplex, or the numbers approaching Grahams' number (or larger than that number) are unwritable.
We don't even need to stray from the integer numbers if we want to get to unwritable numbers. Plenty of numbers too large to be ever written down even if we use every single atom in our universe (or any such combination of atomic-states with all the atoms in our universe).
"X cannot be written" seems to be a rather arbitrary limit.
Why does the existence or absence of a finite name affect the reality of a number? You can put every real number in a one-to-one correspondence with itself. Or a one-to-one correspondence of f(x).
This is a really interesting thought experiment, and I think in some ways reframes the problem of "exists" (which is nebulous) to "has a description which is finite (or countably infinite, but not uncountable)".
But I think if you could prove a way to countably label an arbitrary real number, you could probably inductively prove that you could do it for any. But you run face-first into the interesting number paradox and/or incompleteness.
If unicorns exist, but humans can never observe them do you really care whether they exist or not?
I generally am reducing questions that are only relevant to philosophy such as "Do real numbers exist?" to questions which you can actually answer.
To make something clear, rationals have a finite description. If you allow infinite descriptions, all real numbers can be described by Cauchy sequences. When I talk about countably many names, I am talking about the set of finite names humans use for numbers. You cannot meaningfully have infinitely many sets of finite names that humans agree on so you cannot cover the real numbers with names.
The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers.
When the sets of "finite names" you discuss requires more space than the universe has atoms and/or plank-lengths, this entire discussion feels incredibly arbitrary.
Consider Graham's number for example. Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely.
Despite this random number being a finite value describable by a finite number of digits, there's simply not enough space in the known universe to actually write down the hypothetical random number chosen. I've chosen numbers so large that its impossible for you to literally describe these finite numbers.
Your insistence here & in the other sub-thread that these numbers are represented as a string of digits suggests that you don't understand the argument being made.
If you don't insist on this, then let me give a quick counterexample to "All the numbers larger than say... a googleplex": googleplex + 1
Sure. But my own understanding of your stance has evolved throughout today. My current counter argument is:
> Generate for me a random number between Googleplex and Graham's number, and describe it to me uniquely
I know you cannot do this, and I presume you also know you cannot do this. Just because there exists a finite representation doesn't mean you can tell me that representation.
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I insist upon this because your requirement of "finite representation" is seemingly arbitrary to me. We can enumerate all important numbers as say... all numbers represented by algebra (addition, subtraction, multiplication, division, roots, exponents, variables, logs, sine, cosine, integral, derivatives, Knuth's notation, and other functions... etc. etc.) that can be described in fewer than 1-million symbols.
And now we have a significant number of "irrational" and "i" numbers, specifically the set that we'll figure out with modern mathematics. Its a finite set (by bounding it by 1-million symbols, we have a finite number of numbers), and arguably the more important set of numbers that represents how modern math functions.
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I dare say that all "important" numbers follows my (relatively arbitrary) definition above (all numbers describable in 1-million modern mathematical symbols or fewer), and is certainly more important than say... most of the numbers between googleplex and Graham's number.
Just to be clear, I'm not the person you were originally replying to. You can describe what you're talking about by writing a program--whether or not it terminates in our lifetime does not change that it is in fact a representation of the number.
I can confidently say that the space to uniquely describe a truly uniformly random number between googleplex and graham's number is so large, THE PROGRAM cannot be written down in this universe even with all the atoms in the universe at our disposal.
Graham's number is very very very large. It is finite, it is an integer, but it is absurdly huge. Graham can describe Graham's number, but arbitrarily / uniformly picking a number close to it at random is basically impossible.
> uniquely describe a truly uniformly random number
This is the hard part. Picking a randomly uniform number "close to" Graham's number.
Graham's number itself is easily described of course: I can just say "Graham's Number". The numbers "close to it", (say, +/- 1% of Grahams number), are impossible to describe.
If you don't believe me, then please ship me the impossible number of hard drives that describes one such number, but you will have had to have picked it truly randomly.
Pick one at random.
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EDIT:
> Graham's number can be computed in a few lines of code using Knuth's up-arrow notation.
Also, this is wrong. The first number of the pyramid can be described in up-arrow notation. But even the 2nd number of the pyramid requires g1 (ie: ~7.6 Trillion) up arrows to describe.
I can safely say that g3 (which requires G2 arrows to describe) has more up-arrows involved than there are atoms in this universe. So g3 already cannot be described by a computer program using up-arrow notation alone. And Graham's number is g64, sitting on top of a huge pyramid of such numbers.
Again, you're implicitly assuming the representation has to be a string of digits. I'm not sure that I can convince you, or even what to convince you of, as you aren't accepting the premises of the argument. The numbers +/- 1% of Graham's number can be trivially represented with a program. Likewise, here's some people codegolfing programs to output Graham's number[1]. If it seems like I'm cheating by using too powerful of a method, that's the point that the original person was making: there exist real numbers that can't be described even like this.
> Again, you're implicitly assuming the representation has to be a string of digits
No. I'm implicitly assuming that g64 different numbers requires at least log(g64)/log(character size) different atoms to describe (assuming each atom in this universe can represent say 256 different states, ie a byte, then all the atoms in the world cannot be lined up to uniquely describe the entire set of numbers of g64 +/- 1%).
Lets say you have a program that accurately describes g64, good job. Now g64-1. Now g64+1. Now... g64+2. Etc. etc. I get that, you can represent some of these numbers in a compact form. But I'm asking for something far more difficult.
How many symbols do you need before you can describe g64 +/- (1% of g64)? Well... a lot, it turns out. The number of programs you'd have to write would fill all the hard drives in the universe before you're done writing even a smidgen of them.
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And my challenge is effectively closer to describing a random number from 50% of g64 to 100% of g64. This will require log(g64) / log(character size), which is easily beyond the number of atoms or even the arrangement of atoms in this universe. That's just the innate limitation of language in general.
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The numbers of g64 +/- 1% are so huge that there's no way any program written on any kind of hard drive can describe those numbers. The shear number of numbers destroys the capabilities of all the hard drives of the world. It doesn't matter what magical notations you use, they're too large to reasonably describe.
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EDIT: Or maybe this is more easy to think about? All programs of size 1MB or smaller can at best, represent 2^(8 * Megabyte) numbers.
All programs of size 1000 Zetabytes or smaller can at best, represent 2^(8 * zetabytes) number of numbers.
As it turns out, 2^(8*zetabyte) is smaller than 1% of g64. So you cannot possibly uniquely represent those numbers (g64 +/- 1%) even with 1000 Zetabytes worth of storage.
If we extend this out to another few dozens sets of magnitude, by multiplying by 10^300 or so, its still smaller than the space of g64 +/- 1%, so even if we used all the atoms of the universe in a giant, intergalactic hard drive, we still can't represent this set of numbers.
> If unicorns exist, but humans can never observe them do you really care whether they exist or not?
I do, actually.
It depends what you are using the unicorns for. If you use them for their blood for extending your lifespan, no they don't exist. If you are using the metaphor of unicorns as "the go-to thing which is used as a counter-example of existence", then they do exist. "Existing" is nebulous.
> The interesting number paradox only works for countable sets unless I am mistaken. You can have "interesting" real numbers.
>The ancients may have understood negative numbers, but IIRC the concept of zero (additive identity) wasn't understood until Arabic Numbers / Algebra. So "zero" isn't real, at least not to the Romans or Greeks.
The idea of the point as a pseudo-algebraic value was very much prominent in euclids elements. The symbol came later, but of course the concept was very much established.
It is preposterous to attribute to a civilization which, in Archimedes, discovered calculus, that they did not understand the concept of zero. They just thought far more geometrically than most people are used to.
Also by "real" he means cauchy sequences of rationals, or some equivalent definition.
> sqrt(-1) occurs in "real" voltages / currents all the time today in Electrical Engineering
I'd say sqrt(-1) describes "real" voltages / currents; but we can find other models which don't involve sqrt(-1). Those models may be more complicated, and have fewer rich connections to pre-existing work, but they're just as valid. This is usually where appeals to Occam's Razor come in, and physicists start appealing to "elegance", "beauty", etc.
You're not going to be able to describe how voltages/currents work in an AC circuit unless you reach into 2-vectors / matrices. In Complex Arithmetic, we call it the "imaginary axis". But reach into Linear Algebra and you'd just call it a "2nd dimension".
When things oscillate (and AC current is oscillating), there's just certain mathematical properties that occur. It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly.
Or, have a 2-vector linear-algebra representation, since its equivalent.
EDIT: I guess there's the differential-equations representation, but I feel that reaches into linear-algebra very, very quickly... or complex numbers. (E^(pi * i) keeps coming up...)
> It doesn't matter if its electricity, a spring/mass system, a bridge, sine-waves, music theory, Fourier Series or anything else, its all going to be represented by complex-math extremely elegantly.
This is an appeal to "elegance", which I mentioned in my last sentence. I don't disagree; but it's not quite the same as proving "existence".
I have never seen complex numbers represented via linear algebra... would love a crash course on this as I find reasoning about complex numbers to be very difficult and find linear algebra to be imminently logical in all cases.
A complex number is "just" a 2-vector. Instead of calling the two dimensions "x and y", consider them "(unnamed) and i".
For example: the vector <1, 1> is equivalent to 1 + i. The complex conjugate is a reflection across the x axis: <1,1> into <1, -1>, or in the Complex representation from 1+i into 1-i.
Addition of numbers progresses just like addition of vectors: <1,1> + <2,2> == <3,3>. Similarly: (1+i) + (2+2i) == (3+3i).
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IIRC, Multiplication of Complex numbers is a rotation across the two axis. The overall concept is pretty difficult to describe succinctly, but it is "obvious" in a simple case.
Multiplication by "i" is just a 90-degree rotation counter-clockwise.
That is: <1, 1> rotated 90-degrees is <-1, 1>. Or 1+i rotated 90-degrees is (i * (1+i)) == (-1 + i).
Rotating again is another multiplication of i, or i * (-1 + i) == (-1 - i) == <-1, -1> in vector form.
Then <-1, 1> and finally <1, 1> again (4x 90-degree rotations == 360 degrees).
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Etc. etc. etc. Everything you do in complex arithmetic has a good interpretation in the vector / geometric space.
A lot of the intuition needs to be built from the polar form of complex numbers: r * e^(theta * i), since the use of complex numbers is largely about rotations about the axis.
It is interesting that the source article from Chaitin it seems to regard its own title as mere wordplay, it does not want to get into the Morpheus Question “What is real? How do you define real?”... But the response seems to take it very seriously and get somewhat upset about it.
Other than those aspects the discussion is kind of snore-fest...
Both this article and that of Chaitin which is cited in it, fail to say clearly what they mean by "physical reality" and they also mention some things that are irrelevant to the title question, such as DNA.
All the physical quantities that can be modeled as continuous, i.e. by real numbers, are derived in some way or another from the positions in space and time. All the physical quantities whose dimensional formulas (in natural units, not in SI, which obscures some of the relationships between physical quantities) do not include space/time quantities, e.g. electric charge, magnetic flux, angular momentum and a few others, are discrete.
So the question whether the real numbers correspond to the physical reality must be reformulated less ambiguously as the question whether the space and time are continuous or discrete.
The traditional model of the space and time as continuous matches successfully all known experiments.
There have been a few attempts of creating models for various parts of the quantum physics by using a discrete space-time, i.e. where the points in space-time form a lattice (i.e. there are minimum distances between points that cannot be subdivided to find an intermediate point) and any physical process progresses by jumps between discrete points, instead of continuous movements.
However, I am not aware of any useful result obtained with such models.
>concludes that these difficulties undermine the common assumption that real numbers underlie physical reality, strongly suggesting that physical reality may in fact be discrete, digital and computational.
Well, it should precise from what population this "common assumption" is supposed to be taken of. Arguably, most humans that ever existed had basic integer arithmetical notions at best.
What humans take for facts are not given but constructed on social interactions, even when protocolled around scientific methods relying on falsifiable models and sound metrics. The best that science can tell us is how much sound our model is in regard of available data and how much resources we need to produce and test an other one.
Note that philosophy of Karl Popper is not itself falsifiable.
> This statement describes a faith, not a scientific principle.
Ultimately, principles lie on some mental faith or an other. There is no irrefutable evidences that solipsism is less sound than believing in an "outside world".
I find it ironic that the "real" numbers are so named because they are definitely less "real" than the integers or rational numbers since they cannot (with p=1) be individually named or specified.
The name was introduced by Descartes in contrast to the disreputable "imaginary" numbers occuring in investigations of roots of polynomials. So not that ironic, more like rain on your wedding day ...
Any mathematician, literally all of them, will tell you there is an injection from the set if rationals and reals, and that your foundational system is likely impoverished if this injection does not behave as the identity
I think it’s worth talking about how the language works here.
We say, “the natural numbers are a subset of the reals,” and this is a sensible thing to say.
We also might say, “you construct each successor number in the natural numbers as n + 1 = n ∪ {n}”. And then we say, “a real number is a Cauchy sequence of rational numbers, or a Dedekind cut of rational numbers.” From a set-theoretic perspective, “the natural numbers are a subset of the reals” is obviously untrue with these definitions, and it’s worth spending a moment to think what the statement actually means, or how you would have to interpret it in order to understand the truth of the sentence.
I might translate the sentence as “there is a left-cancellative morphism from natural numbers to real numbers,” but then I’d have to define what category I’m using, and what the morphisms are—which is usually implied. You end up having to stand on top of a surprisingly tall stack of proofs in order to say “the natural numbers are a subset of the reals” and actually explain what you mean by that, rigorously, from foundations.
Or, put another way, it’s sometimes useful to understand what you mean by “behave as the identity”.
Retrofitting the set theoretic definitions is trivial. Take the set of cauchy real numbers, remove all reals equal to a real-that-would-be-a-rational, i.e. constant sequences. Union together this set with the set of rationals defined as ratios of integers. Repeat as needed with smaller sets.
Obviously these reals are isomorphic to the cauchy reals. The reality, of course, is that no one actually works with foundations, they work with the intuitive understanding that 1:N is 1:Z is 1:Q is 1:R.
If your real numbers are Cauchy sequences of rational numbers, and your rational numbers are a subset of real numbers, then your rational numbers are a subset of Cauchy sequences of rational numbers (which violates the axiom of foundation).
This is not as trivial as it sounds, which is why we invented all these different tools for explaining what “is” or “equals” means in mathematics without resorting to set equality (equality, isomorphism, equivalence, etc).
What? Of course, the sets of integers, rationals and reals respectively are not identical , but the integers are a subset of the rational and the rational a subset of the real numbers.
"In material set theories, the elements of a set X have an independent identity, apart from being collected together as the elements of X. Frequently, they are also
sets themselves.
These are also called “membership-based” set theories.
In structural set theories, the elements of a set X have no identity independentof X, and in particular are not sets themselves; they are merely abstract “elements”
with which we build mathematical structures.
Worse, if you take "Imaginary Numbers" to be only the necessary complement to make polynomials factoring complete... And "Imaginary Numbers" are definitely a necessity to describe reality...
Meanwhile almost all "Real Numbers" are nothing but phantasmagorical numbers which we cannot name nor talk about them...
It seems beyond the pale that we still are teaching these bizarre axioms instead of the Computable Reals which don't have all these phantom numbers...
"God made the integers; all the rest is the work of Man." ~ Leopold Kronecker, when criticizing Cantor's work [0]. See also Stephen Hawking's Anthology by a similar name. [1]
Rational numbers are pretty easy to understand. There are easy to understand proofs that at least some irrational numbers (famously square root of 2) exist. That is, the square root of 2 cannot be represented as a rational number. I'm sure there are other such proofs.
My take is that Kronecker and Hawking were either making an ironic statement or just expressing irony or exasperation.
I think you are missing the point. Kronecker does not deny that irrational numbers exist, he states his belief that the fundamental number system which is inherent to reality are the natural numbers. That (by divine authority) are the irreducable thing which other number system must take from. E.g. the rationals are pairs of natural numbers, the reals cauchy sequences of rational numbers etc.
> I'm sure there are other such proofs.
The reals are uncountable.
DNA is digital, but life is made from its environment, not only from DNA. However, it does not answer the original question, anyways.
I think Borel's number is not very well defined, and the examples given there make that easy to see, because an infinite sequence will not reach all possible questions. However, that is not the only problem with Borel's number, anyways.
The numbers, people, etc are not "existence" except by relation of other things (e.g. you can define additions/multiplications of numbers, greater/less/equal of numbers, and physical objects moving by the laws of physics according to the numbers, physical objects collide with other objects and interfere with them by gravity and other forces (i.e. they are not in isolation), etc). The relations between everything is "real", rather than the objects themself. Due to this, it can occur by many ways, and they are ultimately related to "you", too. Essentially, all existence must be by all relations of everything by everything.
There is problem of what is "is" and what is "reality"; they are not so clearly defined. "Real numbers" can be explained by mathematics, but the meaning of the words in English (or other languages) is not really as precise as it can really be. (And then, even with mathematics, there will be "abuse of notation", etc.)
But, anyways, absolute Truth is inexpressible, so, due to this, you cannot possibly explain everything. You only explain certain concepts, which are descriptions of some things, but whatever is explained, there will be everything else which is not explained and cannot be explained.
A real number might be given a finite description, but it is not necessarily the case. However, sometimes by a part of it, you can figure out some of the other relations, but not all of them, e.g. if you have 100 digits then you can figure out if it is less or greater than another number with 100 different digits, but if they are the same then you will not figure out. But, that is only one way. It is not the only way to make a description, finite or infinite or otherwise, even if there will be one. Even so, that does not mean that such a description necessarily "is" such a "real number", anyways (especially since, as I mentioned above, absolute Truth is inexpressable).
By mathematics, you can prove mathematical things, but you will not make the "set of every possible things mathematically"; rather, it will have to be specifically the theories which are being handled/proven, which are going to be done by mathematics.
Is the integral a "thing"? - No, it, along with the reals, is the method (we call "analysis") that we humans invented to solve a certain class of problems in science and technology.
If you want to understand the epistemic status of any formal object related to "infinity" in Mathematics, the best three starting points are Religion, Sociology, and the History of Mathematics. I think a good History of World Religions course plus some focused reading on notions of infinity is likely more useful than an Analysis course for really understanding questions like "Are Real Numbers Real?"
I do not think it is possible to divorce even modern notions of infinity from the deep impact that superstitious thinking has had on the direction of Mathematical thinking -- and thinking in general about anything remotely metaphysical, really -- for the last few thousand years (with only very recent counter-examples).
Yes, but unfortunately they require some digestion. This thesis isn't, for example, fleshed out in a single pop sci book or even dissertation or academic book afaict. It's more like: reading a lot about the history of philosophy, mathematics, and religion. (That said, I think most people in those three fields -- philosophy, history of mathematics, and religious studies at least as an academic field -- will agree in one way or another with the general sense that mathematical notions of infinity have definitely influenced and been influenced by religion and spirituality.)
The closest single source that comes to mind, at least for one region of the world, might be Russell's history of western philosophy, which commits more time to both theology and mathematical history that is typical for that type of text. But, it's a big tome and only here and there provides nuggets of insight on this topic.
Thinking about other sources, if you are interested in this thesis, I think you'd get a lot out of reading the works of people like Descarte, Leibniz, or even Godel. Also biographies about the same, because the writings by the men themselves are more like primary sources than summative/synthetic expositions. By which I mean, you have to study the text, its cultural context, and its impact on the history of mathematics (rather than just read and comprehend the words on the page). So reading the texts plus the biographies plus knowing enough about mathematics to understand how their ideas do and do not permeate the field will, I think, give you a decent sense for the thing I'm saying above.
Sorry I don't have a single easily digestible podcast/book/audio book :(
if you put 1 out then there is no reason to believe it is uncountable because you can sort the list such that the diagonal is all 0s (if you know an argument that shows [0,1) is uncountable I'd be happy to know)
It is a superset of (0,1), which is uncountable because it is an open interval. All open intervals except the empty one are uncountable. What kind of proof would you like?
There is also no such thing as an uncountable list. Lists can be indexed by integers, which makes them all countable.
the proofs that I've read shows that the list of infinite binary strings between [0,1] are uncountable because the invert of the diagonal of the list is not on the list, do you know any proof that does not refer to the invert of the diagonal?
you can't prove something just by assuming it that would be trivial, if we can build or convince ourselves how to build a wormhole or a time machine then we can believe in Real numbers
What Koshkin says is correct and has already been proven by others. You want to get a book that covers sets, maps and cardinal numbers, and read it carefully if you are interested in this sort of stuff.
I can see that you can symbolically talk about the power set of (0,1) and say it's cardinality is bigger than the set itself (although I haven't studied sets and cardinalities deeply enough) but I can't see what Real numbers offers against the computable Universe hypothesis
Of the three linguistic entities in this sentence, only one is well defined. Let's rephrase from a question to a statement and look at the veracity: "Real numbers are real".
- "Real Numbers" - this is the one thing we can formally define, but there is implied baggage - does the concept of real numbers exist, or does at least one real number exist? The former is just ∃x(x ∈ ℝ), √2 proves that, the set ℝ must exist, no major problem there. Really this question asks "For ∃x(x ∈ ℝ), is x 'real, existing in reality'?" For that we need to look to the other two words.
- "are" - the copula (is/to be) is a doozy and a perennial semantic footgun, put a pin in that
- "real" - this is a nebulous concept, and where the weight of the discussion lies. Basically it is asking can something with infinite information be? And there's that pesky copula.
Is/be is tricky, particularly in English, because its usage is heavily overloaded. It is used to express
- Strict equality: "Two plus two is four"
- Categorization: "A frog is an amphibian" (all frogs are amphibians, not all amphibians are frogs)
- Description: Grass is green (grass has the property green, but green is not grass).
- Predicates (in general): "It is raining" (in many languages, there is no copula, and the sentence is closer to "Raining." or "It rains." see Zero Copula [0]) (arguably, all the above are involved in predicates, definitions are hard)
In "Real numbers are real", I would contend we are talking about the description usage or predicate. Real numbers have the property of "existence". This is where a messy problem gets ultra messy. But at the same time, I believe the problem more broadly collapses to "Does X Exist?" [1] for any X. Chairs are made up of components which do not individually have anything chair-like about them, but the arrangement of matter (the process of producing the chair) nonetheless has the description of chair-like-ness. Real numbers comprise a process which produce them (stating a mathematical formula and asking what the answer is), and thus the process defines their existence. Whether or not an object with infinite information exists is irrelevant if the process to produce that object is finite, and that object can be used to produce another answer (e.g. we can use √2 to define the dimensions of the A Series of paper, or broadly solve for the dimension of a 45° triangle with side length 1). Its existence can no more be denied than the existence of chairs, or heaps, or other intangible ideas.
(I suppose this argument only applies to constructable/algebraic numbers, which means the original question is really only talking about transcendental reals with no "nice" expression. But that gets into Hilbert's seventh problem, and whether you can derive an expression of any arbitrary transcendental using two algebraics).
"Reality" as a linguistic concept is also highly relative. Are dreams real? They are experienced, they can be measured with medical equipment, and they can have lasting psychological side-effects; they are "real". But the events of the dream do not themselves have the observed side-effects in "consensus reality"; they are not "real".
tl;dr - language is nebulous, and nebulosity is challenging [2]
Also, I realized there may indeed be a class of reals that are a subset of the reals, which might not actually exist. Within the reals, you have:
- Reals equal to integers and rationals: Exist insofar as integer/rationals exist
- Algebraic vs Transcendental: Algebraic reals exist insofar as finite algebraic expressions exist. I believe √2 is approximately as real as 2.
- Special-case transcendentals like pi, e: Same "realness cardinality" as √2.
- closed-form transcedentals like 2^√2 (Gelfond–Schneider constant): Also same "realness cardinality" as √2.
From there, I think there is a "realness cardinality" transition, much like the countable to uncountable transition. These "boring transcendentals" have absolutely nothing notable about them, no way to label them using finite information, they are only expressible by the full, infinite description of themselves. these bad boys are ones that might not exist a priori, and the real heart of the question in TFA.
I still contend the "boring transcendentals" do exist, because they allow dense cover of the [0,1] interval. In the same way that you can have a "lazy list" of the primes and can do a containment check of said list without having to generate all the primes in between, you could do a containment check of any piece of information on such a lazy set of transcendentals, including an infinite string of decimals, and it would always return True. So in a sense, every real must exist in that container (that's kind of how we arrived at the set of reals - we can prove their absence would be a contradiction in the rules of algebra).
>In Plato and the Nerd, I review Claude Shannon's channel capacity theorem [Shannon, 1948], which states that any noisy observation of anything conveys only a finite number of bits of information. Borel's know-it-all number cannot be encoded with finite number of bits unless the list all possible yes-no questions is finite, which it is not. It is easy to construct an infinite sequence of valid yes-no questions. For example, let the first question be "Is one a whole number?" Let the second question be "Is the answer to the first question YES?" Let the third question be "Is the answer to the second question YES?" And so on. As a consequence, Borel's number cannot reveal itself to us unless we invent some noiseless way of observing a thing in itself.
It is true that he showed an example of an infinite list of valid yes/no questions, which would lead to a number with an infinite amount of digits. But despite it being infinite, by the way it was constructed you already know all the digits (they will be all 1s), meaning that that number's entropy is remarkably low and that number can be expressed with a finite number of bits of information. So Shannon's theorem isn't a problem here.