Well, in the current theory of the physical world (standard model, or whatever variant of it you prefer) all atoms of the same element are supposed to be EXACTLY the same (indistinguishable, even in principle, with absolute precision), right? — Mephist

If you mean electron microscope photos of a lattice of atoms, those are still subject to the quantum and classical measurement problems. To clarify what I said earlier:

* In quantum theory, nothing has an exact position at all. Before it's measured, it doesn't have a position. Sometimes that's expressed by saying that it's in a "superposition" of all possible positions. Then when you measure the particle, it (somehow -- nobody understands this part) acquires a position drawn randomly from a probability distribution.

This applies to all objects, large and small, though the effect is much more pronounced when an object is small.

For example you yourself are where you are in space right now because that's the most likely place for you to be. It is statistically possible that you might suddenly find yourself in a statistically improbable place. For example all the air molecules in your room could move to the corner of the room and you'd have no air. That is extremely unlikely, but it has a nonzero probability. It could happen.

So even if all instances of a given particle are the same, you still have no idea exactly where it is, or exactly how long a line made up of these particles is.

Atoms, by the way, are way too large and they're all different. I don't even know if two hydrogen atoms are exactly the same.

However it's interesting that every electron in the universe is (as far as we know) exactly the same. Why is that? It's another thing nobody understands.

There's a famous theory of Wheeler and Feynman, "not to be taken seriously" but evidently mathematically possible, that the reason all elecrons are the same is that there is only one electron in the universe. It moves rapidly backward and forward in time; and that's why whenever we see it, it appears to be in a different place. Like a point moving up and down from below a flat plane to above it. Every time it crosses the plane, you'd see an instance of the point. You'd think there are lots of points, when in fact it's only one point traveling perpendicular to your reality.

https://en.wikipedia.org/wiki/One-electron_universe

* And even in classical physics, a measurement is only an approximation.

So now I'd like to re-ask your question but pertaining to electrons, which are all exactly the same. But electons are very small and extremely subject to quantum effects. You simply can't say exactly where an electron is at any time. Only where it's statistically likely to be. One, because nothing is exactly anywhere at all in quantum physics; and even when it is, after a measurement, the measurement itself is subject to classical approximation error. You made the measurement in a particular lab with a particular apparatus, built and operated by humans. It's imperfect and approximate from the getgo.

The most accurate physical theory in the world, Feynman's quantum electrodynamics, has predicted some quantity or other to 12 decimal places. I read that somewhere. 12 decimal places is pretty good. But mathematically, it's not exact at all. If you had 12 decimal places of pi it wouldn't be pi.

Let me add that modern physics no longer thinks about "particles" like electrons and atoms. Rather, everything is interacting probability waves. An electron isn't a pointlike thingie. An electron is a probability wave, smeared all over the universe. When we observe it, we find that it appears to be in a particular location defined by a probability distribution.

There's even a current thread on this site on that very subject. "Fieldism versus materialism." We don't have particles or things or objects anymore. Just probability waves. Very strange, what the wise physicists are up to lately.

https://thephilosophyforum.com/discussion/7414/modern-realism-fieldism-not-materialism/p1

So the short answer to your question is, no. We can never know or measure an exact distance in the physical universe.

You are right, we will never be able to check if this theory is correct with absolute precision, not even in principle, because all physical measurements must necessarily have a limited precision.
Nevertheless, in principle (if you have enough computing power and the model is complete and consistent - I know, that's a big if) you can use the mathematical model to make predictions about the result of experiments with arbitrary precision. — Mephist

Well there are no computers with arbitrary precision. That's the problem with the computational theory of the universe. There's too much it can't account for.

It's those pesky noncomputable numbers again, one of my favorite topics. If the universe is "continuous", in the sense that it's modeled by something like the real numbers; then it is most definitely not a computer or an algorithm. Because algorithms can't generate noncomputable numbers.

So, in a model of the physical world where all distances have to be multiple of a given fixed length (I don't know if such a model exists, but let's assume this as an hypothesis), there cannot be squares
made of unit lengths. — Mephist

Yes ok. I happen to have visited a world like that once. Manhattan. It's composed of a grid of mutually perpendicular streets and avenues. (Not entirely, but for purposes of discussion).

How far is it from 1st street and 1st Avenue to 2nd street and 2nd avenue? Well it's not $\sqrt 2$, because you can't drive or walk diagonally through the buildings. Rather, the distance is 2. You have to walk one block north and one block west.

There's a name for this: The taxicab metric. In the taxicab metric, the unit circle is a square. Next time some philosopher tells you there are no square circles, you can go, "Oh yeah? There are in the taxicab metric!" and thereby confound him.

But you know I still don't agree with you about squares. Of course there are square blocks in New York City. Actually they're rectangles because the streets are closer together than the avenues, but let's ignore that for sake of discussion. There are square blocks. You just can't walk along the diagonal! Your distance is the sum of your vertical and horizontal travel.

So in your hypothetical world there would be squares and if you want to go from (0,0) to (1,1) you simply have to move 2 units, one unit right and one unit up. You can't travel along the diagonal because at the finest level of the lattice, you can't move diagonally. I have no idea what that means physically but I think you are overthinking this or underthinking it. It's kind of tricky, which is a problem for the theory.

I don't know what these unit lengths are made of: they are simply the building blocks of my model, the same as the "strings" of string theory or the "material points" of Newtonian mechanics! — Mephist

Does the taxicab metric help your thinking?

By the way, to be clear, I don't believe in this theory! — Mephist

Some people do! There are some discrete or quantized theories of reality around, like loop quantum gravity. From the article: "The structure of space prefers an extremely fine fabric or network woven of finite loops."

The ultimate nature of our physical world is wide open to speculation, informed and otherwise. Even our wisest don't know.

But I don't speculate about the physical world. Math is so much simpler because it doesn't have to conform to experiment! In math if you want a square root of 2, you have your choice of mathematically rigorous ways of cooking up such a thing.

To state the problem succinctly, set theory allows that two distinct things have the same identity, in the same way that we might say two distinct things are equal. The faulty premise is that things with the same value "2" for example, are the very same thing. In other words set theory premises that, "2" refers to an object, rather than a value assigned to an object. It is a category mistake to treat what "2" refers to, as a particular object, rather than as a universal principle. — Metaphysician Undercover

I see your point.

You are saying that there are 2 books and two fish and 2 schools of thought; but there is no 2 in the abstract.

Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction.

The invention of the concept of number was a great leap forward for mathematics and also for civilization. That let us study 2 + 2 without having to say 2 fish plus 2 fish and then having to re-calculate 2 elephants plus 2 elephants, and then still not being sure about 2 birds plus 2 birds.

It's the power of abstraction that allows us to handle all these cases at once.

You reject abstraction. You're not wrong. It's just a nihilistic philosophy of math and of civilization. Everything about our lives is abstraction. We can't live without abstraction. How do you live without abstraction? How do you function, not believing in numbers?

Now if you want to say, "Yes but you admit numbers aren't real, they're only an abstraction!" I respond: Yes exactly. And traffic lights aren't real either, they're only an abstraction. Law is an abstraction. Government is an abstraction. Science is an abstraction. The Internet is an abstraction. Humans have the power of abstraction. It's how we crawled out of caves and built all this.

"The square root of two" has no valid meaning in the rational number system. This means that taking a square root is not a valid operation. — Metaphysician Undercover

Let me restate your quote as a formal argument.

P: The square root of two" has no valid meaning in the rational number system.

C: This means that taking a square root is not a valid operation.

The conclusion doesn't follow from the premise. A valid conclusion would be, "This means that taking a square root is not a valid operation in the rationals. And of course that is correct.

One can, however, conceive of and build, with logical correctness. systems of numbers in which there IS a square root of 2.

I have an example on my mind, I'll toss it out there.

Do you know modular arithmetic, or the "integers mod 5" and so forth? Telling time is just the integers mod 12. If it's 11 now, what time will it be in 2 hours? The answer is 1. We add "mod 12," which means first do normal addition, then subtract out the largest multiple of 12 we can. In fact you've alluded to this phenomenon. When we divide two integers we get a quotient and a remainder. In modular arithmetic, we only care about the remainder.

We can do the same trick with any modulus, as it's called. Consider the integers mod 7. They consist of the symbols 0, 1, 2, 3, 4, 5, and 6, with addition and multiplication mod 7.

In the integers mod 7 we can add, subtract, and multiply. In general we can't divide. So the integers mod 7 are a perfectly valid system of numbers, not unlike the integers, but not like them either. [They're a quotient of the integers if you took abstract algebra].

Now, what is 3 x 3 in the integers mod 7? Well, 3 x 3 = 9 normally; and in the integers mod 7, the number 9 corresponds to the number 2.

So 3 x 3 = 2. That is, 3 is a number that, when squared, results in 2. So in the integers mod 7, 3 is the square root of 2. Just to startle people I'd go as far as to write

$3 = \sqrt 2$

Like every statement in math, its truth value depends on the context. In the context of the integers, the statement is false. In the context of the integers mod 7, the statement is true.

[By the way what about -3? Well in the integers mod 7, we have -3 = 4. That's because 3 + 4 = 0. Now 4 x 4 = 16 = 2, after we've subtracted off 14. So even in the integers mod 7, it's true that if x is a square root of something then so is -x. That's a general rule that you can deduce just from the laws of basic arithmetic. If you took abstract algebra, we're talking about the ring axioms].

Now that's interesting, but it doesn't solve the problem of having a square root of 2 that also knows about the rationals. But there's a perfectly good number system called $\mathbb Q[\sqrt 2]$ that is:

* A number system where we can add, subtract, multiply, and divide (except by 0); and

* It contains an exact copy of the rational numbers; and

* It contains a square root of 2.

There's no question that such an object exists in mathematics. It has mathematical existence by virtue of the fact that we can (1) characterize it axiomatically; and (2) construct it out of bits and pieces of set-theoretic operations. And even though you don't like set theory we can do the same thing in category theory or homotopy type theory or without any foundation at all simply by writing down the ring axioms and modding out the ring of polynomials having integer coefficients, by the ideal generated by the polynomial $x^2 - 2$. I know you don't like technical stuff by I'm pointing out that I don't need set theory to build a square root of 2.

Now if we are talking about mathematics; and an object has mathematical existence, by what right do you require some other standard of existence?

By the way in the integers mod 5, we have

2 x 2 = 4 = -1.

So the integers mod 5 have a square root of -1.

Now if you want to say, "Yes but you admit numbers aren't real, they're only an abstraction!" I respond: Yes exactly. And traffic lights aren't real either, they're only an abstraction. Law is an abstraction. Government is an abstraction. Science is an abstraction. The Internet is an abstraction. Humans have the power of abstraction. It's how we crawled out of caves and built all this. — fishfry

Indeed, and one needs the abstraction of itself abstraction in order to complain about abstraction in the first place.

It's those pesky noncomputable numbers again, one of my favorite topics. — fishfry

Now those are strange entities, unlike the essentially finite square root of 2 (as you've already noted.) A dark ocean of infinitely informative numbers that can't be named is far more poetic and disturbing than little old $\sqrt 2.$

Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO).

The way I reason about it (ie, as a software engineer), real numbers specify the convergence characteristic of approximation processes that deal with real world problems. What you are saying is that people should study the numerical methods that approximate real world solutions, but shouldn't study analysis of this essential characteristic, which seems to me questionable. Maybe your point relates to the general debate in society - whether engineers should study only constructions and hands-on skills and not analysis (how to derive properties of those constructions), but even then I am leaning towards the usefulness of theoretical understanding.

In Eastern Europe, software engineers studying for bachelor's degree have real analysis, abstract algebra, differential equations, etc, as mandatory subjects. That much is indeed true and many are dissatisfied with the curriculum for being too math heavy.

You are saying that there are 2 books and two fish and 2 schools of thought; but there is no 2 in the abstract.

Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction. — fishfry

Right, my argument is that there is no such thing as an abstract object represented by "2". I replace this supposed object with something closer to the truth, "what 2 means". The symbol "2" has meaning. The key point I make, which some argue against, is that the meaning of "2" varies according to circumstances, context. This variance, or difference, indicates that it is impossible that "2" signifies an object. The Platonic realist argues that this is a difference which doesn't make a difference, but in making this argument, the realist has already invoked contradiction. This contradiction supports the realist's category mistake of failing to distinguish between a particular object, and a universal (meaning).

Well, imagining or mentally conjuring up a "thing" that is 2, by itself, is one of the greatest intellectual leaps of humanity. As I've noted before, you appear to reject the very concept of abstraction.

The invention of the concept of number was a great leap forward for mathematics and also for civilization. — fishfry

I would say that you call this a "leap forward" as determined in relation to a pragmatist perspective. This move serves a purpose. But in relation to a true ontology, it is a blatant falsity put forward for a purpose. Therefore it is a move of deception. The pragmatist, from the perspective of the metaphysician, is a deceiver, a sophist.

That let us study 2 + 2 without having to say 2 fish plus 2 fish and then having to re-calculate 2 elephants plus 2 elephants, and then still not being sure about 2 birds plus 2 birds. — fishfry

Such a study is the study of meaning, it is not a study of objects. When it is presented as a study of objects it is a deceptive presentation.

t's the power of abstraction that allows us to handle all these cases at once.

You reject abstraction. You're not wrong. It's just a nihilistic philosophy of math and of civilization. Everything about our lives is abstraction. We can't live without abstraction. How do you live without abstraction? How do you function, not believing in numbers? — fishfry

I do not reject abstraction, I take it for what it is, and that is not a process of creating objects, it is a process of generalization. You don't seem to understand what "abstraction" means. Fundamentally, abstraction replaces particulars with generalizations, universals. If the universal (the product of abstraction) is presented as a particular (object), what is proposed is clearly a false proposition.

So 3 x 3 = 2. That is, 3 is a number that, when squared, results in 2. So in the integers mod 7, 3 is the square root of 2. Just to startle people I'd go as far as to write — fishfry

This is not true. It's false to claim that 3 is the square root of 2, just because you've taken seven away, just like it would be false to claim that one o'clock in the afternoon is the same as one o'clock in the morning, just because you've taken twelve hours away. You've just presented a mathemagician's trick, pure sophistry.

Like every statement in math, its truth value depends on the context. In the context of the integers, the statement is false. In the context of the integers mod 7, the statement is true. — fishfry

Yes, context is key, as I stated above, and the importance of context is reason why a numeral cannot refer to a number which is an object. The numeral has a meaning which is context dependent. The numeral "2" has a different meaning in mod 7 from the meaning it has in the rational numbers. In your example, you are conflating two distinct concepts. Your use of "the square root of two" is right out of context, because "2" has a different meaning in mod 7. Therefore your use of "2" is irrelevant to any common use of "the square root of two", because "2" has a different meaning in mod 7, like "one" has a different meaning as "one o'clock" from the meaning it has as a rational number. If you deny this, you arguing by equivocation.

Your objections apparently boil down to a demand that mathematicians revise their well-established technical terminology (existence, object, etc.) because some of the definitions conflict with those employed in your peculiar metaphysics. Good luck with that!

Do I strike you as a person who expects people to do what I suggest?

I think your thesis "stick to finitism when teaching basic math" misses the obvious point of how incredibly messy and complex finitism is, both as a mathematical approach and as a practical application. The overwhelming majority of mathematical applications are based on the continuum - physics, engineering, etc. — SophistiCat

I'm not sure what this analysis. I have never heard a criticism of finitism that it is complex for applied applications. The whole point of finitism is that it aligns with practical application.

Engineering is not based on the real number continuum, it is largely based on differential equations that can be setup just as easily in a finitist framework.

Since I am concerned with high-school in this thread, can you give an example of an applied high-school level problem that cannot be addressed in a finitist framework of "arbitrary precision"?

As far as I know, all applied maths problem have precision constraints of their input data which results in precision limits of the output data of the algorithm (whether machine or human) solving the problem.

Even if there was a theorem that has no proof in the finite regime but does have a proof in the infinite regime (that we cannot prove the theorem for arbitrary precision, but we can prove the theorem for "all integers" or "all real numbers"), it is easy to borrow that theorem in a rigorous way by simply having the computer check the theorem up to some limit that we intend to use.

(Indeed, lot's of "theorems", i.e. conjectures, are used in applied math that have no pure math proof because they have been checked numerically to over the range in question; obviously, such a numerical check to some bound has nothing to do with the real number line.)

And this is how, in practice, applied maths work; people look up a theorem and use it, and if it works to solve the problem then that's the end of the thinking on that.

Of course, understanding what math actually is, is to understand proofs. But my whole point here is that simply positing the real numbers without a construction nor framework of rules that contains the paradoxes that otherwise occur with a naive approach, leads to a mystification of maths and not understanding of rigor. If the setup isn't rigorous, it is not a proof students are learning, but rather the applied method of "we're doing it because it works".

The way I reason about it (ie, as a software engineer), real numbers specify the convergence characteristic of approximation processes that deal with real world problems. What you are saying is that people should study the numerical methods that approximate real world solutions, but shouldn't study analysis of this essential characteristic, which seems to me questionable. Maybe your point relates to the general debate in society - whether engineers should study only constructions and hands-on skills and not analysis (how to derive properties of those constructions), but even then I am leaning towards the usefulness of theoretical understanding. — simeonz

My whole point is that students are not actually understanding the mathematical analysis if they do not actually understand real numbers. For me, simply positing the definition of real numbers in terms of some basic rules, does not lend any understanding of what real numbers are.

To repeat what I answered above to SophistiCat, the real numbers are not required to define convergence. Without the real numbers, convergence is to an arbitrary precision rather than a real number on the real number line. Arbitrary precision is perfectly adequate for any real world problem.

Because of this, the radical finitists desire to get rid of the real number system all together even at the university level. The ultra finitist dispute even "arbitrary precision" which in principle goes up to numbers with complexity that cannot be represented in the real world.

I am neither a finitist, much less ultra-finitist, for higher level pure maths. The real number system is, at the least, an interesting mathematical idea that lot's of effort has gone into developing and lot's of theorems are proven in the framework of ZFC that we have no reason to just throw away, and for me, pure maths is about the general question of "systems of symbols and rules" whatever they maybe, and so working with systems of rules that have unintuitive consequences is a good thing for the student of mathematics, as it opens the minds as to what can be done with this general "rigorous proof" based on rules and symbols.

However, for students encountering calculus for the first time, understanding the real number system is essentially impossible and a waste of time to attempt in anycase. In my experience (and it seems the experience of many posters here), teachers at the high-school level don't understand the real number line anyways and simply change from finitist "arbitrary precision" explanations while dropping in tidbits of the bizarre characteristics of normally distributed infinite decimal representations at best, and at worst provide wrong answers. I would wager most teachers and most students understand the real number line as just "numbers with decimals", such as the calculator provides, which is not the case; the calculator provides integers and fractions in decimal representation.

However, if we want to get into the discussion of the limits of this "numerical regime" approach in applied maths, it seems to be everywhere in practice.

For instance, I have always understood re-normalization in quantum physics to be exactly this "get rid of the infinities through the numerical regime" by simply measuring things and replacing divergent functions with constants. The justification of calculations outside the bounds of the reference experiments is basically a numerical regime game of "how far can the error be from this measured constant over here".

Likewise, the invention of "quanta" was due to abandoning the real number line and simply having a finite step (a numerical approach) which got rid of ultraviolet divergence and reproduced experimental result.

If physicists were not committed to a real continuum at this time, this would have simply been the obvious approach to define some "accuracy step" and then narrow in on the right value that matches experiment. And this is basically the argument of finitism in physics: there is no real continuum and so using the real numbers causes the confusions of the above kind, slowing theoretical advancements. I'm not sure if this causes confusions or not for physicists, but I have never heard a counter criticism of some example of a prediction that cannot be made without the real number line. As I mention, I'm not a finitist in higher maths, but if there's a criticism that some physics can't be done with finitist framework I have yet to hear it. The counter argument, is that keeping the real numbers around makes everything easier for both historical and abstraction reasons; just like in principle we could do physics without imaginary numbers, but no one advocates that because it would be clearly more confusing and harder to do physics without them.

As far as I know, physicists at the highest level do not need the real number line for making any prediction nor any theory, and it's simply historical accident that classical theories where developed with the motivation of a "no gaps" continuum, and it's important for physicists to learn these classical theories.

Ultimately, physicists do not justify the theorems they use with pure-maths proofs, but rather they borrow from pure maths "whatever works" and justify that in relation to experiment. Hence the "shutup and calculate" motto of modern physics (sometimes a pure maths proof extends the theory in a way that makes both intuitive sense and matches experiment ... and sometimes not, in which case no bother we'll just ignore that or say the theory breaks down at those energies). Which is why, as far as physicists I've talked to, this issue about real numbers they simply don't care about; it won't change how they calculate to get answers (unlike imaginary numbers, which would change a whole bunch of things and they would "care" about an argument to get rid of imaginary numbers, as it's clearly insane to do so).

However, regardless of whether a physicist thinks real numbers are a help or not, high school students, the subject matter of this thread, are neither learning rigorously about real numbers nor the numerical regime and, if what they are doing is not rigorous then it is not really understanding what mathematics is.

A non-computable real number r refers to a truly random infinite process, and yet the distinction between a truly random infinite process and a pseudo-random infinite process isn't finitely testable, since any finite prefix of r is computable. Since r cannot be finished, at any given time r can be equally interpreted as referring to an under-determined pseudo-random process. Yet any process we specify ourselves is fully determined. Therefore r can only be interpreted as referring to a process of nature that we are observing but that we ourselves haven't specified and have only incomplete knowledge of and control over.

Therefore when a physicist makes the observation x = 0.14 +/- 0.0001, he could be equally described as stating an interval of rational numbers or as stating an interval of real-numbers. If this sounds wrong, "because the real numbers are uncountable, whereas the rational numbers are countable", recall Skolem's Paradox that the set of real-numbers actually possesses a model in which they are countable. The only important thing to know is whether the physicist fixed the result or whether he measured the result, for constructing a certain number is different to measuring an uncertain number - this difference isn't easy to express in either classical or constructive mathematics.

I think that we might come from different understanding of what the real numbers "should" represent in modern mathematics. My real analysis textbook introduced real numbers as corresponding to points on the "real line". I was confused by this explanation and got stuck on researching fundamentals in other textbooks. Almost failed the class due to mismanagement of my time. In retrospect, the definition was characteristic of the old-school soviet style of math textbooks. It assumed that all mathematical objects should be considered metaphors for physical systems. This is not the contemporary view, in my opinion. And it is not my view anymore. For me, mathematical objects are pure concepts.

For real numbers, I consider two interpretations to be pedagogically correct. The first - algebra over totally ordered equivalence classes of Cauchy sequences/processes. This is the concrete/applied way to interpreting them. The fact that those equivalence classes have order and algebra defined over them does not imply that they stand on equal footing with the approximating elements of the sequences themselves. But nonetheless, they do act amorphously enough to allow us to think of them as "quantities". We call them numbers, but we also call complex numbers such, and they are not even totally ordered. So, to some extent, it is just a matter of nomenclature.

We deal with algebras on equivalence classes of Cauchy sequences, because we are interested in the convergence properties of the sequences, as well as how convergence interacts with transformations of various kinds - i.e. whether functions are continuous or not, whether operators are defined when acting on maps for those classes, etc. But ultimately, it still boils to our interest in the concept of convergence, not the reals for the sake of the reals. The real number in this sense is just a specification of the approximation process, whose behavior we need to analyze. Specifications can be, but need not be physically represented.

Alternatively, the notion of a real number from abstract algebra is one of a complete ordered field. Ultimately, it is the same concept. The properties are the same, except that the approach to the investigation is leaning more heavily towards non-constructivism. Which is fine, because this is what abstract algebra is all about. In fact, in some sense, the abstract definition is the proper definition, and the constructive one serves as an illustration. The latter is pedagogically necessary, but once understood, is not essential anymore.

As I said before, I cannot see how the study of complete ordered fields, being equivalent (up to isomorphism) to the ordered algebra of equivalence classes of Cauchy sequences can be replaced with something else, without reducing the scope of the theory. You either have analysis of your objects, or ad-hoc usage, but the latter is just a trial and error.

Now those are strange entities, unlike the essentially finite square root of 2 (as you've already noted.) A dark ocean of infinitely informative numbers that can't be named is far more poetic and disturbing than little old 2‾√. — mask

Yes indeed. I think of them as the "dark matter" of the real number line. We can't name them, we can't compute them, we can't use them for anything. But without them, there aren't enough reals to be Cauchy-complete. The reals lose their essential property. And without dark matter, the galaxies would fly apart. There are more things in heaven and Earth, Horatio, / Than are dreamt of in your philosophy -- Shakespeare.

Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO). — mask

Please explain this to @Metaphysician Undercover! I've had no luck.

I think of them as the "dark matter" of the real number line. We can't name them, we can't compute them, we can't use them for anything. But without them, there aren't enough reals to be Cauchy-complete. — fishfry

I first read about them in Chaitin, before I had the training in math to really understand. It was clear even then that the real numbers had a certain magnificent unreality or ideality. When I studied some basic theoretical computer science (Sipser level), I saw the 'finitude' of now relatively innocent computable numbers like pi, which are of course countable. The measure of R is 0 without those unnameable numbers. Each is an oracle answering an infinite number of yes/no questions unpredictably. So does one believe in them? Within the mainstream game, of course! As pieces of the game that are one of its strangest features.

There are more things in heaven and Earth, Horatio, / Than are dreamt of in your philosophy -- — fishfry

Right! And there are more things in mathematics that are dreamed of by outsiders. I have taught math, and I occasionally hint at a world of strangeness awaiting those who wade in more deeply. It's a poetic enterprise, though one has to learn the grammar and spelling before one can understand the poetry.

Please explain this to Metaphysician Undercover! I've had no luck. — fishfry

I know! I've followed your conversation. We human beings are sometimes stubborn as mules. We don't always want to know. Sometimes we'd prefer to 'win' an argument, however alone we are with that sense of victory. It's basically ridiculous to do philosophy of math without training in math: sex advice from virgins, marital advice from bachelors.

I always follow your posts. You know much more set theory than me, so I learn something. Though I agree that the rational numbers are synthetic, I find them intuitively satisfying enough to function as a foundation from the which the reals can be built as cuts or Cauchy sequences. I like cuts for not being equivalence classes. It's an aesthetic preference. Cuts are beautiful ('liquid crystal ladders').

Was it Cantor who said the rational numbers are like the stars in the night sky and the irrationals are like the darkness in the background? Perhaps this has been posted before.

Was it Cantor who said the rational numbers are like the stars in the night sky and the irrationals are like the darkness in the background? Perhaps this has been posted before. — jgill

I don't know, but I'm familiar with that excellent analogy.

the reals can be built as cuts or Cauchy sequences. I like cuts for not being equivalence classes. It's an aesthetic preference. Cuts are beautiful ('liquid crystal ladders'). — mask

Actually, yes. Dedekind cuts are another constructive approach. Not too different in spirit, I would say.

Was it Cantor who said the rational numbers are like the stars in the night sky and the irrationals are like the darkness in the background? Perhaps this has been posted before. — jgill

Rational numbers are actually quite nasty, if you want to work with them in computations. They are pleasant, if you are performing a finite number of arithmetic operations. But assuming a "fraction" representation, once you start evaluating some recursive formula, the numerator and denominator become unmanageable quickly. I am not sure how fast the periodic part in the repeating decimal representation grows, but I wouldn't want to work with that either. That is why software uses only fractions with denominator in exponent form (*) and represents the rest approximately. Correspondingly, algebraic numbers, and even computable transcendentals, are not that bad, when compared to arbitrary fractions.

I am only stating this, because rationals are looked upon so favorably, but I find that their simplicity is somewhat overstated.

* finite digits after the decimal point in appropriately chosen base;

Actually, yes. Dedekind cuts are another constructive approach. Not too different in spirit, I would say. — simeonz

One can construct the positive real numbers as a simpler version of the cuts. In the version I like we have a ray of rational numbers that starts from zero (a subset of Q that is closed downward with no maximum.) I like this for its intuitive connection to magnitude/length. The square root of 2 in this system of positive reals is just the rationals whose square is less than 2. The analogy is strong. We hold up some piece of Q^+ as a ruler, and we get all that Q itself leaves out. (I'm sure you already know this, so I'm just hyping the charms of this construction for the intuition. )

I am only stating this, because rationals are looked upon so favorably, but I find that their simplicity is somewhat overstated. — simeonz

IMV, the rationals are quite difficult. We know how students hate fractions. But I like the idea of 1/n as a kind of flexible unit. Then m/n is just m of those units. We can adjust n to increase or decrease resolution. And we can do various conversions. So it's difficult but still (after much work and thinking) ultimately intuitive. At least for me.

One can construct the positive real numbers as a simpler version of the cuts. In the version I like we have a ray of rational numbers that starts from zero (a subset of Q that is closed downward with no maximum.) I like this for its intuitive connection to magnitude/length. — mask

This is exactly the geometric interpretation that got me into trouble. :) It assumes that rays have points corresponding to every non-negative real number (or lines have points corresponding to all real numbers.) To which, I remember my brain screamed, how do you know? Of course, if this is just analytic geometry, it would be true by construction, but then the argument becomes circular. So, I was asking essentially, how do we know that lines/rays, as they appear in real life, are complete. They could be, or they might not be, but how would a mathematical textbook use something like that, that we know very little about (i.e. space), and which is not axiomatic in nature, and use it to define a mathematical concept. At least for me, it didn't work, and caused me some difficulty.

I suppose this might be related to the objection that the OP has.

P.S. What I mean is - the definition (edit: interpretation) works if the person reading it has the right attitude. But I am rejecting its use on methodical grounds, of being informal. (And untested.)

P.S.(2): The reason why I got so heavily stuck on this interpretation was not so much because it prevented me from making sense of the mathematics presented, but because I started to question what was the goal - were numbers lexical entities or geometric properties? What was that we were trying to define - quantities, computations, geometrical facts? How could we validate them? At this point I didn't know much about algebraic structures and axiomatic systems. I went completely on a pseudo-philosophical tangent and refused to learn anything whose methodological grounds I did not understand completely.

IMV, the rationals are quite difficult. We know how students hate fractions. But I like the idea of 1/n as a kind of flexible unit. Then m/n is just m of those units. We can adjust n to increase or decrease resolution. And we can do various conversions. So it's difficult but still (after much work and thinking) ultimately intuitive. At least for me. — mask

I understand. After all, this is the rationals' whole gimmick - they are dense. Of course, the finite decimal fractions rationals I talked about earlier are also dense, and easier to compute. But they are not a field. So starting with integers, you cannot scale yourself arbitrarily.

If you mean electron microscope photos of a lattice of atoms, those are still subject to the quantum and classical measurement problems. To clarify what I said earlier:

* In quantum theory, nothing has an exact position at all. Before it's measured, it doesn't have a position. Sometimes that's expressed by saying that it's in a "superposition" of all possible positions. Then when you measure the particle, it (somehow -- nobody understands this part) acquires a position drawn randomly from a probability distribution.

This applies to all objects, large and small, though the effect is much more pronounced when an object is small.

For example you yourself are where you are in space right now because that's the most likely place for you to be. It is statistically possible that you might suddenly find yourself in a statistically improbable place. For example all the air molecules in your room could move to the corner of the room and you'd have no air. That is extremely unlikely, but it has a nonzero probability. It could happen.

So even if all instances of a given particle are the same, you still have no idea exactly where it is, or exactly how long a line made up of these particles is.

Atoms, by the way, are way too large and they're all different. I don't even know if two hydrogen atoms are exactly the same.

However it's interesting that every electron in the universe is (as far as we know) exactly the same. Why is that? It's another thing nobody understands. — fishfry

The even more interesting thing (that's why I talked about atoms) is that this is true not only for elementary particles as electrons, but even for atoms (of any element), and even for entire molecules, and this has been verified experimentally. Two atoms in the ground state (https://en.wikipedia.org/wiki/Ground_state) are EXACTLY IDENTICAL (as mathematical objects in the mathematical model of QM) if the ground state is not degenerate (https://en.wikipedia.org/wiki/Degenerate_energy_levels).
The tricky thing to realize experimentally is to obtain a non-degenerate ground state for a complex object as an atom: very low temperature, external magnetic field, confined position in a very little "box" (usually a laser-generated periodic electromagnetic field). But this is possible, and in this state the whole atom is COMPLETELY DESCRIBED from by one integer number: the energy level.
In this state you can put a bunch of atoms one over the other, if they are bosons (https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate) and the theory says that you can have N IDENTICAL objects all in the same IDENTICAL place.

The result is obtained by purely mathematical considerations on objects made of complex number functions (the states are the eigenvalues of the system's wave function), but the effects predicted using a purely mathematical abstract model generate real physical predictions in the form of measurable quantities. That seems very strange if mathematical objects are only symbols subject to arbitrary rules. In some way, the rules that we invented for the symbols correspond exactly to some of the "rules" of the physical (real) world.

* And even in classical physics, a measurement is only an approximation.

So now I'd like to re-ask your question but pertaining to electrons, which are all exactly the same. But electons are very small and extremely subject to quantum effects. You simply can't say exactly where an electron is at any time. Only where it's statistically likely to be. One, because nothing is exactly anywhere at all in quantum physics; and even when it is, after a measurement, the measurement itself is subject to classical approximation error. You made the measurement in a particular lab with a particular apparatus, built and operated by humans. It's imperfect and approximate from the getgo. — fishfry

Yes, but the indeterminacy is only for the product position * momentum, and not the position alone (for example an electron emitted from the nucleus of an atom has an indeterminacy of initial position of the size of the nucleus from which it was emitted). And the curious thing is that the wave function, if you want the path-integral over the trajectories to be accurate enough, must be described with a much finer granularity of space than the size of the atom. The wave equation works the best if it's defined on the (mathematically imaginary) real numbers (at least for QED). The renormalization of electron's self-energy (https://en.wikipedia.org/wiki/Renormalization) is a mathematical theorem based on a mathematical model where space is the real euclidean space (real in the mathematical sense: vector space defined on real numbers) (I know the objection: it works even on a fine-enough lattice of space-time points, if you make statistics in the right way, but the lattice of positions have to be much smaller of the wavelength of the electron - that for "normal" energies is comparable with the size of an atom).

Well there are no computers with arbitrary precision. That's the problem with the computational theory of the universe. There's too much it can't account for.

It's those pesky noncomputable numbers again, one of my favorite topics. If the universe is "continuous", in the sense that it's modeled by something like the real numbers; then it is most definitely not a computer or an algorithm. Because algorithms can't generate noncomputable numbers. — fishfry

Yes, however in same cases, the system is symmetric enough that you can use analysis to compute the results instead of making simulations, so you can get infinitely precise answers, (such as for example in the case of hydrogen atom's electronic
orbitals) that however you'll be able to verify experimentally only with finite precision.

So in your hypothetical world there would be squares and if you want to go from (0,0) to (1,1) you simply have to move 2 units, one unit right and one unit up. You can't travel along the diagonal because at the finest level of the lattice, you can't move diagonally. I have no idea what that means physically but I think you are overthinking this or underthinking it. It's kind of tricky, which is a problem for the theory. — fishfry

Well, that was a simple example that doesn't have much sense as a real theory of physics (and I absolutely don't believe that it can be a good model of physical space), but it's still a mathematical model suitable to be used to make predictions on the physical space (well, you should say how big are the sticks: surely there are a lot of missing details). However, as a model, you can decide to make it work as you want: in our case, the squares made with sides of one stick can't have a diagonal (so, let's say, nothing can travel along the diagonal trajectory, as in the Manhattan's metrics), and big "squares" can have diagonals but can't have right edges, or straight angles.

Some people do! There are some discrete or quantized theories of reality around, like loop quantum gravity. From the article: "The structure of space prefers an extremely fine fabric or network woven of finite loops." — fishfry

Yes, but in loop quantum gravity loops are only "topological" loops: they are used to build the metric of space-time, not defined over a given metric space.

But I don't speculate about the physical world. Math is so much simpler because it doesn't have to conform to experiment! In math if you want a square root of 2, you have your choice of mathematically rigorous ways of cooking up such a thing. — fishfry

I agree with you on the square root of 2, of course! But I am not so convinced that mathematical objects are only cooked-up fictions not related to physical reality.

I agree with you on the square root of 2, of course! But I am not so convinced that mathematical objects are only cooked-up fictions not related to physical reality. — Mephist

I will allow myself to interject, although the physics involved in your discussion appears beyond my competence. In any case, just because something is not physical, doesn't make it purely fictitious.

For a lame example, if I define a geographical location called "not in London", which has the property that any statement exclusive to the London area is untrue for that abstract location, and all other statements remain undefined, this area would not properly represent any part of the universe. Why? Because many statements can be made about the Universe that are not specific to any location, and remain true for "not in London" in practice, but are not included in my structure/axiomatic system. However, I am constructing "not in London" not to express specific knowledge, but to express my lack of specific knowledge. I am creating an abstract entity which expresses my epistemic stance.

This is the way I look at mathematical objects in general, and real numbers in particular. They can be physically represented, if they happen to be. But generally, they are specifications more so then anything. As all specifications, they express our epistemic stance towards some object, not the properties of the object per se. Real numbers signify a process that we know how to continue indefinitely, and which we understand converges in the Cauchy sense. Does the limit exist (physically)? Maybe. But even if it doesn't, it still can be reasoned about conceptually.

This is exactly the geometric interpretation that got me into trouble. :) It assumes that rays have points corresponding to every non-negative real number (or lines have points corresponding to all real numbers.) To which, I remember my brain screamed, how do you know? — simeonz

Good point. The rays I mentioned have 'melting tips.' In some ways we are just sweeping the problem under the rug. What I like about the Cauchy sequence approach is that the real number is like a program that spits out rationals (closer and closer together in the long run.) There's another version where positive real numbers are increasing sequences of rationals that are bounded above. These have their advantages. If we hobble ourselves and just think in terms of computable functions from N to Q (whose outputs get closer together), we get maximum clarity but lose most of the line. (And we have no more reals than rationals cardinality-wise.)

But maybe there's no perfectly satisfying way to capture our intuition of the line. And sometimes I'm tempted to just enjoy the axioms of R as descriptive of intuitions of space.

I started to question what was the goal - were numbers lexical entities or geometric properties? What was that we were trying to define - quantities, computations, geometrical facts? How could we validate them? At this point I didn't know much about algebraic structures and axiomatic systems. I went completely on a pseudo-philosophical tangent and refused to learn anything whose methodological grounds I did not understand completely. — simeonz

Ah, the curse of being a philosopher! I feel you. For me it was slightly different. I read a boatload of philosophy of math before studying math. I was vaguely anti-foundationalist by the time I was learning real analysis. With basic real analysis (pre-measure theory and Riemann integral), my intuition felt more or less satisfied. As I learned measure theory, all the sets of measure zero were a bit of a turnoff. More and more equivalence classes. Blah. I don't mind them in algebra, but in analysis there like film on a bathtub that needs cleaning. I'm a lapsed intuitionist. Math only has beauty to the degree that it corresponds (at some anchored point) to intuition. 'God intuition created the integers. The rest is the work of man.'

They could be, or they might not be, but how would a mathematical textbook use something like that, that we know very little about (i.e. space), and which is not axiomatic in nature, and use it to define a mathematical concept. At least for me, it didn't work, and caused me some difficulty. — simeonz

It does seem to depend on an intuition of space, a space independent of physics' space (an ideal space.) We can prove (for the intuition) the commutative law for multiplication by turning a rectangle 90 degrees. At some point I'd like to see how many of the axioms of the real number system can be intuitively supported this way.

I know you don't like technical stuff by I'm pointing out that I don't need set theory to build a square root of 2. — fishfry

Especially considering the algebraic approach that you presented. I find it intuitively satisfying without considering set theoretic foundations. And the Dedekind cut is satisfying if one can admit sets of rational numbers (intuitively self-supporting, IMO). — mask

Alternatively, the notion of a real number from abstract algebra is one of a complete ordered field. Ultimately, it is the same concept. The properties are the same, except that the approach to the investigation is leaning more heavily towards non-constructivism. Which is fine, because this is what abstract algebra is all about. In fact, in some sense, the abstract definition is the proper definition, and the constructive one serves as an illustration. The latter is pedagogically necessary, but once understood, is not essential anymore. — simeonz

:up:

Great discussion. I don't really know if this contributes much to it, but I want to throw it among people I'm interested in reading talk about maths.

Something I find very interesting about these structures (and maybe this is part of what you were alluding to with "non-constructivism" @simeonz?) is that they need not be derived from more fundamental stuff (like set theory) in order to be understood in much the same way as if they were constructed from a more foundational object. Nevertheless, how you stipulate or construct the object lends a particular perspective on what it means; even when all the stipulations or constructions are formally equivalent.

I remember studying abstract algebra at university, and seeing the isomorphism theorems for groups, rings and rules for quotient spaces in linear algebra and thinking "this is much the same thing going on, but the structures involved differ quite a lot", one of my friends who had studied some universal algebra informed me that from a certain perspective, they were the same theorem; sub-cases of the isomorphism theorems between the objects in universal algebra. The proofs looked very similar too; and they all resembled the universal algebra version if the memory serves.

Regarding that "nevertheless", despite being "the same thing", the understandings consistent with each of them can be quite different. For example, if you "quotient off" the null space of the kernel of a linear transformation from a vector space, you end up with something isomorphic to the image of the linear transformation. It makes sense to visualise this as collapsing every vector in the kernel down to the 0 vector in the space and leaving every other vector (in the space) unchanged. But when you imagine cosets for groups, you don't have recourse to any 0s of another operation to collapse everything down to (the "0" in a group, the identity, can't zero off other elements); so the exercise of visualisation produces a good intuition for quotient vector spaces, the universal algebra theorem works for both cases, but the visualisation does not produce a good intuition for quotient groups.

If you want to restore the intuition, you need to move to the more general context of homomorphisms between algebraic structures; in which case the linear maps play the role in vector spaces, and the group homomorphisms play the role in group theory. "mapping to the identity" in the vector space becomes "collapsing to zero" in both contexts.

There's a peculiar transformation of intuition that occurs when analogising two structures, and it appears distinct from approaching it from a much more general setting that subsumes them both.

Perhaps the same can be said for thinking of real numbers in terms of Dedekind cuts (holes removed in the rationals by describing the holes) or as Cauchy sequences (holes removed in the rationals by describing the gap fillers), or as the unique complete ordered field up to isomorphism.

Nevertheless, how you stipulate or construct the object lends a particular perspective on what it means; even when all the stipulations or constructions are formally equivalent. — fdrake

I very much agree. For someone who insists on math being beautiful, it has to sing for the intuition. For example, when learning group theory I really liked thinking of groups of permutations. Those were the anchor for my intuition. The operation on the group was 'really' functional composition, which is why groups weren't automatically commutative. The theory doesn't care. Epistemologically it's a non-issue. But it matters for motivation. Another approach is just to understand math as a chess of symbol manipulation. In some contexts this is satisfying enough. But one gets lost in real analysis without intuition guiding the construction of a semi-formal proof.

I also like this approach: https://en.wikipedia.org/wiki/Interval_arithmetic

The problem of irrational numbers arose from the construction of spatial figures. That indicates a problem with our understanding of the nature of spatial extension. So I suggested a more "real" way of looking at spatial extension, one which incorporates activity, therefore time, into spatial representations. Consider that Einsteinian relativity is already inconsistent with Euclidian geometry. If parallel lines are not really "parallel lines", then a right angle is not really a "right angle", and the square root of two is simply a faulty concept. — Metaphysician Undercover

If you consider geometric spatial figures as real physical objects, there are a lot of "problems" with them: first of all, they are 2-dimensional (or 1-dimensional, if you don't consider the internal surface), and all real physical objects are 3-dimensional. The usual "trick" to make some sense of this kind of model is that they are so small that are not directly observable. Einstenian general relativity is the same as Euclidean geometry in this respect: world lines are just a mathematical abstraction to represent trajectories in space-time. They are not real objects, and there is no problem with the distinction between finite or infinitesimal distances: it works even if you consider space-time as discrete. In fact, in practice it's very common in GR simulations to approximate space-time as a 4-dimensional discrete grid of points.

The main point to keep in mind with physical models is that they don't have to be considered the real thing: they simply have to WORK as the real thing. So you have to choose which characteristics (or properties, or attributes) of the model correspond to characteristics of physical real objects and which ones are only mathematical approximations. For GR, the trajectories are only abstract 1-dimensional "lines": what's important (measurable) is only their length, and the angle between them, but only up to a certain approximation. Physical bodies can be represented as "points", or "spheres", just to make calculations simpler: the small-scale details are not important for the model, so they don't correspond to anything "real".

Now, if you think that the distinction between measures expressed with rational or with real numbers is essential in your theory (represents some important characteristics of the real physical space), I don't see any other way other than making lengths become discrete at the microscopical level.
If this is not your idea, in which way the use of rational numbers instead of real numbers could make a difference? I know that you think that real numbers do not exist, but what's the difference if they exist or not, if your model doesn't care of what happens at the smaller scales?

The operation on the group was 'really' functional composition, which is why groups weren't automatically commutative. — mask

I agreed very hard on this in my heart. Tutorials and seminars in abstract algebra mixed between people who preferred algebra and people who preferred analysis; it's a shock to the intuition whenever an algebra lover presents the group operation "the other way around", and vice versa. Seeing groups as transformations was how I imagined them; but as for imagining $\{\mathbb{R},+\}$ as a group in the same sense it didn't work; those intuitions were tied to quantities and magnitudes, but they happened to correspond to translations along the "real line" of a given length, and that intuition could be passed up to vector spaces of low dimension (magnitude + direction, parallelogram rule).

For someone who insists on math being beautiful, it has to sing for the intuition. For example, when learning group theory I really liked thinking of groups of permutations. Those were the anchor for my intuition — mask

In general I have found that working over formalisms is one necessary part of developing understanding for a topic; don't just read it, fight it. Follow enough syllogisms allowed by the syntax and you end up with a decent intuition of how to prove things in a structure; what a structure can do and how to visualise it. Those syllogisms aren't the whole story, the visualisation matters.

What I want to pick a bone with, though perhaps this is a misreading on my part or a difference in emphasis, is whether such intuition development (associating a mental image or a shorthand for forming expectations regarding a structure) is merely aesthetic. We're quite well trained to think of mathematical objects as formal objects, symbol pushing, or as physically rooted (or obversely grounding reality in mathematical abstraction), but what of the required insight to, as you put it, anchor the intuitions of a structure?

Developing such anchors and being able to describe them seems a necessary part of learning mathematics in general; physical or Platonic grounding deflates this idea by replacing our ideas with actuality or actuality with our ideas respectively. In either case, this leaves the stipulated content of the actual to express the conceptual content of mathematics. This elides consideration of how the practice of mathematics is grounded in people who use mathematics; and whether that grounding has any conceptual structure; how is actual mathematics understood by actual people and does that have any necessary structure? Put another way; what is the structure of the conceptual content of mathematics?

Whitehead alludes to something similar regarding philosophical projects:

Every philosophy is tinged with the colouring of some secret imaginative background, which never emerges explicitly into its train of reasoning.

Why should this background of mathematics remain a secret? And is it merely aesthetic in nature (a consideration of mathematical beauty alone)?



This looks cool, the statistician in me likes including uncertainty into the operations of arithmetic, but dislikes characterising uncertainty as the range of a set.