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April 2019
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Reasons for Irrational Numbers

A demonstration of Pythagras' Theorem followed by Hippasus's Proof, the points I make being from my concluding questions: Seeing any need for √2, the most interesting question is how does mathematics envelop new numbers?  After answering that, what extrensic reason is there for enveloping √2 into a number system?  Hippasus' indirect proof merely demonstrated, contrary to our intuition, that √2 cannot be expressed as a rational number p/q; it says nothing about what should replace the assumption from line 8.  And I am not satisifed that we have solved this problem for reasons other than the ideal consistency of the real numbers.

(Disclaimer: intended for a general audience)

Pythagoras' Theorem

I stumbled onto irrational numbers after convincing myself of the veracity of the Pythagorean Theorem and reflecting on the use of the word “irrational”. This reflection led to fundamental questions about how and why mathematics enveloped irrational numbers and whether the rational dilemma presented by Hippasus to the Pythagoreans was thoroughly resolved.

There are many proofs of the Pythagorean Theorem. A common form for this theorem is:

1. c2 = a2 + b2

I used the “areas of plane figures” because I saw the relationship between the common formula and squares; another uses similarity theorems of Euclidean geometry. (Verdina, 130) The Pythagorean Theorem is a good exercise in demonstrating mathematical concepts and the theorem's veracity. Demonstrating the Pythagorean theorem could be made trivial: one needs at least one known, and one unknown as a function of this known; for simplicity's sake, let a = 3 and b = 4. ... Oh this is trivial! I would find the unknown terms and the equation would be satisfied! Rather than trivialize common acceptance of this theorem, ask, “how did Pythagoras arrive as this theorem? what was his rational?

Representing the theorem with no net change made his rational clearer, as:

1'. 0 = a2 + b2 – c2

I reflect on the algebraic and geometric definitions of these symbols and terms. Let the terms a and b be the length of square sides forming a isosceles angle at one end, and c be the length of a square side which forms the secant length of a diagonal drawn from the opposite ends of the sides a and b. When the squares A and B with sides of length a and b respectfully are overlain, the resulting square C has sides of length c. I found that Mathworld calls this method proof by shearing, and can be done as a “Kitchen Experiment” by cutting paper squares A and B and overlaying them to yield square C or by various, online and interactive demonstrations. The "shearing proof" demonstrates that the addition of the area of the squares A and B with sides of length a and b yields the area of the square C with sides of length c: namely, equation 1.

But one cannot rationalize this overlaying process without understanding area, and I found area to be difficult to explain. It merely came down to me defining it: let the fundamental area be a unit square, 1 x 1, and all other areas be the repetitive addition of this fundamental area until the unit square fills the space we confine with geometric lines. For example, a 2 x 3 rectangle has 6 unit squares, calculated by row and column addition of 1 x 1 squares. A fascinating digression is that my definition expresses area pragmatically; it is only valid for whole number areas, not fractional sides. More on that later. Suffice to say, it sounds better to say that we “reasonably” demonstrate the Pythagorean Theorem instead of “rationally” demonstrate it because the terms can be misunderstood as referring to types of numbers. For example, today we express the area of circles “irrationally” (π is involved) but the demonstration that circles contain areas is not irrational.

Eitherway, Pythagoras' rational depends on properties of a triangle, namely the length of any square-side triangle (isosceles triangle) through square areas. These triangular properties are based on Euclidean geometry, where this geometry can be said to be merely the system of postulates used to demonstrate concepts we use in drawing, like side, length, area, etc. (Verdina 290) Perhaps this is boring because of its verbosity but allow this defense: by reasoning through Pythagoras' problem we notice the theorem depends on applying algebraic definitions alongside a geometric demonstration, a geometry which itself is defined. The most interesting algebraic trick was "no net change" seen in 1', which is a fascinating digression in the Theory of Equality, and an interesting relationship between the algebraic representation of a square's area, s2, to the drawing of three squares.

Hippasus' Proof

It occurred to me later that another problem, more perplexing than that of defining area, would arise. Similar to the unit square, 1 x 1, that I used to defined the area for squares with sides of length that are whole numbers, I considered the “unit triangle”. Where the unit square S has sides of length one, S: 1 x 1, I say the isosceles, “unit triangle” has legs of length one. According to Pythagoras' theorem,

1. c2 = a2 + b2

where a and b represent the legs of this triangle and c its hypotenuse. Now the problem: given that a = b= 1, find the hypotenuse c.

2. c2 = 12 + 12

3. c2 = 1*1 + 1*1

4. c2 = 1 + 1

5. c2 = 2

2. through 5. use normal algebraic rules that many take for granted, even if acknowledging them here. According to the definition of squares,

6. c * c = 2

6. asks: what number, multiplied by itself, is 2. Using normal rules of addition and multiplication, 1 * 1 = 1, a number less than 2, but 2 * 2 = 4, a number greater than 2, so the value of c shoule be somewhere between 1 and 2.

Another approach to resolving the problem of this unit triangle's hypotenuse is by reflecting on the meaning of symbols, namely, what numbers represent. Given 5. above, we can also denote that:

6'. c = √(2)

This radical sign has become common notation without much elaboration about what it means. This symbol means the square root of a number n, where we define square root as something whose square yields the number n, called the radicand.

SQRT. √(n)*√(n) = n.

For our example, 6', √(2)*√(2)=2. People would later notice problems when n < 0, or when “taking the square root of a negative number”, and these were settled elsewhere (by imaginary numbers) We will exclude such problems by confining our analysis to the expected length of c. Dividing a number line of domain [ 1 .. 2 ] demonstrates that there are an infinite amount of fractional numbers between the whole numbers 1 and 2, so the Greeks expected to find the value of c somewhere in there. Any fractional number is represented as a ratio of p to q:

7. RN: p/q.

where p and q are non-factorable integers, and with the normal rules for division apply. So if p=9 and q=8, then 9/8 falls between 1 and 2 because, represented as a mixed number, 9/8 = 1 1/8 where 1 1/8 is greater than 1 and less than 2. It is important to reiterate that p/q is reduced, or cannot be simplified by a common factor. For example, a/b=19/9 is not reduced because both a=19 and b=9 have a common factor of 9, reducing the ratio to 2 1/9.

Expecting the hypotenuse to be such a fraction, we can represent c as some integer p divided by some integer q. Let us assume c is such a fractional number on the number line between 1 and 2,

8. c = p/q, c: [1 .. 2]

Substitution of 8. into 5. yields,

9. (p/q)2 = 2

10. p2/q2 = 2

11. p2 = 2*q2

Because the coefficient of the q term is now 2, the p term must be even. For example: let a be unknown and b = 3 for the equation a = 2*b. Solving the equation, 6 = 2 * 3, we notice a is an even number, 6. This is not a proof, although one can be written, but merely exemplifies why the p term in 11. must be even. Also, by assuming 5. and the definition of SQRT, we can infer that q will be odd.

To include the deduction that p is even, one can represent p as a factor of 2: given that any integer r multiplied by the factor 2 is even, then 2*r is even. So, let p = 2 * r for 11. above as:

12. (2 * r)2 = 2*q2

13. 4 * r2 = 2*q2

14. 2*r2=q2

Now, intuition might tell you that something is amiss. 14. looks like 11; this is because they are saying the same thing, although about different terms. As already stated, 11. identifies p as the even term. Using the same, deductive logic, 14. identifies q as an even term. But, if both p and q are even, then the fraction c has a factor of 2. We explicitly demanded in our assumption for 8. that c be non-factorable. This assumption is necessary because of the expected value of c, or in math-lingo, the domain limits the value of c. If there is a common factor, say 2, for p/q, then c is not where we expect to find it. c would be outside the bounded number line [ 1 .. 2 ]. So we have a problem.

The Pythagorean's Dilemma

Does one abandon 5. by insensibly allowing the hypotenuse of the triangle to be longer than geometrically demonstrable? or does one maintain that both the deduction of 11. where p is even and expected q to be odd, andthe deduction of 14 that says the converse are both valid? The Greek response was: either the hypotenuse of the unit triangle is longer than drawn OR q is both odd and even.

The latter possibility – that q is both odd and even -- is, by the Law of (Non-)Contradiction, irrational. The Law of (Non-)Contradiction is a fundamental aspect of rational thinking because it ensures against holding opposite conclusions. It assures against answering, “Yes and No” to a “Yes or No” question. The former possibility – that the hypotenuse of the triangle is longer than expected -- is insensible, meaning impractical for construction of whatever. Even another approach to demonstrating the Pythagorean Theorem -- one I did not see as evident, namely the proof by similar triangles -- more closely depends on the postulates of Euclidean geometry, and these postulates have been proven, if for any other geomatric shape, consistent for such a triangle on a plane.

So the Pythagorean dilemma was: does one abandon Euclidean geometry or the Law of (Non-)Contradiction?

This dichotomy is a sizable quandary in the relationship between what one expects, as drawn, against what one expects, as deduced. Hence the Greeks called the length of this hypotenuse “incommensurable”, meaning it was not congruent with measuring tools. “This result disturbed the logic-loving Greeks, who felt that the existence of such a number was proved by geometry and disproved by algebra. Some historians conjectured that this paradox of the 'incommensurable', as it was called, retarded Greek science, and consequently world science, for centuries.” (187, Denbow and Goedicke) “The early Greeks believed that every measurable quantity had to be a rational number. However, this idea was overturned in the fifth century B.C. by Hippasus of Metapontum * who ... using geometric methods, [] showed that the length of the hypotenuse of the isosceles triangle could not be expressed as the ratio of two integers.” (A1, Anton, Bivens and Davis) This proof was the very one demonstrated above, or at least a contemporary version of it. “According to legend, Hippasus made his discovery at sea and was thrown overboard by fanatic Pythagoreans because his result contradicted their doctrine.” (ff A1, Anton, Bivens and Davis), see also (129 Smith) So the stakes were high when the dilemma arose and how did we resolve them?

Should we consider another case: Neither? We neither abandon Euclidean geometry nor deduction and the Law of (Non-)Contradiction by restating that the dilemma is not a “Yes or No”, bipolar question?

Denbow and Goedicke in Foundations of Mathematics explain the eventual solution surmounted the intuitive notion of the number line and our expectations of it, or as I would like to say, the "fix" recognized the demands this problem placed upon the mathematical system. As they illustrate with the number line [1 .. 2] mentioned above: ”We can then select any two of these [number line] points, no matter how close, and insert an infinite number of further points between them ... If we depend on our intuition we are inclined to believe that this process will give us all the points on the line. ... This intuitive conclusion is false.” (184)  Seeing any need for √2, the most interesting question is how does mathematics envelop new numbers?  After answering that, what extrensic reason is there for enveloping √2 into a number system?  Hippasus' indirect proof merely demonstrated, contrary to ourintuition, that √2 cannot be expressed as a rational number p/q; it says nothing about what should replace the assumption from line 8.  And I am not satisifed that we have solved this problem for reasons other than the ideal consistency of the real numbers. And that's where it should have stayed.

I enjoyed it anyway, though. To place such 'difficult' ideas like irrational numbers or complex numbers into a category of things that must somehow be justified or else be thrown out as 'unnatural' makes distinctions in mathematics that are more emotional than should be. the roots of an equation simply are, whether they be rational, irrational, or complex. No points on the Cartesian Plane have any significance beyond how they are used; all particular distinctions in mathematics are arbitrary.

They certainly need to be justified in the sense that we require proper deductions of their existence, e.g., when someone tells you to "justify your solution" to a problem.

You are right, though, that it is the use of mathematical objects that need justification in the usual sense, rather than the objects themselves. Wow!

So-so! But if you knew Russian! How many problems would be free off! I like how you've (deliberately?) conflated the meanings of "rational" and "irrational" between philosophy and mathematics. I guess the answer to your question a few posts back, "What is the most fundamental property of a rational argument?" is "both the numerator and denominator contain whole numbers".

;-)

Just an aside, I'm sure you knew this: Pythagoras' theorem doesn't open up the numbers to all irrationals, just the square root of all prime numbers. It took Cantor's diagonal to assrape the number system into accepting the uncountable infinitude of the rest of the irrationals. I guess the answer to your question a few posts back, "What is the most fundamental property of a rational argument?" is "both the numerator and denominator contain whole numbers"

Yea, I was actually looking into any relationships between the philosophical and mathematical definitions, although I already knew "rational" in math was derived from "ratio". Still I wanted to see what, if anything, was there.

Pythagoras' theorem doesn't open up the numbers to all irrationals.

Yea, I needed only one instance of an irrational to see the problem and how it was solved.

We also have something called analytic functions in math. Would you like to explore the possible connections between them and Kant's analytic a prior?

(Hint: There is none.) Dude, chill. Just because I waste my time on learning something from searching for something that was not there, why go around bashing the waste? My stupidity isn't harming you. it's a priori, not a prior.

Typo-nazi. I'm reading the comments before the post, but these comments echo one of my own interests. That, yeah, a proper rational argument would have whole numbers in the numerator and denominator.

Bur obviously we can construct mapping conventions that open up ideas which are not rational in this sense.

You choose the unit square, and in math it is set against the unit circle. This is one of the more fundamental considerations.

I'll probably have more to say after I get through the post proper. Yea, maybe there was a notion of rational thinking to rational numbers that later absorbed irrational numbers without changing its label. Eitherway, its interesting that we changed the approach to whatever was, prior to √2, rational. For the pythagoreans, do not forget, this was a devastating revelation, and it is oft repeated they killed to keep the idea supressed to such ideas and knowledge of irrationals limited to their own small enclave...

Now there is something ideed. Math worth killing for. A veritable math war...

Such is belief... I was wondering about the conclusion of the Greeks: "either the hypotenuse of the unit triangle is longer than drawn OR q is both odd and even."

Why both? Wouldn't "neither" be more appropriate? There is more than one way for a number to be not-even, and being odd is just one of them. Of course, the conventions of "integers", which at the time were supposed to represent all of the numbers--either a number is prime, or a composite product of several primes, or a ratio of such composites--and the ontology* of that set of numbers (the whole set in this case, not the proper subset, since they hadn't introduced the irrationals yet) excluded those middle-composites that were neither odd nor even was what forced the contradiction in the reductio and the conclusion that 2q2 as well as 2r2 had to be even.

Maybe I'm focusing on nothing, but the "both", as contrasted with "neither", in the Greek's conclusion to the reductio seems to be something worth investigating.

*Not the right word, but I'll stick with it. Why both? Wouldn't "neither" be more appropriate?

This is precisely my inquiry. It's easy to waft the Greeks aside by saying "we know better than them", but I'm not one that easily buys into superior intelligences.

There is more than one way for a number to be not-even, and being odd is just one of them. Of course, the conventions of "integers", which at the time were supposed to represent all of the numbers--either a number is prime, or a composite product of several primes, or a ratio of such composites--and the ontology* of that set of numbers (the whole set in this case, not the proper subset, since they hadn't introduced the irrationals yet)

According to the responses over on , these "conventions" and "set of numbers" needed to be rigorously defined to ensure consistency within mathematics. I anticipated this as an ideal reason for us but I find it no more ideal than the Greeks demanding consistency of all numbers being rational. There is something else there ... something demanding a more fundamental consistency, something that pointed to choosing the extension/enveloping of irrationals, and I still find it interesting that the contemporary, more rigorous mathematics is still based on systems of axioms that promotes this ideal. Hence, the connection to rational arguments.

the "both", as contrasted with "neither", in the Greek's conclusion to the reductio seems to be something worth investigating.

I concur, otherwise we're dogmatically demanding consistency and other intrinsic ideals for no other reason than "it must be".

I concur, otherwise we're dogmatically demanding consistency and other intrinsic ideals for no other reason than "it must be".

You keep on saying crap like this, but I've yet to see quotes from any mathematician. Stop it, it's dishonest and lazy.

We insist on consistency because in an inconsistent system there exists a deduction proving every proposition that can be formulated in that system. You keep on saying crap like this, but I've yet to see quotes from any mathematician.

I incorporated your suggestion in my synopsis of the discussion on .

For this post, you just demonstrated an intrinsic ideal:

We insist on consistency because in an inconsistent system there exists a deduction proving every proposition that can be formulated in that system.

There's nothing good or bad about intrinsic ideals, they simply differ from doing things for reasons outside of mathematics or for reasons that are merely practical. Fundamentally, your statement is more about logic than it is about mathematics, and if logic is where the ideals are codified, then we can churn through them outside of . The other "problem" identified by the reductio, that "the hypotenuse of the unit triangle is longer than drawn" and "impractical for construction of whatever", seems suspect to me as well, now that I think of it. Consider a slight change to Pythagoras' theorem:

a2 + b2 = c2 + δ

Where δ is some very small, << c2, but nonzero number that adds the tiniest bit of shit to sqrt(2) and produces a rational number that is, for all practical purposes, equal to the length of the hypotenuse of the right triangle. In the "real world" (e.g. in a "shearing" proof that actually uses folded/cut paper) we would never be able to detect this δ.

I'm wondering what problems this δ causes if we slip it in there to avoid irrational numbers? I'm wondering what problems this δ causes if we slip it in there to avoid irrational numbers?

OH my! Einstein's cosmological constant rings a bell! LOL.

The other "problem" identified by the reductio, that "the hypotenuse of the unit triangle is longer than drawn" and "impractical for construction of whatever"

I was reflecting on the geometry problem again. Perhaps it is best to give a counter-example: draw a "unit triangle" (a=b=1) where the hypotenuse (c) is rational. Err...

I believe that geometry is far too intrinsic to "give up". So fudging it with δ ... nah, I see you weaseling in there ;) Whether by analytical rigor or Hippasus' proof, it is far easier to "change" the abstract -- the algebra --, especially when those changes don't affect daily life, like adding up how many apples one buys at the grocery.

Of course, these reasons are unsatisfactory for modern mathematics. I'm not dissing analytical rigor; just saying that Euclidean or not, the set of postulates comprising a geometry can be seen, like a "unit triangle" ... although, the number line can be seen too, but it didn't change in apperance too much after filling in the irrationals. *gasp* Here's the rub, in my eyes: Mathematicians are always so proud of the fact that a system derived from complete consistency and an unyielding embrace of the law of noncontradiction can be used to "describe the real world", and they will say things like "that's remarkable, no?" when a particularly clever formulation of mathematics gives us a clue as to how to test a certain phenomenon and what to expect if some theory is true or alternately if the theory is falsified.

What they forget is that the moment one walks into the "real world", irrational numbers simply disappear. To see what I mean, construct your unit-right triangle out of paper and make the two sides at a right angle precisely (precisely, not exactly) 1 meter long, and then measure the hypotenuse. I absolutely, 100% guarantee, without-a-doubt, can assure you that what you come up with is a rational number for the length of the hypotenuse. That is the real world. I think it is important to realize where the ideal world of mathematics stops and where the real world of physical phenomena starts.

Another place where this crops up is when we compare our a priori intuitions that lead us to the continuity of the number line, and hence to the continuity of geometry, and this runs smack into the discreteness of physical reality.

Now, I'm not saying we should insert the δ into mathematics, as I'm sure it will eventually produce a contradiction, even if we avoid, for a brief shining moment, the scourge of irrational numbers... ;-) What I'm trying to say, I guess, is that the tiniest δ is all that stands between geometry and the real world, and while it is small, it is not insignificant. Where it becomes significant is where we should take notice.

I think this is why I'm such an pain-in-the-ass rabid anti-platonist. Just to follow up, because I had an inclination that my previous claim included adding an irrational number to get to a rational sqrt(2), I came up with this example:

12 + 12 = 1.41422 + 0.00003836

This might go with a particularly precise measurement of a paper "shearing" proof, where you could measure the diagonal to the nearest tenth of a millimeter. Notice that the equation, as it stands, has no need for an irrational "c" or an irrational "δ".

Just something I was playing with--I'm sure there is a branch of chaos theory that can amplify the 0.00003836 into an error of magnificent proportions... The problem is that you aren't really measuring the length. You're giving an appoximation. That approximation is a guess, probably rounded to a nearby rational number (can't say nearest because that doesn't exist).

The chances of cutting a piece of construction paper so that the sides are exactly a rational number are zero because the set of all the rational points on a piece of paper with axes imposed on it will have measure zero. Almost all distances are irrational.

Rationals, not irrationals, only appear when you're counting things or leaving the real world.

More importantly, you're coming up with a strawman by saying that mathematicians forget that real world objects aren't neat and shiny and easy to measure. Mathematicians (applied mathematicians) are happy when they can find something that works pretty well in the real world. If you're doing applied math, it doesn't really matter that pi is irrational. What matters is that your Fourier series converges to the right thing (for instance), near enough to be practical. The rigor of pure math makes sure you're doing the right thing to begin with (you can't develop Fourier analysis without some pretty advance, uh, analysis), and then you can sit back and be proud of the mp3 player you end up with. No one is being smug, certainly not smug enough to deserve derision for "forgetting that irrationals don't exist."

Plenty of areas of math, particularly those very useful to chemists and physicists, don't even need to involve numbers at all, anyway. Group theory and manifolds come to mind. 1. You are correct in that I should not have used the generic "mathematician" in my complaint. It is not a strawman since I've run into a few people who actually believe these things, although I admit it does not include those wonderful creatures called "applied mathematicians" or "statisticians". Point taken.

2. "The problem is that you aren't really measuring the length. You're giving an appoximation. That approximation is a guess, probably rounded to a nearby rational number." No. This is simply a mistake on your part. Measurement is a process of comparison, where A<C and you can say both that "B is longer than A and shorter than C". In scientific measurement, neither A, nor B, nor C are irrational numbers--they are all rational. Measurement takes the form of "the side is between 999.9mm and 1000.1mm in length" or "the side is precisely 1000(+/-0.1)mm in length" ("precisely" should be contrasted with "exactly"). If anybody says "the side is {1000*sqrt}(+/-0.1)mm in length", we can confidently state that they simply have made a category error, even if 1000*sqrt confidently lies roughly in the middle of the way between the two rational numbers A and C.

3. "Almost all distances are irrational." Wrong again. You are assuming that the continuity of the number line can be mapped onto the physical reality of space. I have seen no evidence that this is true, and in fact, quantum physics posits a discrete universe with a minimum Δx, as contrasted with dx, of a single Planck length. In the ideal world where one can treat space as continuous (in the abstract, a priori, sense), then you are correct to state that nearly all distances are irrational, and you can play with the difference between countable and uncountable infinities and have your fun with the unreasonable efficacy of Cantor's diagonal argument. In the real world, however, of physical space and scientific measurements, infinity simply doesn't exist, except perhaps as an approximation. 2. That's the definition of measurement I know too. Some dislike the uncertainty involved. ;) I doubt that you could develop quantum theory and any ideas about Planck lengths without using mathematical concepts that require numbers you're doubting exist in the real world. I think for example that you need them to handle waves, which aren't just distributions of pixels.

(I say this despite being more or less a finitist in mathematics -- there's a lot you can do by just satisfying yourself with inequalities rather than asserting the existence of an exact but irrational number. I think people tend to be too quick to think Planck's lengths solve, e.g. Zeno's paradoxes.) Numbers need not exist in the real world in order to be used in theories about the real world. There can still be correspondence between idealized numbers and real world without becoming a platonist about the reality of said idealizations.

Distances and the relationship between Pythagoras' theorem to real-world triangles that you cut from paper or whatnot are interesting, but I don't think that we need to say that a right triangle cut from paper really has an irrational number that forms the ratio of hypotenuse to one of the other two sides, or that a circle drawn on paper really has a ratio of circumference to diameter that is irrational. I'm fine with saying, "ideally, such and such is the case", but I still think a useful distinction can be made between real and ideal--one not apparent to many people.

I think the distinction between real and ideal is what leads us from Zeno's paradox of the impossibility of motion, in ideality when we hold fast to the principles of noncontradiction and consistency, to the reality of our experience of motion.

That said, you are right when you say that Planck-length discreteness in space is often too-quickly used to dismiss Zeno's paradoxes. I think that Zeno's paradoxes are a conflation of real and ideal, whether or not space is discrete as quantum theories posit, or if it is fundamentally continuous as some variations of string theory posit.

That is, just because I think I can imagine infinity and use it in approximations (e.g. I can assume homogeneity in a gas, which is essentially approximating a mole, or 6x1023, as ∞) of the real world does not mean that infinity is infallibly and universally applicable. Zeno's paradox strikes at the heart of why this is so, whether you solve the paradox by introducing finite discreteness or show that the infinite sum converges to a finite number. and you can play with the difference between countable and uncountable infinities and have your fun with the unreasonable efficacy of Cantor's diagonal argument

Unreasonable?

and you can play with the difference between countable and uncountable infinities and have your fun with the unreasonable efficacy of Cantor's diagonal argument

A quick perusal of Wikipedia tells me that this is (probably) not true.

Measurement takes the form of "the side is between 999.9mm and 1000.1mm in length"

You could just as easily say "between 999.9 + d and 1000.1 - d, where d is a really small irrational number" and you'd be just as correct. You lost me on the first two, although it seems you misinterpreted "unreasonable" and assumed the most unflattering of connotations of the word.

And you are wrong on your last point. It violates parsimony to do what you are doing there. Not that parsimony is the end-all be-all of science (certainly not in mathematics), but in this case I don't think one can justify throwing in an tiny semi-arbitrary "d" irrational number just to convert rationals into their nearest irrational approximation/equivalent.

Your argument is beginning to sound like you disbelieve the existence of rational numbers--certainly if we grant the continuity of the number line and look at it probabilistically, the ratio of the quantity of rationals to irrationals is zero.

Are you prepared to say that "2+2=4" does not exist, and that it should be "(2+d1)+(2+d2)=(4+d3)"? How messy does your mathematics get if we go down that road? 