How the square root of 2 became a number
quantamagazine.org
quantamagazine.org
Imagine not knowing about irrational numbers. You assume all numbers are just integers and fractional ratios between integers. It would be weird (terrifying?) that something as simple as a right triangle would require a whole category of numbers you can't express.
The first people whose thinking was preserved until the present.
Which is still noteworthy, but a different thing.
If you are drawing a diagram for a building and you need a distance equal to the diagonal of a square, you set your compass to the two points and use that distance. No need to determine that it can't be represented by a comfortable multiple of the sides.
But before Pythagoras, it was still an open question if for any two reals A, B there might be a rational Q such that QA = B. Whereas we now know that for "most" reals there is no such Q, thanks to Pythagoras.
While the Pythagoreans did come up with many things (eg the first rigorously documented scientific experiments) my understanding was that it is known for certain that they definitely were not responsible for Pythagoras’ theorem (or this rationality corolary you're talking about), and that the earliest formulation of it that is currently known is from Babylon where it's documented to do with sizing of farm plots[1] about a thousand years before Pythagoras. The proof of the irrationality of the square root of two was so terrifying to the Pythagoreans that the legend has it they threw the dude who produced it off a boat into the sea to drown because it was such a heresy (although I believe that is also known not to be true and the pythagoreans knew that root 2 was irrational).
In that sense it's like Euler's number (first documented by Napier), Lambert's W function (Invented by Euler to solve a family of equations Lambert couldn't solve), Lagrange's notation for calculus (used by Lagrange yes but first also invented by Euler) etc etc.
[1] https://www.researchgate.net/publication/222892801_Methods_a...
The point in any case is that incommensurability as a concept was not widely accepted at the beginning of the 5th century BC and was widely accepted at the end, and the name "Pythagoras" gets attached to the mathematicians who discovered that.
But like for that matter Plato's name wasn't "Plato"; that was a nickname his wrestling coach gave him.
Pythagoras was also a successful wrestling coach. In fact, he coached the most winningest Olympic athlete of all time: Milo of Croton. Milo won 5 consecutive Olympics over a 20 year period.
Pythagoras himself was thrown out of the boys Olympics at age 16 for being too effeminate (long hair), but then entered the men’s Olympics and won. Supposedly, he introduced some new kind of martial arts technique.
This is documented in Thibodeau, 2019 “The Chronology of the Early Greek Natural Philosophers.” Happy to share the refs. Ok, back to maths…
Like my teacher always says: "If at the end of the class, you haven't learned something new, come see me"
I searched for "long hair" and found only this:
>The man with long hair at Samos’: They say there was a Samian boxer with long hair who went to Olympia and won after being mocked by his opponents for looking like a woman; he became proverbial. Eratosthenes says that Pythagoras of Samos won with long hair during the 48 th Olympiad; Duris represents this as Pythagoras being excluded, challenging the men, and beating many of them.
Like many stories concerning Pythagoras, I wonder if this was some local fable onto which his name later became plastered.
But I tend to think that Pythagoras had an even more incredible life than the legends.
Clearly Pythagoras would have sick martial arts moves, if they’d ever make a movie about him.
The theorem is a logical statement about all right triangles, and it has a proof that the statement holds. Pythagorean triples are specific instantiations of the relation for some known triangles and probably would have served as evidence that the statement was even provable.
Historically we probably had triples long before we had a proof of the theorem, just like many of the theorems proved by Euclid were probably already known as rules of thumb.
Compare with an open problem today, like the Riemann hypothesis.
Oh, how I wish there was a book on the history of science and math. Like how they went from the Four Humours or Phlogiston and Spontaneous Generation and Luminferous Ether theories to what they had later. Like how scientists all thought the earth was 100 million years old for a couple centuries until the discovery of radioactivity.
I want a book that would speak about how people made fun of Ignaz Semmelweis for washing hands in hospitals, until Pasteur in France and John Snow in England showed evidence for the germ theory of disease. How people used leeches and bloodletting, and when / why they stopped. (And maybe anecdotes like How Washington Roebling building the Brooklyn Bridge died from a gangrene because he thought pouring water over his wound was enough, and his son finished it)
I want a book that would explain the experiments that led to the theories, like Michelson-Morley that challenged the Lumeniferous Ether model. Or how people first discovered X-rays and didn’t know what to make of them.
How, indeed, did people prove to others that atoms existed? I don’t mean Democritus’ theories 2500 years ago, I mean what made people convinced the world was made from atoms?
And then the experiments that led to the standard model, how was it developed? The word Quark, where it came from, the reactions of scientists to Quantum theory etc.
Our science comes shrinkwrapped, showing only the end result, not the history of thought and the places where (eg Andalusia in the 1100s or China in 20 AD). To me it is very interesting when people weren’t sure yet but got the right result based on circumstantial reasoning, didn’t have high-powered electron microscopes and still somehow realized about atoms and molecules (Gregor Mendel etc). Or how Darwin and Mendeleev developed theories genetics, why they so thoroughly discounted Lamarckian evolution and were they correct etc.
I want a book that discusses the history including the ERRONEOUS theories, the time frames, the experiments that challenged prevailing theories, the controversies, and what made people find the correct theories.
And maybe the people (eg Newton avoided women, and studied Hebrew to reconstruct Biblical history, Michael Faraday was a devout Christian etc.)
Is there such a book? Can you guys recommend ??
That is hard enough already. Showing the history of science properly is easily a hundred-fold more monumental of a task. It is a subject which is studied, and taught -- but for the specialists in the history and philosophy of science. There are many books dedicated both to cursory high level overviews of the history of science, and to some specific episodes in this history, going into nuances of what happened and how it happened.
For example, for the history of mathematics, one can consult "The History of Mathematics: An Introduction" https://www.amazon.com/History-Mathematics-Burton-Professor-...
Here is a much more specialized book, showing how much trouble relatively modern physicists had in correctly conceptualizing "heat" and "temperature": "The tragicomical history of thermodynamics, 1822–1854" https://scholar.google.com/scholar?q=history+of+thermodynami...
Beware! The history of evolution of ideas in science and technology is a vast, vast and an extremely fascinating field, with many dangerous rabbit holes!
Alpha Beta Gamma: History of Radioactivity https://www.youtube.com/watch?v=-G_CtD2UnAc
First Law of Thermodynamics: History of the Concept of Energy https://www.youtube.com/watch?v=a9c7u-FM-Wc
This attitude to numbers can be felt even now when we are taught to solve straightedge and compass construction problems. Greeks had no issues dealing with square root of 2 with geometry or "geometric algebra" how Van der Warden names it.
https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Hipparch...
“Because rationals are dense —between any two rationals there are infinitely many other rationals—there are actually vastly more spaces between rational numbers, than rational numbers themselves. These spaces-between are the real numbers.”
“Because every finite text document can be converted to UTF-8 and thus then an integer, it is only possible to describe 0% of the real numbers between 0 and 1 with text.”
“Since most numbers are indescribable, there are (discontinuous) functions which have the value 2 for almost all numbers, but any number that you actually can describe and try to evaluate the function on, gives 1 and not 2.”
You start to appreciate that logic itself is this Lovecraftian eldritch-horror abomination, and that we only live in the Bliss of Sanity because we live in ignorance, never staring into its depths lest the abyss stare directly back into our souls.
Oh poppycock. We are the eldritch horror. We are the universe experiencing itself. Humans are space orcs, if Reddit is to be believed.
I think you mean irrational :)
That is, the numbers are a subset of the rationals, but it does not follow that we can't describe a rational with a number. In fact the rationals between [0, 1) have a well known numbering,
[ 0/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5,
1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, ... ]
For that reason, quite a few mathematicians view the real numbers as a useful, but ultimately absurd set. Much more sane is the set of computable numbers, that is the set of numbers for which you can find an algorithm that computes the number to arbitrary precision. (More formal: A number x is computable if there exists a Turing machine that gets as input a natural number n, terminates on all inputs, and outputs a rational number y such that |x-y|<10^-n .) Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers.
But perhaps still not as sane as one may hope. It would be very sane to be able to compute, for any two numbers, which one is larger (or whether they're equal), but sadly this is not computable for the computable numbers.
> Every number you ever thought of is computable, but as a mathematician, working with the set of computable numbers is much more tedious than working with real numbers.
I mean, I've thought of noncomputable reals like Chaitin constants.
I'd like to understand - Can you explain this? It seems like it would be easy to have a Turing machines that uses the other two Turing machines, adding one digit at a time until it finds a difference.
> I mean, I've thought of noncomputable reals like Chaitin constants.
Heh, but how many digits can you actually provide? Not too many. So have you really thought of the number in any meaningful sense when you barely know any of its digits?
Also interesting that computer languages themselves are countable, so while it's hard to specify the digits algorithmically for the Chaitin constant of any computer language, you already know that the set of ALL Chaitin constants are countable.
I assume it runs into problems when you try to check if 2 > 2.
I’ve never thought through very many digits of pi either. Or even 1/3 for that matter!
It is possible to describe 100% of the rational numbers with text. You describe the numerator, make a space, then describe the denominator. The length of the text document depends on the number described and can be arbitrarily long.
The easy semantics for intuitionist logic is that every statement is about provability: “A or B or C” is a statement that one or more of these 3 proofs has been supplied.
Where this gets a little bit funky is, you can still take an open mathematical problem and still encode it into the reals: “The nth bit of this number r is 1 if n is a counterexample to the conjecture, or else 0 if not.” If for each n that problem is decidable in a finite number of steps, then this is a perfectly good intuitionistic predicate with which to define a number, and so Goldbach’s conjecture for example can be phrased as “is the Goldbach real equal to 0?” You can do that in the classical approach and Brouwer doesn't limit this too much.
But, now you want to assert that “the Goldbach real number is either 0 or positive.” Because you know it is on the range [0, 1] by construction, right?! But no no no no no, if you want to stay that it is either 0 or positive, you have to furnish me with either a proof that it is 0 (solving the Goldbach conjecture in the affirmative), or a proof that it is positive (solving the Goldbach conjecture in the negative). So you have to come up with alternative ways to talk about the order on the real numbers because ordering statements are this classic example where people love to use the very law of the excluded middle that Brouwer has forbidden.
So you really would have to change the basis of logic to get rid of the strangeness of Cantor's sets, huh? That's quite a leap, I wonder if that would lose other properties of mathematics as well, some that would be nice to actually have?
I mean if you insisted infinities don't exist, you'd get rid of a lot of funky stuff, but lose analytic derivatives and integrals, which definitely is not a good trade-off in my opinion. Some people actually advocate for this, all working on discrete math of course.
Short answer is yes.
Long answer is that the accepted framework of the "basis of logic", i.e. ZF set theory, is a direct result of Cantor's program -- Cantor was not working from an axiomatic basis, he was creating the formalism for set theory. Where things went wrong were not so much that he was an idiot or anything; clearly he is a tremendous genius and saw implications of his programme that led to incredibly strange and counterintuitive spaces. But instead of revisiting the basis, he found himself drawn to this verdant landscape.
Intuitionists (and its various offshoots and cousins) don't reject the notion of infinity per se; even finitists, the most extreme class, still accept that there is an infinity in the form of a repeated process -- that the positive integers are "infinite" in the sense that you can always produce a larger one than any proposed maximum. Integration and differentiation still exist, but are much easier to formalize, because, essentially, the behavior of any constructible function is completely defined by its behavior on rational numbers (or any other constructive version of dense number systems, like binary or decimal expansions).
In re: "different infinities", this is the big red herring of Cantor's work. This requires that you accept that a 1:1 correspondence of infinite sets yields a class of sets that you can group by into "cardinality", and those cardinality classes have an interesting meaning. But this defies the operational use of infinity -- there are more natural numbers than even numbers if we're thinking about strict subsets, but when we're summing series, they are effectively the same size. So it's not necessary to choose some definition of the "size" of an infinite set; you can just choose what operational characteristic you are looking at in the context you're working in.
But you can just ignore this and treat infinity in an operational sense rather than try to have a general definition. The problem with the uncountable argument is that not only are the rational numbers countable, but so are the algebraic numbers, and so, in fact, are any numbers that can be constructed from a finitely expressed constructive process. Unfortunately, even after you leave all those subsets behind, you are left with an uncountable number of elements, none of which can be constructed. These phantoms are the "everywhere" in "almost everywhere".
The short version, I guess, is that those phantom numbers are artifacts of a formalism that allows you to deduce the existence of something by disproving its non-existence rather than constructing an element. The core of most intuitionist mathematics is that the law of the excluded middle (that something must be either true or false) is disallowed under most circumstances. That pretty much ends up taming most of the crazy counterintuitive junk. Still plenty of unexpected and exciting results, and places where intuition breaks down, but far fewer instances of "this clearly is not true" stuff.
Poincare says it better in the book though.
(edit: oh, I see the article makes that clear. Well here's a link anyway)
For example, for the longest time we thought of numbers as 1 dimensional and refused to consider 2-dimensional numbers. Even know we try to shunt them off to the side as much as possible ("complex", "imaginary") even though they model reality more closely.
Might a hypothetical alien race begin with 2D numbers? or something entirely different?
How would they define angles, coordinates etcetera?
What is outside it?
The set of constructs we built are those which our brain sees as being logical and in tune with our observations. The brain is a construct of space time, and its purpose is to model, understand and predict events in space time.
For instance, 0.123123123... is checked to be the same as 123/999, a fraction -- hence rational. Similarly, 0.abcdabcdabcd... is the same as abcd/9999. This works for repeating blocks of digits of any length.
If an irrational number has to have infinite digits without repeating (or stopping), and we can't calculate infinite digits, how can we ever know that a number is actually irrational? What if it's just an absurdly specific rational number, and we just haven't gotten to the end?
https://en.wikipedia.org/wiki/Proof_that_e_is_irrational
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrationa...
The "third" fundamental constant of mathematics, Euler's constant, is thought to be irrational but there is as yet no proof (so it's conceivable it could be rational!)
10^nx-x
infinite periods will cancel each other due to the subtraction, you'll number with a finite amount of digits, say y, so:
10^nx - x = y or x = y/(10^n - 1).
y is rational (finite amount of digits), 10^n - 1 not just rational but integer, so x is rational.
Dedekind had a great idea that at its heart was a constructive notion of what a "number" was in terms of our ability to approximate it. That's the core of intuitionism, and after a long dark interval has finally come back into prominence in modern mathematics.
That's the key. The problem isn't whether the number exists or not. It does exist as a point on the number line. The issue is our inability to describe its position using our number system. Adopt a different numbering system, a different language for describing locations on the number line, and one can avoid the debate altogether.
All the issues such as Banach-Tarski disappear once you apply mathematics to real-world things. Meanwhile, classical mathematics remains insanely practical.
People like you, who call Cantor's work "hot garbage", are giving constructive mathematics, which by itself can be a very useful additional way of doing maths, a bad name. Cantor's diagonal argument, for example, doesn't just disappear in a constructive framework.
Physicists and engineers are notoriously imprecise with their use of mathematics -- "all functions are integrable" and "all matrices are invertible", etc., and that's where all the real-world uses of mathematics have yielded results.
I would love to hear examples where non-constructive techniques have yielded anything of interest. There have been places, for sure, where mathematicians working in those spaces have emerged with real and interesting work (Turing and von Neumann come to mind) but their work ultimately fits well within the bounds of intuitionism.
That said, few working mathematicians are working in a constructive framework, so almost everywhere modern mathematics is successfully used in practice (e.g probability theory), it is based on classical methods, no matter whether it could be theoretically proved constructively or not. Classical proofs are often shorter and more elegant.
Yes, it's true that many classical proofs are shorter and more elegant than constructive proofs. My impression (my intuition?) is that working mathematicians generally have a strong intuition as to whether a non-constructive proof can lead to a constructive argument as opposed to being a purely abstract axiom-of-choice trick of formalism. That is, you get the feeling of whether it's a question of "can be constructed, but I don't know how just yet" vs. "absolutely almost by definition cannot be constructed" and the proofs are weighted accordingly.
There's a huge amount of discussion as to when the law of the excluded middle can be applied constructively which is very interesting as well; with discussions about narrowing what a "proof by contradiction" vs "proof by negation" means in particular that are really trying to explore how many classical proofs are intrinsically constructive, to hopefully narrow the gap in proof elegance over time.
Probability theory remains an area where classical methods still hold pretty firm sway, despite the fact that Kolmogorov himself was pretty strongly in favor of intuitionism. It remains one of the bastions of measure theory, which remains the most pervasive intrusion of non-constructive mathematics into practical mathematics.
I think one big reason Lean is much more popular among mathematicians than Coq ever was is that Lean is pretty unabashedly classical (that, and mathlib). That said, for people who care about program verification, Coq might be the better choice. I'm not anti-constructivism where it makes sense.
Put another way, why don’t we reject out-of-hand the notion that sqrt(2) cannot be calculated, given that right isosceles triangles do exist in reality and their hypotenuse has a definite length?
Put yet another way, why not just say sqrt(2) equals 1.41 (or however much precision you need) + some infinitesimal amount?
What does that even mean? If we're rejecting the notion of an irrational, then the statement, "some infinitesimal amount" might as well be "some gorkly boggleboop".
Sure, we can approximate to whatever precision is required for building a wall or calculating an orbit, but math itself would be hobbled by trying to make discoveries with the handicap of only allowing rationals.
> given that right isosceles triangles do exist in reality and their hypotenuse has a definite length?
I might be agreeing with you in a sideways manner, but right isosceles triangles don't exist in reality. Nor do any of the simple shapes like squares and circles. We have physical things that approximate those ideal shapes, but even the most precise triangle will not have a perfect right or 45 deg angle. Nor will the real-world hypotenuse be precisely sqrt(2). These physical items are made of a countable amount of molecules each of which is in some quantized state. Hell, the length of each side of the most perfect triangle we can make will be in constant flux.
So for practical everyday purposes, sure. We can't work directly with irrationals, and there's no need to. But for making new discoveries in math, we must work out how to deal with "weird new" classes of numbers, like 0, or the negatives, or the complex, etc.
Each one of those classes of numbers has survived because it has proven useful. If you can identify the "something wrong with how we think about this issue", you would probably win a big old prize for that :)
No one is really inventing numbers like 2^(0.5). They just are what they are. Calling them irrationals is just naming something that always existed.
Let me try it! If I make a square with sides 1 inch long, then measure the diagonal with a tape measure I get... 1 and 6/16ths! Only whole numbers and the ratios between them involved there, so I guess that's all the Greeks needed after all.
Don't we all start out doing math by counting/measuring real-world objects? It was all "Sally has x apples" and rulers when I started school but some kids do math using other tools like cubes and cuisenaire rods which are also used to do math through measuring/counting real world objects.
And that's what the article says the Greeks believed they could do using whole numbers and fractions. In the case of a real world square with sides of length 1 it seems that you can get away with describing the length of its diagonal in those terms which made it seem odd that it was what caused them to abandon their belief/aspiration. Maybe the author just described their dilemma poorly/strangely.
I'm not at all suggesting that math has to be limited to the real world or that irrational numbers don't have their place. We're certainly better off with them.
In some cases there is such an answer and you can reason it out without conducting a series of increasingly precise measurements (though you could also find it with a sufficiently precise measurement), but in other cases there is no such answer. Isn't that interesting?