▲ 136 points
back
106 comments
It's funny that everybody remembers Pythagoras for the right-triangle theorem, but that wasn't what made him important at the time. The right-triangle square equality had been known empirically for centuries. What he proved that was so completely earth-shaking is that for some right triangles A, B, C there is no rational Q for which QA = C. That was what was so important about his proof.
I'm not sure you've got that right. We know almost nothing about the historical Pythagoras and none of his actual writings survived. We only know of him through the Pythagorean brotherhood and other people (eg Plato) who he influenced.
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 documentatary evidence is fragmentary but there is significant evidence that a 5th-century BC Greek mathematician proved the incommensurability of the side of a square with its diagonal (it was apparently trivially known a century later since it appears in Plato's "Meno"). Whether that person was named Pythagoras or Hippasus or something else is really neither here nor there since he was pretty clearly part of the Pythagorean tradition that got associated with one name.
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.
I haven't read the paper you posted yet, but just to clear up a common confusion, Pythagorean triples are not the Pythagorean theorem.
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.
This is exactly right!
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 ??
Stigler's Law of Eponymy
I thought the Pythagoreans threw one of their members over board for proving the root of 2 to be irrational
This is a myth, it's covered in https://en.wikipedia.org/wiki/Hippasus
There's multiple stories about this; one of the better-attested is that for a time the brotherhood swore each other to secrecy (with threats of drowning) about it because it ran against the Parmenedian epistemology of the time.
That would seem to imply that for A =/= 0, C/A is irrational. This seems counterintuitive.
What is counter-intuitive? In a triangle with sides A=1, B=1, then C=root(2), so C/A is irrational. That's what was so impactful about the discovery.
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 3/4/5 triangle is rational. The unit right triangle is not. You’ve dropped the “some” from the parent.
For "most" right triangles, yes, C/A is irrational. In fact the triangles for which C/A is rational are vanishingly rare (though Pythagoras proved many important things about them[1])
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.
According to Van der Warden "Science awakening"[1] Ancient Greeks treated numbers as some kind of "dirty" (Real) model of a platonic Ideal of a quantity. Numbers were invented by filthy traders and accountants while wise philosophers used geometry to reason about quantities.
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.
Hipparchus must have been the greatest accountant of ancient times
https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Hipparch...
I never learned of these formal definitions in high school mathematics. Nor in the lower level college ones that I took. There’s a beauty to this perspective— irrational numbers are what rationals are not.
Yes, but it also messes with a lot of normal intuitions. Some examples:
“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.
For what it's worth, this is only true in Cantor's horrifying paradise. In the world of the intuitionists, none of this is true. Reject Cantor and embrace Brouwer and you can once again live in a world without these horrors, and all you lose is absurd statements about things true "almost everywhere" that are never true, and crazy results like Banach-Tarski that get an impossible result by doing two impossible things to set it up.
> it is only possible to describe 0% of the rational numbers between 0 and 1 with text
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.
> 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 that's at the heart of mathematics - deceptively simple definitions that capture the essence of something.
I happen to be reading Poincare's Science and Hypothesis right now and he introduced a way of defining the square root of two that I found enlightening. In less articulate form: consider two sets, one of which contains all numbers whose square is less than two and one of which contains all numbers whose square is greater than two. The square root of two is then the symbolic name for the element that divides those two sets.
Poincare says it better in the book though.
That's also the definition given in the article, attributed to Richard Dedekind.
I think you're describing a https://en.wikipedia.org/wiki/Dedekind_cut
(edit: oh, I see the article makes that clear. Well here's a link anyway)
You can do the same with all numbers, and you get the construction of the surreal numbers.
Irrational numbers have become a constant nuisance for me in Isabelle. I really wish that the Greeks' initial hypothesis that everything could be expressed in rationals was actually correct.
Something I always love to ponder: Would aliens who perceive reality vastly differently than humans come up with different number systems?
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?
Ultimately mathematics is describing space-time. You can definitely think to start with the intuition of having 2D numbers, but to define that you'd have to define what 1D number is anyway.
I have pondered if a hypothetical race of liquid or gaseous aliens living in a fluid world invented maths that were not rooted in counting numbers, what would that look like.
Interesting timing for this video that talks about bias in stem history (for example how naming discoveries/inventions is done in different cultures)
How do you prove irrational numbers doesn’t repeat down the line?
That's a great question, and the answer is by direct inspection that repeating digits cause the number to be rational.
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 a number start repeating from some point, you can do a nice trick: multiply this number by 10^n choosing n to be a length of a period (effectively shifting the decimal points n places to the right). Then subtract:
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.
I'm with Dedekind on this one -- Cantor's work, by and large, was hot garbage and led and continues to power some of the most naval-gazing mathematics ever invented.
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.
It's really hard to see the history of the sciences especially in the last couple of centuries as anything but a resounding success of modern mathematics. Whatever qualms some people may have about classical mathematics, nobody has shown it to entail a contradiction, nor has any practical result obtained in physics, engineering or anywhere else been shown to be erroneous for mathematical reasons (modelling errors, of course, happen all the time, no matter what mathematics you use).
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.
>> constructive notion of what a "number" was in terms of our ability to approximate it.
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.
Why is it better to invent a weird new class of numbers (irrationals) rather than just identify that there is something wrong with how we think about this issue?
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?
> 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 :)
How would that help you? Being able to reason about irrational numbers is useful.
>invent a weird new class of numbers (irrationals)
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.
We probably have not discovered the unified theory of numbers yet, and these are all patches to the current system
lol finally a quantamag math article where I can follow the math
> The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction.
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.
So then you notice that if you measure more precisely you get a different fraction... and even more precisely another different fraction. It's natural to ask what ratio you'd get if you measured with endless precision.
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?
You can’t do math by measuring real-world objects.