What never ceases to amaze me when reading ancient mathematics is how much harder everything is without proper notation.
For example, you don't have rational numbers, so you are forced to represent everything as awkward sums. Even computing, say, three fifths of a number is a challenge under that system!
You don't have \pi to represent "ratio of circumference to diameter", so you have to use ad-hoc approximations.
You don't have notation for polynomials, so even basic equations are ad-hoc problems.
Another example: (almost) the entire work of Diophantus, the man diophantine equations are named after, is basically obsoleted by \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Mind you, it's not that these people were dumb. That they were solving those problems with limited tools is proof enough that they were very smart.
It's like trying to write a program in 80s Basic, at a certain point the incidental complexity is eating you alive.
> at a certain point the incidental complexity is eating you alive.
I had a similar deep insight when studying both theoretical physics and computer language / compiler design: for some areas of human endeavour, the only limit is the human brain. The machines ceased being the limiting factor a long time ago, but the human brain is essentially unchanged over the timescale of millennia.
In areas like this, incidental complexity is the key limit holding us back.
Once you've reached 100% of the brain's capacity, the only way to "get more out of it" is to waste less of its capacity on non-useful things.
You mentioned pi as a significant advancement. Yes, it is. But tau would be better. Every time I (or anyone) mentions this, there's dozens of people coming out of the woodwork saying that it doesn't matter, that it's just notation, that there's an established standard, blah, blah, blah.
Meanwhile, basically nothing in trigonometry really clicked for me until I saw Vihart's rant about why we should use tau. None of it. Not really, because the pi notation is nonsense, an accident of history, nothing more. It filled my brain with incidental complexity: that factor of two everywhere that made no sense to me.
> …for some areas of human endeavour, the only limit is the human brain. … But tau would be better. … [people dismiss this because “it’s just notation”…]
The case for me of this is current flowing from - to +. Franklin didn’t know better and had a 50/50 chance of getting it right, but happened to get it “backwards”.
A small thing but one that always makes me skip a beat while following a circuit.
> In areas like this, incidental complexity is the key limit holding us back.
> Once you've reached 100% of the brain's capacity, the only way to "get more out of it" is to waste less of its capacity on non-useful things.
Queue modern day social media... It has its uses, but I feel like the brainrot and lessening of attention span it induces (and not to mention everything else like polarization, cult-like behaviours etc) is basically not worth the small amount of benefit it actually provides.
Tau is better for some equations (most of them?) and worse for some others. It would probably be a net positive to adopt it if you discount the huge chore of transitioning during decades. But what this fails to account is that most mathematicians regard 2pi as an important quantity on its own already; it's not a single symbol but it's close enough. So 2pi is expected to appear in many equations and any other value, like 3pi, looks fishy.
Really, the only problem with 2pi is that if you multiply it by a constant, it disappears. so people might write 2*2pi as 4pi. Sometimes you may not want to do that.
Anyway you don't need to fully transition to tau in order to benefit from it. Just like the planck constant has a reduced planck constant (which is probably more useful), and both are used, tau can be already used when it makes sense, without ever fully replacing pi.
The question this should inspire: what do we find hard that would be trivial if we developed better notation?
Einstein tensors are a great example of notation as an enabling technology - it’s a mistake to think we have thought of all the notation tricks we might need.
I wonder if, for example, in attempting to reason about neural networks in terms of ‘gradients’ and ‘matrix operations’ we might look, to future mathematicians, like poor ancients struggling to handle arithmetic without a clean notation for rational fractions…
I think quantum computing is at the start of a notation transition, from using mainly quantum logic circuits to using mainly ZX graphs. It makes things that used to look different obviously the same, so you can easily translate results.
https://arxiv.org/abs/2206.12780 (using ZX to build a 4-body parity measurement out of 5 2-body parity measurements instead of 6, and to show the previous constructions are easily related)
The hard part of course is that genuinely new notation (i.e. not just a more convenient way to write the same thing as before) requires genuinely new mathematics to go along with it.
People had been doing derivatives (at least sort of) before Newton and Leibniz, Cardano (sort of) used complex numbers before Euler, algebra before Galois, categories before Mac Lane.
But these people "connected the dots", as it were. It's not merely a trick -- it's a fundamental contribution.
Can’t we view it as new notation creates new mathematics? A placeholder symbol vs an actual number doing the same thing and more (discovery of zero as a number).
The new mathematics at least could be initially tiny, and then the notation provides the leg work for huge forthcoming discoveries.
Can new notation also create new physics, not just mathematics?
Thinking about Feynman diagrams.
I am neither physicist nor mathematician, so I don't know if this makes sense.
Wondering if Feynman diagrams were not just a crisp articulation of existing concepts, but did they also pave the way for new ideas in physics?
Heavyside's vector notation compressed Maxwell's twenty-some component equations into four. Thats elegant for teaching and teeshirts. However when I want to compute them on a computer grid, I use something closer to Maxwell's original equations.
And if you use a formalism accepting general multivectors, you get 1 equation.
The fully expanded coordinate version is based on an arbitrary choice of spacetime coordinates.
Ideally you'd use a computer language/environment with a rich enough basic formalism to let you write the 1 formula in a human-legible way and have your compiler expand out all of the coordinate version as appropriate for the particular problem for the sake of doing floating point arithmetic.
Diophantus is credited with inventing some new algebraic notation.
The quadratic formula wasn't interesting to Diophantus because he didn't believe in negative or irrational or imaginary numbers; he considered those "impossible".
> (almost) the entire work of Diophantus, the man diophantine equations are named after, is basically obsoleted by \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
This is nowhere close to an accurate statement, even if we only consider works that have survived to the present (most of his work has been lost). Have you tried to read the work of Diophantus or any modern summary?
> 1, 2, 6, 7, 8 and 9 loaves of bread (respectively, in each problem) are divided among 10 men. In each case, represent each man's share of bread as an Egyptian fraction.
I guess it shouldn’t be surprising, but wow, this form of question really goes back.
I'm a bit in two minds about these things. On the one hand it feels wrong it was removed from the country of origin. On the other hand would it have been as well preserved if it hadn't been?
On the third hand, this is clearly a cultural concern as had I been born 100 years ago I wouldn't have cared in the slightest about removing a document from primitive people. And had I been born 100 years in the future I wouldn't have the slightest regard for the biological humans' concerns.
I think my answer to this would be "thanks for preserving this for us; we're ready to take it back now". These artifacts absolutely belong to their countries of origin, but we have a collective duty to ensure that they are protected -- and museums should ensure that conditions are in place for them to be properly handled before returning them.
World heritage objects (over 1000years old?) should be under multilateral supervision and spread across the world, with every region having a sampling of artifacts from the whole world.
The people who live in Egypt now are not any closer to the writers of the Rhind Papyrus than the people who live in Norway or Argentina. The Rhind Papyrus belongs to the world heritage, not to the Egypt heritage.
Egypt (like probably every other place on Earth) experienced countless migrations. Egypt was under Persian rule for 300 years, then under Greco-Roman rule for 1000 years. After that (1400 years ago) it was conquered by the Arabs, and to this day it considers itself an Arab state. For many hundreds of years during this period (roughly from 1200 to 1800) Egypt imported lots of slaves that became soldiers (the "mamluks"). Most of these slaves were of Slavic origin, thus resulting in the English word "slave" coming from "Slav". Bottom line, Egypt was a huge melting pot for at least 3000 years, and most likely before too. People who live now in Egypt can probably trace most of their ancestors to regions outside of Egypt. They are not more heirs of the writers of the Rhind Papyrus than you and me.
But let's say that they are. What did the inhabitants of Egypt do about the Pharaonic era historical artifacts? They looted them time and again. Looted and sold them to the highest bidder.
Maybe you want to claim that people from Egypt don't "loot" the Ancient Egypt artifacts, they only "take possession" of them, since they are the rightful owners.
But if you claim that, then you are not in luck, my friend. This is what happened to the Rhind Papyrus. It was found (or looted) by the locals, who sold it to the highest bidder, who happened to be a guy from the Great Britain. The papyrus was not stolen, it was simply bought.
> One of the modern applications of Egyptian fractions is the request of a specific resistance value needed in the design of an electrical circuit, a problem called in the literature the 2- Ohm problem. College students know well from their physics class, that the equivalent resistance R of two parallel resistances and is given from a law very easy to deduce, based on equating the current passing through the fictitious equivalent resistance R with the two currents passing through both resistances while maintaining same potential. One direct application of this, suppose an engineer wishes to incorporate in one of his designs a resistor of so many ohms which the manufacturer does not produce; for it is impossible that the latter displays in the market all possible ohm-values for his resistors. First, the market cannot possibly sustain it, but more important, one cannot feasibly produce resistors with values as elements of a dense subset of the real line, being, as analysis taught us, an uncountable set. Rather, manufacturers display only in the market what they call an "E12 series", i.e. resistors in sets of 12 different values, namely
> Now suppose an engineer needs in one of his designs a resistor of 7 ohms, then he would resort to a parallel combination from the fraction 1/7=1/10+1/56+1/100+1/120+1/150 in which all the resistors belong to the E12 series, i.e. he will replace his 7 ohms resistor with 5 parallel resistors; this he reaches using a special software (computer programmes exist for such designs, yet the exact solution is by no means trivial). What I did myself instead, is to resort to Ahmes 2/n table and wrote 2/7=1/4+1/28 or that 1/7=1/8+1/56. Decomposing further 1/8 into 1/12+1/24=1/12+1/48+1/48, but then I shall have to use instead of the 48 ohms resistor a resistor of 47 ohms from the E12 table. My final fraction is 1/7=1/12+1/ 47+1/47+1/56, i.e. my resistor of 7 ohms will be simulated by 4 parallel resistors instead of 5 (I am accepting equal fractions). My solution is both minimal and optimal based on Ahmes table. The relative error of my design will not exceed 0.6 per cent, being negligible; especially that any manufactured resistor will itself be subject to some allowed tolerance of the same order.
https://web.archive.org/web/20130625181118/http://weekly.ahr...
So, a thousands of years ago? Ancient Egypt and before? :)
Also:
> The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.
> one cannot feasibly produce resistors with values as elements of a dense subset of the real line, being, as analysis taught us, an uncountable set.
Thisnis hyperbolic.
The number of abstract mathematically plausible values is countable, not uncountable, and the number of practical values under real world tolerances is finite. Real numbers have nothing to do with it, and Egyptian fractions are useful for compressing a large finite set into a small finite basis. They do nothing thay helps compress infinite sets.
For example, you don't have rational numbers, so you are forced to represent everything as awkward sums. Even computing, say, three fifths of a number is a challenge under that system!
You don't have \pi to represent "ratio of circumference to diameter", so you have to use ad-hoc approximations.
You don't have notation for polynomials, so even basic equations are ad-hoc problems.
Another example: (almost) the entire work of Diophantus, the man diophantine equations are named after, is basically obsoleted by \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Mind you, it's not that these people were dumb. That they were solving those problems with limited tools is proof enough that they were very smart.
It's like trying to write a program in 80s Basic, at a certain point the incidental complexity is eating you alive.