In an earlier post I talked a bit about complex numbers and the great, great, mathematician Augustin-Louis Cauchy (1789 - 1857), who pretty much single-handedly invented the discipline of complex analysis (he was very prolific and did lots of other great stuff too).
Gerolamo Cardano, in the 16th century, is credited with the ‘invention’ of complex numbers. The original idea behind them, at least as it is recounted in the textbooks, does make it seem like an invention; something just ‘magicked’ up out of thin air.
There were certain equations that, at the time, couldn’t be solved.
The simplest example is
which we can ‘read’ as : what number, when multiplied by itself, gives the result minus one?
It was ‘obvious’ that there was no solution at the time. So Cardano just ‘invented’ one. This led to the idea that these numbers were ‘imaginary’ and so the nomenclature stuck. Cardano just plucked the answer seemingly out of nowhere and just said there exists a number i such that
We call this the imaginary number i. At this stage of the game, it does seem to be just a tad on the fanciful side - more useless pratting about by those weird mathematician creatures. We can’t solve this equation, so let’s just pretend one exists1.
It does seem a bit that way doesn’t it?
I want to try to convince you otherwise - that mathematicians are not just making it up as they go along (at least as far as complex numbers go).
Remember the Pandemic™ ?
Remember how it wasn’t just any old disease progression? It was, hold breath and look frightened, described as rising exponentially. The dreaded exponential was wheeled out to make it sound not just terrifying, but terrifying with technical bits.
Oy, Mildred. The Scientists™ say it’s spreading exponentially. I think I’ve just wet myself
So, even though we might not know all the uses of the exponential function we know of at least one use; that authoritarian and manipulative governments can make use of it to scare the living Bejesus out of their population.
But what IS the exponential function? What are they talking about?
There are various ways to define the exponential function, but the one I’m going to pick is the following
This is a good one to pick because it captures the behaviour of the function. The dy/dx piece here can be (kind of) read as : the change in y divided by the change in x. So if you change x a little bit and this changes y a lot, then dy/dx is going to be big. If, on the other hand, if you change x a lot and only see a small change in y then this dy/dx thing is going to be small.
The jargon for this process is differentiation. We differentiate a function to find out how quickly it changes with respect to some input variable.
In simple terms, we can say that for the exponential function ‘how quickly the something changes’ depends on ‘how much of the something there is’.
In virus terms, then, suppose we have a patient zero - the person at Wuhan who was a bit lax with safety and fancied some bat for tea perhaps - and this person infects 2 people, who then each go on to infect another 2 people. What we see then is a rise in the number of cases that looks like this
And we see that if we work out the change for each step (so 2-1, 4-2, 8-4, 16-8 and so on) we just recover the original sequence.
This is the kind of thing The Scientists™ mean when they talk about exponential growth.
The dy/dx = y equation above can be read as the mathematical question : what function, when I differentiate it, gives me the function again ?
We can think of differentiation as an ‘operation’. We take some function, operate on it and, for this particular operation of differentiation, it leaves the function unchanged. The exponential function just doesn’t care; you can differentiate it till the cows come home and it will remain blissfully unaffected. It’s the only function with this property with respect to the operation of differentiation - which is why this equation is a defining equation for the exponential.
I think most of us might agree that, even if we don’t fully understand it, the exponential function has its uses and is somehow ‘real’ and not just plucked from thin air, so to speak. It can be used to describe processes that we might observe. We could, perhaps, think of it as a mathematical ‘model’ when it’s used to describe real-world processes.
So, the exponential is a rather special function and, as a consequence, was extensively studied.
One possibly surprising thing that was discovered was that you could ‘make up’ this exponential function by adding together other functions. You could start with 1, then add an x, then a bit of x squared, then a bit of x cubed, and so on - and if you added up ‘enough’2 of these terms, you’d get the exponential function.
It turns out you can do this with other functions too. So, you can take sin(x) and write it as the sum of powers (it has odd powers only) and cos(x) and do the same thing (it has even powers only). So the ‘wiggly’ functions - the sine and the cosine - which look very different to the exponential function, can also be ‘expanded’ as a sum of powers.
This expansion is called a series. The particular kind of series here is known as the Taylor Series for the exponential function after the English mathematician Brook Taylor who first introduced the general method3 in 1715.
This technique of ‘expanding’ functions like this is very important and gives us another insight into why mathematicians are interested in things like infinity - and seemingly abstract questions like “does this blasted expansion thing actually converge to the right value?”
But what, you may ask, does this have to do with complex numbers?
We have to backtrack a wee bit and recall that the hypothesis of ‘solution’ that Cardano used was a bit more general than just i. It was found that solutions to various, previously unsolvable, equations could be written as a new kind of number that looked like this
These new numbers, usually given the symbol z these days, were called complex numbers. It was found that some equations had solutions only involving the x bit (the real bit), but some equations needed both the real AND the y bit (the imaginary bit) in order to solve them.
Mathematicians, being the playful, happy-go-lucky imaginative souls that they are, started to have fun with these new numbers. What happens, they asked, if we take one of these functions we’ve been pratting around with and, instead of just using x (a real number with no imaginary bits) as the input variable, we use this new complex number z as the input?
Then a kind of magic happens - it’s just one part of the magic of complex numbers.
If we set x = 0, so we have only the imaginary bit y, our complex number would look like z = iy. What happens when we stuff this into the exponential function? What does exp(iy) look like?
If we bung this number z = iy into our series for the exponential, something really interesting happens. After a bit of work we get the following
In other words, we find that
So we have this function, the exponential function, which (we thought) skyrockets away and doesn’t ‘wiggle’ at all has, hiding in there, the wiggly sine and cosine functions!
So we’ve gone from this ‘imaginary’ idea, plucked like the proverbial rabbit out of a mathematician’s Trilby, to a connection between exponentials, sines and cosines that is only apparent when we use complex numbers.
It’s like the exponential was ‘constrained’ by only using it with real numbers. When we plug in a complex number (and real numbers are just complex numbers with zero imaginary part) we get a much richer behaviour. The real numbers and functions of real numbers, then, are only giving us a partial slice of a far richer mathematical ‘reality’.
As you delve a bit deeper into the subject of complex numbers you see more and more examples of this magic. Problems that weren’t properly understood4 when using just real numbers suddenly became explicable when you shunted everything into the complex number world.
This is why I love maths (although I’m not a mathematician). You ‘invent’ stuff and, seemingly out of nowhere, all these ‘hidden’ structures and surprising connections emerge. It’s an interesting (and unresolved) question whether mathematics is invented or just discovered.
*Note : as I was reading a bit about Taylor I learned of the Indian Kerala School of Astronomy and Maths. I’m, therefore, going to revise a bit my earlier statements about how much we might ‘owe’ to non-European mathematicians. Possibly more than I thought - but it’s also not certain to what extent people in Europe knew about these ideas. They may have got their inspiration from them, but also took this inspiration to new and dizzying heights (they didn’t just ‘copy’ stuff, in other words, but massively built on things - always assuming they knew about this prior work in the first place).
It’s also a bit puzzling why this earlier flourishing seemed to die out. It’s like other places got to a certain point and then just stopped (which would be a real shame). I’m not aware of a continuing and independent tradition of advanced mathematics anywhere else that rivals the explosion of the subject in Europe from around the 16th century onwards.
One of my scientific heroes is Ibn al-Haytham (Iraq, 10th to 11th centuries) who was known as the Father of Optics in Europe in this period of European mathematical and scientific progress. Al-Haytham was the first, to my knowledge, to state and implement the scientific method; he built equipment to test out his theories. He should rightly be known as the Father of Science in my view.
Scholarship in the Muslim world around the time of al-Haytham was much more advanced than anything that existed in Europe at the time, but it kind of fizzled out and never really went anywhere. I don’t know why.
Although The Ferg™ is not strictly a mathematician we can draw a parallel with his work : there isn’t a real pandemic, so let’s just pretend there is one. Look, I have a model.
You need to add an infinite number of such terms. But you can get a pretty good approximation to the exponential function by, say, only going up to terms to the 10th power in x. You can even work out the error term - which gives you an idea of how much error you introduce by truncating things.
It is evident that the Indian Kerala school of mathematics had developed Taylor series for specific functions back in the 14th century. Taylor, however, is credited with the development of the general method for their derivation for any (suitable) function. Whether Taylor knew about this earlier work is not clear. Newton had also developed a general method earlier (at least before 1671) but Newton, like Gauss, didn’t always publish his stuff.
For example, the Taylor series expansion for 1/(1-x) only converges to the correct value for values of x between (but not equal to) -1 and +1. It’s obvious why it doesn’t work when x = 1, because then we have 1 divided by zero (which is ‘infinite’). But why should this series ‘blow up’ when x = -1? The value of the function at x = -1 is just 1/2, so why does the series not work here? The answer to this conundrum only becomes apparent when you examine the complex function 1/(1-z) of which our original function involving just the real x is only a ‘slice’ of.
Well, that gave me terrible flashbacks to 1st year Pure Maths at uni (way back in the Cretaceous Period). Invent i to be the square root of minus one. As for lemmas where you assume what you want to prove and then prove it by assuming the proof (I think that is how those things went but it was so long ago. I fled Maths after that and stuck to English Lit and Psychology and Philosophy. A weird species, mathematicians!
The reason the budding scholarship in the islamic world petered out was because it came into conflict with the priesthood, to put it bluntly.
To the priests, even a thousand years ago, the Quran contained all necessary knowledge since it was the word of god by way of the angel who dictated to the prophet who, being illiterate, repeated it to a scribe.
(The kadis and imams of the time had obviously never heard of 'chinese whispers'.)
This set-in-stone approach was very popular with the secular leaders of the islamic world, and mainly being arabs or influenced by arabs and therefore lacking any scientific or even civilisational basis (the arabs' achievements were mainly repossessed greco-roman-egyptian ones in new packaging, and being smart enough to use clever conquered peoples wok to the benefit of the state) above tribal level, they opted for the priests' line of thinking, leading to islamic scholars either renouncing their idas, getting murdered or moving to China or Europe.
(There's a lesson in there for parlour-pinks, wokesters and sundry.)
The above is also the reason why Europe started surpassing the islamic world after the Black Plague had done its job: here, new ideas were welcome as long as they didn't challenge papal dogma, and the catholic church virtually killed its own moral authority by its - even at the time! - well-known depravity, until Luther popped the zit called Pope. Had the islamic world produced a similar figure as Martin Luther... but islamic-arabic culture is incapable of that it seems, being firmly rooted in clan-structures and inbreeding (yes, marrying first cousins for ten centuries is inbreeding, it's not racism but genetics and genetics doesn't care about feelings).
Again, there's a lesson: don't muck about with anything that has to do with fertility, hormones and heritability.
Speaking of exponential growth: a wild sow can have 2-3 litters per year, of 6-8 "boarlets". Without predators, almost all of them will reach maturity inside a year and start reproducing.
Q: Starting with 100 females and 100 males (Yes, Virginia, there are only two sexes) and assuming no more than 10% of "boarlets" die before reproducing, and that all survivors reach at least 5 years of age; when do the population of boar reach 100 000 adult animals?
The above is too difficult for 100% of all feminists, LGBTP, Greens, woke, liberals, progressives and modern-day po-mo marxists I've asked.
Which is the reason Sweden have a very real problem with far more than 100 000 boars destroying crops and gardens.
Science doesn't care about fee-hee-hee-lings.