I’ve always been interested in the great scientific figures of history. Partly, I think, it’s because, in comparison, I’m as thick as two fridge doors1 and I want to understand how come they could do it and I can’t?
We often hear statements along the lines of “Western mathematics owes much to the East”. It’s a kind of olive branch thrown out to assuage the gods of diversity. But is it true?
Whilst it’s true that some debt is owed for things like the concept of ‘zero’ and the use of numerals and some other basic bits and bobs along those lines, it’s a very misleading statement overall.
I’m not much of a historian, in general, and so I’m a bit sketchy on the details and the dates, but sometime around the 15th century, and onwards, there was an explosion of intellectual endeavour in the West, and specifically in Europe. It’s hard to compare the art and music of one culture with another. Is a haiku “superior” to Byron?
But when it comes to science and mathematics that comparison isn’t hard at all. There was simply nothing else anywhere in the world that came even remotely close to the explosion of understanding and invention that happened in Europe in these domains.
The woke may moan and pontificate that it was all “built” on slavery and exploitation, that it’s been put thought several wash cycles that wash it “whiter than white”, but the simple fact of the matter is that nowhere else on the planet did we see such a flourishing in maths or the sciences.
When it comes to the results of this endeavour, I don’t care about the politics or the morality of the participants. I am supremely indifferent to these things, as far as the results go. Newton might have had umpteen links to the slave trade, been the most despicable rancid racist, and had a small band of Asian adolescents in the back room that he mercilessly whipped for some light relief from his intellectual endeavours.
I simply don’t give a shit when it comes to his scientific and mathematical achievements.
If you happen to walk across a golf course as a short cut on your way to work then your “lived experience” will likely contain more than a few headaches as those golf balls fly according to Newton’s Laws of Motion. No ire at “colonialism”, justified or not, will alter that fact. Nor will it alter the fact that Newton was the first to really get to grips with the burgeoning science of mechanics.
Did these advances give the west the technical superiority that allowed them to subjugate and colonize much of the rest of the world? Undoubtedly so, but let’s not kid ourselves that had these advances happened elsewhere there wouldn’t have been, shall we say, similar expansionist tendencies to rail against today. Humans, everywhere, can be pretty crappy, as the historical record shows.
Humans, everywhere, given half a chance, will do the shittiest things to one another and especially if that other belongs to another ‘tribe’ or can be characterized as different in some regard. It’s the sad reality of our existence.
Slavery is an appalling evil, a terrible stain on humanity and, again, it has been another of those depressing human “constants” throughout history. African tribes did it, Muslims did it, we did it (to name just 3 of a very long list) - all of it inexcusable.
But, in my view, it was the philosophical constructs of the ‘west’, and the ability to openly argue against slavery as codified in the notion of freedom of speech, that ultimately allowed us to end the vile practice. Would the opponents of slavery been denounced as “anti-slaver conspiracy theorists” and had their opinion removed as “misinformation” had today’s technology existed back then?
Food for thought.
Anyway, we shouldn’t confuse the intellectual achievements of the west with the morality of those doing the achieving. Nor should we reject those achievements simply because they were sometimes put to somewhat morally questionable ends, or achieved by people who benefitted from their existence within a certain political system (you know, the one said to be steeped in whiteness).
When many people think of “mathematics” they tend to think of arithmetic - things like 2 + 2 = 4. Although arithmetic is critical for developing an understanding of number and might, legitimately, be said to be the foundation that all subsequent mathematical understanding is built on, we have to recognise that the Taj Mahal is not the same as the foundation stones upon which it has been built.
Let’s give an example of the kind of thing I mean.
When we first learn about “maths” it’s based on counting. We probably remember2 math problems along the lines of : if Jemima makes 3 rape allegations against Donald, and Penelope makes 4, how many allegations in total have been made against Donald?
We learn to count using what we call the positive integers. A bit later on we might get zero thrown into the mix. Later still, we try to get our head round negative integers. Somewhere along the way we might get introduced to the idea of a fraction where we divide one integer by another. After working things out like the fraction of the number of rape allegations divided by the number of fires started during mostly peaceful protests, we gradually begin to get a ‘feel’ for what numbers are about and the various operations (add, subtract, multiply, divide) we can use to combine them to give another number. We learn that not all numbers are integers.
A bit later on we may learn that not all numbers can be represented as an integer or a fraction (one integer divided by another). At this point we will probably get confused by one of those oddities of math terminology that has stuck. Numbers that can’t be represented as one integer divided by another integer are called irrational numbers. Of course it makes sense because of the word “ratio”, but it still sounds odd.
At each stage our conception of what a number IS gets expanded. We might even begin to see that the integers are “nested” within the larger group of the rational numbers (the fractions created by dividing two integers) which are themselves “nested” within a larger group of numbers known as real numbers3.
Now we’re getting mathematical - and notice I haven’t talked about “getting the answer right” at all yet.
At this point we haven’t really got much further than the Greeks did.
Then along comes the Italian mathematician (and gambler, apparently) Cardano (1501 - 1576) who hit upon the idea of something we call a complex number. The set of numbers we had wasn’t “big enough” - there was a piece missing. The real numbers were themselves “nested” inside this larger group of complex numbers.
In simple terms he was trying to find the solution of equations. Let’s suppose we have an equation that reads as
which we can read as the question : what number, when we square it, gives us the result 1?
Well, the number 1 works. One times one (one squared) is equal to one. So does the number -1. Minus one times minus one also equals one. There are two numbers (and only two) which are solutions to this equation.
The absolutely pillockish ‘debate’ around 2 + 2 = 5 would utterly fall apart when confronted with the question of the objectivity of the solutions to this equation. There are only two real number solutions; one and minus one. It’s not open to subjective “negotiation”.
Cardano, however, wanted to answer a different question and to find solutions to
which is asking what number, when we square it, gives us the result minus 1?
Cardano had to invent a new kind of number - the number i - which is the solution to this problem. When we square i we get minus one. When we square minus i, we get minus one. So, again, there are two solutions.
A number that could be represented as some real number times i became known, in another one of those unfortunate examples of mathematical terminology that has stuck, as an imaginary number. There’s nothing “imaginary” about it at all. It’s actually a fundamental part of Schrödinger’s equation which we believe (with good reason) governs all matter.
A complex number is a number that has a ‘real’ bit and an ‘imaginary’ bit and is usually given the symbol z so that
where x and y are real numbers. A real number, then, is just a complex number in which the imaginary bit (the y bit) has been set to zero.
There’s no “subjectivity” whatsoever about the fact that the real numbers are a subset of the complex numbers. Once you have defined what a real number is and what a complex number is, then it is objectively true that the reals are a subset of the complex.
I’ve tried to find the “white supremacist thinking” in these equations, and how they perpetuate “whiteness”, but it’s my white fragility and white privilege that just can’t recognize where, and how, these are infused with whiteness4 and deeply racist, colonialist and capitalist in nature - despite the solutions having been discovered by a white man.
We’ve now had some element of getting the answer ‘right’. There is only one right answer to the question of what numbers, when squared, give the result 1. The number tuple of (1, -1) is the right answer. Unequivocally.
Sort of “hidden” in there is another sense of being ‘right’. If we have a polynomial equation in which the highest power is 4, say, an example of which might be the quartic equation
then there will be 4 solutions (values of x) that work. If the highest power is n, then there will be n solutions. No ifs, no buts, no subjectivity; the right answer to “how many solutions are there to a quartic equation?” is 4.
We can see that the concept of “right” in mathematics extends way beyond, for example, being able to multiply 6 and 7 together and get the right answer.
So, let’s finish off with a spectacular theorem (it is also right) about the complex numbers. Here it is; the Cauchy Integral Formula
Oh my word, doesn’t that just look fearsome!
Without going into all the technical details which specify the conditions under which this is true, the theorem says something quite remarkable.
A function can be thought of as just some rule which takes a number input and spits out another number - the “f” in the above expression is a function. Here we’re thinking of a function that takes a complex number as input (a and z are complex numbers in the above) and spits out another complex number.
For example, the function “x squared” takes some number x as input and spits out the square of that number - so put in the number 3 and the function spits out the number 9. We can do exactly the same thing with complex numbers and so the complex function, “z squared” would take a complex number as input and spit out the square of that number (which would be another complex number).
So what does this really scary looking set of symbols actually say?
It says that if we have some region of inputs for our function, and we go round a closed loop (boundary) in that region, then we can work out the value (what it spits out) at every point inside the loop simply by looking at what’s happening at the boundary.
Amazing, beautiful, stuff.
This kind of thing is what I mean when I talk about mathematics.
It’s so far beyond mere ‘counting’ or being able to accurately divide one number into another.
It was also discovered (invented?) by another one of those dead white men (allegedly) steeped in white supremacist thought and with colonialism coursing through every vein.
There is a profound reason why mathematics has been “Eurocentric” - because it was the people in Europe, and nowhere else, doing this stuff to this depth and inventiveness.
As the European ‘culture’ spread elsewhere, admittedly mostly through conquest and colonization, we begin to see others taking up this mathematical mantle. Srinivasa Ramanujan in India (1887 - 1920), for example, although hampered by his lack of formal training, could well have become a mathematician to equal the likes of a Gauss or a Riemann had he lived longer. Sad to say, but it’s probably only because of the British empire do we even know about him at all - or that his amazing mathematical gifts would have surfaced.
Ultimately I don’t give the slightest fuck what the ‘race’ or culture of any great thinker is. I respect them for their achievements quite independent of the culture, or race, or sex, or sexuality, they were born into.
To dismiss, as the woke do, the achievements of one group simply because they happen to have lived in a particular region, with a particular skin colour, at a particular time in history, is a great evil we must strive to avoid and, in their words, to dismantle and resist.
I’m being over generous to myself here. In reality, the comparison should be made using (at least) 10 fridge doors.
I think the problems I had were phrased in terms of things like apples and oranges, but I can’t properly remember now. As I went to a Catholic primary school run by nuns, I suspect that adding up the total number of rape allegations was not part of the curriculum.
The real numbers are the union of the disjoint sets of the rationals and the irrationals
There’s also no sense in which a real number can ‘feel’ like it has been born into the wrong number system and that, deep inside, it’s really an imaginary number. But perhaps I shouldn’t “poke the woke” too much.
Pitnicking:
"African tribes did it, Muslims did it, we did it..." should read "African tribes are still doing it, Moslems are still doing it, we stopped doing voluntarily and tried stopping them from doing it..." for historical accuracy.
Also, slavery is not the same all over. The way africans were used in the US, Brazil, Argentine et cetera was an anomaly in Occidental history - normally, the various forms of indentured service, thralls, serfs et cetera had different judicial and societal (and class-based) implications and were governed by laws regulating the whats and hows and so on.
Egypt, Rome, Persia, Babylon, Sumeria, Greece all had laws regulating what slavery was and what the duties of the owner/master were. Note: duties, as in the master was bound by law to act in a certain manner.
Among africans and arabs/moslems, only brute force counted - slaves were/are essentially things to be used and abused as it pleases the owner. A quintessential cultural difference.
And at least here in the North it was consindered unmanly and foolish to mistreat thralls - they were your labour working for you, so by mistreating them you were in effect shooting yourself in the foot (and by abusing someone unable to defend themselves, you showed that your were a coward). Thralls could also attain freeman status via work, deeds or the decree of their master.
if Jemima makes 3 rape allegations against Donald, and Penelope makes 4, how many allegations in total have been made against Donald? Answer: If Donald was a Muslim these “Racist” attacks would never have made it to court. If this was Saudi Arabia the woman would have been beheaded. So the answer varies depending on which model of maths you use …. East or West.