History of Mathematics.
You might wonder why I chose this subject to be first; after all, what could possibly be interesting about the stories behind how people came to work out that the cross-product of the tangent and normal is the binormal when taking the derviatives of a curve in R3 or some such incomprehensible mumbo-jumbo??
Good point. So I won't be talking about that, but rather how it all got started in the first place, which reveals something about human nature (ancient Greek human nature, anyway) and most importantly why mathematicians across history are the people who have most shaped the world.
For a full history, covering the beginning of maths until the present day, there are several books available, but by about the year 1700 the concepts discussed are well into university-level maths, so my *short* history ends in around 800AD, with many big gaps before then.
Mathematics begins, it seems, with problem solving. It's all very well saying, "I'd like a big city built here in my name, guys, get right on it!", but how will you know how many slaves/ materials/ days to finish/ food suppliers etc., will be needed to finish the job?
So early maths solves problems like:
- We're sending a 5,000-man army to conquer that city across the desert. How much water and food will we need, if it takes 10 days to get there and the siege will last at least 20 days?
- When will the Nile next flood?
- How should we divide up the late rich farmers' fields among his 12 sons, given that their shares should be not equal but in relation to their position in the family?
(yes, these examples all relate to Ancient Egypt, an important early centre of these sorts of things.)
And so on. The methods for solving these become remarkably advanced, but - and here's a thing that's rather interesting - we don't know how they worked out how to sovle these problems. This means that much of early maths is a set of instructions for how to solve a problem given these numbers, and very often the same basic problem can be restated ten or more times with different starting values (e.g 4,000 men and 200 camels instead of 5,000 men and just a single, solitary camel).
Dry stuff, so far, eh? Well done getting this far, but there's not that much interesting at the moment about maths of 4,000 years ago - except for a couple of things that suggest maybe a couple of people were doing Maths for fun.
Maths, you see, is about going beyond the real and entering the most abstract concepts in the world. So these problem-solvers, while very good, weren't really doing anything other than what they had to do. Given any
problem, they would look up the method and find you the answer, then sit back and count their wages.
Maths as we know it today starts, then, with the Greeks. For some reason, it seems that these guys just loved to argue with each other, for no reason at all other than they just enjoyed it [as explained by my History of Maths Lecturer). This eventually led to schools of argument being set up, e.g. the Sophists (hence "sophisticated"). One of
these blokes, a guy named Parmenides, steps up in around 600-530 BC and declares, "You can't have sure and certain knowledge of the world, because I can create uncertainty in it - see this stick? Well, *breaks stick" it's in two bits now - bet you didn't see that coming!!!" Or words to that effect.
Ouch.
The big deal of this is that you'd quite like to be able to know some things are absolutely true, otherwise it's not very easy to argue - you may be triumphantly hammering home your point when someone points out, "But you can't know all this so everything that comes after it is just a theory!!"
This is where mathematics steps in, in the form of Geometry. Here, the Greeks spend a long time essentially working on establishing facts in Geometry that you can't argue with no matter how hard you try - in other words, they try to prove things beyond all possible doubt. Out of this time you get Archimedes, Euclid, Plato and Aristotle (philosophers by trade, but Plato at least made some big steps in Maths on-the-side), and a couple of others no-one seems to have heard of - Eudoxus and Hippocrates of Chios. Look them up to see what they did, if you like. Over this time, the idea of something having been "proved" develops, from the simple, "Well man, it's obvious innit?" to the more formal language of today.
So mathematicians now started to churn out results that are actually rather incredible - Ptolemy much later managed to construct a model solar system that fitted the known facts and, though it's wrong, it's still an incredible achievement since, if my History of Maths professor is right, one of his predictions is that the Earth (which he places at the centre of the solar system) isn't quite at the centre of the Solar system. In other words, his model essentially predicts ellipses long before Kepler came along.
Essentially, the History of Maths can be explained like this: "Yes we CAN know certainly things about the world, see??", and they look for more and more things to prove even today.
The final part of this little tale is about why we have to thank the Middle East for our modern way of life:
- All the stuff that came before them would be lost were it not for an incredible amount of time they spent collecting, preserving and combining it all into their libraries.
- The Arabs invented modern algebra, without which we could go nowhere in maths.
Just to show how important algebra is:
Suppose you have a garden 20 metres long and 50 metres wide. Then you'd like a path around the edges of it to get to the shed at the far side. You'd like, of course, some garden left and the house owner wants about 800 square metres left after the path is put in. How much concrete will we need to make the path, given that it takes two bags per square metre and that the path will have the same width throughout?
Oo-erg??? After frantic measuring and alot of adding together rectangles you might find the answer, but:
width of paths is, area wanted to be left is ab - (a-2x)(b-2x) [ab=1000, ab
- (a-2x)(b-2x) = 800, a=50, b=20]
=> x2 - 35x + 50 = 0 => x = 1.5m
area of path = 2x((a)+(b-x)) [a=20, b=50] =>. area is 204m2
area*n [n=2] = 408 bags.
Four lines of working solving any problem of this nature. Brilliant. You may not be able to follow it, but you gotta admire how quick that was.
Hope you enjoyed that!! It's a bit of a crash course, and there's a lot more out there about this period in Maths and what follows.