Cartesian conceptual question, can I please get some help?

Username99

Hi, I had been giving a bit of thought to differentiation from first principles lately and I am finally happy with how the whole thing works, but I had been reading up a bit on Rene Descartes and there is a old wives tale that he came up with the Cartesian co-ordinate system when he was lying sick in bed looking up at a fly crossing the room just beneath the tiled ceiling.

Now all along I had thought of the Cartesian system as these rigid set of co-ordinates almost like solid iron structures that could never be moved. I then started to think about how I would explain Differentiation to someone who would like to learn. I could not come up with a single idea that did not include some function of time etc... where it did not involve a case where it made no sense to talk about negative t (time) values (as to my understanding there was never a negative time), it just made sense to speak about initial time and final time i.e. t = 0s; t = ks (k is some constant).

However I then began to think about finance and how a company director might stand at the top of a boardroom in front of a room of chief executives and point to a presentation which displayed the company profits over lets say a 10 year (120 month) period. But again I was stumped when I couldn't decide where to put (0,0), but this got me thinking it was kind of arbitrary where I put (0,0). I could put it at the initial stage of the 120 month period and just take everything in the positive sense or I could land (0,0) at the exact middle of the 60th month and take everything that came before it i.e. 0 - 60 in a negative sense and everything that comes ie. 60 - 120 in the positive sense depending on which mood I am in or what aspect I want to analyse a piece of information from. This then got me thinking about why we use variables, not I know why we use variables obviously but it really got me thinking about the whole thing. Once we have a forumla we can decide how we want to look at this information and the forumla itself will tell us where (0,0) is?

Please any advice at all will be greatly appreciated, I know this is very trivial for most on here I struggle immensely trying to allow my mind think freely regarding this stuff.

Username99

Take the quadratic in the attachment as an example if I were shown this graph and told it was data collected whilst monitoring a sprinters training regime. (Let x - axis = t - axis (time axis) and let y - axis = s- axis (distance axis) ). The data tells me where he was at any point in time. My first instinct would be to negate all values for negative t as there was never a negative time, from the data I would then deduce that he initially started from 3 metres behind what we will call the start line, he started fast but decelerated for 3 seconds. After 3 seconds he was approx (I am guessing of the graph, I havn't worked this problem out) 15 metres behind the start line. He then stops for a 'split second' ( dy,dx = 0 ) {He is 12 metres from where he started} and sprints back towards the start line, he crosses the start line 4 seconds later or 7 seconds into his run, and he is evidently some sort of magical runner who accelerated for the rest of time :-), maybe someone told us that he stopped running after 10 seconds therefore we can negate all values after 10 seconds ( there was a fault with the machine and it made up its own data after that. Then I could restrict the domain to the closed interval [ 0 ,10 ].

****Now lets shift the axis****
But we could change it up and say that (0,0) is now where we said t was = 3s. Lets use the closed interval [-4, 7]. Now at t = -4s, the man gets out of his car and wants to do a warm up jog, he jogs back towards town ("negative direction") then at t = 0s (the sprint starts) he turns on his heal (Obviously stopping for just a fraction of a second) and sprints for the next 7 seconds, he ends up 34 metres away from the car or you could also say that he is 34 metres further away from the town than the spot he parked the car in.

Can anyone correct this story for me, is it right, wrong, stupid, what story would you tell?

bnt

As you say, the choice of origin point in economics can be arbitrary - and you could say that it has to be, since the data form a continuum, often with no clear start or end.

Take the Case-Shiller Index of USA house prices. which shows how they have varied over the years, adjusted for inflation.



The x-axis "origin" in this index is the year 1890, and the y-axis represents house prices relative to the price in 1890 rebased to start at 100. Why 1890? I don't know. I suspect it has a lot to do with the availability of data. Which might sound like a problem, but is it? It depends on whether you care about the absolute value of the index, or its deviation from 100, or even its rate of change.

If anything, assuming good data, I think the rate of change is most important in this case, because of what it means: prices can rise over time for multiple reasons including population growth. But what the data shows is that not only was the index increasing, its rate of change was increasing too, as per this (not very good) chart:



The solid line is the rate of change, which kept increasing between 1993 and 2005. After 2005, the rate of change dropped to zero over the next two years, though the index value was still increasing, and only fell after 2007. In other words, it was the fluctuations in the rate of price change that set off alarm bells, even before prices started dropping.

So, it was on the basis of this data that Shiller concluded that there was a house price boom in the USA back in the mid-2000s, and correctly predicted that a bust was coming. For this and other empirical work, Shiller shared the Nobel Prize in Economics in 2013.

bnt

Re your 2nd post - I'm not really sure what you're asking, so apologies if I'm going over stuff you already know. Your attachment shows the function f(x) = x² - 6x - 7. You say you can't have negative time - why not? It depends on where you put the origin. Imagine I was standing there with a stopwatch in hand, but didn't start it until the point where x=0 on the chart. Does that mean you only started running then? No, the previous bit of the run was in negative time, relative to the time shown on the stopwatch. You don't have real timing data for that negative time.

But since you are saying that your position followed a mathematical function, we can use that function to extrapolate back in time to see where you were, even though the stop watch wasn't running. That's the beauty of functions: they let us extrapolate in to regions without data, but there are risks involved in doing that. More than one economist has felt the consequences of unwisely fitting a function to data.

if you differentiate that function you get f'(x) = 2x - 6, which is the function for the rate of change of the function w.r.t. x. Now imagine shifting the whole function up along the y-axis, so it becomes g(x) = x^2 - 6x + 20. The differential g'(x) = 2x - 6 = the same as before.

So, by differentiating, you have actually lost some of the original function's information. This is made explcit if you integrate the differentiated function:
g'(x) = f'(x) = 2x - 6 -> g(x) = x² - 6x + c, where c is a placeholder to show that there is some value there, but you can't know what it is without more information ("initial conditions"). To put it another way: if all you have is a rate of change in a variable, you can't find the origin from that alone.

You also see vectors used to express this concept. A vector of (4, 3) expresses a change in (x, y) co-ordinates, divorced from any origin. Sometimes you do have an origin, but you can't assume that.

Often, that's OK, because e.g. all motion is relative. When we hear about a Boeing 777 cruising at 900 km/h, we assume that means airspeed, because that is what matters in the physics of plane flight. Yet I've been in a 777 over the Atlantic, watching the speed indicator showing over 1200 km/h. Was the plane exceeding the speed of sound? No, because that figure was ground speed (via GPS), courtesy of the jetstream pushing us in an easterly direction. You can view the plane's airspeed and the jetstream as vectors being added together to form a third vector: its ground speed.

But we still choose to ignore the speed of the Earth spinning on its axis, or the Earth moving around the Sun, because they are irrelevant w.r.t. the immediate problems: (a) keeping the plane in the air and (b) navigating to its correct destination on the ground. The jetstream doesn't matter for (a), but it does matter for (b). If it wasn't for navigational aids, the pilot might not even know there was a jetstream at all, and end up well off course.

tl;dr: time and motion are relative, and your choice of origin depends on what you're trying to measure. You may not even need an origin.