Until today, when I completely randomly came across this truly awesome page on the mathematical structures known as torsors. What in the world is a torsor, and who cares? If you look up Wikipedia, you'll get something along these lines:
A G-torsor is a set X for a group G such that for any x, y in X, there is a unique g in G such that
x . g = y.Extremely enlightening for the mathematical geniuses among us; for me, a pile of gibberish. What the heck does any of this mean?
Well, if you read John Baez's excellent page on torsors, you'll find out exactly what it means. Torsors are essentially mathematical structures that are almost, but not quite, mathematical groups. What's a group? A set of elements that you can combine in a well-defined way, usually symbolized by addition or multiplication, and which must include an identity element. Examples of groups:
- The integers, Z, under the operation of addition, with the identity element being 0
- The real numbers, R, under the operation of addition, with the identity element being 0
- The real numbers excluding 0, R \ {0}, under the operation of multiplication, with the identity element being 1 (why not 0? essentially because you can't divide by 0)
So to go back to the two things that confused me (and which Baez thoroughly addresses):
- Points are different from vectors in that points are elements of a torsor of the vector space; in other words, you can't add points to each other, but you can take the difference of two points and get a vector. If, however, you pick some arbitrary point and decide to label it the "origin", you've now converted your torsor into a vector space.
- Energies are also elements of a torsor, which means that their differences are what we can play with and what really matter to us, computationally.
F' - G' = 0.But if you integrate that, you'll get
F - G = some real number!In other words, F and G are what you get if you pick specific values for the integration constant C. But then what about F + G? You do get a new function H, but it will not be true that H' = f. In fact,
H' = (F + G)' = F' + G' = 2 . fWhat this means, as Baez points out, is that the indefinite integral of a function is not merely a function, but actually forms an R-torsor (since the difference between any two antiderivatives of the function is a real number, and the addition of a real number to a particular antiderivative creates another antiderivative, but the sum of two antiderivatives of a function is not generally an antiderivative of the function itself).
What other entities can be torsors? Baez lists musical intervals and quantum phases. I don't know much about finance, but I would tentatively add stock market indices (not stock prices) to the list. Why? Because, in a certain sense, the exact value of a stock market index does not matter. What is more important is its relative performance. I could be totally wrong about this, and would gladly welcome corrections.
Why does this matter, you may ask? I'll let Baez have the last word on the subject:
I'll admit it: if you've survived this far without torsors, you can probably continue to survive without them. You can always pretend a torsor is a group. But, it involves an arbitrary choice! This means that when you do this, you're imposing some structure on the situation which is not really there already. Through long experience in mathematics, I've learned that making arbitrary choices causes problems. I could list specific technical problems, but the basic problem is that it distorts your thinking. So, the idea of torsors is to avoid pretending something is a group when it doesn't come naturally equipped with an identity element.
Maybe I'm having difficulty with your nomenclature, but I'm not sure what you mean by, "The exact value of a stock market index does not matter. What is more important is its relative performance."
ReplyDeleteI should have been clearer with my wording. What I meant to say was this: If the Dow hits 10,000 tomorrow, then that number "10,000" does not itself refer to some sort of real "value". What it says is that relative to an arbitrarily picked point in time when the Dow's value was set to be 1000, the current value is ten times that value. At least, that is what my limited understanding of the stock market index tells me.
ReplyDeleteIf my understanding of the stock index is right, this is a bit like a Consumer Price Index, which expresses the value of a basket of goods relative to the basket's value in some other year that is arbitrarily picked to be the origin. And regardless of what year you arbitrarily pick to be your index, you can take ratios of the CPI from different years (assuming the basket of goods stays the same) to produce useful information. However, you can't multiply CPIs from different years to produce a new CPI.
In other words, what we think of as the CPI for a year is in fact the ratio of the value of the basket for that year to the value of the basket in some arbitrarily picked year. Sounds to me like the textbook definition of a torsor with some element arbitrarily picked to be the identity.
I think the same principle applies to the stock index (not to stock prices, which are real entities in and of themselves), but I could be totally wrong.
interesting, but i can't understand the concept completely.
ReplyDeleteHe says that temperature was once treated as an R-torsor since only differences could be measured. But when abslt 0 was discovered, it was revealed to be R itself.
So, today, a statement like "It's twice as hot as the other" does make sense. but it wouldn't have made sense before the discovery of absolute zro, right?
Basically: can i conclude that ratios of torsor element don't make sense unless u define an origin/zero, etc?
like for eg, when you say that energy is an element of torsor and tht it's only the diff that we can play with...a statement like "This body has twice the energy of the other" does make sense, right? Have we defined a scale of energy that begins from abs 0?
or am I missing something huge? :S
It would not really have made sense before the existence of absolute zero. Consider this: if the temperature of one object is 20 K and another is 40 K, then today we can legitimately say that the second object is twice as hot as the first. If, on the other hand, we measure two cups of water and find that one is 20 degrees C and the other is 40 degrees C, it's not really accurate to claim that the second cup is "twice as hot" (except in a purely numerical sense); proof being that the absolute temperatures of the two cups are 293 K and 313 K. It is true, however, to claim that the second object is "twice as far" from an ice-cube as the first.
ReplyDeleteWhat this means is that your conclusion, as far as I can tell, is not entirely warranted. The point is that you can always take differences of elements in the torsor to get meaningful entities without having to arbitrarily define an origin. In other words, you can always say that Boston is 150 miles from New York without arbitrarily having to consider the distance of Boston and New York from Chicago. It is therefore meaningful to say things like "Boston is closer to New York than San Francisco is to Los Angeles" without having to define an arbitrary center of the cosmos. But in the absence of a(n arbitrarily picked) center of the universe, it is meaningless to merely say "Boston is 100 miles away" [from where??].
I hope that helped; I don't know if it just made things even more confusing!
(There is one bit of terminological looseness that I should straighten out: the differences between elements of the torsor don't necessarily mean subtraction; it could also include division, etc., all depending on the particular group structure in question. So temperature or gravitational energy would involve subtraction; consumer price indices would involve division, etc. And in the case of points in space, the "subtraction" operation is really a vector operation too, or a translation if you prefer.)
Another example that just struck me: solar years, as a way to measure the passage of time, form a Z-torsor. A solar year is simply one revolution of the earth around the sun, and as such it is meaningless to "add" one revolution to another revolution; you can't produce a new revolution that way. But you can take the "difference", so to speak, for that will be an integer that tells you the number of complete revolutions between two particular revolutions. (Let's ignore finer-grained distinctions for now.)
ReplyDeleteAnd once you pick some arbitrary revolution to be the "first" revolution, you can then point to a particular year and say, "this is the 2009th revolution that has taken place since our arbitrary starting point; let's call it 2010!"
hey, thanks for the quick reply!
ReplyDeleteu're right: i still haven't understood the concept of torsors.
i find it ok to accept when the 'difference' is subtraction.
let me put my question this way:
If we don't know the absolute 0 of a measurand, then how can we talk about ratios?(differences is fine, but how ratios?)
for eg:
1) Temperature: u confirmed tht the statement 'this is twice as hot as the other' wouldn't have made sense before discovery of abs 0. but a difference wud have made sense.
2) Position: We cannot talk about ratio of absolute positions of boston and new york tho we can talk about the difference in their positions: distance.
i didn't comprehend the eg of cpi cos i don't know a thing abt economics!
This is a tricky question to answer because the mathematical terminology is a bit misleading. In a certain, abstract mathematical sense, division and subtraction are the "same": they are ways of making certain quantities "smaller," to put it loosely. To put it differently, they are both the inverse operations of multiplication and addition, respectively, which are also the "same" in the sense that they are ways of "building up" certain quantities. Furthermore, both addition and multiplication have an associated "identity element," which is an element that does not build up any other element.
ReplyDeleteFor addition (and for subtraction), this is called zero, by convention. For multiplication, this is called one. These are not obviously the 0 and 1 we're accustomed to dealing with in Z or R. If you take the set of n-by-n matrices, for instance, zero is the matrix all of whose entries are 0; one is the identity matrix, whose diagonal consists of 1s and which has 0s elsewhere.
The point I'm making here is that the choice of words "absolute zero" relies on the fact that we already know temperature is an additive operation, and hence its identity element will be 0. If we were in a different sort of torsor where the operation is multiplicative (as in the consumer price index, or in the quantum phase example that Baez gives), then if it turned out that there were indeed some sort of absolute configuration, then we'd probably call it "absolute one" or something.
Either way, it doesn't matter. All that matters is that torsors rely on an operation that "reduces" quantities in a well-defined way, and have arbitrarily picked identity elements. So when we find out that some set of quantities is not in fact a torsor but a group, then we can label the particular element that we now know is the actual identity element to be the "absolute identity element," so to speak.
I hope that makes sense! I'm still trying to work through this business for myself.