The two meanings of mathematical terms

byJamesCole10y15th Jun 200980 comments


[edit: sorry, the formatting of links and italics in this is all screwy.  I've tried editing both the rich-text and the HTML and either way it looks ok while i'm editing it but the formatted terms either come out with no surrounding spaces or two surrounding spaces]

In the latest Rationality Quotes thread, CronoDAS  quoted  Paul Graham: 

It would not be a bad definition of math to call it the study of terms that have precise meanings.

Sort of. I started writing a this as a reply to that comment, but it grew into a post.
We've all heard of the story of  epicycles  and how before Copernicus came along the movement of the stars and planets were explained by the idea of them being attached to rotating epicycles, some of which were embedded within other larger, rotating epicycles (I'm simplifying the terminology a little here).
As we now know, the Epicycles theory was completely wrong.  The stars and planets were not at the distances from earth posited by the theory, or of the size presumed by it, nor were they moving about on some giant clockwork structure of rings.  
In the theory of Epicycles the terms had precise mathematical meanings.  The problem was that what the terms were meant to represent in reality were wrong.  The theory involved applied mathematical statements, and in any such statements the terms don’t just have their mathematical meaning -- what the equations say about them -- they also have an ‘external’ meaning concerning what they’re supposed to represent in or about reality.
Lets consider these two types of meanings.  The mathematical, or  ‘internal’, meaning of a statement like ‘1 + 1 = 2’ is very precise.  ‘1 + 1’ is  defined  as ‘2’, so ‘1 + 1 = 2’ is pretty much  the  pre-eminent fact or truth.  This is why mathematical truth is usually given such an exhaulted place.  So far so good with saying that mathematics is the study of terms with precise meanings. 
But what if ‘1 + 1 = 2’ happens to be used to describe something in reality?  Each of the terms will then take on a  second meaning -- concerning what they are meant to be representing in reality.  This meaning lies outside the mathematical theory, and there is no guarantee that it is accurate.
The problem with saying that mathematics is the study of terms with precise meanings is that it’s all to easy to take this as trivially true, because the terms obviously have a precise mathematical sense.  It’s easy to overlook the other type of meaning, to think there is just  the  meaning of the term, and that there is just the question of the precision of their meanings.   This is why we get people saying "numbers don’t lie".  
‘Precise’ is a synonym for "accurate" and "exact" and it is characterized by "perfect conformity to fact or truth" (according to WordNet).  So when someone says that mathematics is the study of terms with precise meanings, we have a tendancy to take it as meaning it’s the study of things that are accurate and true.  The problem with that is, mathematical precision clearly does not guarantee the precision -- the accuracy or truth -- of applied mathematical statements, which need to conform with reality.
There are quite subtle ways of falling into this trap of confusing the two meanings.  A believer in epicycles would likely have thought that it must have been correct because it gave mathematically correct answers.  And  it actually did .  Epicycles actually did precisely calculate the positions of the stars and planets (not absolutely perfectly, but in principle the theory could have been adjusted to give perfectly precise results).  If the mathematics was right, how could it be wrong?  
But what the theory was actually calcualting was not the movement of galactic clockwork machinery and stars and planets embedded within it, but the movement of points of light (corresponding to the real stars and planets) as those points of light moved across the sky.  Those positions were right but they had it conceptualised all wrong.  
Which begs the question of whether it really matters if the conceptualisation is wrong, as long as the numbers are right?  Isn’t instrumental correctness all that really matters?  We might think so, but this is not true.  How would Pluto’s existence been predicted  under an epicycles conceptualisation?  How would we have thought about space travel under such a conceptualisation?
The moral is, when we're looking at mathematical statements, numbers are representations, and representations can lie.

If you're interested in knowing more about epicycles and how that theory was overthrown by the Copernican one, Thomas Kuhn's quite readable  The Copernican Revolution  is an excellent resource.