[FOM] Mathematics and rigour

[Timothy Y. Chow]

Vladimir Sazonov <V.Sazonov at csc.liv.ac.uk> wrote:
> Precision is too vague and does not distinguish mathematics from other 
> sciences and human activities at all (virtually any science tries to be 
> precise, and what does it mean "sufficiently precise"? what is 
> sufficient and what is not? it is highly non-specific!)

Not to try to change your mind on the topic, but to clarify my own point 
of view: Every science indeed tries to be sufficiently precise, but is not 
*characterized* by the threshold of precision that it uses.  Physics is 
characterized as the study of the physical world; there is no threshold of 
precision with the property that everything that passes that threshold 
automatically counts as physics.

As for the complaint that the term "sufficiently precise" is not 
sufficiently precise, I deliberately avoided the temptation to make the 
term more precise by exhibiting a particular threshold, because that would 
be a "partisan" move, effectively pledging my allegiance to some 
particular standard philosophy of mathematics.  My main observation is 
that everyone seems to answer the question "is X mathematics?" by judging 
whether X is sufficiently precise; they may differ in their verdicts on 
any particular X, but they are all trying to do the same thing.

> Also it is not a precision what distinguishes various views on the 
> foundation of mathematics (Intuitionism, Predicativism, etc.)

Although it is true that precise formal versions of intuitionism, 
predicativism, etc., have been proposed and gained widespread acceptance, 
the point I am making is that typically these philosophies are *motivated* 
by some sort of argument like this: The law of the excluded middle, or 
impredicativity, or what have you, is dubiously unclear, and therefore 
should not be admitted into the realm of mathematics.  They fall afoul of 
some threshold of precision.

As I constructed that last paragraph, I found myself wondering whether 
*clarity* is a better word than precision.  It might be.

Tim