Science Forums 2

[Don Blazys]

To: Ughaibu,

"Hyper-dimentional geometry, sets of infinite cardinality, algorithmic randomness, incompleteness theorems, etc." are all different, unique and extraordinarily interesting fields of study and I would be loath to even consider any of them "inferior". It would be highly illogical for anyone to categorize an entire branch of mathematics as being either "superior" or "inferior", and if our mothers were prone to such generalizations, then none of us would be here because all of us would have been thrown out with the bath water a long time ago!

The question of what constitutes superior or inferior mathematics is rather complicated, and therefore requires a great deal of specificity.

For example, the results of Lindemann, (the gentleman who proved that (Pi) is both irrational and transcendental) are clearly superior to the "results" of the "circle squarers" that came both before and after him.

Then, there is the question of "completeness". It is, after all, possible to make a pretty good argument that non-Euclidian geometry is, in a sense, "superior" to Euclidian geometry in that the latter is but a special case of the former. However, to imply that Euclidean geometry is therefore "inherently inferior" would be somewhat unfair to Euclid, because in actuality, both geometries are perfectly self consistent and will therefore stand the test of time.

Then again, if we consider "usefullness" as a criteria, we can also take the position that Riemann's elliptic geometry is "superior" to Bolyai's and Lobachevski's hyperbolic geometry on the grounds that the former actually describes the physical universe as a whole, (Einstein used it to develop his theory of relativity) while the latter is confined to more incidental cases resembling "pseudospheres". Again, this wouldn't be entirely fair to Bolyai and Lobachevski because their invention also has it's uses.

Sometimes, a well known mathematical construct is simply inadequate in representing a particular idea such as the concept of a "common factor". For instance, if we are dealing with non-negative integers, then the term on the right in the equation:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

is not only a true "mathematical miracle" in that it prevents us from prematurely "crossing out" the cancelled T's, but it is also clearly superior to the term on the left in that its variables are much better defined.

Perhaps most importantly, at T=1, it clearly shows that the very concept of a "unit common factor" is exactly as ridiculous as a division by zero!

I didn't "create" the incredible term on the right, but simply "observed it on my mental blackboard" while in a very, very deep state of what I call "creative meditation".

The previous post by Thunderbird is very eloquent and echoes perfectly my philosophy on this matter.

Don.