TITLE: Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?
QUESTION [20 upvotes]: This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that:

$L_4(2)$ and $L_3(4)$ both have order $20160$
$O_{2n+1}(q)$ and $S_{2n}(q)$ have the same order for $q$ odd, $n > 2$

I think this means that for each integer $g$, there are $0$, $1$ or $2$ simple groups of order $g$.
Do we need the full strength of the Classification of Finite Simple Groups to prove this, or is there a simpler way of proving it?

REPLY [5 votes]: There are many mathematicians outside finite group theory who asked whether important infinite fragments of the classification were possible without the entire classification.  I believe the favorite questions has always been : Can you prove the finiteness of the sporadics without the full classification? 
There is a good chance that third generation proof technology will reduce the entanglement between different portions of the classification because one knows the unipotent primes during earlier arguments where current methods only reveal semi-simple structure.  There has been one remarkable success in this direction :
Theorem (Altinel, Borovik, Cherlin).  A simple group of finite Morley rank containing an infinite elementary abelian 2-group is a Chevalley group over an algebraically closed field of characteristic 2.  
There is no known proof that simply groups of finite Morley rank even have an involution, much less that groups with odd characteristic looking Sylow 2-subgroup are also algebraic.  In consequence, there is a conjecture by Borovik that basically proposes one might classify finite simple groups who's 2-rank vastly exceeds the p-rank for any other p prime.
The final proof of [ABC] weighs in around 500 pages, but any finite analog would require many thousands of pages to deal with issues like twisted groups of Lie type and alternating groups, even assuming you find some trick for avoiding all the sporadics.  
In short, there are an awful lot of interesting results that depend upon the full CFSG for the foreseeable future because only funky asymptotic fragments look even vaguely realistic as stand alone results and even those look extremely difficult.