[FOM] Finitism and the Gold Standard

[Andrew Boucher]

Harvey Friedman wrote:

<quote>
In fact, the same thing has been said by finitists:

2'. It doesn't make any difference anyways, since mathematicians do  
not use
infinitary arguments, or they do not use infinitary arguments for
"important" results by "important" people, or the infinite methods  
for the
"important" results are easily removed.

There are important counterexamples to ... 2'.
<end of quote>

I'm afraid I can't come to terms with this remark.  In which sense of  
"important" do finitists say this?  I am certainly surprised by the  
reference to "important" people.

Not that I am a finitist, but it would seem to me that finitists  
would be on strongest ground if "important" were interpreted along  
the lines of "important to science" or "scientifically  
applicable".    Are there any counterexamples if 2' is rewritten as:

2".  Mathematicians do not use infinitary arguments, or they do not  
use infinitary arguments for scientifically applicable results, or  
the infinite methods for the scientifically applicable results can be  
removed, or a result which needs infinite methods and which is  
scientifically applicable can be reformulated to another result which  
does not use infinite methods and which maintains the same level of  
scientific applicability.

If there are any, what are these counterexamples?