FOM: constructive platitudes

Peter Schuster pschust at
Mon May 8 13:00:50 EDT 2000

Let me try to add some constructivist's platitudes to some of the questions 
put forward to FOM by Mohammad Sal Moslehian <MSALM at> 
on Mon, 8 May 2000 10:41:24 GMT+330. 

>(A1). Is there anything (so called "reality" as an unseen world) beyond 
>human experiences (so called "appearances" or "phenomenon" as a sensible 

Wittgenstein says "The borderline of my thinking is the borderline of 
my world. ... What we cannot speak about thereof we must be silent". 
When reality is said to lie beyond human thinking, how can we have 
any knowledge of it? 
>(A2). Is there any distinction or separation between "phenomenon" and 
>"reality"? why?

That's very similar to (A1), isn't it? Couldn't things said to be two only if 
they can be distinguished, which in turn would require to know each of them?  

>(A3). Why should we assume that there is "reality"? Isn't it superfluous?

Laplace is said to tell Napoleon about God that he (Laplace) doesn't need this 
hypothesis (God) any longer: "Je n'ai besoin de cette hypothe`se."  
Do we need reality more than just as a comfortable way of speaking, or as a 
sometimes consoling belief? 

>What is "reality"? What is a "phenomenon" (or an "appearance")?

Confer what was added to (A2). 

>Suppose that there is "reality".
>Is "reality" essentially of mathematical form, in other words, is it 
>necessary to use the mathematics for perceiving, explaining, justifying or 
>describing "reality"?

Presumably not, but do we have anything better? At least it works in most cases, 
whoever might know why. 

>(i) If the answer is No, why do we study the mathematics?

Because it brings us to the borderline of human thinking and thus improves our 
knowledge of this borderline from inside, yet of another piece of "reality". 
A good example are such extremely subtle areas like inaccessible cardinals. 

>(ii) If the answer is Yes, how does the mathematics help us to improve our 
>understanding of "reality"?

This seems to be just the dilemma of any kind of Platonism, which says that 
there is mathematical reality out there but does not tell us how mathematicians 
can get a hand on it. 
Formalism/logicism avoids this problem by separating truth from intuition, 
by not caring at all about any external reality other than that of signs 
on the paper, or of patterns of language. 
Constructivism/intuitionism avoids this applicability problem by concentrating 
on the internal reality, whatever this means, but immediately earns new problems 
such as the objectivity problem and the communicability problem. 

So all these "old philosophical questions" seem to live forever. 

Name: Peter M. Schuster
Position: Wissenschaftlicher Assistent
Instituition: University of Munich, Mathematics Department
Research interest: constructive mathematics

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