[FOM] Brouwer on mathematical operations: operationalism in mathematics

[T.Forster@dpmms.cam.ac.uk]

The question of operationalism in mathematics is a very interesting one. It 
seems to me clear that there is a very operationalist streak in a lot of 
modern mathematical practice. In fact i am even trying to write a book 
about it. Category theory is thoroughly operationalist in conception, and 
much of the thinking in theoretical computer science is operationalist. 
They don't use the word, but that's what it is. All the Abstract-Data-Type 
stuff is operationalist. One of my current projects is to buttonhole the 
surviving TCS people of that generation and find out what explicit 
philosophical take they had on their subject matter. In Cambridge we have 
Robin Milner and Tony Hoare. I think we have John McCarthy on this list. I 
would be very interested in what he has to say.

Operationalism in physics has a bad reputation. But perhaps operationalism 
about mathematics might be more sensible. It might be the correct 
development of the views of Hume and the Wiener Kreis. It's very striking 
that not of the philosophers who take an interest in philosophy of 
mathematics ever consider operationalist views of mathematics. I suspect if 
they looked more closely at what they call *structuralism* they would 
discover that quite a lot of what they think is structuralism is actually 
operationalism.

On Dec 27 2008, Lucius Schoenbaum wrote:

>Dear FOMers,
>
>"One cannot inquire into the foundations and nature of mathematics  
>without delving into the question of the operations by which the  
>mathematical activity of the mind is conducted. If one failed to take  
>that into account, then one would be left studying only the language  
>in which mathematics is represented rather than the essence of  
>mathematics."  L.E.J. Brouwer
>
>I am only a student, but I am inclined to agree with Brouwer.  Does  
>anyone else agree also?  If I venture to call intuitionism an  
>"operational" approach in foundations, can anyone think of anything  
>similar existing today in foundations?  What operations do you think  
>Brouwer had in mind?
>
>By the way, I would be much obliged to anyone who could tell me the  
>source of this quotation.
>
>Sincerely,
>
>Lucius Schoenbaum
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