[lit-ideas] Can You Imagine 2 + 2 = 5?

Thanks to Ramos for comments and confession:

>Actually, I've always thought that it was impossible  to imagine a logical 
impossibility (e.g., >2+2=5). The statement is  one of those facts in 
philosophy that has gone unexamined. Everyone  >thought it was true. 
 
Let's revise the source of the mistake: McEvoy. He  writes:
 
>I did two philosophy options at Oxford - 
>philosophy of mind being one. ... I had 
>not known what I had let myself in for.
>The questions in finals things  like:-

Can I imagine  '2+2=5'? 
         Can I  imagine '2+2=4'?

>This is about a number of things - but the 
>questions are so framed that it is  nigh-on
>impossible to say anything about them from 
>non-linguistic turn point of view
>They are 'clever-clever' or smart-arse ways  -
>but an answer to this effect would be punished 
>_as not answering the
>question_, if not seen as  just insulting polemic.

O, I pity you. And how did _you_ answer, if you  recall. Because apparently 
you passed 'with flying colours', as they say in  UC/Berkeley.
 
I suppose by "I" the questions means _you_  McEvoy. Or does it mean, she, the 
professor?
 
Or is it the neutral "Man" of German, 
 
                   Can _one_ imagine that 2 + 2 = 5
 
I would pose the query more as a  statement.

Criticise the following statement:
 
                  2 + 2 = 5, I imagine?
 
                   2 + 2 = 4, I imagine.
 
Note that "?" indicates doubt on the part of the stater. On the other hand,  
the dot, '.', indicates a falling tone, and it's a statement-statement. But 
you  knew that. 
 
Andreas is right that people can imagine lots of things. The Greeks  imagined 
centaurs, tritons, nymphs, and flying horses. -- and 'irrational  numbers' 
(more to the point).

E. Dodds wrote this beautiful book, "The Greeks and the Irrational"  which is 
all about irrational numbers, if you've seen one. 
 
But this arithmetic summations are not _that_ animal. 
 
One problem with imagining 2 + 2 = 5, or 2 + 2 = 4 has to do with Hilbert's  
Heritage. As A. Palma writes:
 
           Hilbert was  adopting a specific view of the 
           notion of  containment (the basis of analyticity.) 
 
Indeed, for Kant, it's the subject-predicate notion. If it can be shown  that 
the concept of "4" is included in the concept of "2", the concept of "+"  
and, again, the concept of "2", _then_ the statement "2 + 2 = 4" would be  
analytic (and tautological). Contrariwise, no
 
Kant's example was, as we know:
 
           "5 + 7 = 12"
 
which was for him _synthetic_ (never mind a priori). So, against Hilbert,  he 
is saying that you do not _deduce_ the concept "12" (if it's deduced that you 
 do with a concept) out of the concepts 5, 7 and +. 
 
Philosophers alla Hume and Ayer, are usually not concerned with analytic  
mathematical statements, since all they want to know is more oriented towards  
what McEvoy calls the causal theory of perception, i.e. whether the pillar box  
_is_ red or just _seems_ so (synthetic a posteriori -- the dogma of 
Empiricism). 
 
Now the transcendental Ego, for Kant, was indeed _rational_ (pure reason),  
so for a purely reasonable (or rational) creature -- poorly reasonable 
creature,  as I'd spell it -- she cannot _imagine_ 2 + 2 = 5. As she cannot 
imagine 
the  quadratura circuli. 
 
The phrasing of the question is slightly irritating, "Can one ...". When I  
rephrased that into 'know', it sounded awkward and clumsy. 
 
"Can you know" -- Can you kennst du John Peele?" sort of thing. "Can" is an  
ability, but it doesn't _force_ anything. It's also disrespectful to ask 'can' 
 questions ("Can you write?"). If taken in theory, of course one can always  
_learn_, so even if I were not able to write shorthand, I would answer, "Yes, 
I  can", meaning, "Yes, I do have the ability -- want to teach me?" (This 
reminds  me of the job interview of Sweet Charity in the eponymous comedy -- 
brilliant). 
 
A further problem is with 'imagine' --With G. E. Moore I have many problems  
imagining apples in a basket. But still, I can imagine an apple in a basket, 
and  2 + 2 apples making 4 apples. But 2 + 2 = 4 _in the abstract_ it's 
*harder* to  imagine. 
 
This is the reason why when Wittgenstein taught mathematics in Switzerland  
-- "The Lost Years" -- he always used apples and pears.  
 
Goedel was a smart guy, and he would have answered your professor's silly  
question in ways which would have blown his socks off, if that's the  
expression.   
 
Of course if he was really into linguistic turn, the expected reply would  
be, "Depends on how you define "2", "+" and "4". It's all conventional and  
artificial -- and it's a formal language, and the assignment of value to  
artificial symbols is arbitrary. So I can imagine a parallel universe (or next  
door, 
actually) where 2 + 2 making 5 is _the_ perfectly sensible thing for 2 + 2  to 
make. 
 
Cheers,
 
J. L. 
    Author of "From Zero to the Infinite" --  and Back.
 
 
 

 



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