[lit-ideas] Can You Imagine 2 + 2 = 5?
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Mon, 19 Nov 2007 21:52:04 EST
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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