The analogy we might be mislead by is this: you ask the question “ You ask, what does it mean that “a rod is 6 inches long”. Someone answers one finds [it|ou]t whether it is so by using a measuring rod divided into equal parts. “What does this mean?” – One divides it into equal parts in such & such a way. I have given a definition in terms of the way of verification. “But isn't this an indirect definition?” Sometimes it is, sometimes it isn't. Sometimes how we look for something may determine what we are looking for, sometimes it doesn't. – We are always liable to think of the fact that we can find out that there is a chair here in many different ways & we forget that this is just conditioned by the particular use we make of the word “chair”, “table” & in general the generic names of physical
17. objects. ⌊⌊
[s|S]upposing we said “what does it mean
to have an enla⌊r⌋ged liver & someone answered
“we ver[y|i]fy it by looking at the
persons eye & seeing
…” ⌋⌋ Describe an object by describing its use. Describe an object by saying what hollow it fits into. More ore less of it's use is expressed by different forms of a prop. Concider this! The sense of a prop is what you must know to understand it. What does understanding a question consist in? What does it mean to understand a mathematical question? Would you understand ‘25 × 25 = ?’ if you didn't know how to calculate it? Would you say you understood the a position of a board game if you had no idea of the way the game was played? ˇAsk How much must you know to understand it? Would you not be inclined to say that you understand it more & more the more you knew about the game? But we could also imagine a case whe in which you would say that you understood the position even though you knew nothing didn't know the exact rules of the game, even if these rules had not yet been given! Now concider such a question as “are there an inf. nr. of primes or not,
18 & if not, how
many?” I show you what we call a prime
nr. & ask you if you
understand this question. Prima
facie you all say yes. Now I want to show you that you
could also look at it from a different & perhaps more
‘exact’ point of view & say
‘no’. We have here a question but we have
not yet got a method of it's solution & I want
you to think of it in terms of 25 × 25 = ? when we
don't know what multiplication is. Now
I'll say (what I've said in a similar case
before): The question for you gets it's
sense by the idea picture of a small finite
nr. of cardinals
etc.. But what does that mean?
In the case of the small nr of
cardinals you have a method, you know what to do, here you
don't know it, so how can what
does it mean, that “the question gets it's
sense from the case in which you do know
it”?
Our mind works on the line that there is
something which hasn't yet been calculated which
could but for our weakness be calculated it takes ˇno
notice of the fact that we have no method & although this
can't alter the facts it determines the way of
expressing: We say we don't yet know so
& so; & if someone shows
19 us
Euler's proof we say
that now we know the answer to the question we have asked.
But are we bound to express ourselves in this way?
Can't I persuade you to adopt a different way of
expressing ourselves of which too we can't say that it is
inconsistent with the usual use of
words? What if I say: This question
i.e. the form of words was indeed suggested by
the finite case, but the analogy just brakes
down because there is no method of solution.
Eulers
“proof”
needn't be conceived as answering the question
“how many primes are
there”. This question might perfectly well be said to be nonsensical. (As e.g. “what colour has visual space.”). “But is it in our power to regard as question as nonsensical or as making sense? And what about Eulers proof?” Need we ask such a question as “how many cardinal numbers are there”? This question which seems in some way to get hold of the infinite i.e. the enormous for this reason might appeal to some of us & might, on the other hand, repell some of us, – e.g. me. Funeral Concider such a prop as: “There are as many squares as there are cardinals”. If we look at it like this: 1, 2, 3, 4, 5, 6, 7, 8, 9 … we should be
20. inclined to say that there
were less squares than cardinals; if we think
of them as 1²,
2², 3², 4², …, we
shou say, there are as many. ⌊⌊
Thinking in terms of an analogy doesnt mean
that this analogy is constantly before our mind. The
Idea of finding out something about the series of
cardinals. ⌋⌋ Why shouldn't we draw the
conclusion that this whole question doesn't make
sense? But we actually say that it does, &
part of it's charm that the answer is in a
way paradoxical adds ˇfor some to it's
charm. (Some + some = some). It
seems to us that we have discovered a new element with utterly
different properties. I should like to get rid of this analogy. – Can “the heptagon be
constructed?” – A
“Now surely this question makes
sense!” – B “Now
surely this question makes no sense!” –
What argument would A use: “you know what it
means ‘to construct the pentagon,
etc. hexagon, so … ,
ˇyou know what a heptagon is, what construction
means”. B: “It
makes no more sense to say ‘can the
7 gon be constructed’,
than to say ‘can the 5-gon be
constructed’. I know what we call ‘a
constructed (as opposed to measured) pentagon’, I know
what we call ‘the
pentagon-construction’. We use this
expression because of certain obvious analogies. But
I don't [k|c]all anything heptagon
construction. So don't ask such a question as can
it be …. Compare this with the statement
“on this animal's forhead there
21. was a heptagon
constr.’ –
answer: I don't know what you're talking
about. “But what about the proof that the heptagon
can't be constructed? As there is such a proof,
it is the answer to that question of which you say it makes no
sense. “Must I conceive of this proof as the
answer to that question?” When the
proof is given I can say: you have now given the
expression 13 gon sense
& you have decided that the expression construction of the
7 gon is
not never to be used at
all’. Further you have now given the
question: “can the
7 gon be constructed”
sense, analogous to that can 735912 be
devided by 19. |

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