WITTGENSTEIN'S WRITING
ON THE INFINITE
Contents
Remarks on the Foundations of Mathematics
Related pages in the Logic Museum
Cantor's Diagonal Argument (original paper, parallel
German-English)
V. 7. Imagine
set theory's having been invented by a satirist as a kind of parody on
mathematics. – Later a reasonable meaning was seen in it and it was
incorporated into mathematics. (For if one person can see it as a
paradise of mathematicians, why should not another see it as a joke?) p. 264
(cf Hilbert, D. Uber
das Unendliche. Mathematische Annalen 95 (1926 In Putnam /
Benacerraf 183-201, p.191) "No one shall drive us out of the
paradise which Cantror has created for us".
II.15 A clever
man got caught in this net of language! So it must be an interesting net.
II.16 The
mistake beings when one says that the cardinal numbers can be ordered in a
series. For what concept has one of this ordering? One has of
course a concept of an infinite series, but here that gives us at most a vague
idea, a guiding light for the formation of a concept. For the concept
itself is abstracted from this and from other series; or: the expression
stands for a certain analogy between cases, and it can e.g. be used to define
provisionally a domain that one wants to talk about.
That, however,
is not to say that the question: "Can the set R be ordered in a
series?" has a clear sense. For this question means e.g.: Can one do
something with these formations, corresponding to the ordering of the cardinal
numbers in a series? Asked: "Can the real numbers be ordered in a
series?" the conscientious answer might be "For the time being I can't
form any precise idea of that". – "But you can order the roots and
the algebraic numbers for example in a series; so you surely understand the
expression!" – To put it better, I have got certain analogous
formations, which I call by the common name 'series'. But so far I
haven't any certain bridge from these cases to that of 'all real
numbers'. Nor have I any general method of of trying whether
such-and-such a set 'can be ordered in a series'.
Now I am shewn
the diagonal procedure and told: "Now here you have the proof that this
ordering can't be done here". But I can reply "I don't know –
to repeat – what it is that can't be done here". Though I can
see that you want to show a difference between the use of "root",
"algebraic number", &c. on the one hand, and "real
number" on the other. Such a difference as, e.g. this: roots are
called "real numbers", and so too is the diagonal number
formed from the roots. And similarly for all series of real
numbers. For this reason it makes no sense to talk about a "series
of all real numbers", just because the diagonal number for each series is
also called a "real number". – Would this not be as if any row
of books were itself ordinarily called a book, and now we said: "It makes
no sense to speak of 'the row of all books', since this row would itself be a
book."
II.17.
Here it is very useful to imagine the diagonal procedure for the production of
a real number as having been well known before the invention of set theory, and
familiar even to school-children, as indeed might very well have been the
case. For this changes the aspect of Cantor's discovery. The
discovery might very well have consisted merely in the interpretation of
this long familiar elementary calculation.
II.18. For
this kind of calculation is itself useful. The question set would be
perhaps to write down a decimal number which is different from the numbers:
0.1246798 …
0.3469876 …
0.0127649 …
0.3426794 …
…………… (Imagine a long series)
The child thinks
to itself: how am I to do this, when I should have to look at all the numbers
at once, to prevent what I write down from being one of them? Now the
method says: Not at all: change the first place of the first number, the second
of the second one &c. &c., and you are sure of having written down a
number that does not coincide with any of the given ones. The number got
in this way might always be called the diagonal number.
II.21 Our
suspicion ought always to be aroused when a proof proves more than its means
allow it. Something of this sort might be called 'a puffed-up proof'.
II.22 … If
it were said: "Consideration of the diagonal procedure shews you that the
concept "real number" has much less analogy with the concept
"cardinal number" than we, being misled by certain analogies,
inclined to believe", that would have a good and honest sense. But
just the opposite happens: one pretends to compare the "set"
of real numbers in magnitude with that of cardinal numbers. The
difference in kind between the two conceptions is represented, by a skew form
of expression, as difference of extension. I believe, and I hope, that a
future generation will laugh at this hocus pocus.
II.23 The
sickness of a time is cured by an alteration in the mode of life of human
beings, and it was possible for the sickness of philosophical problems to get
cured only through a changed mode of thought and of life, not through a
medicine invented by an individual.
Think of the use
of the motor-car producing or encouraging certain sicknesses, and mankind being
plagued by such sickness until, from some cause or other, as the result of some
development or other, it abandons the habit of driving.
V.46 Hence
the issue whether an existence proof which is not a construction is a real
proof of existence. That is, the question arises: Do I understand
the proposition "There is …" when I have no possibility of finding
where it exists? And here there are two points of view: as an English
sentence for example I understand it, so far, that is, as I can explain it (and
note how far my explanation goes). But what can I do with it? Well,
not what I can do with a constructive proof. And in so far as what I can
do with the proposition is the criterion of understanding it, thus far it is
not clear in advance whether and to what extent I understand it.
The curse of the
invasion of mathematics by mathematical logic is that now any proposition can
be represented in a mathematical symbolism, and this makes us feel obliged to
understand it. Although of course this method of writing is nothing but
the translation of vague ordinary prose.
48.
"Mathematical logic" has completely deformed the thinking of
mathematicians and of philosophers, by setting up a superficial interpretation
of the forms of our everyday language as an analysis of the structures of
facts. Of course in this it has only continued to build on the
Aristotelian logic.
p. 282
"The expression "and so on" is nothing but the expression
"and so on" (nothing, that is, but a sign in an calculus which
can’t do more than have meaning via the rules that hold of it; which can't say
more than it shows)".
"For the
sign "and so on", or some sign corresponding to it, is essential if
we are to indicate endlessness - through the rules, of course, that govern such
a sign. That is to say, we can distinguish the limited series "1,
1+1, 1+1+1" from the series "1, 1+1, 1+1+1 and so on". and
this last sign and its use is no less essential for the calculus than any
other." (p. 283)
"... the
sign "1, 1+1, 1+1+1 ..." is to be taken as perfectly exact; governed
by definite rules which are different from those for "1, 1+1, 1+1+1",
and not a substitute for a series "which cannot be written
down"." (p. 284)
The expression
"the cardinal numbers", "the real numbers", are
extraordinarily misleading except where they are used to help specify
particular numbers, as in the "the cardinal numbers from 1 to 100",
etc. There is no such thing as "the cardinal numbers", but only
"cardinal numbers" and the concept, the form "cardinal
number". Now we say "the number of the cardinal numbers is
smaller than the number of the real numbers" and we imagine that we could
perhaps write the two series side by side (if only we weren't weak humans) and
then the one series would end in endlessness, whereas the other would go on
beyond it into the actual infinite. But this is all nonsense. If we
can talk of a relationship which can be called by analogy "greater"
or "smaller", it can only be a relationship between the forms
"cardinal number" and "real number". I learn what a
series is by having it explained to me and only to the extent that it is
explained to me. A finite series is explained to me by examples of the
type 1, 2, 3, 4, and [an?] infinite one by signs of the type "1, 2, 3, 4,
and so on" or "1, 2, 3, 4 …" (p. 287)
§ 19 p. 332. A cardinal number is an
internal property of a list.
The sign for the extension of a concept is a
list. We might say, as an approximation, that a number is an external
property of a concept, and an internal property of its extension (the list of
objects that fall under it). A number is a schema for the extension of a
concept.
I use such a
list when I say "a, b, c, d fall under the concept F(x)": "a, b,
c, d," is the list. Of course this proposition says the same as
FA.Fb.Fc.Fd; but the use of the list in writing the proposition shows it
relationship to "Exist,y,z,u).Fx.Fy.Fz.Fu" which we can abbreviate to
"(E||||x.F(x)."
What arithmetic
is concerned with is the schema ||||. – But does arithmetic talk about
the lines that I draw with pencil on paper? -Arithmatic doesn't talk
about the lines, it operates with them.
p.381 ‘Philosophical
clarity will have the same effect on the growth of mathematics as sunlight has
on the growth of potato shoots. (In a dark cellar they grow yards long).
p. 406 "We
are not saying that f(1) holds and when f(c+1) follows from f(c), the
proposition f(x) is therefore true of all cardinal numbers, but:
"the proposition f(x) holds for all cardinal numbers" means
"it holds for x=1, and f(c+1) follows from f(c)".
p.406
"This proposition is proved for all numbers by the recursive
procedure". That is the expression that is so very misleading.
It sounds as if here a proposition saying that such and such holds for all
cardinal numbers is proved true by a particular route, or as if this route was
a route through a space of conceivable routes. But really the recursion
shows nothing but itself, just as periodicity too shows nothing but
itself".
p. 402
…what is the correct way to use the expression "the proposition (n)
f(n)"? What is its grammar?
§ 39 p.
457. In mathematics description and object are equivalent. "The
fifth number of the number series has these properties" says the same
as "5 has these properties". The properties of a house do not follow
from its position in a row of houses; but the properties of a number are the
properties of a position.
§ 40
p. 461.
After all I have already said, it may sound trivial if I now say that the
mistake in the set-theoretical approach consists time and again in treating
laws and enumerations (lists) as essentially the same kind of thing and
arranging them in parallel series so that one fills in gaps left by the other.
The symbol for a
class is a list.
p. 462.
Human beings are entangled all unknowing in the net of language.
p. 465.
[The attempt to correlate a class with its proper subclass]
So, Dedekind
tried to describe an infinite class by saying that it is a class which
is similar to a proper subclass of itself. … I am to investigate in a
particular case whetehr a class is finite or not, whether a certain row of
trees, say, is finite or infinite. So, in accordance with the definition,
I take a subclass of the row of trees and investigate whether it is similar
(i.e. can be co-ordinated one-to-one) to the whole class! (Here already
the whole thing has become laughable.) It hasn’t any meaning; for, if I
take a "finite class" as a subclass, the attempt to co-ordinate it
with the whole class must eo ipso fail: and if I make the attempt with
an infinite class – but already that is a piece of nonsense, for if it is
infinite, I cannot make an attempt to co-ordinate it. – What we call the
"correlation of all the members of a class with others" in the case
of a finite class is something quite different from what we, e.g., call a
correlation of all cardinal numbers with all rational numbers. The two
correlations, or what one means by these words in the two cases, belong to
different logical types. An infinite class is not a class which contains
more members than a finite one, in the ordinary sense of the word
"more". If we say that an infinite number is greater than a
finite one, that doesn't make the two comparable, because in that statement the
word "greater" hasn’t the same meaning as it has say in the
proposition 5 > 4!
p. 465. The form of expression "m=2n
correlates a class with one of its proper subclasses" uses a misleading
analogy to clothe a trivial sense in a paradoxical form. (And instead of
being ashamed of this paradoxical form as something ridiculous, people plume
themselves on a victory over all prejudices of the understanding). It is
exactly as if one changed the rules of chess and said it had been shown that
chess could also be played quite differently.
p. 468. When "all apples" are spoken
of, it isn’t, so to speak, any concern of logic how many apples there
are. With numbers it is different; logic is reponsible for each and every
one of them. (cf Remarks § 126)
Mathematics consists entirely of calculations.
In mathematics everything is algorithm and nothing
is meaning; even when it doesn't look like that because we seem to be
using words to talk about mathematicl things. Even these
words are used to contruct an algorithm.
p. 469. When set theory appeals to the human
impossibility of a direct symbolisation of the infinite it brings in the
crudest imaginable misinterpretation of its own calculus. It is of course
this very misinterpretation that is responsible for the invention of the
calculus. But of course that doesn’t show the calculus in itself to br
something incorrect (it would be at worst uninteresting) and it is odd to
believe that this part of mathematics [set theory] is imperilled by any kind of
philosophical (or mathematical) investigations. (As well say that chess
might be imperilled by the discovery that wars between two armies do not follow
the same course as battles on the chessboard.) What set theory has to
lose is rather the atmosphere of clouds of thought surrounding the bare calculus,
the suggestion of an underlying imaginary symbolism, a symbolism which
isn’t employed in its calculus, the apparent description of which is really
nonsense. (In mathematics anything can be imagined, except for a part of
our calculus.)
§ 119
The symbol for a
class is a list
A cardinal
number is an internal property of a list.
It is nonsense
to say of an extension that is has such and such a number, since the number is
an internal property of the extension. But you can ascribe a
number to the concept that collects the extension (just as you can say this
extension satisfies the concept).
§ 123. …
there is no path to infinity, not even an endless one.
The situation
would be something like this: We have an infinitely long row of trees, and so
as to inspect them, I make a path beside them. All right, the path must
be endless. But if it is endless, then that means precisely that you
can’t walk to the end of it. That is, it does not put me in a
position to survey the row. (Ex hypothesi not.) That is to
say, the endless path doesn’t have an end "infinitely far away", it
has no end. (cf Grammar § 39 p. 455)
§ 124. It
isn't just impossible "for us men" to run through the natural numbers
one by one; it's impossible, it means nothing.
Nor can you say,
"A proposition cannot deal with all the numbers one by one, so it has to
deal with them by means of the concept of number", as if this were a pis
aller: "Because we can’t do it like this, we have to do it
another way." But it's not like that: of course it's possible to
deal with the numbers one by one, but that doesn’t lead to the
totality. For the totality is only given as a concept.
… you can’t talk
about all numbers, because there's nop such thing as all numbers.
§ 125. An
"infinitely complicated law" means no law at all. How are you
to know it's infinitely complicated? Only by there being as it were
infinitely many approximations to the law. But doesn't that imply that
they information act approach a limit? Or could the infinitely
many descriptions of intervals of the prime number series be called such
approximations to a law? No, since no description f a finite interval
takes us any nearer to the goal of a complete description.
§ 126.
There's no such thing as "all numbers" simply because there are
infinitely many.
… It's, so to
speak, no business of logic how many apples there are when we talk of the
apples. Wheras it's different in the case of the numbers: there it
[logic] has an individual responsibility for each one of them.
§ 129. I
have always said you can't speak of all numbers, because there's no such
thing as "all numbers". But's that's only the expression of a
feeling. Strictly, one should say, … "In arithmetic we never are
talking about all numbers, and if someone nevertheless does speak in
that way, then he so to speak invents something – nonsensical – to supplement
the arithmetical facts." (Anything invented as a supplement to logic must
of course be nonsense).
§ 133. In
philosophy it's always a matter of the application of a series of utterly
simple basic principles that any child knows, and the – enormous – difficulty
is only one of applying these in the confusion our language generates.
It's never a question of the latest results of experiments with exotic fish or
the most recent developments in mathematics. But the difficulty in
applying the simple basic principles shakes our confidence in the principles
themselves.
§ 135. Has
an odd & difficult passage on infinite disjunctions.
"We only
know the infinite by description". Well then, there's just the
description and nothing else.
§ 138
Experience as
experience of the facts gives me the finite; the objects contain the
infinite. Of course not as something rivalling finite experience, but in
intension. Not as though I could see space as practically empty, with
just a very small finite experience in it. But, I can see in space the
possibility of any finite experience. That is, no experience could be too
large for it or exhaust it: not of course because we are acquainted with the
dimensions of every experience and know space to be larger, but because we
understand this as belonging to the essence of space. – We recognise this
essential infinity of space in its smallest part.
Where the
nonsense starts is with our habit of thinking of a large number as closer to
infinity than a small one.
As I've said,
the infinite doesn’t rival the finite. The infinite is that whose essence
is to exclude nothing finite. The word "nothing" occurs in this
proposition and, once more, this should not be interpreted as the expression
for an infinite disjunction, on the contrary, "essentially" and
"nothing" belong together. It's no wonder that time and time
again I can only explain infinity in terms of itself, i.e. cannot explain it.
§ 139 How
about infinite divisibility? Let's remember that there's a point to
saying we can conceive of any finite number of parts but not of an
infinite number; but that this is precisely what constituties infinite divisibility.
Now,
"any" doesn’t mean here that we can conceive of the sum total
of all divisions (which we can't, for there's no such thing). But
that there is the variable "divisibility" (i.e. the concept of
divisibility) which sets no limit to actual divisibility; and that
constitutes its infinity.
– the infinite
film, strip
§ 140 Is
primary time infinite? That is, is it an infinite possibility? Even
if it is only filled out as far as memory extends, that in no way implies that
it is finite. It is infinite in the same sense as the three-dimensional
space of sight and movement is infinite, even if in fact I can only see as far
as the walls of my room.
§ 141 Does
the relation m = 2n correlate the class of all numbers with one of its subclasses?
No. It correlates any arbitrary number with another, and in that way we
arrive at infinitely many pairs of classes, of which one is correlated with the
other, but which are never related as class and subclass. Neither
is this infinite process itself in some sense or other such a pair of classes.
In the
superstition that m = 2n correlates a class with its subclass, we merely
have yet another case of ambiguous grammar. (cf Philosophical Grammar p.
465)
§ 142 … A
searchlight sends out light into infinite space and so illuminates everything
in its direction, but you can't say it illuminates infinity.
Generality in
mathematics is a direction, an arrow pointing along the series generated by an
operation. And you can even say that the arrow points to infinity; but
does that mean that there is something – infinity – at which it points, as at a
thing? Construed in that way, it must of course lead to endless nonsense.
§ 143 … That we
don’t think of time as an infinite reality, but as infinite in intension, is
shown in the fact that on the one hand we can't imagine an infinite time
interval, and yet see that no day can be the last, and so that time cannot have
an end.
We are of course
only familiar with time – as it were – from the bit of time before our
eyes. It would be extraordinary if we could grasp its infinite extent in
this way (in the sense, that is to say, in which we could grasp it if we
ourselves were its contemporaries for an infinite time.)
§ 144 The
infinite number series is only the infinite possibility of finite series of
numbers. It is senseless to speak of the whole infinite number
series, as if it, too, were an extension.
Infinite
possibility is represented by infinite possibility. The signs themselves
only contain the possibility and not the reality of their repetition.
Doesn’t it come
to this: the facts are finite, the infinite possibility of facts lies in the
objects. That is why it is shown, not described.
If I were to say
"If we were acquainted with an infinite extension, then it would be all
right to talk of an actual infinite", that would really be like saying,
"If there were a sense of abracadabra then it would be all right to talk
about abracadabraic sense perception".
§ 145 … what is
infinite about endlessness is only the endlessness itself (p. 167).
§ 151 (p. 176)
has remark about Weyl and Brouwer
§ 173 … the
expressions "divisible into two parts" and "divisible without
limit" have completely different forms. This is, of course, the same
case as the one in which someone operates with the word "infinite" as
if it were a number word; because, in everyday speech, both are given as
answers to the question 'How many?'
§ 174 Set theory
is wrong becauses it apparently presupposes a symbolism which doesn't exist
instead of one that does exist (is alone possible). It builds on a
fictitious symbolism, therefore on nonsense.
§ 181 … The
usual conception is something like this: it is true that the real numbers have
a different multiplicity from the rationals, but you can still write the two
series down alongside one another to begin with, and sooner or later the series
of real numbers leaves the others behind and goes infinitely further on.
But my
conception is: you can only put finite series alongside one another and in that
way compare them; there's no point in putting dots after these finite stretches
(as signs that the series goes on to infinity). Furthermore, you can
compare a law with a law, but not a law with no law. (p. 224)
I'm temped to
say, the individual digits are always only the results, the bark of the fully
grown tree. What counts, or what something new can still grow from, is
the inside of the trunk, where the tree's vital energy is. Altering the
surface doesn’t change the tree at all. To change it, you have to
penetrate the trunk which is still living.
Thus it's as
though the digits were dead excretions of the living essence of the root.
Just as when in the course of its vital processes a snail discharges chalk, so
building onto its shell.
If we ask
whether the musical scale carries with it an infinite possibilty of being
continued, then it's no answer to say that we can no longer perceive vibrations
of the air that exceed a certain rate of vibration as notes, since it might be
possible to bring about sensations of higher notes in another way.
Rather, the finitude of the musical scale can only derive from its internal
properties. For instance, from our being able to tell from a note itself
that it is the final one, and so that this last note, or the last notes,
exhibit inner properties which the notes in between don't have.
(Philosophical Remarks § 223 p. 280).
§ 188 There is no number outside a
system. The expansion of pi is simultaneously an expression of the nature
of pi and of the decimal system. (p. 231).
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