WITTGENSTEIN'S WRITING ON THE
INFINITE

Remarks on the Foundations of Mathematics

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. 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).

THE
LOGIC MUSEUM Copyright © E.D.Buckner
2005.