Remarks on the Foundations of Mathematics

Philosophical Grammar

Philosophical Remarks



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Cantor on Set Theory

Cantor's Diagonal Argument (original paper, parallel German-English)

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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.




Philosophical Grammar


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.)

Philosophical Remarks


§ 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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