GÖDEL, CANTOR AND PLATO

My third Gödel paper was to have this title, starting with the Continuum Hypothesis and ending with remarks on what Gödel meant by mathematical Platonism. It crashed in my computer before I could complete it, and I have meanwhile received a copy of David Fowler’s The Mathematics of Plato’s Academy, which will enable me to wind my paper up by arguing that one could have been a mathematician working under Plato without being what anybody would call a Platonist nowadays. So my new paper will retain the old one’s title but will, I hope, cut the cackle somewhat, inspired by my rediscovery of Natanson’s Theory of Functions of a Real Variable, recommended to me long ago by Imre Lakatos, who must have studied under Natanson in Moscow. His chapter (XIV) on transfinite numbers is an exemplary lesson in conciseness --- achieved, admittedly, by some simplification, for which Natanson excuses himself on the opening page of his first chapter.

Cantor’s continuum hypothesis posits very simply that the set of real numbers, in effect the set of ordered sets of natural numbers, and intuitively the set of points on a straight line, has as its cardinal number Aleph one, the second transfinite cardinal, the first being Aleph zero, the cardinal number of the set of natural numbers. Any infinite subset of the set of real numbers has either the cardinality Aleph zero or Aleph one. There is no intermediate cardinality for it to have (and in general there are no cardinal numbers other than Alephs), nor has the full set of real numbers a higher cardinality than Aleph one. So far the hypothesis (with some non-hypothetical details associated with it).

Gödel’s 1940 (in the second volume of the Collected Works, edited at Princeton and published by Oxford) establishes that the continuum hypothesis, and likewise the axiom of choice, is consistent with the standard axioms of set theory. In other words, it cannot be disproved. Moreover Paul J Cohen, in his 1963 and 1964, proved it to be independent of the standard axioms. In other words it cannot be proved either, and putting the two together it is undecidable. It is remarkable that an hypothesis which mathematicians therefore have every right to take or leave alone should still seem to call out for an intuitive explanation of what exactly it is that one can take or leave alone.

A concept fundamental to all set theory is that of denumerability. A set is denumerable if it is capable of being put into one-one correspondence with the set of finite integers (beginning with zero) or natural numbers (beginning with one) --- it makes no serious difference, but there does turn out to be a superior elegance in starting with one. Choosing this beginning, then, the set of natural numbers becomes the paradigm example of a denumerable set.

Natanson, on his pages 119 and 120, proves with extreme simplicity that the set of all transfinite numbers which relate to denumerable sets is itself non-denumerable. This proof is a simplification of Cantor’s first proof of this proposition in III 4 Nr 5, dated 1883 (the complete III 4 is dated 1879-1884), and makes use of Cantor’s own simplifications in his second presentation of the principles of set theory, III 9, dated 1895-1897. (These codes refer to the Abhandlungen, as below.)

The proof of this set’s non-denumerability is liable to be skipped over and taken for granted by amateur readers, but it is no formality and one form or another of it must be faced. Its counter-intuitiveness can be seen from something that Natanson does after his proof, on page 121, but Cantor does before his first proof, on pages 195-197 of the Olms edition of his Abhandlungen. This is to set out schematically the transfinite numbers which follow the natural numbers, the first such number being called w.

w, w + 1, w + 2 . . . w + w, w + w + 1, w + w + 2 . . . are set out with startling ramifications which you must look up in a standard text book to save me from revealing the inadequacies of my computer. They form a sequence of infinite sequences, and in his very first contribution to set theory, III 1 in the Abhandlungen, dated 1874, Cantor had proved that the set of algebraic numbers formed such a sequence of sequences and that it was, as such, denumerable, while the set of real numbers was not. (It was in the second of these early papers, III 2, dated 1878, that Cantor first expressed his belief that the non-denumerable set of real numbers had the next higher cardinality beyond that of the paradigm denumerable set, these cardinalities not being called Aleph one and Aleph zero respectively until III 9.)

Seeing this sequence of sequences of transfinite numbers set out schematically, one is only too likely to conclude that, like the algebraic numbers, the cardinality answering to them will be denumerable. For on the face of it, this sequence of sequences is what Cantor called the second number class, that is to say the complete set of all transfinite numbers which relate to denumerable sets.

The clue comes from his using the letter a for members of this number class, and distinguishing between calling an arbitrary member of it a and calling the set of all such numbers {a}. If one were to mean by {a} the set of all such transfinite numbers up to an arbitrary a, it would indeed be denumerable. If one means by it the set of all such transfinite numbers at all, it will be more than denumerable.

To give a sketch, then, of Natanson’s proof without further ado. If a set S is a denumerable set of numbers of the second number class and g is the smallest number greater than all members of S, then g can be proved to be also a member of the second number class. Now take S to be the set of all members of the second number class which form a denumerable set. The smallest number greater than all members of S (S’s g, in other words) has therefore to be a member of the second number class, but it cannot be a member of the denumerable set S. Therefore the second number class always contains at least one number outside denumerability. It cannot therefore be arranged in one-one correlation with the paradigm denumerable set. The second number class is therefore non-denumerable.

The first thing that needs explaining is what kind of numbers w and its successors are. So far the only transfinite numbers that have been explained are the cardinals Aleph zero and Aleph one. As I have said, Cantor did not use these terms until his second exposition of set theory, III 9, of 1885-1887, though his concept of cardinality (called Mächtigkeit at first), based on one-one correspondence, had been expounded in III 4. In III 9 Cantor calls the numbers from w onwards ordinal numbers or ordinals. In III 4 he had used a German word in a way that seems to be at odds with its colloquial origin. To understand this point one must go back to primitive finite arithmetic and, forgetting what grammarians mean by ordinals, consider how children count. One, two, three, . . . six, seven --- I’ve got seven apples. The first “seven” was uttered as the last apple was observed or touched, while the second “seven” was intended to cover the total set of apples. The first “seven” can properly be called an ordinal number-word, even though it lacks the grammarians’ “th”, and the second “seven” a cardinal number-word, and the number denoted by it a cardinal number. In normal German there is a less academic-sounding distinction: the number denoted by “sieben” used in the first sense is a Zahl, while the number denoted by a “sieben” in the second sense is an Anzahl.

In his III 4 Nr 5 § 2 (on page 168) Cantor distinguishes “Zahl” and “Anzahl” in a way that appears to contradict my distinction from everyday German but in fact is subtly consistent with it. You must imagine that the child who says it has seven apples does so not only in reference to the total set of apples but also retaining the memory of the order in which it had counted them individually. That combination is what provides Cantor with his new Anzahl concept, and in respect of infinite sets it has the result that while a set keeps its cardinality however it is ordered (being able to be put in one-one correlation with a standard set is all that matters), reordering it can well result in a different Anzahl.

To qualify for a Cantorian Anzahl a set must not only have some order but be well-ordered. This means that the set as a whole and every subset of it must have a first element, which entails that every element must have a next-following element provided there is any element that follows it at all. It does not entail that every element has a next-preceding element. For an example of a reordering that gives a different Anzahl consider the set of natural numbers reordered so that all the odd numbers are followed by all the even numbers. The number 2 will now have no immediate predecessor. That does not prevent the rearranged set’s having an Anzahl, but there is clearly something different about the order, so it must have a different Anzahl. Now put that rearranged set into its reverse order and neither half of it will have a first member, and thus it will have no Anzahl. A simpler case is to put the original set of natural numbers into its reverse order. That gives us two sets which have no Anzahl yet have distinct order characteristics which one can easily specify.

In Cantor’s later terminology these Anzahl-less orderings are said to correspond to order-types, while the order-types which correspond to well-ordered sets are distinguished as ordinal numbers or ordinals. The four examples I have given have, respectively, the ordinals w and w + w and the order-types w* + w* and w*. In some books (and in Cantor’s III 9), order-types and ordinals are said to be abstractions from the sets that correspond to them (and cardinals are said to be further abstractions). It is better to bring these concepts down to earth by ordaining for any given order-type or ordinal number (or for that matter cardinal number) a specific paradigm set which has that type or number. This paradigm set can then be deemed to be the type or number. Nevertheless, all mathematicians agree that the notation which Cantor invented to go with his concept of abstraction is extremely convenient, and no-one proposes to get rid of it.

In III 4, Cantor was clearly oppressed by the prospect of a set of numbers which not only had no end but appeared to have no way of being wound up into a totality. One could, he said, have the impression that these numbers would lose themselves into boundlessness, as it were run away into the sand. He had already proposed two principles of generation, first to move from any given number to its successor, and second to allow an infinite (and thus endless) set to be followed by a new number beyond all its members. w is such a number, coming beyond all finite numbers.

One can perhaps give an intuitive image of this by supposing the finite numbers to be tapped out at an increasing rate which enables all tapping to be completed within a finite period even though there has been no last tap. Then there is a pause and a new series of accelerating taps begins, starting with w, w + 1 etc. This series will lead to a further pause, and then one will have to imagine that these pauses themselves accelerate, and there will clearly be no limit to the need for the series of such processes to accelerate. That is the nightmare of ins Grenzenlose hin sich verlieren, and to overcome it Cantor invents a third principle which he mysteriously calls an inhibiting or limiting principle (Hemmungs- oder Beschränkungsprinzip), to enable him to move beyond the second number class. In fact this principle is not required, since the nightmare only exists in the realm of intuitive explanations, and in III 9 he does not call on it, relying on the simple and clear distinction between the set of all numbers of the second number class up to any arbitrary a, and the set {a} of all such numbers at all. (See pages 331-333.)

Cantor does still use the first two generative principles, but Natanson, of course, uses none, set theory being properly formalised by his time (even though he explicitly exempts himself from attending to all its formal details).

Only in a letter to Dedekind of 1899 (the first printed in the Abhandlungen but by no means the first written) does Cantor introduce a concept which begins to throw light on the question of whether every cardinality is an Aleph, namely the concept of an ‘inconsistent’ plurality, the word “Vielheit” being chosen expressly to avoid the word “Menge” (set) and its implication that what is under consideration can be viewed as a unit. Cantor insists that he is only speaking of definite pluralities, but he gives as an example the totality of everything conceivable (“Inbegriff alles Denkbaren”), which one would not think definite in the least. However, he moves on to set-theoretical examples like the totality of all ordinals or the totality of all cardinals, which lead to contradictions if they are treated as sets. This distinction is what gave rise to the John von Neumann version of axiomatic set theory, revised by Bernays and by Gödel in his 1940, in which all sets are classes but some classes are not sets, distinguished as ‘proper classes’ because as sets they would lead to contradictions. In this letter Cantor allows the possibility that two ‘inconsistent’ pluralities can be in one-one correspondence, which in his earlier terminology would have meant that they had the same cardinality. He also allows the possibility that a well-ordered plurality (which he also calls a series) might fail to count as a set, in which case (contrary, again, to his earlier terminology) it will not be allowed to have an ordinal number.

Since ordinally similar well-ordered sets are necessarily cardinally equivalent, and since they are by definition (as sets) not inconsistent, it follows that when Cantor specifies that the cardinalities of such sets are Alephs, he is in effect restricting Alephs to being the cardinalities of well-ordered sets, and by implication of what he has already written in this letter he is allowing a concept of equicardinality (one-one correlation) to pairs of pluralities which have no cardinal number.

One might put this by saying that a cardinal number that is not an Aleph is what would count as a plurality’s cardinality if it were allowed to have one, except that, being inconsistent, it isn’t. As Zermelo points out in his note [1] on page 451, Cantor has unconsciously and instinctively relied on (but not formulated) an axiom of choice, which provides a justification for declaring that any set can be well-ordered somehow or other. Since this is an axiom, a set theory can be propounded that does not accept it, and so the space for cardinal numbers that are not Alephs coincides quite simply with ‘non-well-orderable’ sets (which being sets are consistent) in a set theory which rejects the axiom of choice. Zermelo, in this footnote, points out (with satisfaction, one suspects) that his own 1904 derivation of the well-ordering proposition from the axiom of choice is carried out without any reference to the concept of inconsistency.

I have often wondered if there is room in set theory for a concept weaker than cardinality, in the same way as the concept of an order type is weaker than that of an ordinal number. One ‘inconsistent’ plurality could then, by being in one-one correlation with another inconsistent plurality, have so to speak the same quasi-cardinality. The problem of cardinalities that weren’t Alephs would then disappear --- they would simply be quasi-cardinalities.

This would not, however, have a bearing on the continuum hypothesis (to return to our muttons at last), for the set of real numbers cannot be dismissed as having only a quasi-cardinality, since it is accepted on all hands as a set. As such, it must have a genuine cardinality, and this must be greater than (since it is different from and cannot be less than) Aleph zero. At least (thanks to Gödel and Cohen), the question is no longer which higher Aleph its cardinality actually is, but how we can organise our set theory to accommodate ranges of Alephs which we can allow it to have.

In the paper that first expounds his continuum hypothesis (almost in passing), III 2 of 1878, Cantor equates real numbers with infinite sets of integers and not with infinite decimal expansions. Nor does he prove the non-denumerability of the set of real numbers by using his famous ‘diagonal procedure’. This is propounded in a short paper of 1890-91, III 8 on pages 278-281. His proof here uses what are tantamount to binary expansions, in the form of infinite series every one of whose members must take one of two values.

As to Gödel’s 1940 itself, my first recommendation is to read the historical summary given by Solovay in Section 2 of his introductory Note to that (in Volume II). This Note also covers three short earlier Gödel papers (1938, 1939 and 1939a) which are helpful summaries, but the most important of these, 1939a, recommended by Gödel himself in note 22 on page 97 of Volume II, is remarkably formal, and most readers will need the help of Section 3 of Solovay’s Note.

In particular, 1939a makes use of the Zermelo Fraenkel version of set theory, and constructs in it, without using the axiom of choice, a model in which not only the model itself obeys the basic axioms but the axiom of choice follows (within that model) as a theorem, and the continuum hypothesis as well for good measure. In contrast, 1940 makes use of the version of set theory originated by John von Neumann and revised by Bernays and Gödel himself, called BG, which I find personally much easier to grasp. For reasons which I cannot explain, the ZF set theory used in 1939a lends itself to Gödel’s formal, abstract and, indeed, rarified treatment in that paper, while the BG formalism enables Gödel to give 1940 a step by step quality (I find Solovay’s term “ad hoc” appropriate) which makes his strategy easier to follow, provided one does so step by step and does not ask for a map. In this longer treatment, the appropriate model is called D and the BG axioms minus the axiom of choice are called S.

Gödel’s own account of his strategy is given (most clearly, in Solovay’s opinion) in a lecture printed in Volume III as *1939b, which he delivered at Göttingen in the first half of the Winter Semester of 1939-1940. A further account is given in a lecture delivered at Brown on Boxing Day, 1940, printed in Volume III as *1940a. For both of these, Solovay’s introductory Note in that volume is extremely helpful.

In brief, the strategy takes up an idea of Hilbert’s for proving the continuum hypothesis and uses it for the more modest purpose of proving its consistency with ZF or BG as the case might be. Although the strategy as thus summarised applies equally to the full proof of 1940, to the three abstracts mentioned and to the Göttingen and Brown lectures, there appears to have been some difference over and above the choice of a version of set theory. In the Brown lecture Gödel refers to giving his proof a new shape. Since 1940 was clearly completed before the Brown lecture, delivered at the very end of 1940, the obvious assumption is that it was 1940 that exemplified this ‘new shape’. My belief, however, is that 1939a represents the improved proof and 1940 a reversion to the earlier one, undertaken because Gödel found that his later and more elegant proof was less suitable for the task of making an academically incontestable presentation with all its i’s dotted and its t’s crossed.

Then there is the question of Gödel’s explanations of the meaning of the continuum hypothesis, given first in a paper requested for the American Mathematical Monthly and printed in Volume II as 1947. Printed as 1964 in the same volume is a revision, reconstructed from his papers, which as well as changes in detail contains two important additions. The first is a supplement, which anticipates the possibility that the hypothesis might turn out to be undecidable and quite contradicts the assumption made by me in my third paragraph, that Cohen’s discovery left mathematicians with the same freedom of choice as had been given to geometers by the independence of Euclid’s fifth postulate. Gödel insists that in set theory, unlike geometry, there is a considerable asymmetry between the alternatives. The second addition is a postscript written in September 1966, acknowledging Cohen’s achievement, without doubt (“no doubt” in Gödel’s faulty English) “the greatest advance in the foundations of set theory since its axiomatisation”.

1940 (taken from lectures delivered at Princeton) was republished in 1951, with minor revisions and considerable extra footnotes, all given scrupulously in the version printed in Volume III. This, however, is not Gödel’s 1951 itself, a cheap paperback which I bought in that year, but a 1970 printing (see the textual notes, page 314). This differs from my memory of 1951 in an important respect, but of course my memory is not infallible, and my copy is lost, so I hope that people who still have the original red-covered paperback will enlighten me.

My memory, then, is that in between the Introduction and Chapter I a single page was inserted on which an an abbreviated schema was printed giving the rules for definition to be found in the middle of Chapter II, on page 45 of Volume II. Of these rules I only remembered one, the first, but it rested in my mind as an expression of the concept of a contextual definition, requiring existence and uniqueness to be proved for whatever is introduced contextually (in this case, special classes). In fact, all four rules govern definitions that are to some extent contextual (even rules 2 and 4 which do not require existence and uniqueness to be proved). Moreover, Gödel frequently uses non contextual proofs, taking them for granted as not needing him to justify them; that is to say, cases where a definiendum is introduced as a simple abbreviation of or replacement for a definiens. Examples are: 1.11 Dfn {x} = {x,x}, 1.12 Dfn <x,y> = {{x},{x,y}} and 1.17 Dfn <x> = x, which can hardly be called an abbreviation since the definiendum is the longer.

As an aside I should like to add that Gödelian contextual definitions do not tell the whole story about what is normally regarded as their problem, the locus classicus for which (as Professor Smiley pointed out to me) is Frege’s Grundlagen der Arithmetik, §§62-69. It is Frege’s problem that Wittgenstein will have had in mind in his aborted attempt to define “Liebespaar” contextually in MS 105, recto page 21. See my article on this website on the Smythies typescript for the reverberations which this failure had for Wittgenstein.

As to the question of what Gödel meant by Platonism, I have been greatly helped by a pre-publication offprint of Was Gödel a Gödelian Platonist? by Michael Potter, to appear in Philosophia Mathematica. I have only two quibbles with Potter, one very minor indeed, so I will mention these first and move on to my agreements. In a paper called The present situation in the foundations of mathematics, printed as *1933o in Volume III, Gödel makes what Potter calls a tantalising remark, referring to “a kind of Platonism, which cannot satisfy any critical mind”. Now this comma is clearly a German one, and if one drops it (as Potter does in his quotation) one does not find that no Platonism can satisfy a critical mind, but that there is some kind of Platonism which cannot. Admittedly, this is still tantalising because the question remains, which kind of Platonism? Previous commentators have always retained the comma, making Gödel appear to have changed from an extreme anti-Platonist (like his teacher Hahn) to a serious Platonist.

Potter uses the plural, “incompleteness theorems”, in pointing out that Gödel regarded both as important to his philosophical views, but he overlooks one of the uses Gödel puts them to, namely establishing the inexhaustibility of mathematics (a word used in the Gibbs lecture, *1951, delivered at Brown). That the first theorem implies this is easy to understand, and most people would be quite satisfied with it for that purpose. In any sufficiently rich mathematical system, a proposition can be found which is undecidable in that system but, according to simple meta-mathematical arguments, clearly true. Since it is clearly true it can be added to the system as an axiom, producing a new system in which there will be a further undecidable proposition, also clearly true by similar arguments; and so on indefinitely.

Gödel’s second so-called theorem is proposition XI of his 1931, which is there introduced as only “in Umrissen skizziert” (sketched in outline) and thus not strictly a theorem at all. He handsomely admits this in the same lecture, saying that it was only properly proved by later mathematicians, the most significant being Turing (Church was one of the others). His formulation in this lecture of this proposition (by then properly called a theorem) is: “for any well-defined system of axioms and rules, in particular the proposition stating their consistency (or rather the equivalent number-theoretical proposition) is undemonstrable for those axioms and rules, provided these are consistent and sufficient to derive a certain portion of the finitistic arithmetic of integers.” He goes on to say that this second theorem is what makes the incompletability of mathematics particularly evident. His argument for this point is not easy to follow, and so I paraphrase it, using my convention of single inverted commas for paraphrases and other things that are not strict quotations.

‘Theorem XI makes the following impossible. Set up a well-defined system of axioms and rules and asseverate: I perceive this system to be correct, and I believe it to contain all of mathematics. For in perceiving the system to be correct you must perceive it to be consistent, but Theorem XI says that the system cannot prove itself to be consistent, and therefore your perception cannot coincide with your system’s limited deductive abilities. You are therefore claiming an insight which (though you had not intended to claim this of it) goes beyond your system. Therefore either your insight is right and your belief that your system contains all of mathematics is wrong, or your insight is a broken reed, in which case you can still less believe that such a faulty system contains all of mathematics.’

As Potter points out, the greater importance for Gödel of his incompleteness theorems was not that they refuted the naive assumption that there could ever be such a thing as “all the mathematics there is”, but rather that they refuted the beliefs of Hahn and the Vienna Circle that mathematics was essentially analytic, or tautologous, or “syntax of language” --- these ideas not being identical and needing delineation, a task which Potter undertakes with more expertise than I can myself, though I shall use this opportunity to comment on Wittgenstein’s Tractatus explanation of tautologies.

Gödel himself concentrates on the idea of syntax, in his his attempts to criticise Carnap under the title “Is mathematics syntax of language?”, of which two drafts are printed in Volume III, as *1953/9, called III and V. In both, Gödel uses the word “semantics” as well, in acknowledgement of the fact that Carnap’s viewpoint had widened to include the meaning of mathematical symbols and not merely the rules specifying their permissible combinations. This widening was (as I tried to show in my second Gödel article on this site) to some extent prompted by Wittgenstein, though this did not lessen the latter’s antagonism to Carnap. As I said in that article, it is by no means clear to me whether Wittgenstein’s accusations of plagiarism relate to a borrowing of his pre-1929 ideas or to post-1929.

As to what meaning it might have to say that mathematics consists of tautologies, I have always felt uncomfortable with Wittgenstein’s Tractatus restriction of the term to the tautologies of propositional calculus. The rediscovery in my old home of my Tractatus first edition led me to check this, and then I found in Ramsey (in the opening pages of his 1925 Foundations of Mathematicsi, printed in his posthumous 1931) something that shone light on a related problem, the picture theory of atomic propositions. Ramsey had known Wittgenstein well when he wrote this paper, and so there is every reason to suppose that these opening pages accord with Wittgenstein’s views at that time. Moreover, they are related (by disagreement) to the change of view expressed in the paper Komplex und Tatsache, which Rhees (in the appendix to Philosophische Bemerkungen) dates to June 1931. It was actually written in MS 110 on the 30th of that month and the first of the next, almost without interruption or change. I infer that this paper meant a great deal to Wittgenstein from the fact that Elizabeth Anscombe gave me a copy of it in 1952, which alas I have long since lost. With such reverberations, I hope that readers expecting Gödel will put up with a somewhat lengthy aside about Wittgenstein.

The basic meaning of the term “tautology” in mathematics and philosophy is that a proposition is tautologous if it is true whatever the facts in the empirical world turn out to be. This is not the normal English meaning, but an example will show how the meanings relate: the tautology “all stallions are horses”. It is tautologous because of the meaning of its words, which preclude the possibility of an animal being a stallion but not a horse. Thus, an explorer who finds an adult male okapi will have to decide whether to refrain from calling it a stallion or to agree to calling an okapi a type of horse. So from the question of the meaning of words, we arrive at a situation where what might actually be found in the world must be considered.

This shift of meaning is an extremely useful one in the philosophy of mathematics. It enables philosophers who wish to, to declare that all pure mathematics is tautologous, and even applied mathematics can come under this rubric. (“Admittedly, applying this system has been known to lead to bridges collapsing, but at least in this system it is true that . . .”)

Wittgenstein clearly had this extended meaning in mind when he confined tautologies to the results of propositional calculus, a tautology being a compound proposition composed of propositional variables joined by truth-functional connectives, if, for every assignment of truth values to the variables, the compound proposition has the value True. That being so, the result of substituting actual propositions for the variables will also have the value True and count as a tautology. What is unsatisfactory about this (set out in 4.46, on page 96 of the 1922 edition, and again in 5.101 on page 104) is that on the face of it, propositional variables are treated as taking values (by substitution) that are given and unanalysable whether the values they take are atomic propositions or ‘molecular’ ones. Wittgenstein takes for granted in the Tractatus that the only way in which atomic propositions can be joined into ‘molecular’ ones is by truth-functional connectives, but this is not self-evident at all. He misses an opportunity for facing this issue at 5.101, where truth-functional tautologies reappear. Setting out the sixteen possibilities for combining two propositions p and q truth functionally, ranging from tautology to contradiction, he concludes “Die jenigen Wahrheitsmöglichkeiten welche den Satz bewahrheiten will ich seine Wahrheitsgründe nennen”. Ogden’s “truth grounds” clearly won’t do for this word, which is standardly translated as truth-conditions --- the T or F assignments to p and q in various combinations. The missed opportunity is to consider the truth circumstances that make each argument p or q true or false individually. So that instead of “der Satz” being made true by assignments of truth values to its ingredients p and q, it could stand as an individual entity for which there will be certain states of affairs that make it true and others that make it false. If all possible circumstances make it true it will count as a tautology in its own right.

Wittgenstein could then interpret 5.11, 5.12, 5.121 and 5.122 as meaning that p follows from q when the truth circumstances of q include those of p (which would allow q implies p to count as a tautology). Another example of missing this opportunity is the parenthesis that concludes 5.1362: (“A knows that p is the case” is meaningless if p is a tautology.). It is clear that Wittgenstein had in mind a tautology of such a form as r Ú ~r, not truth circumstances for p that allowed it to be a tautology ‘in its own right’. And in 5.1311 he had agreed that fa follows from (x).fx, but nowhere acknowledges that (x).fx ® fa is an example of a tautology.

Incidentally, in 5.14 he says that where p follows from q, q says more than p, which does not contradict 5.121 because because the proposition with wider truth-conditions (or in my suggestion, truth circumstances) says less than the narrower one --- the narrower one is the more specific. From this, it would follow that a proposition whose truth circumstances were so wide that no state of affairs could falsify it would say nothing at all (as is clearly the case, and in Wittgenstein’s view too, with a truth functional tautology which admits all truth-conditions).

My personal view that such width of truth circumstances could lead to a proposition that was completely bland without being meaningless will not appeal to logicians (it would have to be meaningful enough for its denial to count as a meaningful self-contradiction too). It would certainly not have appealed to Wittgenstein --- but to save me from the embarrassment of defending it, Wittgenstein has suggestions of his own to make in the right-hand pages of his MS 105 (the first of what Rhees called the ‘Siamese twin’ volumes, MSS 105-6), written in the early months of 1929. The suggestions are made on the successive pages 78 and 80, and are of two types. The first arises out of a colour problem of a few pages earlier. This is introduced by a counting problem. If, as one normally does, one says that there are four apples on a table intending to exclude there being five, one can say “four apples and no more” to make oneself clear. With colour, similar confusion can arise if one says “a touch of blue” without saying how intense a touch or what other colours there are touches of. This leads later to problems about atomic propositions which I discuss in my soon-to-be-internetted chapter on the 1929-1930 notebooks. Here, he asks how, if a first colour proposition has used unspoken implications while a second is explicit, the second can contradict the first if its implications turn out to have been to a different effect. After all, he exclaims, at the end of that paragraph on right-hand page 78, two atomic propositions cannot contradict each other --- but he does not explain why, in these very peculiar circumstances, either of them should be supposed to be atomic.

The stage is now set for the paragraph starting at the bottom of that page, discussing propositions of an apparently related kind, such as: one material point can have only one velocity at one time. These propositions are all “Selbsrverständlichkeiten” and their contradictions “Widersprüche”, which in the following paragraph become “Tautologien” and “Contradiktionen”. In a subparagraph he considers “Two particles cannot occupy the same place at the same time” and interprets this as following from the definition of “one particle”, and considers this a different case from the examples of velocity etc.

In my opinion he is right to think them different but wrong in assigning the difference. The second case is a description of something that, empirically, never happens (there is no need to add that the workings of the universe do not allow it to hapen, but you are welcome to if you wish). The other cases (velocity etc) would come under Wittgenstein’s later terminology of a grammatical observation. He uses a similar sounding phrase not so very much later in referring to what Mach had called a thought experiment, on page 284 of MS 107, 6.2.30, as eine grammatische Betrachtung, but it has a different meaning, and the later meaning of eine grammatische Bemerkung is what applies here. (See Investigations, § 248, where “One plays patience on one’s own” is a paradigm example.)

An example of what I consider to be something that does not in fact happen comes in the right-hand pages of MS 106, written when Wittgenstein was preparing his Nottingham paper (which in the end he replaced by a lecture on infinity, prepared later in the right-hand pages of MS 106 and then in its left-hand pages). In the preparations for the paper that was printed but not read, he writes out a truth table for the atomic propositions “a is green” and “a is red” and their conjunction --- which will be true if both are true. In his table, the line giving “true” for both and “true” for their conjunction is crossed out, leaving three lines false, because the conjunction never takes place (why it does not is left in the air by an unexplained “unmöglich”). In actual fact, a red light shone on the same place as green light gives an appearance of yellow. Then he gives a similar account of “a is green implies that a is not red” by crossing out the line for green true and red true and the combination false, leaving the other three lines giving “true”. We thus have a conjunction which in actual fact is never true and an implication which in actual fact is always true --- referred to by Wittgenstein respectively as a contradiction and a tautology.

Finally, there are two paragraphs in which Wittgenstein might, on such grounds, have used the words “tautology” and “contradiction” but did not. These come in MS 107, written on 8.11.29 in Cambridge, and in MS 108, written on 13.12.29 in Vienna. The first is fascinating because it asks if two people can share the same body, arguing, one would think decisively, that the impossibility of this rests on our bodies’ being our principle of individuation --- but then Wittgenstein claims to give a meaning to its possibility, along lines used in the Blue Book. I can imagine myself to have a pain in another body, and so with alittle more inventiveness I can imagine that my whole body becomes insensitive and motionless and my pains and other sensations transfer completely to the other body. This body would then have two people in it, and the imagined possibility would have to fall in the category of things that never happen.

The Vienna paragraph faces this issue more directly, citing the fact that, with rare exceptions like amputated limbs, my sensations do not extend beyond my body, so that one person can have only one body and not two. This is an example of “sehr interessante ganz allgemeine Sätze von grosser Wichtigkeit, wirkliche Sätze die also auch eine wirkliche Erfahrung beschreiben, die also auch hätte anders können aber nun einmal so ist”: in other words, empirical truths that happen to be the case. The question asked in Cambridge was, similarly, “eine ungemein wichtige und interessante Frage”. From now on, “interessant” becomes a Wittgenstein code-word for general facts that tempt us to think of them as necessities but are actually what happens to be the case. I do not think Wittgenstein ever again used the word “tautology” for facts of this kind or “contradiction” for their opposites. (And in Vienna, to to give an example of something non-empirical, he chooses “I cannot remember the future” which, like its contrary, is meaningless. It could of course also be considered as a significant grammatical observation about the way we use words.)

I now have to explain what all that has to do with Ramsey and the recantation implied by Komplex und Tatsache. In his 1925 Ramsey specifies that fa is an atomic (and therefore unanalysable) proposition, in so far as “f” and “a” both, equally, denote simples. Such propositions can then be combined by using truth-functional (propositional calculus) connectives, and the result will be a composite proposition, analysable by propositional calculus means alone.

The sense of “complex” in Komplex und Tatsache, in which Wittgenstein retracts his picture theory, is not that of “composite”. An atomic proposition would also have been a complex in the picture theory sense, as shown by the street accident analogy. If each object in that depiction was a simple (say a piece of wood representing a vehicle but showing nothing of its structure) then a number of such objects could be arranged in a configuration, and a proposition depicting the configuration would still count as atomic, symbolised by, say, f(a, b, c, ...), being a configuration indicator but still, like a, b, c etc, a simple. After fifty years I believe that the problem here explains the fundamental superstition of the Tractatus (indeed, I hope to show that it resolves into two superstitions). It also explains why Wittgenstein opens the Tractatus by declaring that the universe consists of facts, not of things. For things correspond to the simple pieces of wood in my version of the Paris accident: only when combined in configurations do they constitute elements of the universe.

The problem I am trying to nail with Ramsey’s help comes when one compares a configuration of two or more elements with what can possibly be meant by an atomic proposition of the form fa. What configuration can a single element form? Ramsey would appear to say that it configures with his other simple, the predicate f, but in 3.1432 Wittgenstein uses the notation “aRb” (rather than f(a,b), which he does use elsewhere) and goes out of his way to say that a relates to b without there being a further entity that could be called the relation R --- the symbol “R”, in other words, has no denotation. This reinforces what he had already said in 2.03, “Im Sachverhalt [by which he understands an atomic circumstance] hängen die Gegenstände ineinander, wie die Glieder einer Kette”, and in 2.032, “Die Art und Weise, wie die Gegenstände im Sachverhalt zusammenhängen, ist die Struktur des Sachverhaltes” --- in other words, the links of the chain are simples but their structure is not a further simple, or any other kind of entity, holding them together. They do that for themselves. (A point made in Komplex und Tatsache, in the third and fourth paragraphs from its end.) Unfortunately, that does not leave anything for a single elementary object to be doing.

There is no doubt that Wittgenstein does allow atomic propositions with only one argument, because in 4.24 he says that he writes an atomic proposition as a function (of a name or names) in the forms fa, f(a,b) etc, that is to say with one argument, with two arguments “etc”. One must also take into account 4.01, saying that a sentence is a picture or model of reality, and 4.011 reiterating this, but one must also consider 4.014, the gramophone record, its grooves constituting a picture of its music but bearing no pictorial resemblence to it.

This brings me to the major superstition. In 1950 I was so impressed by the latter point that I took the whole picture / model image as a mere metaphor. Only in recent years, reading Wittgenstein’s recantations, have I come to appreciate the fact that it was never a metaphor --- it was meant “auch im gewöhnlichen Sinn” (in 4.011). I then decided that this doctrine, namely that if one took an everyday sentence and expanded it into its atomic constituents, these would form, im gewöhnlichen Sinn, a picture of the state of affairs that the original described, was the fundamental superstition underlying the Tractatus. And I was encouraged in this by my memory of Wittgenstein’s recantations, not only in Investigations in general but in particular in Komplex und Tatsache.

Now, putting all that together with what Ramsey wrote and reading Komplex und Tatsache afresh, I find that the superstition was a double one, the second being how atomic functions of a single variable were interpreted. For Wittgenstein must have tried to have his cake and eat it. He makes way for doing so in 2.0131, where “Der Raumpunkt ist eine Argumentstelle”, and so is a point in musical-note space. Individual simples can be assigned positions in these spaces --- one simple to such and such a coordinate reference in space in the normal sense, another to such and such a colour, another to such and such a pitch. That is all the Tractatus allows an atomic proposition with a single argument to do.

In the framework of the Tractatus that might seem quite elegant and allowable, until one realises, from Komplex und Tatsache, that Wittgenstein had thought of an individual simple and its place in some descriptive space as two simples bearing a mutual relationship, just as Ramsey had implied. For example, this circle’s being red or my being tired had been (and this is precisely what he is beating his breast about) complexes with the constituent parts circle and redness, or me and tiredness. So that is quite definitely a second superstition which would never have occurred to one if Wittgenstein’s recantations had not pointed it out. What is intriguing is that in the fourth paragraph from the end, which I have already mentioned, Wittgenstein uses a point from the Tractatus to argue against his old viewpoint. “Auch die Kette besteht aus ihren Gliedern, nicht aus ihnen und deren räumlichen Beziehungen”. He has hoist himself on his own petard.

There is an echo of the penultimate paragraph in MS 127, where “der verwirrende Gebrauch des Wortes ‘Gegenstand’” becomes “der sprachwidrige Gebrauch”. Wittgenstein wrote this when he was going through the Tractatus with Bachtin, and the notebook deserves study not only for its paragraphs of Tractatus-critique but for its composition as Wittgenstein was getting his breath for revising Investigations.

Returning to Gödel and what he meant (in his later years) by his Platonism, I do not want to argue that he did not seriously believe in a realm of mathematical objects created in advance for us to discover, since he makes this quite clear. He does so in particulat in his Gibbs lecture delivered at Brown, printed in Volume III as *1951. At the very end of this he quotes an expressly ‘creationist’ passage from Hermite, and a few paragraphs earlier he expresses his own credo: “The truth, I believe, is that these concepts form an objective reality of their own, which we cannot create or change but only perceive and describe”. I believe, rather, that his arguments for this belief are compatible with a much more ‘modern’ interpretation. Concepts, of course, can well form an objective reality without being objects, and elsewhere in this lecture he expresses himself more mildly. It opens with his tribute to Turing when he discusses the inexhaustibility of mathematics Towards the end there is a passage marked by double vertical lines, ║ . . . ║, which he apparently did not deliver at Brown, and following it is the credo I have quoted.

My own way of propounding the interpretation that I believe compatible with Gödel’s arguments may, however, seem far from modern to most mathematicians, so I will get it off my chest for a start.

Ever since I began to teach mathematics, first in schools and then in a teachers’ training college, and when all I knew of Gödel was his basic work on incompleteness and the continuum hypothesis, my view has been that a denoting grammar has such convenience that the sensible thing is to devise a plausible excuse for keeping it rather than try to outgrow it. One day (and it still hasn’t come) the concept of a value given to a variable (or to any other symbol), taken as primitive, might give us a grammar that we could honourably call non-denoting, but until then we should be honest about the matter. We need a realm of objects for our symbols to denote, so we simply declare them to exist, and then take steps to ensure that our adopting them does not lead us into contradictions. What I am prepared to claim now is that the very arguments used by Gödel to defend his belief in these objects’ real existence can equally be applied to their defence as objects of our mere deeming-to-exist.

The article in which Gödel sets out his arguments in a manner coming half-way towards my view is his drafted criticism of Carnap, printed as *1953/9 in two sample drafts which I have already mentioned. Gödel quotes Carnap’s distinction between Realwissenschaft and Formalwissenschaft, and approves of the fact that in drawing this distinction Carnap was less ‘left wing’ than Schlick and Hahn had been (see the unpublished essay printed as *1961/? in Volume III for this distinction). This softening of Carnap’s consisted rather undramatically in the fact that, in accepting mathematical objects as denoted by formulae, he claimed that their real existence had no cognitive content and was thus meaningless, whereas in his Schlick-Hahn days he had called it false. Gödel, however, insists that what is sauce for Realwissenschaft is sauce for Formalwissenschaft too. If verifiable consequences follow from the assumption of objects’ existence, then they exist as really in one case as in the other. That this is not a mere ‘façon de parler’ follows from the fact that there is a risk of falsifying consequences too, if inconsistencies are derived from our assumptions.

My personal view here is that existing as a façon de parler is quite sufficient if it has consequences that accord with ascertainable facts, while if instead they lead to contradictions that shows that they had not been any use even as a façon de parler.

The earlier of these Carnap-critique drafts includes what seems to me an intriguing thought-experiment, though it is far from clear to me on which side of the fence Gödel is left by it. Assume that the reason with which we draw conclusions from our positing of mathematical objects were a further sense observing them. This sense would show us a reality completely separated from space-time, so regular that it could be described by a finite number of laws. Then nothing could prevent us from simply deeming this new reality to be an illusion. Or if we deem it to be real, we face the criticism that this is a mere matter of deeming. However, the truth is that we are not dealing with a further sense but with reason, and consequently . . . but what the consequence is, is not clear to me at all.

Gödel, however, seems to deliver his judgement in the second of these Carnap drafts. On page 361 of Volume III there is an admission of a distinction between space-time descriptions of objects (let us say iron filings) and such “abstract physical objects” as a field of force posited as determining their arrangement. Consequently, a positivist has no more reason to reject mathematical objects than fields of force, and none to deride mathematical objects as pseudo objects.

Gödel then lists, very fairly, three plausible reasons for taking mathematics to be void of content, of which I leave readers to look up the second. The first and third reinforce each other, but, in their upshot, give Gödel, as he interprets them, grist to his mill. What we mean by content is the admission of certain states of affairs and the exclusion of others. A logically true proposition excludes no states of affairs, and therefore has no content. However, admitting or excluding states of affairs is a semantic matter, that is to say one of meaning. Consequently we have a kind of paradox: logical propositions which are devoid of content because of not excluding anything, but nevertheless true and meaningful because it is a semantic rule that declares their lack of content. That, at least, is what I take his point to be, and he goes on to make a more telling one. What we regard as content depends on our intentions, and logical words such as “not” and “or” do interest us, and this fact opens the way for formal truths to have metamathematical and metalogical content.

The crux of Gödel’s leaning towards Carnap while criticising him seems to me to come in the distinction between syntax and semantics. After all, as I said when I first mentioned them, the title given to the Carnap drafts is “Is mathematics syntax of language?”, but in the course of them Gödel discusses semantics, that is to say the meaning of the symbols which syntax merely allows us to combine in certain specified ways. For our meanings surely influence the syntactical rules we devise for our symbols to obey. Moreover, our meanings determine relations between concepts, so that it is entirely reasonable to assume that mathematical propositions have “sound objective content” (see the Gibbs-Brown lecture following the ║ . . . ║ omission) even though this cannot be content in the sense of accepting-or-rejecting-circumstances --- because, once meanings have been decided on, the relations between them become objective. Nevertheless, since other other concepts and meanings could have been chosen, this is certainly not objectivity in the Platonic sense.

In a nutshell, I believe that the passages where Gödel’s arguments can be taken as giving most support for a ‘reasonable, modern’ interpretation are to be found, along with the Carnap drafts, in the Gibbs-Brown lecture after the ║. . . ║ omissions, precisely where he expresses himself in the least ‘modern’ fashion.

Potter cites Parsons as saying that Gödel upheld mathematical intuition in widely differing senses, and while agreeing wholeheartedly with this I have two comments to make. The first is that I hope no-one will think I believe that my own “realm of intuitive explanations”, devised for teaching mathematics to children, could ever come within the width of Gödel’s views. The second is that the Chihara quotation at the beginning of Potter’s paper is nothing but a caricature of Gödel’s understanding of intuition. No-one who attends to the detail of Gödel’s mathematical and metamathematical arguments can fail to appreciate their down-to-earth sanity, whatever quibbles and, indeed, exasperations one might have with his extra-mathematical prose.

All that remains is for me to keep my promise to say something about Fowler’s The Mathematics of Plato’s Academy (Oxford, 1999). This impressive book is based on his discovery that a passage in Plato’s Theaetetus can be interpreted as suggesting that Theaetetus himself (who may have died young soon after or lived considerably longer) used what came to be called Euclid’s algorithm to provide what in effect was a substitute for a real number system. The relevance of the Theaetetus-Theodorus passage in the dialogue is explained in sections 10.3 b-c of the Appendix to the second edition (1999). Fowler calls calculations using the ‘Euclid’method anthyphairetic. For a simple example, he imagines Meno’s slave boy to be asked for the relationship in quantity between a heap of twenty six and one of sixty (presumably identical) stones. The former can be taken twice out of the latter leaving eight over. The eight can be taken three times out of the twenty six leaving two over, and finally the two can be taken four times out of the eight leaving none over, giving what we should regard as the ordered set of numbers [2,3,4] (exactly the same as if the original heaps had been thirteen and thirty). Applied to indefinitely divisible quantities such as lines or areas or volumes, this process can either come to an end when remainders are no longer practically detectable, or continued if we have theoretical criteria that enable us to persist. This does indeed put us in the modern realm of real numbers, and we can conveniently express our results either by continues unit fraction or as ordered sets of natural numbers (it makes an instructive exercise to ask what would happen if a zero appeared among them).

Fowler quotes Cherniss (The Riddle of the Early Academy, 1945, reprinted by Garland, 1980) as arguing that the members of Plato’s Academy were under no obligation to subscribe to his metaphysical theory, and moreover that formal instruction there bore no resemblance whatever to his dialogues, being restricted to mathematics. Nor did Plato himself supply any of that instruction, acting merely as a prompter, not even a supervisor, of members’ research (a status one could quite fancy if one were distinguished enough to deserve it).

Fowler points out (in Chapter 7) that the analogue-thinking Greeks had very little contact with the digital-thinking Babylonians before Alexander’s conquests. Had they had more, they could well have used our modern concept of fractions to express ratio and proportion. How they might have managed without, he explains in great detail. And in his chapter 9, he explains in detail how the posited technique of Theodorus and Theaetetus relate to our modern use of continued fractions --- except that we do not use them as much as we ought to. They provide an insight into the nature of real numbers that reinforces the incomplete insight offered by decimal expansions. This chapter of Fowler’s would be useful on its own to any mathematician wanting to learn more about them (or even learn about them from the beginning), and it also gives an historical account of their discovery by Euler and their elaboration by Gauss, and of their subsequent comparative neglect. Natanson too is helpful on them, and his American editors refer one to Hardy and Wright’s Theory of Numbers if one still needs help.

As Fowler mentions, the modern analysis of real numbers was originated by Dedekind, who records that he discovered his ‘cuts’ on 24.11.1858. This was the first historical moment when real numbers could properly be said to have an ontology. Before Dedekind, Cauchy had provided a technique of limits, and my instinct is to deny this the status of an ontology, but it would need someone with the talent and scholarship of Imre Lakatos to settle the point. Without an ontology, the question “did they exist in objective reality?” ought not to be asked. Now it is conceivable that for mathematicians between Theodorus and Eudoxus some equivalent ontology had been considered, or that later Greek mathematicians with, in effect, a concept of numerical fractions and Eudoxus’s technique of so-called exhaustion at their disposal (now accepted as equivalent to the limit concept) also had something they could call an ontology, enabling them to ask that question. However, I think that in both cases that is extremely unlikely. They had their respective techniques, and that is all. My conclusion, then, is that there was no ground in which any kind of mathematical Platonism could grow.

As I have said above, the sense in which mathematical concepts can be said to have objective reality is subject to our assigning meanings to them. Then the separate question of mathematical objects arises. There is a possibility (which I do not personally think would be worth the enormous grammatical effort) of construing mathematics as having no objects at all. Without going so far as that, we have the option of calling mathematical objects into fictional existence and submitting them to the exigencies of the concepts to which we relate them. These exigencies are dauntingly restrictive. They fully deserve to be typified as giving objective reality to mathematics. If you accept that, and think it reasonable, thereby, to call yourself a mathematical Platonist, then do so; but personally I find the term goes with too many dubious overtones for me to want to embrace it.