- [Körner1968]
- Körner, Stephan.
*The Philosophy of Mathematics: An Introductory Essay*. Hutchinson & Co (London: 1960, 1968). Unabridged and unaltered republication by Dover Publications (New York: 1968). ISBN 0-486-25048-2 pbk. -

I would say there are four steps onto the slippery slope and off the precipice into the deeps of mathematics:

0. Accepting the principle of an unbounded but always-extendable sequence as being the essence of the natural numbers

1. Adopting the principle of mathematical induction

2. Taking an always-extendable sequence as the axiomatic basis for the existence of infinite sets and the progression of transfinite cardinals

3. Realizing functions over transfinite sets as transfinite sets

The first and third seem to require remarkably little commitment. The middle two are disturbing in progressive degrees to different people, depending on the nature of their concerns over the foundation of set theory.

Whatever my own predilections to walking those steps, I must somehow contend with them. They lie at the heart of what it is we profess to know about the limitations of computers and the prospects for approaching universal computation. In seeking clarity on the foundations of computation, I am led to explore these knotty topics. I begin by looking at what Körner has to offer. -
*-- Dennis E. Hamilton*

Seattle, 2002 July 31

**Last updated
2002-08-05-22:20 -0700 (pdt)**

These are undistilled notes from reading the book. They are mostly as quotations, with an organization along my theme, not Körner's. These notes are also not digested, which will be the second stage of the note. I have found, in the reading, that I have found some clarity, and now want to refine it to a summary of what I make of it, rather than so much of what Körner has said in this place and that. --dh:2002-08-05.

"Since the philosophy of mathematics is mainly concerned with the exhibition of the structure and function of mathematical theories, it would seem to be independent of any speculative or metaphysical assumptions. Yet it may be doubted whether such autonomy is even in principle possible ... ." from the Introduction, p.11.

"As regarding pure mathematics I propose to argue that the concepts and statements of any (existing) mathematical theory are--in a precise sense of the terms--purely exact, that they are disconnected from perception, and that in so far as a mathematical theory contains existential statements these, unlike,

e.g., empirical and theological statements of existential import, are not unique. As regards applied mathematics I shall, roughly speaking, argue that the 'application' of pure mathematics consists in interchanging perceptual and purely exact statements in the service of a given purpose." The Nature of Pure and Applied Mathematics, p.159."Each of the philosophies of mathematics which I discussed declares the whole or part of classical mathematics to be in some way defective, proclaims the need for replacing the defective by sound mathematical theories, and tries to meet the need by actual construction. All agree that the set-theoretical antinomies are not only obvious defects of classical mathematics, but also symptoms of deeper lying defects which they diagnose in different ways. The arguments used in the diagnosis are ... mainly philosophical arguments,

i.e., arguments which belong neither to natural science or to logic." The Nature of Pure and Applied Mathematics, p.185.

I did not expect to find so much here on this aspect. But there is a big question on what it is that mathematical language is language about, and also its relationship to perceptions in the world. I don't think I need to make so much of it, although the historical perspective is extremely valuable, and something to explore more deeply as occasions permit. I think there is something about where things start. That is, it seems perfectly possible to theorize for its own sake, and not with any connection to the world or perceptions. This may not matter here. -- dh:2002-08-05

"One of the strange features of mathematics, at least since Leibniz, is that, in spite of the certainty of its truths, it is by no means generally agreed of what, if anything, the true propositions of mathematics are true." Some Older Views, p.16.

"For Plato ... applied mathematics describes empirical objects and their relations, in so far as they approximate to (participate in) the mathematical Forms and their relations." Some Older Views, p.18.

"According to Aristotle, the form or essence of any empirical object, an apple or a plate, constitutes a part of it in the same way as does its matter. ...

"Aristotle distinguishes sharply between the possibility of abstracting (literally 'taking away') unity, circularity and other mathematical characteristics from objects and the independent existence of these characteristics or their instances, i.e., units and circles. He frequently emphasizes that the possibility of abstraction does by no means entail the independent existence of that which is, or can be, abstracted. The subject matter of mathematics is those results of mathematical abstraction which Aristotle calls 'mathematical objects.'" Some Older Views, p.18."Indeed it would not even be possible for [Aristotle] to speak of a true or false idealization, but rather of one which would be more or less adequate for some given purpose." Some Older Views, p.19.

"Aristotle also pays more attention than Plato did to the structure of whole theories in mathematics as opposed to isolated propositions. Thus he distinguishes clearly between (i) the principles which are common to all sciences (or, as we might nowadays say, the principles of formal logic which are presupposed in the formulation and deductive development of any science); (ii) the special principles which are taken for granted by the mathematician engaged in the demonstration of theorems; (iii) the definitions, which do not assume that which is defined exists, e.g., Euclid's definition of a point as that which has no parts; and (iv) existential hypotheses, which assume that what has been defined exists -- independently of our thought and perception. Existential hypotheses in this sense would seem not to be required for pure mathematics." Some Older Views, p.20.

"[Leibniz] anticipated modern movements, in particular modern logicism, by bringing logic and mathematics together. ... One the one hand he presents a philosophical thesis concerning the differences between truths of reason and truths of fact, and their mutually exclusive and jointly exhaustive character. On the other hand he introduces the methodological idea of using mechanical calculation in aid of deductive reasoning, not only within those disciplines which belong traditionally to mathematics, but also beyond them. This means, in particular, the introduction of calculation into logic." Some Older Views, p.22.

"Another difficulty concerning truths of fact arises from the principle of sufficient reason 'which affirms that

nothing takes placewith out sufficient reason, that is to say nothing happens without its being possible for one who should know things sufficiently, to give a reason which is sufficient to determine why things are so and not otherwise.' This is for Liebniz not merely a general injunction to look for sufficient reasons to the best of one's ability, but in some ways, like the principle of contradiction, it is a principle of inference and analysis." Some Older Views, p.23."Mathematical propositions to [Liebniz] are like logical propositions in that they are not true of particular eternal objects or of idealized objects resulting from abstraction or indeed of any kind of object. They are true because their denial would be logically impossible. Despite every prima facie appearance to the contrary, a mathematical proposition is as much or as little 'about' a particular object or class of objects as is the proposition 'If anything is a pen it is a pen' about my particular pen, or about the class of pens, or the class of physical objects or any other class of objects. We might say that both propositions are necessarily true of all possible objects, of all possible states of affairs, or, using the famous Liebnizian phrase,

in all possible worlds." Some Older Views, p.23."[Liebniz] must have found ... that the discovery of a symbolism for the representation of statements and demonstrations on the one hand, and the insight into their logical structure on the other, though separable in thought, are very rarely separable in fact.

"The concrete representation, in suitable symbols, of a complicated deduction is, in [Leibniz's] words, a 'thread of Ariadne' which leads the mind. Leibniz's programme is first of all to devise a method of so 'forming and arranging characters and signs, that they represent thoughts, that is to say that they are related to each other as are the corresponding thoughts.'" Some Older Views, p.25."Frege [introduces] his own notion of an

analyticproposition: A proposition isanalyticif it can be shown to follow merely from general laws of logicplusdefinitions formulated in accordance with them." Mathematics as Logic, p.33."According to Frege numbers are logical objects which it is the task of a philosophy of mathematics to point out clearly. To define them is not to create them, but to demarcate what exists in its own right. Contextual definition of logical objects will not do because it does not exhibit their character as independent entities." Mathematics as Logic, p.36.

"But if arithmetic is deducible from logic, then the deduced arithmetical propositions can hardly be assertions about objects of any kind; not, at any rate, if, as Russell holds, logic has no subject matter. It (a) must be shown that the phrases which seem to represent entities (the so-and-so, the class of all things such that ...), whenever they occur in the deduction of arithmetic from logic, are embedded in contexts, which do not imply the assumption that such mental objects exist; and (b) these contexts must be defined." Mathematics as Logic, p.35.

"Russell's way of dealing with the impression--correct or incorrect--that in talking about classes we are not talking about entities is similar to his theory of descriptions. Again, 'the class of objects which ...', which seems to name an entity, is absorbed into contexts, each of which is defined as a whole and is free from the existential implications which Russell desires to avoid." Mathematics as Logic, p.36

"Whether or not we accept this view--that all true propositions denote (are names of) the truth-value T and all false propositions denote (are names of) the truth-value F--is not the issue. The point is that in so far as we regard a proposition as a truth-function of its components, we must take account only of their truth or falsehood and ignore any other information which it or its components may convey." Mathematics as Logic, p.41.

"Because of its importance for logic and the philosophy of logicism the point is worth emphasizing that truth-functions are a very special and abstract kind of compound proposition, a type which in some respects represents

certainfeatures of English and other natural languages ... but which in other respects are idealizations and simplifications ... ." Mathematics as Logic, pp.41-42."The last group of postulates needed to carry out the Frege-Russell programme concern the use of the terms 'all' and 'some' in mathematics. These were the last part of the logical apparatus to be formalized. The need for them becomes very clear when we consider the transition from statements in which a property is asserted of one individual object, to statements in which the same property is asserted of a finite number of objects, and thence to statements in which it is asserted of an actually infinite number." Mathematics as Logic, p.47.

"The search, however, for such a general characteristic ... as would cover both logic and mathematics has, so far, been unsuccessful--which is perhaps the least surprising, in view of the pragmatic principles included among the postulates of logicist systems, especially principles which are almost indistinguishable from empirical hypotheses about the universe." Mathematics as Logic: Criticism, p.57.

"The formalist not only needs the assurance that his formalism formalizes a consistent theory, but also that it completely formalizes what it is meant to formalize. A formalism is complete, if every formula which--in accordance with its intended interpretation--is provable within the formalism, embodies a true proposition, and if, conversely, every true proposition is embodied in a provable formula." The Science of Formal Systems, pp.76-77.

"Since the concepts of metamathematics and the statements in which these concepts are applied are not empirical, their subject-matter is also not empirical. The strokes on paper and the operations upon them are just as little the subject-matter of metamathematics as figures and constructions on paper are the subject-matter of Euclidean geometry. Both types of marks and constructions are diagrammatic; and diagrams, however useful and practically indispensable, are 'representations' which are neither identical nor isomorphic with that they are used to 'represent'. ... (I have put 'represent' in quotes in order to indicate that I am not using the term in its (now perhaps) dominant sense implying isomorphism between representing and represented system. The inexactness of the empirical and the exactness of the non-empirical concepts precludes isomorphism between the instances and relations of the two systems.)" The Science of Formal Systems: Criticism, pp.104-105.

"If no two people can live through the

sameexperience, then the experience, not being intersubjective, cannot ever validate the intersubjective statements of any science. For example, there can be no intersubjective science of introspective psychology, and no intersubjective science of mathematics as reporting intuitive constructions." Intuitive Construction: Criticism, p.137."We have said that the postulation of objects for internally consistent concepts is the foundation of mathematical existence-propositions. This does not imply any answer to the question whether one should actually postulate such objects in a given case. ... What has been said about existence-propositions in mathematics seems very much in line with the notions of working mathematicians. Indeed their use of the term 'existence-

postulate' suggests quite clearly the non-uniqueness of mathematical existence-propositions." The Nature of Pure and Applied Mathematics, p.176.On applied mathematics, there is a quotation of Dirac, from

The Principles of Quantum Mechanics, 1956, "The new scheme becomes a precise physical theory when all the axioms and rules of manipulation governing the mathematical quantities are specified and when in addition certain laws are laid down connecting physical facts with the mathematical formalism, so that from any physical conditions equations between the mathematical quantities can be inferred and vice versa. .... The justification for the whole scheme depends apart from internal consistency on the agreement of the final results with experiment." (Körner p.177)"The procedure of theoretical physics and of applied mathematics in general is to

replaceempirical propositions by mathematical ones, to deduce mathematical consequences from the mathematical premisses, and to replace some of these consequences by empirical propositions. That this procedure can be, and has often been, highly successful depends on the world's being what it is. That satisfiable rules governing--more or less strictly--the interchange of exact and inexact concepts (before and after mathematical deduction) have been found, depends on those features of the world which go under the name of human ingenuity." The Nature of Pure and Applied Mathematics, p.180."In choosing a criterion of the soundness of a physical or mathematical theory one chooses a programme for the construction of theories. ... In the case of mathematical theories the control by experience, if any, is at most quite indirect; and the choice is determined more by metaphysical convictions, allegedly based on insights into the nature of 'reality', or on sound practice and tradition. These become effective as regulative principles,

i.e. as rules of conduct--the area of conduct being the construction of mathematical theories." The Nature of Pure and Applied Mathematics, p.184.

"[Aristotle] distinguishes between the possibility of adding a further unit to the last member of any sequence of numbers, such as in particular the sequence of natural numbers, 1, 2, 3, ... and the possibility of making always another subdivision of, say, a line between two points, which had previously been subdivided any number of times. Here the possibility of going on

ad infinitumis what may be meant by calling the sequence infinite, or the line 'infinitely' divisible (consisting of infinitely many parts). This is the notion of potential infinity." Some Older Views, p.20."Kant's account of the propositions of pure arithmetic is similar to his account of pure geometry. The proposition that by adding 2 units to 3 units we produce 5 units describes--synthetically and a priori--something constructed in time and space, namely the succession of units and their collection. ... The logical possibility of alternative arithmetics is not denied. What is asserted is that these systems would not be descriptions of perceptual space and time." Some Older Views, p.29.

"Kant's notion of construction as providing the instances of mathematical concepts, the internal consistency of which is granted, assumed, or, at least, not called into question, has many recognizable descendants in later developments in the philosophy of mathematics." Some Older Views, p.29.

"In a mathematical sequence or progression a rule tells us how to take each step after the previous one has been taken. Kant will not allow the assumption that when such a rule is given the totality of all the steps is necessarily in some sense also given. The issue is particularly important in cases when there is no last step or where there is no first step." Some Older Views, p.30.

"According to [Aristotle] not only are there no instances of actual infinity within sense-experience; it is logically impossible that there should be." Some Older Views, p.30.

"Let us consider a concept '

nis a Natural Number' which is so defined that it doesnotentail 'nhas a unique immediate successor'. In other words, we admit the possibility, envisaged by Russell, that the number-sequence comes to an end. Moreover if there is a last Natural Number we assume it to be so great that no one--whether scientist or grocer--needs to be perturbed about it." Mathematics as Logic: Criticism, p.59"Let us consider next the concept '

nis a natural number' which is so defined that itdoesentail 'nhas a unique immediate successor', and, therefore, ''nhas infinitely many successors'.Thisconcept may not be applicable to groups of perceptual objects." p.59"Moreover, the hypothesis of the infinite sequence of natural numbers by which the concept 'natural number' is defined and provided with its infinite range admits of no empirical falsification or confirmation. It leaves room for further 'hypothesese' of a similar kind, one of which 'assures us' that the class of natural numbers is completely given, another that in addition the class of its subclasses is also given. But there are also hypotheses which assure us to the contrary. This freedom of defining mutually inconsistent concepts and of

providing them by definitionwith different ranges shows us that none of these concepts is empirical. The Natural Numbers on the other hand are empirical concepts, characteristics of perceptual patterns, such as groups of strokes or of temporally separated experiences. They and their relationship to each other are found, not postulated." p.60."My aim in this section has been to show that the logicist account of applied mathematics implies an illegitimate conflation of mathematical number concepts and of corresponding empirical ones. Ignoring the difference between the corresponding concepts, logicism cannot, and does not, say anything about the nature of this correspondence." pp.61-62.

Here, this is basically a question around rules of inference that lead to use of universal quantification on all of the numbers. It is not singled out (so far).

"Again the principle of induction, the most characteristic of all the principles of arithmetic, is, in the words of Hilbert and Bernays not an 'independent principle' but 'a consequence which we take from the concrete construction of the figures [of numbers]." The Science of Formal Systems, p.78.

This is an odd idea, except we would agree that the expressions in formal systems about arithmetic tend to exhibit the quality of number in their forms. However, Körner raises some questions about finite methods. This is at the heart of it, since we formalize using structures that are similar to the natural numbers, and are left with a problem there about any "reaching beyond," or even to the boundary. Here is how Körner sets it out: -- dh:2002-08-05

"Elementary arithmetic is the paradigm of mathematical theory. It is an apparatus which produces formulae, and which can be entirely developed by finite means. This statement, however, ... is still needlessly imprecise, and requires an actual and explicit characterization of what is to be meant by 'finite methods'.

"First, every mathematical concept or characteristic must be such that its possession or non-possession by any object can be decided by inspection of either the actually constructed object or the constructive process which would produce the object. ... Thus one is reasonably content with a process of construction which is 'in principle' performable. Indeed it is at this point, namely when the choice arises between making the formalist programme less strict or sacrificing it, that some relaxation of the finite point of view may be expected.

"Secondly, a truly universal proposition--a proposition about all stroke-expressions for example--is not finite: no totality of an unlimited number of objects can be made available for inspection, either in fact or 'in principle'. It is, however, permissible to interpret any such statement as being about each constructed object. Thus, that all numbers divisible by four are divisible by two means that if one constructs an object divisible by four, this object will have the property of being divisible by two. Clearly this assertion does not imply that the class of all numbers divisible by four is actually and completely available.

"Thirdly, a truly existential proposition--to the effect,e.g., that there exists a stroke-expression with a certain property--is equally not finite: we cannot go throughallstroke-expression (of a certain kind) to find one which has the property in question. But we may regard an existential proposition as an incomplete statement to be supplemented by an indication of either of a concrete object which possesses the property or of the constructuve process yielding such an object. ...

"Fourthly, the law of excluded middle is not universally valid. ...

"Even in elementary arithmetic there is occasion for using, in a restricted way, transfinite methods, in particular the principle of excluded middle. ... " The science of formal systems. pp.78-80.

Aristotle's arguments for rejecting the notion of actual infinity have a central idea behind them: "that a method for a step-by-step procedure,

i.e.for making the next step if the preceding step has been taken, does not imply that there is a last step, either in thought or in fact.""The rejection of the notion of actual infinity is regarded as being of little important for the mathematician who, so Aristotle holds, needs only that of potential infinity for the purposes of mathematical demonstration. ... It might possibly be argued that Aristotle admits the possibility of the consistent use of actually infinite sets in a purely mathematical system which is not applicable to the physical universe." Some Older Views, p.21.

Kant's analysis of infinity "reminds one in many ways of Aristotle's doctrine, except that in Kant the distinction between actual and potential infinity is worked out still more clearly." Some Older Views, pp.29-30.

"Kant does

notregard the notion of actual infinity as logically impossible. It is what he calls an Idea of reason, that is to say an internally consistent notion which is, however, inapplicable to sense-experience since instances of it can be neither perceived nor constructed. Kant's view is that ... we can neither perceive nor construct an actually infinite aggregate." Some Older Views, p.30."This transition from the notion of potential, constructive infinity to the notion of actual, non-constructive infinity is in Kant's view the main root of confusion in metaphysics. Whether it is required, desirable, or objectionable or indifferent

within mathematicsis a question which divides the contemporary schools of philosophy of mathematics perhaps more radically than any other problem." Some Older Views, p.31."In [Frege's famous definition of 'number'] the role placed by the familiar notion of parallel lines is played by the less familiar notion of 'similar' concepts. Two sets of objects are similar if, and only if, a one-to-one correspondence can be established between their members. ... Every concept determines, as we have seen, a set of things, namely the set of things falling under it--the extension of the concept. If the sets of things falling under two concepts ... are similar we shall say that the concepts themselves are similar."

Frege says we may "define the number of a concept, saya, as the range of values (extension) of the concept 'ξis a concept similar toa'." Mathematics as Logic, p.38."Now, some of the most important propositional functions in pure mathematics, such as '

xis an integer' or 'xis an irrational number', are of ranges which are not finite. They must be regarded as infinite, at least potentially. The philosophical mathematicians who adopt the Russellian programme regard the ranges of both 'xis an integer' and 'xis an irrational number' as actually infinite, and they regard the latter range as being in a clearly definable sense greater than the former. ...

"Postulates providing the infinite ranges covered by the logicist equivalent of the phrases 'for every integer ...' or 'for every real number ...' are susceptible to various interpretations. They may be regarded as merely technical devices, admissible so long as they are demonstrably innocent of leading to contradictions. This is essentially the view of Hilbert and his school. They may be regarded, again as inadmissible because misrepresenting the nature of mathematics. This is essentially the view of Brouwer and his disciples. They may be regarded, lastly, as empirical assumptions about the world. ... To say that there is an infinite number of individuals in the world would according to Russell be to make an empirical statement which may be true or false but which in Principia Mathematica is assumed to be true." Mathematics as Logic, pp.48-49."The logistic account of the sequence of natural numbers involves the assumption of actual infinities. But although logicism, following Cantor, employs this notion most liberally by developing a mathematics of infinities of various sizes and various internal structures, its mathematical theory is not backed by any philosophical theory or analysis." Mathematics as Logic: Criticism, p.53.

"[Frege and Russell] are not ... compelled to hold that the similarity or lack of it between two classes,

i.e.the presence or absence of a one-to-one correspondence between their members, can be established in every case. But they do assume it to be true for any two classes; either that they are or are not similar, even if there is no possible way of fining out. The nature of this assumption is, at best, obscure and in need of justification." Mathematics as Logic: Criticism, p.59.The historical importance of the 'naive' transfinite mathematics, created by Cantor and incorporated almost wholly into

Principia Mathematica, can hardly be overestimated."For proving the consistency of a system, two methods are available: the direct and the indirect. In some cases it can be shown by combinatorial means that inconsistent statements are not deducible in a given theory. In other cases, the direct method proceeds by exhibiting a perceptual model of the theory. More precisely it consists (i) in identifying the objects of the theory with concrete objects, (ii) in identifying the postulates with exact descriptions of these objects and their mutual relations, and (iii) in showing that an inference within the system will not lead to any other than exact descriptions. Since mathematics abounds in concepts of actual infinities which cannot be identified with perceptual objects, the use of the direct method is restricted to certain small parts of mathematics.

"A theory involving actual infinities can--at leastprima facie--be tested for consistency only by the indirect method. One proceeds in this by establishing a one-to-one correspondence between (a) the postulates and the theorems of the original theory and (b) all or some of the postulates and theorems of a second theory, which is assumed to be consistent. ... But none of these theories can have a concrete model.

"Amongst indirect proofs of the consistency of any geometical or physical theory the most common are based on arithmetization,i.e.on representing the objects of these theories by real numbers or systems of such. ... But the reducibility to arithmetic of physical and mathematical theories which contain ideal notions, and which cannot be proved consistent by the direct method, raises the question of the consistency of arithmetic itself." The Science of Formal Systems, p.75."We now turn to non-elementary arithmetic. The subject-matter of this arithmetical

theoryis, of course, no longer finite. But may be possible to construct an arithmeticalformalism--with statement-formulae and theorem-formulae corresponding ... to statements and theorems of the theory; and this formalism could then be the subject-matter of a metatheory. Since the subject-matter, namely formula-construction, would be finite, the metatheory would be just as finite as elementary arithmetic, from which it would differ only by being about a different kind of perceptual construction. If a formalism corresponding, in the required manner, to the theory of non-elementary arithmetic can be constructed, then we can again, by demonstratingformalconsistency of the formalism,eo ipsoestablish logical consistency of the theory. ... Our next task, therefore, must be to consider the formula-constructing activities, or formalism--both formalisms considered by themselves and formalisms which are at the same time formalizations of theories." The Science of Formal Systems, p.84.

Considered there to be an absolute, unchanging reality separate from the world of appearances.

"Precision, timelessness and--in some sense--independence of their being apprehended is certainly, for Plato, characteristic of mathematical statements, and the view that numbers, geometric entities and the relations between them have an objective, or at least an intersubjective, existence is plausible." Some Older Views, p.15.

"Plato certainly held that there are mind-independent, definite, eternal objects which we call 'one', 'two', 'three', etc., the arithmetic Forms." Some Older Views, pp.15-16.

Aristotle gave a detailed formulation of the problem of mathematical infinity. "[Aristotle] was the first of many to see the two main ways of analyzing the notion of infinity as actual and as merely potential; and he was the first who made a clear decision in favor of the second alternative." Some Older Views, p.20.

"Leibniz's more radical [than Aristotle] logical position, that the predicate of every proposition is 'contained in' the subject, is paralleled, on his side, by the famous metaphysical doctrine that the world consists of self-contained subjects--substances or monads which do not interact." Some Older Views, p.21.

"Kant does not agree with the view of pure mathematics which would make it a matter of definitions and of postulated entities falling under them. To him pure mathematics is not analytic; it is synthetic

a priori, since it is about (describes) space and time." Some Older Views, p.28.

"Cantor, the founder of the general theory of classes or sets, defined a set (

Menge) as 'a collection of definite, well-distinguished objects of our perception or thought--the elements of the set--into a whole' [Cantor1895]. ...

"One of the most important and fruitful events in the history of mathematical logic and the philosophy of mathematics was the discovery that Cantor's logic of classes, by admitting as a class any collection, however formed, leads to contradictions. Their occurrence ... makes it necessary to distinguish between admissible and inadmissible classes ... . The region of thought where such distinctions are imperative is a little like some notorious swamp which no one could drain, and which, consequently, it was imperative to bridge, by whatever artificial means available." Mathematics as logic, pp.44-45.

"Frege's explanation of the analytic character of arithmetic presupposes that the general laws of logic, which he lists and uses as premisses, are such that would generally and at once be recognized.

"These laws are propositions which he simply enumerates. He does not characterize them by any common feature, such as all analytic propositions might be presumed to possess though not always being immediately seen to possess it. ...

"The path from the listed initial propositions, by inferential steps, to the theorems of arithmetic can be expected to be long, especially if every step is to be open to thorough inspection. ... Frege and his followers adopted and extended the symbolic representation of deductive reasoning used by mathematicians. ... The extension consists on the one hand of not only symbolizing the notions used in the traditional branches of mathematics but those used in all deductive reasoning; and on the other hand of formulating explicitly the permissible rules of inference." Mathematics as Logic, pp.33-34.

"According to Russell, definition is a purely notational device. ... Yet, it is maintained that in at least two ways, definitions, theoretically superfluous though they are, often convey most important information. They imply 'that the

definiensis worthy of careful consideration' and further that 'when what is defined is (as often occurs) something already familiar, such as cardinals or other ordinal numbers, the definition contains an analysis of a common idea, and may therefore express a notable advance'.[Russell1925:p.11ff]" Mathematics as Logic, pp.34-35.

"[Quine] claims that the notions of arithmetic can be defined in purely logical terms; that 'the notions of identity, relation, number, function, sum, product, power, limit, derivative, etc. are all definable in terms of our three notational devices: membership, joint denial, and quantification with its variables' [Quine1981: p.125ff] Definition here may be both explicit or contextual, and does not imply the existence, in any sense, of objects falling under the defined concepts." Mathematics as Logic, p.50.

"According to Curry, who is quite explicit on this question, we must distinguish between the

truthof a formula within a formal system--i.e. the statement that is derivable within the system--and theacceptabilityof the system as a whole. The former is 'an objective matter about which we can all agree; while the latter may involve extraneous considerations'. Thus he holds that 'the acceptability of classical analysis for the purposes of application in physics is ... established on pragmatic grounds and neither the question of intuitive evidence nor that of a consistency proof has any bearing on this matter. The primary criterion of acceptability is empirical; and the most important considerations are adequacy and simplicity'. ... The domain of formal theories and the propositions about their formal properties are, Curry holds, cleary demarcated." The Science of Formal Systems, p.87.

"According to Brouwer the principles of classical logic are linguistic rules in that those who 'linguistically follow' them may but need not 'be guided by experience'. This means that the rules of classical logic are employed in description and communication but not in the activity itself of constructing; as they are not employed, except as inessential aids, in the activity of mountain climbing. Mathematics is essentially independent, in this sense, not only of language but also of logic." Intuitive Constructions, p.122.

- [Cantor1895]
- Cantor, Georg. Beiträge zur Begründung der transfiniten
Mengenlehre I. Mathematische Annalen (1895). Translated
and reprinted in [Cantor1915].

- [Quine1981]
- Quine, Willard Van Orman.
*Mathematical Logic*. revised edition. Harvard University Press (Cambridge, MA: 1940, 1951, 1979, 1981). ISBN 0-674-55451-5 pbk. - [Russell1925]
- Whitehead, Alfred North., Russell, Bertrand. Principia Mathematica, vol. 1. ed.2. Cambridge University Press. (Cambridge: 1927). See [Whitehead1997].

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