Abstract
The basic tenets of intuitionism are the rejection of the law of excluded middle and the view that a judgement is correct if it is knowable, indicating a reversal in priority between the objective and the subjective. Intuitionism revived the age-old problem of universals, and the controversy between nominalism, conceptualism, and realism, now represented by formalism (nominalism), intuitionism (conceptualism), and set-theoretical Platonism (realism). In the old controversy, moderate realism, i.e., the Aristotelic-Thomistic school, came out on top, with its simultaneous rejection of conceptualism and exaggerated realism, on the grounds that the former leads to subjectivism, and the latter is epistemologically untenable. This paper takes a similar stance in the modern foundational debate: set-theoretical Platonism, is rejected on epistemological grounds, and pure conceptualism is rejected on the grounds that if fails to account for the objective nature of mathematics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Cf., Diogenes, Lives of Eminent Philosophers, Ch. 7, in particular n. 32, sqq.; and Cicero, De Fato, n. 1.
- 2.
From oratio to ratio, to use two common Latin translations of the word λόγος.
- 3.
Cf., Aristotle, :̧def:̧def Perih., Ch. 1
- 4.
To aid the understanding of the first part of this quotation, it should be added that Cajetan’s interpretation of Aristotle’s point of view is that words are signs of concepts and that concepts are signs of things (cf., ibid., Ch. 1, 16a4).
- 5.
Cajetan, In:̧def:̧def Praed., Ch. 1
- 6.
Wittgenstein, Tractatus, § 7: “Wovon man nicht sprechen kann, darüber muss man schweigen.” The exact contrapositive of Cajetan’s point is that, if there are no words for a thought, then it cannot be communicated. This is tantamount to Wittgenstein’s dictum.
- 7.
Note that ‘On Concept and object’ is Geach’s translation of the title of Frege’s article ‘Über Begriff und Gegenstand’. Cf., Maritain, The Degrees of Knowledge, Ch. 3, § 10, § 24; De Morgan, Formal Logic, Ch. 2; Husserl, Log. Unt. II, Pt. 1, Inv. 1, § 33.
- 8.
Another example, due to Descartes, is that it may happen that we remember something that somebody told us, without remembering in which language it was spoken (‘The World or Treatise on Light’, Ch. 1, n. 4).
- 9.
Author’s translation of a table from the Editors’ preface of Aquinas, ‘In:̧def:̧def Perih.’, p. ix. Simple apprehension is the scholastic term used, e.g., by Gredt, Elem. Phil. n. 6. Perception is a modern equivalent used, e.g., by Locke, An Essay Concerning Humane Understanding, Bk. 2, Ch. 9. Cf., Arnauld, The art of thinking, p. 29, Kant, Kritik der reinen Vernunft, and Bolzano, Wissenschaftslehre.
- 10.
Cf., Bocheński, Ancient Formal Logic.
- 11.
Since the definition of logic is a controversial matter, we have avoided it completely. Cf., Husserl, Log. Unt. I, § 3; Mill, A System of Logic, § 1; and Gredt, Elem. Phil. n. 4.
- 12.
The syntactic-semantic method of logic is associated with Martin-Löf. It should be noted that the words formal and syntactic also have modern senses, originating with Hilbert and Carnap respectively, according to which only that is formal or syntactic which treats of words without regard to meaning or content.
- 13.
In ‘The theory of algorithms’, nn. 5–7, p. 2, Markov makes the same distinction between what he calls elementary signs and the corresponding forms, which are called abstract elementary signs.
- 14.
This principle is commonly attributed to Frege, even though it was not explicitly formulated by him.
- 15.
Cf., Cocchiarella, ‘Conceptual Realism as a Formal Ontology’.
- 16.
Husserl, Log. Unt. II, Pt. 1, Inv. 1, § 13 (Author’s translation). Cf., the parallel place in Aquinas, ‘Summa:̧def:̧def Theol.’, Pt. 1, q. 13, a. 1: “voces referuntur ad res significandas, mediante conceptione intellectus”: words refer to things signified, through the intellect’s concept (Author’s translation).
- 17.
The Greek word ὅρος became terminus in Latin.
- 18.
Aristotle, An. :̧def:̧def Pr., Bk. 1, Ch. 1, 24b17. Cf., Boëthius, ‘De syllogismo categorico’.
- 19.
This use of the word term can be motivated as follows: the ancients speak about the three terms of a syllogism; equivocation is the fallacy of using an equivocal middle term in a syllogism, as in the argument “no light is dark; all feathers are light; therefore, no feathers are dark” (by Celarent); “an utterance is not called equivocal because it signifies many external things but because in signifying those many external things, there correspond to it different concepts in the soul.” (Buridan, Summulae de Dialectica, Treatise 3, Ch. 1, § 2).
Fig. 1.2 
Meaning, referent, and term added to the triangle picturing the threefold correspondence between object, concept, and expression
- 20.
Cf., Gredt, Elem. Phil. n. 16a.
- 21.
Cf., Martin-Löf, Intuitionistic Type Theory, p. 7.
- 22.
To use the jargon of mathematical category theory, the diagram presented in Fig. 1.3 is commutative, i.e., following any chain of arrows from one point to another gives the same result.
- 23.
Maritain, The Degrees of Knowledge, Ch. 3, § 24.
- 24.
Ibid., App. 1, § 4, p. 419. That is, communication does not only consist in an exchange of words, but also of their meanings.
- 25.
Poinsot, Material Logic, p. 421. Cf., Husserl, Log. Unt. II, Pt. 1, Inv. 1, § 34; and Gredt, Elem. Phil. n. 16c.
- 26.
Cf., Aristotle, :̧def:̧def Metaph., Bk. 7, Ch. 1, for the various senses of the word being.
- 27.
Strictly speaking, things which are actual are also called possible (ab actu ad posse valet illatio) so real being and possible being amount to the same; but, when real being is divided into actual and possible, possible has to be taken to exclude actual.
- 28.
A parte rei: on the side of things.
- 29.
Aquinas, In Metaph., Bk. 4, Les. 1, n. 12: “quam dicimus in ratione esse, quia ratio de eis negociatur quasi de quibusdam entibus, dum de eis affirmat vel negat aliquid” (trans. Rowan).
- 30.
After Maritain, The Degrees of Knowledge, Ch. 2, fn. 43.
- 31.
Cf., Husserl, Log. Unt. II, Pt. 1, Inv. 1, § 32: “Ideality in the ordinary, normative sense does not exclude reality” (trans. Findlay).
- 32.
Cf., Maritain, The Degrees of Knowledge, Ch. 2, § 33, p. 144
- 33.
Aquinas, ‘De ente et essentia’, Ch. 2: “Unde si quaeratur utrum ista natura sic considerata possit dici una vel plures, neutrum concedendum est, quia utrumque est extra intellectum humanitatis et utrumque potest sibi accidere. Si enim pluralitas esset de intellectu eius, nunquam posset esse una, cum tamen una sit secundum quod est in Socrate. Similiter si unitas esset de ratione eius, tunc esset una et eadem Socratis et Platonis nec posset in pluribus plurificari.” (Trans. Klima).
- 34.
Aquinas, ‘De Veritate’, q. 2, a. 3, arg. 19. Author’s translation: nothing is in the intellect that was not previously in the senses. Cf., Coffey, The Science of Logic, p. 7.
- 35.
Cf., e.g., Gredt, Elem. Phil. n. 114.
- 36.
Cf., Husserl, The Crisis of European Sciences and Transcendental Phenomenology, Pt. 2.
- 37.
Blancanus, ‘A Treatise on the Nature of Mathematics along with a Chronology of Outstanding Mathematicians’, pp. 179–180.
- 38.
Husserl, Log. Unt. II, Pt. 1, Inv. 1, § 35.
- 39.
Cf., Gredt, Elem. Phil. n. 7.
- 40.
Cf., Maritain, The Degrees of Knowledge, Ch. 4, § 6, pp. 152–154.
- 41.
Husserl, Log. Unt. II, Pt. 2, Inv. 6, § 12. Cf., ibid., Pt. 1, Inv. 1, § 15.
- 42.
Maritain calls absurd beings of reason the “thieves and forgers” among beings of reason (The Degrees of Knowledge, Ch. 2, § 33, p. 143).
- 43.
Cf., Bernays, ‘Mathematische Existenz und Widerspruchsfreiheit’.
- 44.
Cf., Maddy, ‘Mathematical existence’.
- 45.
Hilbert, ‘On the infinite’, p. 370. Cf., what von Neumann reportedly said to a colleague who didn’t understand the method of characteristics: “Young man, in mathematics you don’t understand things. You just get used to them.”
- 46.
Cf., Becker, ‘Mathematische Existenz’.
- 47.
This view was expressed by Kronecker in the famous sentence “Die ganze Zahl schuf der liebe Gott, alles übrige ist Menschenwerk” (Cajori, A History of Mathematics, p. 362).
- 48.
This notion was introduced by Brouwer. Kant is the likely source of his terminology: “Philosophical knowledge is the knowledge gained by reason from concepts; mathematical knowledge is the knowledge gained by reason from the construction of concepts.” Kritik der reinen Vernunft, Pt. 2.1.1, p. 469 (B 741) (trans. N. K. Smith).
- 49.
Frege, Grundgesetze der Arithmetik II, § 105 (trans. Black).
- 50.
Weyl, ‘Comments on Hilbert’s second lecture on the foundations of mathematics’, p. 484.
- 51.
Bourbaki, Elements of Mathematics, p. 336.
- 52.
Simpson, ‘Logic and mathematics’, § 3.2.
- 53.
Skolem, ‘Some remarks on axiomatized set theory’, pp. 300–301.
- 54.
Bishop et al., Constructive Analysis, Ch. 1, p. 7.
- 55.
Sentence is that which was λόγος in Greek and became oratio in Latin.
- 56.
We prefer the word correct to the word true to avoid confusion with true propositions, discussed later.
- 57.
Aristotle, :̧def:̧def Metaph., Bk. 6, Ch. 4, § 2. Cf., Moore, ‘The nature of judgement’, p. 179.
- 58.
Cf., Martin-Löf, ‘On the meanings of the logical constants and the justifications of the logical laws’, p. 24 and ibid., p. 19.
- 59.
Aristotle, :̧def:̧def Metaph., Bk. 4, Ch. 7, § 1.
- 60.
Author’s translation of “veritas est adaequatio rei et intellectus”, Aquinas, ‘Summa:̧def:̧def Theol.’, Pt. 1, q. 16, a. 2.
- 61.
One concept is conceptually prior to another if the definition of the latter involves the former, and one thing is ontologically prior to another if the latter cannot be conceived as existing without the former existing also.
- 62.
Martin-Löf, ‘A Path from Logic to Metaphysics’; Sundholm, ‘Inference versus Consequence’. Cf., also Aristotle, An. :̧def:̧def Pr., Bk. 1, Ch. 1; An. Post., Bk. 1, Ch. 10; Top., Bk. 1, Ch. 1.
- 63.
Cf., Aristotle, An. :̧def:̧def Post., Bk. 1, Ch. 2; Aquinas, ‘In An. :̧def:̧def Post.’, Bk. 1, Lect. 5; Poinsot, Material Logic, p. 461.
- 64.
Ibid., p. 462.
- 65.
This last explanation of what constitutes an immediate inference is due to Sundholm, ‘Inference versus Consequence’, p. 35. As an aside, in contrast to Whitehead and Russell, we do not think that an axiom can be accepted on purely practical grounds (cf., Principia Mathematica, Intro., Ch. 2, § 7, p. 62). The argument that “things have been taught to be self-evident and have yet turned out to be false” (ibid.) has little force, since, clearly, they were not self-evident after all: errare humanum est.
- 66.
Cf., Aquinas, ‘Summa:̧def:̧def Theol.’, Pt. 1, q. 2, a. 1.
- 67.
Cf., Aristotle, An. :̧def:̧def Pr., Ch. 1.
- 68.
Cf., Poinsot, Material Logic, p. 462.
- 69.
Subsequently we will prefer the word assertion to the word judgement. This choice differs from that of Martin-Löf, ‘On the meanings of the logical constants and the justifications of the logical laws’, who chooses judgement as the primary word, but it agrees with that of Russell, e.g., ‘The Theory of Implication’, § 1.1.
- 70.
Geach, Logic Matters, p. 255. But, as pointed out by Klima, the Frege point was recognised long before Frege, for example, by Buridan, in Summulae de Dialectica, Treatise 5, Ch. 1, § 3, p. 308: “a syllogism has an additional feature in comparison to a conditional in that a syllogism posits the premises assertively, whereas a conditional does not assert them.”
- 71.
These connectives are called conjunction, disjunction, implication, and falsum (or absurdum) respectively. The word connective applies strictly speaking only to the first three, since they connect A and B, but the meaning of the word is often extended to include falsum too (as well as negation and equivalence, see below). The symbol & is a ligature for the Latin word et meaning and; the symbol ∨ is just a stylised abbreviation of the Latin word vel meaning or; the symbol \(\supset \) is due to Peano (Arithmetices Principia Nova Methodo Exposita, Log. Not., n. 2), in fact, \(\subset \) is a stylised C abbreviating is a consequence of, so \(B \subset A\) means that B is a consequence of A, or, equivalently, that A implies B; finally, the symbol \(\Lambda \) for falsum is due to Peano (ibid.), and it is a V for verum turned upside down.
- 72.
Aristotle, :̧def:̧def Perih., Ch. 1 (cf., ibid., Ch. 4, 17a2).
- 73.
Boole, ‘The Calculus of Logic’. Cf., Martin-Löf, ‘On the meanings of the logical constants and the justifications of the logical laws’, p. 14.
- 74.
Cf., Sundholm, ‘Inference versus Consequence’, p. 26.
- 75.
I.e., the jumping into a different domain or science. The phrase is derived from Aristotle, An. :̧def:̧def Post., Bk. 1, Ch. 7, 75a38, which is concerned with the impossibility of proving facts in one science using the methods of another, e.g., to prove a geometrical fact by appeal to optics.
- 76.
Husserl, Log. Unt. II, Pt. 2, Inv. 6, § 30.
- 77.
Cf., Geach, ‘The law of excluded middle’, pp. 71–73.
- 78.
Martin-Löf, Intuitionistic Type Theory, p. 11, cf., Brouwer, ‘The Unreliability of the Logical Principles’.
- 79.
These definitions are copies of Martin-Löf’s definitions (Intuitionistic Type Theory, p. 11) with the word proof replaced by the word cause.
- 80.
To know is to have cognizance of the thing through causes. This dictum is derived from Aristotle, An. :̧def:̧def Post., Bk. 1, Ch. 2, 71b9, sqq. Cf., Metaph., Bk. 2, Ch. 1, n. 5, sqq. Other formulations are the poetic “Felix, qui potuit rerum cognoscere causas” (Virgil, Georgics, Bk. 2, l. 490) and “Vere scire, esse per causas scire” (Bacon, Novum Organum, Bk. 2, Ch. 20). With respect to the division of causes (Aristotle, :̧def:̧def Metaph., Bk. 5, Ch. 2; Phys., Bk. 2, Ch. 3), the kind of cause we have in mind here could be called a logical cause (cf., An. Post., Bk. 2, Ch. 11).
- 81.
Author’s translation of Leibniz, ‘Principia Philosophiæ’, § 32: “vi cujus consideramus, nullum factum reperiri posse verum, aut veram existere aliquam enunciationem, nisi adsit ratio sufficiens, cur potius ita sit quam aliter, quamvis rationes istæ sæpissime nobis incognitæ esse queant.”
- 82.
It is difficult to determine to what extent Leibniz identified ratio with causa; cf., Di Bella, ‘Causa Sive Ratio’.
- 83.
In this setting, it also makes sense to call the cause or reason a truth-maker, since, in a sense, it is the cause that makes the proposition true. Cf., Sundholm, ‘Existence, Proof, and Truth-Making: A Perspective on the Intuitionistic Conception of Truth’.
- 84.
Martin-Löf, Intuitionistic Type Theory, p. 12. This interpretation is called the BHK interpretation after its discoverers Brouwer (in many of his works), Heyting (‘Sur la logique intuitionniste’), and, independently, Kolmogorov (‘Zur Deutung der intuitionistischen Logik’). It should be mentioned that there is direct line of thought from Husserl to the BHK interpretation: Becker, one of Husserl’s students, interpreted propositions as expectations (‘Mathematische Existenz’), and influenced Heyting who interpreted propositions as problems (cf., Mancosu, From Brouwer to Hilbert, pp. 275–285). This leads to the identification of: (1) the cause of a proposition, (2) the fulfillment of an expectation, and (3) the solution of a problem.
- 85.
The symbol \(\sim \) for negation is due to Russell (‘Mathematical Logic as Based on the Theory of Types’, § 6).
- 86.
But, cf., Bishop et al., Constructive Analysis, pp. 10–11.
- 87.
The symbol \(\supset \!\subset \) for equivalence is due to Heyting (‘Die formalen Regeln der intuitionistischen Logik’, § 2).
- 88.
An incomplete assertion, e.g., A true, constitutes an incomplete communication (unvollständige Mitteilung) in that the speaker suppresses certain information (cf., Hilbert and Bernays, Grundlagen der Mathematik, p. 33; and Kleene, ‘On the Interpretation of Intuitionistic Number Theory’, § 1). Also, what we call an incomplete assertion was called a judgement abstract (Urteilsabstrakt) by Weyl (‘Über die neue Grundlagenkrise der Mathematik’, p. 54).
- 89.
This important step of bringing the causes into the language of logic, i.e., of naming them, was first taken by Martin-Löf, ‘An intuitionistic theory of types’, p. 77, under the guise of proof objects. Cf., Martin-Löf, ‘Analytic and synthetic judgements in type theory’, where the distinction between the complete assertion \(c:\mathop{\phantom{()} \mathrm{cause}}(A)\) and the incomplete assertion A true is related to the Kantian distinction between analytic and synthetic judgements.
- 90.
Peano, Arithmetices Principia Nova Methodo Exposita, § 1, with the difference that, as is now customary, the first number is zero instead of one. It is more natural to start the number series in the sense of Peano at zero since, if starting at one, there are two different formalizations of the unit, the starting point one, and the \(\mathop{\phantom{()}\mathrm{s}}\) for the successor.
- 91.
With mention of the causes, the definition of a < b becomes: there is a cause of \(a <\mathop{\phantom{()} \mathrm{s}}(a)\), and if there is a cause of a < b, then there is a cause of \(a <\mathop{\phantom{()} \mathrm{s}}(b)\).
- 92.
Geach, ‘Identity’, p. 3. Cf., Quine’s dictum: “no entity without identity” (Theories and Things, p. 102).
- 93.
See Klev, ‘Categories and Logical Syntax’ for a comprehensive analysis of the development of the notion of category from Aristotle to Kant, and beyond.
- 94.
Martin-Löf, ‘About models for intuitionistic type theories and the notion of definitional equality’, p. 93.
- 95.
We use the standard equality sign = for definitional equality. This sign was introduced by Recorde, The Whetstone of Witte, in 1557: “And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicause noe 2 thynges, can be moare equalle.” (there are no page numbers in this work, but the quoted passage stands under the heading “The rule of equation, commonly called Algebers Rule” which occurs about three quarters into the work). This use of the equality sign seems to me most natural since we use it when we make abbreviatory definitions in mathematics. Thus we had to use another sign for propositional equality. In the type-theoretic literature, there are several suggestions, including ‘I’ (Martin-Löf, Intuitionistic Type Theory, p. 59), and ‘Id’ vs. ‘Eq’ (with a slight difference in meaning, Nordström et al., Programming in Martin-Löf’s Type Theory, Ch. 8). According to Cajori (‘Mathematical Signs of Equality’, p. 116), the most popular notation, both before Recorde and in competition with him, was to write equality in words, i.e., something like “æquales”, “égale”, “gleich”, or the abbreviation “æq”.
- 96.
Thus, properly speaking, this is not an axiom, but a theorem, of intuitionistic type theory.
- 97.
The first detailed analysis of the notion of presupposition was given by Duns Scotus, ‘De rerum principio’.
- 98.
These two questions are the type-theoretic equivalents of what Quine calls soundness and completeness for a system of logic (‘A proof procedure for quantification theory’, p. 145).
- 99.
Gödel, ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’. More specifically, fixing a collection of forms of expression and their corresponding inference rules, containing the expressions and rules of arithmetic, there are arithmetic propositions which cannot be demonstrated using only inference rules from this collection, but which are demonstrable, and hence correct, using valid inference rules outside of the collection.
- 100.
Cf., Martin-Löf, ‘On the meanings of the logical constants and the justifications of the logical laws’, p. 37. Note that, unless we fix our inference rules, the answer to (2) cannot be no. To answer no we have to know an assertion to be correct but not demonstrable, but the only way to come to know that an assertion is correct is though a demonstration.
- 101.
Modus tollendo ponens: mood which by denying affirms.
- 102.
We have chosen to take the inference rule modus ponendo ponens as meaning determining for implication. In doing so we are faithful to the natural formulation of the BHK interpretation of \(A \supset B\), namely that a cause of \(A \supset B\) consists of a method taking a cause of A into a cause of B. Another interpretation which, prima facie, seems equivalent but which, in fact, is not, is that a cause of \(A \supset B\) consists of a cause of B provided that a cause of A is given: this is the interpretation given by Kolmogorov, ‘Zur Deutung der intuitionistischen Logik’, p. 59, with the only difference that his interpretation is formulated in terms of problems and solutions instead of in terms of propositions and causes.
- 103.
Cf., Aristotle, An. :̧def:̧def Pr., Bk. 1, Ch. 1; An. Post., Bk. 1, Ch. 2; Gentzen, ‘Untersuchungen über das logische Schließen I & II’; and Sundholm, ‘Inference versus Consequence’.
- 104.
Cf., Boëthius, ‘De hypotheticis syllogismis’.
- 105.
Carroll, ‘What the Tortoise said to Achilles’.
- 106.
As demonstrated by Gentzen, ‘Untersuchungen über das logische Schließen I & II’. Cf., Granström, Treatise on Intuitionistic Type Theory, Ch. II, § 7.
- 107.
This was first demonstrated by Brouwer, ‘Intuitionistische Zerlegung mathematischer Grundbegriffe’, p. 253.
- 108.
Cf., Aristotle, An. :̧def:̧def Pr., Bk. 2, Ch. 14.
- 109.
Kant, Kritik der reinen Vernunft, Pt. 2.1.4, p. 513 (B 817) (trans. N. K. Smith).
- 110.
For a comprehensive treatment of this topic, the reader is referred to the first two chapters of Mancosu’s book Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century.
- 111.
Aristotle, An. :̧def:̧def Post., Bk. 1, Ch. 13; and ibid., Bk. 2, Ch. 1.
- 112.
Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, p. 12.
- 113.
Ibid., p. 13.
- 114.
Ibid., p. 17.
- 115.
Ibid., p. 25.
- 116.
Nardi, quoted in ibid., p. 63.
- 117.
As discussed in Moore’s 1941 Howison lecture ‘Certainty’ and Wittgenstein’s book On Certainty.
- 118.
This terminology was suggested by Sundholm (personal communication).
- 119.
Heyting, Intuitionism: An Introduction, p. 18.
- 120.
Cf., ibid., Th. 1, p. 17.
- 121.
Sebestik, ‘Bolzano’s Logic’.
- 122.
Kolmogorov, ‘On the principle of excluded middle’. Cf., Glivenko, ‘Sur quelques points de la logique de M. Brouwer’; Gödel, ‘Zur intuitionistische Arithmetik und Zahlentheorie’; and Gentzen, ‘Die Widerspruchfreiheit der reinen Zahlentheorie’. A direct (as opposed to metamathematical) demonstration of the logical laws for irrefutable propositions is given by the Author in Treatise on Intuitionistic Type Theory, Ch. VI, § 1.
- 123.
Also called proof per contradictionem or per impossibile, reductio ad absurdum or ad impossibile.
- 124.
Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, p. 25.
- 125.
Though not explicitly stated in this form, this insight is due to Brouwer, ‘The Unreliability of the Logical Principles’, p. 110.
- 126.
Whitehead et al., Principia Mathematica, Intro., Ch. 2, § 1, p. 40.
- 127.
Cf., Mancosu, From Brouwer to Hilbert, pp. 278–280.
- 128.
Church, ‘On the law of excluded middle’, p. 77.
- 129.
According to which a proposition is interpreted as a truth value, i.e., as an element of the set {true, false}; the truth of a proposition A is interpreted as A being equal to ‘true’; and negation and disjunction have their usual Boolean definitions.
- 130.
Empiricus, Outlines of Pyrrhonism, Bk. 1, Ch. 1.
- 131.
Cf., Hilbert, ‘Mathematical problems’, p. 445.
- 132.
Cf., Church, ‘An unsolvable problem of elementary number theory’ and Turing, ‘On Computable Numbers’.
- 133.
The use of the word oracle in this connection was introduced by Turing, ‘Systems of logic based on ordinals’, § 4, p. 172.
- 134.
Cf., Martin-Löf, ‘Verificationism Then and Now’, Third Law, p. 16.
- 135.
Cicero, De Fato, Ch. 1.
- 136.
Ibid., Ch. 10, beginning.
- 137.
Ibid., Ch. 10, n. 21.
- 138.
This is the most natural interpretation of the principle of bivalence, since the assertion A true can be expanded into I know a logical cause of A, in which the now is implicit.
- 139.
Aristotle, :̧def:̧def Perih., Ch. 9, 19a37–19b5.
- 140.
Cicero, De Fato, Ch. 16, nn. 37–38. I have changed the translation to conform with standard terminology in logic by replacing the word contrary with the word contradictory and removing Cicero’s comment giving an explanation of his unusual sense of the word contrary.
- 141.
Cf., Geach, ‘The law of excluded middle’, pp. 71–73.
- 142.
Kneale et al., The Development of Logic, p. 114.
References
Aquinas, S.T.: Summa Theologiae. In: Opera omnia. Iussu impensaque Leonis XIII P. M. edita, vols. 4–12. Ex typographia polyglotta S. congregationis de propaganda fide, Rome (1888–1906)
Aquinas, S.T., Cathala, M.-R., Spiazzi, R.M. (eds.): In duodecim libros Metaphysicorum Aristotelis expositio. Marietti, Turin (1950)
Aquinas, S.T.: De Veritate. In: Spiazzi, R. (ed.) Quaestiones Disputatae, vol. 1. Marietti, Turin (1953)
Aquinas, S.T.: In Peri Hermeneias. In: In Aristotelis Libros Peri Hermeneias et Posteriorum Analyticorum Expositio, 2nd edn. Marietti, Turin (1964)
Aquinas, S.T.: In Posteriorum Analyticorum. In: In Aristotelis Libros Peri Hermeneias et Posteriorum Analyticorum Expositio, 2nd edn. Marietti, Turin (1964)
Aquinas, S.T.: De ente et essentia. In: Opera omnia. Iussu impensaque Leonis XIII P. M. edita, vol. 43, pp. 315–381. Editori di San Tommaso, Rome (1976)
Aristotle: Metaphysics (Trans. by H. Tredennick). Loeb Classical Library, vol. 271/287. Harvard University Press, Cambridge (1933/1935)
Aristotle: Analytica priora (Trans. by H. Tredennick). Loeb Classical Library, vol. 325, pp. 181–531. Harvard University Press, Cambridge (1938)
Aristotle: Perihermenias (Trans. by H.P. Cooke). Loeb Classical Library, vol. 325, pp. 111–179. Harvard University Press, Cambridge (1938)
Aristotle: Analytica posteriora (Trans. by H. Tredennick). Loeb Classical Library, vol. 391, pp. 1–261. Harvard University Press, Cambridge (1960)
Aristotle: Topica (Trans. by E.S. Forster). Loeb Classical Library, vol. 391, pp. 263–739. Harvard University Press, Cambridge (1960)
Aristotle: The Physics (Trans. by P.H. Wicksteed and F.M. Cornford). Loeb Classical Library, vols. 228 and 255. Harvard University Press, Cambridge (1986/2006)
Arnauld, A.: The Art of Thinking (Trans. by J. Dickoff and P. James). Bobbs-Merrill, New York (1964)
Bacon, F.: Novum Organum. J Billius, London (1620)
Becker, O.: Mathematische Existenz. Zur Logik und Ontologie mathematischer Phänomene. Jahrb. für Philos. und phänomenologische Forsch. 8, 441–809 (1927)
Bernays, P.: Mathematische Existenz und Widerspruchsfreiheit. Etudes de Philosophie des sciences en hommage á Ferdinand Gonseth, pp. 11–25. Griffon, Neuchatel (1950)
Bishop, E., Bridges, D.: Constructive Analysis. Springer, Berlin (1985)
Blancanus, J.: A treatise on the nature of mathematics along with a chronology of outstanding mathematicians. In: Mancosu, P. (ed.) Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Trans. by G. Klima), pp. 178–212. Oxford University Press, Oxford (1996)
Bocheński, I.M.: Ancient Formal Logic. North-Holland, Amsterdam (1951)
Boëthius, S.: De syllogismo categorico. In: Migne, J.-P. (ed.) Patrologiae cursus completus. Series Latina, vol. 64, pp. 793C–832A. Garnier, Paris (1847)
Boëthius, S.: De hypotheticis syllogismis. In: Obertello, L. (ed.) Logicalia, vol. 1. Brescia, Parma (1968)
Bolzano, B.: Wissenschaftslehre. Seidel, Sulzbach (1837)
Boole, G.: The calculus of logic. Camb. Dublin Math. J. 3, 183–198 (1848)
Bourbaki, N.: Elements of Mathematics. Theory of Sets. Addison-Wesley, Boston (1968)
Brouwer, L.E.J.: Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresber. d. Deutsch. Math. Vereinig. 33, 251–256 (1925)
Brouwer, L.E.J.: The unreliability of the logical principles. In: Heyting, A. (ed.) Collected Works. Philosophy and Foundations of Mathematics, vol. 1, pp. 107–111. North Holland, Amsterdam (1975) (original from 1908)
Buridan, J.: Summulae de Dialectica (Trans. by G. Klima). Yale University Press, New Haven (2001)
Cajetan, T.: Commentaria in Praedicamenta Aristotelis. Angelicum, Rome (1939)
Cajori, F.: A History of Mathematics. Macmillan, New York (1919)
Cajori, F.: Mathematical signs of equality. Isis 5(1), 116–125 (1923)
Carroll, L.: What the Tortoise said to Achilles. Mind 4(14), 278–280 (1895)
Church, A.: On the law of excluded middle. Bull. Am. Math. Soc. 34, 75–78 (1928)
Church, A.: An unsolvable problem of elementary number theory. Am. J. Math. 58(2), 345–363 (1936)
Cicero, M.T.: De Fato (Trans. by H. Rackham). Loeb Classical Library. Harvard University Press, Cambridge (1942)
Cocchiarella, N.B.: Conceptual realism as a formal ontology. In: Poli, R., Simons, P. (eds.) Formal Ontology, pp. 27–60. Kluwer, Dordrecht (1996)
Coffey, P.: The Science of Logic, 2nd edn., vol. 1. Peter Smith, New York (1938)
De Morgan, A.: Formal Logic. Taylor and Walton, London (1847)
Descartes, R.: The world or treatise on light. In: The Philosophical Writings of Descartes (Trans. by R. Stoothoff). Cambridge University Press, Cambridge (1985)
Di Bella, S.: Causa Sive Ratio. Univocity of reason and plurality of causes in Leibniz. In: Dascal, M. (ed.) Leibniz: What Kind of Rationalist? Logic, Epistemology, and the Unity of Science, vol. 13. Springer, Dordrecht (2008). Chap. 32
Diogenes, L.: Lives of Eminent Philosophers (Trans. by R.D. Hicks). Loeb Classical Library, vols. 1/2. Harvard University Press, Cambridge (1925)
Empiricus, S.: Outlines of Pyrrhonism (Trans. by R.G. Bury). Loeb Classical Library, vol. 1. Harvard University Press, Cambridge (1961)
Frege, G.: Begriffsschrift. Louis Nebert, Halle (1879)
Frege, G.: Grundgesetze der Arithmetik II, vol. 2. Hermann Pohle, Jena (1903)
Frege, G.: On concept and object. In: Translations from the Philosophical Writings of Gottlob Frege (Trans. by P.T. Geach), pp. 42–55. B Blackwell, Oxford (1960)
Geach, P.T.: The law of excluded middle. Supp. Proc. Arist. Soc. 30, 59–90 (1956)
Geach, P.T.: Identity. Rev. Metaphys. 21(1), 3–12 (1967)
Geach, P.T.: Logic Matters. Blackwell, Oxford (1972)
Gentzen, G.: Untersuchungen über das logische Schließen I & II. Math. Zeit. 39, 176–210 & 405–431 (1935)
Gentzen, G.: Die Widerspruchfreiheit der reinen Zahlentheorie. Math. Ann. 112, 493–565 (1936)
Glivenko, V.: Sur quelques points de la logique de M. Brouwer. Acad. R. de Belg. Bull. 15, 183–188 (1929)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Math. und Phys. 38, 173–198 (1931)
Gödel, K.: Zur intuitionistische Arithmetik und Zahlentheorie. Ergeb. eines math. Kolloqu. 4, 34–38 (1933)
Granström, J.G.: Treatise on Intuitionistic Type Theory. Logic, Epistemology and the Unity of Science. Springer, Dordrecht (2011)
Gredt, I.: Elementa Philosophiae Aristotelico-Thomisticae, 4th edn., vol. 1. Herder, Freiburg (1925)
Heyting, A.: Die formalen Regeln der intuitionistischen Logik. Sitzungsberiche der Preussischen Akademie der Wissenschaften, 42–56 (1930)
Heyting, A.: Sur la logique intuitionniste. Acad. R. de Belg. Bull. 16, 957–963 (1930)
Heyting, A.: Intuitionism: An Introduction. North-Holland, Amsterdam (1956)
Hilbert, D.: Mathematical problems (Trans. by M.W. Newson). Bull. Am. Math. Soc. 8(10), 437–479 (1902)
Hilbert, D.: On the infinite. In: van Heijenoort, J. (ed.) From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931, pp. 367–392. Harvard University Press, Cambridge (1967)
Hilbert, D., Bernays, P.: Grundlagen der Mathematik, vol. 1. Springer, Berlin (1934)
Husserl, E.: Logische Untersuchungen, 3rd edn., vol. 1. M Niemeyer, Halle (1922)
Husserl, E.: Logische Untersuchungen, 5th edn., vol. 2. M Niemeyer, Tübingen (1968)
Husserl, E.: The Crisis of European Sciences and Transcendental Phenomenology (Trans. by D. Carr). Northwestern University Press, Evanston (1970)
Kant, I.: Kritik der reinen Vernunft. Kants Werke, vol. 3. de Gruyter, Berlin (1968)
Kleene, S.C.: On the interpretation of intuitionistic number theory. J. Symb. Log. 10(4), 109–124 (1945)
Klev, A.: Categories and logical syntax. PhD thesis, Leiden University (2014)
Kneale, W., Kneale, M.: The Development of Logic. Oxford University Press, Oxford (1962)
Kolmogorov, A.: Zur Deutung der intuitionistischen Logik. Math. Zeit. 35, 58–65 (1932)
Kolmogorov, A.: On the principle of excluded middle. In: From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931 (Original in Russian from 1925), pp. 416–437. Harvard University Press, Cambridge (1967)
Leibniz, G.W.: Principia Philosophiæ. Petrus Conrad Monath, Frankfurt (1728)
Locke, J.: An Essay Concerning Humane Understanding, 2nd edn. Thomas Basset, London (1694)
Maddy, P.: Mathematical existence. Bull. Symb. Log. 11(3), 351–376 (2005)
Mancosu, P.: Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford University Press, Oxford (1996)
Mancosu, P. (ed.): From Brouwer to Hilbert. Oxford University Press, New York (1998)
Maritain, J.: The Degrees of Knowledge (Trans. by R. McInerny). University of Notre Dame Press, Notre Dame (1995)
Markov, A.A.: The theory of algorithms. Am. Math. Soc. Transl. Ser. 2 15, 1–14 (1960)
Martin-Löf, P.: An intuitionistic theory of types: predicative part. In: Rose, H.E., Shepherdson, J. (eds.) Logic Colloquium ’73, pp. 73–118. North-Holland, Amsterdam (1975)
Martin-Löf, P.: About models for intuitionistic type theories and the notion of definitional equality. In: Proceedings of the Third Scandinavian Logic Symposium. Studies in Logic and the Foundation of Mathematics, vol. 82, pp. 81–109. North-Holland, Amsterdam (1975)
Martin-Löf, P.: Intuitionistic Type Theory. Studies in Proof Theory. Bibliopolis, Napoli (1984)
Martin-Löf, P.: A path from logic to metaphysics. In: Atti del Congresso Nuovi Problemi della Logica e della Filosofia Scienza, Viareggio, 1990, vol. 2, pp. 141–149. Clueb, Bologna (1991)
Martin-Löf, P.: Analytic and synthetic judgements in type theory. In: Parrini, P. (ed.) Kant and Contemporary Epistemology, pp. 87–99. Kluwer (1994)
Martin-Löf, P.: Verificationism then and now. In: De Pauli-Schimanovich, W., Köhler, E., Stadler, F. (eds.) The Foundational Debate; Complexity and Constructivity in Mathematics and Physics, pp. 187–196. Kluwer (1995)
Martin-Löf, P.: On the meanings of the logical constants and the justifications of the logical laws. Nord. J. Philos. Log. 1(1), 11–60 (1996)
Mill, J.S.: A System of Logic. Harper & Brothers, New York (1846)
Moore, G.E.: The nature of judgement. Mind 8(30), 176–193 (1899)
Moore, G.E.: Certainty. In: Philosophical Papers. Allen and Unwin, London (1959)
Nordström, B., Petersson, K., Smith, J.M.: Programming in Martin-Löf’s Type Theory. Oxford University Press (1990)
Peano, G.: Arithmetices Principia Nova Methodo Exposita. Fratelli Bocca, Turin (1889)
Poinsot, J.: The Material Logic of John of St. Thomas. Basic Treatises (Trans. by Y.R. Simon, J.J. Glanville, and G.D. Hollenhorst). The University of Chicago Press, Chicago (1955)
Quine, W.V.: A proof procedure for quantification theory. J. Symb. Log. 20(2), 141–149 (1955)
Quine, W.V.: Theories and Things. Harvard University Press, Cambridge (1981)
Recorde, R.: The Whetstone of Witte. Da Capo Press, Amsterdam (1969)
Russell, B.: The theory of implication. Am. J. Math. 28(2), 159–202 (1906)
Russell, B.: Mathematical logic as based on the theory of types. Am. J. Math. 30(3), 222–262 (1908)
Scotus, J.D.: De rerum principio. In: Hibernicus, M. (ed.) Quaestiones Subtilissime Scoti in Meta-physicam Aristotelis. Eiusdem de Primo Rerum Principio Tractatus Atque Theoremata. Bonetus Locatellus, Venice (1497)
Sebestik, J.: Bolzano’s logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Stanford (2007)
Simpson, S.G.: Logic and mathematics. In: Rosen, S. (ed.) The Examined Life, Readings from Western Philosophy from Plato to Kant, pp. 577–605. Random House, New York (2000)
Skolem, T.A.: Some remarks on axiomatized set theory. In: van Heijenoort, J. (ed.) From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931, pp. 290–301. Harvard University Press, Cambridge (1967)
Sundholm, G.: Existence, proof, and truth-making: a perspective on the intuitionistic conception of truth. Topoi 13, 117–126 (1994)
Sundholm, G.: Inference versus consequence. In: Childers, T. (ed.) The LOGICA Yearbook 1997, pp. 26–35. Filosofia, Prague (1998)
Turing, A.M.: On computable numbers, with an application to the entscheidungsproblem. Proc. Lond. Math. Soc. 2(42), 230–265 (1936)
Turing, A.M.: Systems of logic based on ordinals. Proc. Lond. Math. Soc. Ser. 2 45(2239), 161–228 (1939)
Virgil: Georgics (Trans. by H.R. Fairclough), 2nd edn. Loeb Classical Library. Harvard University Press, Cambridge (1999)
Weyl, H.: Über die neue Grundlagenkrise der Mathematik. Math. Zeit. 10, 39–79 (1921)
Weyl, H.: Comments on Hilbert’s second lecture on the foundations of mathematics. In: van Heijenoort, J. (ed.) From Frege to Gödel. A Source Book in Mathematical Logic, 1879–1931, pp. 480–484. Harvard University Press, Cambridge (1967)
Whitehead, A.N., Russell, B.: Principia Mathematica, vol. 1. Cambridge University Press, Cambridge (1910)
Wittgenstein, L.: Tractatus Logico-Philosophicus (Trans. by C.K. Ogden and F.P. Ramsey). Routledge & Kegan Paul, London (1922)
Wittgenstein, L., Anscombe, G.E.M., von Wright, G.H.: On Certainty (Trans. by D. Paul and G.E.M. Anscombe). Basil Blackwell, Oxford (1969)
Acknowledgements
I would like to express my great appreciation to Prof. Shahid Rahman for suggesting and encouraging the composition of this summary of the Author’s book Treatise on Intuitionistic Type Theory, and to Springer for permission to reuse parts of the book. The book version of this paper was written under supervision of Prof. Per Martin-Löf, and I would like to offer him my special thanks for his signficant time investment and his constant precision of thought and expression.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Granström, J.G. (2016). Perennial Intuitionism. In: Redmond, J., Pombo Martins, O., Nepomuceno Fernández, Á. (eds) Epistemology, Knowledge and the Impact of Interaction. Logic, Epistemology, and the Unity of Science, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-26506-3_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-26506-3_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-26504-9
Online ISBN: 978-3-319-26506-3
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)