Construction and Constitution in Mathematics

van Atten, Mark

doi:10.1007/978-94-024-1132-4_12

Mark van Atten³

Part of the book series: Synthese Library ((SYLI,volume 384))

594 Accesses
6 Citations

Abstract

I argue that Brouwer’s notion of the construction of purely mathematical objects and Husserl’s notion of their constitution by the transcendental subject coincide. Various objections to Brouwer’s intuitionism that have been raised in recent phenomenological literature (by Hill, Rosado Haddock, and Tieszen) are addressed. Then I present objections to Gödel’s project of founding classical mathematics on transcendental phenomenology. The problem for that project lies not so much in Husserl’s insistence on the spontaneous character of the constitution of mathematical objects, or in his refusal to allow an appeal to higher minds, as in the combination of these two attitudes.

Originally published as Mark van Atten. 2010. “Construction and constitution in mathematics”. The New Yearbook for Phenomenology and Phenomenological Philosophy 10:43–90. Copyright ©2010 Acumen Publishing. Reprinted by permission, which is gratefully acknowledged. Corrections and the occasional additional note in this reprint (marked as such) are those of the reprint in van Atten (2015a).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic

$34.99 /Month

Get 10 units per month
Download Article/Chapter or eBook
1 Unit = 1 Article or 1 Chapter
Cancel anytime

Buy Now

Chapter: USD 29.95; Price excludes VAT (USA)

eBook: USD 129.00; Price excludes VAT (USA)

Softcover Book: USD 169.99; Price excludes VAT (USA)

Hardcover Book: USD 169.99; Price excludes VAT (USA)

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Construction and Constitution in Mathematics

Reconciling the Realist/Anti Realist Dichotomy in the Philosophy of Mathematics

Constitution and Construction

Notes

1.
‘F¨ur den Intuitionisten ist diese Erkenntnis [daß die Arithmetik nicht aus der formalen Logik herzuleiten ist] nicht verwunderlich, sondern nur eine Folge seiner Auffassung, daß die reine Mathematik ¨ uberhaupt auf eine Reihe von intuitiv zu erkennenden Akten und kategorialen Gegenst¨andlichkeiten ber¨uht, wie Kolligieren, Ordnen, Zuordnen, Vertauschen usw., die sogar eine Ausdehnung ins Endlose gestatten.’ (Becker 1927, 23; see also 69 and 196)
2.
In Sect. 31 of Formal and Transcendental Logic, Husserl describes the ideal of the definite manifold, and attempts to characterise it precisely by saying that such a manifold indeed is described by a syntactically complete axiom system. The actual claim that pure mathematics is a definite manifold is implied in a text from 1920–1921: ‘The ideal total extension of the purely rational objects and the extension of the objective objects [objektive Gegenst¨ande] cognizable to every possible rational I, coincide. It turns out that this extension is a definite manifold, and infinitely constructible system’ (Husserl 1974, 388). But Gödel’s first Incompleteness Theorem, established shortly after Formal and Transcendental Logic was published, shows that in any consistent theory containing arithmetic there are undecidable sentences. A fortiori, the same holds for set theory and the whole of pure mathematics. See also Lohmar (1989, 197n27). Clearly, for Gödel, this fact was not an essential obstacle to embracing Husserl’s transcendental-phenomenological approach to the foundations of mathematics, and, more generally, in Gödel’s foundational project there hardly seems to have been a role for Husserl’s theory of formal systems; given Gödel’s realist conception of mathematical objects, this is not surprising. Note that Dietrich Mahnke had raised the question whether pure mathematics is one definite manifold in his Neue Monadologie of 1917, and conjectured the answer is yes, while realising that an argument is needed (Husserl marked this passage in his own copy): ‘That not all, indeed even rather few, manifolds in the actual world have this property of being definite, is obvious. But also in formal mathematics it is still a big question whether its totality is a heap of infinitely many different and unrelated theories of manifolds, or rather can be organised into one big, definite system. The concept of mathematics seems to demand that the latter is the case. Yet a proof is still to be found’ (Mahnke 1917, 32). ‘Dass nicht alle, ja sogar nur herzlich wenige Mannigfaltigkeiten der wirklichen Welt diese Eigenschaft der Definitheit haben, liegt auf der Hand. Aber auch in der formalen Mathematik ist noch eine grosse Frage, ob ihre Gesamtheit ein beziehungsloses Nebeneinander von unendlich vielen verschiedenen Mannigfaltigkeitslehren ist oder vielmehr selbst in ein einziges, grosses definites System geordnet werden kann. Der Begriff der Mathematik scheint zu erfordern, dass das letztere der Fall ist. Doch steht der Nachweis daf¨ur noch aus.’
3.
See also Husserl (1974, 82), Husserl (1975, 245, 247), Husserl (1976a, 26–27), Husserl (1979, 166), a text from 1903, Husserl (1985a, 52, 55, 61, 78, 167), and Husserl (2002b, 266).
4.
‘Die gesamte mathesis universalis ist also Analytik m¨oglicher Kategorialien, Theorie ihrer Wesensformen und Wesensgesetze.’ (Husserl 1976a, 143)
5.
‘Die formale Ontologie [kann] auch von vornherein direkt als Aufgabe gestellt werden, ohne von der Idee einer Wissenschaftslehre auszugehen. Ihre Frage ist dann: was kann man innerhalb der Leerregion Gegenstand-¨uberhaupt aussagen? Rein apriori in dieser formalen Allgemeinheit stehen die syntaktischen Gestaltungen zur Verf¨ugung, durch die aus irgendwelchen als m¨oglich vorgegeben gedachten Gegenst¨anden (Etwas-¨uberhaupt) immer neue Kategorialien erzeugt gedacht werden k¨onnen. Man wird dabei auch auf den Unterschied m¨oglicher Erzeugungen kommen, die bloß deutliche Meinungen liefern, aber als widerspruchsvolle nicht zu m¨oglichen Gegenst¨anden selbst f¨uhren k¨onnen usw. Offenbar erw¨achst dann die ganze formale Mathesis.’ (Husserl 1974, 153–154, emphasis mine)
6.
‘Ein Kreuz ist f¨ur mich das Verh¨altnis von Mathematik und Ontologie als formaler Ontologie. Nachdem ich aber die Idee des ‘sachhaltigen’, der ‘Realit¨at’ in der Bedeutungssph¨are untergebracht habe, so gewinne ich ja die Idee des Nicht-Sachhaltigen, und das ist das Ont < ologische > im formalen Sinn’ (Husserl 1995a, 343). Note the contrast with Alexander Pfänder, who, working in the period 1928–1935, saw mathematics not as part of formal ontology, but as a material region (Pfänder 1973, 42). In his lectures on ‘first philosophy’, Husserl speaks of both the ‘formal region’ and the ‘formal-ontological quasi-region’; (Husserl 1956a, 187n1).
7.
Husserl’s description of the mathematics of non-contradiction as ‘the mathematics of the mathematicians’ (die Mathematik der Mathematiker, Husserl 1974, 146) is problematic. On the one hand, it is true that only a mathematician will engage in the study of formal systems for their own sake; on the other hand, after criticism by Henri Poincaré and Brouwer, David Hilbert acknowledged in 1922 that study requires accepting at least part of pure mathematics as contentual (Hilbert 1922, 165, 174). (Husserl knew this paper; see Lohmar 1989, 216n10.) Roughly, that part is arithmetic including the principle of induction; for elaborate discussion see, e.g., Kreisel (1965), Tait (1981), and Parsons (1998). In other words, pure mathematics cannot be limited to the logic of consequence. James Dodd (in an article that I in many respects agree with) speaks of ‘Brouwer’s polemical accusation, invited one must say by Hilbert himself, that finitism amounts to a mere empty game with symbols devoid of all sense, and which in the end has no genuine connection to mathematical objectivity at all’ (Dodd 2007, 268). (The reference given in Dodd’s footnote to this sentence is not, as one would have expected, to a passage where Brouwer makes the alleged accusation but to Husserl 1974, Sects. 33–34.) But Brouwer never did, and never would, equate finitism with empty symbolism devoid of all sense: finitism’s characteristic acceptance of the mathematics of natural numbers as contentual is precisely due to Brouwer’s influence on Hilbert. (For a discussion of this influence, with full references, see Brouwer (1928A2). Husserl owned a copy of the latter; see van Atten 2007, 128n7).
8.
In 1908 Brouwer both showed the consistency of the principle of the excluded middle (as its double negation is true, the principle itself cannot be false) and justified his doubts about its truth. ‘Consequently the theorems which are usually considered as proved in mathematics, ought to be divided into those that are true and those that are non-contradictory’ (Brouwer 1975, 110n2) (‘Men behoort dus in de wiskunde de gewoonlijk als bewezen geldende stellingen te onderscheiden in juiste en niet-contradictoire’, Brouwer 1908C, 158n2). Brouwer characterises the difference between a true proposition and a merely non-contradictory one by the presence, in the former case, of a mathematical construction that proposition adequately describes. This corresponds to Husserl’s characterisation of truth-logic as the one that, unlike consequence-logic is concerned with the existence of the objects. Note that, before Gödel proved his Incompleteness Theorem, Brouwer was quite optimistic about Hilbert’s program to establish the consistency of classical mathematics; in his first Vienna lecture of 1928, he said that ‘An appropriate mechanisation of the language of this intuitionistically non-contradictory mathematics should therefore deliver exactly what the formalist school has set as its goal’ (Brouwer 1929A, 164). (‘Eine geeignete Mechanisierung der Sprache dieser intuitionistisch-nichtkontradiktorischen Mathematik m¨ußte also gerade das liefern, was die formalistische Schule sich zum Ziel gesetzt hat.’) Of course, he at the same time insisted that would have no value for mathematics as such, given its concern, in his view, with constructions.
9.
‘alle kl¨arende und im Rahmen der axiomatischen Klarheit vollzogene ontologische Einsicht, die nicht direkt ph¨anomenologisch ist, wird dazu durch eine bloße Blickwendung, wie umgekehrt im All der ph¨anomenologischen Einsichten solche auftreten m¨ussen, die durch bloße Blickwendung zu ontologischen werden.’ (Husserl 1952, 105) Husserl also comments on the phenomenological clarification of geometry by tracing it back to its constitution in nexuses of consciousness: ‘That is an application of phenomenology, not phenomenology itself. The predicate ‘phenomenological’ carries over, of course, to the applications’ ‘Das ist Anwendung der Ph¨anomenologie, nicht Ph¨anomenologie selbst. Das Pr¨adikat “ph¨anomenologisch” ¨ ubertr¨agt sich nat¨urlich auf die Anwendungen’, Husserl (1952, 83). As we will see in the main text, later Husserl expressed an even stronger view.
10.
See also Husserl (1988, 18–19), and Husserl (2002c, 300–301).
11.
‘gegen¨uber der nur scheinbar universalen Ontologie in der Positivit¨at die wahrhaft universale’ (Husserl 1962, 297).
12.
‘F¨ur die apriorischen Disziplinen, die innerhalb der Ph¨anomenologie zur Begr¨undung kommen (z.B. als mathematische Wissenschaften) [kann es] keine “Paradoxien”, keine “Grundlagenkrisen” geben.’ (Husserl 1962, 297)
13.
Noted added in this reprint: Even though it does not affect the content of my argument, here I committed a howler. All the lines I quote from Hill’s article here are actually translations of sentences written by Husserl. Hill presents these without quotation marks or indentation, but at the end of her paragraph there is a reference to the corresponding pages in Husserliana XXIV (411, 422, 423; and 425 must be added). I thank Dr. Hill for bringing this to my attention, and apologise for this mistaken attribution to her. The reason my argument is not affected is that the sentences Hill quotes come from a group of texts of 1908, and what I document here is precisely how, as Husserl developed his transcendental phenomenology over the following years, he came to oppose the view that a priori ontology is not part of transcendental phenomenology.
14.
‘hinsichtlich aller mir geltenden Objektivit¨at’ (Husserl 1959, 445).
15.
In a text from 1923, Husserl writes of the formal Apriori and the purely formal objects (Husserl 1959, 225n): ‘But here the matter is quite different from that of a geometrical, nature-ontological etc. Apriori; here, on the ontological side the specifications and correlations that have an influence on the sense are lacking, only the general constitutive relation remains’ (‘Aber hier ist die Sache doch anders als bei einem geometrischen, naturontologischen usw. Apriori; hier fehlen auf ontologischer Seite die sinn-mit-bestimmenden Besonderungen und Korrelationen, es bleibt nur die allgemeine konstitutive Korrelation’).
16.
‘Die idealen Bedingungen der M¨oglichkeit kategorialer Anschauung ¨ uberhaupt sind korrelativ die Bedingungen der M¨oglichkeit der Gegenst¨ande kategorialer Anschauung und der M¨oglichkeit von kategorialen Gegenst¨anden schlechthin.’ (Husserl 1984b, 718–719)
17.
This point I have argued for in greater detail in van Atten (2002).
18.
‘¨Uber allen Wissenschaften steht eine Mathesis universalissima, und nicht als eine naive Mathematik, die noch weit ¨ uber die Leibniz’sche Mathesis universalis hinaus das formalontologische Apriori systematisch geordnet konstruiert und in Theorien entfaltet, sondern als eine Mathematik von Erkenntnisleistungen, deren noetisches und in der reinen Subjektivit¨at vollzogenes Studium das Mathematische als noematisches Gebilde der Vernunft und somit als Bewußtseinskorrelat begreift.’ (Husserl 1959, 249)
19.
For further discussion of the relation between phenomenology and mathematics, see Gödel (*1961/?) and Yoshimi (2007).
20.
For recent introductions to intuitionism and its history, see Troelstra and van Dalen (1988), Hesseling (2003), and van Atten (2004b). Also the following articles in the on-line Stanford Encyclopedia of Philosophy: van Atten (2008), Iemhoff (2009), Moschovakis (2008), van Atten (2009b), and Bridges (2009).
21.
Dodd writes that ‘Brouwer’s revolution [has] run aground on the insuperable technical difficulties in re-establishing classical analysis on exclusively intuitionist principles’ (Dodd 2007, 300–301). For an intuitionist, the impossibility to re-establish classical analysis is of no particular philosophical importance. Moreover, a number of theorems of intuitionistic analysis formally contradict classical analysis. That fact also shows that, while previously established theorems of classical analysis may of course serve as a heuristic in the search for new intuitionistic theorems, this is only so up to a point.
22.
On p. 14 of Eley (1969), there is an interesting reference to ‘H.-R. Brennecke, Untersuchungen zur Genealogie der Menge. Das Vorpr¨adikative und Pr¨adikative in der Begr¨undung der Mengenlehre bei E. Husserl und L.E.J. Brouwer (K¨oln 1968).’ Unfortunately, it turns out that this was a projected dissertation by an assistant at the Husserl Archive in Cologne that was never finished; moreover, at the Archive, not even fragments of it are known. I thank Matthias Wille, Dirk Fonfara, and Dieter Lohmar (all in Cologne) for their clarification of this matter.
23.
‘in het intellect …niet gevoelde, doch waargenomen wiskundige volgreeksen’ (Brouwer 1907, 81).
24.
Husserl (1962, 25–26), Husserl (1974, 86, 87, 176, 267, 270, 323), and Husserl (1985b, 301). Also, e.g., Husserl (1950a, 87–88), Husserl (1966, 291), and Husserl (2003b, 147).
25.
‘Was das Ich hier in seinem Tun erzeugt, sind eben nur die Darstellungen von ihm, nicht aber der Gegenstand selbst.’ (Husserl 1985b, 301)
26.
‘Hingegen wird im spontanen Erzeugen der Sachverhalt selbst erzeugt und nicht eine Darstellung von ihm.’ (Husserl 1985b, 302)
27.
‘Daß auch der [Wahrnemungs-Gegenstand selbst unter transzendentalem Gesichtspunkt Produkt der Konstitution ist, kann im Rahmen dieser Kontrastierung, die einen ontischen Unterschied betrifft, außer Betracht bleiben.’ (Husserl 1985b, 301)
28.
‘Ich bilde meine Zahlvorstellungen …aber …Zahlvorstellung ist nicht Zahl selbst.’ (Husserl 1976a, 48–49)
29.
‘So ist auch im spontanen Abstrahieren nicht das Wesen, sondern das Bewußtsein von ihm ein erzeugtes.’ (Husserl 1976a, 50)
30.
E.g., Tieszen (1989, 178–179), Tieszen (1992, 189)l, and Tieszen (2010).
31.
‘Es gibt reale und ideale Erzeugnisse. Eine Erzeugung eines “Kentauren” in der Phantasie ist Erzeugung eines Idealen und nicht Erzeugung des psychischen Aktes (der verwechselt wird mit seinem noematischen “Gegenstand”). Ebenso ist Wesen ein ideales Erzeugnis.’ (Husserl 1976b, 482–483)
32.
‘Jede ideale Gegenst¨andlichkeit ist, was sie ist, nur als actus verus, als immerw¨ahrende Potenz, die ich jederzeit aktualisieren k¨onnte une eventuell jetzt wirklich aktualisiere.’ (Husserl 2002c, 280)
33.
‘wie die Subjektivit¨at in sich selbst rein aus Quellen ihrer Spontaneit¨at Gebilde schaffen kann, die als ideale Objekte einer idealen “Welt” gelten k¨onnen’ (Husserl 1974, 267, emphasis mine).
34.
Note also that if the transcendental Husserl had been a Platonist in any sense that is not compatible with constructive mathematics – there is a compatible sense: one might, perhaps somewhat redundantly, hold that the objects that exist independently correspond exactly to what we can mentally construct – one would not have expected him to be as reserved about the validity of the principle of the excluded middle in truth-logic as he is in Formal and Transcendental Logic (Husserl 1974, Sects. 77, 79).
35.
‘Eben dieser Umstand, daß ideale Gegenst¨ande uns gegebenenfalls als subjektive Gebilde im bildenden Erleben und Tun entgegentreten, war die Quelle der damals fast allgemeinen Psychologisierung der idealen Gegenst¨ande gewesen. Wenn nun auch evident gemacht wurde, daß ideale Gegenst¨ande, trotzdem sie zur Bildung im Bewußtsein kommen, ihr eigenes Sein, Ansich-sein haben, so bestand hier doch eine große und nie ernstlich gesehen und in Angriff genommene Aufgabe: n¨amlich die, diese eigent¨umliche Korrelation zwischen idealen Gegenst¨anden der rein logischen Sph¨are und subjektiv psychischem Erleben als bildendem Tun zum Forschungthema zu machen.’ (Husserl 1962, 25–26) See also Husserl (1974, 177–178).
36.
‘Ja wir sehen uns sogar gedr¨angt zu sagen: erzeugt werden die Zahlen in Z¨ahlen, erzeugt werden die Urteilss¨atze im urteilenden Tun. …‘Wiederhole’ ich ein gleiches Erzeugen, ¨ ube ich noch einmal gleiche z¨ahlende, pr¨adizierende, schließende Aktionen, so sind sie zwar seelisch ein neues Faktum, aber ich kann evident erkennen, daß, was da geworden ist, identisch dieselbe reine Zahl, identisch dieselbe Wahrheit usw. sei.’ (Husserl 1962, 25–26).
37.
‘[Ideale Gegenst¨andlichkeiten] sind, was sie sind, nur “aus” urspr¨unglicher Erzeugung. Das sagt aber keineswegs, sie sind, was sie sind, nur in und w¨ahrend der urspr¨unglichen Erzeugung. Sind sie “in” der urspr¨unglichen Erzeugung, so sagt das, sie sind in ihr als einer gewissen Intentionalit¨at von der Form spontaner Aktivit¨at bewußt, und zwar im Modus des originalen selbst. Diese Gegebenheitsweise aus solcher urspr¨unglichen Aktivit¨at ist nichts ander als die ihr eigene Art der “Wahrnehmung”.’ (Husserl 1974, 176)
38.
‘Die Irrealit¨at des Satzes als Idee einer synthetischen Werdenseinheit ist Idee von Etwas, das an jeder Zeitstelle auftreten kann, an jeder notwendig zeitlich und zeitlich werdend auftritt, und doch “allzeit” dasselbe ist.’ (Husserl 1985b, 311)
39.
‘Geistige Sch¨opfungen sind die Zahlen, sofern sie Resultate von T¨atigkeiten bilden, die wir an konkreten Inhalten ¨ uben; aber was diese T¨atigkeiten schaffen, das sind nicht absolute Inhalte, die wir irgendwo in der ‘Aussenwelt’ wiederfinden k¨onnen, sondern es sind eigent¨umliche Relationsbegriffe, die immer wieder nur erzeugt, aber keineswegs irgendwo fertig vorgefunden werden k¨onnen.’ (Husserl 1970, 317)
40.
‘bestaan in de wiskunde beteekent: intu¨ıtief zijn opgebouwd …De wiskunde is een vrije schepping, onafhankelijk van de ervaring.’ (Brouwer 1907, 177, 79)
41.
In the original, op te bouwen for opgebouwd; see van Dalen (2001a, 134n(g)).
42.
Aktivit¨at der Menschheit; der Wille zum Leben des einzelnen Menschen; das intellektuelle Urph¨anomen; passive Einstellung; Willesakt.
43.
In this context, it is interesting that Husserl decided to use the term ‘syntactic’ in two distinct senses: with respect to linguistic form, and with respect to the categorial form of an object: Husserl (1976a, Sect. 11), Husserl (1974, Sect. 42b–d), Husserl (1974, Beilage I), and Husserl (1985b, 247n).
44.
‘de formuletaal, die de wiskunde begeleidt als het notenschrift een symfonie van Bach of een oratorium van H¨andel’ (Brouwer et al. 1937, 262). For a reason unknown to me, the (partial) translation ‘Signific Dialogues’ (Brouwer 1975, 447–52), which as far as I know is not by Brouwer, substitutes ‘Formal language accompanies mathematics as the weather-map accompanies the atmospheric processes’ (Brouwer 1975, 451); that simile is not inappropriate, but the original is better, because it is a richer analogy.
45.
[[Note added in this reprint: Here I was mistaken. As is now evident to me, the translation in Brouwer (1975) was made from the publication of the ‘Signifische Dialogen’ in book form Brouwer et al. (1939), not from the prior publication in the journal Synthese (Brouwer et al. 1937); the bibliographical details as stated in Brouwer (1975), p. x, do not mention there are differences between the two. But already in the book, the musical image had been replaced by ‘…de formuletaal, die de wiskunde begeleidt als het weerkaartje het atmosferisch gebeuren’ – which the English quoted above renders correctly. In the preface to the book, Mannoury says that this edition is ‘more complete’, and he thanks the editorial board of Synthese, but he does not say that changes have been made, let alone by whom or why. Be that as it may, I apologise to the translators.]]
46.
‘kan ik mij wiskundige juistheid denken, die nooit in enig formulesysteem kan worden vastgelegd …Maar ook áls de intu¨ıtieve wiskunde en de formalistiek overeenstemmen, of juister uitgedrukt: parallel lopen, dan ligt het exakte in de intu¨ıtie, maar nooit in de formule.’ (Brouwer et al. 1937, 262–263)
47.
‘Diese gedanklichen, im allgemeinen unendlich viele Glieder aufweisenden mathematischen Beweisf¨uhrungen d¨urfen mit ihren endlichen, notwendigerweise inad¨aquaten, mithin nicht zur Mathematik geh¨orenden sprachlichen Begleitungen nicht verwechselt werden.’ (Brouwer 1927B, 64n8)
48.
‘mein Hauptargument gegen die Anspruche der Hilbertsche Metamathematik’.
49.
For a different view, see Tieszen (1995, 453). For extensive discussion of intersubjectivity and intuitionism, see Placek (1999) and van Atten (2004b, Chap. 6).
50.
‘Prinzipiell aber bildet die Mathematik und reine Logik ¨ uberhaupt …ein[en] Fonds reiner Verstandeswahrheiten …Sie gr¨unden vielmehr durchaus und rein in den bloßen ‘Denkformen’, in dem Wesen der formalen Bedeutungs- und Gegenstandsgedanken, die Schablonen gleichen, in die erst ein Stoff eingef¨ullt sein muß, damit sachhaltige Gedanken mit Beziehung auf sachhaltige Gegenst¨ande resultieren k¨onnen.’ (Husserl 1985a, 61)
51.
‘Er zijn elementen van wiskundige bouwing, die in het systeem der definities onherleidbaar moeten blijven, dus bij mededeeling door een enkel woord, klank of teeken, weerklank moeten vinden; het zijn de uit de oer-intu¨ıtie of continuumintu¨ıtie afgelezen bouwelementen; begrippen als continu, eenheid, nog eens, enzoovoort zijn onherleidbaar.’ (Brouwer 1907, 180)
52.
‘…dat dus in dezen opbouw, onder de verplichting, zorgvuldig acht te geven, wat de intu¨ıtie veroorlooft te stellen en wat niet, de eenig mogelijke grondvesting der wiskunde is te zoeken’ (Brouwer 1907, 77).
53.
This also provides an answer to the following objection to intuitionism, formulated by Lohmar (1989, 212): ‘Other doubts in turn are directed at the view that the objects of mathematics are produced in the mathematician’s acts. However understandable this view is as a counter-reaction to Platonism, and in spite of its pointing to the contribution of actions to the constitution of mathematical objects, it conceals the fact that cognition and itself-givenness of mathematical connections are founded on something which, in the activity that leads up to them, occurs passively. In mathematics, too, all we can do is to bring ourselves to the point where cognition either takes place or not.’ (‘Andere Bedenken richten sich wiederum auf die Ansicht, daß die Gegenst¨ande der Mathematik im Handeln des Mathematikers erzeugt werden. So verst¨andlich dies als Gegenreaktion zum Platonismus ist und so klarsichtig hiermit auf den Anteil an Handlungsaktivit¨at hingewiesen wird, der in der Konstitution mathematischer Gegenst¨andlichkeiten enthalten ist, so wird damit doch ¨ uberdeckt, daß Erkennen und Selbstgegebenheit mathematischer Zusammenh¨ange auf etwas beruht, das sich in der Aktivit¨at des Heranf¨uhrens passiv einstellt. Auch in der Mathematik gilt, daß wir nur an den Punkt heranf¨uhren k¨onnen an dem sich Erkennen einstellt oder nicht.’)
54.
‘Ist nun das Zeitbewusstsein die Urst¨atte der Konstitution von Identit¨atseinheit oder Gegenst¨andlichkeit, und dann der Verbindungsformen der Koexistenz und Sukzession aller bewusst werdenden Gegenst¨andlichkeiten, so ist es doch nur das eine allgemeine Form herstellende Bewusstsein. Bloße Form ist freilich eine Abstraktion, und so ist die intentionale Analyse des Zeitbewusstseins und seiner Leistung von vornherein eine abstraktive. Sie erfasst, interessiert sich nur f¨ur die notwendige Zeitform aller einzelnen Gegenst¨ande und Gegenstandsvielheiten, bzw. korrelativ f¨ur die Form der Zeitliches konstituierenden Mannigfaltigkeiten.’ (Husserl 1966, 128) Also Husserl (1966, 312); Husserl (1974, Beilage II 2c, 318), Husserl (1950a, 99).
55.
[[Note added in this reprint: For a detailed discussion of the role of inner time awareness in the constitution of sets, see van Atten (2015b).]]
56.
‘En daar deze samenvalt met de bewustwording van den tijd als verandering zonder meer, kunnen we ook zeggen:
Het eenige aprioristische element in de wetenschap is de tijd.’ (Brouwer 1907, 99)
57.
‘Natuurlijk wordt hier bedoeld de intu¨ıtieve tijd, wel te onderscheiden van de wetenschappelijke tijd, die, wel zeer a posteriori, eerst door de ervaring blijkt, als met een eenledige groep voorziene eendimensionale co¨ordinaat geschikt te kunnen ingevoerd tot het katalogizeeren der verschijnselen.’ (Brouwer 1907, 99n1)
58.
‘Die idealen Bedingungen der M¨oglichkeit kategorialer Anschauung ¨ uberhaupt sind korrelativ die Bedingungen der M¨oglichkeit der Gegenst¨ande kategorialer Anschauung und der M¨oglichkeit von kategorialen Gegenst¨anden schlechthin.’ (Husserl 1984b, 718–719)
59.
‘…dat de door de zelfontvouwing der oerintu¨ıtie ontstaande taallooze constructies, uit kracht van hun in de herinnering aanwezig zijn alleen, exact en juist zijn, dat echter het menschelijk herinneringsvermogen, dat deze constructies heeft te overzien, ook als het lingu¨ıstische teekens te hulp roept, uit den aard der zaak beperkt en feilbaar is’ (Brouwer 1933A2, 58).
60.
‘Wie kann eine Unendlichkeit erfahren sein? Nur so, dass ein endlicher Bestand jeweils direkt in die Erfahrung f¨allt und zugleich Tr¨ager ist einer Horizontpr¨asumtion, einer Verweisung auf einen subjektiv m¨oglichen Fortgang zu neuer Erfahrung usw.’ (Husserl 2001b, 107)
61.
Husserl’s historical claim here is not quite correct: e.g., Brouwer in his dissertation (1907) and Ludwig Wittgenstein in his Tractatus logico-philosophicus (1921) had thematised the notion ‘and so on’. Brouwer (Brouwer 1975, 80n): ‘The expression “and so on” means the indefinite repetition of one and the same object or operation, even if that object or that operation is defined in a rather complex way’ (‘Waar men zegt “en zoo voort”, bedoelt men het onbepaald herhalen van eenzelfde ding of operatie, ook al is dat ding of die operatie tamelijk complex gedefinieerd’, Brouwer 1907, 143n); Wittgenstein: ‘The concept of the successive application of an operation is equivalent to the concept “and so on”.’ (‘Der Begriff der successiven Anwendung der Operation ist ¨ aquivalent mit dem Begriff “und so weiter”.’, Wittgenstein ((1921) 2013, 52 (5.2523))
62.
‘Ich erinnere nur noch an die von den Logikern nie herausgehobene Grundform des “Und so weiter”, der iterativen “Unendlichkeit”, die ihr subjektives Korrelat hat im “man kann immer wieder”. …Die Mathematik ist das Reich unendlicher Konstruktionen, ein Reich von idealen Existenzen, nicht nur “endlicher” Sinne, sondern auch von konstruktiven Unendlichkeiten. Offenbar wiederholt sich hier das Problem der subjektiven konstitutiven Urspr¨unge als der verborgenen, zu enth¨ullenden und als Norm neu zu gestaltenden Methode der Konstruktionen, der Methode, in der das “und so weiter” verschiedenen Sinnes und die Unendlichkeiten als neuartige kategoriale Gebilde …evident werden.’ (Husserl 1974, 196)
63.
See also Husserl (1985b, 258–259).
64.
For Brouwer’s rejection of psychological interpretations of intuitionism in a letter to Van Dantzig from 1949, see van Atten (2004b, 75–76).
65.
‘De wiskunde is zeker geheel onafhankelijk van de materieele wereld.’ (Brouwer 1907, 177)
66.
See on this idealisation also Husserl (2001b, 200–201), Husserl (1975, 188–189), Husserl (1984b, Sect. 64), and Becker (1927, 285, 287, 292f, 304, 320f).
67.
unsterblich, die Zukunft unendliche Zeit bedeutet, ein ewiges Sein im Werden (Husserl 1966, 378, 379, 381).
68.
hypothetische mensen met onbeperkt herinneringsvermogen (Brouwer 1933A2, 59).
69.
het scheppend subject (Brouwer 1948A).
70.
‘Der Husserlsche Ansatz der Philosophie kennt eine thematisierende Subjektivit¨at, hingegen nicht eine konstruktive im Sinne der Intuitionisten.’ (Husserl 1970, xxvi) Note that Eley, in his Metakritik der formalen Logik, is sympathetic with Brouwer’s thought (Eley 1969, 14, 15, 64, 264n1, 329n2, 332).
71.
‘Admittedly, in Kant’s thought the categorial (logical) functions play an important role; but he does not arrive at the fundamental extension of the concepts of perception and intuition over the categorial realm’ (‘In Kants Denken spielen zwar die kategorialen (logischen) Functionen eine große Rolle; aber er gelangt nicht zu der fundamentalen Erweiterung der Begriffe Wahrnehmung und Anschauung ¨ uber das kategoriale Gebiet.’, Husserl 1984b, 732).
72.
A detailed discussion of this matter can be found in van Atten (2012).
73.
‘van de theorie van Kant de aprioriteit der ruimte prijs te geven, doch aan de aprioriteit van de tijd des te vastberadener vast te houden’ (Brouwer 1912A, 11).
74.
In 1929, in his second Vienna lecture, Brouwer expresses his ‘fundamental [im wesentlichen] agreement’ with Kant and Schopenhauer on the specific point of taking the continuum to be given in a priori intuition (Brouwer 1930A, 1, 6). That formulation does not imply agreement on the details.
75.
It is true that in a context of multiple subjects, a choice sequence is owned by the subject that creates it. But all that this subject knows about the sequence can be communicated by it to other subjects.
76.
That is the translation Tieszen uses. ‘Mathematische und sonstige irreale Gegenst¨ande “gibt es”, die noch niemand konstruiert hat. Ihr Dasein erweist freilich erst ihre Konstruktion (ihre “Erfahrung”), aber die Konstruktion der schon bekannten er¨offnet voraus einen Horizont weiter entdeckbarer, wenn auch noch unbekannter. Solange sie nicht entdeckt sind (von niemandem), sind sie nicht faktisch in der raum-Zeitlichkeit, und sofern es m¨oglich ist (dar¨uber wie weit dies m¨oglich ist, braucht nicht entschieden zu werden), daß sie nie entdeckt worden w¨aren, h¨atten sie ¨ uberhaupt keine Weltwirklichkeit. (Husserl 1985b, 312).
77.
‘waarvan niet anders dan een aftelbare groep welgedefinieerd is aan te geven, maar waar dan tevens dadelijk volgens een of ander vooraf gedefinieerd wiskundig proces uit elke zoodanige aftelbare groep nieuwe elementen zijn af te leiden, die gerekend worden eveneens tot de verzameling in kwestie te behooren’ (Brouwer 1907, 148–149).
78.
Nor, for that matter, is it Kant’s. E.g., in his letter to Johann Schultz of November 25, 1788, there is this well-known passage: ‘Time has, as you very well remark, no influence on the properties of numbers (as pure determinations of magnitude), as it does for example on the property of any alteration (as of a quantum), which itself is possible only relative to a specific property of the inner sense and its form (time), and the science of number is, regardless of the succession that any construction of magnitude requires, a pure intellectual synthesis, which we represent to ourselves in thought’ (‘Die Zeit hat, wie Sie ganz wohl bemerken, keinen Einflus auf die Eigenschaften der Zahlen (als reiner Gr¨oßenbestimmungen), so wie etwa auf die Eigenschaft einer jeden Ver¨anderung (als eines Qvanti), die selbst nur relativ auf eine specifische Beschaffenheit des inneren Sinnes und dessen Form (die Zeit) m¨oglich ist, und die Zahlwissenschaft ist, unerachtet der Succession, welche jede Construction der Gr¨oße erfodert, eine reine intellectuelle Synthesis, die wir uns in Gedanken vorstellen’, Kant 1900–, 10:556–557). I take it that when in the Critique of Pure Reason, A142–43/B182, Kant speaks of number as a schema, he is speaking of number in so far as that concept is constructible by us.
79.
‘Wir sehen also, die Zeit spielt f¨ur unsere Begriffe nur die Rolle einer psychologischen Vorbedingung und dies in doppelter Weise:
1. (1)
  Es ist unerl¨aßlich, daß die in der Vorstellung der Vielheit bzw. Anzahl geeinigten Teilvorstellungen zugleich in unserem Bewußtsein vorhanden sind.
2. (2)
  Fast alle Vielheitsvorstellungen und jedenfalls alle Zahlvorstellungen sind Resultate von Prozessen, sind aus den Elementen sukzessive entstandene Ganze. Insofern tr¨agt jedes Element eine andere zeitliche Bestimmtheit an sich.
Wir erkannten aber, daß weder die Gleichzeitigkeit noch die Aufeinanderfolge in der Zeit in den Inhalt der Vielheits- und somit auch der Zahlvorstellungen irgendwie eintreten.’ (Husserl 1970, 32)
80.
See van Atten 2007, 72–74 for further discussion.
81.
A somewhat revised version has been published as an appendix in van Atten (2007).
82.
Note that the qualification ‘Fichtean’ is not Brouwer’s. It is of course a separate question to what extent that qualification is applicable; I will not go into it here.
83.
Also Heyting: ‘It is, as a matter of principle, impossible to devise a system of formulas that would be equivalent to intuitionistic mathematics, because the possibilities of thinking do not admit a reduction to a finite number of rules that can be set up in advance.’ (‘Es ist prinzipiell unm¨oglich, ein System von Formeln aufzustellen, das mit der intuitionistischen Mathematik gleichwertig w¨are, denn die M¨oglichkeiten des Denkens lassen sich nicht auf eine endliche Zahl von im voraus aufstellbaren Regeln zur¨uckf¨uhren’, Heyting 1930a, 3.)
84.
Note that Becker in his Grundlagen der Mathematik in geschichtlicher Entwicklung of 1954 describes the horizon phenomenon and the possibilities of reflection, but without reference to his work of 1927. He says: ‘And thus one can endlessly proceed through the series of indices numbered by transfinite ordinal numbers – in so far as the ordinal numbers used can be defined constructively, and hence can be univocally and exactly named.’ (‘Und so kann man unbegrenzt fortfahren in der Reihe der durch transfinite Ordnungszahlen numerierten Indices – soweit sich die verwendeten Ordnungszahlen konstruktiv definieren und infolgedessen eindeutig und exakt bezeichnen lassen’, Becker 1954, 386.) For a clear recent note on constructive transfinite ordinals, see Jervell (2006).
85.
These numbers are the infinite ordinals ɛ such that ɛ = ω^ɛ. The smallest is
86.
‘Waar men zegt “en zoo voort”, bedoelt men het onbepaald herhalen van eenzelfde ding of operatie, ook al is dat ding of die operatie tamelijk complex gedefinieerd’. (Brouwer 1907, 143n)
87.
For Gödel’s philosophical views, in particular in their relation to phenomenology, see, e.g., Tragesser (1977), Tieszen (1992), Føllesdal (1995), Parsons (1995), and van Atten and Kennedy (2003).
88.
‘It has some plausibility that all things conceivable by us are denumerable’ (Gödel 1946, 152). In his introduction to that paper, Parsons notes that this is ambiguous between ‘For any x, if x is conceivable by us, then x is denumerable’ and ‘Only denumerably many things are conceivable by us’, and says that the latter reading seems more likely (Gödel 1990, 148). I agree: at the point in the text where Gödel makes this remark, it is used to support an objection to accepting ‘undenumerably many sets.’
89.
As quoted on p. 466 of van Atten and Kennedy (2003).
90.
This is Cantor’s hypothesis that the power of the (classical) continuum, $2^{\aleph _{0}}$, is equal to the first uncountable cardinal, $\aleph _{1}$.
91.
See Sect. 12.2.1 above, and van Atten (2002).
92.
In Formal and Transcendental Logic, Husserl says that the then prevailing understanding of purely formal mathematics among mathematicians is that of ‘a pure analytics of consistency’, and adds that on that understanding, the concept of truth remains unthematised (Husserl 1974, 15–16).
93.
With respect to mathematics, Gödel and Brouwer are both what may be called ontological descriptivists; see Sundholm and van Atten (2008, 71).
94.
See also Husserl (1974, 66–67n1), Husserl (1985b, 346), and Husserl (2009, 217).
95.
See also Lohmar (1989, 141–143).
96.
Note that my objection does not apply to Husserl’s appeal to Kantian ideas in the context of geometry (which is not part of purely formal mathematics), e.g., in Sect. 74 of Ideas I. There, in an act of ideation an ideal geometrical notion is given through the elements of an incompletable infinite series of sensuous intuitions, as its ideal limit. But as that notion is not a (mereological) composition out of these elements, for our present purpose there is no relevant analogy with the constitution of higher-order categorial objects.
97.
[[Note added in this reprint: An exception would seem to be a passage in a manuscript probably from the early 1920s:

Finally, there is the concept of a Kantian idea, which needs its own clarification. This comprises real and ‘ideal’, irreal objects, e.g., the number sequence ‘1, 2, 3 and so on’ is a truly existing object, seeable as such, with this ‘and-so-on’. A law of iteration and iterative construction is given to me with evidence, together with the indeterminate ‘idea’ of an open multiplicity. (‘Endlich haben wir den Begriff der Kantischen Idee, der seiner eigenen Kl¨arung bedarf. Umspannt sind hier reale und ‘ideale’, irreale Gegenst¨ande, z.B. die Zahlenreihe ‘1, 2, 3 usw.’ ist ein wahrhaft seiender, als das erschaubarer Gegenstand, mit diesem ‘Und-so-Weiter’. Ich habe ein Gesetz der Iteration und iterativen Konstruktion einsichtig gegeben, neben der unbestimmten ‘Idee’ einer offenen Vielheit’, Husserl 2012, 79.)

But I argue that the sequence as a potentially infinite one is given adequately here; it is only the sequence as an actually infinite one that might be given as a Kantian idea, inadequately. But in that case, it would be inconsistent with Husserl’s ideas about categorial objects to think of this sequence as one of them. Be that as it may, it should be noted that, in its dependence on an iterative construction that is given to us with evidence, Husserl’s use of the notion of Kantian idea is not one that would generalise to the non-denumerable.]]
98.
‘Es gibt andere Welten und vern¨unftige Wesen einer anderen und h¨oheren Art.’ Transcription Robin Rollinger, according to whom the word ‘einer’ is actually hard to make out on the microfilm.
99.
Compare Gödel’s remark on Frank Ramsey’s idea of using propositions of infinite length to provide a foundation of a classical theory of classes: ‘Ramsey’s viewpoint is, of course, everything but constructivistic, unless one means constructions of an infinite mind’ (Gödel 1944, 145; see also 142).
100.
In the period of his intensive study of Leibniz in the 1940s, one idea of Leibniz that appealed to Gödel was that the objects of mathematics exist in God’s mind; see van Atten (2009a, 7–8).
101.
Gödel’s argument was first published in Wang (1974, 182); for criticism, see Parsons (1977) 1983 and Hallett (1984, 220).
102.
For expositions and defences of that conception see, e.g., Boolos 1971; Wang 1974, Chap. 6; Shoenfield 1977, 322–327.
103.
‘Oft verwechselt fortw¨ahrend concepts und sets und ausserdem versteht er “Idealisierung” nicht genug weit’.
104.
It should be noted that, on another occasion, Gödel admitted that arguments that depend on idealising human mental (constructive) capacities are not always the most evident ones. In a letter to Paul Cohen of August 13, 1965, he wrote:

As far as the axiom of the existence of inaccessibles^{Footnote 105} is concerned I think I slightly overstated my view. I would not say that its evidence is due solely to the analogy with the integers.^{Footnote 106}But I do believe that a clear analogy argument is much more convincing than the quasi-constructivistic argument in which we imagine ourselves to be able somehow to reach the inaccessible number. On the other hand, Levy’s principle^{Footnote 107} might be considered more convincing than analogy. (Gödel 2003, 386)
105.
Note MvA. A cardinal number κ is (strongly) inaccessible if it is uncountable and neither the sum nor the product of κ numbers smaller than κ. Such an inaccessible, itself a set, can be used to build a model of the axioms of ZFC, and thereby establish the consistency of that theory. From Gödel’s second Incompleteness Theorem it follows that, if ZFC is consistent, it cannot prove the existence of inaccessible numbers, because otherwise ZFC could prove its own consistency. The question of the existence of inaccessibles therefore concerns the acceptability of a new axiom.
106.
Note MvA. All numbers smaller than the countable infinite cardinal $\aleph _{0}$ are finite, but $\aleph _{0}$ is neither the sum nor the product of finitely many finite numbers. If one believes that $\aleph _{0}$ is not very special among the infinite cardinals, then there should exist an uncountable cardinal with the analogous property.
107.
Note MvA. This is a specific form of the general reflection principle. The latter (roughly) says that, if the universe of all sets has a certain property, then there is a set in the universe that also has it; the property of the universe is reflected in that set. Note that the reflection principle does not yield a construction of that set. The argument Gödel refers to uses reflection (roughly) as follows: If the cardinality of the universe is inaccessible, then by reflection the same is true of a set in the universe. For a discussion of Gödel’s justification of the reflection principle by an analogy to Leibniz’ monadology, see van Atten 2009a.
108.
Errett Bishop, who saw constructive mathematics as the mathematics that finite beings are capable of, said: ‘If God has mathematics of his own that needs to be done, let him do it himself’ (Bishop 1967, 2).
109.
‘Mann kann den Gedanken erw¨agen, dass wie der Mensch intellektuell h¨oher steht als die Mineralien oder ¡die¿ Qualle, so es in Wirklichkeit Wesen geben mag, die dem Menschen gegen¨uber intellektuell h¨oher entwickelt sind, und zwar so, dass sie ¨ uber ganz neue, prinzipiell neue Erkenntnisarten verf¨ugen. …Die allgemeine Rede von m¨oglichen Erkenntniswesen, die durchaus nicht unsere sind und mit unseren gar nichts zu tun haben, ist sinnlos, ist in der Tat widersinnig: da nichts vorhanden ist, was die Einheit des Begriffs der Erkenntnis aufrecht erh¨alt. Soll von solchen M¨oglichkeiten sinnvoll die Rede sein, so muss es sich um Erlebnisarten handelen, die einsehbar wesensidentisch ihrem Gattungscharacter nach sind mit den unseren; und wenn auch aus faktischen Grunden, aus empirisch psychologischen, in unserer Seele nie solche Erkenntnisarten auftreten und wirklich vorstellbar sein k¨onnen, so m¨usste es a priori die M¨oglichkeit einer Erweiterung unserer Erkenntnis bestehen als ideale M¨oglichkeit, durch die unsere Erkenntnis selbst zu der Erkenntnis jener in indirekt-leerer Vorstellung gedachten h¨oheren Intellekte w¨urde.’ (Ms. K II 4, 109a/b, October 1909; quoted from Kern 1964, 129–130.)
110.
The original reads: ‘Die h¨oheren Wesen sind durch Analogie, nicht durch Komposition mit den anderen verbunden.’ Transcription Robin Rollinger.
111.
For further discussion of Husserl’s denial of essentially higher minds, see Kern (1964, 125–134); also of interest here is the correspondence between Becker and Mahnke (2005), together with Mancosu’s introduction to it (Mancosu 2005).
112.
‘Muss jedes Subjekt ¨ uberhaupt, in eidetischer Allgemeinheit gesprochen, jede eidetische Gegenst¨andlichkeit (und ebenso jede Sinnesgegenst¨andlichkeit, jede ideale Gegenst¨andlichkeit im weitesten Sinn) erschauen k¨onnen? Die Frage ist zu bejahen. Es handelt sich hier um ein Apriori. Nicht jedes Subjekt braucht nat¨urlich (nicht jedes wirkliche oder, in der Einstellung reiner M¨oglichkeitsbetrachtung, jedes m¨ogliche, d.i. als m¨ogliche Wirklichkeit angesetzte) jeden eidetischen Gegenstand wirklich zu erkennen (oder als ihn erkennend gedacht zu werden).’ (Husserl 2003b, 147, from 1918 at the latest)
113.
‘Der ideale Gesamtumfang der rein rationalen Gegenst¨ande und der Umfang der einem jeden m¨oglichen Vernunft-Ich erkennbaren objektiven Gegenst¨ande deckt sich.’ (Husserl 1974, 388, from 1920–1921)
114.
‘Jeden mathematischen Schritt, den ein anderer macht, muß ich in mir selbst originaliter nachmachen k¨onnen. …Die Feststellung [ist] zu machen, dass eine einzelsubjektive Mathematik eo ipso intersubjektiv sei und umgekehrt keine intersubjektiv m¨oglich ist, die nicht schon voll und ganz einzelsubjektiv begr¨undet ist.’ (Husserl 1974, 344, November 1926)
115.
‘Es muß eingesehen werden, daß, was ein Ich denkt (die Denkmaterie aber vorausgesetzt), jedes Ich denken k¨onnte. Dieses K¨onnte besagt aber, daß mit jedem logischen Gebilde und jeder Wahrheitsbegr¨undung jede m¨ogliche Hemmung vertr¨aglich ist und daß kein Wesensgesetz in ein logisches hineingreifen kann, wie in den Logischen Untersuchungen gezeigt ist: Wie jede Materie frei variabel ist, so ist auch jede Hemmung “variabel”. Das sagt: Die Wesensgesetze der Freiheit setzen zwar eine Unfreiheit (“niedere” Psyche) voraus, aber keine durch Setzen und Eingreifen.’ (Husserl 1974, 386n1, from 1920–1921)
116.
For a brief discussion of this matter, see van Atten and Kennedy (2003, 454–455).
117.
In the intuitionistic literature, ‘intensional identity’ (and the related ‘intensional equivalence’) is a standard notion, both in philosophical and mathematical discussions (e.g., Dummett 2000a, 16–17, and Troelstra 1977, 5, respectively). Applied to choice sequences, ‘extensional’ means ‘in terms of the numbers chosen in the sequence’, ‘intensional’ means ‘in terms of other aspects of how the sequence is given to us’. In particular, the moment at which a sequence was begun is such an intensional aspect. So are any of the restrictions that we may have imposed on our choices.
118.
It is worth pointing out that the identity criterion for choice sequences that Null himself suggests (‘CIP’, Null 2008, 127), is a criterion in which protentions play no role whatsoever: for n < ω, CIP operates only on initial segments (which are finite sequences and as such given to us without an open horizon (van Atten 2007, 90)), and for n = ω, we are dealing with sequences of infinitely many given numbers, which are not given as open-ended either. So CIP is not suitable as an identity criterion for Brouwer’s choice sequences; they are not even among the objects it operates on.
119.
On this difference see also the remarks in Bernet et al. (1989, 94–95).
120.
This is discussed in van Atten (2008, 70–71).
121.
Quoted in van Atten (2008, 11–12).
122.
Incidentally, characterisation (ii) is explicitly stated in my book to be the appropriate one (e.g., van Atten (2008, 1, 24)).
123.
Note MvA. Husserl here means spatio-temporal locality; see Husserl (1985b, 311).
124.
‘Daß ein Subjekt ein Satz evident denkt, das gibt dem Satz Lokalit¨at, und als gedachtem dieses Denkers etc. eine einzige, aber nicht dem Satz schlechthin, der derselbe w¨are als zu verschiedenen Zeiten etc. gedachter.’ (Husserl 1985b, 312–313)
125.
‘Und gehen auch die zeitkonstituierenden Akte der Unterstufe mit ein, so brauchen sie es doch nicht so zu tun, dass die Zeiten wie der Gegenst¨andlichkeiten selbst in die h¨oher konstituierten Gegenst¨andlichkeiten eingehen.’ (Husserl 1985b, 310)
126.
‘eine solche Irrealit¨at [hat] das zeitliche Sein der ¨ Uberzeitlichkeit, der Allzeitlichkeit, die doch ein Modus der Zeitlichkeit ist.’ (Husserl 1985b, 313)
127.
There are other infelicitous claims in Null’s review, of which I here briefly mention the following two (which, moreover, are incompatible with each other). (1) Discussing the tree he set up on p. 126 (‘Fig. 1a’), Null claims of the paths through it that ‘each of these 2 to the $\aleph _{0}$ many sequences is an ideal object of the sort accepted by Husserl, Weyl, Becker, and Kaufmann’ (Null 2008, 127). This is not correct. On the one hand, these sequences are all predeterminate (in each sequence the n-th element is fixed from the beginning for each n); on the other hand, by laws one can specify at most denumerably many sequences. As a consequence, most of these $2^{\aleph _{0}}$ many sequences are predeterminate but not given by a law. But that combination is possible in neither of the respective varieties of constructivism by which Weyl, Becker and Kaufmann defined their philosophical positions. Husserl is not explicit about the matter, but, as I argue in the present paper, accepting non-constructible objects in mathematics is not an option in his framework either. (2) Null claims that Husserl’s formal objects should be ‘countable’, which, as defined by Null, implies decidability of equivalence. But although classical mathematics, intuitionistic mathematics, recursive analysis, Bishop’s constructive mathematics all have different conceptions of real number, in none of them is equality of real numbers decidable. This would leave Husserl’s phenomenology incapable of founding any of them. But I do not think that Husserl anywhere actually poses or implies Null’s countability condition.
128.
Except where required otherwise,
1. 1.
  Brouwer’s writings are referred to according to the scheme in Van Dalen’s bibliography (van Dalen 2008);
2. 2.
  Gödel’s writings are referred to according to the scheme in the Collected Works (Gödel 1986, 1990, 1995, 2003,a);
3. 3.
  Husserl’s writings are referred to in the Husserliana edition (Husserl 1950a–).

References

M. van Atten, Why Husserl should have been a strong revisionist in mathematics. Husserl Stud. 18(1), 1–18 (2002)
Article Google Scholar
M. van Atten, Intuitionistic remarks on Husserl’s analysis of finite number in the philosophy of arithmetic. Grad. Fac. Philos. J. 25(2), 205–225 (2004b)
Google Scholar
M. van Atten, Review of Gnomes in the Fog. The reception of Brouwer’s intuitionism in the 1920s, by Dennis Hesseling. Bull. Symb. Log. 10(3), 423–427 (2004c)
Google Scholar
M. van Atten, Brouwer Meets Husserl: On the Phenomenology of Choice Sequences (Springer, Dordrecht, 2007)
Book Google Scholar
M. van Atten, Luitzen Egbertus Jan Brouwer, in Zalta 1997 (Winter 2008). http://plato.stanford.edu/archives/win2008/entries/brouwer
M. van Atten, Monads and sets: On Gödel, Leibniz, and the reflection principle, in Primiero and Rahman 2009 (2009a), pp. 3–33
Google Scholar
M. van Atten, The development of intuitionistic logic, in Zalta 1997 (Summer 2009b). http://plato.stanford.edu/archives/sum2009/entries/intuitionistic-logic-development
M. van Atten, Construction and constitution in mathematics, in The New Yearbook for Phenomenology and Phenomenological Philosophy, vol. 10 (2010), pp. 43–90
Google Scholar
M. van Atten, Kant and real numbers, in Dybjer et al. (2012), pp. 3–23
Google Scholar
M. van Atten, Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer (Springer, Dordrecht, 2015a)
Google Scholar
M. van Atten, On the fulfillment of certain categorial intentions, in The New Yearbook for Phenomenology and Phenomenological Philosophy, vol. 13 (2015b), pp. 173–185
Google Scholar
M. van Atten, P. Boldini, M. Bourdeau, G. Heinzmann (eds.), One Hundred Years of Intuitionism (1907–2007): The Cerisy Conference (Birkhäuser, Basel, 2008)
Google Scholar
M. van Atten, J. Kennedy, On the philosophical development of Kurt Gödel. Bull. Symb. Log. 9(4), 425–476 (2003)
Article Google Scholar
J. Barwise (ed.), Handbook of Mathematical Logic (North-Holland, Amsterdam, 1977)
Google Scholar
O. Becker, Mathematische Existenz: Untersuchungen zur Logik und Ontologie mathematischer Phänomene. Jahrbuch für Philosophie und phänomenologische Forschung 8, 439–809 (1927)
Google Scholar
O. Becker, Grundlagen der Mathematik in geschichtlicher Entwicklung (Alber, Freiburg/München, 1954)
Google Scholar
O. Becker, D. Mahnke, Briefwechsel mit Dietrich Mahnke. With an introduction by Bernd Aust and Jochen Sattler, in Peckhaus (2005), pp. 245–278
Google Scholar
P. Benacerraf, H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, 1st edn. (Cambridge University Press, Cambridge, 1964)
Google Scholar
P. Benacerraf, H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, 2nd edn. (Cambridge University Press, Cambridge, 1983)
Google Scholar
R. Bernet, I. Kern, E. Marbach, Edmund Husserl: Darstellung seines Denkens (Meiner, Hamburg, 1989)
Google Scholar
E. Beth, H. Pos, J. Hollak (eds.), Proceedings of the 10th International Congress of Philosophy, Amsterdam 1948, vol 2, bk 1 (1949)
Google Scholar
E. Bishop, Foundations of Constructive Analysis (McGraw-Hill, New York, 1967)
Google Scholar
L. Boi, P. Kerszberg, F. Patras (eds.), Rediscovering Phenomenology (Springer, Dordrecht, 2007)
Google Scholar
G. Boolos, The iterative concept of set. J. Philos. 68, 215–231 (1971)
Article Google Scholar
P. Bossert (ed.), Phenomenological Perspectives (Martinus Nijhoff, Den Haag, 1975)
Google Scholar
D. Bridges, Constructive mathematics, in Zalta 1997– (Summer 2009). http://plato.stanford.edu/archives/sum2009/entries/mathematics-constructive/
L.E.J. Brouwer, Archive (Noord-Hollands Archief, Haarlem, The Netherlands)
Google Scholar
L.E.J. Brouwer, Over de grondslagen der wiskunde. PhD Dissertation, Universiteit van Amsterdam. English translation in Brouwer 1975, 1907, pp. 11–101
Google Scholar
L.E.J. Brouwer, Die mögliche Mächtigkeiten (1908A). Lecture, published in Castelnuovo 1909, pp. 569–571. Facsimile reprint in Brouwer 1975, pp. 102–104
Google Scholar
L.E.J. Brouwer, De onbetrouwbaarheid der logische principes. Tijdschrift voor Wijsbegeerte 2, 152–158 (1908C). English translation in Brouwer 1975, pp. 107–111
Google Scholar
L.E.J. Brouwer, Intuïtionisme en formalisme (Clausen, Amsterdam, 1912A). English translation in Benacerraf and Putnam 1983, pp. 77–89
Google Scholar
L.E.J. Brouwer, Intuitionism and formalism. Bull. Am. Math. Soc. 20, 81–96 (1913C). Facsimile reprint in Brouwer 1975, pp. 123–138
Google Scholar
L.E.J. Brouwer, Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten: Erster Teil: Allgemeine Mengenlehre. KNAW Verhandelingen 5, 1–43 (1918B). Facsimile reprint in Brouwer 1975, pp. 150–190
Google Scholar
L.E.J. Brouwer, Über Definitionsbereiche von Funktionen. Math. Ann. 97, 60–75 (1927B). Facsimile reprint in Brouwer 1975, pp. 390–405. English translation of Sects. 1–3 in van Heijenoort 1967, pp. 457–463
Google Scholar
L.E.J. Brouwer, Intuitionistische Betrachtungen über den Formalismus. KNAW Proc. 31, 374–379 (1928A2). Facsimile reprint in Brouwer 1975, pp. 409–414. English translation in Mancosu 1998, pp. 40–44
Google Scholar
L.E.J. Brouwer, Mathematik, Wissenschaft und Sprache. Monatshefte für Mathematik und Physik 36, 153–164 (1929A). Facsimile reprint in Brouwer 1975, pp. 417–428. English translation in Mancosu 1998, pp. 45–53
Google Scholar
L.E.J. Brouwer, Die Struktur des Kontinuums (Komitee zur Veranstaltung von Gastvorträgen ausländischer Gelehrter der exakten Wissenschaften, Wien, 1930A). Facsimile reprint in Brouwer 1975, pp. 429–440. English translation in Mancosu 1998, pp. 54–63
Google Scholar
L.E.J. Brouwer, Willen, weten, spreken, in Brouwer et al. 1933 (1933A2), pp. 45–63. English translation of Sect. 3 in Brouwer 1975, pp. 443–446. Full English translation in van Stigt 1990, pp. 418–431
Google Scholar
L.E.J. Brouwer, Essentieel negatieve eigenschappen. Indag. Math. 10, 322–323 (1948A). English translation in Brouwer 1975, pp. 478–479
Google Scholar
L.E.J. Brouwer, Consciousness, philosophy and mathematics, in Bethetal 1949, pp. 1235–1249. Facsimile reprint in Brouwer 1975, pp. 480–494
Google Scholar
L.E.J. Brouwer, Historical background, principles and methods of intuitionism. S. Afr. J. Sci. 49, 139–146 (1952B). Facsimile reprint in Brouwer 1975, pp. 508–515
Google Scholar
L.E.J. Brouwer, Points and spaces. Can. J. Math. 6, 1–17 (1954A). Facsimile reprint in Brouwer 1975, pp. 522–538
Google Scholar
L.E.J. Brouwer, Philosophy and Foundations of Mathematics. Vol 1 of Collected Works, ed. by A. Heyting (North-Holland, Amsterdam, 1975)
Google Scholar
L.E.J. Brouwer, Brouwer’s Cambridge Lectures on Intuitionism. ed. by D. van Dalen (Cambridge University Press, Cambridge, 1981A)
Google Scholar
L.E.J. Brouwer, J. Clay, A. de Hartog, G. Mannoury, H. Pos, G. Révész, J. Tinbergen, J. van der Waals Jr. (eds.), De uitdrukkingswijze der wetenschap: Kennistheoretische openbare voordrachten gehouden aan de Universiteit van Amsterdam gedurende de kursus 1932–1933 (Noordhoff, Groningen, 1933)
Google Scholar
L.E.J. Brouwer, F. van Eeden, J. van Ginneken, G. Mannoury, Signifische dialogen. Synthese 2(5,7,8), 168–174, 261–268, 316–324 (1937)
Google Scholar
L.E.J. Brouwer, F. van Eeden, J. van Ginneken, G. Mannoury, Signifische dialogen (Erven J. Bijleveld, Utrecht, 1939). English translation in Brouwer 1975, pp. 447–452
Google Scholar
R. Butts, J. Hintikka (eds.), Logic, Foundations of Mathematics and Computability Theory (D. Reidel, Dordrecht, 1977)
Google Scholar
G. Castelnuovo (ed.), Atti del IV Congresso internazionale dei matematici, Roma, 6–11 aprile 1908: Comunicazioni delle sezioni III-A, III-B e IV (Tipografia della Reale Accademia dei Lincei, Roma, 1909)
Google Scholar
D. van Dalen, L.E.J. Brouwer en de grondslagen van de wiskunde (Epsilon, Utrecht, 2001a)
Google Scholar
D. van Dalen, A bibliography of L.E.J. Brouwer, in van Atten 2008, pp. 343–390
Google Scholar
M. Davis (ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions (Raven Press, Hewlett, 1965)
Google Scholar
J. Dodd, Husserl between formalism and intuitionism, in Boi et al. 2007, pp. 261—265
Google Scholar
M. Dummett, Is time a continuum of instants? Philosophy 75, 497–515 (2000a)
Google Scholar
M. Dummett, Elements of Intuitionism, 2nd rev edn. (Clarendon Press, Oxford, 2000b)
Google Scholar
P. Dybjer, S. Lindström, E. Palmgren, G. Sundholm (eds.), Epistemology versus Ontology: Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf (Springer, Dordrecht, 2012)
Google Scholar
L. Eley, Metakritik der Formalen Logik (Martinus Nijhoff, Den Haag, 1969)
Book Google Scholar
D. Føllesdal, Gödel and Husserl, in Hintikka 1995, pp. 427–446
Google Scholar
K. Gödel, Russell’s mathematical logic, in Schilpp 1944, pp. 123–153. Reprinted, with original page numbers in the margin, in Gödel 1990, pp. 119–141
Google Scholar
K. Gödel, Remarks before the Princeton bicentennial conference on problems in mathematics (1946). Lecture, first published in Davis 1965, pp. 84–88. Page references are to the reprint in Gödel 1990, pp. 150–153
Google Scholar
K. Gödel, “What is Cantor’s continuum problem? Am. Math. Mon. 54, 515–525 (1947). Reprinted, with original page numbers in the margin, in Gödel 1990, pp. 176–187
Google Scholar
K. Gödel, The modern development of the foundations of mathematics in the light of philosophy (*1961/?). Lecture draft in German, published, with an English translation, in Gödel 1995, pp. 374–387. The English title is Gödel’s
Google Scholar
K. Gödel, What is Cantor’s continuum problem? in Benacerraf and Putnam 1964, pp. 258–273. Revised and expanded version of Gödel 1947. Reprinted, with original page numbers in the margin, in Gödel 1990, pp. 254–270
Google Scholar
K. Gödel, Publications 1929–1936. Vol 1 of Collected Works, eds. by S. Feferman, J. Dawson Jr., S. Kleene, G. Moore, R. Solovay, J. van Heijenoort (Oxford University Press, Oxford, 1986)
Google Scholar
K. Gödel, Publications 1938–1974. Vol 2 of Collected Works, eds. by S. Feferman, J. Dawson Jr., S. Kleene, G. Moore, R. Solovay, J. van Heijenoort (Oxford University Press, Oxford, 1990)
Google Scholar
K. Gödel, Unpublished Essays and Lectures. Vol 3 of Collected Works, eds. by S. Feferman, J. Dawson Jr., W. Goldfarb, C. Parsons, R. Solovay (Oxford: Oxford University Press, 1995)
Google Scholar
K. Gödel, Correspondence A–G. Vol 4 of Collected Works, eds. by S. Feferman, J. Dawson Jr., W. Goldfarb, C. Parsons, W. Sieg (Oxford University Press, Oxford, 2003)
Google Scholar
K. Gödel, Correspondence H–Z. Vol 5 of Collected Works, eds. by S. Feferman, J. Dawson Jr., W. Goldfarb, C. Parsons, W. Sieg (Oxford University Press, Oxford, 2003a)
Google Scholar
M. Hallett, Cantorian Set Theory and Limitation of Size (Clarendon Press, Oxford, 1984)
Google Scholar
M. Hartimo (ed.), Phenomenology and Mathematics (Springer, Dordrecht, 2010)
Google Scholar
M. Heidegger, Prolegomena zur Geschichte des Zeitbegriffs. Heidegger Gesamtausgabe, vol 20, ed. by P. Jaeger (Vittorio Klostermann, Frankfurt am Main, 1979)
Google Scholar
J. van Heijenoort (ed.), From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931 (Harvard University Press, Cambridge, 1967)
Google Scholar
D. Hesseling, Gnomes in the Fog: The Reception of Brouwer’s Intuitionism in the 1920s (Birkhäuser, Basel, 2003)
Book Google Scholar
A. Heyting, Die formalen Regeln der intuitionistischen Logik. Pt. 1. Sitzungsberichte der Preussischen Akademie der Wissenschaften 42–56 (1930a). English translation in Mancosu 1998, pp. 311–327
Google Scholar
A. Heyting, Intuitionism: An Introduction (North-Holland, Amsterdam, 1956)
Google Scholar
A. Heyting, L.E.J. Brouwer, in Klibansky 1968, pp. 308–315
Google Scholar
D. Hilbert, Neubegründung der Mathematik (Erste Mitteilung). Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 1, 157–177 (1922). English translation in Mancosu 1998, pp. 198–214
Google Scholar
C. Hill, Husserl on axiomatization and arithmetic, in Hartimo 2010, pp. 47–71
Google Scholar
J. Hintikka (ed.), From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics (Kluwer, Dordrecht, 1995)
Google Scholar
E. Husserl, Cartesianische Meditationen und Pariser Vorträge. Husserliana, vol. 1, ed. by S. Strasser (Martinus Nijhoff, Den Haag, 1950a)
Google Scholar
E. Husserl, Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie: Drittes Buch: Die Phänomenologie und die Fundamente der Wissenschaften. Husserliana, vol. 5, ed. by M. Biemel (Martinus Nijhoff, Den Haag, 1952)
Google Scholar
E. Husserl, Erste Philosophie (1923/1924): Erster Teil: Kritische Ideengeschichte. Husserliana, vol. 7, ed. by R. Boehm (Martinus Nijhoff, Den Haag, 1956a)
Google Scholar
E. Husserl, Erste Philosophie (1923/1924): Zweiter Teil: Theorie der phänomenologischen Reduktion. Husserliana, vol. 8, ed. by R. Boehm (Martinus Nijhoff, Den Haag, 1959)
Google Scholar
E. Husserl, Phänomenologische Psychologie. Husserliana, vol. 9, ed. by W. Biemel (Martinus Nijhoff, Den Haag, 1962)
Google Scholar
E. Husserl, Analysen zur passiven Synthesis (1918–1926). Husserliana, vol. 11, ed. by M. Fleischer (Martinus Nijhoff, Den Haag, 1966)
Google Scholar
E. Husserl, Philosophie der Arithmetik.Husserliana, vol. 12, ed. by L. Eley (Martinus Nijhoff, Den Haag, 1970)
Google Scholar
E. Husserl, Cartesian Meditations, trans. by D. Cairns (Martinus Nijhoff, Den Haag, 1973d)
Google Scholar
E. Husserl, Experience and Judgment, trans. by J. Churchill, K. Ameriks (Routledge & Kegan Paul, London, 1973e)
Google Scholar
E. Husserl, Formale und transzendentale Logik. Husserliana, vol. 17, ed. by P. Janssen (Martinus Nijhoff, Den Haag, 1974)
Google Scholar
E. Husserl, Logische Untersuchungen: Erster Band. Husserliana, vol. 18, ed. by E. Holenstein (Martinus Nijhoff, Den Haag, 1975)
Google Scholar
E. Husserl, Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie: Erstes Buch. 1. Halbband: Text der 1.–3. Auflage. Husserliana, vol. 3/1, ed. by K. Schuhmann (Martinus Nijhoff, Den Haag, 1976a)
Google Scholar
E. Husserl, Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie: Erstes Buch. 2. Halbband: Ergänzende Texte (1912—1929). Husserliana, vol. 3/2, ed. by K. Schuhmann (Martinus Nijhoff, Den Haag, 1976b)
Google Scholar
E. Husserl, Aufsätze und Rezensionen (1890–1910). Husserliana, vol. 22, ed. by B. Rang (Martinus Nijhoff, Den Haag, 1979)
Google Scholar
E. Husserl, Logische Untersuchungen: Zweiter Band, 2. Teil. Husserliana, vol. 19/2, ed. by U. Panzer (Martinus Nijhoff, Den Haag, 1984b)
Google Scholar
E. Husserl, Einleitung in die Logik und Erkenntnistheorie: Vorlesungen 1906/07. Husserliana, vol. 24, ed. by U. Melle (Martinus Nijhoff, Den Haag, 1985a)
Google Scholar
E. Husserl, Erfahrung und Urteil, ed. by L. Landgrebe (Meiner, Hamburg, 1985b)
Google Scholar
E. Husserl, Aufsätze und Vorträge (1922–1937). Husserliana, vol. 27, ed. by T. Nenon, H. Sepp (Martinus Nijhoff, Den Haag, 1988)
Google Scholar
E. Husserl, Logik und allgemeine Wissenschaftstheorie: Vorlesungen 1917/18. Mit ergänzenden Texten aus der ersten Fassung 1910/11. Husserliana, vol. 30, ed. by U. Panzer (Kluwer, Den Haag, 1995a)
Google Scholar
E. Husserl, Natur und Geist: Vorlesungen Sommersemester 1927. Husserliana, vol. 32, ed. by M. Weiler (Kluwer, Dordrecht, 2001b)
Google Scholar
E. Husserl, Zur phänomenologischen Reduktion: Texte aus dem Nachlass (1926–1935). Husserliana, vol. 34, ed. by S. Luft (Kluwer, Dordrecht, 2002b)
Google Scholar
E. Husserl, Einleitung in die Philosophie: Vorlesungen 1922/23. Husserliana, vol. 35, ed. by B. Goossens (Kluwer, Dordrecht, 2002c)
Google Scholar
E. Husserl, Philosophy of Arithmetic: Psychological and Logical Investigations with Supplementary Texts from 1887–1901.Edmund Husserl Collected Works, vol. 10, ed. and trans. by D. Willard (Kluwer, Dordrecht, 2003a)
Google Scholar
E. Husserl, Transzendentaler Idealismus: Texte aus dem Nachlass (1908–1921). Husserliana, vol. 36, ed. by R. Rollinger in collaboration with R. Sowa (Kluwer, Dordrecht, 2003b)
Google Scholar
E. Husserl, Untersuchungen zur Urteilstheorie: Texte aus dem Nachlass (1893–1918). Husserliana, vol. 40, ed. by R. Rollinger (Springer, Dordrecht, 2009)
Google Scholar
E. Husserl, Zur Lehre vom Wesen und zur Methode der eidetischen Variation: Texte aus dem Nachlass (1893–1918). Husserliana, vol. 41, ed. by D. Fonfara (Springer, Dordrecht, 2012)
Google Scholar
R. Iemhoff, Intuitionism in the philosophy of mathematics, in Zalta 1997– (Winter 2009). http://plato.stanford.edu/archives/win2009/entries/intuitionism/
H. Jervell, Constructing ordinals, in Cahier spécial 6: Constructivism: Mathematics, Logic, Philosophy and Linguistics. Philosophia Scientiae, ed. by G. Heinzmann, G. Ronzitti (2006), pp. 5–20
Google Scholar
I. Kant, Gesammelte Schriften, 29 vols., ed. by A. der Wissenschaften (Reimer, Berlin, 1900–). (from 1920 De Gruyter)
Google Scholar
I. Kern, Husserl und Kant (Martinus Nijhoff, Den Haag, 1964)
Book Google Scholar
S. Kierkegaard, Fear and Trembling, ed. by S. Evans, S. Walsh (Cambridge University Press, Cambridge, 2006)
Google Scholar
R. Klibansky (ed.), Logic and Foundations of Mathematics. Vol. 1 of Contemporary Philosophy. A Survey (La Nuova Italia editrice, Firenze, [Johannes de Silentio, pseud.]. (1843) 1968)
Google Scholar
G. Kreisel, Mathematical logic, in Saaty 1965, pp. 95–195
Google Scholar
G. Küng, Das Noema als reelles Moment, in Bossert 1975, pp. 151–153
Google Scholar
D. Lohmar, Phänomenologie der Mathematik: Elemente einer phänomenologischen Aufklärung der mathematischen Erkenntnis nach Husserl (Kluwer, Dordrecht, 1989)
Book Google Scholar
D. Lohmar, On the relation of mathematical objects to time: are mathematical objects timeless, overtemporal or omnitemporal? J. Indian Counc. Philos. Res. 10(3), 73–87 (1993)
Google Scholar
D. Lohmar, The transition of the principle of excluded middle from a principle of logic to an axiom: Husserl’s hesitant revisionism in the field of logic, in The New Yearbook for Phenomenology and Phenomenological Philosophy, vol. 4 (2004), pp. 53–68
Google Scholar
D. Mahnke, Eine neue Monadologie. Kantstudien Ergänzungsheft, vol. 39 (1917)
Google Scholar
P. Mancosu (ed.), The Debate on the Foundations of Mathematics in the 1920s, in From Brouwer to Hilbert (Oxford University Press, Oxford, 1998)
Google Scholar
P. Mancosu, Das Abenteuer der Vernunft: O. Becker and D. Mahnke on the phenomenological foundations of the exact sciences, in Peckhaus 2005, pp. 229–243
Google Scholar
J. Mensch, After Modernity: Husserlian Reflections on a Philosophical Tradition. Chap. Intersubjectivity and the constitution of time (State University of New York Press, Albany, 1996), pp. 57–66
Google Scholar
J.R. Moschovakis, Intuitionistic logic, in Zalta 1997– (Fall 2008). http://plato.stanford.edu/archives/fall2008/entries/logic-intuitionistic/
G. Null, Entities without identities vs. temporal modalities of choice: review of Mark van Atten, Brouwer Meets Husserl. Husserl Stud. 24, 119–130 (2008)
Google Scholar
C. Parsons, (1977) What is the iterative conception of set? in Parsons 1983, pp. 268–297. Originally in Butts and Hintikka 1977, pp. 335–367
Google Scholar
C. Parsons, Mathematics in Philosophy: Selected Essays (Cornell University Press, Ithaca, 1983)
Google Scholar
C. Parsons, Platonism and mathematical intuition in Kurt Gödel’s thought. Bull. Symb. Log. 1(1), 44–74 (1995)
Article Google Scholar
C. Parsons, Finitism and intuitive knowledge, in Schirn 1998 (Oxford, 1998), pp. 249–270
Google Scholar
V. Peckhaus (ed.), Oskar Becker und die Philosophie der Mathematik (Wilhelm Fink Verlag, München, 2005)
Google Scholar
A. Pfänder, Philosophie auf phänomenologischer Grundlage: Einleitung in die Philosophie und Phänomenologie, ed. by H. Spiegelberg (Fink, München, 1973)
Google Scholar
T. Placek, Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for Intuitionism (Kluwer, Dordrecht, 1999)
Book Google Scholar
G. Primiero, S. Rahman (eds.), Judgement and Knowledge: Papers in honour of B.G. Sundholm (College Publications, London, 2009)
Google Scholar
G.E. Rosado Haddock Review of mathematical intuition by Richard Tieszen. J. Symb. Log. 56(1), 356–360 (1991)
Article Google Scholar
G.E. Rosado Haddock, Husserl’s philosophy of mathematics: its origin and relevance. Husserl Stud. 22(3), 193–222 (2006)
Article Google Scholar
G.E. Rosado Haddock, Platonism, phenomenology and interderivability, in Hartimo 2010, pp. 23–46
Google Scholar
T. Saaty (ed.), Lectures on Modern Mathematics, vol. 3 (Wiley, New York, 1965)
Google Scholar
P.A. Schilpp (ed.), The Philosophy of Bertrand Russell. The Library of Living Philosophers, vol. 5 (Nortwestern University Press, Evanston, 1944). 3rd edn (Tudor, New York, 1951)
Google Scholar
M. Schirn (ed.), The Philosophy of Mathematics Today (Oxford University Press, Oxford, 1998)
Google Scholar
K. Schuhmann, Reine Phänomenologie und phänomenologische Philosophie (Martinus Nijhoff, Den Haag, 1973)
Google Scholar
J. Shoenfield, Axioms of set theory, in Barwise 1977, pp. 321–344
Google Scholar
B. Smith, D. Smith (eds.), The Cambridge Companion to Husserl (Cambridge University Press, Cambridge, 1995)
Google Scholar
W. van Stigt, Brouwer’s Intuitionism (North-Holland, Amsterdam, 1990)
Google Scholar
G. Sundholm, M. van Atten, The proper interpretation of intuitionistic logic: on Brouwer’s demonstration of the Bar theorem, in vanAtten 2008, pp. 60–77
Google Scholar
W. Tait, Finitism. J. Philos. 78(9), 524–546 (1981)
Article Google Scholar
R. Tieszen, Mathematical Intuition: Phenomenology and Mathematical Knowledge (Kluwer, Dordrecht, 1989)
Book Google Scholar
R. Tieszen, Kurt Gödel and phenomenology. Philos. Sci. 59, 176–194 (1992)
Article Google Scholar
R. Tieszen, Mathematics, in Smith and Smith 1995, pp. 438–462
Google Scholar
R. Tieszen, Mathematical realism and transcendental phenomenological idealism, in Hartimo 2010, pp. 1–22
Google Scholar
R. Tragesser, Phenomenology and Logic (Cornell University Press, Ithaca, 1977)
Google Scholar
A. Troelstra, Choice Sequences: A Chapter of Intuitionistic Mathematics (Oxford University Press, Oxford, 1977)
Google Scholar
A. Troelstra, On the origin and development of Brouwer’s concept of choice sequence, in Troelstra and van Dalen 1982, pp. 465–486
Google Scholar
A. Troelstra, D. van Dalen (eds.), The L.E.J. Brouwer Centenary Symposium (North-Holland, Amsterdam, 1982)
Google Scholar
A. Troelstra, D. van Dalen, Constructivism in Mathematics: An Introduction, 2 vols. (North-Holland, Amsterdam, 1988)
Google Scholar
H. Wang, From Mathematics to Philosophy (Routledge/Kegan Paul, London, 1974)
Google Scholar
H. Wang, Reflections on Kurt Gödel (MIT Press, Cambridge, 1987)
Google Scholar
H. Wang, A Logical Journey: From Gödel to Philosophy (MIT Press, Cambridge, 1996)
Google Scholar
H. Weyl, Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis (Veit, Leipzig, 1918)
Google Scholar
L. Wittgenstein, Tractatus logico-philosophicus, ed. by J. Schulte (Suhrkamp, Frankfurt am Main, (1921) 2013)
Google Scholar
J. Yoshimi, Mathematizing phenomenology. Phenomenol. Cogn. Sci. 6(3), 271–291 (2007)
Article Google Scholar
E. Zalta (ed.), The Stanford Encyclopedia of Philosophy (The Metaphysics Research Lab, CSLI, Stanford University, Stanford, 1997–). http://plato.stanford.edu

Download references

Acknowledgements

This text (except for the appendix) grew out of my talk at the conference ‘Phénoménologie discrète: Le parcours intellectuel de Gian-Carlo Rota entre mathématiques et philosophie’, Lille, November 8 and 9, 2009, of which a later version was presented in the ‘Ideals of Proof Seminar’, Paris, March 24, 2010. I thank the organisers for their invitations, and the audiences for their questions and comments. I am indebted to the Institute for Advanced Study for permission to quote from the Kurt Gödel Papers, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, on deposit at Princeton University. I am grateful to Robin Rollinger for his transcriptions from Gödel’s Gabelsberger shorthand. Mirja Hartimo kindly provided photocopies of some of the items in the bibliography. Special thanks for discussion and comments are due to Robert Tragesser.

Author information

Authors and Affiliations

SND (CNRS/Paris IV), 1 rue Victor Cousin, 75005, Paris, France
Mark van Atten

Authors

Mark van Atten
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark van Atten .

Editor information

Editors and Affiliations

Institut für Philosophie, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany
Stefania Centrone

Appendix: Null on Choice Sequences

Gilbert Null’s review of my Brouwer Meets Husserl is centred around his attribution to me of two claims (Null 2008, 119, 128):

1.
choice sequences are objects without identity criterion;
2.
choice sequences are real (reell) parts of inner time.

Not only are these claims made nowhere in the book, they are directly and explicitly contradicted in it.The passage that Null quotes from p. 36 of my book to support the attribution of claim (1) denies only that (non-lawlike) choice sequences have an identity criterion that is extensional. It does not at all say that choice sequences can have no identity criterion whatsoever. Indeed, as I argue extensively in Chap. 6, in particular on pp. 92–93 (which Null nowhere refers to), the principle of individuation (in terms of the moment of the beginning of a choice sequence) provides the identity criterion, and this criterion I identify explicitly as intensional (e.g., p. 90).^{Footnote 117}

Null further maintains that my book ‘notes …but leaves the need for a closer analysis of choice event protentions …unfulfilled’ (Null 2008, 129). Not only is their role made fully explicit in the analysis on pp. 92–93, but this role is discussed at many other places as well because, as is explicit on p. 92, protentions are essential to the constitution of choice sequences as open-ended (pp. 6, 15, 36, 92, 97, 98, 105). In this light, it is incomprehensible that Null makes it seem as if my book, and ‘Brouwerians’ in general, neglect ‘any present (actual) choice event’s inner horizon of openly possible contrary futures’ (Null 2008, 129).^{Footnote 118}

Claim (2) is denied outright on p. 92 of my book, for reasons given on p. 91. Null’s mistaken attribution of claim (2) to me may well stem from my expressing agreement with Guido Küng’s thesis (Küng 1975) that the noema is a real (reell) moment (p. 70 and note 162). But whatever the merits of the thesis of the noema as a real moment, I do consider it a thesis about the noema in its fullest extent; and I state just as explicitly (p. 71 and pp. 89–90) that I consider the noematic essence and the noematic nucleus as ideal objects that are omnitemporal.^{Footnote 119} On p. 128 of his review, Null says that I am also committed to the thesis that a number can occur in a choice sequence only once. But since it is in virtue of the noematic nucleus that an intentional act has whatever object it has, also on my conception of the noema it is perfectly possible to choose the same number more than once in a choice sequence. One might think that, if the full noema is taken to be a real (reell) moment of the noesis, then the noematic nucleus cannot be omnitemporal, as the concrete noesis can, by definition, not be repeated in time. But it does not follow from this that the noematic nucleus cannot be omnitemporal. According to Sect. 64c of Experience and Judgment, Husserl calls an object ‘omnitemporal’ (allzeitlich) if in its constitution, which always takes place at a specific moment in time, this moment in time does not ‘enter into’ (eingehen) that object, that is, this temporal determination is not part of what makes the object the object it is. But then it is not at all excluded that some appropriate part of a given noesis can be constituted at a different time as identically the same: namely, if in the constitution of that part the moment in time does not ‘enter into it’, is not part of what makes it the object it is.^{Footnote 120} Not only is this not at all excluded, it is exactly parallel to Husserl’s explanation how categorial objects are, ontically, productions, yet omnitemporal (Husserl 1985b, 311)^{Footnote 121}; see the discussion of point C2 above.

This also bears on page 124 of Null’s review. He there says that a choice sequence contains either (i) the senses or (ii) the referents of choice events in an associated choice process.^{Footnote 122} The problem with (ii), according to Null, is that in that case ‘the available Husserlian approach [sic] leaves Van Atten’s characterization of choice sequences as intratemporal prima facie unsupported’, because ‘numbers are omnitemporal if anything is’ (Null 2008, 124). If this argument is to work, then Null must hold that, if a higher-order object (here, a choice sequence) is founded on omnitemporal objects (here, numbers), then that higher-order object must also be omnitemporal. As Null does not supply any argument for this idea, he is begging the question against those who defend that choice sequences are intratemporal objects.

One may ask, of course, whether an argument such as Null fails to supply nevertheless exists. This is not the case, because there are counterexamples (independently from the one that, as I argue, choice sequences are). A specific act in which I judge, with full evidence, that the number 2 is an omnitemporal object, is founded on the number 2: for if the number 2 did not exist, neither could this specific act of judgement with full evidence. But this act, when objectified, is not an omnitemporal object, as it exists only for a certain stretch of time.

Husserl makes a closely related point: ‘That a subject conceives a proposition with evidence, lends the proposition locality,^{Footnote 123} and, as the thought of this thinker etc. a unique one, but not to the proposition as such, which would be the same when thought at different times’.^{Footnote 124} In Husserl’s example, the particular thinker’s particular thought episode of the proposition as evident is founded on the proposition, but does not share the temporal characteristics of the latter.

More generally, Husserl had already remarked, speaking of the constitution of a higher-order object on the foundation of lower-order ones, ‘And even when the time-constituting acts of the lower level also enter, they need not do this in such a way that the times enter, like the objects themselves, into the objects constituted at the higher level’.^{Footnote 125} (Recall that Husserl says that ‘such an irreality has the temporal being of supertemporality, of omnitemporality, which however is a mode of temporality’.^{Footnote 126})

In view of the above, Null’s ascription of claims (1) and (2) to me is entirely mistaken, and, since his review turns on them, his discussion is, in effect, not about my book.^{Footnote 127}

^{Footnote 128}

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Cite this chapter

van Atten, M. (2017). Construction and Constitution in Mathematics. In: Centrone, S. (eds) Essays on Husserl's Logic and Philosophy of Mathematics. Synthese Library, vol 384. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1132-4_12

Download citation

DOI: https://doi.org/10.1007/978-94-024-1132-4_12
Published: 08 September 2017
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-024-1130-0
Online ISBN: 978-94-024-1132-4
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)

Publish with us

Policies and ethics

Construction and Constitution in Mathematics

Abstract

Access this chapter

Subscribe and save

Buy Now

Similar content being viewed by others

Construction and Constitution in Mathematics

Reconciling the Realist/Anti Realist Dichotomy in the Philosophy of Mathematics

Constitution and Construction

Notes

References

Acknowledgements

Author information

Authors and Affiliations

Corresponding author

Editor information

Editors and Affiliations

Appendix: Null on Choice Sequences

Rights and permissions

Copyright information

About this chapter

Cite this chapter

Download citation

Publish with us

Subscribe and save

Buy Now

Navigation

Construction and Constitution in Mathematics

Abstract

Access this chapter

Subscribe and save

Buy Now

Similar content being viewed by others

Construction and Constitution in Mathematics

Reconciling the Realist/Anti Realist Dichotomy in the Philosophy of Mathematics

Constitution and Construction

Notes

References

Acknowledgements

Author information

Authors and Affiliations

Corresponding author

Editor information

Editors and Affiliations

Appendix: Null on Choice Sequences

Appendix: Null on Choice Sequences

Rights and permissions

Copyright information

About this chapter

Cite this chapter

Download citation

Share this chapter

Publish with us

Search

Navigation