Mark Van Atten
email: vanattenmark@gmail.com

Eric Audureau

Matthias Baaz

Paul Benacerraf

Ju- Lien Bernard

Marc Bezem

Paola Cantù

Pierre Cassou-Noguès

Thierry Coquand

Gabriella Crocco

Dirk Van Dalen

Cheryl Dawson

Michael Detlefsen

Igor Douven

Jacques Dubucs

Eva-Maria Engelen

Fernando Ferreira

Juliet Floyd

Jaime Gaspar

Warren Goldfarb

Yannick Grannec

Leon Horsten

Piet Hut

Shinji Ikeda

Nuno Jerónimo

Aki Kanamori

Juliette Kennedy

Roman Kossak

Georg Kreisel

Nico Krijn

Per Paolo Mancosu

Amélie Mertens

Mitsu Okada

Marco Panza

Charles Parsons

Jan Von Plato

Adrian Rezuş

Robin Rollinger

Gian-Carlo Rota

Philippe De Rouilhan

Rudy Rucker

Wil- Fried Sieg

Hourya Sinaceur

Göran Sundholm

Steven Tainer

Richard Tieszen

Robert Tragesser

Anne Troelstra

Jouko Väänänen

Albert Vis- Ser

This is a collection of most of the essays on Kurt Gödel that I have authored or co-authored. In their publication over the past decade, these essays have been dispersed, as they address various topics for various primary audiences: philosophers of logic and mathematics, phenomenologists interested in science, and historians of modern philosophy. The rationale for bringing them together here is that, not so much in spite as because of this variety, they show a coherence predicated on that of the many-sided project of Gödel's that they collectively analyse: the project of using Husserl's phenomenology to reconstruct and develop Leibniz' monadology as an axiomatic metaphysics, and then to provide a Platonistic foundation for classical mathematics starting from the metaphysics thus obtained. Brouwer's intuitionism serves as a foil. In choosing the title of this book, I have preferred descriptive accuracy to other, perhaps greater, qualities.

At times these essays go into issues internal to Leibniz', Husserl's, or Brouwer's thought that Gödel made few explicit comments on, or none at all. But in an evaluation of Gödel's reception of other philosophers this is only to be expected: A judgement to what extent the various ideas Gödel appeals to fit together and suit the purpose he has for them de-

pends not on the web of Gödel's remarks, but on the web of ideas to which those remarks point.

I have chosen to leave the papers in their original form, and not to rework them into one continuous narrative. Naturally, this entails some repetition and overlap, but I hope that this at the same time facilitates access to the book as a whole. My consideration has been twofold. First, the continuous narrative would have been the form of choice for the more organic and comprehensive analysis that an intellectual biography of Gödel calls for; but, although the essays collected here may be read as preparatory steps for a biography of that type, they cannot, in their limitation to this one particular project, be more than that, and more should not be suggested. (In the Introduction, I give some examples of how these essays refrain from establishing connections to other parts of Gödel's life of the mind.) Second, it is intrinsic to Gödel's project to be of interest from different perspectives and to different audiences; a form of presentation that explicitly responds to these differences is therefore not inappropriate.

For the occasion of their reprint in this volume, the essays have been recast in a uniform format, including uniform bibliographical references, and citations and translations have silently been added where missing. Spelling and punctuation have been standardised to British where appropriate. But otherwise I have followed what I consider to be the good practice of not revising papers when collecting them. Occasionally I have added a content footnote to these reprints, flagged as such; and the citation footnotes that were required by some journals have been deleted in favour of citations in the main text. As a consequence, the footnote numbering in these reprints in general diverges from that in the original publications. To be able to make the chapters available separately, as is required for the electronic edition, each comes with its own list of references; with an eye on the paper edition, a cumulative bibliography has been added, as well as a subject index and an index of authors and citations. In quotations, translations are my own, while emphasis stems from the author quoted, except where noted otherwise.

Generous institutional support during the writing of these essays came from the Department of Philosophy at the University of Leuven; the Institut d'Histoire et de Philosophie des Sciences et des Techniques (CNRS / Paris I / ENS), Paris; the Institute for Advanced Study, Princeton; and Sciences, Normes, Décision (CNRS / Paris IV), Paris.

Many thanks are due to Shahid Rahman and John Symons for accepting this volume in their series Logic, Epistemology, and the Unity of Science. At Springer, Ties Nijssen and Christi Lue were helpful and efficient editors. Springer also engaged a reader who commented on the manuscript as a whole, which I much appreciated. I prepared the manuscript for printing using Donald Knuth's typesetting system T E X and Leslie Lamport's extension L A T E X, editing my files with Rob Pike's editor Acme. I am grateful to the authors of these very useful and interesting programs, and to the internet communities dedicated to them for their advice and examples.

Without the love and patience of my wife and son, this book could not have been completed.

This volume is dedicated to the memory of Leen Stout, who, in his history class at the Erasmiaans Gymnasium in Rotterdam, oversaw my first writing on Gödel.

Saint-Germain-en-Laye, MvA. May 2014 2 Introduction I have been trying first to settle the most general philosophical and epistemological questions and then to apply the results to science.

Gödel to Cohen, 19671

There is no definite knowledge in human affairs.

Even science is very prejudiced in one direction. Knowledge in everyday life is also prejudiced. Two methods to transcend such prejudices are:

(1) phenomenology;

(2) going back to other ages.

Gödel to Wang, early 1970s2 Abstract. After a statement of the subject and aim of the book, three aspects of Gödel's philosophical methodology are discussed: Gödel's commitment to phenomenology from about 1959 to the end of his life, the religious component in phenomenology, and the pragmatic value of Husserl's and Gödel's historical turns in philosophy. Finally, an overview is provided of the essays that follow.

1. [START_REF] Gödel | Correspondence A-G[END_REF], 386. 2. Wang 1996, 308. Wang does not give the exact date; p. 326 suggests it is 1971. But if comments 9.3.22 and 9.3.23 (the one quoted here) were made in the same session, then it seems it should be 1972, as Gödel made the suggestion to change 'structural factualism' to 'factual substantialism' in 1972 and on p. 144 Wang says these comments were made after that renaming. Of course, the appropriateness of the motto does not hinge on this.

Subject and aim

Far from considering past philosophers irrelevant to actual systematic concerns, Kurt Gödel3 embraced the use of historical authors to frame his own philosophical perspective and work. The subject of this book is a project of his defined by reference to Leibniz and Husserl, consisting of two stages:

1. Use Husserl's transcendental phenomenology to reconstruct and develop Leibniz' monadology into an axiomatic metaphysics, 2. Apply the metaphysics thus obtained to develop a Platonistic foundation for classical mathematics.

By 'Platonism', I here mean the view that, to adopt one of Gödel's own formulations, 'mathematical objects and facts (or at least something in them) exist independently of our mental acts and decisions';6 I will not 3. For readers wishing to form a picture of Gödel as a person, I refer to the biographies [START_REF] Dawson | Logical Dilemmas: The Life and Work of Kurt Gödel[END_REF][START_REF] Yourgrau | A World Without Time: The Forgotten Legacy of Gödel and Einstein[END_REF]; the accounts by [START_REF] Kreisel | Kurt Gödel: 28 April 1906-14 January 1978[END_REF] and Rucker (1983, 164-171) of their respective personal contacts with him;4 and the extant snippet of film footage of Gödel at https://www.youtube.com/watch?v=aq9X-ERgnuY (Gödel appears from 1:05 to 1:25).5 There are now also various dramatisations in novels or theatre plays; the one whose portrayal of Gödel I prefer is Grannec 2012 (whose main character is actually Gödel's wife, Adele). 4. In his blog, Rudy Rucker has now made available scans of his handwritten conversation notes: http://www.rudyrucker.com/blog/2012/07/31/conversatons-with-kurtgodel/ (note the misspelled 'conversatons'). There is also a reprint of the text of his 1983 account at http://www.rudyrucker.com/blog/2012/08/01/memories-of-kurtgodel/ 5. According to the text at http://www.cosmolearning.com/videos/einstein-diracgodel-selberg-harish-chandra-in-princeton-1947-1125/ (paired to a likewise impressive, but different video), the film was made in 1947 by the mathematician Abe Gelbart. Gelbart was a member of the IAS in 1947-1948. 6. Gödel, *1951, 311, 311. For a discussion of Gödel's Platonistic views throughout his career, see [START_REF] Parsons | Platonism and mathematical intuition in Kurt Gödel's thought[END_REF] be concerned with the familiar question to what extent Plato's own views were Platonistic in this sense.7

The aim of this book is to analyse historical and systematic aspects of this project of Gödel's, and to assess its feasibility. The emphasis is on its second stage, and, to that end, also the discussion of the first stage is oriented towards pure mathematics; correspondingly, my assessment will be a partial one, as I will have relatively little to say about the feasibility of the first stage.

Towards the end of his life, Gödel was willing to admit in print that he had not succeeded in completing the first stage. In a text of 1976 called 'Some facts about Kurt Gödel', Hao Wang's written record of an account that Gödel gave him of his own intellectual development, and that Gödel permitted Wang to publish after his death, we read that In philosophy Gödel has never arrived at what he looked for: to arrive at a new view of the world, its basic constituents and the rules of their composition. (Wang 1987, 46) Moreover, Wang reports elsewhere that Gödel did not think that he himself had come close to attaining the ideal of an axiomatic theory of metaphysics. He said several times that he did not even know what the primitive concepts are. (Wang 1996, 294) The fact that, at a given point, one has not attained the ideal, is of course no argument that this cannot be done. As Gödel had remarked to Wang a few years before, in 1972: 7. For discussion of that question, see, e.g., [START_REF] Burnyeat | Platonism and mathematics[END_REF]Moravcsik 1992, Chap. 7;[START_REF] Pritchard | Plato's Philosophy of Mathematics[END_REF] A wider-ranging book-length discussion of Platonism in contemporary mathematics is Panza and Sereni 2013.

It is not appropriate to say that philosophy as a rigorous science is not realizable in the foreseeable future. Time is not the main fact [factor]; it can happen any time when the right idea appears. (Wang 1996, 143) Indeed, perhaps the right idea will appear, and the first stage of Gödel's project will be completed. The upshot of my final chapter, however, will be that the second stage is bound to fail. The reasons for that failure will turn out to be internal to transcendental phenomenology as Husserl developed it, rather than due to some twist that Gödel put on it. This means that, if my argument there is correct, its range of application is wider than just the case of Gödel's project.

The essay in this volume that would serve well as an extended introduction to Gödel's project, and hence also to this book, is 'On the philosophical development of Kurt Gödel' (written with Juliette Kennedy). The reader looking for such a longer introduction is advised to start there. If I have nevertheless not put that paper before the others, it is because I have preferred the alternative of an arrangement congruent with the inner logic of Gödel's project: Start with Leibniz (Part II), modify and develop using Husserl (Part III), compare with Brouwer (Part IV), and, finally, assess (Part V). But not much is at stake here, as the essays can be read in different orders.

The comparison with Brouwer is occasioned by the fact that his intuitionism is, to my mind, the principal foil for Gödel's project: the close affinities between phenomenology and intuitionism -conceptual affinities, and to some extent also historical ones -set the bar for Gödel's attempt to use phenomenology in quite the opposite, non-constructivistic direction. Ample attention is given therefore not only to Gödel's reception of Leibniz and Husserl, but also to his lifelong, vivid interest in Brouwer's intuitionism and the challenge that that alternative foundation poses to his project.

Gödel obviously had many other philosophical interests besides the project under discussion, e.g., Plato,8 medieval philosophy,9 Kant,10 other varieties of post-Kantian German Idealism than transcendental phenomenology, the relation of the monadology to (modern) physics and biology, and Sheldon, Royce, and Hartshorne.11 I make no attempt here to analyse any of these interests; in a more general work on Gödel and philosophy that would of course be required.

And although I occasionally make essential use of material from Gödel's pre-phenomenological Max-Phil notebooks (Gödel Papers,, filled from the late 1930s to 1946 and then, much more sporadically, until about 1955, I make no systematic effort to relate these notebooks to the project under discussion. It is clear from the partial transcription of Max-Phil that is presently available that various ideas recorded there remained dear to Gödel, who for example in the 1970s repeated them in conversations with Hao Wang and with Rudy Rucker. But it is, certainly in hindsight, evident that Gödel had grown discontented with the philosophical approach he had taken in those notebooks, and that to some extent the turn to phenomenology was meant as a new start.12 These is-8. See, e.g., Toledo's notes of her conversations with Gödel [START_REF] Toledo | Sue Toledo's notes of her conversations with Gödel in 1972-5[END_REF]) and the comments on them in Franks 2011. Also Yourgrau 1989, 394-395;397-403;405;407-408;and Yourgrau 1999, 196-200. 9. See, e.g., Engelen 2013. One particular aspect of Gödel's interest in medieval philosophy will play a role in 'Monads and sets ' (Chap. 3,Sect. 3.4.3). 10. See, e.g., Yourgrau 1999, Chap. 5;[START_REF] Kovač | Gödel, Kant, and the path of a science[END_REF][START_REF] Parsons | Gödel and philosophical idealism[END_REF]. A work that one would have expected to find notes to in Gödel's archive is Husserl und Kant [START_REF] Kern | Husserl und Kant[END_REF], which appeared as Vol. 16 of the series Phaenomenologica, available at the Princeton University Library. (Gödel's archive does contain reading notes to Vols. 2 and 4 of that series, together with a note on Vols. 1 to 23; Gödel Papers 9c/22.) 11. These are the three he mentions when asked by Wang 'to name some recent philosophers whom he found congenial' (Wang 1996, 141). 12. Gödel had clearly been looking for such a new start. In a letter of April 20, 1967, Kreisel said to Gödel that the 'pregnancy' of the latter's formulations in a letter on Feferman's work had reminded him of a conversation in 1956, in which Gödel had mentioned to him that he was going to write a book on philosophy (Gödel Papers 01/90, 011233).

sues can probably not be tackled before the transcription of Max-Phil has been completed.13 Finally, I should mention Gödel's philosophical correspondence with Gotthard Günther (Gödel 2003, 456-535). It took place precisely in between the period of the Max-Phil notebooks and Gödel's turn to phenomenology [START_REF] Husserl | Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie[END_REF][START_REF] Gödel | /? The modern development of the foundations of mathematics in the light of philosophy[END_REF]Gödel's last letter is from 1959). In his Introduction to that correspondence in the Collected Works, Charles Parsons arrives at the following conclusion: [Gödel] was evidently prepared to entertain the possibility that post-Kantian idealism, to which he had apparently not had a lot of exposure, would be a source of illumination. He found Günther a clear expositor of ideas from that tradition. But he does not seem to have been disposed to work out himself a line of thought in which self-consciousness is a central concept, and when Günther did not pursue what Gödel thought the most promising direction, he lost interest. Not long after his last letter he began his study of Husserl, whose version of idealism he seems to have found much more satisfactory. (Gödel 2003, 475-476) As I think that that conclusion is correct, and that Günther's thought did not significantly contribute to Gödel's project either as a source or as a foil, I will not treat of it here. That is not to say, however, that I think that Günther's work and his exchange with Gödel are without systematic and historical interest for idealistic philosophy.

13. That work is currently being done in a group led by Gabriella Crocco (Université d'Aix-Marseille), of which I am a member.

Gödel's commitment to phenomenology

The project described here was central to Gödel's philosophical thought from about 1959, when he began his serious study of Husserl,14 until the end of his active career. The best known of Gödel's (implicit or explicit) recommendations of phenomenology is of course the posthumously published essay of 1961, 'The modern development of the foundations of mathematics in the light of philosophy' [START_REF] Gödel | /? The modern development of the foundations of mathematics in the light of philosophy[END_REF]. Others are:

1. His remark in a letter to Bernays of August 11, 1961, about Kreisel's work purporting to show that 0 is the exact limit of finitary mathematics:

I find this result very beautiful, even if it will perhaps require a phenomenological substructure in order to be completely satisfying. (Gödel 2003, 193)15

2. Gödel's remark in a draft for the supplement to the 1964 reprint of his Cantor paper, left out from the published version:

Perhaps a further development of phenomenology will, some day, make it possible to decide questions regarding the soundness of primitive terms and their axioms in a completely convincing manner. (Gödel Papers 4/101,040311,12) 14. That is the year he mentions to Wang in 1976(Wang 1987, 46;Wang 1996, 88). Among the first items Gödel studied was the 1959 volume of the Zeitschrift für philosophische Forschung, of which the first two issues contained a number of contributions on Husserl, on the occasion of the latter's 100th birthday. The library request slip in Gödel's archive is stamped October 21, 1960 (Gödel Papers 5/22,050111). 15. 'Ich finde dieses Resultat sehr schön, wenn es auch vielleicht eines phänomenologischen Unterbaus bedürfen wird, um voll zu befriedigen.' (Gödel 2003, 192) 3. His recommendation to some logicians in the 1960s, reported by Wang, 'that they should study the sixth investigation in Logical Investigations for its treatment of categorial intuition' (Wang 1996, 164).

4. Kreisel's strongly expressed wish, in a letter to Gödel of June 12, 1969 (Gödel Papers 1/92, 011266), that, in one of their future conversations, Gödel give him examples of detailed phenomenological analyses. This indicates that in their exchanges at the time Gödel had continued to advocate phenomenology.16

5. Gödel's arrival at the view of the Dialectica Interpretation as an application of phenomenology in the late 1960s. Details are presented in 'Gödel and intuitionism' (Chap. 11). The work on the revision of the Dialectica paper is surely Gödel's deepest response to his reading of Husserl. (Note that this work had begun before Kreisel's letter mentioned in the previous item; apparently, but if so, not uncharacteristically, Gödel had refrained from bringing it up in their conversations.)

6. Gödel's statement in a draft letter to Gian-Carlo Rota of 1972 that I believe that his [i.e., Husserl's] transc [endental] phen[omenology], carried through, would be nothing more nor less

16. An advocacy that had not been lost on Kreisel. In that same year, Kreisel published his one recommendation of phenomenology that I know of:

What this shows is, at most, that the notions considered [of subset and powerset] are difficult to analyze, not that they are dubious … Coming back to set theory, probably the first step is: to recognize the objectivity of the basic notions (subset, powerset) mentioned above; and then, if possible, to give a phenomenological analysis of these notions. (Kreisel 1969, 97) It is clear that in fact the whole paper is strongly influenced by Gödel.

than Kant's critique of pure reason transformed into an exact science, except for the fact that the result (of the 'critique') would be far more favourable for human reason. (Gödel Papers 1/141, 012028.7)

7. Wang's report that 'in his discussions with me in the 1970s he repeatedly urged me to study Husserl's later work' (Wang 1996, 164).17

8. Gödel's statement to Wang in these same discussions that Husserl's is a very important method as an entrance into philosophy, so as finally to arrive at some metaphysics. Transcendental phenomenology with epoche as its methodology is the investigation (without knowledge of scientific facts) of the cognitive process, so as to find out what really appears to beto find the objective concepts. (Wang 1996, 166) Even when Gödel acknowledged to Wang that Phenomenology is not the only approach. Another approach is to find a list of the main categories (e.g., causation, substance, action)

17. In From Mathematics to Philosophy (Wang 1974, 189), Wang also writes:

With regard to the task of setting up the axioms of set theory (including the search for new axioms), we can distinguish two questions, viz (Toledo 2011, 206) But such qualms are perfectly compatible with a commitment to phenomenology as a body of thought. Also the occasional criticism of Husserl does not change this, as when, for example, Gödel says to Toledo in 1972 that His analysis of the objective world (e.g., p. 212 of From Formal to Transcendental Logic [sic]) is in actuality universal subjectivism, and is not the right analysis of objective existence. It is rather an analysis of the natural way of thinking about objective existence. (Toledo 2011, 202)19

Likewise, the fact that Gödel never published anything on (or using) phenomenology does not, by itself, indicate a reservation on Gödel's part about the validity of phenomenology. [START_REF] Kennedy | Review of Richard Tieszen, After Gödel: Platonism and Rationalism in Mathematics and Logic[END_REF] sees the fact that the 1961 essay does not appear on either of Gödel's two lists 'What I could publish' (Was ich publizieren könnte) found in the archive20 as one step towards the view that 'judgement on this point [i.e., Gödel's commitment to phenomenology] should perhaps be left open'. But that fact is wholly consistent with a characteristic trait of Gödel's, here exemplified for his views on mind and matter (Wang 1996, 5):

In commenting on a draft of this paper,21 Gödel asked me to add the following paragraph: ing mind and matter which he believed are contrary to the commonly accepted views today. The reasons for his convictions are of a very general philosophical nature and the arguments he possessed are not convincing to people with different convictions. Hence, he had chosen to state only those parts or consequences of his convictions which are definite even without reference to his general philosophy.

All things considered, then, there is no question but that from 1959 until the end of his active career Gödel was not only studying phenomenology, but moreover was committed to it, seeing it not as a finished doctrine laid down in any single text, but as a research program, to be developed, applied, and modified in the light of further reflection and experience. This is of course the only way in which commitment to a body of thought can make philosophical sense. Husserl himself had seen it that way, as Gödel, and any attentive reader of Husserl's work, was well aware:

According to Gödel, Husserl just provides a program to be carried out. (Wang 1996, 164) Given Gödel's strong commitment, it is regrettable that, much as Gödel scholarship owes to Hao Wang,22 Gödel did not, as far as I have been able to determine, discuss phenomenology also with someone who was more interested in it and better prepared. I am thinking of William Howard in particular, who recounts:

In the fall of 1972 I am having lunch in the Institute cafeteria, and in walks Hao Wang. We know each other from ASL meetings in past years.

Hao Wang: 'I come in once every two weeks, from New York, for a meeting with Gödel. He is making me read various parts of Husserl's writings, which I don't particularly want to do,23 and then, at the meetings, he makes me discuss what I have read.' I told this to my friend Tennenbaum, and he said, 'Gödel is one of the greatest living authorities on Husserl'. I decided that I should take advantage of this, so I went to the Princeton University bookstore and looked for books by Husserl. There was one called Cartesian Meditations. I decided that, since I was an expert on meditation, this was down my alley. I was right. So I studied this and also the two books by Husserl that Hao Wang told me Gödel was making him read.24 When I felt I was sufficiently prepared (early spring of 1973), I tried to get Gödel to talk about Husserl.

No dice! Gödel had decided that Husserl was not on the agenda for any of our meetings, and that was that. (Howard, Now and then Gödel mentioned things of interest to me which seemed related to what we had discussed on some previous occasion. When I asked him why he had not said these things before, he would reply, 'But you did not ask me'. I interpret this response to imply that, since he had so many ideas on so many things, he preferred to limit his remarks to what was strictly relevant to the immediate context. One consequence of this was that he avoided topics and views on which he did not believe there was a shared interest, or even some empathy.

Gödel's reply to Wang reminds one of a dictum of the 14th-century monk Kenkō, which suggests a different explanation than Wang's, or, depending on Gödel's psychology, a complementary one:

It is impressive when a man is always slow to speak, even on subjects he knows thoroughly, and does not speak at all unless questioned. [START_REF] Kenkō | Essays in Idleness: The Tsurezuregusa of Kenkō[END_REF] 1967, 69)

Howard's case, however, was the opposite of Wang's, for he was interested in phenomenology, and he did ask. Gödel's refusal must have been a frustrating experience for Howard, as it is, indirectly, for me today.26 More generally, the regrettable fact seems to be that there is no record 26. In an email of March 7, 2013, William Howard adds:

I did not feel that he was brushing me aside; it was just that he had a list of topics that he wanted to discuss with me, and Husserl was not on the list. When I say 'list', I mean it literally: When I arrived for our meetings, he would have a sheet of paper on his desk before him, a sort of memorandum to himself concerning the topics for our meeting. I had a list of topics that I wanted to discuss (not on a sheet of paper but firmly in my mind); but he had his own questions, which he would ask me one after the other; it was hard for me to get any of my questions in edgewise! left of anyone's conversation with Gödel on phenomenology at the level of expertise that his remaining reading notes and bibliographical memoranda on the topic indicate he was capable of.

The religious component in phenomenology

In Reflections on Kurt Gödel, Hao Wang suggests a difference between Gödel and Husserl that, to my mind, would strongly limit the extent to which Husserl's phenomenology could be used to realise the first stage Gödel's project, if it indeed exists:

In addition, G looks for an exact or axiomatic theory in philosophy and thinks that it is also Husserl's aim. But G's conception of metaphysics as first philosophy includes centrally the concepts of God and soul.27 It appears clear that this religious component is not part of Husserl's conception of philosophy. (Wang 1987, 161) 27. Note MvA. There is of course Gödel's remark in 1970, reported by Oskar Morgenstern in a diary note for August 29, 1970, that he feared that publishing his ontological proof of God's existence would lead people to think 'that he actually believes in God, whereas he is only engaged in a logical investigation (that is, showing that such a proof with classical assumptions (perfection, etc.), correspondingly axiomatized, is possible.)' (Gödel 1995, 388, translation modified).28 But, besides to Wang (e.g., Wang 1996, 88), over the years Gödel expressed an unequivocal belief in God in a number of places. To mention three: the Max-Phil notebooks; a series of letters to his mother in 1961 (Gödel 2003, 428-439); and a draft reply of 1975 to a questionnaire of the sociologist Burke Grandjean where he specified that 'My belief is theistic not pantheistic (following Leibniz rather than Spinoza)' (Gödel 2003, 448). It seems safe to say, then, that Gödel believed in God more often than not. See also Chap. 10, footnote 4 in this volume. 28. 'Über sein ontologischen Beweis -er hatte das Resultat vor einigen Jahren, ist jetzt zufrieden damit aber zögert mit der Publikation. Es würde ihm zugeschrieben werden daß er wirkl[ich] an Gott glaubt, wo er doch nur eine logische Untersuchung mache (d.h. zeigt, daß ein solcher Beweis mit klassischen Annahmen (Vollkommenheit usw.), entsprechend axiomatisiert, möglich sei)', as quoted in Dawson 1997, 307. Does the last sentence of this quotation reflects Gödel's view or Wang's?29 It is, in any case, not difficult to see what may have suggested that view.

In his published works Husserl speaks about God very rarely, and when he does so in Sects. 51 and 58 of Ideas I [START_REF] Husserl | Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie: Erstes Buch[END_REF], it is to say that God can neither be a mundane object nor have His being as an episode in consciousness (Sect. 51), that God transcends both the world and absolute consciousness (Sect. 58), and that God therefore falls outside the scope of phenomenology as Husserl defines it there.

Yet, in a letter to William Ernest Hocking of July 7, 1912, so from the time Husserl was writing Ideas I, Husserl writes that Even if I have made it my life's task to found a philosophy 'from below' at least for myself, to my satisfaction (which is very difficult to gain!), I nevertheless strive unceasingly from this 'below' upwards into the heights. In the last years, metaphysical considerations, and especially the idea of God, have entered ever more powerfully into the horizon of my studies. (Brainard 2002, 251-252n80)30 29. In the articles on Husserl in the volume of the Zeitschrift für philosophische Forschung that Gödel borrowed in 1960, mentioned in footnote 14 above, it is made very clear that Husserl believed in God and that this plays a central role in his later philosophy. See in particular Diemer 1959, 248-250 (who also notes the contrast between Husserl's published work and his correspondence) and Ingarden 1959, 462. There are various other such places in the early literature on Husserl after 1945 that Gödel is likely to have seen, but the present example is documented and already strongly suggests that the view Wang states in this quotation is not Gödel's. Note that, in a perceptive comment on Gödel's 1961 essay, Wang remarks that 'His proposed solution appears to be Husserl's phenomenology, and he says nothing explicitly about its relation to religious concepts … Elsewhere he suggests that Husserl's method may be applicable to metaphysical or religious concepts as well' (Wang 1996, 162). 30. 'Habe ich es mir zur Lebensaufgabe gemacht eine Philosophie "von unten" mindestens für mich, zu meiner (sehr schwer zu gewinnenden!) Befriedigung zu begründen, so strebe ich doch unablässig von dem "Unten" hinauf in die Höhen. In den letzten Jahren sind metaphysische Erwägungen und ist insbesondere auch die Indeed, in research manuscripts and correspondence from 1908 (predating Ideas I ) until the end of his life,31 kept reflecting on God and metaphysics in relation to phenomenology, against the background of Leibniz. (Gödel would have been sensitive to this contrast between Husserl's published and unpublished writings.) For example, a text from 1908 has the title 'Teleology, God, the possibility of an all-consciousness, transcendentalphenomenologically founded metaphysics and teleology'.32 In it, Husserl presents a conception of God as the universal consciousness, which creates the finite monads and unifies all the contents of their consciousnesses. That can certainly be seen as an interpretation of Leibniz' concept of the central monad; similarly, a later text, probably from 1922, considers, as its title indicates, 'The possibility of fusion of monads; the possibility of a highest (divine) monad'.33 But it is important to note that for Husserl these are questions, possibilities and convictions that he ponders as such; he does not present full phenomenological analyses leading to conclusions. And to the philosopher Husserl, as distinct from the faithful Christian that he also was,34

Gottesidee immer stärker in den Kreis meiner Studien getreten.' (Husserl 1994a, 3:160) 31. A highly interesting report on conversations with Husserl on religion in the last years of his life is Jaegerschmid 1981a and Jaegerschmid 1981b. For a biographical perspective on Husserl's religiosity, see Karl Schuhmann's introduction to Husserl's correspondence (Husserl 1994a, 10:33-36). For systematic considerations, the most important reference here is of course part III of Husserl 2013, 'Metaphysik: Monadologie, Teleologie und Philosophische Theologie'. See also [START_REF] Hart | A précis of a Husserlian philosophical theology[END_REF][START_REF] Iribarne | Husserls Gottesauffassung ind ihre Beziehung zu Leibniz[END_REF][START_REF] Lo | Die Gottesauffassung in Husserls Phänomenologie[END_REF], and Ales Bello 2009. 32. 'Teleologie, Gott, Möglichkeit eines All-Bewusstseins. Transzendentalphänomenologisch fundierte Metaphysik und Teleologie' (Husserl 2013, 160-168). 33. 'Möglichkeit der Verschmelzung von Monaden. Möglichkeit einer (göttlichen)

Übermonade' (Husserl 1973, 300-302). 34. Husserl was a Jew by birth, but was not raised as a practising one. As a student he read the New Testament, decided to convert to Christianism, and was baptised Lutheran (as Gödel would be, and Leibniz had been). But Husserl (again like Gödel, and like Leibniz) was not a churchgoer. Brouwer (presumably) was baptised Dutch it remained essential to follow the right methodology.35 As he writes in a letter of 1933,

The philosophical problems disclose themselves in their genuine meaning as transcendental-phenomenological ones in an essential systematic series of steps. On these occasions it becomes manifest that the religious-ethical problems are problems of the highest level. … This is precisely the reason why in my writings I kept silent about the problems of philosophy of religion. (Spiegelberg 1981, Reformed, as Wang 1996, 27 andDawson 1997, 4-6;on Husserl, Husserl 1994a, 3:432;on Leibniz, Guhrauer 1846, 1:1, 2:332. 35. When Husserl received a copy of Rudolf Otto's The Holy (Das Heilige, Otto 1918), he wrote in a letter to its autor (and his friend and former colleague in Göttingen) of March 5, 1919, that he much appreciated the book for its description of religious phenomena, but criticised its philosophical elaboration as follows: 'The metaphysician (theologian) in Mr Otto has carried, so it seems to me, the phenomenologist Otto away on his wings and for an image I think here of the angels who with their wings cover the eyes' (Husserl 1994a, 7:207;'Der Metaphysiker (Theologe) in Herrn Otto hat scheint es mir den Phänomenologen Otto auf seinen Schwingen davongetragen u[nd] ich denke dabei als Gleichnis an die Engel, die mit ihren Schwingen die Augen verdecken.'). Husserl's reference is to Isaiah 6:2.

182)36,37

Moreover, in a letter of a few months earlier, Husserl did estimate that he had made progress in being able to frame the questions in the right manner. About the question of God as 'indeed the "highest and final question" in the system-building of the phenomenological method'38 he writes:

I am grateful enough that I have been able to develop the method and explicitly carry on with it to see the theoretical locus of the problem as a phenomenological one: first of all as the problem of the possibility of the transcendental Totality.39 Indeed, Husserl wrote in probably Summer 1934, this is where philosophy intrinsically leads:

An autonomous philosophy, such as the Aristotelian was and such as remains an eternal demand, will necessarily arrive at a philosophical teleology and theology -as a non-confessional way to God.40 36. Husserl to E.P. Welch, June 17/21, 1933; 'Die philosophischen Probleme erschliessen sich mit ihrem echten Sinn als transcendental-phänomenologische in einer wesensmässigen systematischen Stufenfolge. Es zeigt sich dabei, dass die ethischreligiösen Probleme solche der höchsten Stufe sind. … Eben darum schwieg ich mich in meinen Schriften über religionsphilosophische Probleme aus' (Husserl 1994a, 6:459). 37. A similar attitude is found in the work of Michael Dummett; see for example his introduction to The Logical Basis of Metaphysics [START_REF] Dummett | The Logical Basis of Metaphysics[END_REF]. 38. 'die in der Tat im Systembau der phänomenologischen Methode "höchste und letzte

Frage" ' 39. Husserl to Father Daniel Feuling, March 30, 1933, in Husserl 1994b, 7:87f. 'Ich bin dankbar genug, dass ich die Methode soweit durchbilden und explizit fortführen konnte, um den theoretischen Ort des Problems als eines phänomenologischen zu sehen: zunächst als des Problems der Möglichkeit der transcendentalen Totalität.' 40. 'Eine autonome Philosophie, wie es die aristotelische war und wie sie eine ewige

The purpose of presenting the quotations from Husserl above is to show that, contrary to what the passage from Wang suggests, a religious component was very much present in Husserl's conception of philosophy; from a systematic point of view there is, ultimately, no mismatch between Gödel's and Husserl's aims in philosophy on this account.

I had hoped to find in Gödel's archive reading notes to Dietrich Mahnke's 'Eine neue Monadologie ' (1917), essentially a rewriting of Leibniz' tractate in phenomenological terms. Gödel thought highly of this work by Mahnke, assessing it as 'sensible!' (vernünftig! ; Gödel Papers 5/25, 050120.1); the paragraphs on God in it are closely related to (but not the same as) Husserl's ideas on the topic.41 The task of broadening the contextualisation of Gödel's project as presented here should start, I believe, with an analysis of Mahnke's work on the monadology. It would be very interesting if it could be determined whether Gödel's thoughts on the matter were, in effect, closer to Husserl's or to Mahnke's.42 Likewise, one could reflect on Gödel's version of the ontological proof in this context. A key question here is whether Gödel saw that proof as making merely a logical point43 or took that proof indeed to establish the existence of God; as Parsons observes, For that, it would be necessary for him to have confidence in the specific conceptual apparatus and premises of the proof. I suspect that if questioned about that, he would have said that he had not developed his philosophical views to a sufficient extent to have that level of certainty. (Parsons 2010, 186) Forderung bleibt, kommt notwendig zu einer philosophischen Teleologie und Theologie -als inkonfessioneller Weg zu Gott.' (Husserl 2013, 259) 41. See also Chap. 6, Sect. 6.4.1 in this volume. 42. For a list of differences between their positions, prepared by Husserl's assistant Eugen Fink in 1933, see Husserl 1994a, 3:519-520. 43. As the remark taken down by Morgenstern suggests; see footnote 27 above.

One way of trying to obtain that level of certainty would be to develop a phenomenological critique of that conceptual apparatus and the premises. It is not clear to me whether or not Husserl at some point meant to go that way. In 1892/93, Husserl lectured on proofs of God's existence (Schuhmann 1977, 34), but no lecture notes seem to remain, and in any case this was long before Husserl's development of the transcendental phenomenology that interested Gödel. A relevant reminiscence about the transcendental Husserl, one of the few pieces of evidence on the topic, can be found in the memoirs of the biologist and philosopher Hans Driesch. On a very long conversation he had with Husserl at a conference in April 1914, he notes:

Of particular interest to me was a specific point in our conversation. I asked H[usserl] whether I was right to see the ontological proof of God -which from the 'essentia', i.e., the conceptual 'essence' of God, wants to conclude to his 'existentia' -as the final goal of his logical investigations. He answered yes to this question, but he has, as far as I know, never gone into it in his writings.44

The qualification 'as the final goal' is crucial; certainly Husserl would not, after his transcendental turn, accept, in a philosophical sense, God's existence without further phenomenological ado. A forceful statement to this effect in relation to proofs of God's existence was made by Husserl almost twenty years later, in a letter of 1932 to Father Erich Przywara. Having stated his methodological priority of an exhibition and transcendental cri-44. Translated from the quotation in Schuhmann 1977, 186: 'Von besonderem Interesse war mir ein bestimmter Punkt unserer Unterhaltung. Ich frug H[usserl], ob ich im Recht sei, wenn ich den ontologischen Gottesbeweis -(der aus der "essentia", d.h. dem begrifflichen "Wesen" Gottes, seine "existentia" ableiten will) -als letztes Ziel seiner logischen Untersuchungen ansahe. Er bejahte diese Frage, ist aber meines Wissens nie schriftstellerisch auf sie eingegangen.' The original is in the posthumous Driesch 1951, 153-154. tique of the evidence again, and having confirmed that this also applies to religious evidence, Husserl continues:

That will help the theologians some day, although at first it will seem as if this results in bad heresies.45 Any wish to commit myself to theism in the Scholastic tradition -in the usual interpretations of its intentions -I decidedly dismiss. Of course this is said against Mr Keilbach and his proofs of God's existence that I (according to things I am supposed to have said in conversation) allegedly hope for.46,47 45. Note MvA. See also this passage in Formal and Transcendental Logic, written three years earlier: 'Even God is for me what he is, in consequence of my own productivity of consciousness; here too I must not look aside lest I commit a supposed blasphemy, rather I must see the problem. Here too, as in the case of the other ego, productivity of consciousness will hardly signify that I invent and make this highest transcendency.' (Husserl 1969, 251) ('Auch Gott ist für mich, was er ist, aus meiner eigenen Bewußtseinsleistung, auch hier darf ich aus Angst vor einer vermeinten Blasphemie nicht wegsehen, sondern muß das Problem sehen. Auch hier wird wohl, wie hinsichtlich des Alterego, Bewußtseinsleistung nicht besagen, daß ich diese höchste Transzendenz erfinde und mache', Husserl 1974, 258). 46. Husserl to Przywara, June 15, 1932, in Husserl 1994a, 7:237. 'Den Theologen wird das einmal helfen, obschon es zunächst scheinen wird, dass dabei arge Ketzereien resultieren. Mich auf den Theismus der Schultradition -in den üblichen Auslegungen seines Sinnes -festlegen zu wollen, lehne ich entschieden ab. Natürlich ist das gegen Herrn Keilbach gesagt und dessen vermeintlich von mir (nach angeblichen Gesprächsäusserungen) erhoffte "Gottesbeweise".' 47. Husserl here has in mind this passage in Keilbach 1932, 213: We also know for a fact, that Husserl in 1926 made the following oral profession: ' … That the solution to the teleological problem can only be found in the theological conception, is something I too believe. But it will take a 100 years before my school can carry through an exact proof of God's existence. ('Wir wissen auch genau, daß Husserl im Jahre 1926 mündlich folgendes Bekenntnis ablegte: " … Daß die Lösung des teleologischen Problems nur im theologischen Begriff gefunden werden kann, das glaube ich auch. Aber es wird noch 100 Jahre dauern, bis meine In absence of evidence on specifically phenomenological-theological ideas such as Gödel may have had, I cannot attempt to develop this theme any further.48 2.4 The pragmatic value of Husserl's and Gödel's historical turn Not addressed in these essays is an aspect of transcendental phenomenology that is nevertheless of direct importance to Gödel's project: the later Husserl's insistence that a turn to history is not only of pragmatic value to systematic philosophy, but is necessary to it, without philosophy thereby becoming a form of historicism. Husserl argues for this position in The Crisis of European Sciences and Transcendental Phenomenology [START_REF] Husserl | Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie[END_REF], a work that Gödel owned and knew well; besides to the Crisis, I refer the interested reader to the analyses by Carr (1987, Chaps. 3 and 4) and [START_REF] Hopkins | The Philosophy of Husserl[END_REF]. I do not have much to add to their discussion, and, as far as I can see, neither would Gödel have had. But I should like to make some comments here on what the pragmatic value of a historical turn consists in; to Gödel, interested in the history of philosophy from his student days,49 Husserl's discussion in the Crisis and related texts will have been a reinforcement and a development of a view he already held.

The study of earlier positions and their development may clarify and sharpen our own ideas in various ways:

1. They may show interesting contrasts to our own position that al-Schule einen exakten Beweis für das Dasein Gottes wird führen können." ') Keilbach does not cite a source. Also noteworthy is that there is no offprint of Keilbach's article in Husserl's personal library at the Husserl Archive in Leuven. 48. Wang says that he is 'sure' that 'Gödel's tentative thoughts about religious metaphysics … did not … make much use of Husserl's method' (Wang 1996, 163). 49. In 1925, Gödel attended Heinrich Gomperz' course 'Übersicht über die Geschichte der europäischen Philosophie'; the notes he took have been preserved (Gödel Papers 3/72.5, 030100.4).

low us to arrive at a richer articulation of it;

2. They may reveal presuppositions of our own position;

3. They may enrich our own position by showing us motivations, arguments and approaches that we had not, or not sufficiently, been aware of may be available to us as well.

The study of an earlier philosophical system may turn not only to the questions and arguments that that system has, as a matter of historical fact, dealt with, but also to alternative questions and alternative arguments, which, for one reason or another, were not actually taken up.

(Such reasons may themselves have been philosophical ones, but need not.) This study of the systematical possibilities and limits of a historical position is nowadays known as 'doing philosophy historically'.50 It leads to a better understanding of a historical position and, through comparison, of our own. When doing philosophy historically, one may, and indeed should, freely use philosophical and other (e.g., mathematical) knowledge that was developed only after the historical position being studied. As Parsons has written to justify his use of modern knowledge of the foundations of logic and mathematics in a paper on Kant, Experience shows that one does not get far in understanding a philosopher unless one tries to think through the problems on their own merits, and in this one must use what one knows; second, if one is today to take Kant seriously as a philosopher of mathematics, one must confront him with modern knowledge. [START_REF] Parsons | Kant's philosophy of arithmetic[END_REF](Parsons [ ] 1983, 110-111) , 110-111) Taking Leibniz and Husserl seriously as philosophers of mathematics this 'Monads and Sets' analyses and criticises Gödel's attempt to justify, by an argument from analogy with the monadology, the reflection principle in set theory. The direct importance of that chapter for my present purpose is that my counterargument proceeds in such a way that it at the same time lends support to the belief embodied in the first stage of Gödel's project, the belief that the monadology needs to be reconstructed phenomenologically.

'On the philosophical development of Kurt Gödel' studies Gödel's reading of Husserl, its relation to Leibniz' monadology, and its influence on his published writings. A much greater influence of phenomenology, however, was overlooked when writing 'On the philosophical development'; this is addressed in the following paper.

'Gödel and intuitionism' discusses how on various occasions Brouwer's intuitionism actually inspired Gödel's work, in particular the Dialectica Interpretation, which Gödel in an unpublished note once characterised as 'a new intuitionistic insight … based on phenomenological reflection'. Although we will see (Chap. 12) that Gödel abandoned this particular attempt to construe the Dialectica Interpretation as intuitionistic in the noetic sense, the shift to the notion of reductive proof employed in the even further and better known revision still depended on phenomenology, and still marked a rapprochement to Brouwerian intuitionism. The work on a revision of the Dialectica paper shows that Gödel was not only studying Husserl and recommending his work to others, but also tried to advance the phenomenology of mathematics by working on a concrete problem that was at the same time of great technical interest. For reasons I explain in the next paper I will mention, I do not think it is a coincidence that Gödel deepest response to his reading of phenomenology lies in an elaboration not of Platonistic, but of constructivistic ideas.

'Construction and constitution in mathematics' addresses the question whether classical mathematics admits of the phenomenological foundation that Gödel envisaged. It proceeds by arguing that phenomenology rather leads to intuitionistic mathematics. That view is not only contrary to Gödel's, but also to that of recent authors on the topic like Føllesdal, Hartimo,Hauser,Liu,Rosado Haddock,and Tieszen.51 What I find wanting in Gödel as well as in the authors mentioned is, briefly, an appreciation of the way the transcendental Husserl developed his thought about categorial objects and categorial intuition in Formal and Transcendental Logic, Experience and Judgement, and related manuscripts from that period, as compared to the earlier Logical Investigations and Ideas I. In this paper, I describe that later doctrine in detail, drawing particular attention to its contentions that the objects of pure mathematics are productions in the ontic sense, and that all possible rational subjects can in principle produce the same purely categorial objects. This means that the kind of mathematics compatible with Husserl's variety of transcendental idealism is constructive mathematics, and that classical mathematics cannot be conceived of as constructive mathematics for a higher (in particular: ideal) mind. Anyone concerned with developing a Platonistic foundation for classical mathematics under reference to Husserl's later works therefore should argue that Husserl was wrong to develop his doctrine of categorial objects and intuitions that way, and strive to give a detailed phenomenological account of Platonism based on a new, altern-51. E.g., Føllesdal on p. 372 of his introduction to [START_REF] Gödel | /? The modern development of the foundations of mathematics in the light of philosophy[END_REF][START_REF] Hartimo | Husserl's pluralistic phenomenology of mathematics[END_REF][START_REF] Hauser | Gödel's Program revisited: The turn to phenomenology[END_REF][START_REF] Liu | Gödel's philosophical program and Husserl's phenomenology[END_REF][START_REF] Haddock | Husserl's epistemology of mathematics and the foundation of Platonism in mathematics[END_REF][START_REF] Tieszen | After Gödel: Platonism and Rationalism in Mathematics and Logic[END_REF] ative development of the doctrine of categorial intuition. None of this is to be found in Gödel or the mentioned recent authors; but to use the later Husserl in support of a metaphysics of mathematical objects that exist independently of our mental constructions is perverse. Clearly, exegesis of Husserl is no idle matter at this point. I will leave in the middle here whether the conclusion of this chapter should be taken as an objection to classical mathematics or to phenomenology. In either case it entails that Gödel's project cannot succeed.

The essays surrounding those four central ones have been included for further details and context.

In the 'Note on Leibniz and infinite wholes', I defend the claim that Leibniz' famous argument against the existence of infinite wholes is not only incorrect, as Russell has shown, but incorrect even on Leibniz' own terms. This is relevant to Gödel's project because it shows that, should there be an obstacle to integrating Cantorian set theory within a Leibnizian philosophy, as Gödel wished to do, it will not be this. 'Gödel's Dialectica Interpretation and Leibniz' shows that there was a direct influence of Leibniz' ideas about proofs on Gödel's revision of his Dialectica Interpretation. Seen together with his description of that work, quoted above, as 'a new intuitionistic insight … based on phenomenological reflection', we see that Gödel at that late stage of his career was willing to experiment with elements adopted and adapted from each of Leibniz, Husserl, and Brouwer.

'Mathematics' is a general discussion, with examples, of the phenomenology of mathematics.

'Gödel, mathematics, and possible worlds' provides a phenomenological unification of Gödel's Platonism and the Leibnizian idea of possible worlds, thus rejecting Hintikka's view on the motivation of Gödel's Platonism. Although this chapter belongs just a much in the part on Gödel and Leibniz, I have chosen to put it in the part on Gödel and Husserl: the question it addresses has its home in the Leibnizian context, but the argument it develops wholly depends on transcendental phenomenology. 'Two draft letters from Gödel on self-knowledge of reason' discusses unpublished remarks from the 1960s in which Gödel connects his Incompleteness Theorem to idealistic philosophy, of which transcendental phenomenology is a particular form. As far as their content is concerned (not necessarily Gödel's intentions), these remarks could be read as notes for a continuation of his 1961 essay.

'Rivaling brothers' looks at Gödel's relation to Brouwer and shows that, besides deep disagreements, there are also deep agreements between their philosophical ideas.

'Mysticism and mathematics' (written with Robert Tragesser) compares Gödel's and Brouwer's explorations of mysticism and its relation to mathematics. It is, of course, the essay farthest removed from the discussion of Gödel's project as such, in which mysticism plays no rôle at all; it has been included because it complements 'Rivaling brothers' and 'Gödel and intuitionism'. chus Sowa at RWTH Aachen, for help in finding details about Husserl and Keilbach. I thank John and Cheryl Dawson for sharing transcriptions with me of the two lists 'Was ich publizieren könnte' in Gödel's archive.

  have not included this passage in the list above because Wang here only reports his impression, not what Gödel said. Its content is similar to that of item 2 in the list.

	and their interrelations,
	he continued
	which, however, are to be arrived at phenomenologically. The task
	must be done in the right manner.
	It is true that Gödel, like many, had some qualms with Husserl's writings.
	For example, Wang recounts (Wang 1996, 320):

. (1) what, roughly speaking, the principles are by which we introduce the axioms, (2) what their precise meaning is and why we accept such principles. The second question is incomparably more difficult. It is my impression that Gödel proposes to answer it by phenomenological investigations. I Even though Gödel usually praised Husserl's work, he did occasionally express his frustration in studying it. I have a record of what he said on one of these occasions.18 I don't like particularly Husserl's way: long and difficult. He tells us no detailed way about how to do it. His work on time has been lost from the manuscripts. And a conversation note by Sue Toledo from 1975 reports that Gödel said about the Husserliana volume Analysen zur passiven Synthesis:

Material of Vol. XI of Husserliana (passive constitution) should have been interesting but doesn't appear to be so.

Work published during Husserl's lifetime appears more interesting.

  Gödel told me that he had certain deep convictions regard-Gödel Papers, 4/108, 040360 and 040361. These have not yet been published, but Cheryl Dawson has transcribed them. Gödel is reported to have sent a third list of this type to Oskar Morgenstern(Gödel 1995, v note a). 21. Note MvA.[START_REF] Wang | In memoriam Kurt Gödel 28 April 1906-14 January 1978. Kurt Gödel's intellectual development[END_REF]

19. Note MvA. Husserl's analysis on p. 212 (in the pagination of the original edition of Formale und transzendentale Logik, Husserl 1929; p. 240 in the translation Husserl 1969) begins as follows: 'Let us start from the fact that for us -stated more distinctly: for me qua ego -the world is constituted as "Objective" (in the above-stated sense: there for everyone), showing itself to be the way it is, in an intersubjective cognitive community.' ('Gehen wir davon aus, daß die Welt für uns, deutlicher gesprochen, daß sie ja für mich als Ego konstituiert is als "objektive", in jenem Sinn der für Jedermann daseienden, sich als wie sie ist in intersubjektiver Erkenntnisgemeinschaft ausweisenden.') 20.

  story 1, p. 80)25One may wonder what Gödel's intentions were at the occasion. Wang has made the following observation on the dynamics of (his) conversation with Gödel: 23. Note MvA. Hao Wang writes, for example: 'Gödel had recommended Husserl's Ideas to me, and I tried to read it. Not being sufficiently motivated, I found it too long-winded.' (Wang 1996, 142) 24. Note MvA. According to prof. Howard (in the email referred to in the next footnote, and a second one of March 9, 2013), the two texts in question were the two that, in Quentin Lauer's English translation, are included in Husserl 1965: 'Philosophy as a Rigorous Science' and 'Philosophy and the Crisis of European Man'. The latter is Husserl's Vienna Lecture of 1935, 'Die Philosophie in der Krisis europäischen Menschheit'(Husserl 1954, 314-348). 25. As related, with minor editorial changes, in an email from William Howard to MvA, March 7, 2013. Later, in a letter of August 2, 1973, in which he requested a fourth meeting, Howard asked Gödel specific questions about intentionality and also about the relation of phenomenology to Indian philosophy.

Note MvA. The date is 1976; seeWang 1996, 168.

For a comprehensive appraisal of Hao Wang as a logician and as a philosopher, see the collection Parsons and Link 2011.

Acknowledgement

In reworking the penultimate version of this Introduction into the final one, I have benefited from discussion with Palle Yourgrau. William Howard kindly granted permission to quote from the reminiscences he generously shared with me; additional material comes from the collection Stories (Howard) that he prepared for Amy Shell-Gellasch, who used a selection for her article Shell-Gellasch 2003. Those notes are now held at the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin, as part of the William Howard Oral History Collection, 1973, 1990-2003. The Archives of American Mathematics holds the copyright; quotations are by permission. I thank Carol Mead of the Archives for her help and advice concerning this material and its use. Dirk van Dalen, director of the Brouwer Archive, kindly permitted to include the quotation from Brouwer's notebook. I am also grateful to Thomas Vongehr at the Husserl Archive in Leuven, and Ro-


Walter Riofrio

Looking at the Beginnings

Keywords: evolvability, far from thermodynamic equilibrium, open-ended evolution, prebiotic world dynamics, self-organized systems

teaching and research institutions in France or abroad, or from public or private research centers.

Dynamic systems are dynamic entities that include different types of activities among their components. Depending on the type of complex system, the different forms of interrelation that exist among its components will produce certain activities that will be subject to observation and measurement [START_REF] Capra | The Web of Life: A New Scientific Understanding of Living Systems[END_REF][START_REF] Mainzer | Thinking in Complexity: The Complex Dynamics of Matter, Mind & Mankind[END_REF][START_REF] Bak | How Nature Works: The Science of Self-Organized Criticality[END_REF]. A characteristic shared by all complex systems is that, at the moment a determined activity is produced, a uniform distribution over time does not follow. In other words, the "timing" in which the determined action or activity is produced is important. Or put a different way, "when" things happen is important [START_REF] Rensing | Biological timing and the clock metaphor: Oscillatory and hourglass mechanisms[END_REF][START_REF] Carr | Timing is everything: Organization of timing circuits in auditory and electrical sensory systems[END_REF][START_REF] Bell-Pedersen | Circadian rhythms from multiple oscillators: Lessons from diverse organisms[END_REF][START_REF] Vanrullen | Spike times make sense[END_REF]. For example, it is not totally true to assume that (a) email users send their messages at uniform intervals in time, (b) an infection by a determined contagious disease can be adequately represented through the Poison distribution, or (c) molecule X, moving on a metabolic pathway, running into molecule A is equivalent, in terms of probability, to it running into molecule B. These reflections force us to face a very important issue in the study of complex systems; of particular interest is discovering the causes that produce their dynamic organization in spontaneous conditions. These issues are especially important when our research is directed to living systems. We know that biological systems are self-organized and self-reproductive, and, in this context, we also understood that the self-organizational dynamic of these systems is composed of many levels and types of processes that are interconnected and interdependent [START_REF] Fontana | The arrival of the fittest: Toward a theory of biological organization[END_REF][START_REF] Furusawa | Origin of multicellular organisms as an inevitable consequence of dynamical systems[END_REF]. One of the most characteristic global behaviors of biological systems is the one that enables an autonomous and agent behavior with respect to their environment [START_REF] Kauffman | The Origins of Order: Self Organization and Selection in Evolution[END_REF]. Another important global behavior leads us to ask whether something exists in the biological system that enables it to feature a great deal of resilience and robustness in the face of adverse conditions or of breakdowns in certain components in its processes: its incredible homeostatic capacity [START_REF] Barkai | Robustness in simple biochemical networks[END_REF][START_REF] Levin | Resilience in natural and socioeconomic systems[END_REF][START_REF] Carlson | Complexity and robustness[END_REF][START_REF] Kitano | Biological robustness[END_REF][START_REF] Stelling | Robustness of cellular functions[END_REF]). Even when the knowledge and isolation of the different molecular components in biological systems continues to accumulate on an increasing basis in the respective data banks, we are still lacking a descriptive scheme that enables us to understand -fundamentally -what the essential properties are for determining and characterizing the organizational dynamic of living systems [START_REF] Quastler | The Emergence of Biological Organization[END_REF][START_REF] Maturana | Autopoiesis: the Organization of the Living[END_REF][START_REF] Barabási | Network biology: Understanding the cell's functional organization[END_REF].

Origin of Prebiotic World

It would seem that a suitable place to consider certain explanations on this topic is at the beginning of the move from an exclusively chemical and physicalchemical dynamic towards a type of dynamic that has more in common with the evolutionary dynamics of living beings. In other words, the search for the core of cellular organization may send our research back to the very origins of biological systems, proposing an alternative approach to what currently is being considered as the only possible explanations [START_REF] Cairns-Smith | Seven Clues to the Origin of Life[END_REF], Wächtershäuser1988, Maynard Smith, Szathmáry 1999[START_REF] Hazen | Life's rocky start[END_REF]. It is possible to hypothesize that the reproductive and catalytic properties observed in all living systems would be most comprehended from other more primitive or more basic capacities. For all of that, we claim that it is still important to continue with more research that is proposing to find alternative explanations about how, in the physical reality, the emergence of biological organization is produced in ways that are self-sustained, robust, autonomous, agent, and homeostatic with the addition of possessing the potential for increasing, throughout time, its levels of structural and organizational complexity.

In this new stage in the development of scientific knowledge, we find ourselves in unsurpassed conditions for broaching, with huge possibilities of success, the problems that remained unresolved in the earlier schemes of scientific research, problems such as the origin of life, which includes that crucial dilemma of the controversy between the "RNA first" or "metabolism first" scenario. We can visualize this dilemma from Eigen's proposal (Eigen1971), which identifies the relationship between biological information and biological function, like a "chicken-egg dilemma". We now understand that this question moves beyond being just a historical problem. Before inquiring which came first, it is more important that we ask ourselves which one of them could have given rise to the prebiotic evolutionary processes. At present, we recognize that the solution for the old dilemma must necessarily pass through the stage of understanding the notion of biological information [START_REF] Wills | Does Information Acquire Meaning Naturally?[END_REF]) and of designing a definition of functions in naturalist terms [START_REF] Collier | Autonomy and process closure as the basis for functionality[END_REF], that is, without falling into epiphenomenic versions or appealing to some form of adscription. We propose an initial working hypothesis with the purpose of contributing to the discussion and improving the understanding or clarification of what more than likely happened in the distant past, in the transition that meant the step from being inanimate to being animate.

Dynamic Organization and Internal Properties

To understand the conditions that make the appearance of living forms possible is to understand that their most basic properties were present at the origin of the prebiotic world. We also want to explain that when referring to the origin of the pre-biotic universe, it is illogical to speak in terms of a Darwinian evolution since neither DNA nor RNA had appeared and neither had anything like "genetic material". These emerged long in the history of the pre-biotic world. Therefore, we cannot speak of fitness and other concepts connected with biological evolution.

There is just open-ended evolution -a type of evolution whose components are related to the idea of evolution by natural selection. Nevertheless, it is possible for us to refer to the capacities that permit the gradual building of the structure and dynamic organization of those systems in terms of their internal properties [START_REF] Bagley | Spontaneous emergence of a metabolism[END_REF][START_REF] Bak | How Nature Works: The Science of Self-Organized Criticality[END_REF][START_REF] Jeong | The large-scale organization of metabolic networks[END_REF]. Likewise, the solution for the problem sketched above (the old dilemma) has to move through a period of establishing the criteria that can differentiate the selforganizing dynamics of the systems that make up the prebiotic world from other types of self-organized systems, and perhaps, through a time of a better understanding of the new science of networks [START_REF] Bray | Molecular networks: The top-down view[END_REF][START_REF] Watts | The "new" science of networks[END_REF], which enables us to understand the transition from the inanimate world to the animate world and to understand the step from random to scale-free networks. Another aspect is the design of a new version of evolvability that could be applied to the distinct changes that were produced in the prebiotic world [START_REF] Kirschner | Evolvability[END_REF]. We are proposing a theoretical construct that visualizes the most basic properties of living systems, originating in the systems that provided the possibilities for its emergence in the universe: in the prebiotic systems. More accurate put, we consider these original properties to be autonomy, information, and function [START_REF] Riofrio | What is life? Again, is this question interesting? Paper presented at the Complexity[END_REF]. The aforementioned theoretical construct is a type of dynamic system that we defend as the one that decisively established a division in the different types of self-organized systems. On one hand, we would have the self-organized systems that would be generating self-organized dynamics with certain degrees of ordering but whose matter and energy sources -which enable them to be in the far from thermodynamic equilibrium statewould strongly depend on their exterior environment. Instead, we affirm that there might have existed a type of self-organized system that could have incorporated as part of its dynamic organization the necessary factors that provided it the ability to be and to maintain itself, by its own dynamic, in the far from thermodynamic equilibrium state. We have named this type of self-organized system the "Informational Dynamic System" and, moreover, we propose that it is this variation of a self-organized system that for the first time opened the doors of the prebiotic world. This is the one that caused the necessary and sufficient conditions to set off the succession of certain phenomena of the universe towards the possibility of the emergence of biological systems. It is also the system that might have been the driving force to the beginning of the move from the inanimate world to the animate one [START_REF] Riofrio | Informational dynamic systems: Autonomy, information, function[END_REF]. Actually, this type of dynamic system would be composed of a group of molecular compounds that are interconnected by networks of processes, where determined types of actions take place and that, on a global scale, will enable this system to maintain and sustain itself in a robust manner over time and in conditions that are far from thermodynamic equilibrium.

To visualize the organization we are hypothesizing for the Informational Dynamic Systems, we need to include the interaction and interdependence among three different groups of molecular processes. The first corresponds to a process that enables the system to behave as a self-organized system for reasons that are systemically internal to it. It has managed to keep itself far from thermodynamic equilibrium since part of its organization is constituted by the group that links the endergonic processes with the exergonic ones and the facilitation that produces the coupling of molecules with "high energy" bonds. The second is a set of processes that constitutes the protoplasmic membrane. In this portion of the system's organization, the tasks of isolating the system from its exterior are done; in addition, it is the part of the system organization that is a complete agent in where solutions to chemical problems, such as the "osmotic crisis", the management of electrochemical gradients, will be generated as well as the work associated with the problems of "recognition" and transportation of matter and energy compounds from the outside to its inside. The third group involves the processes which are linked to the tasks of reproduction, repair, and control of the internal dynamics. These processes interact among each other as well as with the ones inside the two groups mentioned above. One of the basic characteristics of the first ancestors of living systems, those that opened the door of pre-biotic world, was the capacity to generate "meaningful information" about their environment and about their dynamical internal milieu. The important thing for our idea is that the information, properly said -The Information -has a meaning (very basic semantic) that is created on the inside of the system. Any type of signal or sign can be a carrier of what might be information (or what we call "potential information"). We consider a signal or sign to be any matter-energy variation (sound wave, electromagnetism, concentration of a chemical compound, change in pH, etc.). The "potential information" carrier must be in the surroundings, on the inside, or have transmitted the information to some system component. So, we need that one sign or signal which is the carrier of potential information must have been incorporated into some process inside the informational dynamic system. Secondly, the possible information becomes information ("information with meaning" for the system) since it has the capacity to produce something (an effect). Third, the effect has a repercussion in the system, influencing its own dynamic organization. The effect of the information that has meaning for the system can be found in the maintenance or the increase of the system cohesion. Such cohesion is constituted by the group of relationships and interdependencies that exist among the processes; it is a relational and dynamic definition that encompasses the nature of the system organization [START_REF] Collier | Entropy in evolution[END_REF][START_REF] Collier | Self-organization, individuation and identity[END_REF]. As well, the effect could produce some level of interference in the system cohesion, possibly interrupting one or more processes. It is clear that meaningful information can be caused by some signal (that carries potential information) coming from the environment like a signal that is generated in the internal dynamic of the system. In all cases, whether an effect in favor of or in contrast to cohesion, the system will develop some type of response that will be correlated to that meaningful information and the process or processes related to the effect.

In the other hand, we consider the function to be located in the action that generates a determined process and what we can understand as 'what contributes' (cooperates, favors, supplies) to the interrelation and interdependence among the processes with a view to maintaining the far from equilibrium state. On the inside of the process, each component will have the mechanism that is proper to its chemical constitution, but that chemical action (or group of actions) will have its raison d'être because of the process. Then, each process that carries out a determined and particular action (made up by the causal nexuses among the component mechanisms that constitute them) will be assimilated into the dynamic organization only if this particular action is maintaining, improving, or achieving the far from equilibrium. And this action is precisely its function in this organizational logic of the Informational Dynamic Systems. Thus, a function is something that contributes or facilitates to maintaining or increasing the far from equilibrium state. A dysfunction, on the other hand, is something that does not contribute or facilitate to maintaining the far from equilibrium state. Succinctly stated, we can say that our theoretical proposal visualizes the emergence of information and biological function in a sort of coordinated origin -at the same time -in the local processes of these types of systems. This visualization of the physical emergence of Information and Function gives us to talk about the notion of "Information-Function"; each time we observe functions happening in these systems' processes, we can be sure that some type of information has been transmitted through them. Likewise, when we observe the transmission of some type of information among the process, we can be sure that some function has been produced among them.


Anna Sinopoli

DYNAMICS AND IMPACT IN A SYSTEM WITH UNILATERAL CONSTRAINTS THE RELEVANCE OF DRY FRICTION

sulla dinamica e sull'urto di un sistema rigido, soggetto a vincoli unilaterali. Tale problema e particolarmente interessante per la comprensione del comportamento meccanico delle strutture monumentali costituite da blocchi lapidei, in presenza di forze est erne variabili. L 'analisi fornisce informazioni sulla evoluzione dinamica del sistema; tali informazioni sono indispensabili per uno studio sulla stabilitd.

SUMMARY. The aim of the present work is to investigate the impact and the dynamics of a rigid system, subject to unilateral constraints. The problem is particularly interesting to understand the mechanical behaviour of stone structures subject to external forces and to give informations about the dynamical evolution; such informations are basic for a stability study.

INTRODUCTION

In the last years particular attention has been devoted to problems with unilateral constraints.

At the same time, studies on masonry structures have also received many contributions. Experimental and numerical researches have been undertaken to understand the mechanical behaviour; moreover, analytical studies have been developed to define constitutive laws.

The problem mentioned above are strictly connected; in fact, it is well known that masonry structures are characterized by a very weak (or null) tensile strenght, due to the presence and the degradation of the mortar.

In particular, stone monumental structures can be modeled as rigid-labile systems with unilateral constraints.

The possibility to define a stability degree for such systems, in the case of monodimensional structures (arches, portals, etc.) with one degree of freedom has been investigated in former works [1], [2], (3]. The most interesting result is that, due to unilateral constraints, the equilibrium configuration is stable, so that it is possible to define a stability degree. Further, the stability definition is strictly connected to the possibility of stable oscillations around the equilibrium configuration, particularly in presence of external forces; this is the case of earthquakes. Therefore, a dynamical analysis of stone structures, under the assumptions above, is mandatory. The oscillations are expected to have a peculiar character, because they are possible by means of inversion of the mechanisms. Then the problem is non linear, due to the fact that the inversion is characterized by an impact.

The free dynamics of particular systems, both in the case of small and large amplitude oscillations has been studied

[1], [2], [4], under the restrictive assumption that such systems are conservative.

The amount of energy dissipation during the impulsive phenomen is, to the author opinion, an open problem.

The dynamical analysis of a column has been undertaken in [5], with a numerical approach, but an «a priori» assumption about the motion after the impact has been imposed.

Such assumption is the same made by Housner [6]. Its fundamental work analyses both the free and forced motion only for very slender stone structures, the behaviour of which is assumed similar to the inverted pendulum.

In this work, the dynamics and the impact of a stone element are investigated, assuming as fundamental parameter the dimensional ratio, which is allowed to vary in a very large range. As a consequence, even if the analytical expression for the solution of the free motion is the same of Hausner, the difference from Hausner periods increases with the dimensional ratio.

The large range of variation of the dimensional ratio permits to define the relevance of dry friction and to identify the only possible motion after the impact.

Both the dynamics and the impact are studied for two extreme cases: a) absence of dry friction; b) dry friction large enough to prevent sliding. Such cases are useful to define the boundary of the behaviour for all the possible values of dry friction.

DYNAMICS WITHOUT FRICTION

Referring to Fig. 1, let us assume that: a) the system is constituted by a rigid bloc, simply supported by a rigid ground; b) h, b are its dimensions; h is the height and b the base length (the base are is b x 1); c) (x, y) is an inertial reference system with its origin initially coincident with point A.

Let us consider the free dynamics of this system, under the assumption that friction is negligible. Moreover, initially, let A be the only point in contact with the ground. This corresponds to consider one of the two possible mechanisms of the system.

Let xG, YG and e be the lagrangian coordinates (respectively the two center of mass translations and the rotation angle), then the equazions of motion are: where m is the mass of the system, X is the constraint reaction, g the gravity acceleration and IG the baricentric inertia moment.

In order to solve system ( 1 ), the constraint equation due to ground impenetrability must also be imposed; this is a unilateral constraint:

(2) If we assume that, during the motion, point A never gets off the ground, eq. (2) can be written as:

dy G + (: sene -~ cos e) de = 0 (3)
From eqs. ( 2) and (3) the dynamical system becomes:

xG = 0 y G = ( ~ cos 8 -% sen e) 0 -( : sen 8 + % cos e) 8 2 A=myG+mg m .. I 8 + -(b 2 cos 2 e + h 2 sen 2 e -2bh sen B cos 8) e + G 4 (4) + ' :!... [bh sen 2 e -bh cos 2 e + (h 2 -b 2 )sene cose]1i 2 - 4 m --g(h sen(}-b cos B)= 0.
2

It is noted that system ( 4) has only two degrees of freedom, corresponding to the lagrangian coordinates x G and B; further, the pertinent equations are uncoupled.

The horizontal motion of the center of mass is characterized by:

(5)

Regarding to rotation, if we assume that 8 is small enough, so that sen 8 ~ e and cos e ~ 1, the corresponding equation can be written as:

( I + m b 2 -m bh e) 0 + G 4 2 (6) ( h2 -b 2 bh ) h b + m --4-e -4 iJ2-mg 2 8 =-mg 2
Let use note that in eq. ( 6), in spite of the small amplitude assumption, a term proportional to 0 2 is present. In this respect some observations both of mathematical and physical character must be made. We are concerned to study the small amplitude motion around the equilibrium configuration corresponding to e = 0.

This configuration is stable only because of ground impenetrability; such constraint, introduced when the bloc reaches the ground, cannot be taken into account by eq. ( 6), but it must be imposed, by means of another relationship. In fact if we consider solutions with the form: e(t) = const, eq. ( 4) ~ives the two equilibrium configurations: e = + arctg t , which are, respectively, stable and unstable.

Regarding to motion near (} = 0, it is far from natural equilibrium configurations of the system and in particular, it can be considered as a large amplitude motion around the unstable configuration. As a consequence, the angular velocity can assume finite values, so that 0 2 cannot be neglected. In proof of this argument, it is worth to remark that the potential energy U(e), for e = 0, exhibits a boundary minimum characterized by ~~ 4= 0.

Eq. ( 6) is a non linear equation, with variable coefficients. In order to find a solution which takes into account infinitesimal values of 8 and finite values of 8 let us consider a solution with the form: (7) that is, the angle 8 i~ obtained as the difference between two finite terms, where e 0 is the unstable equilibrium con- figuration corresponding to: e = arctg ~ . Substituting (7) in ( 6), and neglecting the first order terms, eq. ( 6) becomes:

.. . 2 ( b2) bh b I +m-e --8 =-mg- G 4 0 4 0 2 (8)
Eq. ( 8) can be easily transformed into a first order differential equation, with the substitution eo= y, so that:

The constants C 1 and C 2 depend on the initial conditions.

If .XG(O) = 0, the motion of the center of mass is along the vertical. At instant tl' when e (t 1 ) = 0, the instantaneous center of rotation is, in a reference system fixed with respect to the bloc, coincident with the point the coordinates of which are (0, 1 ).

The motion, at the same instant, is characterized by:

. Fort> tl' the motion is undetermined; it will depend on the impact between the rigid bloc and the ground. This aspect will be investigated in the following.

DYNAMICS WITH DRY FRICTION

Now let us assume that dry friction is present. Moreover the amount of friction is large enough to prevent sliding of point A in Fig. 1. Then the dynamical equations are:

mxG = A. 1 mjiG = A. 2 -mg
with the constraints equations:

dxG + (: sen (} + : cos(}) d(J = 0 dy G + (: sen (} -: cos (}) d(} = 0 (11) (12)
where A. 1 and A. 2 , represent, respectively, the horizontal and vertical constraint reactions.

The motion, in this case, is a rotations around the fixed point A. Then, let us assume the rotation angle (}, as lagrangian coordinate. The corresponding dynamical equation is:

.. mg

IA (} -- (h sen 8 -b cos(}) = 0 2
where fA is the inertia moment with respect to A. (13) With the assumption of small values of(}, eq. ( 13), on the contrary of eq. ( 6), does not contain 8 terms, so that it can be solved and the solution of the first order approximation is:

b 8 (t) = -+ C 1 cosh (ext+ C 2 ) h where: (14)
It is particularly interesting to calculate A. 1 /A. 2 , that is the ratio between the horizontal and the vertical reaction. Such ratio, in fact provides information about our assumption of friction large enough to prevent sliding of point A.

From eq. ( 11), ( 12) and ( 13), it can be found, at the zero order approximation, that:

(15)

If fa represents the limit value of the dry friction coefficient, during the motion, point A does not slide only if: (16) Neglecting first order terms, eq. ( 16) corresponds to:

'2 g 2 2 (b-hfa)(} (t);;;. 2 2 [3bh-_fo(b + 4h )] 2(h +b) (17)
Inequality ( 17) has been investigated by means of parametric analysis on t and fa. The results are reported in fig. 2, where the lines fa= 0, 75 and fa= t , useful for algebraic considerations, have been drawn. It can be said that, in region for which *;;;.fa and fa> 0,75, enequality ( 17) is verified for any initial conditions and at any instant.

On the contrary, for } ;;;. fa and fA.;:;; 0,75, ( 17) is satisfied only for particular values of the ratio * .

In tab. 1 such values are reported in correspondence to different values of fa. The region where the friction is not sufficient to prevent sliding has been dashed in Fig. 2.

THE IMPACT PROBLEM

The problem of the impact, which occurs when the rotation angle is zero, is difficult to solve. In fact, using only the fundamental equations of the impulsive phenomenon, the motion after the impact is undetermined, due to the unilaterality of the constraints and to the finite extension of the surface involved.

Moreover, the experimental law of Newton cannot be used, because the impact is not central and direct.

An alternative approach to solve the problem is the one which refers to the symbolic equation of impulsive motion and uses some principle of minimum. The symbolic inequality of impulsive motion for discrete systems with unilateral constraints is:

n [ U;-mi /::,vi) X olj ~ 0 (18) i= 1
where I; is the external impulse applied to the i-th mass point, /::,vi is the corresponding velocity variation and oPi the infinitesimal displacement congruent with the constraints.

For our system, the symbolic inequality, when the bloc has reached the ground, imposes a restriction on the velocities distribution.

But, in the same time, it allows an infinite number of motions after the impact, congruent with the unilateral constraint.

The determination of the unique natural motion is possible by using a varied form of the Gauss principle of least performance, furnished by the constraint reactions. This varied form due to Robin for systems with bilateral constraints, states that the unknown motion after an impact is the one which minimizes the G function between any motion congruent with the constraint system.

In the case of null external impulse such G function is the kinetic energy due to the sudden variations of the velocities.

The validity of the Robin principle for systems with unilateral constraints has been verified by Mayer [9], so that it is used to study the impact in both the cases with and without dry friction.

The impact without friction

Let us consider again the system of Fig. 1. At the instant tl' when (J (! 1 ) = 0, the absolute instaneous center of rotation is coincident with point H (Fig. 3).

Then, the motion before the impact is characterized by:

x 0 (t 1 ) =canst. b . _y-u) = -e- o 1 2
The unilateral constraint introduced by the impact is:

oyp~o
where Pis any point of the bloc.

Eq. ( 20) can be written as:

oy 0 + (xp-x 0 ) oe ~ o that is: (19) (20) (21) (22)
Eq. ( 22), as function of the velocities implies that the permitted motion are only those for which:

(23) A consequence of inequality ( 23) is: (24) because xp can assume any value between zero and b. In virtue of the Robin principle, the natural motion is the one which minimizes the kinetic energy, G, due to the sudden

H G l A Fig. 3.
velocities variations. In our case the expression of G is:

of which we must find the minimum value according to (24). Regarding to eq. ( 25) we can note that: a) x 0 is an independent parameter of the motion; b) from ( 24), the minimum value for G is the one corresponding toy J = ~ 18+ I• because 8-< 0 so that-.Y(; > 0 and.YJ;;;. 0. Then, .eq. ( 25) can be written as:

G= ~ m[<xJ-ic(;) 2 +(i8+i_8-) 2 : 2 ]+ 1 . . + -I 0 (e+ -e-) 2 .
2

The minimum of G can be found requiring that:

a a -= 0 a •+ XG a a -. =0.
ae+

The first of eqs. ( 27) corresponds to:

(26)

(28) so that the horizontal motions before and after the impact are the same. On the contrary, the presence of an absolute value for 8+, in the second of eqs. ( 27), requires that two cases must be considered, respectively, for iJ + < 0 and 8+;;;. 0.

The consequent conditions are: and Eq. ( 29) implies that:

h 2 -2b 2 8+ = 8- 4b2 + h 2 (29)
for e+;;;. 0 (30)

where for 1 0 has been assumed the expression:

JG = ;2 m(b2 + h2).
Because both iJ+ and ir are negative, eq. ( 31) is valid only if ~ > v2. In this case, the motion after the impact is a rototranslation charaterized by an angular rate reduced in value but with the same sign. Regarding to eq. ( 30), we can observe that is implies an absurdity; in fact, if t}+;;;. 0, it is impossible that 8+ = ir < 0. Neverthless, some considerations can be made.

The G function does not have a minimum characterized dG by dO+ = 0, but because it is an increasing function of responding to 8+ = 0.

Then, for t . ; ; ; ; ; .../2, the motion after the impact can be a pure translation; its occurence depends on the value of xJ = x 0 . In fact, if x 0 = x 0 (0) = 0, after the impact the system is at rest.

It is interesting to note that, if x 0 (0) = 0, for any value of : , the kinetic energy depends only on the angular rate 8, so that the amount of dissipated energy can be evaluated.

With reference to the ratio e between the values of kinetic energy after and before the impact, it can be found that: Let us consider the same system in presence of dry friction; let the value of } and fa be the ones (Fig. 2) which permit to point A to act as an hinge.

At instant t 1 , the motion of the system, before the impact, is: Now it is espected that eq. ( 20) is satisfied as an equality for a particular point P*, belonging to the base AB, for which:

  Fig. I

  XG(tl) = O b yG(tl) = -B(tl) ;(I,)~-2(:•) ~ '"'{.:: h' (10)

h

  For f. < 0.75 and ->!.a• a In order to verify inequality (17) in the region where } <f 0 , initial conditions of the motion must be specified.It we consider ones corresponding to 8 (0) = (} 0 and 8 (0) = 0, no contraints must be imposed to verify (17).

Fig

  Fig. 2.

b

  The diagram of ratio e is reported in Fig.4as function of !!.. b 3.2. The impact in presence of dry friction

  Fig. 4.

Table l .

 l

			h
	0.1	b -.;;; 0.13 h	b and -;;. 29.9 h
	0.2	b -.;;;9.27 h	b and -;;. 14.7 h
	0.3	b -.;;;0.42 h	h and -> 9.58 b
	0.4	b -.;;;0.57 h	h and -;;. 6.92 b
	0.5	b -.;;;0.76	h

b fa range of values for -

Eq. ( 33), for any pointP, is equivalent to: Then, from ( 23) and ( 24), the Robin function, for 8+-:/= 0, becomes: (35) where, even in this case, we have assumed that:y~ = % lfi+ I•

The minimum value of G is the one corresponding to:

and for: o+ > 0.

(37) Regarding eq. ( 36), if * > Jf , the motion is a rotation around point B, with reduced angular velocity. In this case it can be easily shown that the dry friction is sufficient to prevent sliding of point B, also after the impact.

On the contrary, for o+ > 0, eq. ( 37) implies as in the previous section that the function cannot have a minimum with di? = 0 de •

Let us consider again eq. ( 35); foro+> 0, G is an increasing function and it approaches its minimum value when iJ+-+ o+. But, for 8+ = 0, the formulation of G becomes:

(38)

In this case, in virtue of the same arguments used above, in order to find the minimum it must be imposed that:

Then for !!_ .,;;; .J2 the motion after the impact is a transla- ' b 2 tion.

Even in this case we can evaluate the amount of dissipated energy. Fort > Jf the expression of e is:

On the contrary, for ~ <; Jf, it becomes:

The diagram of this coefficient as function of 1can be seen in Fig. 4. Surprisingly enough, it is worthwile to remark that, with the same initial conditions, the impact dissipates more energy in the case without friction than in the case with friction.

[5] GIANNINI R., Analisi dinamica dei sistemi di blocchi sovrapposti, Pubbl. n. 219 1st. Scienza Costruzioni-Facolti di Architettura- -Roma, 1984. (6]