Blockchain Papers

Maja Korolija

Abstract Two ideological views on science dominated the Cold War era: one of a free and apolitical science, and the other emphasizing partisanship in science, associated with the Western and Eastern Blocs, respectively. This study offers a specific perspective of important elements belonging to these scientific positions, as it reveals their entanglement with geopolitical and socio-economic processes of the (semi)peripheral Yugoslav socialist system during the Cold War period. After the Second World War, and before its break with the USSR in 1948, Yugoslavia tended to emulate Soviet ideology in all aspects of society, including science. In the period following this break, the Yugoslav socialist regime, at least initially, leaned heavily toward the Western Bloc. By comparing Yugoslav science before and after the break with the USSR, this study provides insight into the consequences of the geopolitical shift and socio-economic transition of the Yugoslav socialist system, primarily in terms of the model of scientific organization, financing, and scientific discourse. Exposed to the dynamics of decentralization and, to a larger extent than before, market forces, Yugoslav socialism after the break with the USSR adopted a specific form, namely Socialist Self-Management. Herein, I show that this led to the emergence of novel organizational and discursive tendencies in Yugoslav science, which were compatible with certain aspects of the perspective of science as ‘pure’, autonomous, and apolitical.

Victor Nascimento, Luiz Carlos Pereira, Elaine Pimentel

Debates concerning philosophical grounds for the validity of classical and intuitionistic logics often have the very nature of logical proofs as one of the main points of controversy. The intuitionist advocates for a strict notion of constructive proof, while the classical logician advocates for a notion which allows non-construtive proofs through reductio ad absurdum. A great deal of controversy still subsists to this day on the matter, as there is no agreement between disputants on the precise standing of non-constructive methods. Two very distinct approaches to logic are currently providing interesting contributions to this debate. The first, oftentimes called logical ecumenism, aims to provide a unified framework in which two "rival" logics may peacefully coexist, thus providing some sort of neutral ground for the contestants. The second, proof-theoretic semantics, aims not only to elucidate the meaning of a logical proof, but also to provide means for its use as a basic concept of semantic analysis. Logical ecumenism thus provides a medium in which meaningful interactions may occur between classical and intuitionistic logic, whilst proof-theoretic semantics provides a way of clarifying what is at stake when one accepts or denies reductio ad absurdum as a meaningful proof method. In this paper we show how to coherently combine both approaches by providing not only a medium in which classical and intuitionistic logics may coexist, but also one in which classical and intuitionistic notions of proof may coexist.

Nicolas Michel

For more than three decades, fierce debates raged both in private letters and across public spaces over a formula expressed in 1864 by the French geometer Michel Chasles. Proofs and refutations thereof abounded, to no avail: the formula was too useful to be abandoned by its defenders, too elusive to be made rigorous for its detractors. The disputes over Chasles’s formula would not be solved by a definitive proof or rebuttal; rather, the core epistemic issues at stake shifted from generality to rigor and from truth to geometrical significance. This essay tracks the main lines of circulation of Chasles’s formula and shows how the disputes to which it gave rise embody conflicting mathematical selves—that is to say, different normative accounts of what being a mathematician entails. This perspective allows for a renewed understanding of what historians have described as the conflicted rise of modernism in mathematics and a firmer rooting of it within broader late nineteenth-century cultural trends.

John H. Smith

Kant, Calculus, Consciousness, and the Mathematical Infinite in Us John H. Smith Der Begriff des Unendlichkleinen, darauf die Mathematik so öfters hinauskommt, wird mit einer angemaßten Dreistigkeit so geradezu als erdichtet verworfen, anstatt daß man eher vermuten sollte, daß man noch nicht genug davon verstände, um ein Urteil darüber zu fällen. Die Natur selbst scheint gleichwohl nicht undeutliche Beweistümer an die Hand zu geben, daß dieser Begriff sehr wahr sei. Denn wenn es Kräfte gibt, welche eine Zeit hindurch kontinuierlich wirken, um Bewegungen hervorzubringen, wie allem Ansehen nach die Schwere ist, so muß die Kraft, die sie im Anfangsaugenblicke oder in Ruhe ausübt, gegen die, welche sie in einer Zeit mitteilt, unendlich klein sein. Es ist schwer, ich gestehe es, in die Natur dieser Begriffe hineinzudringen; aber diese Schwierigkeit kann allenfalls nur die Behutsamkeit unsicherer Vermutungen, aber nicht entscheidende Aussprüche der Unmöglichkeit rechtfertigen.1 [The concept of the infinitely small, which comes up so often in mathematics, is rejected straight out with presumptuous audacity as a fiction, instead of assuming that it is not well enough understood to form a judgment about it. Nature itself seems, however, to provide clear proofs that there is truth to this concept. For if there are forces that work continuously through time in order to produce movements, as it would appear is the case with gravity, then the force that effects this movement in an initial instant (or rest) must be infinitely small as opposed to the one that imparts movement in time. It is difficult, I confess, to penetrate into the nature of these concepts; but this difficulty can at most justify cautiously avoiding unfounded suppositions and not claiming decisively that it is impossible.] The question that I address in this essay is simple: what is the mathematical infinite doing at crucial moments in Kant’s philosophy? By doing I mean the way that specific notions of the infinitely small—the infinitesimal, the differential, infinite approximation, continuity—serve as metaphors in a strong Aristotelian sense; that is, they attempt to “bring before the eyes” something abstract and thereby contribute a kind of intuition to the unintuitable.2 And yet, there is something doubly ironic going on in this effort. On the one hand, although mathematics and the mathematization of nature have been criticized by Husserl for abstracting modern thought from the concrete life-world of experience, I will look to places where there is a turn to the mathematical infinite in order to be concrete.3 On the other hand, the clear [End Page 95] intuitions associated with the mathematical infinite can easily get caught up in contradictions that undermine the initial clarity.4 By exploring these moments of metaphoric and paradoxical representation, we can see how major thinkers grappled with a notion of the “immanence of the infinite” that characterizes the epochal turn of modernity.5 First, let me give a general sense of what is at stake in the notion of the mathematical infinite and the issues of representation associated with infinitesimal calculus, which was up through the nineteenth century arguably the most powerful tool for understanding the physical world. Although Kant rarely addresses calculus as such, this tool has particular relevance for him given the fundamental role that both its inventors, Leibniz and Newton, play in the development of his critical project. Consider the problem of a line tangent to a curve—that is, a straight line that touches any curved line at precisely one point. As I hope and expect has just occurred in the mind of any reader at this point, a somewhat-clear image has taken shape. This image can be rendered mathematically precise thanks to the technique of calculus; indeed, calculus was developed in large part to address this problem. The reason is that such a tangent can be considered the rate of change of the curve at any instant, which can be calculated given the curve’s function. Thus, say, a soaring baseball defines a parabolic arc at a changing rate of speed (fast at first, slowing at the peak, then ever faster as it returns to the ground). The tangents at any point indicate...

R. G. Collingwood

The argument of this paper may be summarized as follows. Causal propositions (propositions of the type “x causes y”) are ambiguous. Such a proposition may have any one of three meanings (possibly more; but three is enough for this paper). The ambiguity, however, is of a rather odd kind. Sense I, which is historically the original sense, is pre-supposed by the others, and remains strictly speaking the one and only “proper” sense. When we assert propositions containing the word cause in senses II and III, we are “saying” one thing and “meaning” another; we are describing certain things as if they were things of a kind which we do not actually believe them to be. This always has an element of danger in it: the danger of inadvertently beginning to “mean” what one had only intended to “say”, i.e., of thinking that things are what we describe them as if they were. This danger is much worse when our “metaphors” get “mixed”. This is what has happened with the so-called “idea of causation” from the time of Kant onwards. It is a confusion of certain characteristics belonging to sense II with certain others belonging to sense III. Nothing can be done, therefore, towards clearing up our minds about causation, by merely analysing the idea as it stands and detailing the various elements it contains; for these elements are mutually contradictory. We must carry the process further, by segregating the elements under different heads, and distinguishing these as different “senses” of the word. But even this is not enough. A further step in the process is needed: namely a critical discussion of each “sense” taken singly. When this is done it will be found that the best way of avoiding confusion will be to restrict our use of the word cause to occasions on which it is used in its “proper” sense, No. I; that on the occasions on which we use it in sense II we should be wise to use instead the terminology of means and ends; and that when we use it in sense III we should do better to speak of “laws” and their “instances”. In the sense of the word that which is is the and of a and and to do it means a for we may to in the kind of in This is the time a and sense of the word with its and in and of the in of in and in A in a in causes of This not that the to the that with and it that on the up to In the sense we that a causes a to a that causes to from an It is not only a sense, it is of In is the word for as when the the of with the and the of have it that this is a use of the word and from what sense III, i.e., the sense in which one thing in is to be the cause of This is not the sense is the of the as can for will the in the and the in and and the cause in the II and III have of it by a process which in the A cause in this sense of a cause and a The is a of things the is a of things to be of these be a cause if the were A to a certain may be to by a about the of that but this not cause to to in the of an a to a not cause to in a certain if in that The is not a of things as it is a of things by the in to a a certain of the cause of is not this as it is this as to the in by The is not a as it is an A is to in a certain way not by to in that way it is to to but by to in that may be to the of but is not of this kind may the of a and in as the that a certain of things as when a of a certain the to a certain This is to the to do a certain to do The however, is an it not be done not be the it of A causes to do an is and not is a in and is for is a which A to by certain certain is the is the proposition that the by and the proposition that a in of and is for the the proposition the in this A is to the for the This not that a is a which be an which do when they speak of It means that is for the A is for the of certain to certain to the When a of a on its do is not is as an This is what when of the and A is to when about the is not on from and are In this in which a is to is not It has a and a But has done for the of the and the which in the is to cause has done for can be to cause as as to do in that the of be an which to an of on the of and has as only will not what they by the word In sense in sense I, the word cause an idea to but the in this is an intended to not but things in In this sense, the of an in is the to by which we can we to to a and it we can certain of our we about for its The is the cause of an means in this can we This sense of the word may be as follows. A cause is an of things which it is in our to and by which we can that cause it is to be. When speak of to occasions as when one a and the of things by the proposition is the with the to the This is an sense in The cause of a is the which a on the cause of is the of a the cause of a is the cause of is their in a the cause of a is that has taken the cause of a in the is that the the cause of is that and The for causes in sense II is in that sense of the in which is what a not for its and but for its for the which it is and is by The of a is the in terminology of The for is but it of i.e., by that to it to the that things are is to that we can and A of is A of time and is on is the cause of we the cause of we should be to that is the of this But should it be that the cause of to to have the cause of but that not in be of any use the cause had not a thing that be Such a be and one that had done what to It be that not what the word cause the of be it in a a proposition of the “x causes the proposition “x is that can be as of the of This sense of the word is not It can be of to the the in the of the these the A in sense II means which is by to the When in this sense we that causes we are about by We are always about in with things which we do not these will not cause to are The the cause and these as has for by the cause of an as its to sense III, not to sense the should the cause and certain are and the in that the should is of sense II and not of sense the cause is not what we have the cause but this the that the cause is the of a of and that what the cause is one of by a of with a that only to the have that this is by means It is to a on a and what the cause not by a that the of a is further from the its and that is to a to the this is what the has is one of the the of and as has one of these and it the has done what we always But an the up a and on three is the cause of the It is the it has not it has as the thing that can which the will had a by on the have to to the as the cause of the not the is a but can it The cause is not it is to a which is in the of the cause in sense It is for that the has a of things from which But this not In the the of any are in that one them for even if In the one to them should that can get a by certain may be that should not be it a were but as get it do not what these to a in one of them to get do not to what they only to what the one is that has this a which of that the of an three and that are three of A is to and only is to and only and is to and only if each of them the cause of each will have to a different is the for and for The may be by that for any the cause of a thing is that one of its which is to a a certain and the of the cause of the and the is that one must the of the cause a and the is that one must the of the cause and the is that one must the of one of these three the for the on one of the others, if for the is the of to of that it we that be done, by any towards that is to of their cause be a that not in But a cause in this sense of the word is by is this means that so-called is not The proposition in which it is in a cause of from of i.e., that one of its which to is which is to but the of in of in which we and they are that is but of the idea of cause as it is not to it should be. A further of the is for a is not to any of its a has cause and any about it will be a the of a that of had that the cause of This of a but one better from a with these to their In sense II of the word only a is with a certain kind of can an about its a are When to the of in time the idea of a the cause of to which in not are of propositions in sense II of the word they will of be in of the various in which the them can to their and of the means by which in each this can be propositions will be In them that their will on of will to a for any proposition is a of our to a certain of things by the use of certain and one what can can do may by and the that can do a thing that one has done in the the proposition “x causes is a of a it is that can be to in that is to the and are not but and the proposition of is a thing is means of of It be in this sense of the word to the cause of any thing as It is a of sense that proposition is a In sense I, proposition is an which be in the In sense III, propositions be the of the sense a is do describe this in do not they them they them they terminology to things is that of means and the must be that they to to things idea which is by the terminology and not by the is this The is not The terminology an idea not only of one to but of one to an idea of what as is this idea from in our in the of to from the of to do by argument the we in a in which can only carry by that and from the of to do it to this to the of in which we our by not of but of things in In to this we must that sense II of the word cause is a sense, and in with the of in the and that the and the an of of what we the as a of each with its and and and In and in the in the of the use of with is an a these are to be taken and to what they to namely that the way in which use what we as means to our is not in different from the way in which we use We use by them to be with of their and them in a way that they to do what is in with our This is them to in the and original sense of that word. is in much the way as we must use it to the and we can to this use of it the word in the sense, as are certain in which things if to is to their to in these and by the of to them not as they but as if is as to in our we do use which taken but that it is and to be taken and will that this about and that are with a in which that is is that we certain means to certain Sense III of the word cause an to it not to but to the characteristics which to this sense if the were and certain which in the to In the to which sense II a cause is in its as for its on in its as for the of its on In the to which sense III a cause is in its as it to in its as its what not are The cause to its by in the cause and is a can be of and of to of to the and of the in sense II from its in sense III by these senses and A cause thing to and to its to the a cause is one with its is of In to the of sense III, will to the the of in as a of a in and the that a cause its in that in is in sense but in sense III. to the of a and the its is a the and and a of But this is by the of the a of and this process is a of the to the is to the the of the and the of the is in the sense, not the in the proposition “x causes the we to use the word cause in the sense, must any it must the of the not its that process the for be an to be but its the is is the cause in sense III and when it is it its not but in sense III is with it is with its in The cause of the is the causes and that is in a and in a the from to is used in sense may be any as it is by a of which are of But if is used in sense III, must if it be any on the of a and be the and be a But the and causes in sense III, is a not cause in sense III. The of in the is a cause in sense III, is in the of sense III. The about sense III is to what is by that a cause its When is used of senses and II we what it in sense it means that a for in sense II that is means of But what it of it in sense are to this which may be the and The means A on this is a and its to its is the of to a When that means on this that and is the kind of which one has the of one of a the of the and the one as to the of one must that in a the of the can be it and even when it on the of a The therefore, that the cause is the not only in the sense that if the cause the actually but that if the cause is the is to to the of a thing can the by thinking the of Nothing in the of is This is in a in the sense if to a of in which the for causes means a for is to from This in what to to be a of and as is a in to the it that the is But the of causation, it may be as the of a be as an of the propositions by it were these propositions have to be as one that they can be by that by the so-called to be in the so-called it is this is that the what causes a and what a cause have to be by and the of a of the type not an of propositions of the type “x causes propositions of the type “x but the of propositions in that kind of and the use of propositions in the of and propositions not propositions be the of in these propositions This not be by that the in which propositions are Such a that the idea of is a idea which when will the different idea of This to be the of the of and if to it do not to but must that it not from the what the idea is that its had to the means of the this one be taken for will that the proposition “x means merely the have to be with and not in the will be with In the for and we from the of to the of the we from the of these propositions to the that the proposition the have to be with is not what we by “x but is only the on the of which we assert the different proposition “x if means of it must be that is that is is we assert on the of of what to that is what should to assert a we must from the of the propositions to the this is we to assert but on the occasions when we do assert them we to be namely to our has by in a of to which have to but and to for it and to is a of a that it is for of its argument This will the In as it to that in sense III a cause and But to of what to on the a proposition to be even that the in to a in if is a of and of may be an even if they are it may not be to them are the of any we that a will always be a that is in the of a the the which have for do not intended to and But if has a as when on for a paper get of it this i.e., to a paper on this is for to a the word is not a of a it to a of in sense I, and means that has to This sense of in which it to is to the sense which is not only in sense and sense II of the word cause but in sense III The that can be and of the that happened only the of on the that are from one original and that be not in the sense of of by but in the sense of a in it be on of and on to the cause of an but it to be that has a even we what its cause and if a is to the that an has it may may not be but it be a of of for it that that when we use the word in sense III we to by it different from and of in the and in the and that this and is in the of that this is We have to the which we when we found the idea in sense and we have to it in the what is this idea It to be that it is from our of occasions on which we have others to in certain by them in their to the that they are in of a kind that only by can they the we them to and occasions in which we have is an idea from our and in what is a way not only to our with things in II of the word but to the which these things have Causal propositions in sense III are of in The we are in the of these of that they are the of the that it up in with the of to which in sense II of the word but that in this the a of a can be to certain in by certain means to them and if is as of the but it will that will be as about certain things in by the of certain a step in the argument if we to it we should is a thing in by as a means of we to be a in of its but as in the of that must be. in that that which is by the word that word not be as a to and be the is for not the but the This not the in a of causes of the but a cause of a as from The a of the on this it that in certain for the of certain these a in certain to with a kind of and these the of and that of to the is and is a kind of one thing in thing in The and are used and to a sense of and for the of things is taken as the of these causes is a of and is a of This the in which our of and in which the to that were in the and when the of and by a the word cause not a it a and its in these when we to and the to we as a of a taken to in a kind of which to one to as to that a cause is of that in which its is if what is is a of a certain that but by the and of the it is that for the idea of is the idea of by thing which if not under this this different by This of one thing in by is the of is that a by and in have to be in it is to and this which is has to be by an on the of the that This the namely of its which it to the as an to not that this of to is the to is that as a which the of are It is to that in is of not only not assert that must have a and this in In the of a its be what as from is for means and means a from to and to the which is its from to is in sense by the of its from to for it is not is that any which to the of is an the of are in the of that is a thing as as may be any the therefore, it were to that is that is to to the as the of that is on it that is and the be of the It have that is the and that idea of causation” is a of a to which it is is what has the in the and it has the of cause In of that we get a and of the of the of that to that as the of has to an as to up the the of the cause in certain as is not a of their in them the word cause is not used in the sense. are and they use the word cause in sense the use of the word in that sense with it a of but this is not by the A of may its The in has a of one the with its on causes in sense the of causes sense with the of and the of the cause as an in time to its the one thing in which has found namely the that what to a not a as the a which has the that has a the cause of is a and this is to not from but a be by that has a It be if were used in sense In sense II it is in that is not if we for of not to is what Kant a have with the idea in a must be used in sense III. But what can be for can only that the of cause with of that and that a of the of The cause of an can be a III, only when cause is used in sense in the proposition “x causes means be an may have but its be The of a the a these not the only that can is an that is with is in a of is a from The of the is an by the Kant has to that in the propositions the word is used in different is therefore, to it a the to the as a of of of and a certain of the in which had by when to sense III and get to sense the of cause is not only but actually the in of a of cause as can be from with with the of cause to and on a of causes a to have that the only sense of the word cause is sense III. But when as of from to causation, for the of i.e., the of the is it for in of sense II in any on the from of as as to the sense in which the word cause is will to paper the of is that has on the the The of this paper by the of on the is to a of the of has for that the of causation” is and and that in propositions about causes have are by propositions about to to and to that from it may be only in however, a a of the of time which cause and we the may this which the if is used in the sense, this proposition is for it means that is for a certain may be which is must be in the sense. But if is to is the in the sense, of are as a is is but that not that is in that sense, its the only to a is from one sense to the when the in and that the cause it the that causes i.e., that they are in way to is that as not that causes are in way to further and that this is of the word by But what in a is not this it is the cause and have this The has by and the confusion of sense II and sense III is of a the of a proposition of the causes is to a proposition of the the must be taken to not to a of of but to the of the of the word that the word is used in sense III, the sense. But the that it is used in sense for only in that sense is it that in a proposition of the type causes and are is with a idea of in which one element is from the characteristics of sense II and from of sense III. It is not therefore, that even to any proposition must be a for a proposition of the as must be in and is that a proposition which is in is is a of which that is thing as from of a that they in but and of a of this has a on The of and its The which is the of this is the of that the that is by what has This is and the for means the belonging to an to a of have a that therefore, a this as belonging to the of The is for the a namely of of that must have a cause and that this cause must be a but that our of this is have only to what by a in to that this is the that Kant on to will not that we a on as an as a which we will is that will of and to the that one is that is of to the that which is to which it to as it to with happened that but them that we of that is a that it be is not do not that things are as they are they were as they were. But do The are one in an of which to be of them has to any Kant the propositions of which this is are that as to we a of it The have in a to which have to by the word in This is that a it and have it the The word to a of the It is not used by a is thinking is to the is this of which are and what are its have the of it when in they are of their but when they to be to the to of it as have the use of is an will are It have any of the have to that the is is the “idea of causation” which for is the which this our to not only their but their

Sébastien Gandon, Brice Halimi

All the articles of this issue come from the international conference ‘Logicism Today’ organized in Besse-en-Chandesse (the very place where the first Bourbaki Congress was held) in June 2011.1 The wish of the organizers was to explore the internal diversity of the logicist movement, and to delineate its coherence and conceptual core: is there something, beyond the allegiance to the same forefathers, Frege, Russell and Carnap, and beyond confronting common adversaries, which would constitute a shared conceptual framework? Or, on the contrary, should one give up trying to find conceptual unity in what is really an archipelago of different programs? This was a risky undertaking since it looked very much like a family meeting, where one's dirty linen can be washed in private. Happily, this did not happen, and we warmly thank Patricia Blanchette, Valentin Sorin Costreie, Bob Hale, Gregory Landini, Øystein Linnebo, Steven Methven, Marco Panza, Robert Trueman, Sean Walsh, Mark Wilson, and all the students attending the talks for having always preferred constructive criticisms to lazy dismissings. What emerged finally was the multiform implementation of logicism according to three main parameters: the founding father taken as a reference (Frege, Russell, Ramsey, …), the connection established between logicism and philosophy of mathematics, the connection established between logicism and philosophy of language. The papers collected in this issue give a fair picture of the diversity of all talks. The first two, Hale's and Wilson's, pertain to properties, extensions, and sets. The last three get back to Russell, Landini's to clarify the lesson usually learned from Russell's paradox, Gandon's and Halimi's to focus on Russell's substitutional theory. Despite its multifarious aspects, logicism always corresponds to some balance sought between speculative constraints (concerning the use of unrestricted variables, the rejection of ontological types, or the status of properties) and mathematical goals (the abstractionist reconstruction of mathematical objects, the logical analysis of mathematical theories, the conceptual grasp of mathematical-physical processes). In ‘Properties and the interpretation of second-order logic’, Bob Hale offers a fresh look at the neo-Fregean abstractionist program in philosophy of mathematics through a discussion of the second-order comprehension scheme and an analysis of the very nature of the properties that are quantified over in that scheme. Hale's paper defends a deflationary approach where a property exists if and only if there stands for it a predicate that can be given appropriately determinate satisfaction conditions. Accordingly, it puts forward a nonstandard semantics for second-order logic, where the range of monadic second-order variables is the collection of all the subsets of the individual domain that are definable in the metalanguage. By doing so, he departs from Frege on two scores: first because he endorses an intensional view of properties and defines identity of two properties through a modal clause; secondly, because he does not take second-order quantification to be unrestricted. Above all, the main thread of the paper is a metaphysical analysis of properties whose consequences for logic and mathematics are only derivative. Hale concludes his analysis with two points: the first is that the usual metalogical features of compactness and completeness still hold in his semantical framework; the second is that categoricity of second-order axiomatization of important mathematical theories fails but can be given up without great loss. Hale finally focuses on the issue of impredicativity of second-order comprehension and argues both that his conception refers to a determinate range of possible values and that it is not inflationary. In ‘Enlarging one's stall or How did all of these sets get in here?’, Mark Wilson claims that two very different reasons led mathematicians to introduce sets in their theories. Set theory was first needed to extend the use of our concepts, ‘to shore up the so-called “free creativity of the mathematician” ’. Foundational programs based on abstraction principles still follow this old path. According to Wilson, however, another fundamental, yet often neglected, rationale has to be recognized behind the introduction of sets, namely ‘the desirability of understanding our computational relationships to the sorts of physical behaviors captured in differential equations (and allied forms of dynamical recipe)’. Wilson takes and deepens several examples of functions introduced by differential equations: the recourse to sets has to be understood in the context of the ‘algorithmic adjustments’ needed to extract reliable information from the equation and to compute as accurately as possible the solutions of the function at stake. Now, the language of set theory has proved to be the most natural to capture ‘our computational place in nature’, i.e., to describe the ‘mildly transcendental circumstances’ under which our ever-improving computations surround some target curve, asymptotically home in on natural processes or, on the contrary, fail to converge to the actual solution. Wilson then underlines that physical structures cannot generally be described as structures for a formal language, because the most important physical features of reality cannot be reduced to a list of primitive predicates. Set theory enters the scene precisely to capture previously unconceptualized properties, i.e., to handle physically relevant quantities to which no available linguistic predicate corresponds. The rest of Wilson's paper turns toward the other motivation for set theory, but illustrates the role of sets as supplementary entities by concrete detailed examples that point once again to geometry and mathematical physics. Wilson's conclusion is that hostility toward sets as abstract objects stems only from the stress put on one aspect of sets at the expense of the other, and from a philosophical blindness to the computational involvement of sets. Which is all grist for the partisan of mathematical naturalism. In ‘Zermelo and Russell's paradox’, Gregory Landini sets out to defuse Zermelo's priority claim, according to which Zermelo would have discovered, before Russell, what has become to be known as Russell's paradox. Through a careful examination of Zermelo's unpublished 1899 proof, Landini argues that Zermelo's result does not amount to Russell's paradox and does not show that there is no universal set: whereas Zermelo presupposed the full truth of the axiom of separation in order to prove that there is no universal set, Russell considered on the contrary the existence of a universal set as granted and diagnosed a ‘very subtle fallacy’ in the proof of Cantor's powerset theorem, due to the falsity of one instance of the axiom of separation. Indeed, as a consequence of a purely logical theorem proved by Russell in 1905 in ‘On some difficulties in the theory of transfinite ordinals and order types’, the introduction of the diagonal set corresponding to the universal set has to be precluded. Of course, Cantor's powerset theorem is true in modern set theory (with the axiom of separation available), but admission of set theory just begs the question. It remains that Zermelo's argument may be construed as a refutation, at least, of the extensional conception of sets (according to which any set is the extension of some propositional function, and any extension, a set). Yet the full refutation of the extensional conception, and the genuine Russell's paradox as well, only come from the Appendix B of the Principles of Mathematics: in short, Russell is the sole genuine author of Russell's paradox. In ‘Variable, structure and restricted generality’ and ‘Structured variables’, Sébastien Gandon and Brice Halimi draw on Russell's 1906 substitutional theory to reflect on generality and variables. Owing to Russell's universalism (his view that variables are ultimately unrestricted), the main problem is how to get a genuinely restricted range of variation. Gandon argues that structuring the variables is the Russellian answer. He goes on to cast a new light on the Julius Caesar problem and on the problem of absolute generality. Russell's substitutional approach, indeed, helps to explain why ‘2 is Julius Caesar’ is meaningless while avoiding any type distinction. It also gives rise to an ‘absolutely restricted generality’, i.e., restricted ranges of significance structurally emerging from within an absolute understanding of everything — whereas Tarski's semantics leans on relatively unrestricted domains (each domain representing a relative interpretation of ‘everything’). In connection with that point, Halimi uses the framework of syntactic fibrations as a platform to compare and partly conciliate Russell's and Tarski's respective conceptions of what it is to be a possible value for a variable. The aim pursued is first to reactivate Russell's substitutional insight in a fibrational setting. It is also to relativize Tarski's semantics as a degenerate case of what can be introduced as ‘Tarski's fibration’, namely the environment obtained when all assignments of values for variables of some language are not considered in one structure at a time, but taken upon the base space of all structures for the language. In Russell's as in Tarski's fibration, the structure of structured variables comes from the constraints set upon all the ways of picking out an item in each fiber. In other words, logical structure is construed as what restricts the available sections of some given logical fibration. The tool of fibrations, coming from category theory, thus allows one to put to the fore variables that are both structured as in Russell and interpreted as in Tarski.

Arthur C. Danto

>The translator into Hungarian of several of my writings on art theory recently sent me a catalog of the work of a young artist from her country, Ágnes Eperjesi.1 The translator—Eszter Babarczy—is a gifted critic in her own right, and I cannot do better than quote from her letter, which explains why she felt I would be especially interested in Eperjesi's art. She thought your ideas about transfiguration and the commonplace applied to her work, and she took inspiration from your essays. Hers is a remarkable and long journey from experimental photography to an absolutely unique venture of collecting wrappings of household products, and taking the humble sign language of ordinary household chores, and recreating them as objects of beauty and irony. The products, like dishwashers and vacuum cleaners—or underwear—were in all likelihood intended for export to countries anywhere in the world, which implies that they will fit into forms of life that must be much the same the world over. So the images must themselves belong to the sign language of globalism, necessarily universal in that they have to be legible to consumers who cannot be counted on to have a common language. The signs show what the consumers need to know about the products they have purchased—which side of a pair of underpants is front and which side is back, for example. In many, though not, all cases, they are pictograms or even isotypes—an acronym for “International System of Typographic Picture Education”—signs of a kind initially invented in 1936 by the Logical Positivist, Otto Neurath, who may have been influenced by Wittgenstein's so‐called picture theory of language. Neurath anticipated globalism in saying: “The visual method becomes the basis for a common cultural life and a common cultural relationship.”2 Just consider their international use in giving traffic directions to drivers who may or may not be able to read the language of the country in which they are traveling, or in guiding us through foreign airports. Isotypes are among the rare practical and positive contributions made by modern philosophy to the common life of humankind. When these pictograms are recycled—or transfigured—into works of art, their implied universality is elevated to a portrait of the society in which the products are to be used. I would not altogether follow Babarszy in calling Eperjesi's works beautiful and, in truth, I am somewhat at a loss to describe them aesthetically. But I can appreciate that in transfiguring the isotype into an artwork, an interesting reversal of Walter Benjamin's famous distinction has taken place: the art of mechanical reproduction has acquired, through transfiguration, an aura, and in virtue of that the images may acquire an aesthetic interest they heretofore lacked. At least, as art, we look at them as critics, and notice their aesthetic qualities, such as they are. Just knowing that an object is intended as a piece of fine art may trigger the sense that it must be beautiful. Or better, we use this standard term of aesthetic commendation in connection with them that we would hardly use in characterizing the pictograms on which they are based. The term ‘beautiful’ may simply be a compliment paid to art as art, without really being descriptive of the art at all. The goal of trans‐cultural communication may mean that features that imply ethnic differences will be erased—the persons are depicted neither as white nor black nor red nor yellow—leaving figures that are abstractly human, which in a way look quite “modern.” But is “looks modern” an aesthetic characterization or a stylistic one? Modernist design strove for a kind of simplification through its distaste for ornament, and this led to stylization in modernist pictography. This does not explain why the isotypical pictograms are stylized as they are, which comes, rather, from globalist imperatives—so “looks modern” is façon de parler. But in any case, the pictogram has undergone some sort of aesthetic transformation in the course of its artistic transfiguration, however we finally describe it. Eperjesi's aesthetic enhancement of workaday images is as distinctive as Andy Warhol's transformations of Polaroids into portraits. As to Eszter Bararcszy's description of the work as “ironic,” that is true—but it is due more to Eperjesi's titles—or perhaps they are captions—than to the images themselves: the covers of The New Yorker Magazine have titles, the cartoons have captions. Either way, the titles/captions express thoughts that the images alone do not. Eperjesi has often selected her images with the intention of using them to convey how women in contemporary Hungary regard themselves, and how they think about the housework for which the product designated by the image is intended to be used—usually by women as a matter of course. One of her images shows a man, a woman, and a child on bicycles. It is difficult to determine what product it stands for, though I would guess that it belongs to what we might think of as the secret life of women. The image could show a woman living a full active life, thanks to the product in question. Whatever its rhetorical aim, Eperjesi captions it this way: “My new boyfriend gets along fine with my daughter. I hope it's not just a show for now” (Figure 1). This is interesting because the woman is evidently a single mother, looking for a new relationship. The irony derives from the implication that while women in contemporary society have been greatly liberated, they remain in a disadvantageous position in relationship to men, as much so in Hungary as in America or western Europe. They are the ones left to do the housework, and to hope that men are not just out to exploit them sexually. Ágenes Eperjesi. The irony goes well beyond the power of isotypes. The three figures could, after all, just represent a family. Interestingly, most of the women are in Western clothing, which has become fairly isotypical: I imagine that the isotype on the door of women's' toilets, shown with a short flaring skirt, would be recognized in airports everywhere in the world, even by women who wear burkhas. In any case, the titles/captions have to be translated, as the artist has done, from Hungarian into the language of the country in which they are to be shown—English, for example, in the catalog I saw. English itself is not, for all it ubiquity, isotypical. Even if it were universally used, as Latin once was, there would be a distinction between pictures and words, which means that while isotypes may validate the picture theory of language, they do so in ways having nothing to do with natural languages as spoken or written. The semantics of sentences in a natural language differ from the semantic of a sentence used pictorially, as in a quotation. But even quotations have to be translated. Tempting as it is to dwell on the artistic semantics of Eperjesi's pictures, my immediate purpose in using her work has to do with its extreme contemporaneity, and to the way it illustrates the pluralistic structure that has increasingly come to define the production of contemporary art, especially since the 1960s when artists first began to explore the possibility of using vernacular imagery. I have already touched on the question of their indeterminate aesthetic qualities, but I want ultimately to discuss this with reference to the problems that artistic pluralism has raised for aesthetic theory, and especially for Kantian aesthetic theory, which more or less dominated discussion until the decade when pluralism became a driving force in art. I understand “Kantian” to mean the view that artistic excellence is one with aesthetic excellence, which is understood to be a matter of pleasure and value distinct from the pleasure of sensual gratification, and internally connected with an ideal of disinterested contemplation. It concerns what the classical aestheticians designated “taste,” and the main features of the theory are laid out in the section of the Critique of Aesthetic Judgment titled “The Analytic of Taste,” which served as the great empowering text for aesthetic theory in modern times, especially, at least in America, in the critical thought and practice of Clement Greenberg. Immanuel Kant's initial interest was in natural beauty, which it was easy enough to relate to the visual arts, understood—the decorative arts apart, which were appraised in terms of “free beauty”—either as accurate representations of natural beauty, or else as beautified representations of natural objects that in reality fell short of beauty. For what would be the point of making pictures of aesthetically repellent or deficient motifs? Greenberg had no such interest, as far as I can tell, and his chief focus was on abstract painting, which could be treated in terms of “free” beauty. This had the advantage for him that he could treat even representationalist art as if abstract, and hence subject to formalist analysis. That is to say, he appreciated painting in terms of what we might call the aesthetics of medium, since painterly excellence is determined by what pertains to properties essential to that medium, namely, in Greenberg's view, relationships between flat forms, irrespective of what the forms may signify. Aesthetic value is what these forms convey to visual perception, in which all concepts are put out of play. Kant of the pleasure taken in an object of any For the alone with he or she may put for the in The of the artist was to from painting not the The was the production of beauty for The of Greenberg's modernist aesthetics on so‐called art in the was is is how pluralism have at all when the had all the power and in the at least so far as contemporary visual art was But for that an I am to the aesthetic began to way in the and became just when Greenberg his most in his The implied by in in the catalog for the at the of was to the between art and would have spoken of the between art and reality so far as were In artistic that the of giving to art out of of had become the it was was not when in the but the kind of aesthetic by the making for what one may think of as a pluralism of aesthetic for a aesthetic that is the of the kind of aesthetic excellence Kant and Greenberg took for It is the aesthetics of and as in he the and in which the work applied as it was used by to an object of use in connection with which and are in or as by what once as is the aesthetic of by and there is question that a for it can be and even by and to young with themselves through a of Kant's main was to what one might call a pluralism of by which I mean the common and somewhat view that beauty is in the of the and that differences in are to differences in Kant to show that beauty is and to be the same for all. This was a kind of aesthetic view that so‐called were simply aesthetically in their was the for the of Western in what to be Greenberg was that his critical practice was by Kant's Critique of Aesthetic which he often the on art. In truth, the of might have in the the remarkable of Greenberg's critical could be taken to have a of on Kant's abstract which a from a of painting hardly in his own of Kant's to Greenberg's which to aesthetic that in the New there was Kant's that of beauty are and that they are universally that they are in no sense Greenberg of beauty. interest was in what he in art, which that his could not be to the aesthetics of which would of course have been of interest to In Greenberg that in art can be neither nor by or alone in this the so to of Greenberg's view is that is what is Greenberg Kant for how it was to be or in of aesthetic not think one had to know of the kind that art concerns itself with in to be or about art. he that had the possibility of of art that we or art has nothing to do with once that though he about art, he would be able to out the or three in a They need not be by the by which themselves such but that was because they were by that had to do with aesthetic as he understood was to say, in front of a piece of art, beyond an But that in no sense that one was as Greenberg's in philosophy would have having come to the view that aesthetic is That it was, on the is by Greenberg's way of his and them when the work to be was in front of the as if a the had to to by way of was what aesthetic Greenberg's was through his in artistic in at a when there was on the of to abstract art as This would have art on the New at the New and at the New the to on abstract art, because they have not taken the to of it. enough to be able to the from the in abstract art, no one has the to be on this were who to with Greenberg on could I have been using for Greenberg was the New to in the that made The of his of had about it the of a It him as well as great power in the Greenberg and, more Kant was in their to that there is a of aesthetic qualities, that to be recognized when of language touched on the of aesthetics at about the same that Greenberg dominated critical in I have in an of much it is to be we could for a while about the beautiful and to the and the this in his in his for as this out the that we to use the of ordinary language some interesting were made by such as who to that aesthetic were not It would have been interesting to out if this was the for aesthetic as a for if not, what if there are. not in Greenberg's was But aesthetics took a in the decade to the philosophy of art, I am to say, with my “The which was by and to a by the work of and especially of the became the of art, and that has more or less been the for the philosophy of art was interesting was how a aesthetics in that as as the aesthetic in artistic production and in the increasingly art world, in the and but in and finally everywhere that art was to the The artists that were and the and the in work aesthetics was of since the of art had to with the and the in which aesthetic were at there was a question of aesthetics had really to do with art at all. This was a that from the it had that aesthetic pleasure was what art was all I was fairly about the of aesthetics for art. In my main work in the philosophy of art, The of the I with what I thought of as for a of art is about and hence and that an its which is what art I this by calling works of art In my The of I more or less that aesthetics is than had been and if there were not a namely, that to be a work of art, has to have some aesthetic not beauty, say, not I the that art need have any aesthetic at all. I a distinction between and beauty, and, by between and when stands for any aesthetic that may is what I by beauty. The beauty of an is when it to the I several of this from contemporary art, for the and in In a I thought a was the use that made of beauty in painting the of as beautiful in which like a from the The beauty of was like the beauty of and the of the painting was that for the who must the of this by their the beauty is not in a work of art it which means that it in Kant's “free and In my was to from the aesthetic of and an aesthetics of It is at this point that one might that the distinction the of aesthetic to which and the ordinary language aestheticians in the us consider once In is a had just the that the work was When first the of in it was a for was so and and made so much he was to It was simply a world from my it was like a that In a from that on was the of a made of and from a Kantian to an as when he applied as his for the that they have the of aesthetic that they neither pleasure nor to the They were from the of of they were aesthetic from the that was to in the when aesthetic became an aesthetic to the of the and itself a matter of the way had been for one might say, an aesthetic of of the kind out of which Ágnes Eperjesi her art. It is not easy to describe this but it is easy enough to and it is due to the of itself due to the of making the images easy to read and to understand everywhere in the to a an recognized aesthetic in the images of the that Andy used in his first in in the of on They were the kind of that were on in for and like of the It is the aesthetics of by the need to the that would to the product But that aesthetic gets to be to the works made of as the aesthetics of images gets to be to the work Ágnes Eperjesi on of work show their and from this is not the of The of this is that the to the question of aesthetics into the of pluralism is and It is if we are of the aesthetic of and disinterested contemplation. It is if we are of the way in which aesthetic qualities, of them to as by Kant and are to the of works of art as In the of pluralism has to the of aesthetic far far than aesthetics was able to I would say, that of these aesthetic is as as Kant beauty is in the of the but in the way in which sense are in the of the just as they are. in in the he but this in no sense it from as and and all qualities, not in but in the The is the same with beauty and virtue and This however no more from the reality of the than from that of the were to in the would or be less or is a in the and of to all these the objects of art and and to have the on life and taken the matter to this I must some to view on works of art a in a section of the the of the that he his of The Kant of is not the Kant of Kantian which is on the of I it to to show how my are to his in this section of his the of which shows how Kant was the in that the of was from that alone was not the when it to of products of which we that they at least in as beautiful art, they are without we nothing to in them on the of he means of the and this he goes on to say, no than the of It is of Kant that he will a kind of in to for a when the one might say, is really is as it an that has been in the way it was used by it to what is to is is that he has that is to sense and we a through the than a or a or a Kant as an one of the which we are to that Kant is as a sort of the when in the its does that often one by The in the of ones life and without through the image of a beautiful This is a quite commonplace use of and has nothing to do with as Kant to it “The is one through as in irony or in that the in this Kant's is about the course of a as a way of about the course of a It is a beautiful which need have nothing to do with the beauty of the The just after from the visual is as an in its and as a as a with The power of is made through the that is not than can be a being of must have The image us more than is all alone the of power that an to the to itself a of representations that more thought than can be in a determined by It is in regard to the of that Kant of and of as from the of and as in reference to the of the This was much in the in the when he his In for example, a of for the of of which he was at the must have to the of an namely, that there are no in It in from this that we cannot the practice of painting on the of or on any of text with a to the of the to in full without and to use means for them from the that shows them the way to this or that of text a from the that his work to the of his work, but it a about art. art a and is not that can be ideas have nothing much to do with the aesthetics of and they are what is from Greenberg's who of in his of in art. In a which has to natural and beauty, has to do with art, which in was in the in of what were felt to be of classical beauty. is of that in his one these are hardly of ideal beauty. the of but I to for this work, in the de he for a to and is taken to be the of there is no using painting for of a to and to be in any Greenberg would have this as having nothing essential to do with art at all. was a term of critical in Greenberg's and in formalist own view is that the relationship of aesthetics to art was and The of pluralism has nothing in this But the theory of art as the of it how aesthetic can to the of the work that This I am is what when he at the of his on why artistic beauty is to natural beauty. It is because natural beauty is Kant could have since for him natural beauty is a of and us the sense that the world is not to for has a kind of as the has recently in his natural beauty, as in the of the was an to this kind of But its in works is to their beauty as art. One can that without for a that itself is in the of and The works of Andy I of convey they have as aesthetic what belongs to through their They convey the of the and the that for a will be and that and will finally come Ágnes Eperjesi in the in which products are are of the society in which products are used. They are or better, as her a picture of what like a in her may in be the of a of a woman with a in a while gets into my I can of my (Figure even image of a woman with a is into a of and a on for the of Ágenes Eperjesi.

Moritz Epple

No abstract is available for this record.

Karine Chemla

The Argument To highlight speculative trends specific to the mathematical tradition that developed in China, the paper analyzes an excerpt of a third-century commentary on a mathematical classic, which arguably contains a proof. The paper shows that the following three tasks cannot be dissociated one from the other: (1) to discuss how the ancient text should be read; (2) to describe the practice of mathematical proof to which this text bears witness; (3) to bring to light connections between philosophy and mathematics that it demonstrates were established in China. To this end the paper defines its use of the word “proof” and outlines a program for an international history of mathematical proof. It describes the sense in which the text conveys a proof and shows how it simultaneously fulfills algorithmic ends, bringing to light a formal pattern that appears to be fundamental both for mathematics and for other domains of reality. The interest in transformations that mathematical writings demonstrate in China at that time seems to have been influenced by philosophical developments based on The Book of Changes (Yi-jing) , which the excerpt quotes. This quotation within a mathematical context makes it possible to suggest an interpretation for a rather difficult philosophical statement.

Johannes Sedlmeir, Steffen Schwalm

Zero knowledge protocols provide a way of proving that a statement is true without revealing anything other than the correctness of the claim. Zero knowledge protocols have practical applications in cryptography and are used in many applications. While some applications only exist on a specification level, a direction of research has produced real-world applications. Zero knowledge protocols, also referred to as zero knowledge proofs, are a type of protocol in which one party, called the prover, tries to convince the other party, called the verifier, that a given statement is true. Sometimes the statement is that the prover possesses a particular piece of information. This is a special case of zero knowledge protocol called a zero-knowledge proof of knowledge. Formally, a zero-knowledge proof is a type of interactive proof.

Jürgen Mittelstraß

No abstract is available for this record.