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Text of the page (random words):
ze syntactic and semantic aspects of the interface between two existents which play no explicit role in sartre s analysis as recognizably demanded by that analysis now the submerged role of this object in constituting the two dimensional lattice of concepts and objects remains unacknowledged in frege but it is precisely the debt incurred in the institution of the transcendental which is paid back inevitably by the diagonalization out of this lattice for frege denying the debt its consequences are nothing short of disastrous the repayment entails recognizing that the elevated object is isomorphic to various objects that are not constructed or at least not bound by the intent of any particular construction once again it s by the diagonal theorems that we know that i the number two in its recognition as classical truth value object is inseparably entangled with everything else about the number two that is not intended in this usage and ii that the latter is strictly inexhaustible inseparability from an inexhaustible excess is one form taken by the repayment of reason s debt note that the language of promotion though prompting apt questions about the contingency and lifespan of a transcendental is not incompatible with recognizing that the promotion is also entirely justified since it is the one and at least some of its immediate neighbors which must function as elements of a truth value object if cognition which is to say compression is to be possible the dialectic here is one of necessity and contingency and especially sartre of the necessity of contingency but not of a simple inversion of logicist necessity into empiricist contingency it s natural to want to appeal to the infinite in order to overcome this necessity of contingency to say for instance that the one etc are only the initial elements in the construction of a transcendental that can be extended into the transfinite the diagonal theorems however anticipate and block this restoration all at once and from the outset 2 a second way of looking aslant at frege s thesis likewise applies the principle that identity is not a one way relation when we say that concepts are functions at least if we want to know what we are saying and not merely to promulgate a riddle we import productively into logic the substantive mathematical problems concerning the nature and behavior of functions now the stock objection is that this is a fallacious conversion the assertion is partititve says the textboook fregean all concepts are functions but of course not all functions are concepts addition for instance is not a concept e g heck and may 2011 pp 126 66 the correct reply as it now seems to me is to grant that we can take the thesis as partitive but only by being forced to recognize that this partitive logic taking the concept as its proper sphere now selects from a pre existing ideal matter of functions becoming a derivative activity rather than a presuppositionless one while any identification of it with first philosophy is blocked so whether we choose a broader or a narrower conception of the concept the systematic importance is the same in recognizing that concepts are functions the first philosopher or ontologist whether or not she decides to call herself a logician is forced to take on the problems of functions as such work on the problems of functions as such will either be classified as belonging to an ontology prior to logic or else as part of a radical expansion of logic a greater logic ironically the function x 1 seems like the most natural choice for a fixed point free map taking over the role of classical negation in more general contexts that is to say since generality is itself a contextually relative notion in contexts which are more general in a particular context diagonalization taken as a name for the whole structure essential to the diagonal theorems is after all about the distribution of fixed points if not over here a fixed point free map then over there in the diagonal map in the narrow sense so our choice is then a curious one we can remain logicists and accept the startling converse all functions are concepts even if we necessarily don t know how to use nearly any of them as concepts even if we therefore don t intuitively recognize or want to recognize most of them as concepts you would expect this to be deleuze s line but as we know it isn t and instead we find the rather inert distinction in what is philosophy between the scientist s creation of functions and the philosopher s creation of concepts or else we can save the letter of the proposition all concepts are functions but not all functions are concepts by losing the systematic significance of concept and logic and demoting logic to a branch of mathematics the latter henceforth the ontologist s primary concern frege s thesis concepts are functions then far from representing the inaugural contribution to logicism he intends and as which it is generally received provokes a dilemma that brings the ideology of logicism to its close at the instant of its historical inception or an instant before whether by expanding logic to include all of its foundation in mathematics or by showing itself to be a founded specialization of the latter logic testifies to the primacy of the mathematical for ontology logicism like the pythagoreanism whose trajectory it wonderfully recapitulates is from the beginning between two deaths though its recognition as such can only be retroactive the first clear articulation of logicism is equivalent to its self refutation again like pythagoreanism and ultimately for exactly the same reason there are too many functions and in this lies the uniqueness and supreme usefulness of pythagoreanism and logicism both for while they are doubtless ideological to the core no mere ideology is capable of self destruction it doesn t matter how many absurdities say catholicism piles one atop the other it will never be capable in recognizing a contradiction as its own being of reduplicating the feat of its god in learning to die but pythagoreanism and fregeanism are guided by a consistency which makes it possible for them to provoke and discover their own impasse in the face of a mathematical real which they can neither assimilate nor reject from one angle they are immediately surpassed ideological formations from another they remain as permanent subsystems of any thinking relation to that real they belong negatively but forever to a dialectic which can be nothing other than their negation note that all functions are concepts doesn t mean contrary to our experience that any given function can be grasped as a concept it both implies and explains the exact opposite most functions cannot be used as concepts in a given situation in exactly the same way in which most numbers are uninteresting but in that case there is a least but being the least uninteresting number is an interesting property etc cf berry s paradox and the examination paradox uninterestingness like indefinability is evanescent in exactly the manner of sartre s contingency though sartre of course does not recognize the computational basis of this phenomenon or that it enables and even demands a computational account of the for itself diagonalization fulfills the promise of sartre s dyadic phenomenology in providing the rigorous solution which sartre anticipates only through his sketch of pure reflection to the problem of how such evanescences can be known to the phenomenologist as i show elsewhere the problem and its solution are isomorphic to those of meno s paradox the right mapping between frege and sartre then between the ideal of the in itself for itself and a system which would bring together the intensional and extensional sides of functions as such not only functions restricted to a domain gerrymandered to get this correspondence to work out and baptized logical we thus wind up shifting our ground from both analytic logicism and phenomenology whose sterile oppositions only make sense from within their joint refusal of mathematical ontology this two sidedness is metalogical difference the negation which holds together the two sides in the greatest intimacy yet without possibility of synthesis is metalogical negation of which diagonalization is the witness 3 how should this account modify our reception of frege s second great idea that concepts about number are second order concepts the content of a statement about number is an assertion about a concept again largely by opening this bridge for two way traffic so that rather than the intended logicist reduction of number we have a permanent contamination of logic and through it of ontology by number the result is a platonism far more radical and i think far more interesting than frege s so called platonism in which first order concepts are flash frozen without dialectical transformation few things give me more joy by the way than reflecting on frege s assertion its grounds and consequences and on the limits of the identity it proposes and which frege would like to keep domesticated between number and some higher concepts though it claims to be able to work from given concepts indifferently mathematical or nonmathematical without transformation it suffices to provide a royal road to the investigations of the mathematical dialectician to call this into question meanwhile immediately it gives to mathematical dialectic the key to the relation between socratic elenchos about aboutness and the difficulty of speaking about the concept and the platonic ontology i e the essential relation of concepts to number which relation is constitutive of any useful theory of forms ideas how does frege block the return traffic by observing that in respect of their unsaturatedness functions differ fundamentally from numbers but this is quite mistaken not only because as i maintain reversibility between the roles of concept and object is permanently characteristic of number in each of its local domains but because an excess of unsaturatedness has been mistaken for a lack we recognize the unsaturatedness of ordinary first order concepts isblue and the like precisely because we see how to compete them that there is one way to fill in the blank makes the blank both conspicuous and manageable but what gives the illusion of common sense to frege s observation about numbers it is that the ways in which they can function as well functions are as numerous as the functions themselves this excessive unsaturatedness disrupts the logicist s implicit or explicit typing scheme by being able to be connected in too many ways to other mathematical objects numbers place second order demands on the indexing of concepts types etc this excess of relatedness in number is misread by the logicist as a lack of relatedness except insofar as the number is taken up by a proper concept but the problem is not restricted to number note that rather there must be a certain unsaturatedness of the object generally expressed in frege s own context principle though it is not directly exchangeable with the unsaturatedness of the concept in absence of a higher function for coordinating the two nevertheless we should not speak of the object as saturated but as unsaturated in a different place in a different order than is what counts as the concept perhaps we should speak of the object as unsaturated to the left and the concept as unsaturated to the right and consequentially of left and right adicity or similar leave a comment september 16 2015 from lautman 1939 a very rough translation really just google plus a first pass at corrections of the closing paragraphs of lautman s half of this 1939 joint presentation with cavaillès to the spdp entitled mathematical thought la pensee mathematique which for whatever reason are edited out of the goldhammer translation in the balibar rajchman boyman ed anthology french philosophy since 1945 which is the only place i know of the text s being available in english whew anyway to me these lines seem important and to differ in minor but interesting ways from the statements of the same theses by lautman elsewhere for these see of course s duffy s translation of mathematics ideas and the physical real so having poked at them a bit i offer them to the stranger gods of the web corrections are entirely welcome probably half the people who d be inclined to read this are more proficient and definitely more efficient francophones than yours truly il me semble que le problème des rapports de la théorie des idées et de la physique pourrait être étudié de la même façon considérons par exemple le problème de la coexistence de deux ou de plusieurs corps c est là un problème purement philosophique dont nous dirons que kant l a plutôt posé que résolu dans la troisième catégorie de la relation il se trouve néanmoins que dès que l esprit essaie de penser ce que peut être la coexistence de plusieurs corps dans l espace il s engage nécessairement dans les difficultés encore insurmontées du problème des n corps considérons encore le problème des rapports du mouvement et du repos on peut poser abstraitement le problème de savoir si la notion de mouvement n a de sens que par rapport à un repos absolu ou si au contraire il n y a de repos que par rapport à certains changements mais tout effort pour résoudre de pareilles difficultés donne naissance aux subtilités de la théorie de la relativité restreinte on peut également se demander à laquelle des deux notions de mouvement et de repos il faut attacher un sens physique et c est un point où s opposent la mécanique classique et la mécanique ondulatoire celle là envisage l onde comme un mouvement physique réel pour celle ci au contraire l équation d onde n apparaît plus que comme un artifice destiné à mettre en lumière l invariance physique de certaines expressions par rapport à certaines transformations il apparaît ainsi que les théories de hamilton de einstein de louis de broglie prennent tout leur sens par référence aux notions de mouvement et de repos dont elles constitueraient la véritable dialectique il se peut même que ce que les physiciens appellent une crise de la physique contemporaine aux prises avec les difficultés des rapports du continu et du discontinu ne soit crise que par rapport à une certaine conception assez stérile de la vie de l esprit où le rationnel s identifie avec l unité il semble au contraire plus fécond de se demander si la raison dans les sciences n a pas plutôt pour objet de voir dans la complexité du réel en mathématiques comme en physique un mixte dont la nature ne pourrait s expliquer qu en remontant aux idées auxquelles ce réel participe it seems to me that the problem of the relations of the theory of ideas and physics could be studied in the same way consider for example the problem of the coexistence of two or more bodies this is a purely philosophical problem that we say that kant posed rather than solved in the third category of relation it is...