It is this intentional content of Ideas that renders them at once transcendent and immanent to the mathematical field. Qua problems posed, relative to liaisons that are susceptible of supporting between them certain dialectical notions, the Ideas of this Dialectic are certainly transcendent in the usual sense in relation to mathematics.
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On the contrary, since every effort to give a response to the problem of these liaisons, by the very nature of things, yields the constitution of effective mathematical theories, we are justified in interpreting the overall structure of these theories in terms of immanence for the logical schema of the solution sought. Lautman traces this all the way into Leibnizian metaphysics. The Kantian tradition of constitution. Now, for Lautman, as we have seen, through the history of mathematics, a Dialectic of the Concept becomes transcendentally constitutive. With such a theoretical gesture come great difficulties in evaluation.
To render it transcendentally constitutive is thus in some way to historicise the a priori and, more precisely, in so far as mathematics exercises a schematising function relative to the categories of diverse regional ontologies, to historicise the schematism. Whereas Hegel affirms contradiction in the concept alone, Lautman affirms the labour of the speculative within the physico-mathematical itself. In Lautman we rediscover the speculative Hegelian conception of contradiction as the life of the concept and the movement of reason. But whereas Hegel affirms contradiction in the concept alone, independently of all relation to Kantian formal objectivity, and thus independently of any mathematics or physics, Lautman on the contrary affirms the labour of the speculative within the physico-mathematical itself.
But he critiques phenomenology in its guise as a philosophy of consciousness reflexively regressing towards a constitutive subjectivity. In order to clarify these various points, let us further develop three particularly delicate motifs. The passage from metamathematics to metaphysics. But through an authentically speculative gesture, Lautman will considerably enlarge the field of the significance of metamathematics.
Now, such concepts are more numerous than might appear. There exist. Thus the dialectical Ideas rethink metamathematics in metaphysical terms and, in doing so, extend metaphysical governance to mathematics. The question of Platonism. For Boutroux, as for Brunschvicg and for the great majority of mathematicians, there exists an objective mathematical real. If, on the other hand, we conceive this objectivity as pure construction, we will, like the logicists, adopt a nominalist position according to which the mathematical real is purely a being of language.
To which we must add a more technical aspect of Platonism, concerning the possibility of mastering mathematical entities in a manner at once ontological and finitary:.
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In the debate opened up between formalists and intuitionists, since the discovery of the transfinite, mathematicians have tended to designate summarily under the name of Platonism every philosophy for which the existence of a mathematical being is assumed, even if this being cannot be constructed in a finite number of steps. As far as we are concerned, the most adequate response to the aporia of Platonism seems to lie in the Husserlian principle of noetic-noematic correlation , which allows that the transcendence of objects is founded in the immanence of acts. If one does not take up a thinking of correlation, one must either make of the noema real non-intentional components of acts, thus ending up with a subjectivist idealism; or hypostasise them into subsistent transcendent objects, thus ending up with an objectivist realism.
In Mathematical Idealities Jean Toussaint Desanti has shown very well, with several examples the construction of the continuum and the Cantorian theory of sets of points , how to develop an analysis of mathematical objects as intentional objects. The objects constructed in this way are not intuitable as such.
Dialectic - New World Encyclopedia
They are rationally authorised objects, axiomatically governed but not given intuitively the critique of Husserlian given intuitions. When an act of positing definitions, axioms, etc. It is the second moment that is essential, in so far as it operates the passage from the first to the third.
Now, qua intentional, this moment is extralogical or extramathematical. We can thus say that, in structural mathematics, the axiomatic formalises intentionality.
The latter is essential for clarifying the profound solidarity between Husserlian phenomenology and Hilbertian axiomatics, and in particular allows us to clarify—and, we believe, even to resolve—the aporia of Platonism. Their dialectic is historical. Ontological difference. In this Heideggerian reinterpretation of Platonism and transcendental logic, we arrive back at historicity. Dialectical Ideas are to mathematical theories what being and the meaning of being are to beings and to the existence of the being ontological difference. Certainly, just as Heidegger conceived metaphysical systems as so many responses to the question of the meaning of being, responses each time oriented towards beings and not towards the comprehension of being, which remained unthought in them the play of the veiling-unveiling of aletheia , so Lautman conceived mathematical theories as so many responses to Ideas, responses always oriented towards mathematical facts and objects and not towards the comprehension of Ideas themselves, which remained unthought in these theories.
And yet, as Barbara Cassin points out, in Heidegger ontological difference cannot be seen as homologous with the opposition between Essence and Existence. For the latter like the opposition between transcendence and immanence is metaphysical. The Heideggerian ontological difference between being and beings cannot be made homologous with any metaphysical difference. One cannot therefore use any such metaphysical difference to speak either of Heideggerian difference or of the relations between it and the history of the systems of responses that it has engendered.
There is no metalanguage capable of speaking adequately of ontological difference. But we must remark that this problem is not pertinent for Lautman. For in so far as he treats of mathematical theories and not metaphysical systems, for him metaphysical languages can, and indeed as we have seen do constitute an adequate metalanguage.
In particular, here we should deepen the analogies between the Hegelian dialectic and Heideggerian historiality. It was the February 4, Six years to the day before Yalta…. He thus upholds four theses. Mathematics has a solidarity—a unity —that prevents any regression to a supposedly absolute beginning this being a critique both of logicism and of a phenomenology of the origin developed within the framework of a philosophy of consciousness. Mathematics develops according to a singular, autonomous, and originarily unforeseeable becoming—thus, an authentically dialectical becoming. The resolution of a problem is analogous to an experiment that is effective, as a programme, through the sanction of rule-governed acts.
Mathematical activity is an experimental activity—in other words, a system of acts legislated by rules and subject to conditions that are independent from them. In mathematics, the existence of objects is correlative with the actualisation of a method. It is non-categorical, 69 and proceeds from the very reality of the act of knowing. As correlates of acts, the objects project into representation the steps of a dialectical development.
Their self-evidence is conditioned by the method itself. To these theses, which he largely shares, Lautman responds by placing the accent on the question of Sense. Lautman repeats:.
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One passes insensibly from the comnprehension of a dialectical problem to the genesis of a universe of mathematical notions, and it is the recognition of this moment when the idea gives birth to the real that, in my view, mathematical philosophy must aim at. Here Dialectics is converted naturally into a research programme, an ambitious programme which Lautman formulates with remarkable simplicity and sobriety by inscribing it into the Platonist, critical, and phenomenological traditions of idealism:.
To remake the Timaeus—that is, to show, within ideas themselves, the reasons for their applicability to the sensible universe. We thus see what the task of mathematical philosophy, and even of the philosophy of science in general, must be. A theory of Ideas is to be constructed, and this necessitates three types of research: that which belongs to what Husserl calls descriptive eidetics—that is to say the description of these ideal structures, incarnated in mathematics, whose riches are inexhaustible.
The spectacle of each of these structures is, in every case, more than just a new example added to support the same thesis, for there is no saying that it might not be possible—and here is the second of the tasks we assign to mathematical philosophy—to establish a hierarchy of ideas, and a theory of the genesis of ideas from out of each other, as Plato envisaged.
It remains, finally, and this is the third of the tasks I spoke of, to remake the Timaeus—that is, to show, within ideas themselves, the reasons for their applicability to the sensible universe. I do not seek to define mathematics, but, by way of mathematics, to know what it means to know, to think; this is basically, very modestly reprised, the question that Kant posed. Mathematical knowledge is central for understanding what knowledge is.
The precise point of our disagreement bears not on the nature of mathematical experience, but on its meaning and its import. That this experience should be the condition sine qua non of mathematical thought, this is certain; but I think we must find in experience something else and something more than experience; we must grasp, beyond the temporal circumstances of discovery, the ideal reality that alone is capable of giving sense and value to mathematical experience. Beyond its moving spiritual significance, this historical debate shows that the knot of the Dialectic consists in making the problematic of the constitution of objective realities equivalent to a hermeneutics of the autonomous historical becoming of mathematics.
How are we to evaluate the Lautmannian conception, both on the plane of mathematical philosophy and that of transcendental philosophy the relation between metaphysics, reality, and mathematics in the framework of a constitutive doctrine of objectivities? He called these generally occulted formations of sense themata , and developed a psycho-historical and sociocultural version of transcendental dialectic. As a system of conflicts between opposing notions—as problematic Ideas—the themata develop an antithetic of objective reason.
They are non-refutable, and manifest a certain stability even if, obviously, the evolution of the sciences leads to considerable variations in their determination. His orientation is thus psychological imagination , sociological controversy , and historical empirical case studies —in short, anthropo-semiotic rather than epistemological and gnoseological.
1. A Philosopher-Mathematician
In this sense, he is rather close to a hermeneutico-communicational Habermasian analysis of beliefs repressed through the formation of consensus. One might then say that, in a context where positivist dogmatism had reigned triumphant, Lautman provided the bases for a thematic analysis of pure mathematics.
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This is in itself already of great importance: i. For the history of ideas; ii. For the study of the mathematical imagination in its relation to diverse sociocultural symbolic formations a surpassing of the traditional opposition between respectively internalist and externalist points of view ; iii. But this remains largely insufficient. It bears not upon the activity of the epistemic subject, but upon the reality of the theoretical object. As we have seen, it has an ontological import, and must be evaluated in transcendental terms.
But to speak in terms of a transcendental logic a logic of the objectivity of the object of knowledge of a dialectic of the concept immanent to the development of objective theories, is to admit an aporetic ground of the real. It is principally the irreducible tension between the antagonistic metaphysical principles of unification and diversification that are found at the origin of irreducible aporias such as the discrete and the continuous, space and matter, etc.
We do indeed find here, implicitly, the Lautmannian concept of problematic dialectical Ideas. But the principal difficulty remains: that of the intersection between objectivity and sense , that is to say between a transcendental thinking of the constitution of objects and a hermeneutic thinking of the historical becoming of theories. It is the embodiment of the coherent ethos of a culture as expressed in its artistic forms.
Plato insisted on moving forward through these stages as he realized that the oral state, where the citizen lived through the memories of the culture was inimical to critical reason. He argued for a new form of cultural participation based on the philosophical method or critical thinking. The first martyr to the primitive mindset was Socrates as prejudice resulted in him bring put to death.
Plato understood the need to evolve from a primitive form of consciousness to a new rational state. Language is the key to reason and as education standards fail, language which provides the vocabulary of reason is lost and we sink into a morass of meaningless MTV culture. The Perennial Wisdom.
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Plato needs to be appreciated as a perrenialist teacher. He transmitting wisdom which comes from a higher state of cognition, he called this technique the dialectic. This approach is different from scholastic study and stands at the essential core of hermeticism, alchemy, Sufism, Neo-Platonism and so on. Sadly since the time of Aristotle Plato was transformed from perrenialist to philosopher and his teachings became distorted beyond recognition.
The teachings of Plato are clearly a mystic transmission, a direct perception of wisdom not a dry logical system. This philosophia was a living system based on teaching a means for direct experience of the nature of ultimate reality. This teaching is transmitted by specific teachers in specific times sadly uninitiated students codify the teachings after their death and their potency and sometimes even their intent is lost.
The Dialectic system of Plato is a practice not just a theoretical system. Since Aristotle was not an initiate platonic the initiatory gnosis was passed via other ancient teachers. Since philosophy is a vein of living esoteric thought, Plato must be studied, meditated on, applied and the perennial wisdom experienced. To apply such wisdom we must know ourselves and free ourselves from delusion. This is a very different view from most modern interpreters of Plato and Livergood makes his case clear and eruditely.
This is also found in other perennial traditions such as the Bhagavad Gita. Platonism and later explorations such as the Cambridge Platonists in England continued the lineage of personal evolution against the dry philosophic structures of Cartesian dualism. Aristotle was used to bolster the power of the church with Platonic thought only found in Sufism and some schools of Jewish mysticism. Later strains of hermetic and platonic thought began to resurface in contradiction to both dogmatism and empiricism. Her philosophical works include critical studies in existential and analytic philosophy, and in her famous works, The Sovereignty of the Good , and Metaphysics as a Guide to Morals , developed a philosophy based on value realism and moral perception, inspired by the Platonism of Simone Weil.
Porphyry of Tyre His was originally named after his father, Malkus, which in Syriac means "King". He was later nicknamed "Porphyry" by his teacher Longinus, which meant "Purple", the colour used by kings and emperors for their presentation to the people. Porphyry was born to an established family in the prosperous Syrian city of Tyre, an ancient city enriched by international wealth and known for their multifarious religious heritage.
In his youth, he travelled widely, throughout Syria, Egypt, and the Near-East in search of spiritual wisdom. Soon, he was drawn to Christianity and quickly studied under the famous Christian theologian Origen, under whom he learned Christian scripture, biblical exegesis, literary criticism, and the Christian principles of salvation and ideology. It's not likely that he was baptized or entered into the official orders of the Church. However, after being assaulted by a group of Christian radicals, he sought for a path of deeper truth.
He masterly studied rhetoric, logic, mathematics, history, mythology, theology, and many other fields. After studying with the esteemed rhetorician Longinus, he met with Plotinus, the great Neoplatonic philosopher in Rome, and stayed with him for several years Simmons, For Porphyry, the one thread of meaning throughout his life was the discovery of a universal method, an all-encompassing spiritual philosophy, which would allow everyone to attain salvation.
Though Christianity promised this, was there a truly universal salvation available to all, a path outside the Abrahamic religion? Porphyry's life quest was to discover such a path.
In the end, after decades of study and personal experience, he expounded a three-stage means: 1 for the general population, those who are simply trapped in the madness and business of life, a rich devotion and theurgic adherence to their ancestral gods will be sufficient for their future life among the gods after their deaths. Porphyry's life and work is a rich ocean of study, but it's often written off as being unoriginal or rationalist by those who have not delved into it deeper.
However, his thought is more mystical, more religious, and more comprehensive than is often appreciated cf. Truly, his life's work is an act of charity, a generous pursuit into trying to find a way for everyone, for each of us, a realistic path in which our souls can find eternal hope and everlasting happiness. Schindler, D. He is man, balanced in virtue and knowledge. After receiving his Ph. He served as editor of Communio: International Catholic Review since Seung, T.