Mathematics

Simone Weil’s Reflections on the Pythagoreans

D.K. Levy read

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Summary of paper for F[ather] P[errin].

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Mathematical truth was initially theological (Philolaus).{257}  Mediation for the non-square, unjust numbers (Aristotle).{256}  Logoi alogoi, scandal, absurdity.{259}  Passage from the Timaeus on geometric mediation.{253} Epinomis.{258} Natural law, function and proportion.{260}

Invention of demonstrative proof (the soul of our science, including the experimental method), due to the Greeks’ need of certainty for divine things, even in their images.{260}  Numbers, choice image

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of divine things on account of their certainty.  But the real number is better.  Certainty and non-representability.   Introduction to faith.{261}

Scale from the least to the most certain and from the most to the least representable.  Mathematics [is] intermediate.  At the same time, epitome of the mechanism that governs matter and image of divine truths.  [And] at the same time, it has mediation at the core.{261}

Astonishing poetry.  Revelation.{261}

This need for certitude in divine things had been lost.{261}

Lately the need for rigour in mathematics has been found again.  That can be a path.  For in mathematics there is no objective.{261}

[ms. 29]

Using mathematics to make felt the possibility of certainty concerning that which one does not understand.  To mould mathematics for this purpose.{262}

(Koan.  It is that which one does not understand that one is certain about. It is the opaque which alone is seen.)

Providential arrangement precluding mathematics from lapsing into technique.{262}

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φιλίαν εἶναi ἐναρμόνιον ἰσότητα. [friendship is an equality composed of harmony]{262}

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1. Trinity, ἁρμονία, δίχα φρονεόντων συμφρόνησις. [harmony, the common thought of those thinking separately]{262}

Proof of the Trinity. God the subject. But also the object and the link between the two in him. And each of these things is “I am”.  Plato, Same and Different.{263}

Equality between one and many, between one and two. {263}

(The one, άρχή [arché] and the first to be put together. Phil[olaus].){263}

(Saint Augustine, aequalitas, connexio.) [equality, connection]{264}

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2. Opposition between the creator and the creature, limiting and unlimited.  Second pair of opposites in God (Philebus).{264}  The most beautiful harmony, the maximum of separation and of unity.  That a divine Person should be a thing, inert matter. (an enslaved man and dying in agony).  Christ’s agony. “Why have you [forsaken] me…”  Christ has many young brothers.{265}

Limit = number.  Philebus: one and many (Trinity), limit and unlimited (creation).  Number, mean proportional.

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Philebus: every study reproduces this hierarchy.  Intelligence, an image of faith.{264}

[Passage to the limit, solution to insoluble difficulties.  E.g. Christ’s agony.] [her brackets]{265}

δίχα φρονεόντων. — “My God, why…?” — συμφρόνησις, one single God.  Love that exceeds all knowledge. [the common thought…of those thinking separately]{265}

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3. Friendship between God and man.  Harmony as mean proportional. Mediation. Timaeus, Symposium, Epinomis, St. John. Friendship, geometric equality in the Gorgias. “No one enters through here unless he is a geometer.” Ὁ θεὸς ἀεὶ γεωμετρεῖ (double meaning). [God is a perpetual geometer.] Geometry, the first of the prophecies.  That is why science has become diabolical.  The true relation of science to charity, analogous to Gregorian chant.{266}

[ms. 30]

Demonstrative rigour is to science as stone is to sculpture.{267}

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4. Friendship between men.{267}  According to nature, either [the] I is at the centre (perspective), or else another who dominates me by brute force is, and the rest are only particles of the universe, {268} save in the exceptional case of natural justice (Thucydides, Athenians and Melians)—Friendship is identical to supernatural justice.  Christ, mediator between men.{269}

“We”, a collective feeling, false friendship, without harmony, since here the terms are of the same kind, the same root, the same level.{271}

Supernatural justice, operation analogous to the one that overcomes perspective.  No centre within the world, only outside the world.  Renouncing through love of God the illusory power that he leaves us to say, “I am.”  Not, as some do, by transferring it onto Him, like Oneone onto Phaedra or Pylades onto Orestes.{269}  For the true I am of God differs infinitely from our, illusory one.  Renunciation without transfer.  That is the love of God.  But since any real human thought is of a concrete object here below, [the love] initially presents itself either as love of the beauty of the world or as love of one’s neighbour.  This renunciation is the abandonment of all one’s goods in order to follow Christ.  Social standing is only a boost to the ability to say “I”.  The acceptance of poverty, “If one wants to become invisible…”{270}

Harmony, here, [is] unity of opposites.  Opposites: me and

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the other. Unity only in God — Justice and love (of one’s neighbour), identical.{270}

God [is] mediation between God and God, God and man, man and man.  One single harmony.{271}

Christ [is] always present between two true friends.  Conversely the statement, “If two or three…” is the promise to his friends of human friendship.{271}

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5. Harmony in things. — 1] between things and God. 2] between things.[her brackets]  In the two cases, mediation is limit, which is also λόγος and ἀριθμός [relation and number] — Things, men, included as natural beings (myself included).{272}

[ms. 31]

The opposition [between] limit-unlimited encompasses all theories of knowledge.{272}

Philolaus. Number gives a body to things.  Gnomon, i.e. invariant and variation group.{272} Cube (Lagneau). The cube is actually the body of the [cuboid] box.  The reality of the perceptible universe is mathematical necessity.  Theory of perception.  Matter, only a support for necessity, which being conditional needs a support.{273}

Necessity: set of laws of variation determined by fixed relations and invariants.{274}

Reality = contact with a necessity — (contradiction: necessity is not tangible. Harmony, mystery.[)]{273}

Cube, transcendent in relation to the appearances of the [cuboid] box.{273}

Necessity, [being] conditional, supported by a matter (Timaeus) of which water (baptism) is the image.  Mer, mère, matière, Marie…[Sea, mother, matter, Mary…] The essence of necessity (variation and invariant) is function. Example. (See margin) [the figure page1image43647392.png is in the margin] Function, proportion, λόγος, ἀριθμός. [relation, number]{274}

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Necessity, enemy for the man who says I.  [It is] a slave in reverie (and through social domination).  A brutal master in affliction.  Natural equilibrium is apparent at the optimum point of methodical action.  (Same 3 relations between men.)  This equilibrium joined to natural justice would be natural happiness.  The legislator’s objective.{274}

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But this equilibrium is [only] an appearance.  Being fatigued makes one sense this.  The human will is also an inert thing.{275}

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In the practical experience of necessity, illusions are always attached |in the service| to the exercise of the will.  Necessity is thought of in a pure way only as theoretical and conditional.  There, man is absent, except as regards the very operation of thought. Any concrete knowledge of facts, even human ones, is a recognition in them of a mathematical, or analogous, necessity.{275}

(Each time in a specific form.  Analogy with embodiment.)

Relation of necessity to man, thus, not of master to slave, or of equals, but of painting to looking.  In that looking is born the supernatural faculty of consent.  One does not consent to force as such (for it compels), but as necessity—Pure intelligence is at the intersection of nature and the supernatural.{276}

[ms. 32]

This consent is a madness that responds to the triple madness of God (Creation, Incarnation, Passion), but primarily to the first.{276}

λόγος, name of Necessity, given to the Beloved — Light and rain in the Gospels, Stoicism.{278}

Necessity, mediator between the natural part of us and supernatural consent.{278}

By analogy, conceived as mediator between matter and God.{278} Creation and ordering action.{279}  Zeus and Bacchus.{280}  Myth of the Chaos.{279}  Necessity, willed to a much higher level by God.{280}

Mediator between thing and thing.  Anaximander.  Disequilibrium oscillating, [is] equilibrium refracted in time.{279}

God mediator between

God and God

God and man

Man and man

God and things

A thing and a thing

Me and me.{289}

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God is mediation, and in itself everything is divine mediation.{289}

Analogically, for human thought, everything is relation—λόγος.[relation]{289}

Relation is divine mediation.  Divine mediation is God.{289}

“All is number.” {289}

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God thinks necessity.  Necessity is, because he thinks it.  God’s thought is his Son.{279}

The order of the world in God is the orderer [of the world]. In God, everything is subject.{279}

In itself, isolated, every phenomenon is an active principle of the destruction of the universal order.  Through its connection this order is totally present within it.{279}

The “I” keeps us confined within necessity just like the vault of heaven and the surface of the earth. We see necessity from the side that is brute domination.  The renunciation of the “I” moves us through to the other side, puncturing the egg of the world.  We then see necessity from the side that is obedience.  [Being] the children of this household, we love the docility of this slave.{280}

Necessity — freedom; obedience is their unity.{280}

Necessity, the obedience of matter to God.{280}

The idea of a miracle, devoid of meaning.{280}

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One consents to necessity before knowing that this consent pierces the [ms. 33] egg [of the world].  God consents for us in us.{280}

The natural part of us remains in the egg.

What a horizontal plane is to its two sides, necessity is to domination and obedience.{281}

As necessity in God is a thinking Person, so in us it is thought in action, a relation conceived, a proof by demonstration (Spinoza).{281}

1 and 1 do not make 2 without addition.{281}

Attention [is] creative of necessary connections.  (Those which do not depend on the attention are not so.){281}

The involvement of necessity in part in constraint, and in part in intelligence, justice, beauty, faith (symbolic language of divine truths — circular and alternating m[o]v[emen]t  — √2 even and odd, etc.).{281}

Attention of the intelligence, the image of Wisdom.{281}

Intelligence, intersection of nature and the supernatural.  Produces a semi-reality (conditional necessity). Love (consent) produces reality.{282}

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Beauty, pure joy: complicity of the body and the natural part of the soul with the faculty of supernatural consent.  Indispensable, even to those who have a vocation for the Cross.{283}

The feeling of beauty, perceptible feeling, in the carnal part of

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the soul and even in the body, that this necessity which is constraint is also obedience to God.{284}

Image of the mysteries of faith in mathematics.  Mathematics, rational and abstract science, concrete science of nature, mysticism.{284}

Universe, a compact mass of obedience with luminous points.  Everything is beautiful.{286}

Each one [is] also (small mass, a point). Water and spirit.{287}

Mysterious influence (but without [the] violation of [any] laws), on nature, from the presence of supernatural love.{287}

Philolaus. The key.{288}

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Philebus.  Anguish and joy. Anguish and revelation of the beauty of the world: Job.{288}

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Beauty — Operation of pure intelligence in the conception of theoretical necessity and [its] embodiment in concrete knowledge of the world and of practical skills — Flashes of justice, of compassion, of gratitude between men — Three supernatural mysteries constantly present right at the core of nature {288} — Three doors open upon the central door which is Christ.{289}

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For us, everything is relation.{289}

In itself, everything is mediation, divine mediation.  God is mediation.  All mediation is God.{289}

Supreme mediation, supreme harmony between the why of Christ (repeated unceasingly by every soul in affliction) and the silence of the Father.  The universe (including us) is the vibration of this harmony.{290}

(One does not truly comprehend the universe and the destiny of men, especially the effect of affliction on the souls of innocents, except by understanding that they were created, the one as the Cross, the others as the brothers of Christ crucified.){290}

To address the danger of pantheistic error in this comparison, comparison of the cube and the cuboid box.{291}

Thus, touching God through all [there is].{292}

‘None Enters Here Unless He is a Geometer’: Simone Weil on the Immorality of Algebra

Aviad Heifetz read

Abstract

The French philosopher Simone Weil (1909-1943) thought of geometry and algebra not as complementary modes of mathematical investigation, but rather as constituting morally opposed approaches: whereas geometry is the sine qua non of inquiry leading from ruthless passion to temperate perception, in accord with the human condition, algebra leads in the reverse direction, to excess and oppression. We explore the constituents of this argument, with their roots in classical Greek thought, and also how Simone Weil came to qualify it following her exchange with her brother, the mathematician André Weil.

About the Author 

Aviad Heifetz is a professor in the Department of Management and Economics at the Open University of Israel.

The Weil Conjectures

Brian Hayes read

Sample: “. . . . Before going further I should make clear that The Weil Conjectures is not a textbook or a scholarly monograph. It is not addressed to an audience of mathematicians. But it raises questions about relations between mathematics and society that may well be of interest to the mathematical community. This issue is commonly discussed in terms of outreach—the challenge of communicating research-level mathematics to the public. In Olsson’s case it also becomes a question of reach: how can we help someone who feels a powerful attraction to mathematical ideas but cannot negotiate the rugged terrain of prerequisite knowledge?

The heart of Olsson’s book is a personal essay, in which she describes her own intense and turbulent encounters with the world of mathematics. That narrative is braided into the stories of the Weil siblings—whose lives were also marked by intensity and turbulence. . . .”

Notices of the American Mathematical Society, vol. 68, no. 2 (Feb. 2021) (book review)

Mathematics and the Mystical in the Thought of Simone Weil

John Kinsey read

On Simone Weil’s “Pythagorean” view, mathematics has a mystical significance. In this paper, the nature of this significance and the coherence of Weil’s view are explored. To sharpen the discussion, consideration is given to both Rush Rhees’ criticism of Weil and Vance Morgan’s rebuttal of Rhees. It is argued here that while Morgan underestimates the force of Rhees’ criticism, Rhees’ take on Weil is, nevertheless, flawed for two reasons. First, Rhees fails to engage adequately with either the assumptions underlying Weil’s religious conception of philosophy or its dialectical method. Second, Rhees’ reading of Weil reflects an anti-Platonist conception of mathematics his justification of which is unsound and whose influence impedes recognition of the coherence of Weil’s position.

Philosophical Investigations, vol. 43, nos. 1-2 (January-April 2020), pp. 76-100.

The Weil Conjectures: On Math and the Pursuit of the Unknown

Karen Olsson

New York: Farrar, Straus & Giroux

Simone Weil’s Spiritual Critique of Modern Science: An Historical-Critical Assessment

Joseph K. Cosgrove read

Simone Weil is widely recognized today as one of the profound religious thinkers of the twentieth century. Yet while her interpretation of natural science is critical to Weil’s overall understanding of religious faith, her writings on science have received little attention compared with her more overtly theological writings. The present essay, which builds on Vance Morgan’s Weaving the World: Simone Weil on Science, Necessity, and Love (2005), critically examines Weil’s interpretation of the history of science. Weil believed that mathematical science, for the ancient Pythagoreans a mystical expression of the love of God, had in the modern period degenerated into a kind of reification of method that confuses the means of representing nature with nature itself. Beginning with classical (Newtonian) science’s representation of nature as a machine, and even more so with the subsequent assimilation of symbolic algebra as the principal language of mathematical physics, modern science according to Weil trades genuine insight into the order of the world for symbolic manipulation yielding mere predictive success and technological domination of nature. I show that Weil’s expressed desire to revive a Pythagorean scientific approach, inspired by the “mysterious complicity” in nature between brute necessity and love, must be recast in view of the intrinsically symbolic character of modern mathematical science. I argue further that a genuinely mystical attitude toward nature is nascent within symbolic mathematical science itself.

Providence College, Philosophy Department Faculty Publications

A 1940 Letter of André Weil [to Simone Weil] on Analogy in Mathematics

Andre Weil read

Notices of the American Mathematical Society, vol. 52, no. 3, 334-341