Mathematics

Modern mathematicians tried to make their discipline as logically watertight as possible against psychological implications, because they regarded the latter as purely subjective, while they thought that mathematical logic concerns a purely objective, true, nonpsychological reality.

Von Franz identifies the foundational project of modern mathematics as a deliberate exclusion of psychological dimensions, a move ultimately destabilized by Gödel’s incompleteness theorems.

von Franz, Marie-Louise, Psyche and Matter, 2014thesis

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My attempt to penetrate into the fundamental problems of mathematics has until now been nearly entirely ignored. However, I am convinced that the next steps of creative scientific thinking will go further in this direction.

Von Franz claims that her project of reintegrating qualitative, temporal number-theory into mathematics represents the direction of future scientific thought, linking synchronicity to the foundations of the discipline.

von Franz, Marie-Louise, Psyche and Matter, 2014thesis

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The mathematicians discarded the symbolic qualities. So we have two streams of science. And there is nothing in between. The only man who seems to have seen a bit deeper is the German mathematician Hermann Weyl.

Von Franz diagnoses a historical split between symbolic-qualitative and formal-quantitative approaches to mathematics, finding only Weyl capable of bridging them.

von Franz, Marie-Louise, Psyche and Matter, 2014thesis

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logical-mathematical laws (originating in the archetype of order that the numbers are) are certainly not the only ones that exist, but there are also special logical laws for the connections between associations in myths and dreams.

Von Franz situates mathematical logic as one of two distinct but historically co-present logics, the other being mythological or imaginal logic, both rooted in archetypal ordering principles.

von Franz, Marie-Louise, Psyche and Matter, 2014thesis

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one of the most important of contemporary ideas and one that is becoming ever more important, perhaps one of the fundamental philosophical models for modern physics, is the idea that ultimately what we are dealing with is a mathematical structure.

Von Franz traces the modern thesis that reality is fundamentally mathematical back to the Pythagoreans, linking it to the archetypal omnipresence of the divine.

von Franz, Marie-Louise, Psyche and Matter, 2014thesis

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how often mathematicians and physicists were convinced of the rightness of a conclusion by its beauty, even though at the time they could not see why it must be correct, and perhaps even knew of evidence against it that only later was found to be mistaken.

McGilchrist argues that mathematical intuition—grounded in aesthetic response—operates as a right-hemisphere faculty that precedes and can override formal left-hemisphere proof.

McGilchrist, Iain, The Matter With Things: Our Brains, Our Delusions and the Unmaking of the World, 2021thesis

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Descartes believed that every field—such as geometry, mathematics, arithmetic, astronomy, and music—was founded on some ‘universal mathematics’ whose basic principles were the serial character of numbers and their proportional relations.

Von Franz shows that Descartes’ vision of a universal mathematics recapitulates Platonic cosmology, reinforcing the argument that mathematical structures originate in archetypal models.

von Franz, Marie-Louise, Psyche and Matter, 2014supporting

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Mathematics, it is said, is concerned with symbols, the truth or falsehood of which can be known without studying the outside world. Mathematical propositions, Bertrand Russell adds in another paragraph, are thus purely formal.

Rudhyar, drawing on Russell, distinguishes mathematics as a purely formal and analytical discipline from empirical sciences, framing the distinction as foundational for understanding astrology’s epistemic status.

Dane Rudhyar, The Astrology of Personality: A Re-formulation of Astrological Concepts and Ideals in Terms of Contemporary Psychology and Philosophy, 1936supporting

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a few symbolical relations (i.e. formulas) are seen to suffice for the ordering of the multitude of the world’s events into a pattern, knowing which man will gain relative mastery over natural elements through the power of foreseeing.

Rudhyar highlights mathematics’ capacity to impose symbolic order on natural complexity, treating mathematical formulas as instruments of divinatory and scientific foreknowledge.

Dane Rudhyar, The Astrology of Personality: A Re-formulation of Astrological Concepts and Ideals in Terms of Contemporary Psychology and Philosophy, 1936supporting

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I felt a downright fear of the mathematics class. The teacher pretended that algebra was a perfectly natural affair, to be taken for granted, whereas I didn’t even know what numbers really were.

Jung’s autobiographical account of mathematical anxiety reveals a psyche unable to accept abstract quantity divorced from imaginal content—an early expression of the symbolic split von Franz later theorizes.

Jung, Carl Gustav, Memories, Dreams, Reflections, 1963supporting

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Pl. Rep. 525c is careful to distinguish the study (‘with the mind only’) of mathematics from its commercial use. This is quite different from Aristoxenus’ derivation of arithmetic from commerce.

Seaford contextualizes the Platonic separation of pure mathematical study from practical commercial reckoning, illuminating the historical roots of the split between formal and applied number.

Seaford, Richard, Money and the Early Greek Mind: Homer, Philosophy, Tragedy, 2004supporting

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For all things to be number, or made of number, it must be the same sort of thing as, and yet ontologically prior to, everything else. The doctrine focuses on the quantitative aspect of things to the exclusion of the qualitative.

Seaford argues that the Pythagorean reduction of reality to number parallels the monetization of value, both privileging abstract quantity over qualitative difference.

Seaford, Richard, Money and the Early Greek Mind: Homer, Philosophy, Tragedy, 2004supporting

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it bears the same relationship to similar statements of philosophy as the statements of applied mathematics bear to pure mathematics.

Bion uses the distinction between pure and applied mathematics as an analogy for the relationship between his theoretical system and empirically testable psychoanalytic hypotheses.

Bion, W.R., A Theory of Thinking, 1962supporting

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these mathematical disciplines are assigned a highly ‘philosophical’ intentionality. This is to ‘compel the soul to reason about abstract number … rebelling against the introduction of visible or tangible objects into the argument’.

Sharpe and Ure document Plato’s pedagogical use of mathematics as a propaedeutic for philosophical abstraction, training the soul to disengage from sensory particulars.

Sharpe, Matthew and Ure, Michael, Philosophy as a Way of Life: History, Dimensions, Directions, 2021supporting

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Another dimension where perhaps more than a simple analogy exists between the concepts of physics and of Jungian psychology is that of ‘absolute knowledge.’

Von Franz proposes that information-theoretic and Jungian notions of absolute knowledge may share more than analogy, gesturing toward a mathematical-physical basis for synchronicity.

von Franz, Marie-Louise, Psyche and Matter, 2014aside

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of all bonds the best is that which makes itself and the terms it connects a unity in the fullest sense; and it is of the nature of a continued geometrical proportion to effect this most perfectly.

Plato grounds cosmic bonding in continued geometrical proportion, establishing mathematical structure as the ontological principle of unity underlying the visible world.

Plato, Plato’s cosmology the Timaeus of Plato, 1997aside

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lines can be expressed as numbers. We have already had a hint of this in the phrase ‘giving them a distinct configuration by means of (geometrical) shapes and numbers’.

The Timaeus commentary shows Plato reducing geometrical elements to numbers, establishing the mathematical reduction of physical structure that becomes foundational for later Pythagorean-Platonic physics.

Plato, Plato’s cosmology the Timaeus of Plato, 1997aside

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