Kant (and mathematics): * see EndNote<A>

Mathematics was an important subject for Kant, on which he wrote & taught.  However, his view of mathematics was limited to arithmetic & geometry; he does not reference calculus.  He claimed that elementary mathematics, like arithmetic, is synthetic, based on a priori intuition, it provides new knowledge but is not derived from experience.  In his writing he uses exclusively examples & proofs based on geometric & arithmetic operations.  Calculus is not part of his mathematics vocabulary or thinking even though it was already nearly a century old when he wrote the Critique of Pure Reason.

 

Baroque thinkers (& analytical maths):

The great mathematic giants of the 17th & 18th centuries (the Enlightenment, Spengler’s Baroque Age) were Leibnitz (1646-1716) & Newton (1642–1726) who independently developed analytic maths or calculus between 1684 and 1687.  Another notable leader in mathematic thinking in this age was Descartes (1596-1650)

 

mutatis mutandis:

Medieval Latin, meaning with things changed that should be changed or having changed what needs to be changed or once the necessary changes have been made

 

pre-Socratics & Plato: * see EndNote<B>

The first pre-Socratics philosopher was Thales (624-545 BC); these thinkers explained the natural world using logic not mythological or religious explanations.  They developed metaphysics, cosmology, ontology & mathematics on the basis of reasoned discourse.  The emergence of the Athenian school (Socrates, Plato & Aristotle) in the 5th Century BC signalled a new approach to philosophy as Ethics emerged as a critical element.  Socrates (470-399 BC) was the first to bring ethics & politics into philosophic discussion & is considered the first moral philosopher.  His most important contribution to thought was his dialectic method of inquiry (the Socratic method).  His ideas were formalized & documented by his student Plato (428-347 BC).  Plato also developed metaphysics with his theory of Forms. 

 

Descartes (Mathematics):

(1596-1650) prior to Descartes algebraic rules were given geometric proofs, Descartes reversed this, he developed Cartesian or analytic geometry & used algebra to describe geometry; he was first to assign a fundamental place for algebra in the system of knowledge, as a method to automate reasoning, particularly about abstract, unknown quantities.  He preceded Leibniz in envisioning a more general science of algebra or "universal mathematics," as a precursor to symbolic logic; he invented the convention of representing unknowns as x, y, and z, and knowns as a, b, and c; he pioneered the standard notation using superscripts to show the powers or exponents;

 

Leibniz:

see above  Baroque thinkers

 

Newton:

see above  Baroque thinkers

 

Gauss:

see Chapter II, pages 59, 88

 

Pythagoras (and mathematics):

(570-495 BC) Ionian Greek philosopher; founder of the Pythagorean religion, credited with many mathematical & scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the 5 regular solids, the Theory of Proportions, the Earth as a sphere & identity of the morning & evening star as Venus; he influenced Plato, whose dialogues, especially Timaeus, reflect Pythagorean teachings.  However according to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application.

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Plato (and mathematics): * see EndNote<C>

Plato (428-348 BC) was a major thinker in Greek mathematics, responsible for the 5 Platonic solids; he was also a great patron of mathematics as a discipline, through his Academy which inspired many later Greek mathematicians. 

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Archytas (and mathematics):

(428-347 BC) Greek philosopher, mathematician, astronomer, statesman, strategist; a scientist of the Pythagorean school, famous as founder of mathematical mechanics; one of his most notable accomplishments was the mathematical solution to The Delian Problem (aka doubling the cube); he also developed the Archytas curve which he used to determine the construction of a cube with a volume of one third of that of a given cube.

 

Archimedes:

see Chapter II, page 59, Chapter III, page 112

 

Kant (and Calculus):

see above- Kant (and mathematics)

 

the axiomatic of Leibniz:

Leibniz developed an approach to questions of modality: namely the principles of Possibility, Contingency: Necessity & Impossibility.  All were considered self-evident or axiomatic.  They served an important function within his general metaphysics, epistemology & philosophical theology

 

Fichte (unmathematical):

Fichte did not publish any complete article, chapter or monograph on the topic of mathematics & few scholars see any theory of mathematics in his works; his thoughts on mathematics are scattered among his many works & these expositions are in no way comparable with the technical sophistication of Kant or Descartes.  Like Kant, for him mathematics meant arithmetic & geometry, not Calculus.  His most detailed discussion is found in his Erlanger Logik lectures of 1805 in which he speaks of the logical function, syllogisms & higher logic.  He also makes a number of digressions which include remarks on aesthetic feeling & philosophical understanding regarding Raphael’s Sistine Madonna and a comparison between the Pythagorean theorem and philosophical insight. 

 

Hegel (unmathematical):

Hegel addresses calculus in his Science of Logic (1812-16) in which he points out the compulsion of mathematicians to neglect the infinitesimal differences that result from calculus equations in order to arrive at a coherent result.  This inexactitude he blames on the failure to distinguish between Quantum as the Quantity that each individual term of a differential co-efficient represents, and the Qualitative nature of their relationship when in the form of a ratio.  He denies mathematics the basic capability to treat the problems of mathematical continuity.  His judgment has not stood the test of time very well.  In 1902 Bertrand Russell stated:

 “Although their [ Hegel et al] arguments were fallacious, a special Providence saw to it that their conclusions were more or less true.  Hegel fastened upon the obscurities in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses.”

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the Romantics (and mathematics):

Romanticism rejected much of the 18th century Enlightenment & its rationalism & materialism as well as the Classical values of order, calm, harmony, balance, idealization & rationality.  Against these it promoted the individual, the subjective, the irrational, the imaginative, the personal, the spontaneous, the emotional, the visionary, the transcendental.  There was little or no room for Mathematics.

 

Zeno (unmathematical): * see EndNote<D>

(334-262 BC), Hellenistic philosopher of Phoenician origin from Citium in Cyprus, founder of the Stoic school of philosophy, which he taught in Athens from 300 BC.  In logic the Stoics are incompatible with Aristotle; in physics & epistemology they are incompatible with Plato.  Their philosophy was unlike the current thinking, its boundaries and methods could not be mapped onto Platonic or Aristotelian schemes.  Yet the Stoics have their own traditions as well.  Based on the moral ideas of the Cynics, Stoicism laid great emphasis on goodness and peace of mind gained from living a life of Virtue in accordance with Nature.  The Stoics did not ignore geometry or mathematics, but their focus was on Ethics, not Logic, Epistemology or Physics, a tendency which became more pronounced in the later centuries of Stoicism.  Although Zeno did follow the Old Academy when he divided philosophy into 3 parts (logic, physics & ethics) much of his thinking was unconventional, following Heraclitus rather than Aristotle or Plato. 

 

Epicurus (unmathematical): * see EndNote<E>

(341–270 BC) Greek philosopher & sage, born on Samos to Athenian parents; influenced by Democritus, Aristippus, Pyrrho & the Cynics, he turned against the dominant Platonism of his day.  In 307 BC he established the Epicurean School.  With regard to geometry & mathematics, in his Letter to Herodotus, he denies infinite divisibility based on his belief in atomism; infinite divisibility is a technical term for the mathematical axiom (namely, that any size can be bisected again and again ad infinitum), it is also a core principle of Euclidean geometry.  Epicurus was a materialist & his atomist belief struck both at mathematics & geometry.

 

Schopenhauer (unmathematical): * see EndNote<F>

Schopenhauer was a realist regarding mathematics.  In metaphysics, realism states that an object exists in reality independently of the person viewing it.  Said object is ontologically independent of someone's conceptual scheme or perceptions, linguistic practices or beliefs.  Mathematical realism holds that mathematical entities exist independently of the human mind.  Humans do not invent mathematics, but discover it & there is really one sort of mathematics that can be discovered. 

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Nietzsche (unmathematical): * see EndNote<G>

His explicit remarks on mathematical topics are not extensive.  His thoughts on mathematics are those of intuitionism; he attaches moral & artistic importance to mathematics; he grants the capacity of mathematical thought to dispel errors which sustain reactive religious and political superstitions and intuitions & suggests the use of mathematics in the construction of higher human cultures.  However he was not a logicist.

 

Gelegenheitsdenker:

German for casual thinker

 

“stone for bread”:

idiom meaning an exchange of no value;  stone has no value whereas bread is valuable

 

Schopenhauer (4th book): * see EndNote<H>

Schopenhauer developed his ideas early in his career (1814–1818); they culminated in the publication of the World as Will and Representation in 1819.  It was broken down into 4 books: epistemology, ontology, aesthetics and ethics, in that order.  In the Preface to the first edition, he states the book conveys " a single thought."  The structure is, "organic rather than chainlike," with all of the book's earlier parts presupposing the later parts "almost as much as the later ones presuppose the earlier."  Each of the 4 books functions as "four perspectives…on the one thought."

 

Kant (pure not applied):

Kant published 3 main works on ethics: Groundwork of the Metaphysic of Morals (1785), the Critique of Practical Reason (1788) & the Metaphysics of Morals (1797).  However his main contribution to philosophy was his Critique of Pure Reason (1781) in which he elaborates his doctrine of transcendental idealism: space and time are mere "forms of intuition" which structure all experience, and therefore that while "things-in-themselves" exist and contribute to experience, they are nonetheless distinct from the objects of experience.  It was this metaphysical doctrine which would dramatically impact subsequent German philosophy, much more so then his contribution to ethics via his “Categorical Imperative”.

 

Aristotle: (before and after):

Philosophy before Aristotle included the pre-Socratics, who focused on Cosmology, Physics, Logic & Metaphysics.  [see pre-Socratics & Plato above]  With Socrates the earlier focus declined while Ethics took new importance.  Aristotle (384-322 BC), a student of Plato, built a complex synthesis of the various philosophies existing prior to him. He established the Lyceum (the Peripatetic school of philosophy) in Athens & was a prolific writer.  Although a pupil of Plato, Aristotle modified Plato’s theory of universals: he placed the universal (the ideal Form) in particulars, things in the world.  For Aristotle, "form" is still what phenomena are based on, but is "instantiated" in a particular substance.  Much of his work survived; he wrote on physics, biology, zoology, metaphysics, logic, ethics, aesthetics, poetry, theatre, music, rhetoric, psychology, linguistics, economics, politics & government.  After him we have the emergence of the great “schools” of the Hellenistic age.  .  All chiefly focused on Ethics and achieving the good life.  Antisthenes (a pupil of Socrates) founded the school of Cynicism; Zeno of Citium (334-262 BC) established the Stoa (the Stoics).  Epicurus (341–270 BC) bought property in Athens for his school, the Epicureans.  These would become the dominant schools of the Roman Empire.

 

Nietzsche: (illogical): *see EndNote<I>

his dismissal of logic is clearly stated; in Human All Too Human (1878) he says: “Logic…rests on assumptions that clearly do not correspond to anything in the real world…”  In The Will to Power he states: “The will to logical truth can be carried through only after a fundamental falsification of all events as assumed….logic does not spring from will to truth” and later “Logic (like geometry & arithmetic) applies only to fictitious entities that we have created.”

 

Schopenhauer (as Pessimist):

In 1803-04 young Schopenhauer was engaged in studies, preparing for a career in business.  However his father, Heinrich, became increasingly dissatisfied with his results.  There were several instances of serious mental health issues on Heinrich’s side of family & at this time Heinrich began getting very fussy & prone to unsociable behaviour, anxiety & depression.  Even his wife started to doubt his mental health.  In 1805, when Schopenhauer was 17, Heinrich died by drowning in a canal by their home in Hamburg.   It was possible that his death was accidental, but his wife and son believed that it was suicide.  Arthur Schopenhauer would show similar moodiness from his youth onwards & acknowledged that he inherited it from his father.