• This philosophy is demonstrated by current practice
• in the way math is taught in schools and universities in Kashmir today.

• Also discussed math and math education with several people privately.
• One of them was a senior mathematician from this university.
• I asked him his opinion about formal/Western mathematics.
• He said "West has given us heaven in math".
• When I asked him about set theory, he said "there are blemishes even in heaven".

• as taught, practiced, and celebrated in Kashmir today.
• Western philosophy of math globalised (not specific to Kashmir)
• but is global = universal?

• A. No!
• Traditional Indian philosophy of गणित is different.
• What is the difference?

• gives ONE Indian notion of proof: (Nyaya sutra 2)
• based on प्रत्यक्ष, अनुमान, उपमान, and शब्द.
• Proof not specific to math. Same notion of proof applied to ALL knowledge (math, science, religion…)
• Examples of use in mathematics

• Different schools of thought disagreed about what constituted acceptable proof
• Buddhists accepted only प्रत्यक्ष and अनुमान.
• Cārvāka/Lokāyata ("people's philosophers") accepted only प्रत्यक्ष.

• All traditional schools of thought in India accepted प्रत्यक्ष.
• Acceptance of प्रत्यक्ष did NOT exclude use of reason/deduction अनुमान (except for Cārvāka)
• any more than acceptance of experimental method (empirical proof) in science excludes reasoning.

• acceptance of reason, but exclusion of the empirical.

• A proof is a sequence of statements in which each statement is
• either an axiom, or
• is derived from preceding statements by some rule of reasoning such as Modus Ponens.
• (MP: \(A,~A \implies B, ~\therefore B\).)

• stated also in the class IX NCERT text on math (which everyone in any Univ. expected to know)
• Empirical is prohibited in formal math: "beware of what you see" (class IX NCERT, page 301).
• i.e, proof (or reasoning) in formal math prohibits the प्रत्यक्ष. (See, e.g., my Hawai'i paper, 2001 in Philosophy East & West)

• So, if one distrusts the empirical, what should one trust?
• Clarification: axiom = postulate = assumption \(\neq\) universal truth

• Proof in traditional Indian gaṇita accepts प्रत्यक्ष but also reasoning/deduction अनुमान
• Proof in formal math accepts reasoning but prohibits the empirical (प्रत्यक्ष)
• Q. Which is better?

• (This question never debated by Indian philosophers. So does blindly imitating the West lead to heaven?)

• West boasts formal math "superior", "rigorous".
• If math used for science, which accepts empirical,
• then nothing gained by prohibiting empirical in math.

an empirical proposition cannot possess the qualities of necessity and absolute universality, which, nevertheless, are the characteristics of all geometrical propositions

• If prohibiting empirical makes math "superior"
• then prohibit empirical (experimental method)also in science, to get "superior" science!😄

• Brazen falsehood: अनुमान used in India gaṇita to deduce earth is round (gola), prove "Pythagorean" theorem, in Yuktibhāṣā.
• "use of reason" = prohibition of empirical?
• Not in "Euclid" whose very first proposition uses empirical as does 4th (SAS).
• Not a single axiomatic proof in "Euclid's" Elements, despite centuries of Western mythical claims.

• Reason for this confusion in Western thought: - no "Greek" definition of (mathematical) proof as in Nyaya sutra 2
• Proclus says: "proofs vary with kinds of being" (re. Plato)
• Hence, "Euclid's" Elements actually a "Neoplatonic" (= Sūfī) religious text.

• In Meno, Plato's Socrates speaks of the soul,
• and connects it his doctrine "all learning is recollection" [of the eternal knowledge in the soul]
• Proclus derives "mathematics" from mathesis.
• Hence empirical deprecated (NOT prohibited) since shutting out the empirical
• as in yogic mūrchā arouses soul by driving mind inward].

• Well known, the church banned this kind of (Neoplatonic) math
• because the related "pagan" notion of soul conflicted with the notion of soul in post-Nicene Christianity
• Hence, Justinian shut down all schools of philosophy and mathematics in the Roman Empire in 532 CE.

• Also well known, "Neoplatonism" came into Islam as the "theology of Aristotle"
• which West believes to have been written by Plotinus, Proclus etc.
• probably derived from Egypt.
• And is strikingly similar to Sufīsm.

• Little-known that the Crusades were an attempt to convert Muslims to Christianity by force,
• the way Europeans were earlier converted to Christianity by force.
• But force failed with Muslims then.
• All the nations of Europe combined under a religious banner were militarily too weak.

• Therefore, church tried TRICKS to persuade Muslims.
• Aquinas (and schoolmen) erected Christian rational theology,
• to contest Islamic rational theology (aql-ī-kalām)
• (not explained in any book on Western philosophy).

• The church accepted reason
• to persuade Muslims who accepted reason (but thought the Bible was corrupted)
• BUT the church could not also accept the empirical
• Briefly, reason + empirical facts = science contra church.

• Realization that only EMPIRICAL against church dogma
• not reason by itself
• Therefore, the church trick was to accept reason but prohibit the empirical

• i.e., church accepted reason MINUS facts as in formal math today.
• Aquinas gave the first axiomatic proof by proving his angel theorem (many angels can fit on a pin)
• from the axiom/postulate that angels occupy no space (Summa Theologica, First Part, Q. 52, article 3)
• POSTULATE needed, since no empirical facts about angels.

• To consolidate this trick, the church said this sort of reasoning (MINUS empirical) was invented by Euclid.
• This false claim (axiomatic proofs in "Euclid" book) was believed in the West for SEVEN centuries, since Crusades
• until it was exposed as FALSE over a century ago by Bertrand Russell and David Hilbert
• Nevertheless, these two foolishly accepted the MYTH about the "superiority" of (church method of) axiomatic proof

• Because of this myth, formal mathematics is based on the same belief:
• that reasoning MINUS empirical is "rigorous", and "superior".
• Actually, formal mathematical theorems are at best "relative truths".
• Relative to (metaphysical) axioms (fantasies of infinity/eternity).

• Axiom ALL laid down by Westerners, must be approved by them.
• E.g. Not ALLOWED to teach calculus EVEN with a
• different set of axioms (based on non-Archimedean arithmetic of Brahmagupta)
• which makes calculus easy.

• Because divorce from the empirical turns formal math into metaphysics,
• mathematical theorems are only "metaphysical relative truths"
• There is no way to verify metaphysics except by reliance on authority (शब्द प्रमाण)
• And the primary teaching of colonial education is to blindly trust Western authority

• So, while the Crusades were military failures
• Crusades + colonialism brought Muslims under the mental control of the West through math
• regarded as essential for science.

• Prohibiting the empirical makes math excessively difficult
• 1+1 = 2 cannot any longer be proved in the kindergarten way since empirical is prohibited.
• Bertrand Russell needed 378 pages to axiomatically prove 1+1 = 2
• And few would understand what is there on that page 378.

• Worse, because numbers no longer directly correlate with the empirical
• there is no unique notion of the number 1.
• The number 1 as a cardinal or natural number *differs& from the number 1 as a rational number or the number 1 as a "real" number.
• E.g. Peano's axioms apply to natural numbers, but NOT to real numbers.

• Hence, my "Cape Town challenge" to prove 1+1 = 2 in real numbers
• from first principles directly from the axioms of set theory,
• and without assuming any theorem of axiomatic set theory.
• This challenge offered in JNU with a prize of ₹10 lakhs]. No one claimed the prize.

• Lack of understanding of elementary math (1+1=2) forces reliance on Western authority
• That reliance on West can be used to trick people in many ways.
• By pushing Western metaphysics through formal math into science.

• Real numbers, or the continuum, contrary to al Ashari's ideas of atomicity and discreteness
• Real numbers "needed" for differential equations of physics makes science anti-Islam
• As does the belief in "laws of nature" contrary to al Ghazali.
• Metaphysics of real numbers forces time in physics to be superlinear, contrary to the belief in quasi-cyclicity needed for Sufi ethics.
• Cannot explain in 10 minutes church tricks which people failed to understand in 10 centuries

• To go back to the point where I started, about my meeting with a Kashmiri Prof. of math
• If "the West has given us heaven in math" (by controlling the mind!)
• Then that is the Christian heaven as described by Dante in Divine Comedy
• Where Paigambar Mohammad is horribly tortured eternally by the Christian God. (Won't show linked image; blame Dante and Wikipedia for blasphemy!)