C. K. Raju

Indian Institute of Education

G. D. Parikh Centre, J. P. Naik Bhavan

University of Mumbai, Kalina Campus

Santacruz (E), Mumbai 400 098

- Colonialism, like racism, boasted of superiority.
- Macaulay's 1835 "Minute on Education" claimed that as regards math and science
- “the superiority of the Europeans becomes absolutely immeasurable”.

- This boast was used to trick the colonised into accepting colonial education
- for “superior” Western math and science.

- But European education until 1870 Elementary education act was 100% CHURCH education
- from primary school to univs like Oxford, Cambridge, and Paris, e.g. Gregory-9 edicts of 1231.

- Church education used for childhood INDOCTRINATION of the colonised
- e.g. “Pythagorean theorem” term repeated 32 times in current Indian class X text

- But no one able to produce evidence for Pythagoras or his connection to the proposition
- Claim based on utterly FALSE history of math (and science)

- Adds no EPISTEMIC value (e.g. my Durban keynote in
*AlterNation*) - deduction MORE fallible than induction, empirical proof.
- SUBTRACTS practical value by making math difficult
- e.g. Bertrand Russell's silly 378 page proof of 1+1=2 useless in a grocery shop

- the false history/myths of math
- (e.g. West “discovered” calculus”)
- AND the bad philosophy/superstitions of math (axiomatic proofs, axiomatic reals “superior”, like White skin)

- 1. Calculus began in India in the 5th c.; was stolen in the 16th c.
- Theft proved beyond reasonable doubt: opportunity, motivation, circumstantial evidence
- Term “stolen” used to emphasize that stolen knowledge is usually poorly understood,
- e.g. Newton’s pitiable flux ions with no understanding of infinitesimals.

- 2. Eventual Western solution to its poor grasp of calculus,
used metaphysical or unreal “real” numbers,
- globalized by the power of colonialism.
- (Decolonisaton asserts this is an INFERIOR solution.)

- 3. Teaching calculus as it first originated
- using Āryabhaṭa’s (5th c.) numerical finite difference methods +
- Brahmagupta’s (7th c.) “non-Archimedean” arithmetic of polynomial (with infinitesimals +
- a secular philosophy of zeroism

- makes calculus very easy
- enables students to solve harder practical problems using calculus
- as demonstrated by pedagogical experiments in last 15 years in 5 univs in 3 countries.

- Accepting this involves great loss of face, power, and money for West (like end of slavery, apartheid)
- and is also contrary to Western religious beliefs linked to math since Plato and Crusading church.
- But if West fails to switch from formal to normal math
- it will fall behind in technology development.

- "To decolonize math stand up to its false history and bad philosophy"
- To reiterate: BOTH false history
- and BAD philosophy of math used to claim supremacy, BUT
- Related article published in
*Conversation*, went viral, THEN censored - since no valid intellectual response to my arguments.

- See Was Euclid a black woman?,
- "Black thoughts matter: Decolonised math, academic censorship…",
*J. Black Studies***48**, no. 3 (2017): 256–78. - Mathematics and censorship,
- Article also in Rhodes must fall-Oxford, chp. 26.

- Stephen Hawking’s co-author G.F.R. Ellis started the conspiracy theory polemic against me in Cape Town
- as also the Bantuization polemic suggesting that my decolonised calculus course involves a dumbing down
- when it actually enables students to solve harder problems not included in usual calculus courses.

- Similar deliberate misrepresentation by John Armstrong of King’s College London
- claiming my decolonized course teaches only software!
- The West has not understood it is fast losing all credibility
- and such “joker responses” only accelerate the loss of credibility.

- My personal interest: to check if the West is at all capable of an intellectual response
- based on facts and arguments.

- I was a formal mathematician, teaching/doing real/functional analysis.
- Was invited by ANU math department
- and lectured on a key DEFECT of CALCULUS as taught today.
- Problem: can't differentiate discontinuous functions such as \[\theta (x) = \begin{cases} 0 & \text {if}\ x < 0 \\ 1 &\text {if}\ x > 0 \end{cases}.\]

- Why do we need to do so?
- E.g. to make sense of the differential equations of physics at a physical discontinuity
- such as a shock wave or blast wave in a fluid.

- BUT, even if one defines the derivative (at a discontinuity), e.g., as a Schwartz distribution
- \(\theta ' = \delta\)
- one is UNABLE to multiply Schwartz distributions (e.g., \(\delta \cdot \delta\))
- (required since differential equations of physics are nonlinear).

- My 1982 product used (Robinson's) non-standard analysis
- a key feature of which is “non-Archimedean” arithmetic
- which has infinities and infinitesimals (hence, no limits),
- very different from (“Archimedean”) real numbers.

- But these infinites and infinitesimals disappear in the final result
- as in my product applied to shocks in fluids, discontinuities in relativity
- and shocks and singularities in general relativity
- (Won't go into technical details. See appendix to my CFM book.)

- My points: (1) Calculus originated in India with 5th c. Āryabhaṭa
- over 1150 years before Newton,
- as a numerical technique (“Euler’s” method of solving ODEs) using finite differences.

- (2) calculus (and infinite series) best understood with “non-Archimedes” arithmetic (+ zeroism)
- which has infinities and infinitesimals (hence no limits)
- (NOT formal real numbers and limits + non-standard analysis)

- this Indian technique since 7th c. Brahmagupta's “non-Archimedean” arithmetic of polynomials.

- Brahmagupta's (7th c.) अव्यक्त गणित = “unexpressed arithmetic” (Brāhmasphuṭasiddhānta vol. 4)
- = arithmetic of polynomials = “unexpressed numbers”
- “unexpressed” since \(2x+3\) acquires a value only when a value is assigned to \(x\).
- Later badly translated as “algebra” (al jabr waal Muqabala) by al Khwarizmi.

- Arithmetic of polynomial fractions or (“rational functions”)
- very similar to arithmetic of ordinary fractions (rational numbers).
- In terms of formal algebra both rational numbers and rational functions constitute an
**ordered field** - EXCEPT that ordering among polynomial fractions is “non-Archimedean”.

- As a
**number**\(x-n\) positive or negative depending on value of \(x\) - but as a
**polynomial**\(x-n>0\) for every \(n\) - since
**number**\(x-n\) positive for sufficiently large \(x\) for any \(n\)(e.g. Moise (1963)^{1}) - hence \(x > n\) for every \(n\), i.e., \(x\) is infinite.

- Formally, since rational functions are a field, we also have
- \(0 < \frac{1}{x} < 1/n\) for every \(n\), so \(\frac{1}{x}\) is infinitesimal.
- No limits possible with infinities and infinitesimals in Brahmagupta’s “non-Archimedean”) arithmetic

- Avyakt gaṇita is NOT non-standard analysis. But works similarly:
- infinitesimals disappear from the final results.
- E.g. sum of infinite geometric series (Nīlakanṭh, 15th-16th c.)
- infinitesimals drop out using a philosophy of inexactitude called zeroism (inexactitude earlier called śūnyavāda).

- So, we already have some fundamental differences between original Indian calculus and current Western calculus:
- (1) Brahmagupta's (“non-Archimedean”) arithmetic vs (“Archimedean”) real numbers
- (2) Zeroism (exactitude impossible in real world) vs Western belief in (metaphysical) exactitude in math,
- West believes in exactitude since it always connected math to religious belief

- In the 16th c. Jesuits in Kochi systematically translated many Indian texts
- on the Toledo model and sent them to Europe.
- Math+astronomy texts went to Clavius-Tycho Brahe-Kepler-Galileo-Cavlieri-Fermat-Pascal-Gregory-Newton-Leibniz
- So the story that “Newton discovered calculus” is COMPLETELY BOGUS history.

- Newton called "discoverer" of calculus
- on the genocidal church dogma of “Christian discovery”.
- in which Newton believed, hence he heavily invested in evil transcontinental slave trade
- “morally justified” by that dogma.

- According to the dogma the first Christian to spot a piece of land
- or knowledge, is declared its “discoverer” hence OWNER
- regardless of prior inhabitation (of land) or prior knowledge. (US Supreme Court judgment.)

- So, Columbus “discovered” America, (regardless of millions of prior inhabitants who could hence be “morally” murdered)
- Vasco “discovered” India, Cook “discovered” Australia,
- Newton “discovered” calculus, regardless of its prior invention by other non-Christians.
- This was brazen theft, Newton was a calculus thief.

**Epistemic test**: Knowledge thieves fail to FULLY understand the knowledge they steal,- like students who cheat in an exam fail to FULLY grasp what they copy
- and can't explain their answer-sheet (They grasp some part.)

- Because calculus was stolen knowledge from India, Europeans failed to FULLY understand it.
- Newton could not explain the infinitesimals he used for his fluxions.
- Europeans did not understand how to sum Indian infinite series for centuries after Newton.

- widely ADMITTED till 19th c.
- e.g. by bishop Berkeley (1734)
- “why not set infinitesimals to zero at the start of a calculation not at the end”

- Karl Marx (ca. 1860),“Newton’s calculus mystical”,

- Dedekind (1871) “calculus lacks rigor”(when he invented Dedekind reals).
- But today Dedekind reals are the socially accepted solution.

- Many superstition e.g. racism too socially accepted in the West for centuries,
- that no reason why WE should accept it.
- My point Dedekind reals a BAD solution
- add NIL practical value
- epistemc value only on Western superstitions.

- On my second visit to Australia and New Zealand, in 2005,
- I already spoke on the Indian origin of calculus at UNSW and at Auckland (1,2),
- but was also invited by Sydney University (and Melbourne U.)to talk on my theory of time.

- On the morning I left, I was attacked by a racist dog,
- in the park in front of Sydney University.
- Took all my knowledge of dog psychology to fend off the dog without even a stone,

- till its master leisurely sauntered up and said
- "he is a good dog, it is just your clothes".
- Both racist master and dog apparently believed that Western clothes are universal:
- the norm is everyone OUGHT to wear them.

- This mandatory “ought to” is what I call “normative universality”
- in my decolonized curriculum on HPS, part 2
- Normative universality is double speak for a racist sense of SUPREMACY
- “you OUGHT to do calculus with reals and limits, as we do”.
- even if our understanding is INFERIOR (like De Morgan’s 19th c. understanding of elementary arithmetic?)

- A couple of personal examples
- on how false history continues to be concocted
- and preserved.
- Show why West will never willingly accept decolonisationm agenda.

- On day of the racist dog, Sir Michael Aliyah, Fields Medalist + Abel Laureate + President of Royal Society,
- gave his 2005 Einstein Centenary lecture to the AMS
- in this lecture he plagiarised the thesis in my 1994 Time book

- that correcting Einstein's mistake
- and using functional differential equations in electrodynamics and relativity
- could explain quantum mechanics
- But did NOT cite my book published a decade earlier,
- instead added "Don't forget that I suggested this".

- it is unethical to claim ignorance of past work even ONCE:
- especially a Springer book ten years in print.
- Yet Atiyah was immediately informed and acknowledged the emails.

- But he AGAIN repeated his claim a 2nd time
- "Don't forget that I suggested this"
- in an article he got published in the Notices of the AMS 53(6) 674-78 (2006)
- prominently reporting Atiyah's “Einstein lecture”.

- Eventually Atiyah was forced to admit my priority,
*Notices of the AMS*54(4) (2007) p. 472. - Atiyah was held prima facie guilty by an ethics committee (case no. 2 of 2007)
- But no apology, BRAZEN pretense of accidental oversight was maintained by
*Notices*which REFUSED to publish my letter pointing out that this plagiarism happened twice- 2nd time after Aliyah was personally informed!

- Clear case of brazen dishonesty at the top
- supported by key math institutions.

- Cultural Foundations of Math
- on Indian origin of calculus and how Europeans (Newton, Leibniz etc.) stole it

- was itself repeatedly stolen from India (from me)
- by the serial plagiarists George Joseph, Dennis Almeda and others.
- 2007 press release from Manchester 2 months after my book on calculus was published
- claiming Joseph and Almeida authored a paper on calculus transmission

- The paper was a near-verbatim copy of my 1999 paper
- submitted to Joseph for a conference he organized in 2000 in Trivandrum.

- Again it was done a third time after two very irregular ethics committees
- one in Exeter had warned one of them in 2003.
- This is clear from the retraction of the 2007 front page news by Hindustan Times.

- Manchester University still keeps on its website the fake news release which made headlines
- though the paper it mentions was NEVER published in 17 years since 2007.
- It could not be published because it VERBATIM plagiarizes most of my paper
- published in the proceedings of a conference organized by Joseph in Trivandrum Jan 2000
- without crediting me.

- Please write to Manchester and ask;
- where is that paper mentioned in your 17 year old fake news?
- Why do you brazenly preserve the fake news on your website
- despite an ethics committee?

- with Stephen Hawking’s co-author G. F. R. Ellis
- responding with an abusive racist mob to my decolonization proposal.

- for West has a long record of systematically dishonest history as a WEAPON to assert supremacy
- such as Western supremacy during colonialism
- (this diagram from my Pretoria-Tübingen keynote "Euclid must fall, part 1")

- Actual history: West was PERSISTENTLY inferior even in elementary arithmetic
- from early Greek times to 19th c.

- Early Greeks/Romans knew calculus
- calculus=pebble=psephos: primitive pebble arithmetic.
- Which early Greeks learnt from their Persian conquerors
- to pay tax to them (tax collector scene from Darius vase −300 CE)

- Hence Greek (and Roman) names for SMALL numbers ape Persian and Indian (Sanskrit) names.
- But Graeco-Roman counting INFERIOR;
- their number names stopped at the puny myriad (Persian beavan) \(10^5\)
- still means uncountable, and connotes it as in "myriad stars" and other literary quotes

- Indian names went on to
*parārdha*(=10^{12}) Yajurveda 17.2, *tallakṣhaṇa*(=10^{53}) and beyond (10^{108}, Buddha, Lalita vistara sutta, chp.12)- because Indians used superior place value.

- Contrary to triple miracle of Archimedes + 1200 year Dark Age + verbatim reappearance fantasy.
- Fact 1: From Athenian inscriptions thru Darius vase to Herodian, Greeks used only Attic numerals with no fractions.
- Fact 2: Graeco-Roman arithmetic with no general fractions CONTINUED till 10th c.

- Took smartest of Europeans over 1000 years to understand simple fact
- that Graeco-Roman pebble arithmetic is primitive/INEFFICIENT
- compared to sophisticated place-value of Indian gaṇita.

- Small number 1888 needs 13 figures MDCCLXXVIII not 4
- Large numbers need place value.

- Then Gerbert made a great innovation of apices
- learning the system of "nine Indian figures" from Spain
- Wrote them on the back of counters/jetons
- i.e., using a counter with 9 at the back instead of using 9 counters!

- Gerbert made a monstrous abacus with 27 columns to represent large numbers
- with a 1000 apices
- Gerbert understood large numbers need place value, BUT
- FAILED to understand efficiency of Indian arithmetic for arithmetic operations
- DESTROYED by his abacus.

- E.g. \(89 + 89\) needs 18 operations on Graeco-Roman abacus but only 3 on gaṇita
- \(89 \times 89\) needs 1602 operations on abacus instead of 8 on place-value gaṇita

- Took another 2 centuries for Christian Europeans to grasp the efficiency of Indian gaṇita.
- Then Florentine merchant Fibonacci wrote
*Liber Abaci*(13th c.). - based on al Khwarizmi's 9th c.
*Hisab al Hind*. - Both (Fibonacci and al Khwarizmi) gave a simplified treatment of Indian gaṇita

- As found in common Indian texts emphasizing commercial uses of gaṇita (instead of astronomy)
- Compare contents of 9th c. Mahavira's गणित सार संग्रह (Gaṇita sāra sangraha),
- with Fibonacci's contents (square/cube roots etc. only in chp. 14)
- Note Fibonacci's chp. 4 on subtraction (only LESSER numbers)

- That is, still mired in the paradigm of pebble arithmetic like Gerbert
- Fibonacci blundered: has no negative numbers (like al Khwarizmi)
- because in primitive Graeco-Roman calculus (pebble arithmetic)
- subtraction means removing pebbles from an existing hoard
- and you can't remove more pebbles than there are on the board!

- OF COURSE Fibonacci numbers stolen from India, he made NO original contribution.
- Fibonacci's influence largely limited to Florence and neighbouring states
- since zero (= sifr=cipher) puzzled Europeans] (and Florence passed a law against zero)
- Finally, in 16th c. Gregor Reisz declared victory of algorismus over abacus

- followed by Adam Riese.
- Most importantly Jesuit general Christoph Clavius again imported gaṇita direct from India
- wrote a text on practical mathematics,
- introduced it in the Jesuit syllabus ca. 1575.

- Conclusion: Western math was PERSISTENTLY INFERIOR,
- even in the matter of primary-school arithmetic
- from early Greek times till end of 19th c.
- contrary to colonial tale of Western supremacy.

- On the principle that “phylogeny is ontogeny”
- these Western difficulties with elementary math are reproduced even in the primary school classroom today.
- Hence, aping the West makes math difficult even in the matter of primary school arithmetic.
- So the decolonial solution is to revert to the original methods of teaching ganita.

- E.g. Assume child has learnt representation of numbers by objects at home.
- START teaching with place value, teach large numbers early.
- Teach practical value of arithmetic as in Mahavira etc.
- not ass. law com. law etc.

- Calculus originated with 5th c. Āryabhaṭa
- who obtained sine values (trans.) and value of π
- accurate to the the first sexagesimal minute
**by solving finite difference equations.**

- Āryabhaṭa used a simple recursive numerical technique falsely called “Euler method” today.
- Euler knew of it since he studied Indian math texts (wrote an article on Indian calendar in 1738)
- hence magically solved Fermat's 1657 challenge problem
- (solve \(Nx²+1=y²\) for \(N=61\)) a century after it was posed

- That problem a solved exercise in Bhaskara II (Bījagaṇita 87, trans. Colebrooke)
- Numbers in solution (\(x\) =

226153980, \(y\) = 1766319049) too large for miracle of “independent rediscovery”

- but Western Christian-chauvinist history is full of such miracles.

- Tycho Brahe's astronomical model a carbon copy of the model of Nīlakanṭha (Āryabhaṭa's commentator)
- Clavius' 1607 sine table, an interpolated version of Madhava's sine table
- will skip “pre-calculus” of Cavalieri, Fermat, Pascal

- “Newton's” sine series carbon copy of Madhava's infinite sine series
- like “Leibniz” series for π
- “Stirling's” formula copies Brahmagupta’s (7th c.) quadratic interpolation formula
- he used to revert to traditional table of 6 sine values 15° apart.

- Vaṭeshwara (904 CE) used PAST (backward) and FUTURE (forward) differences to get
- Newton-Gauss formula and sine values precise to seconds.
- Western history of math chock full of miracles.
- But such a long series of improbable “coincidences” or “independent rediscovery” has probability zero.

- I see this as impeccable circumstantial evidence that
- Indian math/calculus texts imported direct from India (not via Arabs)
- were circulating in Europe since Clavius and Tycho).

- Note: Like calculus, West also stole probability, and sampling theory
- along with theory of permutations and combinations from India.

– See my article “Probability in Ancient India”, *Handbook on Philosophy of Statistics*
(2011) 1175–96. Elsevier.

- My 1999/2001 Hawai’i paper gave proof beyond reasonable doubt on criminal law (see extract)
- there was opportunity, motivation, circumstantial, and also documentary evidence

- Roman Catholic missionaries in Kochi (Kerala) since 1500
- set up a mission school + college to try to convert local Syrian Christians
- Using their help to locate and translate local texts in Toledo mode
- and sending the translations back to Rome.

- Europeans had a major navigational problem
- e.g. British law of 1711 constituting board of longitude.
- Solution of longitude problem needed precise trigonometric values (hence Clavius' 1607 trigonometric tables).

**Epistemic test**: BUT, though Europeans (Clavius etc) stole from India and lied they were not smart enough- to USE these precise trigo values to calculate the radius of the earth!
- A value found in every Indian math text.(Next talk on rajju gaṇita).

- Hence, Europeans could not use Indian techniques of celestial navigation for longitude
- Brahmagupta: "भूव्यासस्य अज्ञानाद् व्यर्थं देशान्तरं" (BSS, 11,15-16)
- "ignorance of earth's radius makes longitude [calculations] futile"

- There is also documentary evidence, though signed confession not expected from thieves.
- Matteo Ricci, Clavius pet student and biographer
- wrote from Kochi in 1580 that he was collecting knowledge of Indian methods of time keeping
- from “an intelligent Brahmin or an honest Moor”

- by opportunity, motivation, circumstantial and even documentary evidence
- that calculus was stolen from India
- and dishonestly attributed by Western historians to Newton etc.

- Clear case of ACTIVE dishonesty because Whish article of 1832 suppressed for two centuries
- like my assertion of calculus theft suppressed for 25 years
- and my courses misrepresented and maligned.
- (But West has lost its iron grip on information in last 20 years.)

- But what difference does that theft of calculus make?
- First a question:
- Q. When Europeans made such persistent asses of themselves with elementary arithmetic (zero, subtraction, negative numbers etc.)
- how could they have understood calculus?

- A. They didn't; as clarified at the beginning of this talk.
- Europeans acknowledged difficulties in understanding calculus until Dedekind and axiomatic reals.

- But Dedekind reals only reflect Western social consensus.
- Many issues on which the West had a social consensus for centuries
- such as “moral” genocide,“moral” slavery, “moral” racism, and “moral” colonialism.
- We won't accept such social consensus which we regard as the height of evil
- in fact, aim of decolonisation is to kick out such remnants of colonialism.

- We, the former colonised, will accept it only if it has practical value:
- but formal reals have none.
- When I helped to build the first Indian supercomputer
- one of my tasks was to implement calculation related to rocket trajectories etc. on it.

- I realized that Dedekind real numbers have no practical value
- computers use floating point arithmetic
- For which even the associative law for addition fails.
- as explained in my Hawai'i article (1999/2001) and this video of the related computer program.

- based on Āryabhaṭa's method of numerical solution of differential equations
- Can't teach calculus just with theory of infinite series
- most applications involve solutions of differential equations.

- Āryabhaṭa used linear interpolation/extrapolation (elementary “rule of 3”)
- Later 11th/12th order interpolation/extrapolation used by Āryabhaṭa school of Kerala.

- Central University of Tibetan Studies (2009)
- Universiti Sains Malaysia (4 groups, 2010)
- CISSC Tehran (2012)
- Ambedkar Univ. Delhi, 2012 (social science post grads)
- SGT Univ Delhi (2017) (Science/Engineering undergrads)

- Claim that calculus was “discovered” by Newton etc. is bogus.
- Europeans didn't understand Indian calculus. just as they failed to understand Indian primary-school arithmetic for centuries.
- That in 19th-20th c. they hence invented the metaphysics of unreal real numbers
- which should be rejected since it adds difficulty but no practical value.

- That axiomatic math, like racism, is based on religious (church) superstitions
- which allow West to control content of math
- by controlling axioms and deciding validity of theorems.
- That limits and real numbers are not needed.

- Calculus about numerically solving differential equations,
- not about formulae for integration and differentiation of elementary function.
- Non-elementary functions arise in the simplest applications such as the simple pendulum.

- Derivative as limit not needed finite differences enough for solution.
- Integral not needed, solution of differential equations enough.
- Seroism and Brahmagupta arithmetic better substitute (even for discontinuities)

- Sample tutorial sheet
- Sample results (UG-pure-math, UG-appl-math, UG-non-math, PG-math)
- Sample software my Calcode (demo)
- Decolonization reading list: https://tinyurl.com/decol-list-new

- Normal math(e.g., Indian gaṇita) accepts empirical proofs + deductive inference
- as does science
- as explicitly stated in Nyaya sutra 2 and elaboration

- So proofs in both formal and normal math use deductive inference
- But formal or axiomatic proof prohibits the empirical in a proof
- (e.g. Mendelson, introduction to mathematical logic, page 34)

- i.e., A mathematical 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

- (e.g.,modus ponens 1. \(A \Rightarrow B\), 2. \(A\), ∴ 3. \(B\).)

- it is about
**prohibiting empirical**in proof suited church (since facts fatal to church dogmas)

- This method of proof was read into the Euclid book (which arrived during the Crusades)

- though “Euclid” book has not a single axiomatic proof
- (if you think it has, accept my prize, show me one, don't tell me stories why it is missing).

- Apart from
**myths**about axiomatic proofs - there is also the
**superstition**that prohibiting the empirical - makes proof infallible.

- Church superstition that prohibition of empirical makes proof infallible.
- if true, prohibit empirical in science too to make it infallible?
- Fallacy: As math teachers do you give 100% marks to all proofs given by your students?
- No! You discriminate between valid and invalid deductive proof.

- Is Russell’s 378 page proof of 1+1=2 valid?
- Not even a typo in those 378 pages? Did you check? Or go by trust?
- During my PhD, I had a 110 page derivation.
- Re-derived repeatedly. Each check took a month.

- So, (a) deductive proofs ARE fallible.
- (b) Validity of a deductive proof can only be decided inductively by repeat checks
- or by authority.
- Therefore deductive proofs MORE fallible than inductive proof or proof by authority.
- Fallacy from Hume a racist who copied al Ghazālī without understanding.

- As in Russell’s silly proof of 1+1=2
**but no practical value**- which comes from calculation
- (or computational math in which the associative law for addition fails for floats)

- silly church SUPERSTITION of Crusading theology of reason
- that 2-valued is universal,
- since it binds even God.
- “God cannot create an illogical world but can create the facts of his choice"

- Indian Nyāya school accepted 2-valued logic
- (from which Aristotelian syllogism derives)
- but Buddhist logic of catuṣkoṭi,
- Jain logic of syādvāda,

- and quantum logic (e.g. AND not distributive over OR)
- are all quasi-truth-functional