"1+1=2"? But formalism did not extend to "+" in 379 pages of proof!
Eurocentric narrations locate all math in ancient Greece, despite Greek notions of perception & inference driving "provable" understanding of reality is a clear echo of Vaisesika.
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Eurocentric narrations locate all math in ancient Greece, despite Greek notions of perception & inference driving "provable" understanding of reality is a clear echo of Vaisesika.
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youtube.com
"Formalism" arose in European attempts to seek axiomatic approach to building proofs. Set theory was proposed as building blocks of logic and formalism. Russell paradox showed the existence of logic that could not be fit in this theory.
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In 1910, Russell set out to formalize math in Principia Mathematica (or usurp 4000+ years of math that preceded Europeans), but was unable to complete proof of "1+1=2", cut short by Godel's Incompleteness Theorem, with the famous "This statement cannot be proved"
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Showing that there are statements that are neither true nor false showed the futility of such logical formalism, making Russell give up on his volume 2.
Should math be rooted in axiomatic formalism or should real-world application drive math?
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Should math be rooted in axiomatic formalism or should real-world application drive math?
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@CKRaju14 has written & spoken extensively on this issue in several works. As somebody who waded thru 3 semesters of "Real Analysis" courses, & still don't have a clue, this work is very interesting:
@c_k_raju/the-church-origins-of-axiomatic-math-e08036dbe29d" target="_blank" rel="noopener" onclick="event.stopPropagation()">medium.com
ckraju.net
link.springer.com
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@c_k_raju/the-church-origins-of-axiomatic-math-e08036dbe29d" target="_blank" rel="noopener" onclick="event.stopPropagation()">medium.com
ckraju.net
link.springer.com
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In the late '80s, at @iiscbangalore, I had a friend researching "automated theorem proving", & spent hours of discussion, trying to make sense of the topic, field, & methods. He tried to discuss Godel & Russell with me - but alas, in vain! The subject was too abstruse for me
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In the early '90s, during my EE Ph.D, I took courses on Real & Fn analysis for a minor in Math, along with Algorithms courses for a minor in CS, grappling with NP-Hard problems on one side, & the incomprehensiveness of Cantor sets & Lebesgue Integrals on the other.
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At @iiscbangalore & @OregonState, working with dynamics of large interconnected nonlinear systems & solving 1000s of differential equations numerically, & working with computer control revealed the complexities of the real world & the non-idealities of implementation.
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While working on formal verification in IC chips in the late 90s & early 2000s, suddenly the world made sense & these academic subjects came alive as I grappled with Boolean satisfiability in a horrendously combinatorial world, & the need to give a guarantee of correctness.
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Over years I researched "P=NP?" in vain, discovering novel algorithms of value in the process. Returning to the Real world (i.e., R^n) in the mid-2000s, melding of complexity, functional analysis, nonlinearities, non-idealities, philosophy & history paved understanding.
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Applied math is what makes the world go around. Abstract math can find new applications - such as Ramanujan's works today in math of Black Holes. Formalism needs to be revisited for the foundational notions infused by its inventors. Math pedagogy needs much revision.
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