Euclid Book 1 Props VI-VIII - a foundation for geometry | Sociology and Pure Maths | N J Wildberger

[@phiarchitect]

00:00 Intro 01:04 Elements Book 1 Prop 6 - If two angles of a triangle are equal, then the sides subtending the equal angles will be equal. 10:20 Elements Book 1 Prop 7 - On the same Right Line cannot be constructed two Right Lines equal to two other Right Lines at different points on the same side, and having the same Ends which the first Right Line has. 18:27 Elements Book 1 Prop 8 - If two Triangles have two Sides of the one equal to two Sides of the other, each to each, and the Bases equal, then the Angles contained under the equal Sides will be equal. 22:25 Logical Issues 26:35 Q: If Euclid's Elements are not really a proper logical foundation for geometry - then what is?

[@OKMathh]

Looking forward to "the continued", thank you Professor, excellent video as usual!

[@rickshafer6688]

The proof by contradiction is 200° . I like the way Euclid dispensed with the lawyer talk about math. A contradiction is a contradiction . This is enlightening. - That is not to have to list every possible way the equation is not equal.

[@infra-cyan]

Giovanni Girolamo Saccheri published a critique of Euclidean geometry in 1733 called Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw)

[@santerisatama5409]

I wellcome the deeply philosophical discussion. The modern division between applied and pure mathematics does not really apply to Greek mathematics, which is an empirical science, without the more or less arbitrary division into applied and pure math, which deserves to to studied as historical phenomenon of sociology of mathematics. And here 'empirical' does not mean Cartesian dualism and amputation of mathematical intuition from empirism ("Introspection"). Plato gives a demonstration of intuitive empirism as 'anamnesis' in one of his dialogues. Euclid is not the first or even the most famous example of proofs by contradiction. Most famous example is Zeno's paradoxes, which are empirical proofs of irreducibility of continuous measurements to numerical counting. Let's also note that Zeno's proofs by empirical contradictions are about temporal processes and durations, not about formal coherence and consistency of static system buidling. Of course we can and should improve on Euclid, but the direction Hilbert took is the wrong one, as most here might agree. Arbitrary axiomatics of formalism are anti-empirical and counter-intuitive by the underlying philosophy of mathematics as arbitrary language games. In modern perspective, pure mathematics can be defined by coherence theory of truth, and applied mathematics by coherence of pragmatic goals, mainly in the world of external senses. The foundational crisis of mathematics has been the clash of contradictions between coherence theory of truth and pragmatic coherence, which has insisted on struggling against Zeno's proofs attempting to reduce continuous measurements to numerical counting. What is implied in both Zeno's and Euclid's proofs by contradiction is the more-less relation. Which is considered by both so intuitively and empirically clear and fundamental, that it does not need a formal definition, but is sufficient as such for a proof by contradition. Our everyday life is ordered by various more-less relations, which are profoundly empirical. Postulation of numerical counting by this or that existential quantifier is not an empirical necessity, it's a metaphysical act. Very interesting such!

[@dsm5d723]

I would seek to pragmatize what you do a good job of theoretically. I see the crisis in the foundations as Russell and Wittgenstein failing to overcome the occulted function of spoken language. Transitivity, ergativity and the Middle Voice of proto Indo-European. Math is an implied STATIC and square computation space, as a mathematical object; the calculators running random samples to compute you or the economy or anything else give time to a static architecture that has electricity flowing through it TO compute. Dynamic Complex systems, adaptive at that, are not definable by such static and non-value relational axioms. So many ways to repeat the obvious. Literacy, is a way to either signal in-group members to food, or make out group members food with deception. Visual numeracy is relying on a machine to compute TIME for you and show you the answer on a screen. Same with the evolved function of spoken language. "Trust me bro, I'm not gonna eat you." BANG upside the head with a rock. DINNER.

[@santerisatama5409]

@@dsm5d723 I loved reading Wittgenstein's commentary on mathematics, it was very liberating. His general tone is also very critical of the arbitrary and post-modern language games of formalism. In my view, already thinking about mereology of durations, aka 'Bergson lattice', has a pragmatical dimension. As that by definition leads to huge mathematical simplification and generalization of CPT-symmetry in fundamentally participatory and creative way. A self-referential dialogue with the Eagle. (cf. dynamic holography).

[@dsm5d723]

@@santerisatama5409 If we reduce everything to my TOE, and it is really Nature's, then there are Universal analogs. Time metrics are not legit if not reversible; done. Down to the RNA->DNA->Protein time stamp, allele switches and protein production. The NEWS now is not conspiracy, they hid the Big Picture. Want the PCR real-time IMAGING techniques, seeing literally the RNA->DNA->Protein expression in any cell or organ system? If not applied to Maximal Complexity first, math looks abstract. I'm working back from the mind-body connection TO the biggest math problem. NJ and e(c) + sq2 + Pi, what IS that? Rationalizing algorithmic decimal expansion FOR dimensionality, which like QCD is NOT infinite.

[@santerisatama5409]

@@dsm5d723 Well I seem to have 2*5 toes, and I'm becoming more and more suspicious that 2*5 adds up to 7^2 rather than 10. And the quantum phi could very well really be the golden phi. The theory of p-high is way too p-acidic for my taste. I like the full meal of revercimal Stern-Brocot coprimes from the continuous Origin better. Aware that only matters of taste can be really argued.

[@dsm5d723]

@@santerisatama5409 I must disagree. e(c) and sq2, to 5 decimal places, are an exact doubling AT each decimal, 0.577, 1.414, the right side rule. The 4th and 5th decimal places resolve to 2-1, and are the Dimensional Gauges of THIS world's erroneous design. 3 places, for the Euclidean mapping points, and then the palindrome, or CODONS in bioinformatics. I finished it all with finding Pi and the secret to integrals and summations. 95th decimal place palindrome of 2-1-1. 3rd Power series, any COHERENT 3D mathematical modeling to precision. If it is from the smallest thing we can use, the 1 GeV of gamma radiation, then the New Standard Model can be applied. I am waiting. Fibonacci is the Mandelbrot expansion in Nature. Many ways in words. That is taste. But Dollard's finish to Tesla, a Cosmic Induction Generator, is proof in 3D math model. Everything is perfectly transformable mathematically. If it is understood.

[@dsm5d723]

First Straw Man Ever? ()__() Subtend is a tough one. "Awaiting beneath" is insufficient because the angle is more to be seen as an emanation point for the shape the angle is a mapping of inductively, so "closer temporally to the dynamic locus of material change" might be better. Projection, light cones and dynamical friction are embedded from Natural observation. But this is not "math" as we think of it. It is PURELY based on observation of Nature. The cultural embed of architecture and the Pythagorean Triples-Infinitude of Primes proof is to be seen as "Natural" by Euclid's time. Complex analysis, the basis for quantum woo garbage, is what followed geometry being used to its limits. Unfortunately, math comes from the mapping of the mind-body connection, and answering your big question about e(c), sq2 and Pi makes a literal 3D linear algebra and it is a gene drive. The bioinformatics people I first queried told me to hush; this is why. Eric Dollard finished Heavyside's work on the actual wave form behavior in a phone line, and working out a direct relationship algebraically. Not even Universal (ha ha) geometry. I think he had to work with complex algebra.

[@antizephdaniel7868]

The question; Which comes first , is it the chicken or the egg? is not very difficult to answer.Proof by contradiction : If it is a wild egg , then it has to be a wild egg hence there was never a chicken.

[@njwildberger]

Your logic appears to be impeckable.

[@antizephdaniel7868]

@@njwildberger Yes!And if I may add that the only true infinity is semantics infinite amount of LAYERS.

[@dsm5d723]

@@antizephdaniel7868 No, combinatorial infinity in Nature. Semantics IS a close second.

[@antizephdaniel7868]

@@dsm5d723 You mean COMB-inatorial infinity? Trust me only semantics can dig a rabbit hole so deep.

[@dsm5d723]

@@antizephdaniel7868 As NJ did point out, set theory and the axiom of choice are only a small corner of combinatorics, and a gene drive does explode to 3D infinity with the laws of Nature, and scaling power laws. TIME is perfectly mapped and computed, but words do get annoying, or rather speakers. I so prefer silence and the resonant harmony of the Spheres.

[@value21value]

this man is gonna find out soon enough that the problem he points out eventually comes down to our nature as human beings..we think we discover math cause we can't actually see any other way to describe anything or think about anything.

[@dsm5d723]

I found the end of mathematics, and it is in his big question about e(c) + sq2 + Pi. "Irrational" decimal expansion has a digit-place embed and error correction code, based on mapping the mind-body connection. Omega Code is what Hollywood and entertainment like to call it. Brain map in bits and a gene drive. Euclidean linear algebra, in 3D.

[@whig01]

Modus tollendo tollens. :)