The continuum, Zeno's paradox and the price we pay for coordinates 117 | Math Foundations

Comments · 14

• JA

  @Jay-Raff · 5 years ago

  I'm in love with your videos, Norman. They are truly appreciated. I have a Bachelors of Philosophy which involved a rigorous study of the Ancient Greeks, and the history of philosophy from Ancient, to Medieval, then to Contemporary times. The notion of infinity and infinite processes always disturbed me because after my philosophical studies I held to the metaphysical view of Moderate Realism (Aristotle's philosophical view) which provides powerful arguments against those philosophical views which permit or entail real infinities. I spent the following years thinking of a way to approach mathematics without reliance on infinities. I was without direction much of the time, given I'm no mathematician. Then, alas, I came across your videos, and they're validating and clarifying so many intuitions and disturbances concerning the contemporary reliance on infinities. I'm now completing a degree in Physics, with the hope of becoming a Theoretical Physicist, however, given my background in the history of philosophy, I can truly understand your discontentment with contemporary Professors and students which take so many dubious ideas dogmatically for granted, being completely unaware of how they originated, and how they carry the same flaws, weaknesses, and arbitrariness found in philosophical theories which birthed them.

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• LE

  @LetsbEnemies · 4 years ago

  Thank you Dr Wildberger

• RU

  @russellchido · 1 year ago

  The comparison here of the Euclidean plane to Descartes' blew my mind. I was able to get a better intuition of how it could be that two things could appear to intersect but do not.

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• CE

  @cedricdamar11002 · 4 years ago

  Good stuff

• MA

  @madforlorn · 1 year ago

  But... The cartesian system is merely implied by the Euclidean geometry.<br>You can have two perpendicular lines in euclidean geometry, and call the intersection the point O, for origin.<br>You can have an arbitrary point on any of the perpendicular lines and call this point &quot;1&quot;,<br>you can map the same point &quot;1&quot; onto the other line call it &quot;1&apos;&quot; i.e. the line length O1 and O1&apos; are equal.<br>You can take a multiple of any such line segment O1 or O1&apos; to produce O2, O2&apos; etc.<br>As you can clearly see the Cartesian system is just implied by the Euclidean geometry.

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• IG

  @Igdrazil · 4 years ago (edited)

  Very courageous and passionate man, as always Norman. Your thoughts are always a warm and causy generous delight of &quot;crual inquisition&quot;, that is actualy, hélas, rare, and that always remind me of Machs free speach, independance of mind, accuracy of thoughts and technics that find the weak point of best armors, and last but not least vast culture systematicaly used to surround as an army general, each targeted &quot;city&quot;! Sort of a Janus soul with sagitarius on one side and winged snake on the other. Grounded but skylike winged, stable but agile, calm but explosive, silent but burning, generous but restless, sinuous but rigid, objective but subjective, one way stream but...vice versa, paradoxphobe but still fighting the Minotaurus in the Labyrinth...<br><br>Your main act of steady genious is to keep the bleeding Heart of the Beast, obsessively open, so that the light of thoughts can no forget and study deeper its Anatomy, and keep any needed surgery at hand reach. This is what Science, alive Science is all about, and &quot;more generaly&quot;, THINKING. <br><br>In this combat, I may share some remarks. Poincaré has writen in 1912, 3 weeks after Solvay famous meeting on Quanta Theory, a crucial PROOF, of the radical incompatibility between the &quot;black body ultra violet catastrophy&quot; (phenomenologicaly solved by Planck in 1900 with his revolutionary hypothesis of QUANTA), and differential equations modelisation. This Poincaré &quot;tour de force&quot;, aside its technical genious, was the decisive Archimede balance point, that convinced the World leading physicists, that were previously strongly doubting of Planck Quanta Theory, to finaly accept it, moved by the scientific reputation and admiration that scientific World had unanimously for Poincaré. Poincaré very clearly pointed out that it was here a more drastic coming revolution, than the 1885-1905 Relativity one, since it potentialy ruined our right to fully rigourously use the Contium Principles and thus differential representation and modelisation, to actual, at least, &quot;microscopic physics&quot;. Planck quantum Action, being a new fundamental constant, that drives the line, between &quot;continuous apparent macroscopic World&quot;, and &quot;discrete actual microscopie World&quot;.<br><br>A revolution indeed! But of course things are not si simple... Cause fields and potentials, probabilistic wave functions, Hilbert Space, Lie algebra, diffeomorphic smooth groups,... have soon after filled again the entire Arena of Physics modelisation, with their CONTINUUM SPECTRA. Discontinuity being controled, master, confined, in the EGEN VALUES of matrices and operators (mainly infinite dimensioned), aside their EGEN VECTORS CONTINUOUS FIELDS. <br><br>Chace the Devil by the door and watch him come back by the windows, stronger than before!<br><br>Besides a would like to recall an other crucial insight of Poincaré, dating of his 1890-1902 Sorbonne lessons, several articles, and sumerised in his 1902 best seller &quot;La Science et l&apos;hypothèse&quot;. Poincaré, after a brilliant critical analysis of &quot;Space&quot; and &quot;Time&quot; and of &quot;space-time&quot;, study in powerful details how we psychologicaly built physiological concepts of Space and Time, by GROUPS INCONSCIOUS BUT EFFICIENT MANIPULATIONS, involving of course reflections, translations, rotations. Coincidance and objects matching distant verification, being operated by our mind through the eyes geometric sophisticated machine, by &quot;INVERSE TRANSFORMATIONS&quot;. One recognise by this mean some actual MOVEMENT of an object, if the eye can &quot;anulate&quot; by an &quot;INVERSE TRANSFORMATION&quot;, the observed apparent MOVEMENT. Thus playing &quot;around&quot; the IDENTITY TRANSFORMATION, as an Archimede balance point, between actual exterior relative MOVEMENT, and inner eye inverse ROTATION ! Thus the impression of continuum of phenomenological World being forged by the remarquable &quot;CONTINUOUS&quot; MOBILITY of our EYES...<br><br>It&apos;s to my knowledge the best and unique such deep study of Space, Time, and the mechanisms by witch we FORGE these mix concepts/feelings. By such breakthrough Poincaré opens the Pandora box of our fundamental beliefs and concepts, in a psychological, physiological, physical, mathematical,...all at once, aproach, sign of his mesmerising universal (massively forgotten) genious.

• MA

  @maxpercer7119 · 4 years ago

  great teacher, and I hope to learn more maths from him , (dont agree with his views on infinity)

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• IG

  @Igdrazil · 4 years ago (edited)

  I&apos;m curious Norman, of how you would analyse my inverted analyse of yours, concerning Zeno&apos;s paradox...<br><br>I claim that Zeno&apos;s paradox can well mean the exact opposit! That infinity indeed, MUST therefore exist, for such paradox to have a chance to be solved!...<br><br>Just as infinity MUST, on this &quot;other side&quot; exist, for its dual ZERO, to exist, in the remarquable Infinite/zero symmetry recalled in the x---1/x duality screwed in the algebraic identity xx*=1 where x* stands for an alternative notation of&quot;1/x&quot;.<br><br>Here at work, the impressive illusion that such paradox makes at first in our mind, comes precisely from the difficulty a priori that we have, dealing with infinity. In such way that in our actual finite extrapolation, we on the contrary illusory and naively inconsciously believe in our dreamed Infinite capacity, and temerously conclude : &quot;oh, since every time Achilles reach the last stand point of the turtoise, this one has moved forward, so not even endless repetition of such looser patern, can ever solve such paradox and find Achilles victorious&quot;. <br><br>Whereas, on the contrary, the paradox in fact burst in our mind because we SUPPOSE that we have first well understood the premices (the Relativity of Time for instance and relative motions of frameworks), which is dubious, and secondly that we can easely IMAGINE Infinite repetition of such primitive patern (sort of Bernoulli binary scheme), and even worse exponentialy growing speed of such repetition, whereas we cant actualy, and may face instead crual illusions in such capacity of pragmaticaly even imagine repeating such patern, &quot;far before&quot; an infinite amount of &quot;time&quot;, not even a billion!!!<br><br>The crucial psychological genious of Zeno is to FREEZE OUR MIND IN ALMOST &quot;FROZEN TIME&quot;! He psychologicaly knows that average untrained minds will have even harder time to mentalise exponentialy speeding time laps between two repetitions of his imposed tricky loosing patern, than to mentalise even the purely arithmetic hypnotic growing number of such patern repetition. Zeno knows that few will survive this double headed infinite Beast, and that most of the athenians that have not follen in Charybde, will fall in scylla...<br><br>But Achilles will overpass the turtle! And that actual FACT, looks to arise THANKS TO INFINITY! If Achilles had only a FINITE amount of jokers to repeat the loosing primitive patern, he would have no chance, to ever overpass the slow turtle! Hundred time zero never lead to escap the absorbant zero rabit black hole ! &nbsp;Only an infinite time zero keeps alive a hope of standing out victorious out of such pitfalls !<br><br>In real life furthermore, the needed time by Achilles to reach the next half of the remaining distance, shrinks to zero, in order to overpass the turtle in FINITE Time before this last wins the race on the finit global distance. Here &quot;FINITE&quot; not being antinomic of INFINITE POTENTIALITY, at least integer, thus rational. <br><br>So shall this zen...O&apos;s paradox, delivers us a standing indirect &quot;proof&quot;, of the NECESSITY OF INFINITY AND INFINITE PROCESSUS?...

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