Posts Tagged ‘Euclid’

Proof Of Existence Of Square Root Of Two For Ten Year Old (Take That, Euclid!)

September 6, 2020

Yesterday my ten year old daughter cried like I have not seen her cry since her grandmother died. Tears were running down her cheeks:”Cela n’a aucun sens!” It makes no sense! It was a telling choice of words: the Greek mathematicians made the exact same observation, they called such numbers “irrational”. I reassured her: she was brutally confronted by what drove ancient Greek mathematicians crazy: the square root of two.

How to define square roots algebraically is simple: x is the square root of a if and only if: xx = a… No problem if a is one, four, nine… But put a = 2, and a mystery arises: one can write down numbers ever closer to x = square root of two, but then what? Their squares are never quite two… Worse: ancient Greek mathematicians knew (positive) integers, and their ratios, say m/n, where both m and n are integers. They could demonstrate, in a few lines, that square root of two was not a fraction. 

Mathematics is the way of thinking that is most powerful in some situations. For example, as I wrote: xx = 2, I used no less than three mathematical notions that the ancient Greeks mathematicians did not have: the notion of equation, the equal sign, and the notion of unknown x.  

When the Greeks were confronted to this, they realized that some things existed that should not exist, according to their (mathematical) system of thought. Obviously, their mathematics came short. They discovered “irrational” numbers, the hard way. Further reasonings were halted by the rise of fascism in the Hellenistic and then Roman world. Do we have similar situations nowadays? Obviously yes. Nonlocality in Quantum Physics is an obvious example.

There is a mysterious relationship between numbers and geometry. Draw a line: that’s the x axis. Some distance mark one (1); it is the unit of distance. It does more: it represents the number one. Draw the perpendicular axis, traditionally called the y axis. There too, mark a one. Use the proverbial compass to make the units on the x and y axis the same length. Now one is facing a two dimensional plane. It naturally defines a unit of area, the square of sides equal to one. Cut that square in two to get the area of surface ½, etc… A bit of playing around shows that any square of side s has area ss…

Consider the rectangular triangle from the origin with sides one. The longest side can be used as the base of a square. A quick look shows that this square has area two (2). Thus its side is square root of two. 

This demonstrates that the square root of two exists. 

Ancient Greek mathematicians went that far. And that drove them nuts, as, for them, any “number” had to be of the form m/n, with m and n integers… And the square root of two was not such. 

The reasoning above is mine: it was designed to be understood by a very curious ten year old who is not fanatical about mathematics. So the Pythagorean theorem is demonstrated, in a particular case (the general proof is a somewhat confusing generalization; similar ad hoc geometrical proofs work for other numbers, for example square root of 5). 

Patrice Ayme

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Why the Greeks knew square root of two was no fraction (= “rational” number):

If (m/n)^2 = 2, with xx = x^2 and m and n having no common factors… Then:

mm = 2 nn… Thus m is even (if m is odd, mm is always odd). So m = 2u. This forces n to be odd, because by hypothesis, m and n don’t have common factors, so can’t both be divided by 2…

Hence, n = 2v + 1, for some integer v… Plugging back in the initial equation mm = 2 nn we get: 2u2u = 2 (2v+ 1) (2v + 1). Dividing by 2, we get:

2uu = (2v + 1) (2v +1) = 4vv + 4v + 1…

Now that latter equation is impossible: the left side is even, and the right side is odd… Thus the initial hypothesis, (m/n) (m/n) = 2 is impossible…

Axiom of Choice: Crazy Math

March 30, 2014

A way to improve thinking is to imagine more, and be more rigorous. What a better place to exert these skills than in mathematics and logic? Things are clearer there.

The crucial Axiom Of Choice (AC) in mathematics has crazy consequences. After describing what it is, and evoking some of its insufferable consequences, I will expose why it ought to be rejected, and why the lack of a similar rejection, at the time, in a somewhat similar situation, may have help in the decay of Greco-Roman antiquity.

This is part of my general, Non-Aristotelian campaign against infinity in mathematics and beyond. The nature of mathematics, long pondered, is touched upon. A 25 centuries old “proof” is mauled, and not just because it’s fun. There is deep philosophy behind. Call it the philosophy of sustainability, or of finite energy.

Intolerably Crazy Math From Axiom of Choice

Intolerably Crazy Math From Axiom of Choice

The Axiom of Choice makes you believe you can multiply not just wine, fish and bread, but space itself: AC corresponds, one can say, to a wasteful mentality.

The Axiom of Choice says that, given a collection C of subsets inside a set S, one can consider that a set exists, made of elements, each one of them is an element in exactly one of the subsets. That sounds innocuous enough, and obvious. And obvious it is, if one thinks of finite sets. However, if C is infinite, it gets boringly complicated.

Moreover, AC has a consequence: given a unit sphere, one can cut it in disjoint pieces, and reassemble those pieces to build two unit spheres. Banach and Tarski, both Polish mathematicians working in what’s now Western Ukraine, the object of Putin’s envy and greed, demonstrated this Banach-Tarski paradox. It’s viewed as an object of wonder in General Topology.

I prefer to view it as an object of horror. (The pieces are not Lebesgue measurable, that means not physical objects. Such non measurable objects had been found earlier by Vitali and Hausdorff)

Punch line? The Axiom Of Choice (AC) is central to all of modern mathematics. Position of conventional mathematicians? The fact that AC is so useful, all over mathematics, proves that AC can be fruitfully considered to be true.

My retort? Maybe what you view as fruitful mathematics is just resting on a false axiom, or, at least one against nature, and thus, is just plain false, or against nature. One may be better off, studying mathematics that is not against nature..

As I showed earlier, calculus survives the outlawing of infinity in mathematics. That pretty much means that useful mathematics survives.

You see a problem with mathematics, even the simplest arithmetic, is that, once one has admitted the infinity postulate, thanks to the Cantor Diagonal process, one can always find undecidable propositions (this is part of the Incompleteness Theorems of mathematical logic: Gödel, etc.).

That means a field such as Euclidean geometry is infinite, in the sense that it has an infinite number of non-provable theorems. Each can be decided both ways: false, or true. Each gives rise to two mathematics.

Yet, even modern mathematicians will admit that studying Euclidean geometry for an infinite amount of time is of little interest. Proof? They don’t do it.

Yet, what’s the difference with what they are doing?

Mathematics is neurology, and neurology can be anything, but infinite. Think about what it means. Yes, mathematics is even cephalopod neurology, with the octopus’ nine brains. Fractals, for example, are part of math, but far from the tradition of equating angles or algebraic expressions.

It’s a big universe out there. The number one consequence to draw from the history of science, is that scientists make tribes. Quite often those tribes go astray… for more than 1,000 years (see notes). Worse: my making science, and, or mathematics, uninteresting, they may lead to a weakening of public intelligence.

I would suggest that effect, making science, and mathematics priestly and narrow minded, contributed to the powerful anti-intellectual tsunami that struck the Roman empire.

Greek mathematicians had excluded all mathematics as unworthy of consideration, but for a strict subset of “Euclid’s Elements” (some of the present Euclid Elements were added later). The implementation of those discoveries were made by others (Indians, and to some extent, Iranians and Arabs).

It turned out that these more practical mathematics, excluded by Euclid, because they were viewed as non rigorous and primitive, led to deeper and more powerful insights.

The irony was that Euclid’s Elements, in the guise of rigor, were using an axiom that was not needed, in general, the parallel axiom. That axiom, by supposing too much, killed the imagination.

I suggest nothing less happening nowadays, with the Axiom of Choice: it’s one axiom too far.

Patrice Aymé

Technical notes:

Up to a recent time, if one was not a Supersymmetric (SUSY) physicist, it was impossible to find a job, except as a taxi cab driver. There was a practical axiom ruling physics: the world had got to be supersymmetric.

Now the whole SUSY business seems to be imploding as the CERN’s LHC came up empty, and it dawned on participants that there was no reason for an experimental confrontation in the imaginable future… I have studied SUSY, and I have a competitive theory, where there are two hints of experimental proofs imaginable (namely Dark Energy and Dark Matter).

I said the AC was one axiom too far, but actually I think infinity itself is an axiom too far. I exposed earlier what’s wrong with the 25 centuries old proof of infinity (it assumes one can use a symbol one cannot actually evoke, because there is no energy to do so!).

The geocentric astronomy ruled from Aristarchus of Samos (who proposed the heliocentric system, 3C BCE) until Buridan (who used inertia, that he had discovered to make the heliocentric system more reasonable; ~1320 CE; Copernic learned Buridan in Cracow, Poland). It could be viewed as an axiom.

Hidden axioms are found even in arithmetic, for example the Archimedean Axiom was used by all mathematicians implicitly, before Model Theory logicians detected it around 1950 (it says, given two integers, A and B, a third one can be found, D, such that: AD > B; if not fulfilled one gets non-standard integers).

Aphorisms March 2014: Putin, Plutos, Malta, Math, Brain

March 29, 2014

Whip Stops Baffled Bear Momentarily:

Wonderful! Dictator Putin suggests he won’t invade anything today, if a number of changes are made to Ukraine’s constitution, friends, hopes.

Specifically, Putin let it be known that he wants demonstrations in Ukraine, which, says Putin, have been disrupting him for six months, to come to a stop. Demonstrations are a bad example to Russians: too many demonstrations make the kleptocrats flee. There was actually 50,000 people demonstrating against Putin, in Moscow, because he had annexed Crimea.

Speaking of stopping, Putin will stop, if, and only if, he is persuaded that horrendous consequences are coming his way, otherwise. It’s not going to be easy, considering that some German minister said this week that Europe could not do without Russian gas, for the foreseeable future.

And considering that Londongrad is a mighty ally of Putin… In a West ruled by plutocrats. So this is not just about Putin going crazy, it’s about him realizing he is confronting weak and divided democracies, rotten from inside.

Thus, as in the 1930s, plutocrats are playing both sides, hoping for the best. Just as in 1930s, the dictator (Hitler then, Putin now) feels in command of enough plutocratic power in the West to get what he wants, without a world war. Hitler was astounded, on September 1, 1939, when the French Republic and Britain gave him an ultimatum. He had come to believe what his Anglo-American plutocratic friends had told him, that it would never happen, because they, the plutocrats, controlled everything.

***

I Bank Therefore I Tank:

How come Putin has momentarily come to his senses? Sanctions. By closing Rossiya, a bank close to Putin, at a distance, the USA left 495,000 Russian clients without a bank. And that was just a warning shot. Visa, MasterCard and company control the essentiality of Russian banking. Also banks need some international cooperation, and that was going down too.

It goes without saying that the USA, under corrupt president Roosevelt not only did nothing of the sort, but the exact opposite. As France and Britain were in total world war against Nazism, and 45 French division tried to crash through the Siegfried Line, the USA was busy aiding and abetting Hitler’s fascist dictatorship. In a crucial way (lead tetraethyl story).

Not only were Hitler and his followers encouraged, but the German generals who wanted to arrest the Nazis, got very confused: was the USA allied to the Guide, or not?

This time, the early, swift opposition of the president of the USA to tank-born fascism, is not just the most important thing Obama did, but the most important thing any government of the USA did, in generations.

***

Heavens For Sale:

Malta has put for sale 1,800 passports and nationality. First condition: pay 650,000 Euros for the head of the family, at least 250,000 Euros of investment on top of that, plus more per family member. The stratagem is expected to bring in more than one billion Euros. Reassuringly, Malta announced that it did not expect the new Maltese citizens to spend the year there (they will be free to roam the European Union).

Malta is notorious for refusing Maltese nationality, even for those who have resided there more than twenty years, and the EC is not happy about that.

***

Your Pain Is Our Ecstasy:

The main problem of the socio-economy is plutocracy, though. Plutocracy wants the starvation of the People’s economic activity. That allows to increase the gap between the haves and have-nots, which is the plutocracy’s raison d’être, and ultimate value.

Hence the obsession fabricated by the Main Stream Media against deficits, without saying they are directly related to the plutocracy not been taxed enough. Or the insistence that the People has no skill (thus, presumably ought to be starved in all ways, including access to public education).

Referring to: http://krugman.blogs.nytimes.com/2014/03/29/the-skills-zombie/

***

Food Madness:

2.4% population growth in the 1960s. That was the excuse for the creation of massive food exportation machinery then. In the 1980s, that over-production was massively exported to poor countries and frozen European chicken destroyed local food production in poor countries.

Population growth is only 1.3% now. That’s still about 100 million added, a year.

That does not mean that shocks to the world food system are not coming. They are, thanks to the global warming and weirding.

***

Mathematics = Physics

Many modern thinkers have wondered at the remarkable efficiency of mathematics in physics. Galileo said physics was written in mathematics, Plato viewed knowing math as a prerequisite to advanced thinking.

The latter point of view is the correct one. Better: thinking, advanced or not, is, intrinsically, mathematical. Neurology is math.

Mathematics is just a more abstract physics. So if physics is hard, so can mathematics be too. The best avenue to explore what these abstract thoughts mean, is the history of Euclidean geometry. One physical simplistic simplification that Euclid made was flatness. When mathematicians realized that flatness did not have to be, Riemann soon got the idea that geodesic distanciation was equivalent to force (vulgar physicists believe Einstein got the idea).

Euclid also made many other simplifying assumptions about the nature of continuity and … And Quantum Physics violate them, starting with the notion of points.

Patrice Aymé


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because all (Western) philosophy consists of a series of footnotes to Plato

Patrice Ayme's Thoughts

Striving For Ever Better Thinking. Humanism Is Intelligence Unleashed. From Intelligence All Ways, Instincts & Values Flow, Even Happiness. History and Science Teach Us Not Just Humility, But Power, Smarts, And The Ways We Should Embrace. Naturam Primum Cognoscere Rerum

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Smile! You’re at the best WordPress.com site ever

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Writer, Editor, Berliner

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because all (Western) philosophy consists of a series of footnotes to Plato

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ianmillerblog

Smile! You’re at the best WordPress.com site ever

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Military and general security

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How to Be a Stoic

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Writer, Editor, Berliner