Posts Tagged ‘Metamathematics’

Universe: Not Just Mathematical

August 14, 2014

Some claim the “Universe is mathematical”. Their logic is flawed. I show why.

Max Tegmark, a MIT physics professor, wrote “Our Mathematical Universe”. I present here an abstract I concocted of an interview he just gave to La Recherche. Followed by my own incisive comments. However absurd Tegmark may sound, I changed nothing to the substance of what he said:

La Recherche (France; Special Issue on Reality, July-August 2014): Max, you said “Reality is only mathematical”. What do you mean?

Tegmark: The idea that the universe is a mathematical object is very old. It goes all the way back to Euclid and other Greek scientists. Everywhere around us, atoms, particles are all defined by numbers. Spacetime has only mathematical properties.

La Recherche: Everything is math, according to you?

Formulation Before Revelation of Mathematization

Formulation Before Revelation of Mathematization

Tegmark: Think about your best friend. Her great smile, her sense of humor. All this can be described by equations. Mathematics explain why tomatoes are red and bananas yellow. Brout, Englert, Higgs predicted a boson giving mass to all other particles. Its discovery in 2012 at CERN in Geneva led to the 2013 Nobel Prize in Physics!

Tyranosopher [unamused]: Notice, Max Tegmark, that the “Nobel” thoroughly excites you. You brandish it, as if it were a deep reality about the universe. But, in truth, the Nobel is strictly nothing for the universe. It’s just a banana offered by a few self-interested apes to other self-fascinated apes. The Nobel has more to do with the nature of apish society, rather than that of the universe. In other words, we ask you about the nature of the universe, and you answer with the Authority Principle among Hominidae. You may as well quote the Qur’an.

Tegmark [unphazed]: There are an enormous number of things that equations do not explain. Consciousness, for example. But I think we will make it. We are just limited by our imagination and our creativity.

La Recherche: According to you, there is no reason that part of the world escape mathematics?

Max Tegmark: None whatsoever. All properties are mathematical! We potentially can understand everything!

La Recherche: As a Platonic mathematician, you consider mathematical concepts are independent of all and any conscious act?

MT: I am an extreme Platonist, as I think that not only mathematical structures are real, but they are all what reality is.

Relativity and Quantum Physics confirmed that reality is always very different from what one believes. Very strange and very different from our intuition. Schrodinger’s equation, the fundamental equation of Quantum Mechanics, shows that a particle can be in several places at the same time. Thus one does not try to describe the motion of this particle, but the probability of its presence in such and such a place.

But, a century later, physicists are still in deep disagreement about what it all means. I think this interpretation keeps dividing people, because they refuse to admit what goes against their intuition.

Tyranosopher: Notice, Max Tegmark, that you presented as a fact (“a particle can be in several places at the same time”) something you admit later is only an “interpretation”. That’s dishonest: an “interpretation” is not a “fact”.

Tegmark [livid]: The strength of mathematics comes from the fact that they have no inhibition. Strangeness does not stop them.

Tyranosopher: Indeed, that’s why, as a trained mathematician, I am very insolent.

La Recherche: Max Tegmark, is it your mathematical approach that makes you defend another controversial idea, that of multiple universes?

Max Tegmark: I really believe that human beings never think big enough. We underestimate our capability to understand the world through mathematics, but also our capacity to apprehend its dimensions. To understand that we live on a planet with a diameter of a bit more than 12,000 kilometers was a first, enormous, step. That this planet is infinitesimal in this galaxy, itself one out of billions, was another enormous step. The idea of multiverses is more of the same. We discover again, and more, that what we understand is only a speck of something much larger. That much larger thing is the Multiverses, of types I, II, III, and IV.

Tyranosopher: La Recherche’s Interview then proceeds further, but let me unleash a fundamental critique here.

I am a deadly enemy of the Multiverse, as I believe that it rests on an ERROR of interpretation of Quantum Physics (the one Tegmark presented as a fact above, before admitting that it was, well, only an interpretation). The fact that it is another desperate scaffolding erected to save the Big bang theory does not make it better.

Now for the notion that the universe being full of math. This is understood to mean that the universe is full of equations. Equations were invented in the Sixteenth Century. Many, if not most, equate mathematics with the art of equating.

What’s an equation? It’s something that says that two things independently defined, one on the left side of the equal sign, the other on the right side, are equal. Great. What could be simpler: what is different is actually the same!

Notice this, though: before you can equate, you must define what you are equating. On both sides.

An equation equates concepts independently defined. Ultimately, definitions are not mathematical (see on the Nature of Mathematics, to follow soon). At best, definition is metamathematical. Our metamathematical universe? End of Mr. Tegmark’s naivety.

When we get down to it, it’s more our philosophical universe, before it’s our mathematical universe: no definitions, no equations.

How can a physicist make such a gross logical mistake? Are they not supposed to be smart? (OK, it’s smart to sell lots of books).

What allows to make that logical mistake? Education, or lack thereof. Many a mathematician will make the same mistake too. The problem is that neither conventional mathematicians, nor, a fortiori, physicists, are trained logicians. They just play some in the media.

Who needs a multiverse? It seems the universe of science is already too large for many physicists to understand.

Patrice Ayme’

NON ARISTOTELIAN

November 5, 2013

It’s a NON ARISTOTELIAN WORLD:

Tyranosopher: Finite Logic should be called Non Aristotelian Logic. As I will show.

Simplicius Maximus, a contradictor: I have two objections to your finite math madness. First it makes no sense, and, secondly, even if it did, it would be pointless. 

Tyranosopher: I love contradictions. I squash them, then drink their juicy parts. OK, bring it on. Let’s start with the contradiction you found. A French contributor, Paul de Foucault, already made the objection that m/0 = infinity. 

Sounds good. However, it violates Peano Arithmetic (PA). PA is the arithmetic common to all metamathematics. But for mine, of course. (I violate much, with glee, including the pairing axiom!)

In PA, a.0 = 0 is one of the two axioms defining multiplication. So we see that if x = m/0, we would have x.0 = m. In other words, m = 0.

That’s not surprising: a number called “infinity” is not defined in PA

Simplicius Maximus: OK, fine. Here is my objection. It’s well known that the square root of two is irrational. Even Aristotle knew this, but you apparently don’t. And then you give the world lessons about everything. You are a charlatan. 

T: What do you mean by irrational?

SM: Ah, you see? It means square root of two cannot be equal to m/n, where m and n are integers. Let’s abbreviate square root two by sqrt(2). Irrational means the expansion of sqrt(2) never ends. 

T: Why? 

SM: Here is the proof. Suppose sqrt (2) were rational. That means: m/n = sqrt (2). Let’s suppose the terms m and n are as small as possible. That’s crucial to get the contradiction. 

T: Fair enough.

SM: Now, square both sides.  

T: That means, more exactly, that you contrive to multiply the left hand side of the equation by m/n and the right hand side by sqrt(2).

SM: Happy that you can follow that trivial trick. That gives us the equation: mm/nn = 2.  

T: As sqrt (2) sqrt (2) = 2. Indeed. By the way, you made an unwarranted assumption, so I view your reasoning as already faulty, at this point

SM: Faulty? Are you going mad? 

T: I will dissect your naïve error later. But please finish, Mr. Aristotle. 

SM: Call me Aristotelian if you wish. Multiplying both sides of the equation by nn, we get: mm =  2 nn. That implies that m is even. Because if m were odd, m = 2u + 1, then mm = 4uu + 4u + 1 , the sum of an even number (4uu + 4u) plus 1… And that, the sum of an even number with one, is odd. Hence m = 2a.

But then 2a2a = 2 nn, or: 2 aa = nn. Thus n is even (same reasoning as before: the square of an odd number cannot be even). So we see that both m and n are even, a contradiction, as we assumed m and n were the smallest integers with a ratio equal to sqrt (2). 

T: This proof is indeed alluded to in Aristotle, and was interpolated much later into Euclid’s elements. The official Greek mathematicians did not like algebra. 

SM: I see that, although you don’t know math, you know historiography.

Tyranosopher: I do know math, I’m just more rigorous than you, august parrot.

Simplicius Maximus: Me, a parrot? Me, and 25 centuries of elite mathematicians who are household names, dozens of Field Medalists are also of the avian persuasion? How can you be so vain and smug? 

Tyranosopher: Because I’m smarter.

SM: Really? Smarter than Aristotle? 

T: That’s an easy one. People like Aristotle spent a lot of time, all too much time, with politics, not enough with thinking. OK, let’s go back to your very first naive mathematical manipulation. You took the square of both sides. 

SM: Of course I did. 

Tyranosopher: You can’t do that.

SM: Of course I can.

Tyranosopher: No. In FINITE math, a = b does not imply that aa = bb

SM: Why?

T: Because aa could be meaningless. It could be too big to have meaning. It’s a added to itself a times. If, as we compute aa, we hit the greatest number, #, we must stay silent, as Wittgenstein would have said. 

In FINITE math, the infinite set of integers N does not exist. Only what can be finitely constructed exist. Because there is no way to construct the set N, as it would be infinite (if it existed; that’s a huge difference between what I propose, and what David Hilbert proposed). In my system, integers and rational numbers are constructed,  according to the principles I exposed in META, layer by layer, like an onion

SM: Wait. There are other proofs of the irrationality of square root of two.

T: Yes, but it’s always the same story: at some point, multiplication is involved, so my objection resurfaces.   

SM: OK, all right. Let me go philosophical. What’s the point of all this madness? Trying to look smarter because you are so vain, at the cost of looking mad? Do you realize that you are throwing out of the window much of modern mathematics?

T: Calm down. Entire parts of math are left untouched, such as topology, category theory, etc. My goal is to refocus all of math according to physics, and deny any worth to the areas that rest on nothing.

All too many mathematicians have engaged in a science as alluring as the counting of angels on a pinhead in the Middle-Ages. 

SM: Dedekind said: “God created the integers, and the rest was man’s creation.” 

T: Precisely, God does not exist, so nor does the infinite set of the integers, N. This will allow mathematicians to refocus on what they can do, and remember that there is a smallest scale, and it would, assuredly change the methods of proof, in many parts.

SM: Such as? 

T: Take the Navier Stokes fluid equation: one has to realize that, ultimately, the math have got to get grainy. This would help physics too, including all computations having to do with infinities. 

SM: You are asking for a mad jump into lala land.

T: We are already in lala land. Finding the correct definitions is even more important than finding the correct theorems (as the latter can’t exist without the former). The reigning axiomatic theory, ZFC (Zermelo Fraenkel Choice) requires an infinite number of axioms. What’s more reasonable? An infinite number of axioms, or my finite onion?

The answer is obvious. It’s a NON ARISTOTELIAN WORLD.

In my not so humble opinion, the consequences are far reaching.

***

Patrice Ayme


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Polyhedra, tessellations, and more.

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GrrrGraphics on WordPress

www.grrrgraphics.com

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The intersection of physics, optics, history and pulp fiction

Footnotes to Plato

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

Patrice Ayme's Thoughts

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Polyhedra, tessellations, and more.

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Writer, Editor, Berliner

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