Archive for the ‘Platonism’ Category

Not An Infinity Of Angels On Pinheads

July 1, 2016

Thomas Aquinas and other ludicrous pseudo-philosophers (in contradistinction with real philosophers such as Abelard) used to ponder questions about angels, such as whether they can interpenetrate (as bosons do).

Are today’s mathematicians just as ridiculous? The assumption of infinity has been “proven” by the simplest reasoning ever: if n is the largest number, clearly, (n+1) is larger. I have long disagreed with that hare-brained sort of certainty, and it’s not a matter of shooting the breeze. (My point of view has been spreading in recent years!) Just saying something exists, does not make it so (or then one would believe Hitler and Brexiters). If I say:”I am emperor of the galaxy known as the Milky Way!” that has a nice ring to it, but it does not make it so (too bad, that would be fun).

Given n symbols, each labelled by something, can one always find a new something to label (n+1) with? I say: no. Why? Because reality prevents it. Somebody (see below) objected that I confused “map” and “territory”. But I am a differential geometer, and the essential idea there, from the genius B. Riemann, is that maps allow to define “territory”:

Fundamental Idea Of Riemann: the Maps At the Bottom Are Differentiable

Fundamental Idea Of Riemann: the Maps At the Bottom Are Differentiable

The reason has to do with discoveries made between 1600 and 1923. Around 1600 Kepler tried to concretize that attraction of planets to the sun (with a 1/d law). Ishmael Boulliau (or Bullialdius) loved the eclipses (a top astronomer, a crater on the Moon is named after him). But Boulliau strongly disagreed with 1/d and gave a simple, but strong reasoning to explain it should be 1/dd, the famous inverse square law.

Newton later (supposedly) established the equivalence between the 1/dd law and Kepler’s three laws of orbital motion, thus demonstrating the former (there is some controversy as whether Newton fully demonstrated that he could assume planets were point-masses, what’s now known as Gauss’ law).

I insist upon the 1/dd law, because we have no better (roll over Einstein…), on a small-scale.

Laplace (and some British thinker) pointed out in the late 18C that this 1/dd law implied Black Holes.

In 1900, Jules Henri Poincaré demonstrated that energy had inertial mass. That’s the famous E = mcc.

So famous, it could only be attributed to a member of the superior Prussian race.

The third ingredient in the annihilation of infinity was De Broglie’s assertion that to every particle a wave should be associated. The simple fact that, in some sense a particle was a wave (or “wave-packet”), made the particle delocalized, thus attached to a neighborhood, not a point. At this point, points exited reality.

Moreover, the frequency of the wave is given by its momentum-energy, said De Broglie (and that was promptly demonstrated in various ways). That latter fact prevents to make a particle too much into a point. Because, to have short wave, it needs a high frequency, thus a high energy, and if that’s high enough, it becomes a Black Hole, and, even worse a Whole Hole (gravity falls out of sight, physics implodes).

To a variant of the preceding, in: Solution: ‘Is Infinity Real?’  Pradeep Mutalik says:

July 1, 2016 at 12:31 pm

@Patrice Ayme: It seems that you are making the exact same conflation of “the map” and “the territory” that I’ve recommended should be avoided. There is no such thing as the largest number in our conceptual model of numbers, but there is at any given point, a limit on the number of particles in the physical universe. If tomorrow we find that each fermion consists of a million vibrating strings, we can easily accommodate the new limit because of the flexible conceptual structure provided by the infinite assumption in our mathematics.

***

I know very well the difference between “maps” and territory: all of post-Riemann mathematics rests on it: abstract manifolds (the “territories”) are defined by “maps Fi” (such that, Fi composed with Fj is itself a differential map from an open set in Rx…xR to another, the number of Real lines R being the dimension… Instead of arrogantly pointing out that I have all the angles covered, I replied:

Dear Pradeep Mutalik:

Thanks for the answer. What limits the number of particles in a (small enough) neighborhood is density: if mass-energy density gets too high, according to (generally admitted) gravity theory, not even a graviton could come out (that’s even worse than having a Black Hole!)

According to Quantum Theory, to each particle is associated a wave, itself computed from, and expressing, the momentum-energy of said particle.

Each neighborhood could be of (barely more than) Planck radius. Tessellate the entire visible universe this way. If too each distinct wave one attaches an integer, it is clear that one will run out of waves, at some point, to label integers with. My view does not depend upon strings, super or not: I just incorporated the simplest model of strings.

Another mathematician just told me: ‘Ah, but the idea of infinity is like that of God’. Well, right. Precisely the point. Mathematics, ultimately, is abstract physics. We don’t need god in physics, as Laplace pointed out to Napoleon (“Sire, je n’ai pas besoin de cette hypothese”). (I know well that Plato and his elite, tyrant friendly friends and students replied to all of this, that they were not of this world, a view known as “Platonism”, generally embraced by mathematicians, especially if they are from plutocratic Harvard University… And I also know why this sort of self-serving, ludicrous opinion, similar to those of so-called “Saint” Thomas, a friend of the Inquisition, and various variants of Satanism, have been widely advocated for those who call for self-respect for their class of haughty persons…) 

The presence of God, aka infinity, in mathematics, is not innocuous. Many mathematical brain teasers become easier, or solvable if one assumes only a largest number (this is also how computers compute, nota bene). Assuming infinity, aka God, has diverted mathematical innovation away from the real world (say fluid flow, plasma physics, nonlinear PDEs, nonlinear waves, etc.) and into questions akin to assuming that an infinity of angels can hold on a pinhead. Well, sorry, but modern physics has an answer: only a finite number.

Patrice Ayme’

 

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The MATHEMATICAL MIND HYPOTHESIS

April 25, 2015

Abstract: A new view is seen (“theo-ry”) for the relationship of mind and universe, and mathematics is central. The Mathematical Mind Hypothesis (MMH). The MMH contradicts, explains, and thus overrules Platonism (the ruling explanation for math, among mathematicians). The MMH is the true essence of what makes the Mathematical Universe Hypothesis alluring.

***

What’s the nature of mathematics? I wrote two essays already, but was told I was just showing off as a mathematician, and the subject was boring. So let me try another angle today.

The nature of mathematics is a particular case of the nature of thinking.

For a number of reasons, deep in today’s physics, as I have (partly) explained in “Einstein’s Error”, many physicists are obsessed with the “Multiverse”, an extreme version of which is the “Mathematical Universe Hypothesis” (MUH), exposed for example by Tegmark, a tenured cosmologist at MIT. Instead of telling people what happened in the first second of the universe, as if I considered myself to be god, I prefer to consider dog:

Dogs LEARN To Choose “y” According To Least Time

Dogs LEARN To Choose “y” According To Least Time

[Dogs can also learn to solve that Calculus of Variation problem in much more difficult circumstances, if the water is choppy, the ground too soft, etc. To have such a mathematical brain allowed the species to catch dinner, and survive.]

The “Multiverse” has its enemies, I am among them. Smolin, a physicist who writes general access books, has tried to say something (as described in Massimo’s Scientia Salon’ “Smolin and the Nature of Mathematics”).

“Smolin,” Massimo, a tenured philosophy professor also a biology PhD, told me “as a counter [to Platonism], offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).”

My reply:

Smolin is apparently unaware of a whole theory of “truth” in mathematical logic, and of the existence of the work of famous logicians such as Tarski. When Smolin was in the physics department of Berkeley, so was the very famous Tarski, in the mathematics department. Obviously, the young and unknown Smolin never met the elder logician and mathematician, as he is apparently still in no way aware of any of his work.

Thus, what does Smolin say? Nothing recent. Smolin says mathematics is axiomatic, and develops like games. That was at the heart of the efforts of Frege’s mathematical logic, more than 115 years ago. (Bertrand Russell shot Frege’s theory down, by applying the 24 centuries old Cretan Paradox to it; interestingly, Buridan had found a rather modern solution to the problem, in the 14C!) To help sort things out, it was discovered that one could depict Axiomatic Systems with sequences of numbers. Could not Axiomatics then be made rigorously described, strictly predictive?

Gödel showed that this approach could not work in any system containing arithmetic. Other logicians had proven even more general results in the same vein earlier than that (Löwenheim, Skolem and contemporaries). Smolin is now trying to reintroduce it, as if Löwenheim, Skolem, Gödel, and the most spectacular advances in logic of the first half of the Twentieth Century, never happened.

Does Mr. Smolin know this? Not necessarily: he is a physicist rather than a mathematician (like Tarski, or yours truly).

Smolin: “Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will.”

Smolin: Math brought into existence by HUMAN WILL. Mathematics as will and representation? (To parody Schopenhauer.)

So how come the minds of animals follow mathematical laws? Dogs, in particular, behave according to very complicated applications of calculus.

http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/do-dogs-know-calculus-of-variation

How come ellipses exist? Have ellipses been brought into existence by Smolin’s “human will”? When a planet follows (more or less) an ellipse, is that a “construction which has been brought into existence by exercises of human will”?

Some will perhaps say that the planet “constructs” nothing. That I misunderstood the planet.

Massimo’s quoted me, and asserted that there was no value whatsoever to the existence of mathematical objects:

I had said: “How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?”

Massimo replied: “They don’t.”

And that’s it. It reminded me the way God talked in the Qur’an. It is, what it is, says Allah, and his apparent emulator, Massimo. Massimo did not explain why he feels that the spiral of a nautilus does not exist (or maybe, he does not feel that way, because it clearly looks like a spiral). According to Smolin, the spiral is just a “construct of human will”.

If the spiral is a construct of human will, why not the mountains, and the ocean?

I am actually an old enemy of mathematical Platonism. However, I don’t throw the baby with the bath.

I agree that the “Mathematical Universe Hypothesis”, and Platonism in general are erroneous. However that does not mean they are deprived of any value whatsoever.

Ideas never stand alone. They are always part of theories. And idea such as Platonism is actually a vast theory.

MUH is: ‘Our external physical reality is a mathematical structure.’

I do not believe in the MUH. Because of my general sub-quantic theory, which predicts Dark Matter. In my theory, vast quantum interactions leave debris: Dark Matter. That process is essentially chaotic, and indescribable, except statistically (as the Quantum is). propose a completely different route: our mind are constructed by (still hidden) laws which rule the universe. Call that the MATHEMATICAL MIND HYPOTHESIS (MMD).

Here is the MMD: Our internal neurological reality constructs real physical structures we call “mathematics”.

This explains why a dog’s brain can construct the neurological structures it needs to find the solutions of complex problems in the calculus of variations.

Dogs did not learn calculus culturally, by reading books. Indeed. Still they learned, by interacting with the universe. (It’s unconscious learning, but still learning. Most learning we have arose unconsciously.)

From these interactions, dogs’ brains learn to construct structures which solve very complicated calculus of variations problems. As explained by the Mathematical Mind Hypothesis, (hidden) physics shows up in neurological constructions we call mathematics. And those structures, constructed with this yet-unrevealed, not even imagined, physics, are not just mathematical, but they are what we call mathematics, itself. That’s why dogs know mathematics: their brain contain mathematics.

Patrice Ayme’

Technical Note: Some may smirk, and object that my little theory ignores the variation in neurological structure from one creature to the next. Should not those variations mean that one beast’s math is not another beast’s math?

Not so.

Why? We need to go back to Cantor’s fundamental intuition about cardinals, and generalize (from Set Theory to General Topology). Cantor said that two sets had the same cardinal if they were in bijection. (Then he considered order, and introduced “ordinals”, by making the bijection respect order.)

I propose to say two neurological structure are mathematically the same if they produce the same math. (Some will say that’s obvious, but it’s not anymore obvious than, say, “Skolemization“.)

[Last point: I use “neurology” to designate much more than the set of all neurons, dendrites, synapses, axons and attached oligodendrocytes. I designate thus the entire part of the brain which contributes to mind and intelligence (so includes all glial cells, etc.). That ensemble is immensely complex, in dimensions and topologies.]

Some Basics Of Natural Philosophy

March 25, 2015

NO MULTIVERSE, NO PLATONISM (Less Air Travel, Too):

Some people go around, and brandish the “Multiverse”. Of course, the “Multiverse” exists, in one’s brain. The brain, among other things, extends all over imagination. Out there, among the galaxies, in the real world, there is no reason to suppose there is a “Multiverse, whatsoever.

It is basically something to sell books with. Or, just as with evil minded religions, for some physicists to claim they are like gods and can believe in something really absurd, and grotesquely self-contradictory:

There is No Universe, But the Universe:

The Universe is all there is. By definition. By philosophical definition. Just by philosophical definition? Not so. Any logic is associated to a universe. If the “logic” is nature itself (“all of the logic”) the associated universe (in the Logic sense), is, well, the Universe.

If something some would want to call the “Multiverse”, whatever that would be, existed, it would be part of the Universe.

Galaxies Used To Be Called "Island Universes". They Collide; This Is A Much Older Universe Than People Understand

Galaxies Used To Be Called “Island Universes”. They Collide; This Is A Much Older Universe Than People Understand

***

Age Of The Universe? Really?

Befuddled physicists go around, telling us about the “First Three Minutes” (Weinberg; Electro-Weak Nobel laureate), or the “History Of Time” (Hawking; remarkable survivor-physicist in a wheelchair).

That rests on their perfect knowledge of how the universe evolved.

This, in turn, depends upon ignoring Dark Energy. Dark Energy shows up as an unpredicted acceleration of the expansion of the Universe.

The old theory of expansion of the Universe was established before Dark Energy was discovered.

So they think they know, but I know they don’t really know.

I don’t know if the Universe has an age. But it is aging, or, at least, let’s be more cautious, the Universe is changing.

***

How Both Physics And Mathematics Became Not Even Wrong:

Mathematics themselves have always been developed in particular directions, in light of what it was felt was needed to understand the physical world. That was certainly true with Buridan, and his students, who developed computational methods, and graphs, to handle what they wanted to do with inertia. That was true with calculus developed for all sorts of engineering and physics explanations.

And so on through the next three centuries. However, in the last three decades, what I personally viewed as extremely erroneous notions in physics became dominant.

Indeed, it had become that clear time was not “relative” (whatever that is supposed to mean). True, time was local, as per Relativity, but it was local in an absolute way. The absoluteness comes from Quantum Theory… And the absoluteness of curvature in cosmology (the focusing of light, by galaxies and galactic clusters is absolute, thus so is time, locally around such focusing objects!).

Efforts were launched towards was felt would be the mathematics of “superstrings” and “field theory”. That would have been wonderful, if the initial meta-axiom motivating the whole enterprise, that nature worked with strings, super, and field mathematics perched on field math, all the way down… had been, roughly, correct.

Mathematics is not “natural”. Or let’s say, not anymore “natural” than the human brain can get contrived. Mathematics is an adventure in what the geometry, the Quantum geometry, of neurology is capable of.

Mathematics is not unreasonably effective (as the famous physicist-mathematician Wigner put it).

Mathematics is reason, manipulated to be effective in a particular way. Correctly determining in advance what the way will be makes the difference between understanding nature, and failing to do so.

Math is just, roughly, neuronal geometry that “works” (“working” here meaning what the brain does, whatever it is, beyond just manipulating electric and chemical signals).

***

Do We Need To Tour Frantically With Jets? 

In other news, after the crash in France of a Lufthansa A320 plane, pundits will surely come, and claim aloud that air travel is the safest mode of travel.

Is it? It depends upon the method of measurement.

The way advertisers come up with the “air travel is the safest form of travel” statement is by dividing number of people killed by distance travelled.

However, another measure would be to divide the number of people killed by the number of travels they engaged in. This is a more significant measure to think about. And air travel looks much good that way: in just one day in Europe, more car travels happen than all the air travel for the entire world, in a year.

Not to say that air travel should be discouraged. It is not exactly like smoking, with no redeeming value, whatsoever. Families ought to be reunited. Getting to know other countries, encouraged. However one week tourism, far away, thanks to plane travel ought, in my opinion, to be discouraged.

Instead, the projections are that air travel will augment considerably in the next few decades.

Between Barcelona and Dusseldorf, one ought to be able to travel just as fast by rail (not all the high speed lines are built, nor will they be built, thanks to plutocratically imposed austerity, and subsidies to… air travel). Electric trains pollute much less, by more than an order of magnitude, and are much safer.

The global CO2 situation is that bad. Besides, look at that entire high school classroom of fifteen year old that went down with the plane… Just for a week in Barcelona?

Patrice Ayme’