Posts Tagged ‘Category Theory’

Going Meta Here, There & Everywhere

April 19, 2016

Samantha Power is one of the great “leaders” of this world, so, endowed with quasi-supernatural powers, she powerfully zooms around in jets and armored motorcades. Such is our world: some think they can “lead” it, and it means they are above the law. So her motorcade zooms in Cameroon; at one hundred kilometer per hour on a dirt road. And runs over a seven-year old. Power was there to power up about security. A world led by such power monsters is not secure. This is what a powerful meta-conceptual analysis shows. (The rest of the day, the motorcade from hell slowed down to a crawl, momentarily chastised, because of bad PR…)

If one thinks carefully about it, one realizes that all and every progress in understanding were obtained by enlarging the scope of one’s inquiry. Typically factors not considered before get integrated, and they add logical dimensions to the logic. Another way of expressing this is by “going meta” (“meta” meaning beyond).

Just as there is metalogic and metamathematics, there is metaphilosophy (arguably metabiology has also arrived, with changes not just in genes, but in the bases of DNA). Plain mathematicians, or logicians do not like the “meta” version of their disciplines (nor do they like each other… except for those tremendous enough to transcend the differences). Indeed, at least once, metalogic studies invalidated a part of analysis.

Number of Books Mentioning Category Theory. For Decades, Most Mathematicians Have Feared And Despised Category Theory. CT Is Itself An Example Of Going “META”.

Number of Books Mentioning Category Theory. For Decades, Most Mathematicians Have Feared And Despised Category Theory. CT Is Itself An Example Of Going “META”.

Category Theory can be used to formulate “Meta”. Fascinatingly, the power of Category Theory is to go “meta” by forgetting all the details, and chasing (literally!) the big picture. One does not know why this is so, except, philosophically speaking, that the bark in your face tends to block sight of the forest beyond. To go meta, forget the bark, turn your head.

The incompleteness theorems are just one example of meta. earlier examples were equations, analytic geometry, euclidean versus non-simply-euclidean geometries, topology (generalizing metric spaces).

When one is  a baby it’s obvious that mom (= the Virgin Mary) and dad (= Zeus, Deus, Allah, etc.) created the universe. It’s a simple, deeply satisfying explanation, which helps provide all what one needs. However, when the baby goes meta, baby discovers there are other powers out there, other factors to take into account, from gravity, to getting air out of one’s stomach.

The rest of life is a long gathering of wisdom, by broadening and deepening one’s understanding, in a succession of conceptual mutations.

Some mathematicians (such as Alain Connes) have also complained that non-standard numbers were beyond their own understanding. Well, boys, I have bad news for you all: I invalidate all and any recourse to infinity as if it were another number. That still leave open the usage of “potential infinity”. But it demolishes… potentially, a large part of mathematics (what gets invalidated, or not, would, itself become a branch of mathematics).

Philosophy is the ultimate questioning of all the bases. “Metaphilosophy” should be a redundant notion. Part of metamathematics, such as Category Theory, have become workhorses of the mainstream… even in theoretical physics. Category Theory started as the ultimate pragmatism: forget about the foundations of the objects at hand, just worry about the rules the morphisms relating these objects can satisfy.

Thus one can safely say that fundamental differences between philosophy, mathematics, science and logic are all illusory. Category Theory provides with an example: it was started with a philosophical point of view on mathematics, and is now a must in some areas of physics.

So why does philosophy have a Public Relation problem? Because philosophy attacks the established order, always, and the new orders brutes always try to impose.

The other day the German Chancellor decided to prosecute a comic who had made (gentle) fun of Erdogan, the Turkish Sultan (aka “Turkish president”). It sounds like something straight out of more than a century ago (the Kaiser was allied to the Sultan). The healthy reaction from the philosopher Massimo Pigliucci was: Massimo Pigliucci‏@mpigliucci Apr 15

“Really Germany? Fuck you, Erdogan. “Germany Turkey: Merkel allows inquiry into comic’s Erdogan insult”

That’s excellent. It is excellent, because it is very wise. This is at the core of what the best philosophers have always done: scold infamy. In another tweet of Massimo, and it was a surprise to me, Hillary Clinton was revealed to have 27 million in income (in 2014, it was 24 million). Sanders had just $205,000 (less than 1% of Hillary’s income).

Philosophy, well done, brings revolutions, all over, not just consolation. Philosophy also brings consolation, yes. Boethius, president of the Roman Senate, wrote the “Consolation of Philosophy”, while waiting for his execution by bludgeoning from the local Erdogan. In the Sixth Century. (Islam did not yet exist, but, with examples like that, it could learn from the worst.)

The elites, lest they want revolution(s), can only view philosophy as a self-defeating endeavor. Elites are rarely for revolutions (although Louis XVI of France was for the American Revolution, in spite of strident critiques from his brother and members of his cabinet, that he was creating a Republic).

The fear and contempt elites have for revolution is the main source of the Public Relation problem of philosophy.

Meantime, humanity will keep on going meta, going beyond what was established before. That’s the genius of the genus Homo.

Patrice Ayme’

CATEGORIZING the MIND

October 27, 2014

What is the mind made of? We have progressed enormously as far as the brain objects are concerned: neurons, axons, dendrites, glial cells, neurohormones, various organs and substructures in the brain, etc.

But is there a broad mathematical framework to envision how this is all organized? There is! Category Theory! It turns out it’s a good first order approximation of mind organization. At least, so I claim.

Category Theory is about diagrams. Category Theory has been increasingly replacing advantageously Set Theory. It’s not only because Category Theory does not have to ponder the nature of objects, elements, sets.

Category Theory was long derided as “abstract nonsense” and “diagram chasing”. But it gives very deep, powerful theorems.

http://en.wikipedia.org/wiki/Category_(mathematics)

I claim the powerful theorems of Category Theory should translate directly into… neurology.

Amusingly, although I accused Aristotle to have demolished democracy and fostered plutocracy through his beloved pets, the mass murdering criminal plutocratic psychopaths, Alexander and Antipater, I recognize humbly that it’s the same Aristotle who invented categories (thus making him a great thinker, and justifying an Aristotle cult among those who need to have cults to feel good about themselves)…

Aristotle’s meta-idea about categories was just to talk about the most fundamental notions:

http://en.wikipedia.org/wiki/Categories_(Aristotle)

The present essay was suggested, and is an extension of what the honorable Bill Skaggs seems to have wanted to say, in Scientia Salon, in his “Identity A Neurobiological Perspective”. (As far as I can comprehend.)

However, forget Theseus’ ship and Hollywood’s Star Trek “Transporter”. As I said in “Quantum Identity Is Strong”, Quantum Identity is not erasable, and makes those time honored examples impossibly disconnected with reality. The notion of identity has thus to be found elsewhere (as we intuitively know that there is such a notion).

According to modern Quantum Field Theory, we are made, at the most fundamental level, of fluctuating fields. They come and go, out of nowhere. So, that way, we are continually been deconstructed and rebuilt. The question naturally arises: what is preserved of me, as a set of Quantum Fields? Well, the most fundamental mathematical structure is preserved.

The same seems to hold, to a great extent, in neurobiology, as neuro circuitry, to some extent, seems to come, go, and come back.

Thus we are all like old wooden Greek ships, perpetually falling apart, and rebuilt.

To some extent, this is what happens to species, through reproduction: cells split, and reproduce themselves, thanks to DNA.

A species has identity. Yet that identity is made of DISCONTINUOUS elements: the individuals who incarnate the species, who are born, and then die. And others appear, just the same, sort of. How is that possible?

A species’ identity is its structure. Just as a neurology, or an elementary particle identity is its structure. Not just a geometric structure, not just a topological structure, but its structure, as the most fundamental notion, as a category.

So what is preserved? Shape. And how to morph said shapes… Naturally (there is a notion of natural transformation, in Category Theory).

Historically, analyzing shape was systematized by the Greeks: Euclidean geometry, cones, etc. Then, at the end of the Nineteenth Century, it was found that geometry studied shapes mostly by studying distance, and yet, even if distance was denied consideration, there was a more fundamental notion of shape, topology. That was the structure of shapes as defined by neighborhoods.

Two generations later, Category Theory arrived. Category Theory is about morphisms, and the structures which can be built with them. Please listen to the semantics: structures, morphing… This is all about shapes reduced to their most basic, simplest symbolic expression. It’s no wonder that it would come in handy to visualize neurological structures.

A morphism is a pair of “objects” (CT leaves unspecified what the “objects” are). To model that neurologically, we can just identify ‘objects’ to neurons (or other neurological structures), and morphisms to axons (although dendrites, and more, could be included, in a second stage, when the categoretical modelling become more precise).

The better model is category theory. When are two diagrams equivalent? When are they IDENTICAL? Cantor defined as of the same cardinal two sets in a bijection (a bijection is a 1 to 1, onto map).

Category Theory defines as identical the same diagram (a drawing reduced to its simplest essence). Say: A>B>C>D>A is the same as E>F>G>H>E.

Thus, when are two diagrams identical in category theory? When they are modelled by the same neuronal network. (Or, more exactly, axonal network: make each arrow “>” above, into an axon.) And reciprocally!

Discussing the mind will involve discussing the most fundamental structures constituting it. What better place to start, than the most basic of maths? Especially if it looks readily convertible in neural networks.

Category Theory is the most fundamental theoretical structure we know of. It is the essence of identity, and identification. In conclusion, two objects are identical, neurologically, and in fundamental physics, if they are so, in category theory.

Time to learn something categorically new!

Patrice Ayme’

***

Note: No True Isolated Rocks: In other news, and to address a point of Bill Skaggs, whether a rock can be truly isolated is an open problem, experimentally speaking.

According to the theory of gravitation of Einstein and company, a rock cannot be isolated. Why? Because the rock is immersed in spacetime. Spacetime is animated by gravitational waves: this is what the Einstein Field Equation implies. Now, according to an unproven, but hoped-for principle of fundamental physics, to each force field is associated a particle. In the case of gravity, that hoped-for particle is called the graviton. “Particle” means a “particular” effect. Thus, an isolated rock, according to established theory, and hoped-for theory, ought to be adorned occasionally with a new particle, a new graviton, thus ought not to be isolated.

In my own theory, Objective Quantum Physics, on top of the preceding standard effect, resolving Quantum Entanglements, ought to create even more particles in “isolated” rocks.