The human civilization stumbled first upon metric geometry (the Greeks), then it invented projective geometry (Italian Renaissance) and finally conquered topology (the XIX century), as the study of continuous transformations. See the master himself, here.
The human being, in his/her development from infancy to adulthood, discovers first topology, stumbles upon projective geometry and finally conquers the metric concepts. Ontogenesis does not match phylogenesis. This question was studied first I believe by Jean Piaget: a debate on that (and other points) was hold in France in 1975 between the Cartesian Universal Grammar camp (Chomsky) and the Darwinian evolutionary epistemology camp (Piaget), see here for an index of that series of talks.
In order to develop spacial intuition in children, the MIT mathematician Seymour Papert came along.
Working under the influence of Piaget, he invented Logo (here a modern implementation client side).
Ideas in one of his books (Mindstorms Children, Computers, And Powerful Ideas) are now embedded in very interesting projects like Arduino or Raspberry Pi and more generally in all the talks about programming being part of children early education.
And now for my own history, similar to a karstic river. I had a very gifted teacher in my primary school, the best one I have ever had, who gave me incredibly advanced pointers. I was introduced to Tangram and Logo at a very early age, but somehow the lesson went lost in the grammar school I later attended, amidst Sophokles and Kant, Tacitus and Augustinus. Sort of Dead Poets Society experience.
But the spark he fired could not be forgotten, and I later recovered my passion in the form of programming and open source software, Backus-Naur grammars and kernel recompiling. Building a career as a numerical analyst and finally reaching for “PI in the Sky” (i.e. math), the true calling.
Years later, I stumbled again upon his name. He was working with Eduknoppix in the Italian school system and he had published a book on science (what else?). Then came to my mind that it would have been only a matter of time. He had engineered the initial conditions for the dynamical systems of our minds such that in time we would have converged. When I landed on Debian Linux by chance in one of the most fruitful encounters of my life that spark started firing again.
Why János Bolyai and Nikolai Ivanovich Lobachevsky landed on non-euklidean geometry at the same time (around 1830 that was), why Norbert Wiener and Andrei Kolmogorov landed on the same stochastic filtering equations at the same time (1941 it was) and so much more examples from science?
Is somehow the more stringent set of boundary conditions imposed on the human mind in the pursuit of mathematical knowledge enough to guarantee convergence if same initial conditions are given?
Thank you, Antonino
The following is from chapter “The Old Mimoid”, in Solaris by Stanislaw Lem
“But you don’t know what I was thinking about! Tell me something. Do you believe
Snow darted an apprehensive glance in my direction:
“What? Who still believes nowadays . . .”
“It isn’t that simple. I don’t mean the traditional God of Earth religion. I’m no expert
in the history of religions, and perhaps this is nothing new — do you happen to know if
there was ever a belief in an . . . imperfect god?”
“What do you mean by imperfect?” Snow frowned. “In a way all the gods of the old
religions were imperfect, considering that their attributes were amplified human
ones. The God of the Old Testament, for instance, required humble submission and
sacrifices, and was jealous of other gods. The Greek gods had fits of sulks and family
quarrels, and they were just as imperfect as mortals . . .”
“No,” I interrupted. “I’m not thinking of a god whose imperfection arises out of the
candor of his human creators, but one whose imperfection represents his essential
characteristic: a god limited in his omniscience and power, fallible, incapable of
foreseeing the consequences of his acts, and creating things that lead to horror. He is a . .
. sick god, whose ambitions exceed his powers and who does not realize it at first. A god
who has created clocks, but not the time they measure. He has created systems or
mechanisms that served specific ends but have now overstepped and betrayed them. And
he has created eternity, which was to have measured his power, and which measures his
Snow hesitated, but his attitude no longer showed any of the wary reserve of recent
“There was Manicheanism . . .”
“Nothing at all to do with the principle of Good and Evil,” I broke in
immediately. “This god has no existence outside of matter. He would like to free himself
from matter, but he cannot . . .”
Snow pondered for a while:
“I don’t know of any religion that answers your description. That kind of religion has
never been . . . necessary. If I understand you, and I’m afraid I do, what you have in
mind is an evolving god, who develops in the course of time, grows, and keeps
increasing in power while remaining aware of his powerlessness. For your god, the
divine condition is a situation without a goal. And understanding that, he despairs. But
isn’t this despairing god of yours mankind, Kelvin? It is man you are talking about, and
that is a fallacy, not just philosophically but also mystically speaking.”
I kept on:
“No, it’s nothing to do with man. Man may correspond to my provisional definition
from some points of view, but that is because the definition has a lot of gaps. Man does
not create gods, in spite of appearances. The times, the age, impose them on him. Man
can serve his age or rebel against it, but the target of his cooperation or rebellion comes
to him from outside. If there was only a single human being in existence, he would
apparently be able to attempt the experiment of creating his own goals in complete
freedom — apparently, because a man not brought up among other human beings cannot
become a man. And the being — the being I have in mind — -cannot exist in the plural,
“Oh, then in that case . . .” He pointed out of the window.
My interpretation of it is in one of my previous posts.
“A squire was determined to shoot a crow which made its nest in the watch-tower
of his estate. Repeatedly he had tried to surprise the crow, but in vain; at the
approach of man the crow would leave its nest. From a distant tree it would
watchfully wait until the man had left the tower and then return to its nest. One
day the squire hit upon a ruse: two men entered the tower, one remained within,
the other came out and went away, but the bird was not deceived; it kept away
until the man within came out. The experiment was repeated in the succeeding
days with two, three, then four men. Yet without success. Finally, five men were
sent: as before, all entered the tower, and one remained while the other four came
out and went away. Here the crow lost count. Unable to distinguish between four
and five it promptly returned to its nest” (Tobias Dantzig’s “Number and the Language of Science”)
One of my all time favorite sentences by legendary MIT linguist (whose opus is now preserved here) is the following (from Introduction to D. Guerin. “Anarchism”). It describes what our ‘practical reason’ target should be in life.
“At every stage of history our concern must be to dismantle those forms of authority and oppression that survive from an era when they might have been justified in terms of the need for security or survival or economic development, but that now contribute to — rather than alleviate — material and cultural deficit”
Another remarkable contribution of his genius is the conference Democracy and Education, which remains (for me) one of the very best description of what an authentic growth process for a human being should be, in the spirit of John Dewey and Bertrand Russell.
The problem is how to link the hope that this is the right way to be alive, the right way to walk amongst fellow human beings, with a theoretically sound precept. It may well be that, in Kant’s words, there cannot be ‘pure’ construction of any practical reason. Nonetheless, Chomsky’s maxim keeps resonating.
The best explanation I know about transhumanism, and the poverty of Antigone’s appeal to the ‘unwritten laws of the gods’, is the novel Golem xiv by great Stanislaw Lem. The frame of reference is Princeton molecular biologist Lee Silver’s marvellous book “Challenging Nature”.
All the laws are created by humans, and the famous lecture delivered by Heidegger on the “Ode on Man” of the greek tragedy, was just a restatement of the concept of ὕβρις, clearly backward. Even the Greeks, one of humankind miracles (to quote B. Russell HWP) had this superstition, let alone inferior metaphysics, trapped in pseudoconcepts like sin.
Stanislaw Lem was making reference to MIT mathematician Norbert Wiener book God and Golem although his pitch is very different. This video is a very enjoyable recap of Lem’s novel. The music of Cliff Martinez (of Soderbergh’s Solaris fame) makes for a special listening.
Suppose one has managed to numerically compute the forward transition density of 2-dimensional Markov process under some finite difference algorithm (or finite elements). Of course at each time slice, there is going to be a surface of data. How about plotting its evolution?
Leveraging Fast Fourier Transform magic (here), after having converted each matrix of data into an image, suppose called test_%d.jpg, simply:
ffmpeg -start_number n -i test_%d.jpg -vcodec mpeg4 transDensity.avi
Code converted with http://hilite.me/
There is a precious little book, which contains some hidden gems. It was written by the late great Nobel prize-winner Rita Levi Montalcini. Its title (“Abbi il coraggio di sapere”) is a bold reminder to the justly famous pamphlet by Kant containing the best definition of Enlightenment that is known to me. It has resisted critiques for more than two hundreds years, always proving more fresh as time goes by.
One of the points of Levi Montalcini is the (philosophical) importance of a stance of Nobel prize neurobiologist Roger Sperry. He argued that the study of the brain would progressively erode the traditional stance of science towards a complete aphasy w.r.t. morality. He argues that the grand design (obviously no reference whatsoever to creationism) of nature would furnish such a foundation.
“The grand design of nature perceived broadly in four dimensions, including the forces that move the universe and created man, with special focus on evolution in our own biosphere, is something intrinsically good that it is right to preserve and enhance, and wrong to destroy and degrade.”
―Roger Wolcott Sperry, Science and the Problem of Values (1972)
See also this and this by him.