Abstract
What is mathematics, and how did it originate? Where does the stream of mathematical ideas flow from? What is the ultimate source of mathematics in the brain? These are reminiscent of the ancient question, “What does the Earth rest on?” with our instincts pushing us toward On-a-Giant-Turtle answers.
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Notes
- 1.
A “mathematical model” is understood here in the physicist’s sense with mathematical rigor being a secondary issue.
- 2.
“Ergo” is not a definite thing but, to a large extent, a certain mindset for directing the study of the mechanics (most of which are hypothetical) of “deep learning” and related structures, such as the body of mathematics. It has nothing to do with wetware or with anything else expressible in a few words.
- 3.
Those who are not attuned to science would find all this more far-fetched than the idea of a Giant Turtle. An intelligent Cro-Magnon hunter-gatherer, for instance, would laugh at a learned scientist trying to teach him/her what his/her Earth is.
- 4.
- 5.
Haldane (1892–1964) was a mathematically minded evolutionary biologist and a famous science populariser.
- 6.
This process of mutilation, euphemistically called natural selection, serves to curb rather than foster evolutionary diversity.
- 7.
Probably, the evolutionary development of most complicated and interesting patterns in behavior of social insects, similarly how it is with human brains, followed the routes transversal to the (stochastic) gradient of unrestricted selection.
- 8.
Richard Feynman , while explaining how the phase cancellation in his integral implies the least action principle, jokes of particles that “smell” neighboring paths to find out whether or not they have more action.
- 9.
A corresponding Neural Darwinism model of brain function was suggested by Gerald Eidelman , probably motivated by the immunological selection mechanism of antibody proteins.
On the other hand, the subliminal ego of Poincaré serves as a precursor to what we call “ergo-brain.” But “ergo” entails, albeit stochastic, a high level of structural organization unlike this “ego.”
- 10.
- 11.
Do not confuse this with the subconscious, which is usually understood as a part of consciousness.
- 12.
This, for instance in the case of eating candies, is an elaborate chain of chemical reactions of the oxidation of acetate derived from carbohydrates into carbon dioxide and intracellular chemical energy in the form of adenosine triphosphate.
- 13.
I must admit that I only briefly browsed through a few randomly chosen papers, e.g. “The mathematics used in mathematical psychology” by Robert Duncan Luce [16], Logical and Mathematical Psychology by Nicolae Mărgineanu [17], Mathematical Psychology: An Elementary Introduction by Clyde Hamilton Coombs, Robyn M. Dawes, and Amos Tversky [2], and “Mathematical Psychology: Prospects for the twentieth Century” by James T. Townsend [25].
- 14.
Most “very human ideas,” are driven by the core behavior programs that originated—let us be generous—in the nervous systems of the worm-like ancestors of animals about 500 million years ago. These programs are invisible to our inner eye.
- 15.
These ideas are much further removed from “the true laws of thinking” than the perception of motion installed into our motor control system is from the Newtonian laws of mechanics.
- 16.
Embryogenesis remains an unresolved mystery of Life. How does a developing organism implement the design that is encoded in the genome?
- 17.
Polypeptides are polymeric chains of amino acids (typically, with 100–300 units in them) that, upon being synthesized in cells, fold into definite 3d-conformations.
(This happens essentially spontaneously in accordance with attraction/repulsion forces between residues; yet, no present day mathematical theory is able to fully account for the dynamics of protein folding, which is a “baby version” of embryogenesis.)
The resulting (properly) folded conformations, called proteins, perform most functions in cells, including the polypeptide synthesis itself—which is the most elaborate chemical process taking place in our Universe.
- 18.
This is recorded in Spalding’s short note “On instinct” [24]. His contribution to fundamental psychology was forgotten for years and revived relatively recently. It remains overshadowed by hordes of experiments, answering “profound questions” of the kind: What percentage of people would steal if certain of impunity? For something more amusing, see http://list25.com/25-intriguing-psychology-experiments/.
- 19.
It is unlikely that Nature tried and rejected “second moving,” “third unmoving”…
- 20.
“Constructive” means having a potential of being turned into a computer program that would function with an input that possesses the same (high) levels of diversity and (low) structural organization as what goes into the human brain.
- 21.
Supernovae seem very different from slow burning stars. They have enormous intensities of energy output, some as bright as 100 billion suns. And they are as rare in the skies as Ramanujans are on Earth—none was observed in our galaxy with 300 billion stars since October 9, 1604. Yet both processes depend on the same general principles of gravitation and nuclear fusion; probably about a billion stars in our galaxy will eventually explode as supernovae.
- 22.
We present a futuristic perspective on Freudian complexes in Sect. 6.7 of our Structures, Learning and Ergosystems [6].
- 23.
The philosopher does not describe any experiment verifying his idea.
- 24.
In his article “Maelzel’s Chess-Player” [20] about the fake chess playing machine invented by Wolfgang von Kempelen in 1769, Poe apparently refers to the Difference Engine described by Charles Babbage in 1822 rather than to the universal computer (Analytic Engine) proposed by Babbage in 1837 (99 years before Alan Turing ).
- 25.
This notwithstanding, Poe’s skepticism, which unlike the argument of Dreyfus was based on lucid thinking, can be justified: Poe clearly saw limitations of sequential computing devices available/imaginable in the nineteenth century.
- 26.
No algorithm can be efficiently applicable to all flows of signals; in fact, our framework of learning does not even admit the mathematical concept of unrestricted all.
- 27.
See also Jeff Hawkins’ lecture on the brain: https://www.youtube.com/watch?v=G6CVj5IQkzk.
- 28.
See references at the end of this text.
- 29.
Only recently, comparable general ideas were developed by people in the vision community.
- 30.
Reading some sections in this book requires a minimal prerequisite in molecular biology. Such a prerequisite, we believe, is also needed for understanding the nature of mathematics by mathematicians.
- 31.
- 32.
- 33.
Representations of these “bodies” by networks G can be characterized by their normalised connectivities that are the ratios:
[the first Betti number of G]/[the number of nodes in G].
- 34.
If you’ve already guessed how this should be designed, you must have heard about it and forgot.
- 35.
Isolating and/or creating such units is the main and often most difficult task faced by an ergo-system.
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Gromov, M. (2017). Math Currents in the Brain. In: Kossak, R., Ording, P. (eds) Simplicity: Ideals of Practice in Mathematics and the Arts. Mathematics, Culture, and the Arts. Springer, Cham. https://doi.org/10.1007/978-3-319-53385-8_9
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