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Footnote 25 addendum: mathematical performance
J. Halberda et al.,
K. Number sense across the lifespan as revealed by a massive Internet-based sample
PNAS 109 11116 (2012)
Population trends suggest that the precision of one’s number sense improves throughout the schoolage years, peaking quite late at ca 30 y. Despite this gradual developmental
improvement, there are very large individual differences in number sense precision among people of the same age, and these differences relate to school mathematical performance throughout adolescence and the adult years.
Footnote 27 addendum: depth of observation records
M. Gelfand, “We Do Not Choose Mathematics as Our Profession, It Chooses Us:
Interview with Yuri Manin,” Notices AMS 56 1268 (2009)
Manin says:
Here I take a position that sets me apart from many good colleagues. I’ve heard many arguments against me on this subject. I must explain to you how I imagine mathematics. I am an emotional Platonist (not a rational one: there are no rational arguments in favor of Platonism). Somehow or other, for me mathematical research is a discovery, not an invention. I imagine for myself a great castle, or something like that, and you gradually start seeing its contours through the deep mist, and begin to investigate something. How you formulate what it is you’ve seen depends on your type of thinking and on the scale of what you have seen, and on the social circumstances around you, and so on.
The article is highly recommended.
Footnote 28 On the difficulty of expression by language
It was written there “ whether something cannot be expressed in language or not can be recognized only after a sizable amount of efforts ...,” but this statement suggests that this can be solved with proper investment of efforts. However, we should clearly recognize that important progress in culture and science lies in the aspect that efforts cannot help [believing in the optimistic idea that efforts can solve everything has something akin to believing in the idea that complex systems can be formed by self-organization alone. Both believe that something qualitatively different can be accomplished by simple accumulation of ordinary things.
To state explicitly in language something everybody seems to know intuitively is a great cultural contribution.
Nonlinguistic judgement
The following paper suggests that overall nonlinguistic judgment is often more accurate or more advantageous than that relying on analytical use of language :
[A. Dijksterhuis, M. W. Bos, L. F. Nordgren, and R. B. van Baaren, “On making the right choice: the deliberation-without-attention effect,” Science 311 , 1005 (2006)].
Implicit cognition
Although it is well known for a long time, it is quite suggestive that explicit and implicit cognitions are distinct. That is, even if one is not conscious about one’s knowing something, one’s body recognizes that (e.g., the skin conductivity changes) . See, for example, D. E. Haydn and M. B. Lewis, “Capgras delusion: a window on face recognition,” Trends in Cognitive Sci. 5 , 149 (2001).
“It was my heart that followed my body”--- Lady Murasaki
“It was my heart that followed my body.”
The core of this statement is the deep emotion that one feels when she recognizes that what is assumed as the `lower level’ is actually the substantial master of the `upper level’, and that the governing by the `upper level’ is a mere illusion.
Free will is quite likely to be an illusion, which, however, contributes decisively and effectively our lives by giving us the will to live. See, for example,
Ruud Custers and Henk Aarts
The Unconscious Will: How the Pursuit of Goals Operates Outside of Conscious Awareness
Science 329 47 (2010).
We are likely to say, if asked, that the decision to act produced the actions themselves. Recent discoveries, however, challenge this causal status of conscious will. Actions are initiated even though we are unconscious of the goals to be attained or their motivating effect on our behavior.
Anthony R. Cashmore
Reply to Hinsen: Free will, vitalism, and distinguishing cause from effect
Proc Natl Acad Sci 107 E150 (2010)
Many biologists are more than willing to question religion, yet relatively few show similar skepticism concerning the question of free will (even Dawkins). Biologists are fooled by correlation between thought processes and behavior, mistaking correlation as causality. The reality is that in this instance, the process of evolution has conned us into believing in free will.
Why then do we continue to base our judicial system on a belief in the existence
of free will?