Dear Howard, Stan, lists, I think Stan is right the line has both qualities - the geometric line is continuous and the arithmetic line is discontinuous. Some higher animals are capable of rudimentary mathematics (subitizing small numbers). But Howard, the claim that human brains do mathematics is not equivalent to the claim that mathematics must be part of the study of brain structures. Compare: television sets process TV series - therefore the study of TV series is the study of TV sets. Or: chemical industries produce pure H2SO4 --> the study of pure H2SO4 is the study of chemical industries. Is'nt the simplest reason why points and motions are processed in different areas of the human brain the fact that points and motions are different? Best F
Den 18/09/2014 kl. 15.57 skrev Stanley N Salthe <ssal...@binghamton.edu<mailto:ssal...@binghamton.edu>> : Howard -- Concerning the diremption between discreteness and continuity, isn't the real line an example of a model that combines both? That is, by picking out one number -- say, 5.765 -- we realize that it is infinitely continuous with smaller and larger numbers, and that our choice of 'significant' cutoffs was merely a practical matter. It seems to me that he real line must be a model combining continuity and discreteness IF we take into consideration the scale of observation. STAN On Thu, Sep 18, 2014 at 8:30 AM, Howard Pattee <hpat...@roadrunner.com<mailto:hpat...@roadrunner.com>> wrote: At 12:07 PM 9/17/2014, Frederik wrote: I think it follows from these observations [that MRI scans require mathematics] that it is a preposterous claim to say that mathematics is the study of neurological structures - or that mathematics could, in any way, be reduced to neuropsychology. HP: I agree that there are too many MRI papers, and that MRI images by themselves explain nothing. But also logic by itself explains nothing. I'll repeat my point that, like other primitive concepts, understanding mathematics requires complementary models. What is preposterous is to claim that any one view of the Foundations of Mathematics<http://en.wikipedia.org/wiki/Foundations_of_mathematics> is the only non-preposterous view. Preposterous views in physics are common, but accepted only if supported by experimental evidence. The evidence I know is that only human brains (and their artifacts) actually do mathematics. I have no evidence that inanimate nature does even simple mathematics. Human math began with geometry and numbers. No one has discovered a point or a triangle or a number, the infinite or the infinitesimal, in Nature. From Zeno and Aristotle (“That which moves does not move by counting”) to Cantor and Dedekind, discreteness and continuity have been impossible to combine in one logical model. Peirce spent many years trying to describe precisely how points could form a continuous line. Like Aristotle, he concluded that they couldn’t. We now believe(from MRI images) that concepts of discrete objects, like points, and concepts of continuous motions and structures, like lines, are formed in different regions of the brain, or at least by different neural codes. It all began 400 million years ago when the earliest sensorimotor controls based on vision required brains to distinguish discrete objects from their continuous motions. Howard
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