<HTML><BODY><p style="margin-top: 0px;" dir="ltr">Dear Krassimir,<br> first of all, thanks for reading all the paragraphs of our most difficult paper!<br> We are grateful to you!</p>
<p dir="ltr">Concerning the BUT (AND ITS NOVEL VARIANTS!) let's recapitulate:</p>
<p dir="ltr">Every feature is embedded in a structure. <br> The structure displays n-dimensions. <br> We call this feature: single description. <br> Single descriptions are points, or lines. <br> Single descriptions are perimeters, or areas. <br> Single descriptions are single points. <br> Single descriptions are functions, or vectors, or tensors. <br> Single descriptions are algorithms, or parameters. <br> Single descriptions are spatial patterns, or images. <br> An illumined surface is a single description. <br> Single descriptions are groups, or range of data. <br> Single descriptions are symbols, or signs. <br> Single descriptions are temporal patterns, or movements. <br> Single descriptions are particle trajectories, or paths. <br> Single descriptions are syntactic, or semantic, constructions. <br> Single descriptions are thermodynamic parameters, or signals. <br> A region is single description. <br> Single descriptions are strings.<br> Single descriptions project onto a n+1 structure. <br> Single descriptions stand for two descriptions with matching features on the n+1 <br> structure.<br> I call the two above matching features: matching description.<br><br></p>
<p dir="ltr">What does it mean? This means that the BUT does not describe just POINTS with matching description, but COUNTLESS other types of matching descriptions! <br> Therefore, it also describes a visual and an auditory inputs, if they come from the same environmental source (e.g., in the case of multisensory integration): this occurs for a MATHEMATICAL concept (not a qualitative, nor inaccurate, nor a metaphysical concept) coming from computational proximity, which is a branch of algebraic topology. </p>
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<p dir="ltr">--<br> Inviato da Libero Mail per Android</p>
</div>sabato, 26 novembre 2016, 10:12PM +01:00 da Krassimir Markov
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<p style="margin: 3pt 0cm 3pt 35.7pt;line-height: 15pt;" align="justify"><span style="font-family: ;line-height: 14pt"><font size="4">Dear Arturo, Gordana, Joseph, and FIS Colleagues,</font></span></p>
<p style="margin: 3pt 0cm 3pt 35.7pt;line-height: 15pt;" align="justify"><span style="font-family: ;line-height: 14pt"><font size="4">The key to our current discussion I found in the newest work of Arturo (I have read it before last letter of Arturo <img style="border-top-style: none;border-bottom-style: none;border-right-style: none;border-left-style: none" alt="Smile" src="cid:6330F6660C6049859032E1080AF689DF@VaioMarkov"> ):</font></span></p>
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<p style="margin: 3pt 0cm 3pt 35.7pt;line-height: 15pt;" align="justify"><b><span style="font-family: ;line-height: 14pt"><font size="4">A TOPOLOGICAL/ECOLOGICAL APPROACH TO PERCEPTION</font></span></b></p>
<p style="margin: 3pt 0cm 3pt 35.7pt;line-height: 15pt;" align="justify"><span style="font-family: ;line-height: 14pt"><font face="Times New Roman"><a href="http://biorxiv.org/content/biorxiv/early/2016/11/11/086827.full.pdf" target="_blank"><font size="4" face="Calibri">http://biorxiv.org/content/biorxiv/early/2016/11/11/086827.full.pdf</font></a> </font></span></p>
<p style="margin: 3pt 0cm 3pt 35.7pt;line-height: 15pt;" align="justify"><span style="font-family: ;line-height: 14pt"><font size="4">or</font></span></p>
<p style="margin: 3pt 0cm 3pt 35.7pt;line-height: 15pt;" align="justify"><span style="font-family: ;line-height: 14pt"><a href="https://www.researchgate.net/profile/Arturo_Tozzi/publication/310006296_A_TOPOLOGICALECOLOGICAL_APPROACH_TO_PERCEPTION/links/5827617808ae254c50832922.pdf?origin=publication_list" target="_blank"><font size="4">https://www.researchgate.net/profile/Arturo_Tozzi/publication/310006296_A_TOPOLOGICALECOLOGICAL_APPROACH_TO_PERCEPTION/links/5827617808ae254c50832922.pdf?origin=publication_list</font></a></span></p>
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<font size="4" face="Calibri">What is important is that there exist a non correct using of the topological theory (concretely the BUT).</font>
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<font size="4" face="Calibri">It is taken as an idea to explain the perception when different stimulus create the same meaning in the consciousness.</font>
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<font size="4" face="Calibri">See the example with ambulance of Figure 5a (visual and sound stimulus) which is connected to the same meaning on Figure 5b (single point).</font>
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<font size="4" face="Calibri">But !!!</font>
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<font size="4" face="Calibri">BUT explicitly proof that (citation from the Arturo’s paper):</font>
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<font size="4" face="Calibri">BUT states that, if a single point on a circumference projects to a higher spatial dimension, it gives rise to two antipodal points with matching description on a sphere, and vice versa (<b>Figure 1A</b>) (Borsuk, 1933; Marsaglia, 1972; Matoušek, 2003; Beyer, 2004). This means that the two antipodal points are assessed at one level of observation in terms of description, while a single point is assessed at a lower level (Tozzi 2016b), i.e., point location vs. point description. Points on a sphere are “antipodal”, provided they are diametrically opposite (Henderson, 1996). </font>
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<font size="4" face="Calibri">Examples of antipodal points are the poles of a sphere. This means, e.g., <strong>that there exist on the earth surface at least two antipodal points with the same temperature and pressure</strong>. BUT looks like a translucent glass sphere between a light source and our eyes: we watch two lights on the sphere surface instead of one. But the two lights are not just images, they are also real with observable properties, such as intensity and diameter.</font>
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<font size="5" face="Calibri"><strong>i.e. the antipodal points have the same characteristics !!!</strong></font>
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<font size="4" face="Calibri"><strong>This is not valid for the sound and vision with the same meaning!</strong></font>
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<font size="4" face="Calibri">Nevertheless, Arturo wrote <strong>very important conclusion</strong> (citation):</font>
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<p style="margin: 0cm 0cm 0pt;line-height: normal;text-autospace: ;text-indent: 0cm;mso-layout-grid-align: none" align="left"><font face="Calibri"><font size="4"><span>Gibson’s work strengthens and brings to the front the <strong>primary question of “what” is perceived</strong>, before questions of </span><span style="font-family: ;mso-fareast-font-family: calibri;mso-fareast-theme-font: minor-latin;mso-ansi-language: en-us;mso-fareast-language: en-us;mso-bidi-language: ar-sa">mechanisms and material implementation are introduced (Rao et al., 1997). </span></font></font></p>
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<p style="margin: 0cm 0cm 0pt;line-height: normal;text-autospace: ;text-indent: 0cm;mso-layout-grid-align: none" align="left"><font size="4" face="Calibri">Finally, I like the conclusion. My remark is to be more precise when we use mathematical theoretical results.</font></p>
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<font size="4" face="Calibri">Friendly regards</font>
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<font size="4" face="Calibri">Krassimir</font>
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<b>From:</b>
<a title="joe.brenner@bluewin.ch" href="//e-aj.my.com/compose/?mailto=mailto%3ajoe.brenner@bluewin.ch" target="_blank">Joseph Brenner</a>
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<b>Sent:</b> Saturday, November 26, 2016 10:05 PM
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<b>To:</b>
<a title="tozziarturo@libero.it" href="//e-aj.my.com/compose/?mailto=mailto%3atozziarturo@libero.it" target="_blank">tozziarturo@libero.it</a> ;
<a title="fis@listas.unizar.es" href="//e-aj.my.com/compose/?mailto=mailto%3afis@listas.unizar.es" target="_blank">fis</a>
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<b>Subject:</b> Re: [Fis] Who may prove that consciousness is an Euclidean n-space???
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<font size="2" face="Arial">Dear FISers,</font>
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<font size="2" face="Arial">At the risk of attracting the anger of all the mathematicians in the group, I will agree with Arturo, <em>contra </em>Krassimir. For a non-mathematician like me, a description of complex dynamic processes such as consciousness and information can be partly mathematical but need not involve proofs and their reduced logic.</font>
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<font size="2" face="Arial">The question I have is whether the field description is itself necessary and sufficient and if incomplete, what is missing. Perhaps it is my intuition that consciousness is both continuous and discontinuous, and so is its opposite, unconsciousness, which still involves high-level nervous functions. In my picture, antipodal points are of little relevance compared to the non-Euclidean multi-dimensionality of this dynamic opposition, </font>
<font size="2" face="Arial">moving between identity and diversity, presence and absence, clarity and vagueness, symmetry and dissymetry, within the same high overall energy level. In any case, perhaps we can agree that everything that is moving here is information!</font>
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<font size="2" face="Arial">Thank you and best wishes,</font>
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<font size="2" face="Arial">Joseph</font>
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<a title="fis@listas.unizar.es" href="//e-aj.my.com/compose/?mailto=mailto%3afis@listas.unizar.es" target="_blank">fis</a>
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<b>Sent:</b> Saturday, November 26, 2016 7:06 PM
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<b>Subject:</b> Re: [Fis] Who may prove that consciousness is an Euclidean n-space ???
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<p style="margin-top: 0px" dir="ltr">Dear Krassimir, <br>Thanks a lot for your question, now the discussion will become hotter!</p>
<p dir="ltr">First of all, we never stated that consciousness lies either on a n-sphere or on an Euclidean n-space.<br>Indeed, in our framework, consciousness IS the continuous function. <br>Such function stands for a gauge field that restores the brain symmetries, broken by sensations. <br>Concerning brain and gauge fields, see my PLOS biology paper: <br><a href="http://journals.plos.org/plosbiology/article?id=10.1371%2Fjournal.pbio.1002400" target="_blank">http://journals.plos.org/plosbiology/article?id=10.1371%2Fjournal.pbio.1002400</a></p>
<p dir="ltr">When consciousness lacks, the inter-dimensional projections are broken, and the nervous higher functions temporarily disappear. </p>
<p dir="ltr">Concerning the question about which are the manifolds where brain functions lie, it does not matter whether they are spheres, or circles, or concave, or flat structures: we demonstrated that the BUT is valid not just for convex manifolds, but for all the kinds of manifolds. <br>See our: <br><a href="http://onlinelibrary.wiley.com/doi/10.1002/jnr.23720/abstract?userIsAuthenticated=false&deniedAccessCustomisedMessage" target="_blank">http://onlinelibrary.wiley.com/doi/10.1002/jnr.23720/abstract?userIsAuthenticated=false&deniedAccessCustomisedMessage</a>=<br></p>
<p dir="ltr">Therefore, even if you think that brain and biological functions are trajectories moving on concave structures towards lesser energetic levels, as suggested by, e.g., Fokker-Planck equations, it does not matter: you may always find the antipodal points with matching description predicted by BUT. </p>
<p dir="ltr">Ciao!</p>
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<p dir="ltr">--<br>Inviato da Libero Mail per Android</p>
</div>sabato, 26 novembre 2016, 06:23PM +01:00 da Krassimir Markov
<a href="//e-aj.my.com/compose/?mailto=mailto%3amarkov@foibg.com" target="_blank">markov@foibg.com</a>:
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<p><font size="4" face="Times New Roman">Dear FIS colleagues,</font></p>
<p><font size="4" face="Times New Roman">I think, it is needed to put discussion on mathematical foundation. Let me remember that:</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font face="Times New Roman"><font size="4">The <b>Borsuk–Ulam theorem</b> (BUT), states that every </font><a title="Continuous function" style="href: 'https://en.wikipedia.org/wiki/Continuous_function'"><strong><font size="4">continuous function</font></strong></a><font size="4"> from an </font><a title="N-sphere" style="href: 'https://en.wikipedia.org/wiki/N-sphere'"><font size="4"><i>n</i>-sphere</font></a><font size="4"> into </font><a title="Euclidean space" style="href: 'https://en.wikipedia.org/wiki/Euclidean_space'"><strong><font size="4">Euclidean <i>n</i>-space</font></strong></a><font size="4"> maps some pair of </font><a title="Antipodal point" style="href: 'https://en.wikipedia.org/wiki/Antipodal_point'"><font size="4">antipodal points</font></a><font size="4"> to the same point. </font></font></p>
<p><font size="4" face="Times New Roman">Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.</font></p>
<p><font size="4" face="Times New Roman">Formally: <strong>if</strong> <span><span style="display: none">f : <em>S<sup> n</sup> → R</em> <sup>n</sup> </span></span> <strong>is</strong> <strong>continuous</strong> then there exists an <span><span style="display: none">x ∈ S n </span></span>such that: <span><span style="display: none">f ( − x ) = f ( x )</span></span>.</font></p>
<p><font size="4" face="Times New Roman">[ </font><a title="https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem" href="https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem" target="_blank"><font size="4" face="Times New Roman">https://en.wikipedia.org/wiki/Borsuk%E2%80%93Ulam_theorem</font></a><font size="4" face="Times New Roman"> ] </font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font face="Times New Roman"><font size="4">Who may proof that consciousness is a </font><a title="Continuous function" style="href: 'https://en.wikipedia.org/wiki/Continuous_function'"><font size="4"><strong>continuous function</strong></font></a><font size="4"> from reflected reality ???</font></font></p>
<p><font face="Times New Roman"><font size="4">Who may proof that consciousness is an </font><a title="Euclidean space" style="href: 'https://en.wikipedia.org/wiki/Euclidean_space'"><font size="4"><strong>Euclidean <i>n</i>-space</strong></font></a><font size="4"> ???</font></font></p>
<p><font size="4" face="Times New Roman">After proving these statements we may think further.</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman">Yes, discussion is interesting but, I am afraid, it is not so scientific.</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
<p><font size="4" face="Times New Roman">Friendly regards</font></p>
<p><font size="4" face="Times New Roman">Krassimir</font></p>
<p><font size="4" face="Times New Roman"></font> </p>
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