<div>Dear FISers:</div><div><br><p class="MsoNormalCxSpFirst" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">Recently introduced versions of the Borsuk-Ulam theorem (BUT) state that
a feature on a n-manifold projects to two features with matching description
onto a n+1 manifold (Peters, 2016; Peters and Tozzi, 2016a, 2016b; Tozzi, 2016;
Tozzi and Peters, 2016a, 2016b).
Starting from this rather simple “abstract” claim, a fruitful general
framework has been built, able to elucidate disparate “real” physical and
biological phenomena, from quantum entanglement (Peters and Tozzi, 2016c), to
brain activity (Peters et al., 2016, 2017; Tozzi and Peters, 2016a, 2016b,
2017), from gauge theories (Tozzi et al., 2017) to pre- big bang scenarios
(Tozzi and Peters, 2016c). Summarizing
this novel topological approach, we may state what follows: if you take into
account projections among functional or real dimensions, you achieve a system
of mappings that fit very well with experimental results and are able to assess
countless systems in far-flung scientific branches. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">One of the main concerns of such a topological approach to systems
features is that it talks in rather general terms, leaving apart the peculiar
features of individuals and of single physical and biological processes. </span></p><p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><br></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">In order to tackle this issue, here we ask: what does it mean “matching
description”? In a topological framework, matching descriptions are termed
"descriptively near sets", i.e., two (or more) features that lie on
the same manifold, but that have no points in common. In a semantic matching framework, a matching
description encompasses all information about the matching process. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">Then we ask: has matching description anything to do with “identity”? In the “classical” BUT, the matching features
are just points, therefore a point is equal to another, and we might easily
state that the two points are “identical”. On the other hand, in the novel BUT
variants, the matching features stand not just for simple topological points,
but also for for more complicated structures, such as shapes of space (spatial
patterns), of shapes of time (temporal patterns), vectors or tensors,
functions, or signals, thermodynamic parameters, movements, trajectories, or
lexical structures (either syntactic or semantic), or general symmetries. </span><span style="font-size:10.0pt;
mso-bidi-font-size:11.0pt;line-height:107%;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US"><o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">Therefore, we ask: are two matching features identical? Do they stand
for the same feature, of for two different features with something in
common? In order to solve the issue, we
“steal” the Martin Heidegger's noteworthy account of the “principle of identity”
(Heidegger, 1957), one of the three tenets of the classical logic.<o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">The principle of identity states that A=A. The formula expresses, in its usual
description, an equality of A and A. One
A is equal to another A. A is therefore
the same of A, because “identical” (from Greek and Latin) means: “the same”. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"> <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">However, in another possible version, the formula A=A speaks of “equality”. A is A.
It does not say that A is the same, but that every A is itself the
same. Or, in other words, each thing
itself is the same for itself with itself.
<o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">It can also be stated that matching description “belongs to” an
identity. In this case, sameness is
interpreted as a “belonging together”. This
means that two interpretations are feasible: a) matching description is
determined by an identity as a feature of that identity; b) identity is
represented as a feature of matching description. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">In “belonging together”, the world “together” means: to be assigned and
placed into the order of a together, to be established in the unity of a
manifold, to be combined into the unity of a system. Such assignement and placing occur thanks to
connexions and mappings of the one with the other. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">However, “belonging together” can also mean: the together is now
determined by the belonging. </span><span style="font-size:10.0pt;mso-bidi-font-size:11.0pt;line-height:107%;font-family:
"Times New Roman",serif;mso-ansi-language:EN-US"><o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">Therefore, the possibilities here are two: a) representing belonging in
terms of the unit of together; b) experiencing this together in terms of
belonging. </span><span style="font-size:10.0pt;mso-bidi-font-size:11.0pt;
line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:EN-US"><o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;mso-bidi-font-size:11.0pt;line-height:107%;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">The issue b) leads us into the psychological standpoint of the
observer. Indeed, “thinking” and
matching description can also be thought as the same, so that thinking and
matching description belong together in the same, and by virtue of the
same. If we attempt to represent
together of thinking and matching description as a coordination, we can
establish and explain this coordination either in terms of thinking and
matching description. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">If thinking and matching description belong to each other, matching
description belongs with thinking to an identity, whose active essence stems
from that “letting belong together” which we call “mental representation”. Identity becomes, in this version, a
functional property of the event of mental representation. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">In sum, identity can be
presupposed as a feature of the matching description, or as a spring that
departs from matching description. In this second account, the principle of
identity becomes a spring into the psychological origin of identity. We can therefore assess matching description
and thinking in terms of that which joins the two, by virtue of the event of
mental representation.<o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;mso-bidi-font-size:11.0pt;line-height:107%;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">In conclusion, the concept of matching description
displays the widest range of possible uses.
In particular, matching description do not assess just “the same” thing,
but also things that are “different”.
This implementation makes the BUT and its variants not just the
standpoint for a novel interpretation of almost all the biological and physical
phenomena, but also a suitable tool in order to evaluate the slight (objective
and subjective) differences that make our world an astonishing realm of rich
heterogeneity. <o:p></o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><b><span style="font-size:10.0pt;line-height:107%;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">REFERENCES<o:p></o:p></span></b></p>
<p class="MsoNormalCxSpMiddle" style="text-align:justify"><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US"><o:p> </o:p></span></p>
<p class="MsoNormalCxSpMiddle" style="margin-left:36.0pt;mso-add-space:auto;
text-align:justify;text-indent:-18.0pt;mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;line-height:107%;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">1)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";"> </span></span><!--[endif]--><span style="font-size:10.0pt;line-height:107%;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Heidegger M. 1957.
Identity and difference. Harper
& Row, Publishers (New York, Evanston, and London). <o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:36.0pt;text-align:justify;text-indent:
-18.0pt;mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:
10.0pt;line-height:107%;font-family:"Times New Roman",serif;mso-ansi-language:
EN-US">2)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;line-height:107%;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">Peters JF. 2016.
Computational Proximity. Excursions in the Topology of Digital Images. Edited
by Intelligent Systems Reference Library. Berlin: Springer-Verlag.
doi:10.1007/978-3-319-30262-1.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpFirst" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">3)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Peters JF, Tozzi A. 2016a. Region-Based Borsuk-Ulam Theorem. arXiv.1605.02987.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">4)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Peters JF, Tozzi A.
2016b. String-Based Borsuk-Ulam
Theorem. arXiv:1606.04031.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">5)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Peters JF, Tozzi A. 2016c. Quantum Entanglement on a Hypersphere. Int J Theoret Phys, 1–8.
doi:10.1007/s10773-016-2998-7.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">6)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Peters JF, Tozzi A. Ramanna S. 2016. Brain Tissue Tessellation Shows Absence of
Canonical Microcircuits. Neuroscience
Letters 626: 99–105. doi:10.1016/j.neulet.2016.03.052.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">7)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Peters JF, Ramanna S, Tozzi A, Inan E. 2017.
Frontiers Hum Neurosci.
BOLD-independent computational entropy assesses functional donut-like
structures in brain fMRI image. doi:
10.3389/fnhum.2017.00038. <o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">8)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Tozzi A. 2016.
Borsuk-Ulam Theorem Extended to Hyperbolic Spaces. In Computational Proximity. Excursions in the
Topology of Digital Images, edited by J F Peters, 169–171.
doi:10.1007/978-3-319-30262-1.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">9)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Tozzi A, Peters JF. 2016a. Towards a Fourth Spatial Dimension of Brain
Activity. Cognitive Neurodynamics 10
(3): 189–199. doi:10.1007/s11571-016-9379-z.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">10)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Tozzi A, Peters JF. 2016b. A Topological Approach Unveils System
Invariances and Broken Symmetries in the Brain.
Journal of Neuroscience Research 94 (5): 351–65. doi:10.1002/jnr.23720.<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">11)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Tozzi A, Peters JF.
2016c. Symmetries, Information
and Monster Groups before and after the Big Bang. Information, 7(4), 73; doi:10.3390/info7040073.
<o:p></o:p></span></p>
<p class="MsoListParagraphCxSpMiddle" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">12)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Tozzi A, Peters JF. 2017. A Symmetric Approach Elucidates Multisensory
Information Integration. Information
8,1. doi: 10.3390/info8010004. <o:p></o:p></span></p>
<p class="MsoListParagraphCxSpLast" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><!--[if !supportLists]--><span style="font-size:10.0pt;
font-family:"Times New Roman",serif;mso-ansi-language:EN-US">13)<span style="font-variant-numeric: normal; font-stretch: normal; font-size: 7pt; line-height: normal; font-family: "Times New Roman";">
</span></span><!--[endif]--><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US">Tozzi A, Sengupta B, Peters JF, Friston KJ. 2017.
“Gauge Fields in the Central Nervous System.” In: The Physics of the Mind and
Brain Disorders: Integrated Neural Circuits Supporting the Emergence of Mind,
edited by Opris J and Casanova MF. New York, Springer; Series in Cognitive and
Neural Systems. <o:p></o:p></span></p><p class="MsoListParagraphCxSpLast" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US"><br></span></p><p class="MsoListParagraphCxSpLast" style="text-align:justify;text-indent:-18.0pt;
mso-list:l0 level1 lfo1"><span style="font-size:10.0pt;font-family:"Times New Roman",serif;
mso-ansi-language:EN-US"><br></span></p><p class="MsoNormalCxSpFirst" style="margin-top:12.0pt;margin-right:0cm;
margin-bottom:12.0pt;margin-left:0cm;mso-add-space:auto;text-align:justify;
line-height:115%;mso-layout-grid-align:none;text-autospace:none"><span style="font-size: 14px; line-height: normal; text-align: start;"><font face="courier new, monospace"><b>Arturo Tozzi</b></font></span></p><p class="MsoNormalCxSpFirst" style="margin-top:12.0pt;margin-right:0cm;
margin-bottom:12.0pt;margin-left:0cm;mso-add-space:auto;text-align:justify;
line-height:115%;mso-layout-grid-align:none;text-autospace:none"><span style="line-height: 115%;"><font face="courier new, monospace">AA Professor Physics, University North Texas</font></span></p><p class="MsoNormalCxSpFirst" style="margin-top:12.0pt;margin-right:0cm;
margin-bottom:12.0pt;margin-left:0cm;mso-add-space:auto;text-align:justify;
line-height:115%;mso-layout-grid-align:none;text-autospace:none"><span style="font-size: 14px; line-height: normal; text-align: start;"><font face="courier new, monospace">Pediatrician ASL Na2Nord, Italy</font></span></p><p class="MsoNormalCxSpFirst" style="margin-top:12.0pt;margin-right:0cm;
margin-bottom:12.0pt;margin-left:0cm;mso-add-space:auto;text-align:justify;
line-height:115%;mso-layout-grid-align:none;text-autospace:none"><span style="font-size: 14px; line-height: normal; text-align: start;"><font face="courier new, monospace">Comput Intell Lab, University Manitoba</font></span></p><p class="MsoNormalCxSpFirst" style="margin-top:12.0pt;margin-right:0cm;
margin-bottom:12.0pt;margin-left:0cm;mso-add-space:auto;text-align:justify;
line-height:115%;mso-layout-grid-align:none;text-autospace:none"><font face="courier new, monospace"><a href="http://arturotozzi.webnode.it/" style="font-size: 14px; color: rgb(5, 68, 126); line-height: normal; text-align: start;">http://arturotozzi.webnode.it/</a><span style="font-size: 14px; line-height: normal; text-align: start;"> </span></font><br></p></div><br>