I am not sure what the need for a mathematical basis is for something that has no mathematical basis to begin with and I have no idea where to find that if it exists which, in theory, it probably doesn't because it can't. But you can ask for it. I suggest Schiff's book (as above) which is also mentioned in the following review. Again, for the I dont know how many times I have said this, this discussion is about considering the explanations and possibilities and not closing ourselves to science we have such a problem understanding that we do not even wish to acknowledge its existence.
If we were to call a spade a spade, here is what we are dealing with. The following two paragraphs are snipped from a much longer unsigned reportage article (not copywritten as far as I can see).
http://mathpost.la.asu.edu/~boerner/suppression.html
Intimidate University officials into investigating their "black sheep" for fraud. Withdraw funding and staff. Undermine the professional credibility of the scientists using any means necessary, including, but not limited to, reviews that seriously distort the original research, sham replication of the experiment in question (which is then immediately published by the same journals that refused, or were extremely reluctant, to publish the original research), and, ultimately, slander and character assassination.
(SG note: or uncontrolled replication on television - a single series of 40 attempts versus thousands done in a reputable
medical school lab by a skeptic of homeopathy, e.g. Prof M. Ennis).
All of those things happened to the prestigious French immunologist Jacques Benveniste who proved experimentally that water can "remember" an active agent that has been diluted so much that not a single molecule remains. In years of diligent, careful experimental work between 1985 and 1989, Benveniste's team at the French National Institute for Health and Medical Research (INSERM) showed that sub-molecular dilutions of anti-immunoglobulin E (aIgE) can affect the staining properties of basophils in the same fashion as actual aIgE- thus validating the basic tenet of homeopathy. Extremely reluctantly, Nature editor John Maddox published the research, but accompanied it by a scathing editorial that expressed his belief that the results had to be incorrect. The paper set off a firestorm of scientific bigotry. The Church Of Science decided that scientific results such as these were a heresy of the highest order, and did not rest until Benveniste was deprived of his professional credibility, career and funding. Nature was eager to publish a subsequent unsuccessful replication, but refused to publish a successful one. The paradigm shift was thus averted, and the high priests of the Church Of Science remain safe in their smug, self-deluded belief that they know everything there is to know about nature. Michel Schiff, a scientist at the French National Center for Scientific Research, has documented the details of this scientific witch hunt in his book The Memory of Water - Homeopathy and the Battle of Ideas in the New Science.
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We also need to come to grips with the following where homeopathy may be concerned:
QUANTUM SMARANDACHE PARADOXES
edited by Gheorghe Niculescu
Str. Muncii, Bl. B1, Ap. 50
2698 Uricani, Hunedoara, Romania
Abstract. In this paper one presents four of the smarandacheian paradoxes in physics found in various physics sites or printed material.
1) Sorites Paradox (associated with Eubulides of Miletus (fourth century B.C.):
Our visible world is composed of a totality of invisible particles.
a) An invisible particle does not form a visible object, nor do two invisible particles, three invisible particles, etc.
However, at some point, the collection of invisible particles becomes large enough to form a visible object, but there is apparently no definite point where this occurs.
b) A similar paradox is developed in an opposite direction. It is always possible to remove a particle from an object in such a way that what is left is still a visible object. However, repeating and repeating this process, at some point, the visible object is decomposed so that the left part becomes invisible, but there is no definite point where this occurs.
Generally, between <A> and <Non-A> there is no clear distinction, no exact frontier. Where does <A> really end and <Non-A> begin? One extends Zadeh's "fuzzy set" term to the "neutrosophic set" concept.
2) Uncertainty Paradox: Large matter, which is under the 'determinist principle', is formed by a totality of elementary particles, which are under Heisenberg's 'indeterminacy principle'.
3) Unstable Paradox: Stable matter is formed by unstable elementary particles (elementary particles decay when free).
4) Short Time Living Paradox: Long time living matter is formed by very short time living elementary particles.
References:
[1] Marie-Helene Boyer, "Re: How are possible the Smarandache Uncertainty, Unstable, etc. Paradoxes?", MAD Scientist, Washington University School of Medicine, St. Louis, Missouri,
http://www.madsci.org/posts/archives/972501333.Ph.r.html.
[2] Chong Hu, "How are possible the Smarandache Uncertainty, Unstable, etc. Paradoxes?", MAD Scientist, Washington University School of Medicine, St. Louis, Missouri,
http://www.madsci.org/posts/archives/972501333.Ph.q.html.
[3] Chong Hu, "How do you explain the Smarandache Sorites Paradox?", MAD Scientist, Washington University School of Medicine, St. Louis, Missouri,
http://www.madsci.org/posts/archives/970594003.Ph.q.html.
[4] Amber Iler, "Re: How do you explain the Smarandache Sorites Paradox?", MAD Scientist, Washington University School of Medicine, St. Louis, Missouri,
http://www.madsci.org/posts/archives/970594003.Ph.r.html.
[5] Leonardo Motta, editor, "A Look at the Smarandache Sorites Paradox", Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics,
University of Craiova, Craiova, Romania, December 21 - 24, 2000;
In the web site at York University, Canada,
http://at.yorku.ca/cgi-bin/amca/caft-04.
[6] Gheorghe Niculescu, editor, "On Quantum Smarandache Paradoxes", Second International Conference on Smarandache Type Notions In Mathematics and Quantum Physics,
University of Craiova, Craiova, Romania, December 21 - 24, 2000;
In the web site at York University, Canada,
http://at.yorku.ca/cgi-bin/amca/caft-20.
[7] Florentin Smarandache, "Invisible Paradox" in "Neutrosophy. / Neutrosophic Probability, Set, and Logic", American Research Press, Rehoboth, 22-23, 1998.
[8] Florentin Smarandache, "Sorites Paradoxes", in "Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry", Xiquan Publishing House, Phoenix, 69-70, 2000.
[9] Louisiana Smith and Rachael Clanton, advisor Keith G. Calkins, "Paradoxes" project, Andrews University,
http://www.andrews.edu/~calkins/math/biograph/topparad.htm.