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AISSQ-11




MATHEMATICS
& REALITY


6-7 October 2018
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Science Facing The Spiritual Traditions

Pier Luigi Luisi

I will focus on the question “what is life?”, emphasizing both the autopoiesis theory, as well as the systems view of life . One first important foundation of established, classic science, is that all what we have in our world, including all forms of life and their expression, is due to natural causes, namely without intervention or help from transcendent forces. This does not mean that all scientists are atheist, but that for them, even if believers, the dominion of science is well separated from the dominion of personal faith. The main pillar of traditional science is the following: that all what we have in the world is due to particles and their mutual interactions. Two additional foundations in large part eliminates the strong reductionist and mechanistic flavour of this first one. One is systems thinking. Accordingly, the properties and the understanding of any complex system formed by many components (an organism, a family, a social body, a machine…), are given by the totality of the interactions of the components, and not by the properties of the single isolated components. You cannot study and understand life by studying one single isolated component of life, be DNA or a lipid, at a time-or the separate wing of a butterfly. The additional general principle is the notion of emergence: accordingly, in a complex system formed by many parts, novel “emergent” properties arise due to their mutual interactions, novel in the sense that they are not present in the single parts. Thus, it is generally assumed that also qualitative properties, like the feelings or even the thinking in human and other animals, arise from a material basis –for example the brain tissues, or the complex structure of the entire organism. Even consciousness is generally seen as an emergent property of the brain. Obviously, all this creates a large body of discussion and possible controversy within the science campus, but in particular with the religious view, both in the case of monotheistic religions and in the case of Hinduism and Buddhism. That reality is based and conditioned by the mutual interaction of the parts is present of course in the old Vedic thinking, and is the very base of Buddhism. But already the notion of emergence-the arising from matter of completely new qualities, like life, mind, the consciousness, creates some problem with the classic science view. Then I will analyse more in detail the question “what life is” based on Maturana and Varela’ work, chosen also because Francisco Varela’ life has been strongly influenced by the Dalai lama and the Buddhist-Tibetan Mind and Life institute. In particular, then, I will be emphasizing Maturana's notion of cognition (all living organism are cognitive, including bacteria); and the concept that the living organism is a system which is thermodynamically open, but operationally closed. Since each organism “sees” the world from within its own closed organization, there are as many worlds as observers. This puts a question mark on the notion of objectivity and highlights the notion of a multiverse of existing realities. All this is still a highly debated subject in modern science, as it also links to the problematic relation between the observer and the object of study.

Convergence in the Philosophy of Mathematics

Edward N. Zalta

The Platonist answer to the question, & What is mathematical language about?& , is that it is about abstract individuals (such as zero, the null set, omega, etc.) and abstract relations (such successor, less than, set membership, group addition, etc.). One way to give a metaphysical foundation for mathematics is to systematize this answer with an axiomatic theory of abstract individuals and abstract relations and an analysis of mathematical language that yields denotations for the terms of mathematical theories and truth conditions for mathematical claims. I & ll review such a theory and then show that the background formalism for expressing the theory is subject to various interpretations. The Platonist interpretation is just one of the many ways of interpreting the formalism and the analysis of mathematics. I & ll show that one can develop fictionalist, structuralist, inferentialist, if-thenist, finitist, and logicist interpretations of the formalism. Since each interpretation offers us a clear, but different, answer to our initial question, the resulting analysis not only offers a way to make these philosophies of mathematics more precise, but also unifies them in a new and unsuspected way.

On Some Remarkable Contributions of Bhaskaracharya

Bhagwati Prasad Chamola

The famous Indian scholar Bhaskaracharya wrote the mathematical treatise “Lilavati” in 12th century as one of the four parts of his great work Siddhant Shiromani. It is renowned for its lucid and poetic presentation. A number of commentaries have been written in English, Sanskrit, Malayalam and many other languages on the text of Lilavati. It contains formulae, theorems, methods and short cuts without proofs for solving various problems related to algebra, arithmetic, mensuration, geometry and related areas. The problems are written in poetic style with direct connection to day to day life and refereeing to the animals and plants around us. The techniques proposed in Lilavati are quite interesting and motivating even to a new student of mathematics. In the present talk, many remarkable concepts of his work with illustrative examples will be discussed along with their significance in the present scenario..

Mathematics and Ultimate Reality in Dutich intuitionism and Marburg Neo- Kantianism

Luc Bergmans

Luitzen Egbertus Jan Brouwer (1881-1966), the founder of Dutch intuitionism in mathematics and the philosophy of mathematics, was a deeply religious man, whose ideas on Ultimate Reality were influenced by Eastern and Western traditions of spirituality and mysticism. In his doctoral dissertation Over de grondslagen der wiskunde (On the foundations of mathematics, 1907) Brouwer boldly characterized mathematics as a free creation. Who is the creator of mathematics, the founder of its truth ? According to Brouwer, it is the mathematician himself, or rather the inner Self of the mathematician, which resists any attempt at objectification. As Brouwer puts it, the foundations of mathematics are not about the psychology of mathematics, which is sometimes mistakenly perceived as capable of providing the standards of mathematical truth through the observation of mathematical practice. To the intuitionist, psychology presupposes mathematics, and mathematics presupposes the workings of the inner Self. Brouwer was a soul mate of Frederik van Eeden (1860-1932), the author of the philosophical poem het Lied van schijn en wezen (The song of appearance and essence, part I, 1901). In this work, images belonging to the Vedic tradition as well as new and original metaphors illustrated the idea of the Self residing in a high retreat beyond all attempts at grasping it with the human mind. Both Brouwer and Van Eeden were members of the Dutch semiotic movement Signifika, which promoted the critical analysis of meaning in philosophy, literature and the sciences, and which insisted on the consideration of the necessary link between, on one hand, the human tendency towards objectification which shows itself in the practice of mathematics, and, on the other hand, the tendency towards widening and deepening of thought which prevails in religion and mysticism. In other words, the Dutch significians believed in the polar nature of the human mind - an idea most poignantly expressed in Mathesis en Mystiek (Mathematics and Mysticism, 1925, translated into French as Les deux pôles de l’esprit (The two poles of the mind) written by L.E.J. Brouwer’s fellow mathematician, Gerrit Mannoury (1867-1956). In our lecture we shall defend the view that many of the ideas expressed by the Dutch school of thought should be studied in the light of the contemporary return to the philosophy of Immanuel Kant (1724-1804), as well as of the radically new interpretations of Plato proposed by the Marburg philosophers, Hermann Cohen (1842- 1918) and Paul Natorp (1854-1924). In the opinion of these Neo-Kantians, Truth could not be conceived of otherwise than as searched for or longed for Truth. Brouwer’s intuitionist refinements of mathematical language deserve to be interpreted along the same lines. Rather than making bold statements about truth and falsehood, Brouwer presents the mathematician as reaching out for truth. Paradoxically it is the recognition of a limitation – the mathematician is indeed not omniscient - , which makes the intuitionist’s judgements about mathematical truth more reliable and precise. Looking at the matter from a Platonic point of view, one can say that we do not have immediate access to the Ideas. It is however within our capacity to talk about ourselves as affected by the Ideas or as actively pursuing the Ideas. It was the philological approach of Plato’s dialogues as well as a fresh look at the semantics of the word idea as used by the ancient philosopher which inspired these views of Cohen an Natorp. In the last part of the lecture we shall explain that the interest which the Marburg philosophers took in in the works of the early Renaissance theologian Nicholas of Cusa (1401-1464) was in many ways related to their own views on Truth as never possessed but tirelessly pursued and longed for by Man. Cusanus illustrated his fundamental notion of docta ignorantia or “wisdom of unknowing” by means of geometrical figures in motion. These figures of Cusanus, which one finds in the margins of various manuscripts, do not show infinity, but they do imply infinity starting from Man’s ignorance of infinity. Our conclusion will be that to a mathematician, intellectual humility, recognition of the limitations of human comprehension as well as avoidance of boldness of expression constitute the gateway to Ultimate Reality.

Temperament in Classical Music

Jordan Bergmans

For millennia, human beings have harnessed sound waves, both to commune with nature and to express themselves. This harnessing, done within a temporal structure is what we call music. Whether or not it is a conscious choice, music and its creation are intrinsically linked to mathematical logic, geometry and arithmetic. Crucially, this art form in inextricable from the spiritual and metaphysical context in which it takes form. This paper will explore the chronological establishment of musical pitches as we know them today in Western classical music within a mathematical context, from Pythagoras to the 20th century. Musical pitch is a concept which went through many alterations and “corrections” throughout this period and decisions made around it shaped the musical world we now live in. We will take particular interest in the work of the pivotal composer J.S. Bach and draw parallels between the structuring of pitch in Indian and Western classical music. Following this, it will be interesting to examine the smallest intervals in each of these musical schools and consider their expressive qualities.

Modern mathematics in Ancient India: Sixteen samples from Sanskrit Sources

V.Kannan

The talk starts with sixteen instances of mathematical formulas that are nowadays taught in high schools and colleges. For each one of them an ancient Sanskrit text where it is stated long ago, is mentioned. While doing so, sixteen different Sanskrit books on Mathematics, one for each, will be introduced.  Actual Sanskrit passages will be quoted. These formulas include: Pythagoras Theorem, Formula for the roots of a quadratic equation, Formula for the area of a triangle, Infinite series for the sine function, trigonometric addition formula, Formula for the volume of a sphere, etc.

L. E. J. Brouwer’s philosophical views and the Bhagavad Gita

Teun Koetsier

The philosophical views of the Dutch mathematician L. E. J. Brouwer had a crucial influence on his mathematical work. His conviction that mathematics at heart consists of language-free and logic-free introspective constructions stems from his epistemological solipsism. This solipsism is embedded in an original world view based on Schopenhauerian ideas. In particular Brouwer’s references to the Bhagavad Gita are striking. The paper sketches the development of Brouwer’s views.

The Science of Life

P. B. Sharma

Purna - the Eternal encased in matter, in form and shape as it considers fit, performing its act and actions as per its chosen level of conscious mental, evolving in shape as well as its mental consciousness till it realizes its eternal character and proclaim “aham brahmn asmi” has been the science of life as perceived by India’s Vedic seers and sages, the scientists of the Vedic age. The present paper is an attempt to decode their wisdom to understand life, its meaning and purpose in the modern age of the knowledge era.

Reality – Explorations from Scientific, Mathematical, Philosophical and Vedantic Research Methodologies and their Synthesis

Prof. Ramgopal Uppaluri

In this presentation, firstly, epistemological and ontological analyses of research methodologies have been addressed in the fields of science, mathematics and philosophy.  This is accomplished by considering the most fundamental concepts associated to academic, scientific and technological research approaches and supplemental/astute mathematical tools that support such academic research. In due course of the analysis, few fundamental limitations have been outlined in the inferences associated to relative understanding of the reality. Thereafter, in dedicated sections several examples have been presented to infer that the reality in an absolute sense is beyond physical, philosophical and experiential capabilities.  Eventually, Vedantic perspectives have been summarised to serve as an effective guide for the evolution of mature understanding of the reality (absolute) from its relative understanding. Finally, based on the new science outlined by Dr. T. D. Singh in his books, a synthesis model has been presented to mature academically driven research as a holistic research framework for the inclusion of subjective and spiritual elements.  In summary, the article attempts to evolve scientific and philosophical temper with elements of spiritual wisdom for a better understanding of the reality (absolute) through a holistic integration of reality perceived through relative approaches of sciences, philosophies and absolute notions of the Vedanta (spirituality).

Meta Mathematics: Intuitionism and Formalism in Mathematics

Bhu Dev Sharma

Feverish Mathematical activities especially during 17 th , 18 th and early 19 th centuries lead to a large body of knowledge consisting of new disciplines like probability theory, analytic & projective geometries, infinitesimal calculus, mathematical analysis, modern algebra, non-Euclidean geometries, etc. Various of these disciplines—each apparently having its own individual new concepts, methods, tools, approaches and styles—hardly appeared to be constituents of a single unified discipline, viz. Mathematics. Questions—about connectivity, coherence, logical consistency and compactness of mathematical knowledge having assets of tools and techniques applicable across the sub-disciplines— started being raised. Foundational questions like the validity of the concepts of infinitesimal were raised. It forced the mathematicians to investigate these (meta-mathematical) questions. To handle these meta-mathematical questions, emerged three alternative approaches: Logicism, Intuitionism, and Formalism. Logicism – pioneered by Gottlieb Frege from 1884, and later developed by Russell,Whitehead and others had the goal of exhibiting that whole of classical mathematics can be considered as a part of logic. It was grounded in the philosophy of platonic realism, according to which abstract concepts exist independent of human mind. However, appearance of a number of paradoxes prevented universal acceptance of the approach. H According to Intuitionism—proposed by Brouwer around 1908, partially in order overcome the problems of paradoxes—mathematics should be defined as mental activity, and not a set of theorems, and hence mathematical artifacts should be constructed starting from the natural number 1, and then 2 and so on, through mental constructions which are inductive and effective as happens in a computer system. However, the approach has failed to attract mathematicians for various reasons, including (i) many of the theorems of classical mathematics cannot be proved by intuitionist’s approach, (ii) proofs of theorems by the approach are generally too lengthy and even incomprehensible. The goal of Formalism (approach), pioneered by Hilbert around 1910, is to express any mathematical theory in terms of symbols, which can even be manipulated by a machine. To start with, undefined terms like ‘point’, ‘line’ and ‘ incidence’ in geometry; and ‘0’, ‘+’ and ‘*’ of number theory may be symbolized. Then recursively, other concepts be symbolized. Whole of mathematics be developed like that. However, the purpose was not to practice mathematics like that, but to answer meta-mathematical questions like those of showing it contradiction-free. But, hope of success of Hilbert’s approach was demolished in 1931 by Gödel’s incompleteness theorem..

Can Computers help to sharpen our understanding of ontological arguments?

Christoph Benzmüller

At the 9th AISSQ conference in 2015 I presented the results of computer-supported analysis of Kurt Gödel's modern, modal version of ontological argument for the existence of God [1]. I reported on the inconsistency the theorem prover LEO-II detected in the original proof script by Gödel, which is avoided in Dana Scott's variant of Gödel's work. Since then I have continued with students and colleagues to work on various further variants of the ontological argument which have been proposed by philosophers in the past decades. Many of these variants preserve the main conclusion, the necessary existence of God, while avoiding what is called the modal collapse, which is a side result of the Gödel/Scott-variants. The modal collapse expresses that there are no contingent truths, and it can hence be interpreted as an attack to free will. In my presentation at the 11th AISSQ conference in 2018 I will summarise the core results of our more recent computer-supported verification studies on those latter variants of the ontological argument. Subsequently, I will reflect on the conducted experiments and argue that the interaction with the computer technology, which we have developed over the past years, can not only enable the formal verification or falsification of ontological arguments, but that it can in fact help to sharpen our conceptual understanding of the notions and concepts involved. (This is joint work with David Fuenmayor and others.) [1] Christoph Benzmüller and Bruno Woltzenlogel Paleo, Experiments in Computational Metaphysics: Gödel's Proof of God's Existence. In Savijnanam: scientific exploration for a spiritual paradigm. Journal of the Bhaktivedanta Institute, volume 9, pp. 43-57, 2017.

Infinity: Philosophical, Theological, Mathematical, Scientific, Computational & Cultural Aspects

Prof. Manohar Lal

In different cultures and religions, the concept has been perceived differently. In the Indic culture and religions (e.g. Vedic, Buddhist and Jain), it was accepted in its modern sense, from quite early times as is indicated by following Mantra from Isha Upnishad (Vedic philosophical text):

Om Puurnnam-Adah Puurnnam-Idam Puurnnaat-Purnnam-Udacyate Puurnnasya Puurnnam-Aadaaya Puurnnam-Eva-Avashissyate || Om Shaantih Shaantih Shaantih ||

The ‘line 2:’ above says that ∞ - ∞ = ∞, which is sort of current definition of ‘Infinity’. The Jains were the first to recognize, in around 400 BC, different classes of infinities: nearly-infinite, truly-infinite, infinitely-infinite . However, in western philosophy, ‘Infinity’ has been problematic concept, particularly, due to Pythagorean’s belief that any aspect of the world could be represented by finite arrangement of whole numbers. They used term Apeiron meaning indeterminate, chaotic etc. with negative connotations. Aristotle later introduced the concept of Potential infinity to answer Zeno’s paradox. Reason for the difference in Indic/Eastern and Greek/Western thought regarding ‘Infinity’ is probably cultural one. Western thinking—emphasizing ontological commitment for acceptance of concepts—has difficulty in accepting the concepts like zero and infinity etc. Eastern thinking being groomed in powerful imaginary has no such problems. Infinity has been subject matter of other disciplines including sciences and Mathematics. Galileo and others made significant contribution to the concept. Cantor made infinity a mathematical object amenable to arithmetic operations. Digital infinity is a technical term used in computer science to denote computations of infinite results through repeated use of finite instructions. Paradoxes form essential component of discussion of ‘infinity’. Some of well-known paradoxes include (i) Russell’s paradox, (ii) Berry’s paradox, (iii) Burali-Forti paradox, and (iv) Cantor’s paradox.


ORGANIZER


BHAKTIVEDANTA INSTITUTE

IMPORTANT DATES

Registration Starts: May 1, 2018

Submission of Abstracts by: June 15, 2018

Submission of Full Papers by: July 30, 2018

Registration Ends: October 4, 2018

Conference Dates: October 6-7, 2018

IMPORTANT CONTACTS

Jitun Kumar Dhal (Organizing Secretary)
E-mail :jdhal[at]binstitute.org
Mobile :+91-9734780602
Landline:06742974602

Siddhartha Tiwari (Joint Secretary)
E-mail :siddhartha.tiwari2k7[at]gmail.com
Mobile :+91-9911102091