DOCSLIB.ORG

The Philosophy of Mathematics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Philosophy of Mathematics REFLECTIONS ON PHILOSOPHY The Philosophy of Mathematics Swami Sarvottamananda (Continued from the previous issue) The Mathematical Method understood the nature of math- ments which are neither true nor false. For example, ematical concepts, we now need to brief- interrogatory and exclamatory statements are nei- Hly examine the mathematical method. ther true nor false. Also there is this classic example What is the method by which we arrive at the truth of a paradoxical self-referential statement, which is or falsity of mathematical statements? neither true nor false: In a mathematical system, we have axioms, P: e statement P is false which are facts taken to be obviously true (‘If a We have referred to the term theorem. Now is the is less than b, then a is not equal to b’ is one such time to dene it. What is a theorem? A theorem is axiom—the axiom of linear order), and some non- nothing but a proposition for which there is a for- facts (which we shall call non-axioms) by the help mal proof. What then is meant by proof? A proof of which we prove or disprove theorems. Proving is simply a sequence of deductive steps governed by theorems means deriving them from known axi- well-dened logical rules that follow from a set of oms. If we are able to deduce a theorem starting axioms. An axiom, of course, is a proposition that is from these basic axioms, then we say that the theo- given to be unconditionally true. e following de- rem is true. duction illustrates the rule of specialization, which However, proving the falsity of a theorem is dif- is one of the many rules of logic: ferent. If we are able to derive a non-axiom from a All men are mortal. proposition, then that proposition is false—a non- Socrates is a man. theorem. So, here we go the other way round—we erefore, Socrates is mortal start from the theorem itself, not from non-axioms. us, a mathematical system is a set of axioms Hence proving the truth and falsity of theorems are and non-axioms with predened rules of deduc- not mirror processes. tion, which are also referred to as rules of inference. e underlying assumption of this method is e rules of deduction or rules of inference are noth- that we cannot derive a non-fact from facts. Such ing but rules that add, remove, modify, and substi- a system is called consistent. If a system is inconsist- tute operators and symbols. ent, it is ‘trivially complete’; that is, every statement, Let us try an exercise to understand how the true or false, is derivable in an inconsistent system. rules of inference work. Suppose we have been giv- An inconsistent system, therefore, is of little prac- en the following rules of addition, removal, and tical use. substitution of symbols and (the other symbol remains there as in the starting axiom). e start- Propositional Logic ing axiom is , and x and y are variables: In the study of mathematical method we also need (i) x → x (Derive from ) to study propositional logic. Propositions play an (ii) x → xx ( Derive important part in mathematical proofs. What is a from ) proposition? A proposition is a statement which is (iii) xy → xy (Derive from ) either true or false. Note that there are certain state- (iv) xy → xy (Derive from ) PB October 2007 e Philosophy of Mathematics 45 Now try constructing the theorem starting participate in the exciting activity of mathematical only with the axiom using the above rules of discovery. inference. Is it possible to derive ? e crux of the matter discussed above is that Important Branches of Mathematics in mathematics, as well as in logic, the operators, Among the important branches of mathematics, the constants, and the functions can all be viewed, number theory, set theory, geometry, and logic are as in this example, as symbols which are added, re- historically very old. e oldest civilizations—the moved, and substituted by predened rules of in- Indian, Greek, Chinese, Egyptian, and Babylo- ference, without ascribing any interpretation to nian—had all developed these branches, in one them. Gödel exploited this fact beautifully in prov- form or other, for general use. is is substanti- ing his famous theorem on incompleteness. ated by the fact that without a fair understanding of geometry the remarkable architectural and civil- Is Mathematics a Uniquely Human Activity? engineering feats for which these civilisations are Since doing mathematics involves intricate reason- famous would not have been possible. Even such ing and abstract thinking, it is oen thought to be elementary constructions as a rectangular wall or a a very creative process requiring a lot of intuition. eld, or the more intricate hemispherical dome, re- Kant was of the opinion that since mathematics re- quire at least a rudimentary knowledge of geomet- quires human intuition it cannot possibly be done rical constructions. Incidentally, ancient Greeks by non-humans. But several later philosophers have gave much importance to geometry, whereas In- shown that it really does not require any human in- dians gave up geometry for abstract mathematics tuition to understand a mathematical proof. Find- during the Buddhist period. ing a proof for an open research problem, though, As far as number and set theories are concerned, might be an altogether dierent matter—comput- no one really knows when humans developed these. ers have failed till date to automatically generate Numbers surely came with the need for counting. proofs for even very simple non-trivial mathemati- Most civilizations seem to have been formally using cal problems. is is not to suggest that proving numbers right from their inception. It was need- mathematical theorems is a uniquely human activ- ed for commerce, and in earlier tribal societies to ity incapable of computer simulation—it is simply quantify one’s possessions. a matter of selective processing power. Comput- Set theory is more fundamental than number ers cannot distinguish between boring mathemati- theory, for it deals with classication rather than cal truths and interesting mathematical results and counting. Formal logic was a later development. keep happily churning out one mathematically un- But its rudiments were probably coeval with the interesting result aer another, ad innitum. development of language—with the need to coher- Mathematical thinking, in fact, is apparently ently and intelligently communicate one’s opinions, not unique to humans. Rudimentary mathemati- arguments, and deductions to others. In fact, logic cal understanding is also seen in other animal spe- and language are so interlinked that many consider cies. And, of course, computers are ‘doing’ math- logic to be merely a linguistic construct. Histori- ematics all the time. If one is to argue that nding cally, both Nyaya and Aristotelian philosophy had and discovering mathematical truths rather than formalized logic for their respective civilizations, understanding proofs constitutes the test of mathe- the Indian and the Greek. matical intelligence—and computers fail this test— then it may be pointed out that this will also place Number Theory the majority of humans at par with machine intel- Let us begin with numbers. We may ask: What is ligence, because the vast majority of humans do not the nature of numbers? Are numbers real? In the PB October 2007 46 Prabuddha Bharata Nyaya and Vaisheshika philosophies, for instance, ‘God created the natural numbers; everything else numbers are real entities, belonging to one of the is man’s handiwork’, the German mathematician seven categories of real entities. However, there Leopold Kronecker had famously observed. e are conceptual di culties if we grant numbers an incorporation of zero as a number was the great objective reality. Consider the following: We have contribution of the Indian subcontinent. e natu- two books. So, we have books and we have also ral numbers are complete as far as the operations of the number two. Let us add another pair of books addition and multiplication are concerned—if we to our collection. Does it destroy the number two add or multiply two natural numbers we get an- and create the number four? Or does the number other natural number. However, the class of natural two transform into the number four? Suppose we numbers is not complete with respect to subtrac- add two notebooks, to distinguish them from the tion (you don’t get a natural number if you subtract original pair of books. en we have got a pair of from ). So if the result is to belong to the set of twos as well as a four. None of the original numbers numbers, we need to extend the list of natural num- is destroyed or transformed and yet a new number bers to include negative numbers. e result is the is created. e ancient Buddhists were therefore set of integers. not wrong in pointing out that numbers are in fact Again, we see that the class of integers is not mental concepts. ey do not have any existence complete with respect to division. So the set of outside the mental world. numbers is further extended to include ratios—ra- Furthermore, mathematicians say that numbers tional numbers. e word rational here is derived can also be thought of as properties of sets, being from ‘ratio’ and not ‘reason’. Next we get surds or their sizes (though the Buddhists would not feel irrational numbers, when we extend the set of num- comfortable with this either). Numbers as proper- bers to include limits, sums of series, square roots, ties of sets were called cardinal numbers by George trigonometric functions, logarithms, exponential Cantor in contrast to ordinal numbers which repre- functions, and so on.
Load more

Recommended publications
  • CONCEPT of UNIVERSAL PROPOSITION (UDHARANA) in NAYAYA PHILOSOPHY Mudasir Ahmad Tantray Lecturer at Dept
    ANVESAK ISSN : 0378 – 4568 UGC Care Group 1 Journal CONCEPT OF UNIVERSAL PROPOSITION (UDHARANA) IN NAYAYA PHILOSOPHY Mudasir Ahmad Tantray Lecturer at Dept. of Philosophy, Govt. Boys College Sopore Email: [email protected] Tariq Rafeeq Khan Lecturer at Dept. of Philosophy, Govt. degree college Women’s Sopore Email: [email protected] ABSTRACT Universal proposition (Udharana) is also known as general proposition or categorical proposition. Universal proposition is defined as the proposition in which the relation between the subject term and the predicate term is without any condition, in which the predicate is either affirmed or denied of the subject unconditionally. In nyaya logic the term vyapti is a universal proposition or invariable relation between the middle term (linga/hetu) and the major term (sadya) . According to the category of relation propositions are divided into categorical and the conditional. Although proposition is a logical entity which is an assertion, either affirm or deny the subject. Truth and falsity are the values of proposition. Universal proposition is of two kinds: universal affirmative proposition and universal negative proposition while universality is the quantity of the subject and affirmation or negation is the quality of the proposition. Anumana (inference) is the knowledge of the objects which follows some other knowledge. In nyaya logic Inference is a combined deductive-inductive reasoning consisting of at least three categorical propositions and in it there are at least three propositions consisting of three terms, viz. the paksa or minor term about which we infer something, the sadhya or major term which inferred object, and the linga(probans) or sadana or middle which is invariably related to the major, and is present in the minor.
    [Show full text]
  • A Survey of Indian Logic from the Point of View of Computer Science
    Sadhana, "Col. 19, Part 6, December 1994, pp 971-983. © Printed in India A survey of Indian logic from the point of view of computer science V V S SARMA Department of Computer Science & Automation, Indian Institute of Science, Bangalore 560012, India Abstract. Indian logic has a long history. It somewhat covers the domains of two of the six schools (darsanas) of Indian philosophy, namely, Nyaya and Vaisesika. The generally accepted definition of In- dian logic over the ages is the science which ascertains valid knowledge either by means of six senses or by means of the five members of the syllogism. In other words, perception and inference constitute the sub- ject matter of logic. The science of logic evolved in India through three ~ges: the ancient, the medieval and the modern, spanning almost thirty centuries. Advances in Computer Science, in particular, in Artificial Intelligence have got researchers in these areas interested in the basic problems of language, logic and cognition in the past three decades. In the 1980s, Artificial Intelligence has evolved into knowledge-based and intelligent system design, and the knowledge base and inference engine have become standard subsystems of an intelligent System. One of the important issues in the design of such systems is knowledge acquisition from humans who are experts in a branch of learning (such as medicine or law) and transferring that knowledge to a computing system. The second important issue in such systems is the validation of the knowl- edge base of the system i.e. ensuring that the knowledge is complete and consistent.
    [Show full text]
  • Brahma Sutra
    BRAHMA SUTRA CHAPTER 1 1st Pada 1st Adikaranam to 11th Adhikaranam Sutra 1 to 31 INDEX S. No. Topic Pages Topic No Sutra No Summary 5 Introduction of Brahma Sutra 6 1 Jijnasa adhikaranam 1 a) Sutra 1 103 1 1 2 Janmady adhikaranam 2 a) Sutra 2 132 2 2 3 Sastrayonitv adhikaranam 3 a) Sutra 3 133 3 3 4 Samanvay adhikaranam 4 a) Sutra 4 204 4 4 5 Ikshatyadyadhikaranam: (Sutras 5-11) 5 a) Sutra 5 324 5 5 b) Sutra 6 353 5 6 c) Sutra 7 357 5 7 d) Sutra 8 362 5 8 e) Sutra 9 369 5 9 f) Sutra 10 372 5 10 g) Sutra 11 376 5 11 2 S. No. Topic Pages Topic No Sutra No 6 Anandamayadhikaranam: (Sutras 12-19) 6 a) Sutra 12 382 6 12 b) Sutra 13 394 6 13 c) Sutra 14 397 6 14 d) Sutra 15 407 6 15 e) Sutra 16 411 6 16 f) Sutra 17 414 6 17 g) Sutra 18 416 6 18 h) Sutra 19 425 6 19 7 Antaradhikaranam: (Sutras 20-21) 7 a) Sutra 20 436 7 20 b) Sutra 21 448 7 21 8 Akasadhikaranam : 8 a) Sutra 22 460 8 22 9 Pranadhikaranam : 9 a) Sutra 23 472 9 23 3 S. No. Topic Pages Topic No Sutra No 10 Jyotischaranadhikaranam : (Sutras 24-27) 10 a) Sutra 24 486 10 24 b) Sutra 25 508 10 25 c) Sutra 26 513 10 26 d) Sutra 27 517 10 27 11 Pratardanadhikaranam: (Sutras 28-31) 11 a) Sutra 28 526 11 28 b) Sutra 29 538 11 29 c) Sutra 30 546 11 30 d) Sutra 31 558 11 31 4 SUMMARY Brahma Sutra Bhasyam Topics - 191 Chapter – 1 Chapter – 2 Chapter – 3 Chapter – 4 Samanvaya – Avirodha – non – Sadhana – spiritual reconciliation through Phala – result contradiction practice proper interpretation Topics - 39 Topics - 47 Topics - 67 Topics 38 Sections Topics Sections Topics Sections Topics Sections Topics 1 11 1 13 1 06 1 14 2 07 2 08 2 08 2 11 3 13 3 17 3 36 3 06 4 08 4 09 4 17 4 07 5 Lecture – 01 Puja: • Gratitude to lord for completion of Upanishad course (last Chandogya Upanishad + Brihadaranyaka Upanishad).
    [Show full text]
  • Parasitism and Disjunctivism in Nyya Epistemology
    Parasitism and Disjunctivism in Nyya Epistemology Matthew R. Dasti Philosophy East and West, Volume 62, Number 1, January 2012, pp. 1-15 (Article) Published by University of Hawai'i Press DOI: 10.1353/pew.2012.0012 For additional information about this article http://muse.jhu.edu/journals/pew/summary/v062/62.1.dasti.html Access provided by San Jose State University (30 Oct 2013 13:43 GMT) PARASITISM AND DISJUNCTIVISM IN NYĀYA EPISTEMOLOGY Matthew R. Dasti Department of Philosophy, Bridgewater State University From the early modern period, Western epistemologists have often been concerned with a rigorous notion of epistemic justification, epitomized in the work of D escartes: properly held beliefs require insulation from extreme skepticism. To the degree that veridical cognitive states may be indistinguishable from non-veridical states, appar- ently veridical states cannot enjoy high-grade positive epistemic status. Therefore, a good believer begins from what are taken to be neutral, subjective experiences and reasons outward — hopefully identifying the kinds of appearances that properly link up to the world and those that do not. Good beliefs, beliefs that are justified (war- ranted, etc.), are those that a believer has consciously arrived at by such reasoning (or, in a weaker version, those that could be consciously arrived at by such reasoning if required). This approach, which I will occasionally call a Cartesian approach, has two important features. First, it considers doubt a legitimate default position in the space of reasons. The burden of proof is upon the believer to defend her belief. In the absence of such a defense, belief is suspect.
    [Show full text]
  • Nyaya-Vaisheshika: the Indian Tradition of Physics
    Nyaya-Vaisheshika: The Indian Tradition of Physics Roopa Hulikal Narayan 1 Introduction This paper is the first in a series on the Indian tradition of physics that while summarizing the earlier review by Kak [1], [2] will set the stage for a more comprehensive analysis to follow in later papers. In ancient India, the schools of Nyaya and Vaisheshika focused on logic and atomic approach to matter. In this paper, the idea of atomicity and other physical ideas given in Vaisheshika are reviewed in light of the central role the observer plays in Indian thought. We provide introduction to ideas that are described in greater detail in Potter’s text [10], where the focus is not on physical ideas but rather on philosophy. The Rigveda, the oldest of the Vedic texts of India, generally assigned to the early second millennium BC or earlier, is seen within the Indian tradition as the source of its approach to reality. The Vedic sages recognized a binding unity among all that constitutes this universe. They made an attempt to reflect this pattern of interdependence among the entities of the universe including the very structure of universe itself. This may be seen in the structure and symbolic purpose of Vedic altars, approach to language, and so on [3],[4],[5]. The observer or the experiencing subject was given a privileged state in physical thought [6-10]. By the end of nineteenth century, the place of the observer also became a part of the mainstream discourse of academic physics and psychology in the consideration of the dichotomous issues of order and disorder.
    [Show full text]
  • Ram-Indlogic
    The monkey on the tree and other dilemmas R. Ramanujam The Institute of Mathematical Sciences Chennai - 600 113 email: [email protected] Colloquium, Hangzhou, May 27, 2019 First words . I Thanks to Wang Yi for this opportunity and to Zhejiang University for the wonderful atmosphere. I Please feel free to interrupt any time to comment or question. I Statutory Warning: I am not a Sanskrit scholar or expert on the ancient Indian systems of logic, am only sharing what I have learned from secondary sources. Acknowledgements: 1 Bimal Krishna Matilal I Bimal Krishna Matilal (1935 | 1991): an influential Indian philosopher who wrote extensively on the Indian philosophical tradition in logic. I From 1977 to 1991 he was the Spalding Professor of Philosophy at University of Oxford. Acknowledgements: 2 Jonardon Ganeri I Jonardon Ganeri is a philosopher whose work spans the philosophy of mind, metaphysics and epistemology. He is the editor of the Oxford Handbook of Indian Philosophy (2017). I The Open Minds magazine named him of one of its 50 global open minds for 2016. He is currently in New York. I Almost everything I am talking of here is from Ganeri's writing, and the Stanford Encyclopaedia of Philosphy. I Epistemic: Under what conditions does knowledge of some facts permit knowledge of another fact. I Dialectic: Under what conditions does the acceptance by someone of some facts require him or her to accept some other fact. I Linguistic: Use the forms of linguistic expressions to identify forms of inferences and arguments. This is the modern method. Indian philosphers seem to have been mostly preoccupied with the first three.
    [Show full text]
  • Nyāya -Sutras and the Modern Decision-Making Process in an Organisation: a Note
    International Journal of Hinduism & Philosophy (IJHP), Vol. 2, Issue 1 April 2021 Nyāya -Sutras and the modern decision-making process in an organisation: A Note Deepraj Mukherjee Department of Economics, Kent State University, USA Indian philosophical traditions have received laconic scrutiny from the West. Most of the Keywords contemporary academic work in the West is based on the Greek philosophical tradition. Nyaya-sutras, Given this gap, this paper will attempt to provide an expository analysis of an ancient Decision-making Indian school of thought in the context of organisational management. Specifically, this process, paper will focus on one specific school in Indian philosophy, namely Nyāya Sutras. In this Indian philosophy, article, we implement the Sutras in solving real-world decision-making steps in an Management organisation. We find that western minds largely ignore Indian Philosophy. The sutras provide the policymakers with executing activities that will enhance the effectiveness and efficiency of the organisation. Corresponding author: Deepraj Mukherjee Email address for the corresponding author: [email protected] The first submission received: 4th October 2020. Revised submission received: 14th December 2020. Accepted: 27th March 2021 1. Introduction Indian philosophical traditions have received laconic scrutiny from the West. Most of the contemporary academic work in the West is based on the Greek philosophical tradition. Given this gap, this paper will attempt to provide an expository analysis of an ancient Indian school of thought in the context of organisational management. Specifically, this paper will focus on one specific school in Indian philosophy, namely Nyāya Sutras. In this paper, we attempt to implement the Sutras in solving real-world decision-making steps in an organisation.
    [Show full text]
  • Marks Dissertation Post-Defense
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Playfighting: Encountering Aviddhakarṇa and Bhāvivikta in Śāntarakṣita's Tattvasaṃgraha and Kamalaśīla's Pañjikā Permalink https://escholarship.org/uc/item/9wj6z2j3 Author Marks, James Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Playfighting: Encountering Aviddhakarṇa and Bhāvivikta in Śāntarakṣita's Tattvasaṃgraha and Kamalaśīla's Pañjikā By James Michael Marks A dissertation submitted in partial satisfaction of the Requirements for the degree of Doctor of Philosophy In Buddhist Studies In the Graduate Division Of the University of California, Berkeley Committee in charge: Professor Alexander von Rospatt, Chair Professor Robert H. Sharf Professor Mark Csikszentmihalyi Professor Robert P. Goldman Professor Isabelle Ratié Spring 2019 Abstract Playfighting: Encountering Aviddhakarṇa and Bhāvivikta in Śāntarakṣita's Tattvasaṃgraha and Kamalaśīla's Pañjikā by James Michael Marks Doctor of Philosophy in Buddhist Studies University of California, Berkeley Professor Alexander von Rospatt, Chair The present study collects, translates, and analyzes the surviving fragments of two lost Naiyāyika authors, Aviddhakarṇa and Bhāvivikta, principally as they have been preserved in the works of the eighth-century Buddhist philosophers Śāntarakṣita and Kamalaśīla. (The present study argues, without coming to a definite conclusion as yet, that there is strong evidence Aviddhakarṇa and Bhāvivikta
    [Show full text]
  • Nyaya-Vaisheshika: the Indian Tradition of Physics
    The Theory of Matter in Indian Physics Roopa Hulikal Narayan Abstract This paper is the second in series of the Indian physics of the Nyaya-Vaisheshika school. It may be read in conjunction with the first paper [14], where its concept of matter as vibratory atoms in combination was introduced; this concept is discussed in greater detail in this paper. It is significant that the school defines matter not in terms of something gross that is anchored to the commonsensical notion of an object, but rather in terms of something that has attributes associated with it. Matter, or padartha , is whatever is knowable within the overarching complex of space and time, each of which is taken to be continuous and infinite. The significant concepts discussed here include how to localize an object, and that of nothingness, that is, vacuum. 1 Introduction This paper presents the physics related to matter in the Indian Nyaya-Vaisheshika school, which will henceforth be called “Indian physics”. Kanada, the originator of Vaisheshika, begins by claiming that “classification of things” is the primary task in his system. The purpose of this classification is to define material things. The understanding is to be arrived at using ‘tattva-jnana’ [15], or ascertainment of attributes of reality by categorizing everything in to one or the other among six predicable padartha , logical categories [3]. Indian physics considers both the objective universe, which is taken to be atomic, and the subjective universe of the experimenter or the observer, which is taken to be non-atomic [7], [[11], and [12]. In other words, it presents a dualistic view where that the observed matter is atomic whereas the observing mind and time and space in which the universe exists is continuous.
    [Show full text]
  • Technical Terms of Navya-Nyāya Language and Its Methodology
    International Journal of Management and Humanities (IJMH) ISSN: 2394-0913 (Online), Volume-4 Issue-10, June 2020 Technical Terms of Navya-Nyāya Language and its Methodology Nibedita Banerjee, Anil Pratap Giri Abstract: The Navya-Nyāya language1 (Neo-logic) exhibit one of ▪ To get scientific structure of this language, this can be the purest languages of human intellect. It has been assists India’s used as a tool to develop information technology. intellectual academic culture and traditions. From the 10th Century AD Navya-Naiyāyikas emphasized one-step further in intelligentsia ▪ To justify that Navya-Nyāya Language can be used as a and came out with a precise technical language which is a special medium of Communication to get an ambiguous free form of Sanskrit-delimits ultimate mining of each and every word concept. and capture reality as it is, destroy Jalpa and Vitaṇdā kind of dialogues in intellectual countering amid different school of thoughts. Since Navya-Nyāya language is a peculiar language based II. OBJECTIVES: on logic-thought and reality invented by Great Indian logicians that Analytical methodology is applied to analysis this paper. is why it has been treated an artificial language as well. Due to unique features of Navya-Nyāya language many traditional Through this analytical methodology, this paper analyses philosophical problems have been scientifically clarified and alphabetical form of logic and its relationship with language resolved. Navya-Nyāya methodology proved to be so multifaceted not and reality. just for philosophers, but also in poetics, linguistics, legal theories, and other domains of medieval Indian thought process. Apart from III.
    [Show full text]
  • The Philosophy of Mathematics Swami Sarvottamananda
    REFLECTIONS ON PHILOSOPHY The Philosophy of Mathematics Swami Sarvottamananda ‘Ah! then yours wasn’t a really good school,’ said of paper away. At the end of the day, he went home the Mock Turtle in a tone of great relief. ‘Now (to the old Cambridge address, of course). When at OURS they had at the end of the bill, “French, he got there he realized that they had moved, that music, AND WASHING—extra.”’ he had no idea where they had moved to, and that ‘You couldn’t have wanted it much,’ said Alice; the piece of paper with the address was long gone. ‘living at the bottom of the sea.’ ‘I couldn’t aord to learn it,’ said the Mock Tur- Fortunately inspiration struck. ere was a young tle with a sigh. ‘I only took the regular course.’ girl on the street and he conceived the idea of ask- ‘What was that?’ inquired Alice. ing her where he had moved to, saying, ‘Excuse me, ‘Reeling and Writhing, of course, to begin with,’ perhaps you know me. I’m Norbert Weiner and the Mock Turtle replied; ‘and then the dierent we’ve just moved. Would you know where we’ve branches of Arithmetic—Ambition, Distraction, moved to?’ To this the young girl replied, ‘Yes Dad- Uglication, and Derision.’ dy, Mummy thought you would forget!’ —Alice’s Adventures in Wonderland e world of mathematics is beautifully reect- following story is told about the reputed ed in Alice’s Adventures in Wonderland—a world mathematician Norbert Weiner: When they of ideas, where absurdity is a natural occurrence.
    [Show full text]
  • The Conflict Between the Buddhist and the Naiyayika Philosophers: a Brief Survey
    THE CONFLICT BETWEEN THE BUDDHIST AND THE NAIYAYIKA PHILOSOPHERS: A BRIEF SURVEY - Dr. Sanjit Kumar Sadhukhan Philosophy is nothing but the human quest for truth. From very remote time Indians are blessed with the spontaneous curiosity about what truth is. The first literature containing the truth realized by the ancient Indians is the Vedas. The philosophy revealed in this literature was more or less tuned with human helplessness together with submission to nature Gods. This went on unchallenged till the Buddha preached his new doctrine which afterwards brought about a head-on collision with the Vedic stricture, but the Buddha denied to give any positive answer to any Vedic approach and consequently in later period a gigantic philosophical system was built up against the Vedic philosophy or more accurately there rose a protest against the unverified doctrine. In the Brahmajiila­ suttaofDighanikiiya.Kathavatthu and the Upanisads we fmd that the philosophy has taken l a challenging attitude by now . The people also were clearly divided into two major groups. On the one hand, the Brahmins were there with the Vedic philosophy and on the other, the Buddhists came forward with their new philosophical doctrines. It was the beginning of the Christian era when such a situation was created that the Brahmin and the Buddhist philosophers considered their respective philosophical vi~ws unsuccessful if those were not directed against the opponent and at the same time not victorious. It is obvious that the introduction of the debate system was largely responsible for the creation of this situation. The fundamental difference in outlooks between realism and idealism led to mutual confrontations which continued in an unbroken line for generations of scholars resulting in the growth of a rich and vigorous literature.
    [Show full text]