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	Foundations of Mathematics. By K.Podnieks
		Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.
		In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).
		Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).
	www.ltn.lv /~podnieks (733 words)

	20th WCP: Semantic Realism: Why Mathematicians Mean What They Say
		What category theory does, as far as our talk of mathematical concepts and relations is concerned, is provide a means for organizing and classifying what we say about 'the structure of the relationship' between various mathematical concepts in various mathematical theories.
		We say that category theory is the language of mathematical concepts and relations because it allows us to talk about their structure in terms of "objects" and "arrows", wherein such terms are taken as syntactic assemblages waiting for an interpretation of the appropriate sort to give them formulas meaning.
		We say that category theory is the language of mathematical theories and their relations because it allows us to talk about their structure in terms of "objects" and "functors", wherein such terms are, again, taken as syntactic assemblages waiting for an interpretation of the appropriate sort to give them formulas meaning.
	www.bu.edu /wcp/Papers/Math/MathLand.htm (3322 words)

	Inconsistent Mathematics
		Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
		Thus dynamical theories determine their own logic of possible propositions, and corresponding theories which may be inconsistent, and are certainly as natural as their incomplete counterparts.
		Projective geometry is a mathematical theory which is interesting because we are creatures with an eye, since it explains why it is that things look the way they do in perspective.
	www.braungardt.com /Mathematica/inconsistent_mathematics.htm (1793 words)

	Indispensability Arguments in the Philosophy of Mathematics
		Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter).
		Her reason for this is that scientists themselves do not seem to take the indispensable application of a mathematical theory to be an indication of the truth of the mathematics in question.
		In essence, he argues that mathematical theories are not being tested in the same way as the clearly empirical theories of science.
	plato.stanford.edu /entries/mathphil-indis (4418 words)

	Is Platonism Dead?
		Both theories, however, have been found wanting, particularly by Paul Benacerraf in his 1973 paper, "Mathematical Truth." In Platonism and Anti-Platonism in Mathematics, Mark Balaguer claims that both views are defensible, but also that there is no fact of the matter as to which of the two views is correct.
		The book begins with Benacerraf's epistemological problem for mathematics, as presented in "Mathematical Truth." Since Benacerraf relies on a causal theory of knowledge to highlight the epistemological difficulties of platonism, platonists are left with a relatively easy way out, namely the rejection of the causal theory.
		His argument for favoring fictionalism is that it provides a straightforward, standard, face-value interpretation of mathematical theory, whereas the others, such as conventionalism or formalism, require a non-standard, counter-intuitive interpretation of what mathematicians are up to.
	www.chass.utoronto.ca /pcu/noesis/issue_v/noesis_v_5.html (3030 words)

	Mathematical Reality (Site not responding. Last check: 2007-10-29)
		In his view, mathematical theories are the description of observed or experienced phenomena constructed by humans.
		When the knowledge of mathematics is viewed as semantic and linguistic, mathematics may not involve ontology, thereby the question of mathematical reality is unnecessary.
		Each theory is important to the debate of the existence of mathematical reality and theoretical distributions; each theory embodies valid and invalid assertions.
	seamonkey.ed.asu.edu /~alex/computer/sas/math_reality.html (4388 words)

	Platonism, Intuition and the Nature of Mathematics. By K.Podnieks
		In VI-V centuries BC the evolution of Greek mathematics led to mathematical objects in the modern meaning of the word: the ideas of numbers, points, straight lines etc. stabilized, and thus they were detached from their real source - properties and relations of things in the human practice.
		The evolution of both theories is going on today: new theorems are proved, new methods and algorithms are developed etc. Nevertheless, both sets of basic principles remain constant (such as they were at the lifetime of their creators).
		A mathematical model of some natural process or technical device is essentially a stable model that can be investigated independently of its "original" (and, thus, the similarity of the model and the "original" is only a limited one).
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt1.html (4118 words)

	T. Koetsier : Lakatos' Philosophy of Mathematics, A Historical Approach
		He defined quasi-empirical theories as theories in which the crucial truth flow is the upward transmission of falsity from the "basic statements" to the axioms.
		Mathematics and logic are part of this theory, differing from natural science in the sense that they assume a very central position.
		Although their subject matter is different, mathematical theories and empirical theories have in common the fact that they are fallible.
	peccatte.karefil.com /Quasi/KoetsierOnLakatos.html (3026 words)

	Brainstorms: Mathematical Impossibility of RM&NS Evolution (Site not responding. Last check: 2007-10-29)
		My point was very simple: If a mathematical model does not veridically represent the entities, processes, and relationships that are defined as relevant by a substantive theory of a domain of inquiry, then conclusions based on the the model's behavior do not answer questions about the substantive theory.
		To claim that one has mathematically disproven evolutionary theory and to assert that evolution by random mutation and natural selection is in principle an inadequate explanation of the observed biological phenomena is also to claim to have adequately considered such questions.
		I'm accustomed to a mode of doing science in which theories have contact with data, systematically gathered in field or laboratory, and in which mathematical models are evaluated in the light of the veridicality of their mapping of the theories they are to model and the match between the model's behavior and the data.
	www.iscid.org /boards/ubb-get_topic-f-6-t-000246-p-2.html (5213 words)

	FotFS IV: Abstracts (Site not responding. Last check: 2007-10-29)
		As in mathematics a subject is described only down to the level of isomorphic instances, it follows that not instances of structures, but structures themselves and types of structures are the subjects of mathematical research.
		Mathematics is often described as the queen and servant of the sciences.
		Namely on one hand scientific theories in their own structur, as formal theories, are not to be involved in any sceptical confutation; and on the other hand sceptic’s challenge to enfeeble and annihilate the logic relations of scientific theories is properly to be dismissed.
	www.math.uni-bonn.de /people/fotfs/IV/abstracts.html (11199 words)

	In: Structures in Mathematical Theories, San Sebastian, 1990, pp (Site not responding. Last check: 2007-10-29)
		Some authors treat a model simply as a mathematical structure, exactly as a set with a system of relations (may be functions) on it (Malzev, 1970; Shoenfield, 1967).
		As mathematical structure M (that is called a model of the language L) the set on which the predicates and functions are defined usually is taken.
		The main progress of contemporary mathematics may be explained as a process of transition from usual (singlenamed) sets to more general cases.
	www.math.ucla.edu /~mburgin/res/math/NSetFouM/NSetFouM.htm (1936 words)

	Platonism, Intuition and the Nature of Mathematics. Part 2. By K.Podnieks
		Intuitive theories cannot develop without such reconstructions normally: the definiteness of intuitive basic principles gets insufficient when the complexity of concepts and methods is growing.
		Therefore, after replacing intuitive theory by the axiomatic one (this replacement may be non-equivalent because of defects in the intuitive theory) specialists learn a new intuition, and thus they restore the creative potency of their theory.
		Secondly, axiomatization allows a detailed analysis of relations between basic principles of a theory (to establish their dependency or independence, etc.), and between the principles and theorems (to prove some of theorems only a part of axioms may be necessary).
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt1a.html (5510 words)

	Mathematical Theories of Politics/Introduction to Game Theory
		Which of these additional texts is best suited to your needs should be a function of your mathematical background and how much you hope to do with game theory.
		Binmore, Kenneth G. 1977. Mathematical Analysis: A Straightforward Approach. Cambridge: Cambridge University Press.
		Students will be expected to complete 4 homework sets (approximately one every two weeks), which will count for 50% of the grade, and a final exam, which will count for the other 50%.
	www.nyu.edu /gsas/dept/politics/faculty/hafer/HaferGameTheory01.html (265 words)

	ScienceDaily: Help Page (Site not responding. Last check: 2007-10-29)
		Speak, Memory: Research Challenges Theory Of Memory Storage (November 14, 2006) -- During sleep, freshly minted memories move from the hippocampus, part of the "old" brain, to the neocortex, or "new" brain, for long-term storage.
		This has been the reigning theory for decades.
		Two Markers Strongly Linked To Prostate Cancer Incidence And Mortality Almost A Decade Prior To Diagnosis (November 14, 2006) -- Increased levels of two markers of inflammation, interleukin-6 (IL-6) and C-reactive protein (CRP), are significantly associated with prostate cancer incidence and mortality almost a decade prior to...
	www.sciencedaily.com /articles/computers_math/mathematical_modeling (1165 words)

	[No title]
		It is designed for graduate students in mathematics education to become familiar with leading theories of mathematical learning, to provide a critical research context where these theories of mathematical learning guide inquiries of important issues in mathematics education, and to consider practical applications of these theories to your own research context.
		You are to select one article from one of the leading research journals in mathematics education (e.g., Journal for Research in Mathematics Education and Educational Studies in Mathematics), digest the article in a critical manner, introduce the article to the whole class, present your critiques of the article, and promote discussions about the article.
		In order to argue in a convincing manner for your position, it would be necessary to first examine comprehensively conceptions from all mainstream theories of mathematical learning on this issue.
	www.uky.edu /Education/EDC/edc777s05xm.doc (985 words)

	Cogprints - A hierarchy of languages, logics, and mathematical theories
		We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory.
		We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e.
		a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered.
	cogprints.org /2875 (370 words)

	NSF Mathematical Sciences Institutes
		The IMA exists to increase the impact of mathematics by fostering interdisciplinary research linking mathematics with important scientific and technological problems from other disciplines and industry.
		The overall mission of the Institute for Pure and Applied Mathematics (IPAM) is to make connections between a broad spectrum of mathematicians and scientists, to launch new collaborations, to better inform mathematicians and scientists about interdisciplinary problems, and to broaden the range of applications in which mathematics is used.
		SAMSI is a national institute whose vision is to forge a new synthesis of the statistical sciences and the applied mathematical sciences with disciplinary science to confront the very hardest and most important data- and model-driven scientific challenges.
	www.mathinstitutes.org (206 words)

	[No title]
		All those theories can be organized in a lattice, according to the partial ordering defined by implication; i.e., if every axiom of theory A is a theorem of theory B, then A is more general than B. But not all mathematical theories are in the upper level.
		Chess, for example, is a mathematical theory, but it would be farther down the hierarchy.
		I have had considerable problems with trying to tie down the nature of pure structure, theory, etc. I tried to apply the term "concept" to this, but was ruled out of order on the basis that a concept had to have been conceived by someone, and by implication, at a given point in time.
	grimpeur.tamu.edu /pipermail/kif/2002-January.txt (7410 words)

	Personally developed mathematical theories...
		I will leave this up, to show my error, but be forewarned that the theories formulated here are mathematically false.
		My justification for the idea of "dead number theory," or division by zero, is the fact that the imaginary number i is widely accepted in the mathematical community as the square root of -1.
		The simple algebraic basis for dead number theory is the indisputable proposition (x/y) * y = x.
	www.angelfire.com /de3/deadmath/theories.html (1074 words)

	Mathematical Methods of Population Genetics
		The mathematical methods of population genetics theory characterize quantitatively the gene distribution dynamics in evolving populations [1-3].
		The mathematical models of population genetics describe the gene frequency distributions in evolving populations.
		The deterministic methods are used to analyze the mean frequency dynamics; the stochastic methods take into account the fluctuations, which are due to the finite population size.
	pespmc1.vub.ac.be /MATHMPG.html (1194 words)

	MATHEMATICAL STRUCTURES RESEARCH
		Research topics include mathematical models and theories in the empirical sciences, models and theories in mathematics, category theory, and the use of mathematical structures in theoretical computer science.
		Rudolph,E. and Stamatescu, I.-O.(eds) Philosophy, Mathematics and Modern Physics.
		"Theories and Theoretical Models"in Humphreys.(eds) Patrick Suppes, Reidel,1984
	www.mmsysgrp.com /mathstrc.htm (365 words)

	Citations: Randomness conservation inequalities: Information and independence in mathematical theories - Levin ... (Site not responding. Last check: 2007-10-29)
		In order to have a uniform framework for these new definitions, we need to modify the definitions of Kt and KT in minor ways that affect none of the theorems proved in the earlier work.
		) due to Solomonov [23] Kolmogorov [16] and Chaitin [4] is rooted in computability theory and specifically in the notion of a universal language (equiv.
		Towards a Universal Theory of Artificial Intelligence based on..
	citeseer.ist.psu.edu /context/86056/0 (2191 words)

	The Musical Octave
		Gurdjieff expressed that hint for his students and they began to elaborate.
		will focus on the acoustic laws behind the musical scales on a global scale, and how numbers and mathematic play a part in creation of the intervals in the octave.
		The One Voice Chord, the Tibetan tantric chanting and balancing the chakras.
	home22.inet.tele.dk /hightower (157 words)

	BookHq: Mathematical Theories of Populations: Demographics, Genetics and Epidemics by ( 0898710170 )
		BookHq: Mathematical Theories of Populations: Demographics, Genetics and Epidemics by (0898710170)
		Mathematical Theories of Populations: Demographics, Genetics and Epidemics
		The 10-digit ISBN# is typically found on the back of your book.
	www.bookhq.com /compare/0898710170.html (127 words)

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