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Topic: Where Mathematics Comes From

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Cognitive science of mathematics

Philosophy of mathematics

George Lakoff

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	Where Mathematics Comes From - Wikipedia, the free encyclopedia
		WMCF emphatically reject the Platonistic philosophy of mathematics.
		WMCF would be a major addition to the philosophy of mathematics and the theory of abstract objects, were it not that its authors say little about those subjects and cite little of the relevant literatures.
		------ and Rafael Núñez, 2000, Where Mathematics Comes From.
	en.wikipedia.org /wiki/Where_Mathematics_Comes_From (2202 words)

	[No title] (Site not responding. Last check: 2007-10-01)
		Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'.
		The specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics, but mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations.
		An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences.
	wikiwhat.com /encyclopedia/m/ma/mathematics.html (1186 words)

	mathematics - Anarchopedia (Site not responding. Last check: 2007-10-01)
		Mathematics (often abbreviated to math or, in British English, maths) is commonly defined as the study of patterns of structure, change, and space.
		In general the philosophy of mathematics one adopts has little effect on mathematical practice: mathematicians all over the world can rely on mathematics as a language even if there are arguments about the meaning or reliability of certain constructs or "words" or "phrases" used in any given "sentence".
		In order to clarify and investigate the foundations of mathematics in the 20th century, the fields of set theory, mathematical logic and model theory were developed, and were partially successful at explaining where mathematics "comes from".
	meta.anarchopedia.org /index.php?title=mathematics&redirect=no (2102 words)

	"Where Mathematics Comes From" Commentary
		Where Mathematics Comes From takes on the idea that all of our mathematical knowledge is built up from very simple math abilities that are inherited and that it is always constructed from metaphor.
		Mathematics that can calculate millions of dimensions of subject matter at the same time must be reduced to a form that can be articulated in a two-dimensional form (a variety of sounds following other sounds in relation to linear time) when applied to the creation of language.
		Intelligence is a form of mathematics that serves to symbolize the collective choosing of the greatest pleasure/health/success of the cells caused by the design of a single cell finding its greatest chance of success that becomes a collective greatest chance of success or pleasure when the cells are connected in a network.
	www.stevenharris.com /theory/069.htm (7128 words)

	Mathematics - Internet-Encyclopedia.com (Site not responding. Last check: 2007-10-01)
		Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
		Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics.
		Mathematics is often abbreviated to math (in American English) or maths (in British English).
	www.internet-encyclopedia.com /ie/m/ma/mathematics.html (2273 words)

	Cognitive science of mathematics - Wikipedia, the free encyclopedia
		The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science.
		This approach was long preceded by the study, in cognitive sciences proper, of human cognitive bias, especially in statistical thinking, most notably by Amos Tversky and Daniel Kahneman, including theories of measurement, risk and behavioral finance from these and other authors.
		This book culminates with a case study on Euler's Identity; the authors argue that Euler's Identity is a concept that reflects a cognitive structure unique to humans, or less specifically to a narrow range of beings similar to humans, e.g.
	en.wikipedia.org /wiki/Cognitive_science_of_mathematics (217 words)

	Amazon.ca: Where Mathematics Comes from: How the Embodied Mind Brings Mathematics Into Being: Books: George ... (Site not responding. Last check: 2007-10-01)
		Their basic premise, that all of mathematics is derived from the metaphors we use to maneuver in the world around us, is easy enough to grasp, but following the reasoning requires a willingness to approach complex mathematical and linguistic concepts--a combination that is sure to alienate a fair number of readers.
		Such a project is definitely worth the effort, for mathematics has to be interpreted in the light of what is known about the mind, or as the authors put it, "it should study precise nature of clear mathematical intuitions".
		Where mathematics comes from seeks to explain how mathematics can be explained as an extended metaphor of the world.
	www.amazon.ca /Where-Mathematics-Comes-Embodied-Brings/dp/0465037712 (2077 words)

	WHERE MATHEMATICS COMES FROM (Site not responding. Last check: 2007-10-01)
		Where Mathematics Comes From is an exception, refreshing both in its obliquely revealing approach to these issues and in its fascinating, sometimes iconoclastic conclusions.
		In mathematical terms, Lakoff and Nunez define a metaphor as a mapping from a source domain, which is familiar, to a target domain, which is less so.
		Whatever one's views on the nature of mathematical entities and truths, however, Where Mathematics Comes From is salutary in its emphasis on the human origin of mathematical concepts.
	www.math.temple.edu /~paulos/lakoff.html (1340 words)

	Direct Textbooks Price Comparison for ISBN 0465037712: Where Mathematics Comes From: How the Embodied Mind Brings ... (Site not responding. Last check: 2007-10-01)
		Mathematical concepts, theorems, and principles are not identified, except usually for historical purposes, with those persons who may have conceived of or discovered them.
		The first is with their technique of "mathematical idea analysis", in which they state that a particular metaphor is being applied in some area of mathematics (between two mathematical domains or between a mathematical domain and some conception of the real world), and then provide an explicit mapping between concepts in the two domains.
		In "Where Mathematics Comes From", Lakoff and Nunez defend their thesis that the only kind of mathematics that humans can know are the kind that are known to human minds.
	www.directtextbook.com /price.php?p=prices&q=0465037712&shippingtime=5 (3369 words)

	What is Mathematics
		Coming full circle is John D. Barrow's 1994 book Pi in the Sky: Counting, Thinking, and Being which has as one of its primary themes the idea that mathematics has much in common with religion.
		Inventionism is a wonderful philosophy for the arts and humanities where we see the fruits of imaginative subjectivity; their nature and practice contrast so drastically with that of mathematics that the objective element seems to have failed to be adequately incorporated into this view of mathematics.
		The mathematics according to the intuitionists was just a part of the ocean of mathematical truths that were accepted by other mathematicians.
	members.cox.net /mathmistakes/what_is_mathematics1.htm (4060 words)

	NEWCITYCHICAGO.COM: Street Smart Chicago
		Most of us think of mathematics as some sort of inaccessible (and sometimes beautiful) language that's hidden within nature, only to be uncovered and plucked out through hours of rigorous and painful scholarship.
		What's being espoused is mind-based mathematics, i.e., a math based on our innate ability to perform rudimentary arithmetic and subitization as infants, on basic cognitive tools like conceptual metaphors, blends, prototypes, frames, etc., and the concept of an embodied mind.
		Mathematics is actually a tool that we've constructed, not the language of the universe.
	www.newcitychicago.com /chicago/words-2000-11-30-842.html (447 words)

	Reply to Bonnie Gold's Review (Site not responding. Last check: 2007-10-01)
		In this case, we use everyday cognitive mechanisms such as the BMI together with the inferential structure of the idea of "speck" (which we hypothesize as the everyday idea that was the inspiration for Leibniz's idea of infinitesimal).
		This is not a mathematical result in the classic sense (i.e., it is not a result obtained by proving a mathematical theorem).
		We are aware that it is not a simple task to try to follow a cognitive analysis of mathematical ideas when one has been trained as a professional mathematician, which is probably the case of most readers of this MAA column.
	www.maa.org /reviews/wheremath_reply.html (2077 words)

	rafael núñez - publications
		Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being.
		The Cognitive Foundations of Mathematics: The Role of Conceptual Metaphor.
		Núñez, R. Educación matemática y ciencia cognitiva contemporánea: Reflexiones sobre la teoría del "embodied cognition" [Mathematics education and contemporary cognitive science: Reflections on the theory of embodied cognition].
	www.cogsci.ucsd.edu /~nunez/web/publications.html (1726 words)

	George Lakoff - Rafael Núñez - Rafael Nunez - Where Mathematics Comes From: How the Embodied Mind Brings ...
		Where Mathematics Comes From represents a unique collaboration between linguist Lakoff and psychologist Nunez as they study ideas and how mathematics has evolved.
		Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history." Drs.
		And they bid fair to bring it off, showing how all mathematical ideas--from simple counting to calculus--can be traced to the discrete workings of the human brain, and not to some transcendent realm of Platonic ideals.
	perso.unifr.ch /rafael.nunez/reviews.html (2404 words)

	Works of Karl Schroeder
		Mathematical thinking, it turns out, has no features that are different from other kinds of thought, it simply organizes these mental activities in a particular way.
		Mathematics never breaks free of its grounding in human existence; it can't (although, once again, calculation can be performed by machines, but calculation is not mathematical reasoning).
		Unless he's come up an account of meaning that commands scientific consensus (which is the holy grail), then he's simply philosophizing with the aid of science, pure and simple, and all he's offering us a possible interpretation of a possible state of affairs.
	www.kschroeder.com /1018641827/index_html (2841 words)

	Read This: Where Mathematics Comes From (Site not responding. Last check: 2007-10-01)
		While their description of how humans develop concepts of mathematics is consistent with the restricted social constructivism of Hersh, it is also consistent with any reasonable version of platonism that distinguishes between mathematical facts and human knowledge of those mathematical facts.
		Indeed, as we begin to describe how human understanding of mathematical ideas is consonant with human understanding of other abstract systems, platonists will be able to respond to the challenge from philosophers of how finite corporeal beings can have contact with, and knowledge of, an infinite abstract branch of knowledge.
		On the philosophical side, she is the author of "What is the Philosophy of Mathematics and What Should It Be" (Mathematical Intelligencer, 1994) and was the co-organizer of a session on the philosophy of mathematics at the January 2001 joint meetings in New Orleans.
	www.maa.org /reviews/wheremath.html (1381 words)

	Where Mathematics Comes From ?
		The fact(!) that understanding & using mathematics is essentially a human activity may be evident to anybody; but not to the most of the mathematicians.
		Jacques Hadamard, "The Psychology of Invention in the Mathematical Field",ISBN 0486201074
		The Cognitive Foundations of Mathematics: The Role of Conceptual Metaphor by Rafael Núñez & George Lakoff.
	www.math.iitb.ac.in /~advait/csm.html (1272 words)

	The Rugged Elegance World Marketplace \| Books : Where Mathematics Comes From: How the Embodied Mind Brings Mathematics ...
		On the whole, I enjoyed the book, and thought it was a nice overview of lots of different mathematical ideas, most of which were familiar to me, and some of which were not.
		As far as I know, no circle has been found or created where if we were to measure it at the atomic level we would find the circumference divided by the width yielding an infinite series of digits that works out to our pi.
		It is a serious and ambitious effort to apply cognitive processes to the origin of mathematical concepts.
	www.ruggedelegantliving.com /cgi-bin/amazon/amazon_products_feed.cgi?Operation=ItemLookup&ItemId=0465037712 (934 words)

	Amazon.com: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being: Books: George ... (Site not responding. Last check: 2007-10-01)
		By attacking the transcendental nature of mathematics, and elaborating the grounding of mathematical thought in the metaphorical mapping of the mind, many important implications arise ranging from the meaning of mathematics, the way mathematics is practiced and proofs are formulated, to the way mathematics should be taught.
		The authors formulate their intention to link the fields of mathematical thought and cognitive sciences to generate the field of mathematical idea analysis.
		While I agree with the author's central dogma of mathematics as one of the human mind's most beautiful and enduring products they sometimes take their argument just a little too far.
	www.amazon.com /Where-Mathematics-Comes-Embodied-Brings/dp/0465037712 (4786 words)

	Graphs in Mathematics [encyclopedia] (Site not responding. Last check: 2007-10-01)
		The origin is also identified as the point (0, 0), where both x and y have values of zero.
		Creating a graph with data points is just the opposite procedure: for each (x,y) pair, read along the x axis to the first value (x) and construct a vertical line through that value on the x-axis; then read along the y-axis to the second value (y) and construct a horizontal line through that value.
		The point where the two lines cross is the position of the data point.
	www.kosmoi.com /Science/Mathematics/Graphs (502 words)

	UCLA Anderson School of Management \| Marschak Colloquium \| GL abs
		Cognitive mathematics is a subdiscipline of cognitive science that seeks to use results from neuroscience, neural computation, cognitive psychology, and cognitive linguistics to understand the nature of mathematical ideas.
		We are born with minimal mathematical capacities - the ability to subitize (tell the number of) and do a bit of baby arithmetic.
		The talk will be an introductory tour through some of what is known about cognitive mathematics, answering many of these questions along the way.
	www.anderson.ucla.edu /x1378.xml (235 words)

	Barnes & Noble.com - Books: Where Mathematics Come from, by George Lakoff, Paperback
		Lakoff and N ez directly challenge the most cherished myths about the nature of mathematical truth, offering instead a fresh, profound, empirically grounded insight into the meaning of mathematical ideas.
		Two cognitive scientists examine the structure of mathematical ideas, what mathematics means, and why.
		They argue that understanding the ideas implicit in mathematics, especially the metaphorical ideas, demystifies mathematics so that it makes more sense.
	search.barnesandnoble.com /booksearch/isbnInquiry.asp?isbn=0465037712 (395 words)

	FQS 3(1) Buchholz, Review Note: George Lakoff & Rafael E. Núnez (2000). Where Mathematics Comes From – How ...
		Abstract: Until recently, mathematics was seldom a topic of interest in cognitive science.
		However, cognitive linguists no longer follow the CHOMSKYian dogmatism and have adapted the view founded by George LAKOFF that human experience is basically embodied.
		Mathematics is perceived as an expression and a part of human interaction and their bodily experience in a new and fascinating way.
	www.qualitative-research.net /fqs-texte/1-02/1-02review-buchholz2-e.htm (141 words)

	the Horn Farm Paste Mob » Blog Archive » GEORGE LAKOFF & RAFAEL NUNEZ - Where Mathematics Comes From
		I can’t criticize the book as a whole until I’ve finished it, but from where I am (maybe 1/3 of the way), its primary argument seems so ludicrous that I am not optimistic they’ll pull it off.
		What L&N seem in no hurry to prove is that any of this constitutes the indictment of mathematical truth’s supposed independence of humanity that they promise it does.
		For that fact to matter, however, we would need an account of how cognition could possibly occur in a brain where the most basic aspects of how we conceive objects as being distinct from one another were replaced with the fact that Fuji apples look pretty.
	www.pastemob.org /wp/?p=938 (1130 words)

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